a bit of improvement

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base.ma
index 4193eabcc9ceb13e268d0ece0e3f561fab82bf23..4e8b785f6abd60f3a362416d7fb191b86ed12175 100644 (file)
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base.ma
@@ -61,7 +61,7 @@ theorem plus_permute_2_in_3_assoc:
 theorem plus_O:
  \forall (x: nat).(\forall (y: nat).((eq nat (plus x y) O) \to (land (eq nat x O) (eq nat y O))))
 \def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq nat (plus n y) O) \to (land (eq nat n O) (eq nat y O))))) (\lambda (y: nat).(\lambda (H: (eq nat (plus O y) O)).(conj (eq nat O O) (eq nat y O) (refl_equal nat O) H))) (\lambda (n: nat).(\lambda (_: ((\forall (y: nat).((eq nat (plus n y) O) \to (land (eq nat n O) (eq nat y O)))))).(\lambda (y: nat).(\lambda (H0: (eq nat (plus (S n) y) O)).(let H1 \def (match H0 return (\lambda (n0: nat).((eq nat n0 O) \to (land (eq nat (S n) O) (eq nat y O)))) with [refl_equal \Rightarrow (\lambda (H1: (eq nat (plus (S n) y) O)).(let H2 \def (eq_ind nat (plus (S n) y) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind (land (eq nat (S n) O) (eq nat y O)) H2)))]) in (H1 (refl_equal nat O))))))) x).
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq nat (plus n y) O) \to (land (eq nat n O) (eq nat y O))))) (\lambda (y: nat).(\lambda (H: (eq nat (plus O y) O)).(conj (eq nat O O) (eq nat y O) (refl_equal nat O) H))) (\lambda (n: nat).(\lambda (_: ((\forall (y: nat).((eq nat (plus n y) O) \to (land (eq nat n O) (eq nat y O)))))).(\lambda (y: nat).(\lambda (H0: (eq nat (plus (S n) y) O)).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (n0: nat).((eq nat n0 O) \to (land (eq nat (S n) O) (eq nat y O))))) with [refl_equal \Rightarrow (\lambda (H1: (eq nat (plus (S n) y) O)).(let H2 \def (eq_ind nat (plus (S n) y) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind (land (eq nat (S n) O) (eq nat y O)) H2)))]) in (H1 (refl_equal nat O))))))) x).
 
 theorem minus_Sx_SO:
  \forall (x: nat).(eq nat (minus (S x) (S O)) x)
@@ -81,7 +81,7 @@ theorem neq_eq_e:
 theorem le_false:
  \forall (m: nat).(\forall (n: nat).(\forall (P: Prop).((le m n) \to ((le (S n) m) \to P))))
 \def
- \lambda (m: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (P: Prop).((le n n0) \to ((le (S n0) n) \to P))))) (\lambda (n: nat).(\lambda (P: Prop).(\lambda (_: (le O n)).(\lambda (H0: (le (S n) O)).(let H1 \def (match H0 return (\lambda (n: nat).((eq nat n O) \to P)) with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def (eq_ind nat (S n) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind P H2))) | (le_S m H1) \Rightarrow (\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S n) m) \to P) H3)) H1))]) in (H1 (refl_equal nat O))))))) (\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(\forall (P: Prop).((le n n0) \to ((le (S n0) n) \to P)))))).(\lambda (n0: nat).(nat_ind (\lambda (n1: nat).(\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) \to P)))) (\lambda (P: Prop).(\lambda (H0: (le (S n) O)).(\lambda (_: (le (S O) (S n))).(let H2 \def (match H0 return (\lambda (n: nat).((eq nat n O) \to P)) with [le_n \Rightarrow (\lambda (H2: (eq nat (S n) O)).(let H3 \def (eq_ind nat (S n) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind P H3))) | (le_S m H2) \Rightarrow (\lambda (H3: (eq nat (S m) O)).((let H4 \def (eq_ind nat (S m) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind ((le (S n) m) \to P) H4)) H2))]) in (H2 (refl_equal nat O)))))) (\lambda (n1: nat).(\lambda (_: ((\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) \to P))))).(\lambda (P: Prop).(\lambda (H1: (le (S n) (S n1))).(\lambda (H2: (le (S (S n1)) (S n))).(H n1 P (le_S_n n n1 H1) (le_S_n (S n1) n H2))))))) n0)))) m).
+ \lambda (m: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (P: Prop).((le n n0) \to ((le (S n0) n) \to P))))) (\lambda (n: nat).(\lambda (P: Prop).(\lambda (_: (le O n)).(\lambda (H0: (le (S n) O)).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (n: nat).((eq nat n O) \to P))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def (eq_ind nat (S n) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind P H2))) | (le_S m H1) \Rightarrow (\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S n) m) \to P) H3)) H1))]) in (H1 (refl_equal nat O))))))) (\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(\forall (P: Prop).((le n n0) \to ((le (S n0) n) \to P)))))).(\lambda (n0: nat).(nat_ind (\lambda (n1: nat).(\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) \to P)))) (\lambda (P: Prop).(\lambda (H0: (le (S n) O)).(\lambda (_: (le (S O) (S n))).(let H2 \def (match H0 return (\lambda (_: ?).(\lambda (n: nat).((eq nat n O) \to P))) with [le_n \Rightarrow (\lambda (H2: (eq nat (S n) O)).(let H3 \def (eq_ind nat (S n) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind P H3))) | (le_S m H2) \Rightarrow (\lambda (H3: (eq nat (S m) O)).((let H4 \def (eq_ind nat (S m) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind ((le (S n) m) \to P) H4)) H2))]) in (H2 (refl_equal nat O)))))) (\lambda (n1: nat).(\lambda (_: ((\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) \to P))))).(\lambda (P: Prop).(\lambda (H1: (le (S n) (S n1))).(\lambda (H2: (le (S (S n1)) (S n))).(H n1 P (le_S_n n n1 H1) (le_S_n (S n1) n H2))))))) n0)))) m).
 
 theorem le_Sx_x:
  \forall (x: nat).((le (S x) x) \to (\forall (P: Prop).P))
@@ -91,7 +91,7 @@ theorem le_Sx_x:
 theorem minus_le:
  \forall (x: nat).(\forall (y: nat).(le (minus x y) x))
 \def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).(le (minus n y) n))) (\lambda (_: nat).(le_n O)) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).(le (minus n y) n)))).(\lambda (y: nat).(match y return (\lambda (n0: nat).(le (minus (S n) n0) (S n))) with [O \Rightarrow (le_n (S n)) | (S n0) \Rightarrow (le_S (minus n n0) n (H n0))])))) x).
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).(le (minus n y) n))) (\lambda (_: nat).(le_n O)) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).(le (minus n y) n)))).(\lambda (y: nat).(match y return (\lambda (_: ?).(\lambda (n0: nat).(le (minus (S n) n0) (S n)))) with [O \Rightarrow (le_n (S n)) | (S n0) \Rightarrow (le_S (minus n n0) n (H n0))])))) x).
 
 theorem le_plus_minus_sym:
  \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus (minus m n) n))))
@@ -106,7 +106,7 @@ theorem le_minus_minus:
 theorem le_minus_plus:
  \forall (z: nat).(\forall (x: nat).((le z x) \to (\forall (y: nat).(eq nat (minus (plus x y) z) (plus (minus x z) y)))))
 \def
- \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((le n x) \to (\forall (y: nat).(eq nat (minus (plus x y) n) (plus (minus x n) y)))))) (\lambda (x: nat).(\lambda (H: (le O x)).(let H0 \def (match H return (\lambda (n: nat).((eq nat n x) \to (\forall (y: nat).(eq nat (minus (plus x y) O) (plus (minus x O) y))))) with [le_n \Rightarrow (\lambda (H0: (eq nat O x)).(eq_ind nat O (\lambda (n: nat).(\forall (y: nat).(eq nat (minus (plus n y) O) (plus (minus n O) y)))) (\lambda (y: nat).(sym_eq nat (plus (minus O O) y) (minus (plus O y) O) (minus_n_O (plus O y)))) x H0)) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) x)).(eq_ind nat (S m) (\lambda (n: nat).((le O m) \to (\forall (y: nat).(eq nat (minus (plus n y) O) (plus (minus n O) y))))) (\lambda (_: (le O m)).(\lambda (y: nat).(refl_equal nat (plus (minus (S m) O) y)))) x H1 H0))]) in (H0 (refl_equal nat x))))) (\lambda (z0: nat).(\lambda (H: ((\forall (x: nat).((le z0 x) \to (\forall (y: nat).(eq nat (minus (plus x y) z0) (plus (minus x z0) y))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).((le (S z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n (S z0)) y))))) (\lambda (H0: (le (S z0) O)).(\lambda (y: nat).(let H1 \def (match H0 return (\lambda (n: nat).((eq nat n O) \to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y)))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S z0) O)).(let H2 \def (eq_ind nat (S z0) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y)) H2))) | (le_S m H1) \Rightarrow (\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S z0) m) \to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))) H3)) H1))]) in (H1 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: (((le (S z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n (S z0)) y)))))).(\lambda (H1: (le (S z0) (S n))).(\lambda (y: nat).(H n (le_S_n z0 n H1) y))))) x)))) z).
+ \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((le n x) \to (\forall (y: nat).(eq nat (minus (plus x y) n) (plus (minus x n) y)))))) (\lambda (x: nat).(\lambda (H: (le O x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (n: nat).((eq nat n x) \to (\forall (y: nat).(eq nat (minus (plus x y) O) (plus (minus x O) y)))))) with [le_n \Rightarrow (\lambda (H0: (eq nat O x)).(eq_ind nat O (\lambda (n: nat).(\forall (y: nat).(eq nat (minus (plus n y) O) (plus (minus n O) y)))) (\lambda (y: nat).(sym_eq nat (plus (minus O O) y) (minus (plus O y) O) (minus_n_O (plus O y)))) x H0)) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) x)).(eq_ind nat (S m) (\lambda (n: nat).((le O m) \to (\forall (y: nat).(eq nat (minus (plus n y) O) (plus (minus n O) y))))) (\lambda (_: (le O m)).(\lambda (y: nat).(refl_equal nat (plus (minus (S m) O) y)))) x H1 H0))]) in (H0 (refl_equal nat x))))) (\lambda (z0: nat).(\lambda (H: ((\forall (x: nat).((le z0 x) \to (\forall (y: nat).(eq nat (minus (plus x y) z0) (plus (minus x z0) y))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).((le (S z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n (S z0)) y))))) (\lambda (H0: (le (S z0) O)).(\lambda (y: nat).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (n: nat).((eq nat n O) \to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S z0) O)).(let H2 \def (eq_ind nat (S z0) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y)) H2))) | (le_S m H1) \Rightarrow (\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S z0) m) \to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))) H3)) H1))]) in (H1 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: (((le (S z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n (S z0)) y)))))).(\lambda (H1: (le (S z0) (S n))).(\lambda (y: nat).(H n (le_S_n z0 n H1) y))))) x)))) z).
 
 theorem le_minus:
  \forall (x: nat).(\forall (z: nat).(\forall (y: nat).((le (plus x y) z) \to (le x (minus z y)))))
@@ -121,7 +121,7 @@ theorem le_trans_plus_r:
 theorem le_gen_S:
  \forall (m: nat).(\forall (x: nat).((le (S m) x) \to (ex2 nat (\lambda (n: nat).(eq nat x (S n))) (\lambda (n: nat).(le m n)))))
 \def
- \lambda (m: nat).(\lambda (x: nat).(\lambda (H: (le (S m) x)).(let H0 \def (match H return (\lambda (n: nat).((eq nat n x) \to (ex2 nat (\lambda (n0: nat).(eq nat x (S n0))) (\lambda (n0: nat).(le m n0))))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) x)).(eq_ind nat (S m) (\lambda (n: nat).(ex2 nat (\lambda (n0: nat).(eq nat n (S n0))) (\lambda (n0: nat).(le m n0)))) (ex_intro2 nat (\lambda (n: nat).(eq nat (S m) (S n))) (\lambda (n: nat).(le m n)) m (refl_equal nat (S m)) (le_n m)) x H0)) | (le_S m0 H0) \Rightarrow (\lambda (H1: (eq nat (S m0) x)).(eq_ind nat (S m0) (\lambda (n: nat).((le (S m) m0) \to (ex2 nat (\lambda (n0: nat).(eq nat n (S n0))) (\lambda (n0: nat).(le m n0))))) (\lambda (H2: (le (S m) m0)).(ex_intro2 nat (\lambda (n: nat).(eq nat (S m0) (S n))) (\lambda (n: nat).(le m n)) m0 (refl_equal nat (S m0)) (le_S_n m m0 (le_S (S m) m0 H2)))) x H1 H0))]) in (H0 (refl_equal nat x))))).
+ \lambda (m: nat).(\lambda (x: nat).(\lambda (H: (le (S m) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (n: nat).((eq nat n x) \to (ex2 nat (\lambda (n0: nat).(eq nat x (S n0))) (\lambda (n0: nat).(le m n0)))))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) x)).(eq_ind nat (S m) (\lambda (n: nat).(ex2 nat (\lambda (n0: nat).(eq nat n (S n0))) (\lambda (n0: nat).(le m n0)))) (ex_intro2 nat (\lambda (n: nat).(eq nat (S m) (S n))) (\lambda (n: nat).(le m n)) m (refl_equal nat (S m)) (le_n m)) x H0)) | (le_S m0 H0) \Rightarrow (\lambda (H1: (eq nat (S m0) x)).(eq_ind nat (S m0) (\lambda (n: nat).((le (S m) m0) \to (ex2 nat (\lambda (n0: nat).(eq nat n (S n0))) (\lambda (n0: nat).(le m n0))))) (\lambda (H2: (le (S m) m0)).(ex_intro2 nat (\lambda (n: nat).(eq nat (S m0) (S n))) (\lambda (n: nat).(le m n)) m0 (refl_equal nat (S m0)) (le_S_n m m0 (le_S (S m) m0 H2)))) x H1 H0))]) in (H0 (refl_equal nat x))))).
 
 theorem lt_x_plus_x_Sy:
  \forall (x: nat).(\forall (y: nat).(lt x (plus x (S y))))
@@ -136,7 +136,7 @@ theorem simpl_lt_plus_r:
 theorem minus_x_Sy:
  \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq nat (minus x y) (S (minus x (S y))))))
 \def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to (eq nat (minus n y) (S (minus n (S y))))))) (\lambda (y: nat).(\lambda (H: (lt y O)).(let H0 \def (match H return (\lambda (n: nat).((eq nat n O) \to (eq nat (minus O y) (S (minus O (S y)))))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S y) O)).(let H1 \def (eq_ind nat (S y) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq nat (minus O y) (S (minus O (S y)))) H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind ((le (S y) m) \to (eq nat (minus O y) (S (minus O (S y))))) H2)) H0))]) in (H0 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to (eq nat (minus n y) (S (minus n (S y)))))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S n) (S n0)))))) (\lambda (_: (lt O (S n))).(eq_ind nat n (\lambda (n0: nat).(eq nat (S n) (S n0))) (refl_equal nat (S n)) (minus n O) (minus_n_O n))) (\lambda (n0: nat).(\lambda (_: (((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S n) (S n0))))))).(\lambda (H1: (lt (S n0) (S n))).(let H2 \def (le_S_n (S n0) n H1) in (H n0 H2))))) y)))) x).
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to (eq nat (minus n y) (S (minus n (S y))))))) (\lambda (y: nat).(\lambda (H: (lt y O)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (n: nat).((eq nat n O) \to (eq nat (minus O y) (S (minus O (S y))))))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S y) O)).(let H1 \def (eq_ind nat (S y) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq nat (minus O y) (S (minus O (S y)))) H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind ((le (S y) m) \to (eq nat (minus O y) (S (minus O (S y))))) H2)) H0))]) in (H0 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to (eq nat (minus n y) (S (minus n (S y)))))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S n) (S n0)))))) (\lambda (_: (lt O (S n))).(eq_ind nat n (\lambda (n0: nat).(eq nat (S n) (S n0))) (refl_equal nat (S n)) (minus n O) (minus_n_O n))) (\lambda (n0: nat).(\lambda (_: (((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S n) (S n0))))))).(\lambda (H1: (lt (S n0) (S n))).(let H2 \def (le_S_n (S n0) n H1) in (H n0 H2))))) y)))) x).
 
 theorem lt_plus_minus:
  \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus x (minus y (S x)))))))
@@ -156,7 +156,7 @@ theorem minus_x_SO:
 theorem le_x_pred_y:
  \forall (y: nat).(\forall (x: nat).((lt x y) \to (le x (pred y))))
 \def
- \lambda (y: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((lt x n) \to (le x (pred n))))) (\lambda (x: nat).(\lambda (H: (lt x O)).(let H0 \def (match H return (\lambda (n: nat).((eq nat n O) \to (le x O))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S x) O)).(let H1 \def (eq_ind nat (S x) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (le x O) H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind ((le (S x) m) \to (le x O)) H2)) H0))]) in (H0 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: ((\forall (x: nat).((lt x n) \to (le x (pred n)))))).(\lambda (x: nat).(\lambda (H0: (lt x (S n))).(le_S_n x n H0))))) y).
+ \lambda (y: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((lt x n) \to (le x (pred n))))) (\lambda (x: nat).(\lambda (H: (lt x O)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (n: nat).((eq nat n O) \to (le x O)))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S x) O)).(let H1 \def (eq_ind nat (S x) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (le x O) H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind ((le (S x) m) \to (le x O)) H2)) H0))]) in (H0 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: ((\forall (x: nat).((lt x n) \to (le x (pred n)))))).(\lambda (x: nat).(\lambda (H0: (lt x (S n))).(le_S_n x n H0))))) y).
 
 theorem lt_le_minus:
  \forall (x: nat).(\forall (y: nat).((lt x y) \to (le x (minus y (S O)))))
@@ -288,20 +288,20 @@ definition blt:
 theorem lt_blt:
  \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq bool (blt y x) true)))
 \def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to (eq bool (blt y n) true)))) (\lambda (y: nat).(\lambda (H: (lt y O)).(let H0 \def (match H return (\lambda (n: nat).((eq nat n O) \to (eq bool (blt y O) true))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S y) O)).(let H1 \def (eq_ind nat (S y) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq bool (blt y O) true) H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind ((le (S y) m) \to (eq bool (blt y O) true)) H2)) H0))]) in (H0 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to (eq bool (blt y n) true))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((lt n0 (S n)) \to (eq bool (blt n0 (S n)) true))) (\lambda (_: (lt O (S n))).(refl_equal bool true)) (\lambda (n0: nat).(\lambda (_: (((lt n0 (S n)) \to (eq bool (match n0 with [O \Rightarrow true | (S m) \Rightarrow (blt m n)]) true)))).(\lambda (H1: (lt (S n0) (S n))).(H n0 (le_S_n (S n0) n H1))))) y)))) x).
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to (eq bool (blt y n) true)))) (\lambda (y: nat).(\lambda (H: (lt y O)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (n: nat).((eq nat n O) \to (eq bool (blt y O) true)))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S y) O)).(let H1 \def (eq_ind nat (S y) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq bool (blt y O) true) H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def (eq_ind nat (S m) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind ((le (S y) m) \to (eq bool (blt y O) true)) H2)) H0))]) in (H0 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to (eq bool (blt y n) true))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((lt n0 (S n)) \to (eq bool (blt n0 (S n)) true))) (\lambda (_: (lt O (S n))).(refl_equal bool true)) (\lambda (n0: nat).(\lambda (_: (((lt n0 (S n)) \to (eq bool (match n0 with [O \Rightarrow true | (S m) \Rightarrow (blt m n)]) true)))).(\lambda (H1: (lt (S n0) (S n))).(H n0 (le_S_n (S n0) n H1))))) y)))) x).
 
 theorem le_bge:
  \forall (x: nat).(\forall (y: nat).((le x y) \to (eq bool (blt y x) false)))
 \def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to (eq bool (blt y n) false)))) (\lambda (y: nat).(\lambda (_: (le O y)).(refl_equal bool false))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((le n y) \to (eq bool (blt y n) false))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((le (S n) n0) \to (eq bool (blt n0 (S n)) false))) (\lambda (H0: (le (S n) O)).(let H1 \def (match H0 return (\lambda (n0: nat).((eq nat n0 O) \to (eq bool (blt O (S n)) false))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def (eq_ind nat (S n) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind (eq bool (blt O (S n)) false) H2))) | (le_S m H1) \Rightarrow (\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S n) m) \to (eq bool (blt O (S n)) false)) H3)) H1))]) in (H1 (refl_equal nat O)))) (\lambda (n0: nat).(\lambda (_: (((le (S n) n0) \to (eq bool (blt n0 (S n)) false)))).(\lambda (H1: (le (S n) (S n0))).(H n0 (le_S_n n n0 H1))))) y)))) x).
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to (eq bool (blt y n) false)))) (\lambda (y: nat).(\lambda (_: (le O y)).(refl_equal bool false))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((le n y) \to (eq bool (blt y n) false))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((le (S n) n0) \to (eq bool (blt n0 (S n)) false))) (\lambda (H0: (le (S n) O)).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (n0: nat).((eq nat n0 O) \to (eq bool (blt O (S n)) false)))) with [le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def (eq_ind nat (S n) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind (eq bool (blt O (S n)) false) H2))) | (le_S m H1) \Rightarrow (\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S n) m) \to (eq bool (blt O (S n)) false)) H3)) H1))]) in (H1 (refl_equal nat O)))) (\lambda (n0: nat).(\lambda (_: (((le (S n) n0) \to (eq bool (blt n0 (S n)) false)))).(\lambda (H1: (le (S n) (S n0))).(H n0 (le_S_n n n0 H1))))) y)))) x).
 
 theorem blt_lt:
  \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) true) \to (lt y x)))
 \def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt y n) true) \to (lt y n)))) (\lambda (y: nat).(\lambda (H: (eq bool (blt y O) true)).(let H0 \def (match H return (\lambda (b: bool).((eq bool b true) \to (lt y O))) with [refl_equal \Rightarrow (\lambda (H0: (eq bool (blt y O) true)).(let H1 \def (eq_ind bool (blt y O) (\lambda (e: bool).(match e return (\lambda (_: ?).Prop) with [true \Rightarrow False | false \Rightarrow True])) I true H0) in (False_ind (lt y O) H1)))]) in (H0 (refl_equal bool true))))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq bool (blt y n) true) \to (lt y n))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((eq bool (blt n0 (S n)) true) \to (lt n0 (S n)))) (\lambda (_: (eq bool true true)).(le_S_n (S O) (S n) (le_n_S (S O) (S n) (le_n_S O n (le_O_n n))))) (\lambda (n0: nat).(\lambda (_: (((eq bool (match n0 with [O \Rightarrow true | (S m) \Rightarrow (blt m n)]) true) \to (lt n0 (S n))))).(\lambda (H1: (eq bool (blt n0 n) true)).(lt_le_S (S n0) (S n) (lt_n_S n0 n (H n0 H1)))))) y)))) x).
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt y n) true) \to (lt y n)))) (\lambda (y: nat).(\lambda (H: (eq bool (blt y O) true)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (b: bool).((eq bool b true) \to (lt y O)))) with [refl_equal \Rightarrow (\lambda (H0: (eq bool (blt y O) true)).(let H1 \def (eq_ind bool (blt y O) (\lambda (e: bool).(match e return (\lambda (_: ?).Prop) with [true \Rightarrow False | false \Rightarrow True])) I true H0) in (False_ind (lt y O) H1)))]) in (H0 (refl_equal bool true))))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq bool (blt y n) true) \to (lt y n))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((eq bool (blt n0 (S n)) true) \to (lt n0 (S n)))) (\lambda (_: (eq bool true true)).(le_S_n (S O) (S n) (le_n_S (S O) (S n) (le_n_S O n (le_O_n n))))) (\lambda (n0: nat).(\lambda (_: (((eq bool (match n0 with [O \Rightarrow true | (S m) \Rightarrow (blt m n)]) true) \to (lt n0 (S n))))).(\lambda (H1: (eq bool (blt n0 n) true)).(lt_le_S (S n0) (S n) (lt_n_S n0 n (H n0 H1)))))) y)))) x).
 
 theorem bge_le:
  \forall (x: nat).(\forall (y: nat).((eq bool (blt y x) false) \to (le x y)))
 \def
- \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt y n) false) \to (le n y)))) (\lambda (y: nat).(\lambda (_: (eq bool (blt y O) false)).(le_O_n y))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq bool (blt y n) false) \to (le n y))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((eq bool (blt n0 (S n)) false) \to (le (S n) n0))) (\lambda (H0: (eq bool (blt O (S n)) false)).(let H1 \def (match H0 return (\lambda (b: bool).((eq bool b false) \to (le (S n) O))) with [refl_equal \Rightarrow (\lambda (H1: (eq bool (blt O (S n)) false)).(let H2 \def (eq_ind bool (blt O (S n)) (\lambda (e: bool).(match e return (\lambda (_: ?).Prop) with [true \Rightarrow True | false \Rightarrow False])) I false H1) in (False_ind (le (S n) O) H2)))]) in (H1 (refl_equal bool false)))) (\lambda (n0: nat).(\lambda (_: (((eq bool (blt n0 (S n)) false) \to (le (S n) n0)))).(\lambda (H1: (eq bool (blt (S n0) (S n)) false)).(le_S_n (S n) (S n0) (le_n_S (S n) (S n0) (le_n_S n n0 (H n0 H1))))))) y)))) x).
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt y n) false) \to (le n y)))) (\lambda (y: nat).(\lambda (_: (eq bool (blt y O) false)).(le_O_n y))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq bool (blt y n) false) \to (le n y))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((eq bool (blt n0 (S n)) false) \to (le (S n) n0))) (\lambda (H0: (eq bool (blt O (S n)) false)).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (b: bool).((eq bool b false) \to (le (S n) O)))) with [refl_equal \Rightarrow (\lambda (H1: (eq bool (blt O (S n)) false)).(let H2 \def (eq_ind bool (blt O (S n)) (\lambda (e: bool).(match e return (\lambda (_: ?).Prop) with [true \Rightarrow True | false \Rightarrow False])) I false H1) in (False_ind (le (S n) O) H2)))]) in (H1 (refl_equal bool false)))) (\lambda (n0: nat).(\lambda (_: (((eq bool (blt n0 (S n)) false) \to (le (S n) n0)))).(\lambda (H1: (eq bool (blt (S n0) (S n)) false)).(le_S_n (S n) (S n0) (le_n_S (S n) (S n0) (le_n_S n n0 (H n0 H1))))))) y)))) x).
 

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta.ma
index 9484ee38b8ea38264333ae829f5cf31064a0981a..d753a1ab371d41c7579f32e657c73f4d718af5d9 100644 (file)
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta.ma
@@ -83,12 +83,12 @@ theorem term_dec:
 theorem binder_dec:
  \forall (t: T).(or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))
 \def
- \lambda (t: T).(T_ind (\lambda (t0: T).(or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))) (\lambda (n: nat).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T (TSort n) (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (TSort n) (THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(\lambda (H: (eq T (TSort n) (THead (Bind b) w u))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) w u) H) in (False_ind P H0))))))))) (\lambda (n: nat).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T (TLRef n) (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (TLRef n) (THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(\lambda (H: (eq T (TLRef n) (THead (Bind b) w u))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) w u) H) in (False_ind P H0))))))))) (\lambda (k: K).(match k return (\lambda (k0: K).(\forall (t0: T).((or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (\forall (t1: T).((or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead (Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead k0 t0 t1) (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead k0 t0 t1) (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))))))) with [(Bind b) \Rightarrow (\lambda (t0: T).(\lambda (_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))).(or_introl (ex_3 B T T (\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u: T).((eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u)) \to (\forall (P: Prop).P))))) (ex_3_intro B T T (\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u))))) b t0 t1 (refl_equal T (THead (Bind b) t0 t1)))))))) | (Flat f) \Rightarrow (\lambda (t0: T).(\lambda (_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead (Flat f) t0 t1) (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead (Flat f) t0 t1) (THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(\lambda (H1: (eq T (THead (Flat f) t0 t1) (THead (Bind b) w u))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead (Flat f) t0 t1) (\lambda (ee: T).(match ee return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) w u) H1) in (False_ind P H2))))))))))))])) t).
+ \lambda (t: T).(T_ind (\lambda (t0: T).(or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))) (\lambda (n: nat).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T (TSort n) (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (TSort n) (THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(\lambda (H: (eq T (TSort n) (THead (Bind b) w u))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) w u) H) in (False_ind P H0))))))))) (\lambda (n: nat).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T (TLRef n) (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (TLRef n) (THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(\lambda (H: (eq T (TLRef n) (THead (Bind b) w u))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) w u) H) in (False_ind P H0))))))))) (\lambda (k: K).(match k return (\lambda (_: ?).(\lambda (k0: K).(\forall (t0: T).((or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (\forall (t1: T).((or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead (Bind b) w u)) \to (\forall (P: Prop).P)))))) \to (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead k0 t0 t1) (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead k0 t0 t1) (THead (Bind b) w u)) \to (\forall (P: Prop).P)))))))))))) with [(Bind b) \Rightarrow (\lambda (t0: T).(\lambda (_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))).(or_introl (ex_3 B T T (\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u)))))) (\forall (b0: B).(\forall (w: T).(\forall (u: T).((eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u)) \to (\forall (P: Prop).P))))) (ex_3_intro B T T (\lambda (b0: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead (Bind b) t0 t1) (THead (Bind b0) w u))))) b t0 t1 (refl_equal T (THead (Bind b) t0 t1)))))))) | (Flat f) \Rightarrow (\lambda (t0: T).(\lambda (_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (_: (or (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t1 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t1 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))).(or_intror (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T (THead (Flat f) t0 t1) (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T (THead (Flat f) t0 t1) (THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(\lambda (H1: (eq T (THead (Flat f) t0 t1) (THead (Bind b) w u))).(\lambda (P: Prop).(let H2 \def (eq_ind T (THead (Flat f) t0 t1) (\lambda (ee: T).(match ee return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) w u) H1) in (False_ind P H2))))))))))))])) t).
 
 theorem abst_dec:
  \forall (u: T).(\forall (v: T).(or (ex T (\lambda (t: T).(eq T u (THead (Bind Abst) v t)))) (\forall (t: T).((eq T u (THead (Bind Abst) v t)) \to (\forall (P: Prop).P)))))
 \def
- \lambda (u: T).(match u return (\lambda (t: T).(\forall (v: T).(or (ex T (\lambda (t0: T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T t (THead (Bind Abst) v t0)) \to (\forall (P: Prop).P)))))) with [(TSort n) \Rightarrow (\lambda (v: T).(or_intror (ex T (\lambda (t: T).(eq T (TSort n) (THead (Bind Abst) v t)))) (\forall (t: T).((eq T (TSort n) (THead (Bind Abst) v t)) \to (\forall (P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TSort n) (THead (Bind Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) v t) H) in (False_ind P H0))))))) | (TLRef n) \Rightarrow (\lambda (v: T).(or_intror (ex T (\lambda (t: T).(eq T (TLRef n) (THead (Bind Abst) v t)))) (\forall (t: T).((eq T (TLRef n) (THead (Bind Abst) v t)) \to (\forall (P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TLRef n) (THead (Bind Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) v t) H) in (False_ind P H0))))))) | (THead k t t0) \Rightarrow (\lambda (v: T).(let H_x \def (terms_props__kind_dec k (Bind Abst)) in (let H \def H_x in (or_ind (eq K k (Bind Abst)) ((eq K k (Bind Abst)) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead k t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda (H0: (eq K k (Bind Abst))).(eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex T (\lambda (t1: T).(eq T (THead k0 t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k0 t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))))) (let H_x0 \def (term_dec t v) in (let H1 \def H_x0 in (or_ind (eq T t v) ((eq T t v) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda (H2: (eq T t v)).(eq_ind T t (\lambda (t1: T).(or (ex T (\lambda (t2: T).(eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t1 t2)))) (\forall (t2: T).((eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t1 t2)) \to (\forall (P: Prop).P))))) (or_introl (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t t1)) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t t1))) t0 (refl_equal T (THead (Bind Abst) t t0)))) v H2)) (\lambda (H2: (((eq T t v) \to (\forall (P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))) (\lambda (t1: T).(\lambda (H3: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) v t1))).(\lambda (P: Prop).(let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t _) \Rightarrow t])) (THead (Bind Abst) t t0) (THead (Bind Abst) v t1) H3) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abst) t t0) (THead (Bind Abst) v t1) H3) in (\lambda (H6: (eq T t v)).(H2 H6 P))) H4))))))) H1))) k H0)) (\lambda (H0: (((eq K k (Bind Abst)) \to (\forall (P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead k t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))) (\lambda (t1: T).(\lambda (H1: (eq T (THead k t t0) (THead (Bind Abst) v t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k t t0) (THead (Bind Abst) v t1) H1) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t _) \Rightarrow t])) (THead k t t0) (THead (Bind Abst) v t1) H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k t t0) (THead (Bind Abst) v t1) H1) in (\lambda (_: (eq T t v)).(\lambda (H6: (eq K k (Bind Abst))).(H0 H6 P)))) H3)) H2))))))) H))))]).
+ \lambda (u: T).(match u return (\lambda (_: ?).(\lambda (t: T).(\forall (v: T).(or (ex T (\lambda (t0: T).(eq T t (THead (Bind Abst) v t0)))) (\forall (t0: T).((eq T t (THead (Bind Abst) v t0)) \to (\forall (P: Prop).P))))))) with [(TSort n) \Rightarrow (\lambda (v: T).(or_intror (ex T (\lambda (t: T).(eq T (TSort n) (THead (Bind Abst) v t)))) (\forall (t: T).((eq T (TSort n) (THead (Bind Abst) v t)) \to (\forall (P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TSort n) (THead (Bind Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) v t) H) in (False_ind P H0))))))) | (TLRef n) \Rightarrow (\lambda (v: T).(or_intror (ex T (\lambda (t: T).(eq T (TLRef n) (THead (Bind Abst) v t)))) (\forall (t: T).((eq T (TLRef n) (THead (Bind Abst) v t)) \to (\forall (P: Prop).P))) (\lambda (t: T).(\lambda (H: (eq T (TLRef n) (THead (Bind Abst) v t))).(\lambda (P: Prop).(let H0 \def (eq_ind T (TLRef n) (\lambda (ee: T).(match ee return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind Abst) v t) H) in (False_ind P H0))))))) | (THead k t t0) \Rightarrow (\lambda (v: T).(let H_x \def (terms_props__kind_dec k (Bind Abst)) in (let H \def H_x in (or_ind (eq K k (Bind Abst)) ((eq K k (Bind Abst)) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead k t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda (H0: (eq K k (Bind Abst))).(eq_ind_r K (Bind Abst) (\lambda (k0: K).(or (ex T (\lambda (t1: T).(eq T (THead k0 t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k0 t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))))) (let H_x0 \def (term_dec t v) in (let H1 \def H_x0 in (or_ind (eq T t v) ((eq T t v) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P)))) (\lambda (H2: (eq T t v)).(eq_ind T t (\lambda (t1: T).(or (ex T (\lambda (t2: T).(eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t1 t2)))) (\forall (t2: T).((eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t1 t2)) \to (\forall (P: Prop).P))))) (or_introl (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t t1)) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t t1))) t0 (refl_equal T (THead (Bind Abst) t t0)))) v H2)) (\lambda (H2: (((eq T t v) \to (\forall (P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead (Bind Abst) t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead (Bind Abst) t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))) (\lambda (t1: T).(\lambda (H3: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) v t1))).(\lambda (P: Prop).(let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t _) \Rightarrow t])) (THead (Bind Abst) t t0) (THead (Bind Abst) v t1) H3) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abst) t t0) (THead (Bind Abst) v t1) H3) in (\lambda (H6: (eq T t v)).(H2 H6 P))) H4))))))) H1))) k H0)) (\lambda (H0: (((eq K k (Bind Abst)) \to (\forall (P: Prop).P)))).(or_intror (ex T (\lambda (t1: T).(eq T (THead k t t0) (THead (Bind Abst) v t1)))) (\forall (t1: T).((eq T (THead k t t0) (THead (Bind Abst) v t1)) \to (\forall (P: Prop).P))) (\lambda (t1: T).(\lambda (H1: (eq T (THead k t t0) (THead (Bind Abst) v t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k t t0) (THead (Bind Abst) v t1) H1) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t _) \Rightarrow t])) (THead k t t0) (THead (Bind Abst) v t1) H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k t t0) (THead (Bind Abst) v t1) H1) in (\lambda (_: (eq T t v)).(\lambda (H6: (eq K k (Bind Abst))).(H0 H6 P)))) H3)) H2))))))) H))))]).
 
 theorem thead_x_y_y:
  \forall (k: K).(\forall (v: T).(\forall (t: T).((eq T (THead k v t) t) \to (\forall (P: Prop).P))))
@@ -233,17 +233,17 @@ inductive iso: T \to (T \to Prop) \def
 theorem iso_flats_lref_bind_false:
  \forall (f: F).(\forall (b: B).(\forall (i: nat).(\forall (v: T).(\forall (t: T).(\forall (vs: TList).((iso (THeads (Flat f) vs (TLRef i)) (THead (Bind b) v t)) \to (\forall (P: Prop).P)))))))
 \def
- \lambda (f: F).(\lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: TList).((iso (THeads (Flat f) t0 (TLRef i)) (THead (Bind b) v t)) \to (\forall (P: Prop).P))) (\lambda (H: (iso (TLRef i) (THead (Bind b) v t))).(\lambda (P: Prop).(let H0 \def (match H return (\lambda (t0: T).(\lambda (t1: T).((eq T t0 (TLRef i)) \to ((eq T t1 (THead (Bind b) v t)) \to P)))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1) (TLRef i))).(\lambda (H1: (eq T (TSort n2) (THead (Bind b) v t))).((let H2 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i) H0) in (False_ind ((eq T (TSort n2) (THead (Bind b) v t)) \to P) H2)) H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0: (eq T (TLRef i1) (TLRef i))).(\lambda (H1: (eq T (TLRef i2) (THead (Bind b) v t))).((let H2 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow i1 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i1])) (TLRef i1) (TLRef i) H0) in (eq_ind nat i (\lambda (_: nat).((eq T (TLRef i2) (THead (Bind b) v t)) \to P)) (\lambda (H3: (eq T (TLRef i2) (THead (Bind b) v t))).(let H4 \def (eq_ind T (TLRef i2) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) v t) H3) in (False_ind P H4))) i1 (sym_eq nat i1 i H2))) H1))) | (iso_head k v1 v2 t1 t2) \Rightarrow (\lambda (H0: (eq T (THead k v1 t1) (TLRef i))).(\lambda (H1: (eq T (THead k v2 t2) (THead (Bind b) v t))).((let H2 \def (eq_ind T (THead k v1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i) H0) in (False_ind ((eq T (THead k v2 t2) (THead (Bind b) v t)) \to P) H2)) H1)))]) in (H0 (refl_equal T (TLRef i)) (refl_equal T (THead (Bind b) v t)))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (_: (((iso (THeads (Flat f) t1 (TLRef i)) (THead (Bind b) v t)) \to (\forall (P: Prop).P)))).(\lambda (H0: (iso (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))) (THead (Bind b) v t))).(\lambda (P: Prop).(let H1 \def (match H0 return (\lambda (t2: T).(\lambda (t3: T).((eq T t2 (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i)))) \to ((eq T t3 (THead (Bind b) v t)) \to P)))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H1: (eq T (TSort n1) (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))))).(\lambda (H2: (eq T (TSort n2) (THead (Bind b) v t))).((let H3 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))) H1) in (False_ind ((eq T (TSort n2) (THead (Bind b) v t)) \to P) H3)) H2))) | (iso_lref i1 i2) \Rightarrow (\lambda (H1: (eq T (TLRef i1) (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))))).(\lambda (H2: (eq T (TLRef i2) (THead (Bind b) v t))).((let H3 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))) H1) in (False_ind ((eq T (TLRef i2) (THead (Bind b) v t)) \to P) H3)) H2))) | (iso_head k v1 v2 t2 t3) \Rightarrow (\lambda (H1: (eq T (THead k v1 t2) (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))))).(\lambda (H2: (eq T (THead k v2 t3) (THead (Bind b) v t))).((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ _ t) \Rightarrow t])) (THead k v1 t2) (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))) H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead k v1 t2) (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))) H1) in ((let H5 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k v1 t2) (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))) H1) in (eq_ind K (Flat f) (\lambda (k0: K).((eq T v1 t0) \to ((eq T t2 (THeads (Flat f) t1 (TLRef i))) \to ((eq T (THead k0 v2 t3) (THead (Bind b) v t)) \to P)))) (\lambda (H6: (eq T v1 t0)).(eq_ind T t0 (\lambda (_: T).((eq T t2 (THeads (Flat f) t1 (TLRef i))) \to ((eq T (THead (Flat f) v2 t3) (THead (Bind b) v t)) \to P))) (\lambda (H7: (eq T t2 (THeads (Flat f) t1 (TLRef i)))).(eq_ind T (THeads (Flat f) t1 (TLRef i)) (\lambda (_: T).((eq T (THead (Flat f) v2 t3) (THead (Bind b) v t)) \to P)) (\lambda (H8: (eq T (THead (Flat f) v2 t3) (THead (Bind b) v t))).(let H9 \def (eq_ind T (THead (Flat f) v2 t3) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) v t) H8) in (False_ind P H9))) t2 (sym_eq T t2 (THeads (Flat f) t1 (TLRef i)) H7))) v1 (sym_eq T v1 t0 H6))) k (sym_eq K k (Flat f) H5))) H4)) H3)) H2)))]) in (H1 (refl_equal T (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i)))) (refl_equal T (THead (Bind b) v t))))))))) vs)))))).
+ \lambda (f: F).(\lambda (b: B).(\lambda (i: nat).(\lambda (v: T).(\lambda (t: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: TList).((iso (THeads (Flat f) t0 (TLRef i)) (THead (Bind b) v t)) \to (\forall (P: Prop).P))) (\lambda (H: (iso (TLRef i) (THead (Bind b) v t))).(\lambda (P: Prop).(let H0 \def (match H return (\lambda (_: ?).(\lambda (t0: T).(\lambda (t1: T).((eq T t0 (TLRef i)) \to ((eq T t1 (THead (Bind b) v t)) \to P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1) (TLRef i))).(\lambda (H1: (eq T (TSort n2) (THead (Bind b) v t))).((let H2 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i) H0) in (False_ind ((eq T (TSort n2) (THead (Bind b) v t)) \to P) H2)) H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0: (eq T (TLRef i1) (TLRef i))).(\lambda (H1: (eq T (TLRef i2) (THead (Bind b) v t))).((let H2 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow i1 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i1])) (TLRef i1) (TLRef i) H0) in (eq_ind nat i (\lambda (_: nat).((eq T (TLRef i2) (THead (Bind b) v t)) \to P)) (\lambda (H3: (eq T (TLRef i2) (THead (Bind b) v t))).(let H4 \def (eq_ind T (TLRef i2) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) v t) H3) in (False_ind P H4))) i1 (sym_eq nat i1 i H2))) H1))) | (iso_head k v1 v2 t1 t2) \Rightarrow (\lambda (H0: (eq T (THead k v1 t1) (TLRef i))).(\lambda (H1: (eq T (THead k v2 t2) (THead (Bind b) v t))).((let H2 \def (eq_ind T (THead k v1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i) H0) in (False_ind ((eq T (THead k v2 t2) (THead (Bind b) v t)) \to P) H2)) H1)))]) in (H0 (refl_equal T (TLRef i)) (refl_equal T (THead (Bind b) v t)))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (_: (((iso (THeads (Flat f) t1 (TLRef i)) (THead (Bind b) v t)) \to (\forall (P: Prop).P)))).(\lambda (H0: (iso (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))) (THead (Bind b) v t))).(\lambda (P: Prop).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t2: T).(\lambda (t3: T).((eq T t2 (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i)))) \to ((eq T t3 (THead (Bind b) v t)) \to P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H1: (eq T (TSort n1) (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))))).(\lambda (H2: (eq T (TSort n2) (THead (Bind b) v t))).((let H3 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))) H1) in (False_ind ((eq T (TSort n2) (THead (Bind b) v t)) \to P) H3)) H2))) | (iso_lref i1 i2) \Rightarrow (\lambda (H1: (eq T (TLRef i1) (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))))).(\lambda (H2: (eq T (TLRef i2) (THead (Bind b) v t))).((let H3 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))) H1) in (False_ind ((eq T (TLRef i2) (THead (Bind b) v t)) \to P) H3)) H2))) | (iso_head k v1 v2 t2 t3) \Rightarrow (\lambda (H1: (eq T (THead k v1 t2) (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))))).(\lambda (H2: (eq T (THead k v2 t3) (THead (Bind b) v t))).((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ _ t) \Rightarrow t])) (THead k v1 t2) (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))) H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead k v1 t2) (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))) H1) in ((let H5 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k v1 t2) (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i))) H1) in (eq_ind K (Flat f) (\lambda (k0: K).((eq T v1 t0) \to ((eq T t2 (THeads (Flat f) t1 (TLRef i))) \to ((eq T (THead k0 v2 t3) (THead (Bind b) v t)) \to P)))) (\lambda (H6: (eq T v1 t0)).(eq_ind T t0 (\lambda (_: T).((eq T t2 (THeads (Flat f) t1 (TLRef i))) \to ((eq T (THead (Flat f) v2 t3) (THead (Bind b) v t)) \to P))) (\lambda (H7: (eq T t2 (THeads (Flat f) t1 (TLRef i)))).(eq_ind T (THeads (Flat f) t1 (TLRef i)) (\lambda (_: T).((eq T (THead (Flat f) v2 t3) (THead (Bind b) v t)) \to P)) (\lambda (H8: (eq T (THead (Flat f) v2 t3) (THead (Bind b) v t))).(let H9 \def (eq_ind T (THead (Flat f) v2 t3) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) v t) H8) in (False_ind P H9))) t2 (sym_eq T t2 (THeads (Flat f) t1 (TLRef i)) H7))) v1 (sym_eq T v1 t0 H6))) k (sym_eq K k (Flat f) H5))) H4)) H3)) H2)))]) in (H1 (refl_equal T (THead (Flat f) t0 (THeads (Flat f) t1 (TLRef i)))) (refl_equal T (THead (Bind b) v t))))))))) vs)))))).
 
 theorem iso_flats_flat_bind_false:
  \forall (f1: F).(\forall (f2: F).(\forall (b: B).(\forall (v: T).(\forall (v2: T).(\forall (t: T).(\forall (t2: T).(\forall (vs: TList).((iso (THeads (Flat f1) vs (THead (Flat f2) v2 t2)) (THead (Bind b) v t)) \to (\forall (P: Prop).P)))))))))
 \def
- \lambda (f1: F).(\lambda (f2: F).(\lambda (b: B).(\lambda (v: T).(\lambda (v2: T).(\lambda (t: T).(\lambda (t2: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: TList).((iso (THeads (Flat f1) t0 (THead (Flat f2) v2 t2)) (THead (Bind b) v t)) \to (\forall (P: Prop).P))) (\lambda (H: (iso (THead (Flat f2) v2 t2) (THead (Bind b) v t))).(\lambda (P: Prop).(let H0 \def (match H return (\lambda (t0: T).(\lambda (t1: T).((eq T t0 (THead (Flat f2) v2 t2)) \to ((eq T t1 (THead (Bind b) v t)) \to P)))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1) (THead (Flat f2) v2 t2))).(\lambda (H1: (eq T (TSort n2) (THead (Bind b) v t))).((let H2 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat f2) v2 t2) H0) in (False_ind ((eq T (TSort n2) (THead (Bind b) v t)) \to P) H2)) H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0: (eq T (TLRef i1) (THead (Flat f2) v2 t2))).(\lambda (H1: (eq T (TLRef i2) (THead (Bind b) v t))).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat f2) v2 t2) H0) in (False_ind ((eq T (TLRef i2) (THead (Bind b) v t)) \to P) H2)) H1))) | (iso_head k v1 v0 t1 t0) \Rightarrow (\lambda (H0: (eq T (THead k v1 t1) (THead (Flat f2) v2 t2))).(\lambda (H1: (eq T (THead k v0 t0) (THead (Bind b) v t))).((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t])) (THead k v1 t1) (THead (Flat f2) v2 t2) H0) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead k v1 t1) (THead (Flat f2) v2 t2) H0) in ((let H4 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k v1 t1) (THead (Flat f2) v2 t2) H0) in (eq_ind K (Flat f2) (\lambda (k0: K).((eq T v1 v2) \to ((eq T t1 t2) \to ((eq T (THead k0 v0 t0) (THead (Bind b) v t)) \to P)))) (\lambda (H5: (eq T v1 v2)).(eq_ind T v2 (\lambda (_: T).((eq T t1 t2) \to ((eq T (THead (Flat f2) v0 t0) (THead (Bind b) v t)) \to P))) (\lambda (H6: (eq T t1 t2)).(eq_ind T t2 (\lambda (_: T).((eq T (THead (Flat f2) v0 t0) (THead (Bind b) v t)) \to P)) (\lambda (H7: (eq T (THead (Flat f2) v0 t0) (THead (Bind b) v t))).(let H8 \def (eq_ind T (THead (Flat f2) v0 t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) v t) H7) in (False_ind P H8))) t1 (sym_eq T t1 t2 H6))) v1 (sym_eq T v1 v2 H5))) k (sym_eq K k (Flat f2) H4))) H3)) H2)) H1)))]) in (H0 (refl_equal T (THead (Flat f2) v2 t2)) (refl_equal T (THead (Bind b) v t)))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (_: (((iso (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)) (THead (Bind b) v t)) \to (\forall (P: Prop).P)))).(\lambda (H0: (iso (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) (THead (Bind b) v t))).(\lambda (P: Prop).(let H1 \def (match H0 return (\lambda (t3: T).(\lambda (t4: T).((eq T t3 (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)))) \to ((eq T t4 (THead (Bind b) v t)) \to P)))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H1: (eq T (TSort n1) (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))))).(\lambda (H2: (eq T (TSort n2) (THead (Bind b) v t))).((let H3 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) H1) in (False_ind ((eq T (TSort n2) (THead (Bind b) v t)) \to P) H3)) H2))) | (iso_lref i1 i2) \Rightarrow (\lambda (H1: (eq T (TLRef i1) (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))))).(\lambda (H2: (eq T (TLRef i2) (THead (Bind b) v t))).((let H3 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) H1) in (False_ind ((eq T (TLRef i2) (THead (Bind b) v t)) \to P) H3)) H2))) | (iso_head k v1 v0 t3 t4) \Rightarrow (\lambda (H1: (eq T (THead k v1 t3) (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))))).(\lambda (H2: (eq T (THead k v0 t4) (THead (Bind b) v t))).((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t) \Rightarrow t])) (THead k v1 t3) (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead k v1 t3) (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) H1) in ((let H5 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k v1 t3) (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) H1) in (eq_ind K (Flat f1) (\lambda (k0: K).((eq T v1 t0) \to ((eq T t3 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) \to ((eq T (THead k0 v0 t4) (THead (Bind b) v t)) \to P)))) (\lambda (H6: (eq T v1 t0)).(eq_ind T t0 (\lambda (_: T).((eq T t3 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) \to ((eq T (THead (Flat f1) v0 t4) (THead (Bind b) v t)) \to P))) (\lambda (H7: (eq T t3 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)))).(eq_ind T (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)) (\lambda (_: T).((eq T (THead (Flat f1) v0 t4) (THead (Bind b) v t)) \to P)) (\lambda (H8: (eq T (THead (Flat f1) v0 t4) (THead (Bind b) v t))).(let H9 \def (eq_ind T (THead (Flat f1) v0 t4) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) v t) H8) in (False_ind P H9))) t3 (sym_eq T t3 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)) H7))) v1 (sym_eq T v1 t0 H6))) k (sym_eq K k (Flat f1) H5))) H4)) H3)) H2)))]) in (H1 (refl_equal T (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)))) (refl_equal T (THead (Bind b) v t))))))))) vs)))))))).
+ \lambda (f1: F).(\lambda (f2: F).(\lambda (b: B).(\lambda (v: T).(\lambda (v2: T).(\lambda (t: T).(\lambda (t2: T).(\lambda (vs: TList).(TList_ind (\lambda (t0: TList).((iso (THeads (Flat f1) t0 (THead (Flat f2) v2 t2)) (THead (Bind b) v t)) \to (\forall (P: Prop).P))) (\lambda (H: (iso (THead (Flat f2) v2 t2) (THead (Bind b) v t))).(\lambda (P: Prop).(let H0 \def (match H return (\lambda (_: ?).(\lambda (t0: T).(\lambda (t1: T).((eq T t0 (THead (Flat f2) v2 t2)) \to ((eq T t1 (THead (Bind b) v t)) \to P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1) (THead (Flat f2) v2 t2))).(\lambda (H1: (eq T (TSort n2) (THead (Bind b) v t))).((let H2 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat f2) v2 t2) H0) in (False_ind ((eq T (TSort n2) (THead (Bind b) v t)) \to P) H2)) H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0: (eq T (TLRef i1) (THead (Flat f2) v2 t2))).(\lambda (H1: (eq T (TLRef i2) (THead (Bind b) v t))).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat f2) v2 t2) H0) in (False_ind ((eq T (TLRef i2) (THead (Bind b) v t)) \to P) H2)) H1))) | (iso_head k v1 v0 t1 t0) \Rightarrow (\lambda (H0: (eq T (THead k v1 t1) (THead (Flat f2) v2 t2))).(\lambda (H1: (eq T (THead k v0 t0) (THead (Bind b) v t))).((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t])) (THead k v1 t1) (THead (Flat f2) v2 t2) H0) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead k v1 t1) (THead (Flat f2) v2 t2) H0) in ((let H4 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k v1 t1) (THead (Flat f2) v2 t2) H0) in (eq_ind K (Flat f2) (\lambda (k0: K).((eq T v1 v2) \to ((eq T t1 t2) \to ((eq T (THead k0 v0 t0) (THead (Bind b) v t)) \to P)))) (\lambda (H5: (eq T v1 v2)).(eq_ind T v2 (\lambda (_: T).((eq T t1 t2) \to ((eq T (THead (Flat f2) v0 t0) (THead (Bind b) v t)) \to P))) (\lambda (H6: (eq T t1 t2)).(eq_ind T t2 (\lambda (_: T).((eq T (THead (Flat f2) v0 t0) (THead (Bind b) v t)) \to P)) (\lambda (H7: (eq T (THead (Flat f2) v0 t0) (THead (Bind b) v t))).(let H8 \def (eq_ind T (THead (Flat f2) v0 t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) v t) H7) in (False_ind P H8))) t1 (sym_eq T t1 t2 H6))) v1 (sym_eq T v1 v2 H5))) k (sym_eq K k (Flat f2) H4))) H3)) H2)) H1)))]) in (H0 (refl_equal T (THead (Flat f2) v2 t2)) (refl_equal T (THead (Bind b) v t)))))) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (_: (((iso (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)) (THead (Bind b) v t)) \to (\forall (P: Prop).P)))).(\lambda (H0: (iso (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) (THead (Bind b) v t))).(\lambda (P: Prop).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t3: T).(\lambda (t4: T).((eq T t3 (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)))) \to ((eq T t4 (THead (Bind b) v t)) \to P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H1: (eq T (TSort n1) (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))))).(\lambda (H2: (eq T (TSort n2) (THead (Bind b) v t))).((let H3 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) H1) in (False_ind ((eq T (TSort n2) (THead (Bind b) v t)) \to P) H3)) H2))) | (iso_lref i1 i2) \Rightarrow (\lambda (H1: (eq T (TLRef i1) (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))))).(\lambda (H2: (eq T (TLRef i2) (THead (Bind b) v t))).((let H3 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) H1) in (False_ind ((eq T (TLRef i2) (THead (Bind b) v t)) \to P) H3)) H2))) | (iso_head k v1 v0 t3 t4) \Rightarrow (\lambda (H1: (eq T (THead k v1 t3) (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))))).(\lambda (H2: (eq T (THead k v0 t4) (THead (Bind b) v t))).((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t3 | (TLRef _) \Rightarrow t3 | (THead _ _ t) \Rightarrow t])) (THead k v1 t3) (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead k v1 t3) (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) H1) in ((let H5 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k v1 t3) (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) H1) in (eq_ind K (Flat f1) (\lambda (k0: K).((eq T v1 t0) \to ((eq T t3 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) \to ((eq T (THead k0 v0 t4) (THead (Bind b) v t)) \to P)))) (\lambda (H6: (eq T v1 t0)).(eq_ind T t0 (\lambda (_: T).((eq T t3 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2))) \to ((eq T (THead (Flat f1) v0 t4) (THead (Bind b) v t)) \to P))) (\lambda (H7: (eq T t3 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)))).(eq_ind T (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)) (\lambda (_: T).((eq T (THead (Flat f1) v0 t4) (THead (Bind b) v t)) \to P)) (\lambda (H8: (eq T (THead (Flat f1) v0 t4) (THead (Bind b) v t))).(let H9 \def (eq_ind T (THead (Flat f1) v0 t4) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) v t) H8) in (False_ind P H9))) t3 (sym_eq T t3 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)) H7))) v1 (sym_eq T v1 t0 H6))) k (sym_eq K k (Flat f1) H5))) H4)) H3)) H2)))]) in (H1 (refl_equal T (THead (Flat f1) t0 (THeads (Flat f1) t1 (THead (Flat f2) v2 t2)))) (refl_equal T (THead (Bind b) v t))))))))) vs)))))))).
 
 theorem iso_trans:
  \forall (t1: T).(\forall (t2: T).((iso t1 t2) \to (\forall (t3: T).((iso t2 t3) \to (iso t1 t3)))))
 \def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (iso t1 t2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).(\forall (t3: T).((iso t0 t3) \to (iso t t3))))) (\lambda (n1: nat).(\lambda (n2: nat).(\lambda (t3: T).(\lambda (H0: (iso (TSort n2) t3)).(let H1 \def (match H0 return (\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n2)) \to ((eq T t0 t3) \to (iso (TSort n1) t3))))) with [(iso_sort n0 n3) \Rightarrow (\lambda (H0: (eq T (TSort n0) (TSort n2))).(\lambda (H1: (eq T (TSort n3) t3)).((let H2 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort n) \Rightarrow n | (TLRef _) \Rightarrow n0 | (THead _ _ _) \Rightarrow n0])) (TSort n0) (TSort n2) H0) in (eq_ind nat n2 (\lambda (_: nat).((eq T (TSort n3) t3) \to (iso (TSort n1) t3))) (\lambda (H3: (eq T (TSort n3) t3)).(eq_ind T (TSort n3) (\lambda (t: T).(iso (TSort n1) t)) (iso_sort n1 n3) t3 H3)) n0 (sym_eq nat n0 n2 H2))) H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0: (eq T (TLRef i1) (TSort n2))).(\lambda (H1: (eq T (TLRef i2) t3)).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n2) H0) in (False_ind ((eq T (TLRef i2) t3) \to (iso (TSort n1) t3)) H2)) H1))) | (iso_head k v1 v2 t1 t2) \Rightarrow (\lambda (H0: (eq T (THead k v1 t1) (TSort n2))).(\lambda (H1: (eq T (THead k v2 t2) t3)).((let H2 \def (eq_ind T (THead k v1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n2) H0) in (False_ind ((eq T (THead k v2 t2) t3) \to (iso (TSort n1) t3)) H2)) H1)))]) in (H1 (refl_equal T (TSort n2)) (refl_equal T t3))))))) (\lambda (i1: nat).(\lambda (i2: nat).(\lambda (t3: T).(\lambda (H0: (iso (TLRef i2) t3)).(let H1 \def (match H0 return (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef i2)) \to ((eq T t0 t3) \to (iso (TLRef i1) t3))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1) (TLRef i2))).(\lambda (H1: (eq T (TSort n2) t3)).((let H2 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i2) H0) in (False_ind ((eq T (TSort n2) t3) \to (iso (TLRef i1) t3)) H2)) H1))) | (iso_lref i0 i3) \Rightarrow (\lambda (H0: (eq T (TLRef i0) (TLRef i2))).(\lambda (H1: (eq T (TLRef i3) t3)).((let H2 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0])) (TLRef i0) (TLRef i2) H0) in (eq_ind nat i2 (\lambda (_: nat).((eq T (TLRef i3) t3) \to (iso (TLRef i1) t3))) (\lambda (H3: (eq T (TLRef i3) t3)).(eq_ind T (TLRef i3) (\lambda (t: T).(iso (TLRef i1) t)) (iso_lref i1 i3) t3 H3)) i0 (sym_eq nat i0 i2 H2))) H1))) | (iso_head k v1 v2 t1 t2) \Rightarrow (\lambda (H0: (eq T (THead k v1 t1) (TLRef i2))).(\lambda (H1: (eq T (THead k v2 t2) t3)).((let H2 \def (eq_ind T (THead k v1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i2) H0) in (False_ind ((eq T (THead k v2 t2) t3) \to (iso (TLRef i1) t3)) H2)) H1)))]) in (H1 (refl_equal T (TLRef i2)) (refl_equal T t3))))))) (\lambda (k: K).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H0: (iso (THead k v2 t4) t5)).(let H1 \def (match H0 return (\lambda (t: T).(\lambda (t0: T).((eq T t (THead k v2 t4)) \to ((eq T t0 t5) \to (iso (THead k v1 t3) t5))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1) (THead k v2 t4))).(\lambda (H1: (eq T (TSort n2) t5)).((let H2 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead k v2 t4) H0) in (False_ind ((eq T (TSort n2) t5) \to (iso (THead k v1 t3) t5)) H2)) H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0: (eq T (TLRef i1) (THead k v2 t4))).(\lambda (H1: (eq T (TLRef i2) t5)).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k v2 t4) H0) in (False_ind ((eq T (TLRef i2) t5) \to (iso (THead k v1 t3) t5)) H2)) H1))) | (iso_head k0 v0 v3 t0 t4) \Rightarrow (\lambda (H0: (eq T (THead k0 v0 t0) (THead k v2 t4))).(\lambda (H1: (eq T (THead k0 v3 t4) t5)).((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k0 v0 t0) (THead k v2 t4) H0) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ t _) \Rightarrow t])) (THead k0 v0 t0) (THead k v2 t4) H0) in ((let H4 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k _ _) \Rightarrow k])) (THead k0 v0 t0) (THead k v2 t4) H0) in (eq_ind K k (\lambda (k1: K).((eq T v0 v2) \to ((eq T t0 t4) \to ((eq T (THead k1 v3 t4) t5) \to (iso (THead k v1 t3) t5))))) (\lambda (H5: (eq T v0 v2)).(eq_ind T v2 (\lambda (_: T).((eq T t0 t4) \to ((eq T (THead k v3 t4) t5) \to (iso (THead k v1 t3) t5)))) (\lambda (H6: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).((eq T (THead k v3 t4) t5) \to (iso (THead k v1 t3) t5))) (\lambda (H7: (eq T (THead k v3 t4) t5)).(eq_ind T (THead k v3 t4) (\lambda (t: T).(iso (THead k v1 t3) t)) (iso_head k v1 v3 t3 t4) t5 H7)) t0 (sym_eq T t0 t4 H6))) v0 (sym_eq T v0 v2 H5))) k0 (sym_eq K k0 k H4))) H3)) H2)) H1)))]) in (H1 (refl_equal T (THead k v2 t4)) (refl_equal T t5)))))))))) t1 t2 H))).
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (H: (iso t1 t2)).(iso_ind (\lambda (t: T).(\lambda (t0: T).(\forall (t3: T).((iso t0 t3) \to (iso t t3))))) (\lambda (n1: nat).(\lambda (n2: nat).(\lambda (t3: T).(\lambda (H0: (iso (TSort n2) t3)).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n2)) \to ((eq T t0 t3) \to (iso (TSort n1) t3)))))) with [(iso_sort n0 n3) \Rightarrow (\lambda (H0: (eq T (TSort n0) (TSort n2))).(\lambda (H1: (eq T (TSort n3) t3)).((let H2 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort n) \Rightarrow n | (TLRef _) \Rightarrow n0 | (THead _ _ _) \Rightarrow n0])) (TSort n0) (TSort n2) H0) in (eq_ind nat n2 (\lambda (_: nat).((eq T (TSort n3) t3) \to (iso (TSort n1) t3))) (\lambda (H3: (eq T (TSort n3) t3)).(eq_ind T (TSort n3) (\lambda (t: T).(iso (TSort n1) t)) (iso_sort n1 n3) t3 H3)) n0 (sym_eq nat n0 n2 H2))) H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0: (eq T (TLRef i1) (TSort n2))).(\lambda (H1: (eq T (TLRef i2) t3)).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n2) H0) in (False_ind ((eq T (TLRef i2) t3) \to (iso (TSort n1) t3)) H2)) H1))) | (iso_head k v1 v2 t1 t2) \Rightarrow (\lambda (H0: (eq T (THead k v1 t1) (TSort n2))).(\lambda (H1: (eq T (THead k v2 t2) t3)).((let H2 \def (eq_ind T (THead k v1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n2) H0) in (False_ind ((eq T (THead k v2 t2) t3) \to (iso (TSort n1) t3)) H2)) H1)))]) in (H1 (refl_equal T (TSort n2)) (refl_equal T t3))))))) (\lambda (i1: nat).(\lambda (i2: nat).(\lambda (t3: T).(\lambda (H0: (iso (TLRef i2) t3)).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef i2)) \to ((eq T t0 t3) \to (iso (TLRef i1) t3)))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1) (TLRef i2))).(\lambda (H1: (eq T (TSort n2) t3)).((let H2 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef i2) H0) in (False_ind ((eq T (TSort n2) t3) \to (iso (TLRef i1) t3)) H2)) H1))) | (iso_lref i0 i3) \Rightarrow (\lambda (H0: (eq T (TLRef i0) (TLRef i2))).(\lambda (H1: (eq T (TLRef i3) t3)).((let H2 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow i0 | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i0])) (TLRef i0) (TLRef i2) H0) in (eq_ind nat i2 (\lambda (_: nat).((eq T (TLRef i3) t3) \to (iso (TLRef i1) t3))) (\lambda (H3: (eq T (TLRef i3) t3)).(eq_ind T (TLRef i3) (\lambda (t: T).(iso (TLRef i1) t)) (iso_lref i1 i3) t3 H3)) i0 (sym_eq nat i0 i2 H2))) H1))) | (iso_head k v1 v2 t1 t2) \Rightarrow (\lambda (H0: (eq T (THead k v1 t1) (TLRef i2))).(\lambda (H1: (eq T (THead k v2 t2) t3)).((let H2 \def (eq_ind T (THead k v1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef i2) H0) in (False_ind ((eq T (THead k v2 t2) t3) \to (iso (TLRef i1) t3)) H2)) H1)))]) in (H1 (refl_equal T (TLRef i2)) (refl_equal T t3))))))) (\lambda (k: K).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t3: T).(\lambda (t4: T).(\lambda (t5: T).(\lambda (H0: (iso (THead k v2 t4) t5)).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead k v2 t4)) \to ((eq T t0 t5) \to (iso (THead k v1 t3) t5)))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1) (THead k v2 t4))).(\lambda (H1: (eq T (TSort n2) t5)).((let H2 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead k v2 t4) H0) in (False_ind ((eq T (TSort n2) t5) \to (iso (THead k v1 t3) t5)) H2)) H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0: (eq T (TLRef i1) (THead k v2 t4))).(\lambda (H1: (eq T (TLRef i2) t5)).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k v2 t4) H0) in (False_ind ((eq T (TLRef i2) t5) \to (iso (THead k v1 t3) t5)) H2)) H1))) | (iso_head k0 v0 v3 t0 t4) \Rightarrow (\lambda (H0: (eq T (THead k0 v0 t0) (THead k v2 t4))).(\lambda (H1: (eq T (THead k0 v3 t4) t5)).((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k0 v0 t0) (THead k v2 t4) H0) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ t _) \Rightarrow t])) (THead k0 v0 t0) (THead k v2 t4) H0) in ((let H4 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k _ _) \Rightarrow k])) (THead k0 v0 t0) (THead k v2 t4) H0) in (eq_ind K k (\lambda (k1: K).((eq T v0 v2) \to ((eq T t0 t4) \to ((eq T (THead k1 v3 t4) t5) \to (iso (THead k v1 t3) t5))))) (\lambda (H5: (eq T v0 v2)).(eq_ind T v2 (\lambda (_: T).((eq T t0 t4) \to ((eq T (THead k v3 t4) t5) \to (iso (THead k v1 t3) t5)))) (\lambda (H6: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).((eq T (THead k v3 t4) t5) \to (iso (THead k v1 t3) t5))) (\lambda (H7: (eq T (THead k v3 t4) t5)).(eq_ind T (THead k v3 t4) (\lambda (t: T).(iso (THead k v1 t3) t)) (iso_head k v1 v3 t3 t4) t5 H7)) t0 (sym_eq T t0 t4 H6))) v0 (sym_eq T v0 v2 H5))) k0 (sym_eq K k0 k H4))) H3)) H2)) H1)))]) in (H1 (refl_equal T (THead k v2 t4)) (refl_equal T t5)))))))))) t1 t2 H))).
 
 inductive C: Set \def
 | CSort: nat \to C
@@ -287,7 +287,7 @@ theorem r_dis:
 theorem s_r:
  \forall (k: K).(\forall (i: nat).(eq nat (s k (r k i)) (S i)))
 \def
- \lambda (k: K).(match k return (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (r k0 i)) (S i)))) with [(Bind _) \Rightarrow (\lambda (i: nat).(refl_equal nat (S i))) | (Flat _) \Rightarrow (\lambda (i: nat).(refl_equal nat (S i)))]).
+ \lambda (k: K).(match k return (\lambda (_: ?).(\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (r k0 i)) (S i))))) with [(Bind _) \Rightarrow (\lambda (i: nat).(refl_equal nat (S i))) | (Flat _) \Rightarrow (\lambda (i: nat).(refl_equal nat (S i)))]).
 
 theorem r_arith0:
  \forall (k: K).(\forall (i: nat).(eq nat (minus (r k (S i)) (S O)) (r k i)))
@@ -322,7 +322,7 @@ definition cle:
 theorem tweight_lt:
  \forall (t: T).(lt O (tweight t))
 \def
- \lambda (t: T).(match t return (\lambda (t0: T).(lt O (tweight t0))) with [(TSort _) \Rightarrow (le_n (S O)) | (TLRef _) \Rightarrow (le_n (S O)) | (THead _ t0 t1) \Rightarrow (le_S_n (S O) (S (plus (tweight t0) (tweight t1))) (le_n_S (S O) (S (plus (tweight t0) (tweight t1))) (le_n_S O (plus (tweight t0) (tweight t1)) (le_O_n (plus (tweight t0) (tweight t1))))))]).
+ \lambda (t: T).(match t return (\lambda (_: ?).(\lambda (t0: T).(lt O (tweight t0)))) with [(TSort _) \Rightarrow (le_n (S O)) | (TLRef _) \Rightarrow (le_n (S O)) | (THead _ t0 t1) \Rightarrow (le_S_n (S O) (S (plus (tweight t0) (tweight t1))) (le_n_S (S O) (S (plus (tweight t0) (tweight t1))) (le_n_S O (plus (tweight t0) (tweight t1)) (le_O_n (plus (tweight t0) (tweight t1))))))]).
 
 theorem clt_cong:
  \forall (c: C).(\forall (d: C).((clt c d) \to (\forall (k: K).(\forall (t: T).(clt (CHead c k t) (CHead d k t))))))
@@ -362,7 +362,7 @@ theorem clt_thead:
 theorem c_tail_ind:
  \forall (P: ((C \to Prop))).(((\forall (n: nat).(P (CSort n)))) \to (((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P (CTail k t c))))))) \to (\forall (c: C).(P c))))
 \def
- \lambda (P: ((C \to Prop))).(\lambda (H: ((\forall (n: nat).(P (CSort n))))).(\lambda (H0: ((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P (CTail k t c)))))))).(\lambda (c: C).(clt_wf_ind (\lambda (c0: C).(P c0)) (\lambda (c0: C).(match c0 return (\lambda (c1: C).(((\forall (d: C).((clt d c1) \to (P d)))) \to (P c1))) with [(CSort n) \Rightarrow (\lambda (_: ((\forall (d: C).((clt d (CSort n)) \to (P d))))).(H n)) | (CHead c1 k t) \Rightarrow (\lambda (H1: ((\forall (d: C).((clt d (CHead c1 k t)) \to (P d))))).(let H_x \def (chead_ctail c1 t k) in (let H2 \def H_x in (ex_3_ind K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c1 k t) (CTail h u d))))) (P (CHead c1 k t)) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H3: (eq C (CHead c1 k t) (CTail x0 x2 x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c2: C).(P c2)) (let H4 \def (eq_ind C (CHead c1 k t) (\lambda (c: C).(\forall (d: C).((clt d c) \to (P d)))) H1 (CTail x0 x2 x1) H3) in (H0 x1 (H4 x1 (clt_thead x0 x2 x1)) x0 x2)) (CHead c1 k t) H3))))) H2))))])) c)))).
+ \lambda (P: ((C \to Prop))).(\lambda (H: ((\forall (n: nat).(P (CSort n))))).(\lambda (H0: ((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P (CTail k t c)))))))).(\lambda (c: C).(clt_wf_ind (\lambda (c0: C).(P c0)) (\lambda (c0: C).(match c0 return (\lambda (_: ?).(\lambda (c1: C).(((\forall (d: C).((clt d c1) \to (P d)))) \to (P c1)))) with [(CSort n) \Rightarrow (\lambda (_: ((\forall (d: C).((clt d (CSort n)) \to (P d))))).(H n)) | (CHead c1 k t) \Rightarrow (\lambda (H1: ((\forall (d: C).((clt d (CHead c1 k t)) \to (P d))))).(let H_x \def (chead_ctail c1 t k) in (let H2 \def H_x in (ex_3_ind K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c1 k t) (CTail h u d))))) (P (CHead c1 k t)) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H3: (eq C (CHead c1 k t) (CTail x0 x2 x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c2: C).(P c2)) (let H4 \def (eq_ind C (CHead c1 k t) (\lambda (c: C).(\forall (d: C).((clt d c) \to (P d)))) H1 (CTail x0 x2 x1) H3) in (H0 x1 (H4 x1 (clt_thead x0 x2 x1)) x0 x2)) (CHead c1 k t) H3))))) H2))))])) c)))).
 
 definition fweight:
  C \to (T \to nat)
@@ -462,7 +462,7 @@ theorem lift_flat:
 theorem lift_gen_sort:
  \forall (h: nat).(\forall (d: nat).(\forall (n: nat).(\forall (t: T).((eq T (TSort n) (lift h d t)) \to (eq T t (TSort n))))))
 \def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq T (TSort n) (lift h d t0)) \to (eq T t0 (TSort n)))) (\lambda (n0: nat).(\lambda (H: (eq T (TSort n) (lift h d (TSort n0)))).(sym_eq T (TSort n) (TSort n0) H))) (\lambda (n0: nat).(\lambda (H: (eq T (TSort n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TSort n)) (\lambda (H0: (lt n0 d)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TSort n) t)) H (TLRef n0) (lift_lref_lt n0 h d H0)) in (let H2 \def (match H1 return (\lambda (t: T).((eq T t (TLRef n0)) \to (eq T (TLRef n0) (TSort n)))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (TSort n) (TLRef n0))).(let H2 \def (eq_ind T (TSort n) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n0) H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2)))]) in (H2 (refl_equal T (TLRef n0)))))) (\lambda (H0: (le d n0)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TSort n) t)) H (TLRef (plus n0 h)) (lift_lref_ge n0 h d H0)) in (let H2 \def (match H1 return (\lambda (t: T).((eq T t (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TSort n)))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (TSort n) (TLRef (plus n0 h)))).(let H2 \def (eq_ind T (TSort n) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef (plus n0 h)) H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2)))]) in (H2 (refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TSort n) (lift h d t0)) \to (eq T t0 (TSort n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TSort n) (lift h d t1)) \to (eq T t1 (TSort n))))).(\lambda (H1: (eq T (TSort n) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t: T).(eq T (TSort n) t)) H1 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def (match H2 return (\lambda (t: T).((eq T t (THead k (lift h d t0) (lift h (s k d) t1))) \to (eq T (THead k t0 t1) (TSort n)))) with [refl_equal \Rightarrow (\lambda (H2: (eq T (TSort n) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H3 \def (eq_ind T (TSort n) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) t1)) H2) in (False_ind (eq T (THead k t0 t1) (TSort n)) H3)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h (s k d) t1)))))))))))) t)))).
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq T (TSort n) (lift h d t0)) \to (eq T t0 (TSort n)))) (\lambda (n0: nat).(\lambda (H: (eq T (TSort n) (lift h d (TSort n0)))).(sym_eq T (TSort n) (TSort n0) H))) (\lambda (n0: nat).(\lambda (H: (eq T (TSort n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TSort n)) (\lambda (H0: (lt n0 d)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TSort n) t)) H (TLRef n0) (lift_lref_lt n0 h d H0)) in (let H2 \def (match H1 return (\lambda (_: ?).(\lambda (t: T).((eq T t (TLRef n0)) \to (eq T (TLRef n0) (TSort n))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (TSort n) (TLRef n0))).(let H2 \def (eq_ind T (TSort n) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n0) H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2)))]) in (H2 (refl_equal T (TLRef n0)))))) (\lambda (H0: (le d n0)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TSort n) t)) H (TLRef (plus n0 h)) (lift_lref_ge n0 h d H0)) in (let H2 \def (match H1 return (\lambda (_: ?).(\lambda (t: T).((eq T t (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TSort n))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (TSort n) (TLRef (plus n0 h)))).(let H2 \def (eq_ind T (TSort n) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef (plus n0 h)) H1) in (False_ind (eq T (TLRef n0) (TSort n)) H2)))]) in (H2 (refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TSort n) (lift h d t0)) \to (eq T t0 (TSort n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TSort n) (lift h d t1)) \to (eq T t1 (TSort n))))).(\lambda (H1: (eq T (TSort n) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t: T).(eq T (TSort n) t)) H1 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def (match H2 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead k (lift h d t0) (lift h (s k d) t1))) \to (eq T (THead k t0 t1) (TSort n))))) with [refl_equal \Rightarrow (\lambda (H2: (eq T (TSort n) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H3 \def (eq_ind T (TSort n) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) t1)) H2) in (False_ind (eq T (THead k t0 t1) (TSort n)) H3)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h (s k d) t1)))))))))))) t)))).
 
 theorem lift_gen_lref:
  \forall (t: T).(\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t)) \to (or (land (lt i d) (eq T t (TLRef i))) (land (le (plus d h) i) (eq T t (TLRef (minus i h)))))))))
@@ -472,32 +472,32 @@ theorem lift_gen_lref:
 theorem lift_gen_lref_lt:
  \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((lt n d) \to (\forall (t: T).((eq T (TLRef n) (lift h d t)) \to (eq T t (TLRef n)))))))
 \def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt n d)).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq T (TLRef n) (lift h d t0)) \to (eq T t0 (TLRef n)))) (\lambda (n0: nat).(\lambda (H0: (eq T (TLRef n) (lift h d (TSort n0)))).(sym_eq T (TLRef n) (TSort n0) H0))) (\lambda (n0: nat).(\lambda (H0: (eq T (TLRef n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TLRef n)) (\lambda (H1: (lt n0 d)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef n) t)) H0 (TLRef n0) (lift_lref_lt n0 h d H1)) in (sym_eq T (TLRef n) (TLRef n0) H2))) (\lambda (H1: (le d n0)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef n) t)) H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in (let H3 \def (match H2 return (\lambda (t: T).((eq T t (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TLRef n)))) with [refl_equal \Rightarrow (\lambda (H2: (eq T (TLRef n) (TLRef (plus n0 h)))).(let H3 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow n | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef (plus n0 h)) H2) in (eq_ind nat (plus n0 h) (\lambda (n: nat).(eq T (TLRef n0) (TLRef n))) (let H0 \def (eq_ind nat n (\lambda (n: nat).(lt n d)) H (plus n0 h) H3) in (le_false d n0 (eq T (TLRef n0) (TLRef (plus n0 h))) H1 (lt_le_S n0 d (le_lt_trans n0 (plus n0 h) d (le_plus_l n0 h) H0)))) n (sym_eq nat n (plus n0 h) H3))))]) in (H3 (refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef n) (lift h d t0)) \to (eq T t0 (TLRef n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef n) (lift h d t1)) \to (eq T t1 (TLRef n))))).(\lambda (H2: (eq T (TLRef n) (lift h d (THead k t0 t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t: T).(eq T (TLRef n) t)) H2 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H4 \def (match H3 return (\lambda (t: T).((eq T t (THead k (lift h d t0) (lift h (s k d) t1))) \to (eq T (THead k t0 t1) (TLRef n)))) with [refl_equal \Rightarrow (\lambda (H3: (eq T (TLRef n) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H4 \def (eq_ind T (TLRef n) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) t1)) H3) in (False_ind (eq T (THead k t0 t1) (TLRef n)) H4)))]) in (H4 (refl_equal T (THead k (lift h d t0) (lift h (s k d) t1)))))))))))) t))))).
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt n d)).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq T (TLRef n) (lift h d t0)) \to (eq T t0 (TLRef n)))) (\lambda (n0: nat).(\lambda (H0: (eq T (TLRef n) (lift h d (TSort n0)))).(sym_eq T (TLRef n) (TSort n0) H0))) (\lambda (n0: nat).(\lambda (H0: (eq T (TLRef n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TLRef n)) (\lambda (H1: (lt n0 d)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef n) t)) H0 (TLRef n0) (lift_lref_lt n0 h d H1)) in (sym_eq T (TLRef n) (TLRef n0) H2))) (\lambda (H1: (le d n0)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef n) t)) H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in (let H3 \def (match H2 return (\lambda (_: ?).(\lambda (t: T).((eq T t (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TLRef n))))) with [refl_equal \Rightarrow (\lambda (H2: (eq T (TLRef n) (TLRef (plus n0 h)))).(let H3 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow n | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef (plus n0 h)) H2) in (eq_ind nat (plus n0 h) (\lambda (n: nat).(eq T (TLRef n0) (TLRef n))) (let H0 \def (eq_ind nat n (\lambda (n: nat).(lt n d)) H (plus n0 h) H3) in (le_false d n0 (eq T (TLRef n0) (TLRef (plus n0 h))) H1 (lt_le_S n0 d (le_lt_trans n0 (plus n0 h) d (le_plus_l n0 h) H0)))) n (sym_eq nat n (plus n0 h) H3))))]) in (H3 (refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef n) (lift h d t0)) \to (eq T t0 (TLRef n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef n) (lift h d t1)) \to (eq T t1 (TLRef n))))).(\lambda (H2: (eq T (TLRef n) (lift h d (THead k t0 t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t: T).(eq T (TLRef n) t)) H2 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H4 \def (match H3 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead k (lift h d t0) (lift h (s k d) t1))) \to (eq T (THead k t0 t1) (TLRef n))))) with [refl_equal \Rightarrow (\lambda (H3: (eq T (TLRef n) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H4 \def (eq_ind T (TLRef n) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) t1)) H3) in (False_ind (eq T (THead k t0 t1) (TLRef n)) H4)))]) in (H4 (refl_equal T (THead k (lift h d t0) (lift h (s k d) t1)))))))))))) t))))).
 
 theorem lift_gen_lref_false:
  \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n (plus d h)) \to (\forall (t: T).((eq T (TLRef n) (lift h d t)) \to (\forall (P: Prop).P)))))))
 \def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d n)).(\lambda (H0: (lt n (plus d h))).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq T (TLRef n) (lift h d t0)) \to (\forall (P: Prop).P))) (\lambda (n0: nat).(\lambda (H1: (eq T (TLRef n) (lift h d (TSort n0)))).(\lambda (P: Prop).(let H2 \def (match H1 return (\lambda (t: T).((eq T t (lift h d (TSort n0))) \to P)) with [refl_equal \Rightarrow (\lambda (H2: (eq T (TLRef n) (lift h d (TSort n0)))).(let H3 \def (eq_ind T (TLRef n) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (lift h d (TSort n0)) H2) in (False_ind P H3)))]) in (H2 (refl_equal T (lift h d (TSort n0)))))))) (\lambda (n0: nat).(\lambda (H1: (eq T (TLRef n) (lift h d (TLRef n0)))).(\lambda (P: Prop).(lt_le_e n0 d P (\lambda (H2: (lt n0 d)).(let H3 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef n) t)) H1 (TLRef n0) (lift_lref_lt n0 h d H2)) in (let H4 \def (match H3 return (\lambda (t: T).((eq T t (TLRef n0)) \to P)) with [refl_equal \Rightarrow (\lambda (H3: (eq T (TLRef n) (TLRef n0))).(let H4 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow n | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef n0) H3) in (eq_ind nat n0 (\lambda (_: nat).P) (let H1 \def (eq_ind_r nat n0 (\lambda (n: nat).(lt n d)) H2 n H4) in (le_false d n P H H1)) n (sym_eq nat n n0 H4))))]) in (H4 (refl_equal T (TLRef n0)))))) (\lambda (H2: (le d n0)).(let H3 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef n) t)) H1 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H2)) in (let H4 \def (match H3 return (\lambda (t: T).((eq T t (TLRef (plus n0 h))) \to P)) with [refl_equal \Rightarrow (\lambda (H3: (eq T (TLRef n) (TLRef (plus n0 h)))).(let H4 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow n | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef (plus n0 h)) H3) in (eq_ind nat (plus n0 h) (\lambda (_: nat).P) (let H1 \def (eq_ind nat n (\lambda (n: nat).(lt n (plus d h))) H0 (plus n0 h) H4) in (le_false d n0 P H2 (lt_le_S n0 d (simpl_lt_plus_r h n0 d H1)))) n (sym_eq nat n (plus n0 h) H4))))]) in (H4 (refl_equal T (TLRef (plus n0 h))))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef n) (lift h d t0)) \to (\forall (P: Prop).P)))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef n) (lift h d t1)) \to (\forall (P: Prop).P)))).(\lambda (H3: (eq T (TLRef n) (lift h d (THead k t0 t1)))).(\lambda (P: Prop).(let H4 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t: T).(eq T (TLRef n) t)) H3 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H5 \def (match H4 return (\lambda (t: T).((eq T t (THead k (lift h d t0) (lift h (s k d) t1))) \to P)) with [refl_equal \Rightarrow (\lambda (H4: (eq T (TLRef n) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H5 \def (eq_ind T (TLRef n) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) t1)) H4) in (False_ind P H5)))]) in (H5 (refl_equal T (THead k (lift h d t0) (lift h (s k d) t1))))))))))))) t)))))).
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d n)).(\lambda (H0: (lt n (plus d h))).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq T (TLRef n) (lift h d t0)) \to (\forall (P: Prop).P))) (\lambda (n0: nat).(\lambda (H1: (eq T (TLRef n) (lift h d (TSort n0)))).(\lambda (P: Prop).(let H2 \def (match H1 return (\lambda (_: ?).(\lambda (t: T).((eq T t (lift h d (TSort n0))) \to P))) with [refl_equal \Rightarrow (\lambda (H2: (eq T (TLRef n) (lift h d (TSort n0)))).(let H3 \def (eq_ind T (TLRef n) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (lift h d (TSort n0)) H2) in (False_ind P H3)))]) in (H2 (refl_equal T (lift h d (TSort n0)))))))) (\lambda (n0: nat).(\lambda (H1: (eq T (TLRef n) (lift h d (TLRef n0)))).(\lambda (P: Prop).(lt_le_e n0 d P (\lambda (H2: (lt n0 d)).(let H3 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef n) t)) H1 (TLRef n0) (lift_lref_lt n0 h d H2)) in (let H4 \def (match H3 return (\lambda (_: ?).(\lambda (t: T).((eq T t (TLRef n0)) \to P))) with [refl_equal \Rightarrow (\lambda (H3: (eq T (TLRef n) (TLRef n0))).(let H4 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow n | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef n0) H3) in (eq_ind nat n0 (\lambda (_: nat).P) (let H1 \def (eq_ind_r nat n0 (\lambda (n: nat).(lt n d)) H2 n H4) in (le_false d n P H H1)) n (sym_eq nat n n0 H4))))]) in (H4 (refl_equal T (TLRef n0)))))) (\lambda (H2: (le d n0)).(let H3 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef n) t)) H1 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H2)) in (let H4 \def (match H3 return (\lambda (_: ?).(\lambda (t: T).((eq T t (TLRef (plus n0 h))) \to P))) with [refl_equal \Rightarrow (\lambda (H3: (eq T (TLRef n) (TLRef (plus n0 h)))).(let H4 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow n | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef (plus n0 h)) H3) in (eq_ind nat (plus n0 h) (\lambda (_: nat).P) (let H1 \def (eq_ind nat n (\lambda (n: nat).(lt n (plus d h))) H0 (plus n0 h) H4) in (le_false d n0 P H2 (lt_le_S n0 d (simpl_lt_plus_r h n0 d H1)))) n (sym_eq nat n (plus n0 h) H4))))]) in (H4 (refl_equal T (TLRef (plus n0 h))))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef n) (lift h d t0)) \to (\forall (P: Prop).P)))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef n) (lift h d t1)) \to (\forall (P: Prop).P)))).(\lambda (H3: (eq T (TLRef n) (lift h d (THead k t0 t1)))).(\lambda (P: Prop).(let H4 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t: T).(eq T (TLRef n) t)) H3 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H5 \def (match H4 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead k (lift h d t0) (lift h (s k d) t1))) \to P))) with [refl_equal \Rightarrow (\lambda (H4: (eq T (TLRef n) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H5 \def (eq_ind T (TLRef n) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) t1)) H4) in (False_ind P H5)))]) in (H5 (refl_equal T (THead k (lift h d t0) (lift h (s k d) t1))))))))))))) t)))))).
 
 theorem lift_gen_lref_ge:
  \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to (\forall (t: T).((eq T (TLRef (plus n h)) (lift h d t)) \to (eq T t (TLRef n)))))))
 \def
- \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d n)).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq T (TLRef (plus n h)) (lift h d t0)) \to (eq T t0 (TLRef n)))) (\lambda (n0: nat).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d (TSort n0)))).(let H1 \def (match H0 return (\lambda (t: T).((eq T t (lift h d (TSort n0))) \to (eq T (TSort n0) (TLRef n)))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (TLRef (plus n h)) (lift h d (TSort n0)))).(let H2 \def (eq_ind T (TLRef (plus n h)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (lift h d (TSort n0)) H1) in (False_ind (eq T (TSort n0) (TLRef n)) H2)))]) in (H1 (refl_equal T (lift h d (TSort n0))))))) (\lambda (n0: nat).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TLRef n)) (\lambda (H1: (lt n0 d)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef (plus n h)) t)) H0 (TLRef n0) (lift_lref_lt n0 h d H1)) in (let H3 \def (match H2 return (\lambda (t: T).((eq T t (TLRef n0)) \to (eq T (TLRef n0) (TLRef n)))) with [refl_equal \Rightarrow (\lambda (H2: (eq T (TLRef (plus n h)) (TLRef n0))).(let H3 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus) n h) | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus) n h)])) (TLRef (plus n h)) (TLRef n0) H2) in (eq_ind nat (plus n h) (\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (let H0 \def (eq_ind_r nat n0 (\lambda (n: nat).(lt n d)) H1 (plus n h) H3) in (le_false d n (eq T (TLRef (plus n h)) (TLRef n)) H (lt_le_S n d (le_lt_trans n (plus n h) d (le_plus_l n h) H0)))) n0 H3)))]) in (H3 (refl_equal T (TLRef n0)))))) (\lambda (H1: (le d n0)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef (plus n h)) t)) H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in (let H3 \def (match H2 return (\lambda (t: T).((eq T t (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TLRef n)))) with [refl_equal \Rightarrow (\lambda (H2: (eq T (TLRef (plus n h)) (TLRef (plus n0 h)))).(let H3 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus) n h) | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus) n h)])) (TLRef (plus n h)) (TLRef (plus n0 h)) H2) in (eq_ind nat (plus n h) (\lambda (_: nat).(eq T (TLRef n0) (TLRef n))) (f_equal nat T TLRef n0 n (simpl_plus_r h n0 n (sym_eq nat (plus n h) (plus n0 h) H3))) (plus n0 h) H3)))]) in (H3 (refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef (plus n h)) (lift h d t0)) \to (eq T t0 (TLRef n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef (plus n h)) (lift h d t1)) \to (eq T t1 (TLRef n))))).(\lambda (H2: (eq T (TLRef (plus n h)) (lift h d (THead k t0 t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t: T).(eq T (TLRef (plus n h)) t)) H2 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H4 \def (match H3 return (\lambda (t: T).((eq T t (THead k (lift h d t0) (lift h (s k d) t1))) \to (eq T (THead k t0 t1) (TLRef n)))) with [refl_equal \Rightarrow (\lambda (H3: (eq T (TLRef (plus n h)) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H4 \def (eq_ind T (TLRef (plus n h)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) t1)) H3) in (False_ind (eq T (THead k t0 t1) (TLRef n)) H4)))]) in (H4 (refl_equal T (THead k (lift h d t0) (lift h (s k d) t1)))))))))))) t))))).
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d n)).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq T (TLRef (plus n h)) (lift h d t0)) \to (eq T t0 (TLRef n)))) (\lambda (n0: nat).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d (TSort n0)))).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (lift h d (TSort n0))) \to (eq T (TSort n0) (TLRef n))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (TLRef (plus n h)) (lift h d (TSort n0)))).(let H2 \def (eq_ind T (TLRef (plus n h)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (lift h d (TSort n0)) H1) in (False_ind (eq T (TSort n0) (TLRef n)) H2)))]) in (H1 (refl_equal T (lift h d (TSort n0))))))) (\lambda (n0: nat).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TLRef n)) (\lambda (H1: (lt n0 d)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef (plus n h)) t)) H0 (TLRef n0) (lift_lref_lt n0 h d H1)) in (let H3 \def (match H2 return (\lambda (_: ?).(\lambda (t: T).((eq T t (TLRef n0)) \to (eq T (TLRef n0) (TLRef n))))) with [refl_equal \Rightarrow (\lambda (H2: (eq T (TLRef (plus n h)) (TLRef n0))).(let H3 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus) n h) | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus) n h)])) (TLRef (plus n h)) (TLRef n0) H2) in (eq_ind nat (plus n h) (\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (let H0 \def (eq_ind_r nat n0 (\lambda (n: nat).(lt n d)) H1 (plus n h) H3) in (le_false d n (eq T (TLRef (plus n h)) (TLRef n)) H (lt_le_S n d (le_lt_trans n (plus n h) d (le_plus_l n h) H0)))) n0 H3)))]) in (H3 (refl_equal T (TLRef n0)))))) (\lambda (H1: (le d n0)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef (plus n h)) t)) H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in (let H3 \def (match H2 return (\lambda (_: ?).(\lambda (t: T).((eq T t (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TLRef n))))) with [refl_equal \Rightarrow (\lambda (H2: (eq T (TLRef (plus n h)) (TLRef (plus n0 h)))).(let H3 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus) n h) | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus) n h)])) (TLRef (plus n h)) (TLRef (plus n0 h)) H2) in (eq_ind nat (plus n h) (\lambda (_: nat).(eq T (TLRef n0) (TLRef n))) (f_equal nat T TLRef n0 n (simpl_plus_r h n0 n (sym_eq nat (plus n h) (plus n0 h) H3))) (plus n0 h) H3)))]) in (H3 (refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef (plus n h)) (lift h d t0)) \to (eq T t0 (TLRef n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef (plus n h)) (lift h d t1)) \to (eq T t1 (TLRef n))))).(\lambda (H2: (eq T (TLRef (plus n h)) (lift h d (THead k t0 t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t: T).(eq T (TLRef (plus n h)) t)) H2 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H4 \def (match H3 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead k (lift h d t0) (lift h (s k d) t1))) \to (eq T (THead k t0 t1) (TLRef n))))) with [refl_equal \Rightarrow (\lambda (H3: (eq T (TLRef (plus n h)) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H4 \def (eq_ind T (TLRef (plus n h)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) t1)) H3) in (False_ind (eq T (THead k t0 t1) (TLRef n)) H4)))]) in (H4 (refl_equal T (THead k (lift h d t0) (lift h (s k d) t1)))))))))))) t))))).
 
 theorem lift_gen_head:
  \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) (lift h d x)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))))
 \def
- \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) (lift h d (TSort n)))).(let H0 \def (match H return (\lambda (t0: T).((eq T t0 (lift h d (TSort n))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead k u t) (lift h d (TSort n)))).(let H1 \def (eq_ind T (THead k u t) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (lift h d (TSort n)) H0) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) H1)))]) in (H0 (refl_equal T (lift h d (TSort n))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) (lift h d (TLRef n)))).(lt_le_e n d (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead k u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def (match H1 return (\lambda (t0: T).((eq T t0 (TLRef n)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (THead k u t) (TLRef n))).(let H2 \def (eq_ind T (THead k u t) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) H2)))]) in (H2 (refl_equal T (TLRef n)))))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead k u t) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 return (\lambda (t0: T).((eq T t0 (TLRef (plus n h))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (THead k u t) (TLRef (plus n h)))).(let H2 \def (eq_ind T (THead k u t) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef (plus n h)) H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) H2)))]) in (H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda (k0: K).(\lambda (t0: T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) (lift h d t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k u t) (lift h d (THead k0 t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k0 t0 t1)) (\lambda (t0: T).(eq T (THead k u t) t0)) H1 (THead k0 (lift h d t0) (lift h (s k0 d) t1)) (lift_head k0 t0 t1 h d)) in (let H3 \def (match H2 return (\lambda (t2: T).((eq T t2 (THead k0 (lift h d t0) (lift h (s k0 d) t1))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))))) with [refl_equal \Rightarrow (\lambda (H2: (eq T (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)))).(let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t) \Rightarrow t])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H5 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) H2) in (eq_ind K k0 (\lambda (k: K).((eq T u (lift h d t0)) \to ((eq T t (lift h (s k0 d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))) (\lambda (H6: (eq T u (lift h d t0))).(eq_ind T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s k0 d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k0 d) z))))))) (\lambda (H7: (eq T t (lift h (s k0 d) t1))).(eq_ind T (lift h (s k0 d) t1) (\lambda (t: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k0 d) z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k0 d) t1) (lift h (s k0 d) z)))) t0 t1 (refl_equal T (THead k0 t0 t1)) (refl_equal T (lift h d t0)) (refl_equal T (lift h (s k0 d) t1))) t (sym_eq T t (lift h (s k0 d) t1) H7))) u (sym_eq T u (lift h d t0) H6))) k (sym_eq K k k0 H5))) H4)) H3)))]) in (H3 (refl_equal T (THead k0 (lift h d t0) (lift h (s k0 d) t1)))))))))))))) x)))).
+ \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) (lift h d (TSort n)))).(let H0 \def (match H return (\lambda (_: ?).(\lambda (t0: T).((eq T t0 (lift h d (TSort n))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead k u t) (lift h d (TSort n)))).(let H1 \def (eq_ind T (THead k u t) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (lift h d (TSort n)) H0) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) H1)))]) in (H0 (refl_equal T (lift h d (TSort n))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) (lift h d (TLRef n)))).(lt_le_e n d (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead k u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def (match H1 return (\lambda (_: ?).(\lambda (t0: T).((eq T t0 (TLRef n)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (THead k u t) (TLRef n))).(let H2 \def (eq_ind T (THead k u t) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) H2)))]) in (H2 (refl_equal T (TLRef n)))))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead k u t) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 return (\lambda (_: ?).(\lambda (t0: T).((eq T t0 (TLRef (plus n h))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (THead k u t) (TLRef (plus n h)))).(let H2 \def (eq_ind T (THead k u t) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef (plus n h)) H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) H2)))]) in (H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda (k0: K).(\lambda (t0: T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) (lift h d t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k u t) (lift h d (THead k0 t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k0 t0 t1)) (\lambda (t0: T).(eq T (THead k u t) t0)) H1 (THead k0 (lift h d t0) (lift h (s k0 d) t1)) (lift_head k0 t0 t1 h d)) in (let H3 \def (match H2 return (\lambda (_: ?).(\lambda (t2: T).((eq T t2 (THead k0 (lift h d t0) (lift h (s k0 d) t1))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))) with [refl_equal \Rightarrow (\lambda (H2: (eq T (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)))).(let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t) \Rightarrow t])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H5 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) H2) in (eq_ind K k0 (\lambda (k: K).((eq T u (lift h d t0)) \to ((eq T t (lift h (s k0 d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))) (\lambda (H6: (eq T u (lift h d t0))).(eq_ind T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s k0 d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k0 d) z))))))) (\lambda (H7: (eq T t (lift h (s k0 d) t1))).(eq_ind T (lift h (s k0 d) t1) (\lambda (t: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k0 d) z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k0 d) t1) (lift h (s k0 d) z)))) t0 t1 (refl_equal T (THead k0 t0 t1)) (refl_equal T (lift h d t0)) (refl_equal T (lift h (s k0 d) t1))) t (sym_eq T t (lift h (s k0 d) t1) H7))) u (sym_eq T u (lift h d t0) H6))) k (sym_eq K k k0 H5))) H4)) H3)))]) in (H3 (refl_equal T (THead k0 (lift h d t0) (lift h (s k0 d) t1)))))))))))))) x)))).
 
 theorem lift_gen_bind:
  \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead (Bind b) u t) (lift h d x)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))))))
 \def
- \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead (Bind b) u t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u t) (lift h d (TSort n)))).(let H0 \def (match H return (\lambda (t0: T).((eq T t0 (lift h d (TSort n))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind b) u t) (lift h d (TSort n)))).(let H1 \def (eq_ind T (THead (Bind b) u t) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (lift h d (TSort n)) H0) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H1)))]) in (H0 (refl_equal T (lift h d (TSort n))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u t) (lift h d (TLRef n)))).(lt_le_e n d (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def (match H1 return (\lambda (t0: T).((eq T t0 (TLRef n)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (THead (Bind b) u t) (TLRef n))).(let H2 \def (eq_ind T (THead (Bind b) u t) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H2)))]) in (H2 (refl_equal T (TLRef n)))))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 return (\lambda (t0: T).((eq T t0 (TLRef (plus n h))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (THead (Bind b) u t) (TLRef (plus n h)))).(let H2 \def (eq_ind T (THead (Bind b) u t) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef (plus n h)) H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H2)))]) in (H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead (Bind b) u t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead (Bind b) u t) (lift h d t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead (Bind b) u t) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0)) H1 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def (match H2 return (\lambda (t2: T).((eq T t2 (THead k (lift h d t0) (lift h (s k d) t1))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))))) with [refl_equal \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u t) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H2) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H2) in ((let H5 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef _) \Rightarrow (Bind b) | (THead k _ _) \Rightarrow k])) (THead (Bind b) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H2) in (eq_ind K (Bind b) (\lambda (k: K).((eq T u (lift h d t0)) \to ((eq T t (lift h (s k d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))) (\lambda (H6: (eq T u (lift h d t0))).(eq_ind T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s (Bind b) d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))))) (\lambda (H7: (eq T t (lift h (s (Bind b) d) t1))).(eq_ind T (lift h (s (Bind b) d) t1) (\lambda (t: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s (Bind b) d) t1) (lift h (S d) z)))) t0 t1 (refl_equal T (THead (Bind b) t0 t1)) (refl_equal T (lift h d t0)) (refl_equal T (lift h (S d) t1))) t (sym_eq T t (lift h (s (Bind b) d) t1) H7))) u (sym_eq T u (lift h d t0) H6))) k H5)) H4)) H3)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h (s k d) t1)))))))))))))) x)))).
+ \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead (Bind b) u t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u t) (lift h d (TSort n)))).(let H0 \def (match H return (\lambda (_: ?).(\lambda (t0: T).((eq T t0 (lift h d (TSort n))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind b) u t) (lift h d (TSort n)))).(let H1 \def (eq_ind T (THead (Bind b) u t) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (lift h d (TSort n)) H0) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H1)))]) in (H0 (refl_equal T (lift h d (TSort n))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u t) (lift h d (TLRef n)))).(lt_le_e n d (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def (match H1 return (\lambda (_: ?).(\lambda (t0: T).((eq T t0 (TLRef n)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (THead (Bind b) u t) (TLRef n))).(let H2 \def (eq_ind T (THead (Bind b) u t) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H2)))]) in (H2 (refl_equal T (TLRef n)))))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 return (\lambda (_: ?).(\lambda (t0: T).((eq T t0 (TLRef (plus n h))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (THead (Bind b) u t) (TLRef (plus n h)))).(let H2 \def (eq_ind T (THead (Bind b) u t) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef (plus n h)) H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H2)))]) in (H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead (Bind b) u t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead (Bind b) u t) (lift h d t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead (Bind b) u t) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0)) H1 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def (match H2 return (\lambda (_: ?).(\lambda (t2: T).((eq T t2 (THead k (lift h d t0) (lift h (s k d) t1))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))) with [refl_equal \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u t) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H2) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H2) in ((let H5 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef _) \Rightarrow (Bind b) | (THead k _ _) \Rightarrow k])) (THead (Bind b) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H2) in (eq_ind K (Bind b) (\lambda (k: K).((eq T u (lift h d t0)) \to ((eq T t (lift h (s k d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))) (\lambda (H6: (eq T u (lift h d t0))).(eq_ind T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s (Bind b) d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))))) (\lambda (H7: (eq T t (lift h (s (Bind b) d) t1))).(eq_ind T (lift h (s (Bind b) d) t1) (\lambda (t: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s (Bind b) d) t1) (lift h (S d) z)))) t0 t1 (refl_equal T (THead (Bind b) t0 t1)) (refl_equal T (lift h d t0)) (refl_equal T (lift h (S d) t1))) t (sym_eq T t (lift h (s (Bind b) d) t1) H7))) u (sym_eq T u (lift h d t0) H6))) k H5)) H4)) H3)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h (s k d) t1)))))))))))))) x)))).
 
 theorem lift_gen_flat:
  \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d x)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z)))))))))))
 \def
- \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Flat f) u t) (lift h d (TSort n)))).(let H0 \def (match H return (\lambda (t0: T).((eq T t0 (lift h d (TSort n))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat f) u t) (lift h d (TSort n)))).(let H1 \def (eq_ind T (THead (Flat f) u t) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (lift h d (TSort n)) H0) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))) H1)))]) in (H0 (refl_equal T (lift h d (TSort n))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Flat f) u t) (lift h d (TLRef n)))).(lt_le_e n d (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead (Flat f) u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def (match H1 return (\lambda (t0: T).((eq T t0 (TLRef n)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (THead (Flat f) u t) (TLRef n))).(let H2 \def (eq_ind T (THead (Flat f) u t) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))) H2)))]) in (H2 (refl_equal T (TLRef n)))))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead (Flat f) u t) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 return (\lambda (t0: T).((eq T t0 (TLRef (plus n h))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (THead (Flat f) u t) (TLRef (plus n h)))).(let H2 \def (eq_ind T (THead (Flat f) u t) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef (plus n h)) H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))) H2)))]) in (H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z)))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z)))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead (Flat f) u t) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t0: T).(eq T (THead (Flat f) u t) t0)) H1 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def (match H2 return (\lambda (t2: T).((eq T t2 (THead k (lift h d t0) (lift h (s k d) t1))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))))) with [refl_equal \Rightarrow (\lambda (H2: (eq T (THead (Flat f) u t) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t) \Rightarrow t])) (THead (Flat f) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H2) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat f) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H2) in ((let H5 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow (Flat f) | (TLRef _) \Rightarrow (Flat f) | (THead k _ _) \Rightarrow k])) (THead (Flat f) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H2) in (eq_ind K (Flat f) (\lambda (k: K).((eq T u (lift h d t0)) \to ((eq T t (lift h (s k d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z)))))))) (\lambda (H6: (eq T u (lift h d t0))).(eq_ind T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s (Flat f) d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))))) (\lambda (H7: (eq T t (lift h (s (Flat f) d) t1))).(eq_ind T (lift h (s (Flat f) d) t1) (\lambda (t: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s (Flat f) d) t1) (lift h d z)))) t0 t1 (refl_equal T (THead (Flat f) t0 t1)) (refl_equal T (lift h d t0)) (refl_equal T (lift h d t1))) t (sym_eq T t (lift h (s (Flat f) d) t1) H7))) u (sym_eq T u (lift h d t0) H6))) k H5)) H4)) H3)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h (s k d) t1)))))))))))))) x)))).
+ \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Flat f) u t) (lift h d (TSort n)))).(let H0 \def (match H return (\lambda (_: ?).(\lambda (t0: T).((eq T t0 (lift h d (TSort n))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z)))))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat f) u t) (lift h d (TSort n)))).(let H1 \def (eq_ind T (THead (Flat f) u t) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (lift h d (TSort n)) H0) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))) H1)))]) in (H0 (refl_equal T (lift h d (TSort n))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Flat f) u t) (lift h d (TLRef n)))).(lt_le_e n d (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead (Flat f) u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def (match H1 return (\lambda (_: ?).(\lambda (t0: T).((eq T t0 (TLRef n)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z)))))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (THead (Flat f) u t) (TLRef n))).(let H2 \def (eq_ind T (THead (Flat f) u t) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))) H2)))]) in (H2 (refl_equal T (TLRef n)))))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead (Flat f) u t) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 return (\lambda (_: ?).(\lambda (t0: T).((eq T t0 (TLRef (plus n h))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z)))))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (THead (Flat f) u t) (TLRef (plus n h)))).(let H2 \def (eq_ind T (THead (Flat f) u t) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef (plus n h)) H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))) H2)))]) in (H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z)))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z)))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead (Flat f) u t) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t0: T).(eq T (THead (Flat f) u t) t0)) H1 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def (match H2 return (\lambda (_: ?).(\lambda (t2: T).((eq T t2 (THead k (lift h d t0) (lift h (s k d) t1))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z)))))))) with [refl_equal \Rightarrow (\lambda (H2: (eq T (THead (Flat f) u t) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t) \Rightarrow t])) (THead (Flat f) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H2) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat f) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H2) in ((let H5 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow (Flat f) | (TLRef _) \Rightarrow (Flat f) | (THead k _ _) \Rightarrow k])) (THead (Flat f) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H2) in (eq_ind K (Flat f) (\lambda (k: K).((eq T u (lift h d t0)) \to ((eq T t (lift h (s k d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z)))))))) (\lambda (H6: (eq T u (lift h d t0))).(eq_ind T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s (Flat f) d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))))) (\lambda (H7: (eq T t (lift h (s (Flat f) d) t1))).(eq_ind T (lift h (s (Flat f) d) t1) (\lambda (t: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s (Flat f) d) t1) (lift h d z)))) t0 t1 (refl_equal T (THead (Flat f) t0 t1)) (refl_equal T (lift h d t0)) (refl_equal T (lift h d t1))) t (sym_eq T t (lift h (s (Flat f) d) t1) H7))) u (sym_eq T u (lift h d t0) H6))) k H5)) H4)) H3)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h (s k d) t1)))))))))))))) x)))).
 
 theorem thead_x_lift_y_y:
  \forall (k: K).(\forall (t: T).(\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift h d t)) t) \to (\forall (P: Prop).P))))))
@@ -552,7 +552,7 @@ theorem lift_weight_add:
 theorem lift_weight_add_O:
  \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (f: ((nat \to nat))).(eq nat (weight_map f (lift h O t)) (weight_map (wadd f w) (lift (S h) O t))))))
 \def
- \lambda (w: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (f: ((nat \to nat))).(lift_weight_add (plus (wadd f w O) O) t h O f (wadd f w) (\lambda (m: nat).(\lambda (H: (lt m O)).(let H0 \def (match H return (\lambda (n: nat).((eq nat n O) \to (eq nat (wadd f w m) (f m)))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) O)).(let H1 \def (eq_ind nat (S m) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq nat (wadd f w m) (f m)) H1))) | (le_S m0 H0) \Rightarrow (\lambda (H1: (eq nat (S m0) O)).((let H2 \def (eq_ind nat (S m0) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind ((le (S m) m0) \to (eq nat (wadd f w m) (f m))) H2)) H0))]) in (H0 (refl_equal nat O))))) (plus_n_O (wadd f w O)) (\lambda (m: nat).(\lambda (_: (le O m)).(refl_equal nat (f m)))))))).
+ \lambda (w: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (f: ((nat \to nat))).(lift_weight_add (plus (wadd f w O) O) t h O f (wadd f w) (\lambda (m: nat).(\lambda (H: (lt m O)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (n: nat).((eq nat n O) \to (eq nat (wadd f w m) (f m))))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) O)).(let H1 \def (eq_ind nat (S m) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq nat (wadd f w m) (f m)) H1))) | (le_S m0 H0) \Rightarrow (\lambda (H1: (eq nat (S m0) O)).((let H2 \def (eq_ind nat (S m0) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind ((le (S m) m0) \to (eq nat (wadd f w m) (f m))) H2)) H0))]) in (H0 (refl_equal nat O))))) (plus_n_O (wadd f w O)) (\lambda (m: nat).(\lambda (_: (le O m)).(refl_equal nat (f m)))))))).
 
 theorem lift_tlt_dx:
  \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).(tlt t (THead k u (lift h d t)))))))
@@ -660,17 +660,17 @@ theorem drop_gen_refl:
 theorem drop_gen_drop:
  \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).((drop (S h) O (CHead c k u) x) \to (drop (r k h) O c x))))))
 \def
- \lambda (k: K).(\lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda (H: (drop (S h) O (CHead c k u) x)).(insert_eq C (CHead c k u) (\lambda (c0: C).(drop (S h) O c0 x)) (drop (r k h) O c x) (\lambda (y: C).(\lambda (H0: (drop (S h) O y x)).(insert_eq nat O (\lambda (n: nat).(drop (S h) n y x)) ((eq C y (CHead c k u)) \to (drop (r k h) O c x)) (\lambda (y0: nat).(\lambda (H1: (drop (S h) y0 y x)).(insert_eq nat (S h) (\lambda (n: nat).(drop n y0 y x)) ((eq nat y0 O) \to ((eq C y (CHead c k u)) \to (drop (r k h) O c x))) (\lambda (y1: nat).(\lambda (H2: (drop y1 y0 y x)).(drop_ind (\lambda (n: nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n (S h)) \to ((eq nat n0 O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) O c c1)))))))) (\lambda (c0: C).(\lambda (H3: (eq nat O (S h))).(\lambda (_: (eq nat O O)).(\lambda (_: (eq C c0 (CHead c k u))).(let H6 \def (match H3 return (\lambda (n: nat).((eq nat n (S h)) \to (drop (r k h) O c c0))) with [refl_equal \Rightarrow (\lambda (H2: (eq nat O (S h))).(let H3 \def (eq_ind nat O (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S h) H2) in (False_ind (drop (r k h) O c c0) H3)))]) in (H6 (refl_equal nat (S h)))))))) (\lambda (k0: K).(\lambda (h0: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H3: (drop (r k0 h0) O c0 e)).(\lambda (_: (((eq nat (r k0 h0) (S h)) \to ((eq nat O O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) O c e)))))).(\lambda (u0: T).(\lambda (H5: (eq nat (S h0) (S h))).(\lambda (_: (eq nat O O)).(\lambda (H7: (eq C (CHead c0 k0 u0) (CHead c k u))).(let H8 \def (match H5 return (\lambda (n: nat).((eq nat n (S h)) \to (drop (r k h) O c e))) with [refl_equal \Rightarrow (\lambda (H4: (eq nat (S h0) (S h))).(let H5 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow h0 | (S n) \Rightarrow n])) (S h0) (S h) H4) in (eq_ind nat h (\lambda (_: nat).(drop (r k h) O c e)) (let H6 \def (match H7 return (\lambda (c0: C).((eq C c0 (CHead c k u)) \to (drop (r k h) O c e))) with [refl_equal \Rightarrow (\lambda (H4: (eq C (CHead c0 k0 u0) (CHead c k u))).(let H6 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c k u) H4) in ((let H7 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c0 k0 u0) (CHead c k u) H4) in ((let H8 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u0) (CHead c k u) H4) in (eq_ind C c (\lambda (_: C).((eq K k0 k) \to ((eq T u0 u) \to (drop (r k h) O c e)))) (\lambda (H9: (eq K k0 k)).(eq_ind K k (\lambda (_: K).((eq T u0 u) \to (drop (r k h) O c e))) (\lambda (H10: (eq T u0 u)).(eq_ind T u (\lambda (_: T).(drop (r k h) O c e)) (eq_ind nat h0 (\lambda (n: nat).(drop (r k n) O c e)) (eq_ind C c0 (\lambda (c: C).(drop (r k h0) O c e)) (eq_ind K k0 (\lambda (k: K).(drop (r k h0) O c0 e)) H3 k H9) c H8) h H5) u0 (sym_eq T u0 u H10))) k0 (sym_eq K k0 k H9))) c0 (sym_eq C c0 c H8))) H7)) H6)))]) in (H6 (refl_equal C (CHead c k u)))) h0 (sym_eq nat h0 h H5))))]) in (H8 (refl_equal nat (S h)))))))))))))) (\lambda (k0: K).(\lambda (h0: nat).(\lambda (d: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (_: (drop h0 (r k0 d) c0 e)).(\lambda (_: (((eq nat h0 (S h)) \to ((eq nat (r k0 d) O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) O c e)))))).(\lambda (u0: T).(\lambda (_: (eq nat h0 (S h))).(\lambda (H6: (eq nat (S d) O)).(\lambda (_: (eq C (CHead c0 k0 (lift h0 (r k0 d) u0)) (CHead c k u))).(let H8 \def (match H6 return (\lambda (n: nat).((eq nat n O) \to (drop (r k h) O c (CHead e k0 u0)))) with [refl_equal \Rightarrow (\lambda (H4: (eq nat (S d) O)).(let H5 \def (eq_ind nat (S d) (\lambda (e0: nat).(match e0 return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (drop (r k h) O c (CHead e k0 u0)) H5)))]) in (H8 (refl_equal nat O)))))))))))))) y1 y0 y x H2))) H1))) H0))) H)))))).
+ \lambda (k: K).(\lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda (H: (drop (S h) O (CHead c k u) x)).(insert_eq C (CHead c k u) (\lambda (c0: C).(drop (S h) O c0 x)) (drop (r k h) O c x) (\lambda (y: C).(\lambda (H0: (drop (S h) O y x)).(insert_eq nat O (\lambda (n: nat).(drop (S h) n y x)) ((eq C y (CHead c k u)) \to (drop (r k h) O c x)) (\lambda (y0: nat).(\lambda (H1: (drop (S h) y0 y x)).(insert_eq nat (S h) (\lambda (n: nat).(drop n y0 y x)) ((eq nat y0 O) \to ((eq C y (CHead c k u)) \to (drop (r k h) O c x))) (\lambda (y1: nat).(\lambda (H2: (drop y1 y0 y x)).(drop_ind (\lambda (n: nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n (S h)) \to ((eq nat n0 O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) O c c1)))))))) (\lambda (c0: C).(\lambda (H3: (eq nat O (S h))).(\lambda (_: (eq nat O O)).(\lambda (_: (eq C c0 (CHead c k u))).(let H6 \def (match H3 return (\lambda (_: ?).(\lambda (n: nat).((eq nat n (S h)) \to (drop (r k h) O c c0)))) with [refl_equal \Rightarrow (\lambda (H2: (eq nat O (S h))).(let H3 \def (eq_ind nat O (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S h) H2) in (False_ind (drop (r k h) O c c0) H3)))]) in (H6 (refl_equal nat (S h)))))))) (\lambda (k0: K).(\lambda (h0: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H3: (drop (r k0 h0) O c0 e)).(\lambda (_: (((eq nat (r k0 h0) (S h)) \to ((eq nat O O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) O c e)))))).(\lambda (u0: T).(\lambda (H5: (eq nat (S h0) (S h))).(\lambda (_: (eq nat O O)).(\lambda (H7: (eq C (CHead c0 k0 u0) (CHead c k u))).(let H8 \def (match H5 return (\lambda (_: ?).(\lambda (n: nat).((eq nat n (S h)) \to (drop (r k h) O c e)))) with [refl_equal \Rightarrow (\lambda (H4: (eq nat (S h0) (S h))).(let H5 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow h0 | (S n) \Rightarrow n])) (S h0) (S h) H4) in (eq_ind nat h (\lambda (_: nat).(drop (r k h) O c e)) (let H6 \def (match H7 return (\lambda (_: ?).(\lambda (c0: C).((eq C c0 (CHead c k u)) \to (drop (r k h) O c e)))) with [refl_equal \Rightarrow (\lambda (H4: (eq C (CHead c0 k0 u0) (CHead c k u))).(let H6 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c k u) H4) in ((let H7 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c0 k0 u0) (CHead c k u) H4) in ((let H8 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u0) (CHead c k u) H4) in (eq_ind C c (\lambda (_: C).((eq K k0 k) \to ((eq T u0 u) \to (drop (r k h) O c e)))) (\lambda (H9: (eq K k0 k)).(eq_ind K k (\lambda (_: K).((eq T u0 u) \to (drop (r k h) O c e))) (\lambda (H10: (eq T u0 u)).(eq_ind T u (\lambda (_: T).(drop (r k h) O c e)) (eq_ind nat h0 (\lambda (n: nat).(drop (r k n) O c e)) (eq_ind C c0 (\lambda (c: C).(drop (r k h0) O c e)) (eq_ind K k0 (\lambda (k: K).(drop (r k h0) O c0 e)) H3 k H9) c H8) h H5) u0 (sym_eq T u0 u H10))) k0 (sym_eq K k0 k H9))) c0 (sym_eq C c0 c H8))) H7)) H6)))]) in (H6 (refl_equal C (CHead c k u)))) h0 (sym_eq nat h0 h H5))))]) in (H8 (refl_equal nat (S h)))))))))))))) (\lambda (k0: K).(\lambda (h0: nat).(\lambda (d: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (_: (drop h0 (r k0 d) c0 e)).(\lambda (_: (((eq nat h0 (S h)) \to ((eq nat (r k0 d) O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) O c e)))))).(\lambda (u0: T).(\lambda (_: (eq nat h0 (S h))).(\lambda (H6: (eq nat (S d) O)).(\lambda (_: (eq C (CHead c0 k0 (lift h0 (r k0 d) u0)) (CHead c k u))).(let H8 \def (match H6 return (\lambda (_: ?).(\lambda (n: nat).((eq nat n O) \to (drop (r k h) O c (CHead e k0 u0))))) with [refl_equal \Rightarrow (\lambda (H4: (eq nat (S d) O)).(let H5 \def (eq_ind nat (S d) (\lambda (e0: nat).(match e0 return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (drop (r k h) O c (CHead e k0 u0)) H5)))]) in (H8 (refl_equal nat O)))))))))))))) y1 y0 y x H2))) H1))) H0))) H)))))).
 
 theorem drop_gen_skip_r:
  \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall (d: nat).(\forall (k: K).((drop h (S d) x (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C x (CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k d) e c)))))))))
 \def
- \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) x (CHead c k u))).(let H0 \def (match H return (\lambda (n: nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n h) \to ((eq nat n0 (S d)) \to ((eq C c0 x) \to ((eq C c1 (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C x (CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k d) e c))))))))))) with [(drop_refl c0) \Rightarrow (\lambda (H0: (eq nat O h)).(\lambda (H1: (eq nat O (S d))).(\lambda (H2: (eq C c0 x)).(\lambda (H3: (eq C c0 (CHead c k u))).(eq_ind nat O (\lambda (n: nat).((eq nat O (S d)) \to ((eq C c0 x) \to ((eq C c0 (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C x (CHead e k (lift n (r k d) u)))) (\lambda (e: C).(drop n (r k d) e c))))))) (\lambda (H4: (eq nat O (S d))).(let H5 \def (eq_ind nat O (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S d) H4) in (False_ind ((eq C c0 x) \to ((eq C c0 (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C x (CHead e k (lift O (r k d) u)))) (\lambda (e: C).(drop O (r k d) e c))))) H5))) h H0 H1 H2 H3))))) | (drop_drop k0 h0 c0 e H0 u0) \Rightarrow (\lambda (H1: (eq nat (S h0) h)).(\lambda (H2: (eq nat O (S d))).(\lambda (H3: (eq C (CHead c0 k0 u0) x)).(\lambda (H4: (eq C e (CHead c k u))).(eq_ind nat (S h0) (\lambda (n: nat).((eq nat O (S d)) \to ((eq C (CHead c0 k0 u0) x) \to ((eq C e (CHead c k u)) \to ((drop (r k0 h0) O c0 e) \to (ex2 C (\lambda (e0: C).(eq C x (CHead e0 k (lift n (r k d) u)))) (\lambda (e0: C).(drop n (r k d) e0 c)))))))) (\lambda (H5: (eq nat O (S d))).(let H6 \def (eq_ind nat O (\lambda (e0: nat).(match e0 return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S d) H5) in (False_ind ((eq C (CHead c0 k0 u0) x) \to ((eq C e (CHead c k u)) \to ((drop (r k0 h0) O c0 e) \to (ex2 C (\lambda (e0: C).(eq C x (CHead e0 k (lift (S h0) (r k d) u)))) (\lambda (e0: C).(drop (S h0) (r k d) e0 c)))))) H6))) h H1 H2 H3 H4 H0))))) | (drop_skip k0 h0 d0 c0 e H0 u0) \Rightarrow (\lambda (H1: (eq nat h0 h)).(\lambda (H2: (eq nat (S d0) (S d))).(\lambda (H3: (eq C (CHead c0 k0 (lift h0 (r k0 d0) u0)) x)).(\lambda (H4: (eq C (CHead e k0 u0) (CHead c k u))).(eq_ind nat h (\lambda (n: nat).((eq nat (S d0) (S d)) \to ((eq C (CHead c0 k0 (lift n (r k0 d0) u0)) x) \to ((eq C (CHead e k0 u0) (CHead c k u)) \to ((drop n (r k0 d0) c0 e) \to (ex2 C (\lambda (e0: C).(eq C x (CHead e0 k (lift h (r k d) u)))) (\lambda (e0: C).(drop h (r k d) e0 c)))))))) (\lambda (H5: (eq nat (S d0) (S d))).(let H6 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow d0 | (S n) \Rightarrow n])) (S d0) (S d) H5) in (eq_ind nat d (\lambda (n: nat).((eq C (CHead c0 k0 (lift h (r k0 n) u0)) x) \to ((eq C (CHead e k0 u0) (CHead c k u)) \to ((drop h (r k0 n) c0 e) \to (ex2 C (\lambda (e0: C).(eq C x (CHead e0 k (lift h (r k d) u)))) (\lambda (e0: C).(drop h (r k d) e0 c))))))) (\lambda (H7: (eq C (CHead c0 k0 (lift h (r k0 d) u0)) x)).(eq_ind C (CHead c0 k0 (lift h (r k0 d) u0)) (\lambda (c1: C).((eq C (CHead e k0 u0) (CHead c k u)) \to ((drop h (r k0 d) c0 e) \to (ex2 C (\lambda (e0: C).(eq C c1 (CHead e0 k (lift h (r k d) u)))) (\lambda (e0: C).(drop h (r k d) e0 c)))))) (\lambda (H8: (eq C (CHead e k0 u0) (CHead c k u))).(let H9 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e k0 u0) (CHead c k u) H8) in ((let H10 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead e k0 u0) (CHead c k u) H8) in ((let H11 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow e | (CHead c _ _) \Rightarrow c])) (CHead e k0 u0) (CHead c k u) H8) in (eq_ind C c (\lambda (c1: C).((eq K k0 k) \to ((eq T u0 u) \to ((drop h (r k0 d) c0 c1) \to (ex2 C (\lambda (e0: C).(eq C (CHead c0 k0 (lift h (r k0 d) u0)) (CHead e0 k (lift h (r k d) u)))) (\lambda (e0: C).(drop h (r k d) e0 c))))))) (\lambda (H12: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T u0 u) \to ((drop h (r k1 d) c0 c) \to (ex2 C (\lambda (e0: C).(eq C (CHead c0 k1 (lift h (r k1 d) u0)) (CHead e0 k (lift h (r k d) u)))) (\lambda (e0: C).(drop h (r k d) e0 c)))))) (\lambda (H13: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((drop h (r k d) c0 c) \to (ex2 C (\lambda (e0: C).(eq C (CHead c0 k (lift h (r k d) t)) (CHead e0 k (lift h (r k d) u)))) (\lambda (e0: C).(drop h (r k d) e0 c))))) (\lambda (H14: (drop h (r k d) c0 c)).(let H15 \def (eq_ind T u0 (\lambda (t: T).(eq C (CHead c0 k0 (lift h (r k0 d) t)) x)) H7 u H13) in (let H16 \def (eq_ind K k0 (\lambda (k: K).(eq C (CHead c0 k (lift h (r k d) u)) x)) H15 k H12) in (let H17 \def (eq_ind_r C x (\lambda (c0: C).(drop h (S d) c0 (CHead c k u))) H (CHead c0 k (lift h (r k d) u)) H16) in (ex_intro2 C (\lambda (e0: C).(eq C (CHead c0 k (lift h (r k d) u)) (CHead e0 k (lift h (r k d) u)))) (\lambda (e0: C).(drop h (r k d) e0 c)) c0 (refl_equal C (CHead c0 k (lift h (r k d) u))) H14))))) u0 (sym_eq T u0 u H13))) k0 (sym_eq K k0 k H12))) e (sym_eq C e c H11))) H10)) H9))) x H7)) d0 (sym_eq nat d0 d H6)))) h0 (sym_eq nat h0 h H1) H2 H3 H4 H0)))))]) in (H0 (refl_equal nat h) (refl_equal nat (S d)) (refl_equal C x) (refl_equal C (CHead c k u)))))))))).
+ \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) x (CHead c k u))).(let H0 \def (match H return (\lambda (_: ?).(\lambda (n: nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n h) \to ((eq nat n0 (S d)) \to ((eq C c0 x) \to ((eq C c1 (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C x (CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k d) e c)))))))))))) with [(drop_refl c0) \Rightarrow (\lambda (H0: (eq nat O h)).(\lambda (H1: (eq nat O (S d))).(\lambda (H2: (eq C c0 x)).(\lambda (H3: (eq C c0 (CHead c k u))).(eq_ind nat O (\lambda (n: nat).((eq nat O (S d)) \to ((eq C c0 x) \to ((eq C c0 (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C x (CHead e k (lift n (r k d) u)))) (\lambda (e: C).(drop n (r k d) e c))))))) (\lambda (H4: (eq nat O (S d))).(let H5 \def (eq_ind nat O (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S d) H4) in (False_ind ((eq C c0 x) \to ((eq C c0 (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C x (CHead e k (lift O (r k d) u)))) (\lambda (e: C).(drop O (r k d) e c))))) H5))) h H0 H1 H2 H3))))) | (drop_drop k0 h0 c0 e H0 u0) \Rightarrow (\lambda (H1: (eq nat (S h0) h)).(\lambda (H2: (eq nat O (S d))).(\lambda (H3: (eq C (CHead c0 k0 u0) x)).(\lambda (H4: (eq C e (CHead c k u))).(eq_ind nat (S h0) (\lambda (n: nat).((eq nat O (S d)) \to ((eq C (CHead c0 k0 u0) x) \to ((eq C e (CHead c k u)) \to ((drop (r k0 h0) O c0 e) \to (ex2 C (\lambda (e0: C).(eq C x (CHead e0 k (lift n (r k d) u)))) (\lambda (e0: C).(drop n (r k d) e0 c)))))))) (\lambda (H5: (eq nat O (S d))).(let H6 \def (eq_ind nat O (\lambda (e0: nat).(match e0 return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S d) H5) in (False_ind ((eq C (CHead c0 k0 u0) x) \to ((eq C e (CHead c k u)) \to ((drop (r k0 h0) O c0 e) \to (ex2 C (\lambda (e0: C).(eq C x (CHead e0 k (lift (S h0) (r k d) u)))) (\lambda (e0: C).(drop (S h0) (r k d) e0 c)))))) H6))) h H1 H2 H3 H4 H0))))) | (drop_skip k0 h0 d0 c0 e H0 u0) \Rightarrow (\lambda (H1: (eq nat h0 h)).(\lambda (H2: (eq nat (S d0) (S d))).(\lambda (H3: (eq C (CHead c0 k0 (lift h0 (r k0 d0) u0)) x)).(\lambda (H4: (eq C (CHead e k0 u0) (CHead c k u))).(eq_ind nat h (\lambda (n: nat).((eq nat (S d0) (S d)) \to ((eq C (CHead c0 k0 (lift n (r k0 d0) u0)) x) \to ((eq C (CHead e k0 u0) (CHead c k u)) \to ((drop n (r k0 d0) c0 e) \to (ex2 C (\lambda (e0: C).(eq C x (CHead e0 k (lift h (r k d) u)))) (\lambda (e0: C).(drop h (r k d) e0 c)))))))) (\lambda (H5: (eq nat (S d0) (S d))).(let H6 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow d0 | (S n) \Rightarrow n])) (S d0) (S d) H5) in (eq_ind nat d (\lambda (n: nat).((eq C (CHead c0 k0 (lift h (r k0 n) u0)) x) \to ((eq C (CHead e k0 u0) (CHead c k u)) \to ((drop h (r k0 n) c0 e) \to (ex2 C (\lambda (e0: C).(eq C x (CHead e0 k (lift h (r k d) u)))) (\lambda (e0: C).(drop h (r k d) e0 c))))))) (\lambda (H7: (eq C (CHead c0 k0 (lift h (r k0 d) u0)) x)).(eq_ind C (CHead c0 k0 (lift h (r k0 d) u0)) (\lambda (c1: C).((eq C (CHead e k0 u0) (CHead c k u)) \to ((drop h (r k0 d) c0 e) \to (ex2 C (\lambda (e0: C).(eq C c1 (CHead e0 k (lift h (r k d) u)))) (\lambda (e0: C).(drop h (r k d) e0 c)))))) (\lambda (H8: (eq C (CHead e k0 u0) (CHead c k u))).(let H9 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e k0 u0) (CHead c k u) H8) in ((let H10 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead e k0 u0) (CHead c k u) H8) in ((let H11 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow e | (CHead c _ _) \Rightarrow c])) (CHead e k0 u0) (CHead c k u) H8) in (eq_ind C c (\lambda (c1: C).((eq K k0 k) \to ((eq T u0 u) \to ((drop h (r k0 d) c0 c1) \to (ex2 C (\lambda (e0: C).(eq C (CHead c0 k0 (lift h (r k0 d) u0)) (CHead e0 k (lift h (r k d) u)))) (\lambda (e0: C).(drop h (r k d) e0 c))))))) (\lambda (H12: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T u0 u) \to ((drop h (r k1 d) c0 c) \to (ex2 C (\lambda (e0: C).(eq C (CHead c0 k1 (lift h (r k1 d) u0)) (CHead e0 k (lift h (r k d) u)))) (\lambda (e0: C).(drop h (r k d) e0 c)))))) (\lambda (H13: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((drop h (r k d) c0 c) \to (ex2 C (\lambda (e0: C).(eq C (CHead c0 k (lift h (r k d) t)) (CHead e0 k (lift h (r k d) u)))) (\lambda (e0: C).(drop h (r k d) e0 c))))) (\lambda (H14: (drop h (r k d) c0 c)).(let H15 \def (eq_ind T u0 (\lambda (t: T).(eq C (CHead c0 k0 (lift h (r k0 d) t)) x)) H7 u H13) in (let H16 \def (eq_ind K k0 (\lambda (k: K).(eq C (CHead c0 k (lift h (r k d) u)) x)) H15 k H12) in (let H17 \def (eq_ind_r C x (\lambda (c0: C).(drop h (S d) c0 (CHead c k u))) H (CHead c0 k (lift h (r k d) u)) H16) in (ex_intro2 C (\lambda (e0: C).(eq C (CHead c0 k (lift h (r k d) u)) (CHead e0 k (lift h (r k d) u)))) (\lambda (e0: C).(drop h (r k d) e0 c)) c0 (refl_equal C (CHead c0 k (lift h (r k d) u))) H14))))) u0 (sym_eq T u0 u H13))) k0 (sym_eq K k0 k H12))) e (sym_eq C e c H11))) H10)) H9))) x H7)) d0 (sym_eq nat d0 d H6)))) h0 (sym_eq nat h0 h H1) H2 H3 H4 H0)))))]) in (H0 (refl_equal nat h) (refl_equal nat (S d)) (refl_equal C x) (refl_equal C (CHead c k u)))))))))).
 
 theorem drop_gen_skip_l:
  \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall (d: nat).(\forall (k: K).((drop h (S d) (CHead c k u) x) \to (ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k d) c e))))))))))
 \def
- \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) (CHead c k u) x)).(let H0 \def (match H return (\lambda (n: nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n h) \to ((eq nat n0 (S d)) \to ((eq C c0 (CHead c k u)) \to ((eq C c1 x) \to (ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k d) c e)))))))))))) with [(drop_refl c0) \Rightarrow (\lambda (H0: (eq nat O h)).(\lambda (H1: (eq nat O (S d))).(\lambda (H2: (eq C c0 (CHead c k u))).(\lambda (H3: (eq C c0 x)).(eq_ind nat O (\lambda (n: nat).((eq nat O (S d)) \to ((eq C c0 (CHead c k u)) \to ((eq C c0 x) \to (ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift n (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop n (r k d) c e)))))))) (\lambda (H4: (eq nat O (S d))).(let H5 \def (eq_ind nat O (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S d) H4) in (False_ind ((eq C c0 (CHead c k u)) \to ((eq C c0 x) \to (ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift O (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop O (r k d) c e)))))) H5))) h H0 H1 H2 H3))))) | (drop_drop k0 h0 c0 e H0 u0) \Rightarrow (\lambda (H1: (eq nat (S h0) h)).(\lambda (H2: (eq nat O (S d))).(\lambda (H3: (eq C (CHead c0 k0 u0) (CHead c k u))).(\lambda (H4: (eq C e x)).(eq_ind nat (S h0) (\lambda (n: nat).((eq nat O (S d)) \to ((eq C (CHead c0 k0 u0) (CHead c k u)) \to ((eq C e x) \to ((drop (r k0 h0) O c0 e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift n (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop n (r k d) c e0))))))))) (\lambda (H5: (eq nat O (S d))).(let H6 \def (eq_ind nat O (\lambda (e0: nat).(match e0 return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S d) H5) in (False_ind ((eq C (CHead c0 k0 u0) (CHead c k u)) \to ((eq C e x) \to ((drop (r k0 h0) O c0 e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift (S h0) (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop (S h0) (r k d) c e0))))))) H6))) h H1 H2 H3 H4 H0))))) | (drop_skip k0 h0 d0 c0 e H0 u0) \Rightarrow (\lambda (H1: (eq nat h0 h)).(\lambda (H2: (eq nat (S d0) (S d))).(\lambda (H3: (eq C (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u))).(\lambda (H4: (eq C (CHead e k0 u0) x)).(eq_ind nat h (\lambda (n: nat).((eq nat (S d0) (S d)) \to ((eq C (CHead c0 k0 (lift n (r k0 d0) u0)) (CHead c k u)) \to ((eq C (CHead e k0 u0) x) \to ((drop n (r k0 d0) c0 e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k d) c e0))))))))) (\lambda (H5: (eq nat (S d0) (S d))).(let H6 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow d0 | (S n) \Rightarrow n])) (S d0) (S d) H5) in (eq_ind nat d (\lambda (n: nat).((eq C (CHead c0 k0 (lift h (r k0 n) u0)) (CHead c k u)) \to ((eq C (CHead e k0 u0) x) \to ((drop h (r k0 n) c0 e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k d) c e0)))))))) (\lambda (H7: (eq C (CHead c0 k0 (lift h (r k0 d) u0)) (CHead c k u))).(let H8 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x h)) (r k0 d) u0) | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 (lift h (r k0 d) u0)) (CHead c k u) H7) in ((let H9 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c0 k0 (lift h (r k0 d) u0)) (CHead c k u) H7) in ((let H10 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 (lift h (r k0 d) u0)) (CHead c k u) H7) in (eq_ind C c (\lambda (c1: C).((eq K k0 k) \to ((eq T (lift h (r k0 d) u0) u) \to ((eq C (CHead e k0 u0) x) \to ((drop h (r k0 d) c1 e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k d) c e0))))))))) (\lambda (H11: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T (lift h (r k1 d) u0) u) \to ((eq C (CHead e k1 u0) x) \to ((drop h (r k1 d) c e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k d) c e0)))))))) (\lambda (H12: (eq T (lift h (r k d) u0) u)).(eq_ind T (lift h (r k d) u0) (\lambda (t: T).((eq C (CHead e k u0) x) \to ((drop h (r k d) c e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k d) c e0))))))) (\lambda (H13: (eq C (CHead e k u0) x)).(eq_ind C (CHead e k u0) (\lambda (c1: C).((drop h (r k d) c e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C c1 (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T (lift h (r k d) u0) (lift h (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k d) c e0)))))) (\lambda (H14: (drop h (r k d) c e)).(let H15 \def (eq_ind_r T u (\lambda (t: T).(drop h (S d) (CHead c k t) x)) H (lift h (r k d) u0) H12) in (let H16 \def (eq_ind_r C x (\lambda (c0: C).(drop h (S d) (CHead c k (lift h (r k d) u0)) c0)) H15 (CHead e k u0) H13) in (ex3_2_intro C T (\lambda (e0: C).(\lambda (v: T).(eq C (CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T (lift h (r k d) u0) (lift h (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k d) c e0))) e u0 (refl_equal C (CHead e k u0)) (refl_equal T (lift h (r k d) u0)) H14)))) x H13)) u H12)) k0 (sym_eq K k0 k H11))) c0 (sym_eq C c0 c H10))) H9)) H8))) d0 (sym_eq nat d0 d H6)))) h0 (sym_eq nat h0 h H1) H2 H3 H4 H0)))))]) in (H0 (refl_equal nat h) (refl_equal nat (S d)) (refl_equal C (CHead c k u)) (refl_equal C x))))))))).
+ \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) (CHead c k u) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (n: nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n h) \to ((eq nat n0 (S d)) \to ((eq C c0 (CHead c k u)) \to ((eq C c1 x) \to (ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k d) c e))))))))))))) with [(drop_refl c0) \Rightarrow (\lambda (H0: (eq nat O h)).(\lambda (H1: (eq nat O (S d))).(\lambda (H2: (eq C c0 (CHead c k u))).(\lambda (H3: (eq C c0 x)).(eq_ind nat O (\lambda (n: nat).((eq nat O (S d)) \to ((eq C c0 (CHead c k u)) \to ((eq C c0 x) \to (ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift n (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop n (r k d) c e)))))))) (\lambda (H4: (eq nat O (S d))).(let H5 \def (eq_ind nat O (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S d) H4) in (False_ind ((eq C c0 (CHead c k u)) \to ((eq C c0 x) \to (ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift O (r k d) v)))) (\lambda (e: C).(\lambda (_: T).(drop O (r k d) c e)))))) H5))) h H0 H1 H2 H3))))) | (drop_drop k0 h0 c0 e H0 u0) \Rightarrow (\lambda (H1: (eq nat (S h0) h)).(\lambda (H2: (eq nat O (S d))).(\lambda (H3: (eq C (CHead c0 k0 u0) (CHead c k u))).(\lambda (H4: (eq C e x)).(eq_ind nat (S h0) (\lambda (n: nat).((eq nat O (S d)) \to ((eq C (CHead c0 k0 u0) (CHead c k u)) \to ((eq C e x) \to ((drop (r k0 h0) O c0 e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift n (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop n (r k d) c e0))))))))) (\lambda (H5: (eq nat O (S d))).(let H6 \def (eq_ind nat O (\lambda (e0: nat).(match e0 return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S d) H5) in (False_ind ((eq C (CHead c0 k0 u0) (CHead c k u)) \to ((eq C e x) \to ((drop (r k0 h0) O c0 e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift (S h0) (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop (S h0) (r k d) c e0))))))) H6))) h H1 H2 H3 H4 H0))))) | (drop_skip k0 h0 d0 c0 e H0 u0) \Rightarrow (\lambda (H1: (eq nat h0 h)).(\lambda (H2: (eq nat (S d0) (S d))).(\lambda (H3: (eq C (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u))).(\lambda (H4: (eq C (CHead e k0 u0) x)).(eq_ind nat h (\lambda (n: nat).((eq nat (S d0) (S d)) \to ((eq C (CHead c0 k0 (lift n (r k0 d0) u0)) (CHead c k u)) \to ((eq C (CHead e k0 u0) x) \to ((drop n (r k0 d0) c0 e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k d) c e0))))))))) (\lambda (H5: (eq nat (S d0) (S d))).(let H6 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow d0 | (S n) \Rightarrow n])) (S d0) (S d) H5) in (eq_ind nat d (\lambda (n: nat).((eq C (CHead c0 k0 (lift h (r k0 n) u0)) (CHead c k u)) \to ((eq C (CHead e k0 u0) x) \to ((drop h (r k0 n) c0 e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k d) c e0)))))))) (\lambda (H7: (eq C (CHead c0 k0 (lift h (r k0 d) u0)) (CHead c k u))).(let H8 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x h)) (r k0 d) u0) | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 (lift h (r k0 d) u0)) (CHead c k u) H7) in ((let H9 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c0 k0 (lift h (r k0 d) u0)) (CHead c k u) H7) in ((let H10 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 (lift h (r k0 d) u0)) (CHead c k u) H7) in (eq_ind C c (\lambda (c1: C).((eq K k0 k) \to ((eq T (lift h (r k0 d) u0) u) \to ((eq C (CHead e k0 u0) x) \to ((drop h (r k0 d) c1 e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k d) c e0))))))))) (\lambda (H11: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T (lift h (r k1 d) u0) u) \to ((eq C (CHead e k1 u0) x) \to ((drop h (r k1 d) c e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k d) c e0)))))))) (\lambda (H12: (eq T (lift h (r k d) u0) u)).(eq_ind T (lift h (r k d) u0) (\lambda (t: T).((eq C (CHead e k u0) x) \to ((drop h (r k d) c e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k d) c e0))))))) (\lambda (H13: (eq C (CHead e k u0) x)).(eq_ind C (CHead e k u0) (\lambda (c1: C).((drop h (r k d) c e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C c1 (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T (lift h (r k d) u0) (lift h (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k d) c e0)))))) (\lambda (H14: (drop h (r k d) c e)).(let H15 \def (eq_ind_r T u (\lambda (t: T).(drop h (S d) (CHead c k t) x)) H (lift h (r k d) u0) H12) in (let H16 \def (eq_ind_r C x (\lambda (c0: C).(drop h (S d) (CHead c k (lift h (r k d) u0)) c0)) H15 (CHead e k u0) H13) in (ex3_2_intro C T (\lambda (e0: C).(\lambda (v: T).(eq C (CHead e k u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T (lift h (r k d) u0) (lift h (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k d) c e0))) e u0 (refl_equal C (CHead e k u0)) (refl_equal T (lift h (r k d) u0)) H14)))) x H13)) u H12)) k0 (sym_eq K k0 k H11))) c0 (sym_eq C c0 c H10))) H9)) H8))) d0 (sym_eq nat d0 d H6)))) h0 (sym_eq nat h0 h H1) H2 H3 H4 H0)))))]) in (H0 (refl_equal nat h) (refl_equal nat (S d)) (refl_equal C (CHead c k u)) (refl_equal C x))))))))).
 
 theorem drop_skip_bind:
  \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h d c e) \to (\forall (b: B).(\forall (u: T).(drop h (S d) (CHead c (Bind b) (lift h d u)) (CHead e (Bind b) u))))))))
@@ -705,22 +705,22 @@ theorem drop_conf_lt:
 theorem drop_conf_ge:
  \forall (i: nat).(\forall (a: C).(\forall (c: C).((drop i O c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le (plus d h) i) \to (drop (minus i h) O e a)))))))))
 \def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (a: C).(\forall (c: C).((drop n O c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le (plus d h) n) \to (drop (minus n h) O e a)))))))))) (\lambda (a: C).(\lambda (c: C).(\lambda (H: (drop O O c a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h d c e)).(\lambda (H1: (le (plus d h) O)).(let H2 \def (eq_ind C c (\lambda (c: C).(drop h d c e)) H0 a (drop_gen_refl c a H)) in (let H3 \def (match H1 return (\lambda (n: nat).((eq nat n O) \to (drop (minus O h) O e a))) with [le_n \Rightarrow (\lambda (H: (eq nat (plus d h) O)).(let H3 \def (f_equal nat nat (\lambda (e0: nat).e0) (plus d h) O H) in (eq_ind nat (plus d h) (\lambda (n: nat).(drop (minus n h) n e a)) (eq_ind_r nat O (\lambda (n: nat).(drop (minus n h) n e a)) (and_ind (eq nat d O) (eq nat h O) (drop O O e a) (\lambda (H0: (eq nat d O)).(\lambda (H1: (eq nat h O)).(let H2 \def (eq_ind nat d (\lambda (n: nat).(drop h n a e)) H2 O H0) in (let H4 \def (eq_ind nat h (\lambda (n: nat).(drop n O a e)) H2 O H1) in (eq_ind C a (\lambda (c: C).(drop O O c a)) (drop_refl a) e (drop_gen_refl a e H4)))))) (plus_O d h H3)) (plus d h) H3) O H3))) | (le_S m H) \Rightarrow (\lambda (H2: (eq nat (S m) O)).((let H0 \def (eq_ind nat (S m) (\lambda (e0: nat).(match e0 return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (plus d h) m) \to (drop (minus O h) O e a)) H0)) H))]) in (H3 (refl_equal nat O)))))))))))) (\lambda (i0: nat).(\lambda (H: ((\forall (a: C).(\forall (c: C).((drop i0 O c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le (plus d h) i0) \to (drop (minus i0 h) O e a))))))))))).(\lambda (a: C).(\lambda (c: C).(C_ind (\lambda (c0: C).((drop (S i0) O c0 a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e a)))))))) (\lambda (n: nat).(\lambda (H0: (drop (S i0) O (CSort n) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (drop h d (CSort n) e)).(\lambda (H2: (le (plus d h) (S i0))).(and3_ind (eq C e (CSort n)) (eq nat h O) (eq nat d O) (drop (minus (S i0) h) O e a) (\lambda (H3: (eq C e (CSort n))).(\lambda (H4: (eq nat h O)).(\lambda (H5: (eq nat d O)).(and3_ind (eq C a (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop (minus (S i0) h) O e a) (\lambda (H6: (eq C a (CSort n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(let H9 \def (eq_ind nat d (\lambda (n: nat).(le (plus n h) (S i0))) H2 O H5) in (let H10 \def (eq_ind nat h (\lambda (n: nat).(le (plus O n) (S i0))) H9 O H4) in (eq_ind_r nat O (\lambda (n0: nat).(drop (minus (S i0) n0) O e a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O c0 a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O (CSort n) c0)) (let H11 \def (eq_ind nat (S i0) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H7) in (False_ind (drop (minus (S i0) O) O (CSort n) (CSort n)) H11)) a H6) e H3) h H4)))))) (drop_gen_sort n (S i0) O a H0))))) (drop_gen_sort n h d e H1))))))))) (\lambda (c0: C).(\lambda (H0: (((drop (S i0) O c0 a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e a))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).((drop (S i0) O (CHead c0 k0 t) a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d (CHead c0 k0 t) e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e a))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c0 (Bind b) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h d (CHead c0 (Bind b) t) e)).(\lambda (H3: (le (plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Bind b) t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S i0) h) O e a)))) (\lambda (H4: (drop h O (CHead c0 (Bind b) t) e)).(\lambda (H5: (le (plus O h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 (Bind b) t) e) \to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda (H6: (drop O O (CHead c0 (Bind b) t) e)).(\lambda (_: (le (plus O O) (S i0))).(eq_ind C (CHead c0 (Bind b) t) (\lambda (c1: C).(drop (minus (S i0) O) O c1 a)) (drop_drop (Bind b) i0 c0 a (drop_gen_drop (Bind b) c0 a t i0 H1) t) e (drop_gen_refl (CHead c0 (Bind b) t) e H6)))) (\lambda (h0: nat).(\lambda (_: (((drop h0 O (CHead c0 (Bind b) t) e) \to ((le (plus O h0) (S i0)) \to (drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 (Bind b) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H a c0 (drop_gen_drop (Bind b) c0 a t i0 H1) e h0 O (drop_gen_drop (Bind b) c0 e t h0 H6) (le_S_n (plus O h0) i0 H7)))))) h H4 H5))) (\lambda (d0: nat).(\lambda (_: (((drop h d0 (CHead c0 (Bind b) t) e) \to ((le (plus d0 h) (S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0) (CHead c0 (Bind b) t) e)).(\lambda (H5: (le (plus (S d0) h) (S i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Bind b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Bind b) d0) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Bind b) d0) c0 e0))) (drop (minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) d0) x1))).(\lambda (H8: (drop h (r (Bind b) d0) c0 x0)).(eq_ind_r C (CHead x0 (Bind b) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Bind b) x1) a)) (drop_drop (Bind b) (minus i0 h) x0 a (H a c0 (drop_gen_drop (Bind b) c0 a t i0 H1) x0 h d0 H8 (le_S_n (plus d0 h) i0 H5)) x1) (minus (S i0) h) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) e H6)))))) (drop_gen_skip_l c0 e t h d0 (Bind b) H4)))))) d H2 H3))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c0 (Flat f) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h d (CHead c0 (Flat f) t) e)).(\lambda (H3: (le (plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Flat f) t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S i0) h) O e a)))) (\lambda (H4: (drop h O (CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus O h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 (Flat f) t) e) \to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda (H6: (drop O O (CHead c0 (Flat f) t) e)).(\lambda (_: (le (plus O O) (S i0))).(eq_ind C (CHead c0 (Flat f) t) (\lambda (c1: C).(drop (minus (S i0) O) O c1 a)) (drop_drop (Flat f) i0 c0 a (drop_gen_drop (Flat f) c0 a t i0 H1) t) e (drop_gen_refl (CHead c0 (Flat f) t) e H6)))) (\lambda (h0: nat).(\lambda (_: (((drop h0 O (CHead c0 (Flat f) t) e) \to ((le (plus O h0) (S i0)) \to (drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 (Flat f) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H0 (drop_gen_drop (Flat f) c0 a t i0 H1) e (S h0) O (drop_gen_drop (Flat f) c0 e t h0 H6) H7))))) h H4 H5))) (\lambda (d0: nat).(\lambda (_: (((drop h d0 (CHead c0 (Flat f) t) e) \to ((le (plus d0 h) (S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0) (CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus (S d0) h) (S i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Flat f) d0) c0 e0))) (drop (minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t (lift h (r (Flat f) d0) x1))).(\lambda (H8: (drop h (r (Flat f) d0) c0 x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (let H9 \def (eq_ind_r nat (minus (S i0) h) (\lambda (n: nat).(drop n O x0 a)) (H0 (drop_gen_drop (Flat f) c0 a t i0 H1) x0 h (S d0) H8 H5) (S (minus i0 h)) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) in (eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Flat f) x1) a)) (drop_drop (Flat f) (minus i0 h) x0 a H9 x1) (minus (S i0) h) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5))))) e H6)))))) (drop_gen_skip_l c0 e t h d0 (Flat f) H4)))))) d H2 H3))))))))) k)))) c))))) i).
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (a: C).(\forall (c: C).((drop n O c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le (plus d h) n) \to (drop (minus n h) O e a)))))))))) (\lambda (a: C).(\lambda (c: C).(\lambda (H: (drop O O c a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h d c e)).(\lambda (H1: (le (plus d h) O)).(let H2 \def (eq_ind C c (\lambda (c: C).(drop h d c e)) H0 a (drop_gen_refl c a H)) in (let H3 \def (match H1 return (\lambda (_: ?).(\lambda (n: nat).((eq nat n O) \to (drop (minus O h) O e a)))) with [le_n \Rightarrow (\lambda (H: (eq nat (plus d h) O)).(let H3 \def (f_equal nat nat (\lambda (e0: nat).e0) (plus d h) O H) in (eq_ind nat (plus d h) (\lambda (n: nat).(drop (minus n h) n e a)) (eq_ind_r nat O (\lambda (n: nat).(drop (minus n h) n e a)) (and_ind (eq nat d O) (eq nat h O) (drop O O e a) (\lambda (H0: (eq nat d O)).(\lambda (H1: (eq nat h O)).(let H2 \def (eq_ind nat d (\lambda (n: nat).(drop h n a e)) H2 O H0) in (let H4 \def (eq_ind nat h (\lambda (n: nat).(drop n O a e)) H2 O H1) in (eq_ind C a (\lambda (c: C).(drop O O c a)) (drop_refl a) e (drop_gen_refl a e H4)))))) (plus_O d h H3)) (plus d h) H3) O H3))) | (le_S m H) \Rightarrow (\lambda (H2: (eq nat (S m) O)).((let H0 \def (eq_ind nat (S m) (\lambda (e0: nat).(match e0 return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (plus d h) m) \to (drop (minus O h) O e a)) H0)) H))]) in (H3 (refl_equal nat O)))))))))))) (\lambda (i0: nat).(\lambda (H: ((\forall (a: C).(\forall (c: C).((drop i0 O c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le (plus d h) i0) \to (drop (minus i0 h) O e a))))))))))).(\lambda (a: C).(\lambda (c: C).(C_ind (\lambda (c0: C).((drop (S i0) O c0 a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e a)))))))) (\lambda (n: nat).(\lambda (H0: (drop (S i0) O (CSort n) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (drop h d (CSort n) e)).(\lambda (H2: (le (plus d h) (S i0))).(and3_ind (eq C e (CSort n)) (eq nat h O) (eq nat d O) (drop (minus (S i0) h) O e a) (\lambda (H3: (eq C e (CSort n))).(\lambda (H4: (eq nat h O)).(\lambda (H5: (eq nat d O)).(and3_ind (eq C a (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop (minus (S i0) h) O e a) (\lambda (H6: (eq C a (CSort n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(let H9 \def (eq_ind nat d (\lambda (n: nat).(le (plus n h) (S i0))) H2 O H5) in (let H10 \def (eq_ind nat h (\lambda (n: nat).(le (plus O n) (S i0))) H9 O H4) in (eq_ind_r nat O (\lambda (n0: nat).(drop (minus (S i0) n0) O e a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O c0 a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O (CSort n) c0)) (let H11 \def (eq_ind nat (S i0) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H7) in (False_ind (drop (minus (S i0) O) O (CSort n) (CSort n)) H11)) a H6) e H3) h H4)))))) (drop_gen_sort n (S i0) O a H0))))) (drop_gen_sort n h d e H1))))))))) (\lambda (c0: C).(\lambda (H0: (((drop (S i0) O c0 a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e a))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).((drop (S i0) O (CHead c0 k0 t) a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d (CHead c0 k0 t) e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e a))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c0 (Bind b) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h d (CHead c0 (Bind b) t) e)).(\lambda (H3: (le (plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Bind b) t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S i0) h) O e a)))) (\lambda (H4: (drop h O (CHead c0 (Bind b) t) e)).(\lambda (H5: (le (plus O h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 (Bind b) t) e) \to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda (H6: (drop O O (CHead c0 (Bind b) t) e)).(\lambda (_: (le (plus O O) (S i0))).(eq_ind C (CHead c0 (Bind b) t) (\lambda (c1: C).(drop (minus (S i0) O) O c1 a)) (drop_drop (Bind b) i0 c0 a (drop_gen_drop (Bind b) c0 a t i0 H1) t) e (drop_gen_refl (CHead c0 (Bind b) t) e H6)))) (\lambda (h0: nat).(\lambda (_: (((drop h0 O (CHead c0 (Bind b) t) e) \to ((le (plus O h0) (S i0)) \to (drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 (Bind b) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H a c0 (drop_gen_drop (Bind b) c0 a t i0 H1) e h0 O (drop_gen_drop (Bind b) c0 e t h0 H6) (le_S_n (plus O h0) i0 H7)))))) h H4 H5))) (\lambda (d0: nat).(\lambda (_: (((drop h d0 (CHead c0 (Bind b) t) e) \to ((le (plus d0 h) (S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0) (CHead c0 (Bind b) t) e)).(\lambda (H5: (le (plus (S d0) h) (S i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Bind b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Bind b) d0) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Bind b) d0) c0 e0))) (drop (minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) d0) x1))).(\lambda (H8: (drop h (r (Bind b) d0) c0 x0)).(eq_ind_r C (CHead x0 (Bind b) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Bind b) x1) a)) (drop_drop (Bind b) (minus i0 h) x0 a (H a c0 (drop_gen_drop (Bind b) c0 a t i0 H1) x0 h d0 H8 (le_S_n (plus d0 h) i0 H5)) x1) (minus (S i0) h) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) e H6)))))) (drop_gen_skip_l c0 e t h d0 (Bind b) H4)))))) d H2 H3))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c0 (Flat f) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h d (CHead c0 (Flat f) t) e)).(\lambda (H3: (le (plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Flat f) t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S i0) h) O e a)))) (\lambda (H4: (drop h O (CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus O h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 (Flat f) t) e) \to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda (H6: (drop O O (CHead c0 (Flat f) t) e)).(\lambda (_: (le (plus O O) (S i0))).(eq_ind C (CHead c0 (Flat f) t) (\lambda (c1: C).(drop (minus (S i0) O) O c1 a)) (drop_drop (Flat f) i0 c0 a (drop_gen_drop (Flat f) c0 a t i0 H1) t) e (drop_gen_refl (CHead c0 (Flat f) t) e H6)))) (\lambda (h0: nat).(\lambda (_: (((drop h0 O (CHead c0 (Flat f) t) e) \to ((le (plus O h0) (S i0)) \to (drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 (Flat f) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H0 (drop_gen_drop (Flat f) c0 a t i0 H1) e (S h0) O (drop_gen_drop (Flat f) c0 e t h0 H6) H7))))) h H4 H5))) (\lambda (d0: nat).(\lambda (_: (((drop h d0 (CHead c0 (Flat f) t) e) \to ((le (plus d0 h) (S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0) (CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus (S d0) h) (S i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Flat f) d0) c0 e0))) (drop (minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t (lift h (r (Flat f) d0) x1))).(\lambda (H8: (drop h (r (Flat f) d0) c0 x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (let H9 \def (eq_ind_r nat (minus (S i0) h) (\lambda (n: nat).(drop n O x0 a)) (H0 (drop_gen_drop (Flat f) c0 a t i0 H1) x0 h (S d0) H8 H5) (S (minus i0 h)) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) in (eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Flat f) x1) a)) (drop_drop (Flat f) (minus i0 h) x0 a H9 x1) (minus (S i0) h) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5))))) e H6)))))) (drop_gen_skip_l c0 e t h d0 (Flat f) H4)))))) d H2 H3))))))))) k)))) c))))) i).
 
 theorem drop_conf_rev:
  \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop j O c1 c2)) (\lambda (c1: C).(drop i j c1 e1)))))))))
 \def
- \lambda (j: nat).(nat_ind (\lambda (n: nat).(\forall (e1: C).(\forall (e2: C).((drop n O e1 e2) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 c2)) (\lambda (c1: C).(drop i n c1 e1)))))))))) (\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop O O e1 e2)).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(let H1 \def (eq_ind_r C e2 (\lambda (c: C).(drop i O c2 c)) H0 e1 (drop_gen_refl e1 e2 H)) in (ex_intro2 C (\lambda (c1: C).(drop O O c1 c2)) (\lambda (c1: C).(drop i O c1 e1)) c2 (drop_refl c2) H1)))))))) (\lambda (j0: nat).(\lambda (IHj: ((\forall (e1: C).(\forall (e2: C).((drop j0 O e1 e2) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop j0 O c1 c2)) (\lambda (c1: C).(drop i j0 c1 e1))))))))))).(\lambda (e1: C).(C_ind (\lambda (c: C).(\forall (e2: C).((drop (S j0) O c e2) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 c))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda (H: (drop (S j0) O (CSort n) e2)).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(and3_ind (eq C e2 (CSort n)) (eq nat (S j0) O) (eq nat O O) (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: (eq nat (S j0) O)).(\lambda (_: (eq nat O O)).(let H4 \def (eq_ind C e2 (\lambda (c: C).(drop i O c2 c)) H0 (CSort n) H1) in (let H5 \def (eq_ind nat (S j0) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) H5)))))) (drop_gen_sort n (S j0) O e2 H)))))))) (\lambda (e2: C).(\lambda (IHe1: ((\forall (e3: C).((drop (S j0) O e2 e3) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e3) \to (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e3: C).(\lambda (H: (drop (S j0) O (CHead e2 k t) e3)).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e3)).((match k return (\lambda (k0: K).((drop (r k0 j0) O e2 e3) \to (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 k0 t)))))) with [(Bind b) \Rightarrow (\lambda (H1: (drop (r (Bind b) j0) O e2 e3)).(let H_x \def (IHj e2 e3 H1 c2 i H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c1: C).(drop j0 O c1 c2)) (\lambda (c1: C).(drop i j0 c1 e2)) (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Bind b) t)))) (\lambda (x: C).(\lambda (H3: (drop j0 O x c2)).(\lambda (H4: (drop i j0 x e2)).(ex_intro2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Bind b) t))) (CHead x (Bind b) (lift i (r (Bind b) j0) t)) (drop_drop (Bind b) j0 x c2 H3 (lift i (r (Bind b) j0) t)) (drop_skip (Bind b) i j0 x e2 H4 t))))) H2)))) | (Flat f) \Rightarrow (\lambda (H1: (drop (r (Flat f) j0) O e2 e3)).(let H_x \def (IHe1 e3 H1 c2 i H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 e2)) (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Flat f) t)))) (\lambda (x: C).(\lambda (H3: (drop (S j0) O x c2)).(\lambda (H4: (drop i (S j0) x e2)).(ex_intro2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Flat f) t))) (CHead x (Flat f) (lift i (r (Flat f) j0) t)) (drop_drop (Flat f) j0 x c2 H3 (lift i (r (Flat f) j0) t)) (drop_skip (Flat f) i j0 x e2 H4 t))))) H2))))]) (drop_gen_drop k e2 e3 t j0 H))))))))))) e1)))) j).
+ \lambda (j: nat).(nat_ind (\lambda (n: nat).(\forall (e1: C).(\forall (e2: C).((drop n O e1 e2) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 c2)) (\lambda (c1: C).(drop i n c1 e1)))))))))) (\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop O O e1 e2)).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(let H1 \def (eq_ind_r C e2 (\lambda (c: C).(drop i O c2 c)) H0 e1 (drop_gen_refl e1 e2 H)) in (ex_intro2 C (\lambda (c1: C).(drop O O c1 c2)) (\lambda (c1: C).(drop i O c1 e1)) c2 (drop_refl c2) H1)))))))) (\lambda (j0: nat).(\lambda (IHj: ((\forall (e1: C).(\forall (e2: C).((drop j0 O e1 e2) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop j0 O c1 c2)) (\lambda (c1: C).(drop i j0 c1 e1))))))))))).(\lambda (e1: C).(C_ind (\lambda (c: C).(\forall (e2: C).((drop (S j0) O c e2) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 c))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda (H: (drop (S j0) O (CSort n) e2)).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(and3_ind (eq C e2 (CSort n)) (eq nat (S j0) O) (eq nat O O) (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: (eq nat (S j0) O)).(\lambda (_: (eq nat O O)).(let H4 \def (eq_ind C e2 (\lambda (c: C).(drop i O c2 c)) H0 (CSort n) H1) in (let H5 \def (eq_ind nat (S j0) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) H5)))))) (drop_gen_sort n (S j0) O e2 H)))))))) (\lambda (e2: C).(\lambda (IHe1: ((\forall (e3: C).((drop (S j0) O e2 e3) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e3) \to (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e3: C).(\lambda (H: (drop (S j0) O (CHead e2 k t) e3)).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e3)).((match k return (\lambda (_: ?).(\lambda (k0: K).((drop (r k0 j0) O e2 e3) \to (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 k0 t))))))) with [(Bind b) \Rightarrow (\lambda (H1: (drop (r (Bind b) j0) O e2 e3)).(let H_x \def (IHj e2 e3 H1 c2 i H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c1: C).(drop j0 O c1 c2)) (\lambda (c1: C).(drop i j0 c1 e2)) (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Bind b) t)))) (\lambda (x: C).(\lambda (H3: (drop j0 O x c2)).(\lambda (H4: (drop i j0 x e2)).(ex_intro2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Bind b) t))) (CHead x (Bind b) (lift i (r (Bind b) j0) t)) (drop_drop (Bind b) j0 x c2 H3 (lift i (r (Bind b) j0) t)) (drop_skip (Bind b) i j0 x e2 H4 t))))) H2)))) | (Flat f) \Rightarrow (\lambda (H1: (drop (r (Flat f) j0) O e2 e3)).(let H_x \def (IHe1 e3 H1 c2 i H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 e2)) (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Flat f) t)))) (\lambda (x: C).(\lambda (H3: (drop (S j0) O x c2)).(\lambda (H4: (drop i (S j0) x e2)).(ex_intro2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Flat f) t))) (CHead x (Flat f) (lift i (r (Flat f) j0) t)) (drop_drop (Flat f) j0 x c2 H3 (lift i (r (Flat f) j0) t)) (drop_skip (Flat f) i j0 x e2 H4 t))))) H2))))]) (drop_gen_drop k e2 e3 t j0 H))))))))))) e1)))) j).
 
 theorem drop_trans_le:
  \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O c2 e2) \to (ex2 C (\lambda (e1: C).(drop i O c1 e1)) (\lambda (e1: C).(drop h (minus d i) e1 e2)))))))))))
 \def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (d: nat).((le n d) \to (\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop n O c2 e2) \to (ex2 C (\lambda (e1: C).(drop n O c1 e1)) (\lambda (e1: C).(drop h (minus d n) e1 e2)))))))))))) (\lambda (d: nat).(\lambda (_: (le O d)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H1: (drop O O c2 e2)).(let H2 \def (eq_ind C c2 (\lambda (c: C).(drop h d c1 c)) H0 e2 (drop_gen_refl c2 e2 H1)) in (eq_ind nat d (\lambda (n: nat).(ex2 C (\lambda (e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h n e1 e2)))) (ex_intro2 C (\lambda (e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h d e1 e2)) c1 (drop_refl c1) H2) (minus d O) (minus_n_O d))))))))))) (\lambda (i0: nat).(\lambda (IHi: ((\forall (d: nat).((le i0 d) \to (\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i0 O c2 e2) \to (ex2 C (\lambda (e1: C).(drop i0 O c1 e1)) (\lambda (e1: C).(drop h (minus d i0) e1 e2))))))))))))).(\lambda (d: nat).(nat_ind (\lambda (n: nat).((le (S i0) n) \to (\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h n c1 c2) \to (\forall (e2: C).((drop (S i0) O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus n (S i0)) e1 e2))))))))))) (\lambda (H: (le (S i0) O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (_: (drop h O c1 c2)).(\lambda (e2: C).(\lambda (_: (drop (S i0) O c2 e2)).(let H2 \def (match H return (\lambda (n: nat).((eq nat n O) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus O (S i0)) e1 e2))))) with [le_n \Rightarrow (\lambda (H2: (eq nat (S i0) O)).(let H3 \def (eq_ind nat (S i0) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus O (S i0)) e1 e2))) H3))) | (le_S m H2) \Rightarrow (\lambda (H3: (eq nat (S m) O)).((let H4 \def (eq_ind nat (S m) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind ((le (S i0) m) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus O (S i0)) e1 e2)))) H4)) H2))]) in (H2 (refl_equal nat O)))))))))) (\lambda (d0: nat).(\lambda (_: (((le (S i0) d0) \to (\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d0 c1 c2) \to (\forall (e2: C).((drop (S i0) O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus d0 (S i0)) e1 e2)))))))))))).(\lambda (H: (le (S i0) (S d0))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (h: nat).((drop h (S d0) c c2) \to (\forall (e2: C).((drop (S i0) O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CSort n) c2)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c2 e2)).(and3_ind (eq C c2 (CSort n)) (eq nat h O) (eq nat (S d0) O) (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (H2: (eq C c2 (CSort n))).(\lambda (_: (eq nat h O)).(\lambda (_: (eq nat (S d0) O)).(let H5 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H1 (CSort n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (H6: (eq C e2 (CSort n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n) (\lambda (c: C).(ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 c)))) (let H9 \def (eq_ind nat (S i0) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H7) in (False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 (CSort n)))) H9)) e2 H6)))) (drop_gen_sort n (S i0) O e2 H5)))))) (drop_gen_sort n h (S d0) c2 H0)))))))) (\lambda (c2: C).(\lambda (IHc: ((\forall (c3: C).(\forall (h: nat).((drop h (S d0) c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c2 e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(\forall (c3: C).(\forall (h: nat).((drop h (S d0) (CHead c2 k0 t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 k0 t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Bind b) t) c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e (Bind b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Bind b) d0) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r (Bind b) d0) c2 e))) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Bind b) t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0 (Bind b) x1))).(\lambda (H3: (eq T t (lift h (r (Bind b) d0) x1))).(\lambda (H4: (drop h (r (Bind b) d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: C).(drop (S i0) O c e2)) H1 (CHead x0 (Bind b) x1) H2) in (eq_ind_r T (lift h (r (Bind b) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Bind b) t0) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)))) (ex2_ind C (\lambda (e1: C).(drop i0 O c2 e1)) (\lambda (e1: C).(drop h (minus d0 i0) e1 e2)) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (x: C).(\lambda (H6: (drop i0 O c2 x)).(\lambda (H7: (drop h (minus d0 i0) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)) x (drop_drop (Bind b) i0 c2 x H6 (lift h (r (Bind b) d0) x1)) H7)))) (IHi d0 (le_S_n i0 d0 H) c2 x0 h H4 e2 (drop_gen_drop (Bind b) x0 e2 x1 i0 H5))) t H3))))))) (drop_gen_skip_l c2 c3 t h d0 (Bind b) H0))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Flat f) t) c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r (Flat f) d0) c2 e))) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f) t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0 (Flat f) x1))).(\lambda (H3: (eq T t (lift h (r (Flat f) d0) x1))).(\lambda (H4: (drop h (r (Flat f) d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: C).(drop (S i0) O c e2)) H1 (CHead x0 (Flat f) x1) H2) in (eq_ind_r T (lift h (r (Flat f) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f) t0) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)))) (ex2_ind C (\lambda (e1: C).(drop (S i0) O c2 e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f) (lift h (r (Flat f) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (x: C).(\lambda (H6: (drop (S i0) O c2 x)).(\lambda (H7: (drop h (minus (S d0) (S i0)) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f) (lift h (r (Flat f) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)) x (drop_drop (Flat f) i0 c2 x H6 (lift h (r (Flat f) d0) x1)) H7)))) (IHc x0 h H4 e2 (drop_gen_drop (Flat f) x0 e2 x1 i0 H5))) t H3))))))) (drop_gen_skip_l c2 c3 t h d0 (Flat f) H0))))))))) k)))) c1))))) d)))) i).
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (d: nat).((le n d) \to (\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop n O c2 e2) \to (ex2 C (\lambda (e1: C).(drop n O c1 e1)) (\lambda (e1: C).(drop h (minus d n) e1 e2)))))))))))) (\lambda (d: nat).(\lambda (_: (le O d)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H1: (drop O O c2 e2)).(let H2 \def (eq_ind C c2 (\lambda (c: C).(drop h d c1 c)) H0 e2 (drop_gen_refl c2 e2 H1)) in (eq_ind nat d (\lambda (n: nat).(ex2 C (\lambda (e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h n e1 e2)))) (ex_intro2 C (\lambda (e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h d e1 e2)) c1 (drop_refl c1) H2) (minus d O) (minus_n_O d))))))))))) (\lambda (i0: nat).(\lambda (IHi: ((\forall (d: nat).((le i0 d) \to (\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i0 O c2 e2) \to (ex2 C (\lambda (e1: C).(drop i0 O c1 e1)) (\lambda (e1: C).(drop h (minus d i0) e1 e2))))))))))))).(\lambda (d: nat).(nat_ind (\lambda (n: nat).((le (S i0) n) \to (\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h n c1 c2) \to (\forall (e2: C).((drop (S i0) O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus n (S i0)) e1 e2))))))))))) (\lambda (H: (le (S i0) O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (_: (drop h O c1 c2)).(\lambda (e2: C).(\lambda (_: (drop (S i0) O c2 e2)).(let H2 \def (match H return (\lambda (_: ?).(\lambda (n: nat).((eq nat n O) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus O (S i0)) e1 e2)))))) with [le_n \Rightarrow (\lambda (H2: (eq nat (S i0) O)).(let H3 \def (eq_ind nat (S i0) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus O (S i0)) e1 e2))) H3))) | (le_S m H2) \Rightarrow (\lambda (H3: (eq nat (S m) O)).((let H4 \def (eq_ind nat (S m) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind ((le (S i0) m) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus O (S i0)) e1 e2)))) H4)) H2))]) in (H2 (refl_equal nat O)))))))))) (\lambda (d0: nat).(\lambda (_: (((le (S i0) d0) \to (\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d0 c1 c2) \to (\forall (e2: C).((drop (S i0) O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h (minus d0 (S i0)) e1 e2)))))))))))).(\lambda (H: (le (S i0) (S d0))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (h: nat).((drop h (S d0) c c2) \to (\forall (e2: C).((drop (S i0) O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CSort n) c2)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c2 e2)).(and3_ind (eq C c2 (CSort n)) (eq nat h O) (eq nat (S d0) O) (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (H2: (eq C c2 (CSort n))).(\lambda (_: (eq nat h O)).(\lambda (_: (eq nat (S d0) O)).(let H5 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H1 (CSort n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (H6: (eq C e2 (CSort n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n) (\lambda (c: C).(ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 c)))) (let H9 \def (eq_ind nat (S i0) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H7) in (False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 (CSort n)))) H9)) e2 H6)))) (drop_gen_sort n (S i0) O e2 H5)))))) (drop_gen_sort n h (S d0) c2 H0)))))))) (\lambda (c2: C).(\lambda (IHc: ((\forall (c3: C).(\forall (h: nat).((drop h (S d0) c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c2 e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(\forall (c3: C).(\forall (h: nat).((drop h (S d0) (CHead c2 k0 t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 k0 t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Bind b) t) c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e (Bind b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Bind b) d0) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r (Bind b) d0) c2 e))) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Bind b) t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0 (Bind b) x1))).(\lambda (H3: (eq T t (lift h (r (Bind b) d0) x1))).(\lambda (H4: (drop h (r (Bind b) d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: C).(drop (S i0) O c e2)) H1 (CHead x0 (Bind b) x1) H2) in (eq_ind_r T (lift h (r (Bind b) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Bind b) t0) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)))) (ex2_ind C (\lambda (e1: C).(drop i0 O c2 e1)) (\lambda (e1: C).(drop h (minus d0 i0) e1 e2)) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (x: C).(\lambda (H6: (drop i0 O c2 x)).(\lambda (H7: (drop h (minus d0 i0) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)) x (drop_drop (Bind b) i0 c2 x H6 (lift h (r (Bind b) d0) x1)) H7)))) (IHi d0 (le_S_n i0 d0 H) c2 x0 h H4 e2 (drop_gen_drop (Bind b) x0 e2 x1 i0 H5))) t H3))))))) (drop_gen_skip_l c2 c3 t h d0 (Bind b) H0))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (c3: C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Flat f) t) c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r (Flat f) d0) c2 e))) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f) t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0 (Flat f) x1))).(\lambda (H3: (eq T t (lift h (r (Flat f) d0) x1))).(\lambda (H4: (drop h (r (Flat f) d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: C).(drop (S i0) O c e2)) H1 (CHead x0 (Flat f) x1) H2) in (eq_ind_r T (lift h (r (Flat f) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f) t0) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)))) (ex2_ind C (\lambda (e1: C).(drop (S i0) O c2 e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f) (lift h (r (Flat f) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (x: C).(\lambda (H6: (drop (S i0) O c2 x)).(\lambda (H7: (drop h (minus (S d0) (S i0)) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f) (lift h (r (Flat f) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)) x (drop_drop (Flat f) i0 c2 x H6 (lift h (r (Flat f) d0) x1)) H7)))) (IHc x0 h H4 e2 (drop_gen_drop (Flat f) x0 e2 x1 i0 H5))) t H3))))))) (drop_gen_skip_l c2 c3 t h d0 (Flat f) H0))))))))) k)))) c1))))) d)))) i).
 
 theorem drop_trans_ge:
  \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O c2 e2) \to ((le d i) \to (drop (plus i h) O c1 e2)))))))))
 \def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop n O c2 e2) \to ((le d n) \to (drop (plus n h) O c1 e2)))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H0: (drop O O c2 e2)).(\lambda (H1: (le d O)).(eq_ind C c2 (\lambda (c: C).(drop (plus O h) O c1 c)) (let H2 \def (match H1 return (\lambda (n: nat).((eq nat n O) \to (drop (plus O h) O c1 c2))) with [le_n \Rightarrow (\lambda (H0: (eq nat d O)).(eq_ind nat O (\lambda (_: nat).(drop (plus O h) O c1 c2)) (let H2 \def (eq_ind nat d (\lambda (n: nat).(le n O)) H1 O H0) in (let H3 \def (eq_ind nat d (\lambda (n: nat).(drop h n c1 c2)) H O H0) in H3)) d (sym_eq nat d O H0))) | (le_S m H0) \Rightarrow (\lambda (H2: (eq nat (S m) O)).((let H1 \def (eq_ind nat (S m) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le d m) \to (drop (plus O h) O c1 c2)) H1)) H0))]) in (H2 (refl_equal nat O))) e2 (drop_gen_refl c2 e2 H0)))))))))) (\lambda (i0: nat).(\lambda (IHi: ((\forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i0 O c2 e2) \to ((le d i0) \to (drop (plus i0 h) O c1 e2))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c c2) \to (\forall (e2: C).((drop (S i0) O c2 e2) \to ((le d (S i0)) \to (drop (plus (S i0) h) O c e2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) c2)).(\lambda (e2: C).(\lambda (H0: (drop (S i0) O c2 e2)).(\lambda (H1: (le d (S i0))).(and3_ind (eq C c2 (CSort n)) (eq nat h O) (eq nat d O) (drop (S (plus i0 h)) O (CSort n) e2) (\lambda (H2: (eq C c2 (CSort n))).(\lambda (H3: (eq nat h O)).(\lambda (H4: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: nat).(drop (S (plus i0 n0)) O (CSort n) e2)) (let H5 \def (eq_ind nat d (\lambda (n: nat).(le n (S i0))) H1 O H4) in (let H6 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CSort n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop (S (plus i0 O)) O (CSort n) e2) (\lambda (H7: (eq C e2 (CSort n))).(\lambda (H8: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n) (\lambda (c: C).(drop (S (plus i0 O)) O (CSort n) c)) (let H10 \def (eq_ind nat (S i0) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H8) in (False_ind (drop (S (plus i0 O)) O (CSort n) (CSort n)) H10)) e2 H7)))) (drop_gen_sort n (S i0) O e2 H6)))) h H3)))) (drop_gen_sort n h d c2 H)))))))))) (\lambda (c2: C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h: nat).((drop h d c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le d (S i0)) \to (drop (S (plus i0 h)) O c2 e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3: C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le n (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) e2))))))) (\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le O (S i0)) \to (drop (S (plus i0 n)) O (CHead c2 k t) e2)))))) (\lambda (H: (drop O O (CHead c2 k t) c3)).(\lambda (e2: C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda (_: (le O (S i0))).(let H2 \def (eq_ind_r C c3 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CHead c2 k t) (drop_gen_refl (CHead c2 k t) c3 H)) in (eq_ind nat i0 (\lambda (n: nat).(drop (S n) O (CHead c2 k t) e2)) (drop_drop k i0 c2 e2 (drop_gen_drop k c2 e2 t i0 H2) t) (plus i0 O) (plus_n_O i0))))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le O (S i0)) \to (drop (S (plus i0 n)) O (CHead c2 k t) e2))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(\lambda (H2: (le O (S i0))).(eq_ind nat (S (plus i0 n)) (\lambda (n0: nat).(drop (S n0) O (CHead c2 k t) e2)) (drop_drop k (S (plus i0 n)) c2 e2 (eq_ind_r nat (S (r k (plus i0 n))) (\lambda (n0: nat).(drop n0 O c2 e2)) (eq_ind_r nat (plus i0 (r k n)) (\lambda (n0: nat).(drop (S n0) O c2 e2)) (IHc c3 O (r k n) (drop_gen_drop k c2 c3 t n H0) e2 H1 H2) (r k (plus i0 n)) (r_plus_sym k i0 n)) (r k (S (plus i0 n))) (r_S k (plus i0 n))) t) (plus i0 (S n)) (plus_n_Sm i0 n)))))))) h)) (\lambda (d0: nat).(\lambda (IHd: ((\forall (h: nat).((drop h d0 (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le d0 (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) e2)))))))).(\lambda (h: nat).(\lambda (H: (drop h (S d0) (CHead c2 k t) c3)).(\lambda (e2: C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda (H1: (le (S d0) (S i0))).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k d0) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k d0) c2 e))) (drop (S (plus i0 h)) O (CHead c2 k t) e2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0 k x1))).(\lambda (H3: (eq T t (lift h (r k d0) x1))).(\lambda (H4: (drop h (r k d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: C).(\forall (h: nat).((drop h d0 (CHead c2 k t) c) \to (\forall (e2: C).((drop (S i0) O c e2) \to ((le d0 (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) e2))))))) IHd (CHead x0 k x1) H2) in (let H6 \def (eq_ind C c3 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CHead x0 k x1) H2) in (let H7 \def (eq_ind T t (\lambda (t: T).(\forall (h: nat).((drop h d0 (CHead c2 k t) (CHead x0 k x1)) \to (\forall (e2: C).((drop (S i0) O (CHead x0 k x1) e2) \to ((le d0 (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) e2))))))) H5 (lift h (r k d0) x1) H3) in (eq_ind_r T (lift h (r k d0) x1) (\lambda (t0: T).(drop (S (plus i0 h)) O (CHead c2 k t0) e2)) (drop_drop k (plus i0 h) c2 e2 (K_ind (\lambda (k0: K).((drop h (r k0 d0) c2 x0) \to ((drop (r k0 i0) O x0 e2) \to (drop (r k0 (plus i0 h)) O c2 e2)))) (\lambda (b: B).(\lambda (H8: (drop h (r (Bind b) d0) c2 x0)).(\lambda (H9: (drop (r (Bind b) i0) O x0 e2)).(IHi c2 x0 (r (Bind b) d0) h H8 e2 H9 (le_S_n (r (Bind b) d0) i0 H1))))) (\lambda (f: F).(\lambda (H8: (drop h (r (Flat f) d0) c2 x0)).(\lambda (H9: (drop (r (Flat f) i0) O x0 e2)).(IHc x0 (r (Flat f) d0) h H8 e2 H9 H1)))) k H4 (drop_gen_drop k x0 e2 x1 i0 H6)) (lift h (r k d0) x1)) t H3))))))))) (drop_gen_skip_l c2 c3 t h d0 k H))))))))) d))))))) c1)))) i).
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop n O c2 e2) \to ((le d n) \to (drop (plus n h) O c1 e2)))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H0: (drop O O c2 e2)).(\lambda (H1: (le d O)).(eq_ind C c2 (\lambda (c: C).(drop (plus O h) O c1 c)) (let H2 \def (match H1 return (\lambda (_: ?).(\lambda (n: nat).((eq nat n O) \to (drop (plus O h) O c1 c2)))) with [le_n \Rightarrow (\lambda (H0: (eq nat d O)).(eq_ind nat O (\lambda (_: nat).(drop (plus O h) O c1 c2)) (let H2 \def (eq_ind nat d (\lambda (n: nat).(le n O)) H1 O H0) in (let H3 \def (eq_ind nat d (\lambda (n: nat).(drop h n c1 c2)) H O H0) in H3)) d (sym_eq nat d O H0))) | (le_S m H0) \Rightarrow (\lambda (H2: (eq nat (S m) O)).((let H1 \def (eq_ind nat (S m) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le d m) \to (drop (plus O h) O c1 c2)) H1)) H0))]) in (H2 (refl_equal nat O))) e2 (drop_gen_refl c2 e2 H0)))))))))) (\lambda (i0: nat).(\lambda (IHi: ((\forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i0 O c2 e2) \to ((le d i0) \to (drop (plus i0 h) O c1 e2))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c c2) \to (\forall (e2: C).((drop (S i0) O c2 e2) \to ((le d (S i0)) \to (drop (plus (S i0) h) O c e2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) c2)).(\lambda (e2: C).(\lambda (H0: (drop (S i0) O c2 e2)).(\lambda (H1: (le d (S i0))).(and3_ind (eq C c2 (CSort n)) (eq nat h O) (eq nat d O) (drop (S (plus i0 h)) O (CSort n) e2) (\lambda (H2: (eq C c2 (CSort n))).(\lambda (H3: (eq nat h O)).(\lambda (H4: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: nat).(drop (S (plus i0 n0)) O (CSort n) e2)) (let H5 \def (eq_ind nat d (\lambda (n: nat).(le n (S i0))) H1 O H4) in (let H6 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CSort n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop (S (plus i0 O)) O (CSort n) e2) (\lambda (H7: (eq C e2 (CSort n))).(\lambda (H8: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n) (\lambda (c: C).(drop (S (plus i0 O)) O (CSort n) c)) (let H10 \def (eq_ind nat (S i0) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H8) in (False_ind (drop (S (plus i0 O)) O (CSort n) (CSort n)) H10)) e2 H7)))) (drop_gen_sort n (S i0) O e2 H6)))) h H3)))) (drop_gen_sort n h d c2 H)))))))))) (\lambda (c2: C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h: nat).((drop h d c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le d (S i0)) \to (drop (S (plus i0 h)) O c2 e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3: C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le n (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) e2))))))) (\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le O (S i0)) \to (drop (S (plus i0 n)) O (CHead c2 k t) e2)))))) (\lambda (H: (drop O O (CHead c2 k t) c3)).(\lambda (e2: C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda (_: (le O (S i0))).(let H2 \def (eq_ind_r C c3 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CHead c2 k t) (drop_gen_refl (CHead c2 k t) c3 H)) in (eq_ind nat i0 (\lambda (n: nat).(drop (S n) O (CHead c2 k t) e2)) (drop_drop k i0 c2 e2 (drop_gen_drop k c2 e2 t i0 H2) t) (plus i0 O) (plus_n_O i0))))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le O (S i0)) \to (drop (S (plus i0 n)) O (CHead c2 k t) e2))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(\lambda (H2: (le O (S i0))).(eq_ind nat (S (plus i0 n)) (\lambda (n0: nat).(drop (S n0) O (CHead c2 k t) e2)) (drop_drop k (S (plus i0 n)) c2 e2 (eq_ind_r nat (S (r k (plus i0 n))) (\lambda (n0: nat).(drop n0 O c2 e2)) (eq_ind_r nat (plus i0 (r k n)) (\lambda (n0: nat).(drop (S n0) O c2 e2)) (IHc c3 O (r k n) (drop_gen_drop k c2 c3 t n H0) e2 H1 H2) (r k (plus i0 n)) (r_plus_sym k i0 n)) (r k (S (plus i0 n))) (r_S k (plus i0 n))) t) (plus i0 (S n)) (plus_n_Sm i0 n)))))))) h)) (\lambda (d0: nat).(\lambda (IHd: ((\forall (h: nat).((drop h d0 (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le d0 (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) e2)))))))).(\lambda (h: nat).(\lambda (H: (drop h (S d0) (CHead c2 k t) c3)).(\lambda (e2: C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda (H1: (le (S d0) (S i0))).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k d0) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k d0) c2 e))) (drop (S (plus i0 h)) O (CHead c2 k t) e2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0 k x1))).(\lambda (H3: (eq T t (lift h (r k d0) x1))).(\lambda (H4: (drop h (r k d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: C).(\forall (h: nat).((drop h d0 (CHead c2 k t) c) \to (\forall (e2: C).((drop (S i0) O c e2) \to ((le d0 (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) e2))))))) IHd (CHead x0 k x1) H2) in (let H6 \def (eq_ind C c3 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CHead x0 k x1) H2) in (let H7 \def (eq_ind T t (\lambda (t: T).(\forall (h: nat).((drop h d0 (CHead c2 k t) (CHead x0 k x1)) \to (\forall (e2: C).((drop (S i0) O (CHead x0 k x1) e2) \to ((le d0 (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) e2))))))) H5 (lift h (r k d0) x1) H3) in (eq_ind_r T (lift h (r k d0) x1) (\lambda (t0: T).(drop (S (plus i0 h)) O (CHead c2 k t0) e2)) (drop_drop k (plus i0 h) c2 e2 (K_ind (\lambda (k0: K).((drop h (r k0 d0) c2 x0) \to ((drop (r k0 i0) O x0 e2) \to (drop (r k0 (plus i0 h)) O c2 e2)))) (\lambda (b: B).(\lambda (H8: (drop h (r (Bind b) d0) c2 x0)).(\lambda (H9: (drop (r (Bind b) i0) O x0 e2)).(IHi c2 x0 (r (Bind b) d0) h H8 e2 H9 (le_S_n (r (Bind b) d0) i0 H1))))) (\lambda (f: F).(\lambda (H8: (drop h (r (Flat f) d0) c2 x0)).(\lambda (H9: (drop (r (Flat f) i0) O x0 e2)).(IHc x0 (r (Flat f) d0) h H8 e2 H9 H1)))) k H4 (drop_gen_drop k x0 e2 x1 i0 H6)) (lift h (r k d0) x1)) t H3))))))))) (drop_gen_skip_l c2 c3 t h d0 k H))))))))) d))))))) c1)))) i).
 
 inductive drop1: PList \to (C \to (C \to Prop)) \def
 | drop1_nil: \forall (c: C).(drop1 PNil c c)
@@ -734,12 +734,12 @@ definition ctrans:
 theorem drop1_skip_bind:
  \forall (b: B).(\forall (e: C).(\forall (hds: PList).(\forall (c: C).(\forall (u: T).((drop1 hds c e) \to (drop1 (Ss hds) (CHead c (Bind b) (lift1 hds u)) (CHead e (Bind b) u)))))))
 \def
- \lambda (b: B).(\lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p) (CHead c (Bind b) (lift1 p u)) (CHead e (Bind b) u)))))) (\lambda (c: C).(\lambda (u: T).(\lambda (H: (drop1 PNil c e)).(let H0 \def (match H return (\lambda (p: PList).(\lambda (c0: C).(\lambda (c1: C).((eq PList p PNil) \to ((eq C c0 c) \to ((eq C c1 e) \to (drop1 PNil (CHead c (Bind b) u) (CHead e (Bind b) u)))))))) with [(drop1_nil c0) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq C c0 e)).(eq_ind C c (\lambda (c1: C).((eq C c1 e) \to (drop1 PNil (CHead c (Bind b) u) (CHead e (Bind b) u)))) (\lambda (H3: (eq C c e)).(eq_ind C e (\lambda (c: C).(drop1 PNil (CHead c (Bind b) u) (CHead e (Bind b) u))) (drop1_nil (CHead e (Bind b) u)) c (sym_eq C c e H3))) c0 (sym_eq C c0 c H1) H2)))) | (drop1_cons c1 c2 h d H0 c3 hds H1) \Rightarrow (\lambda (H2: (eq PList (PCons h d hds) PNil)).(\lambda (H3: (eq C c1 c)).(\lambda (H4: (eq C c3 e)).((let H5 \def (eq_ind PList (PCons h d hds) (\lambda (e0: PList).(match e0 return (\lambda (_: ?).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H2) in (False_ind ((eq C c1 c) \to ((eq C c3 e) \to ((drop h d c1 c2) \to ((drop1 hds c2 c3) \to (drop1 PNil (CHead c (Bind b) u) (CHead e (Bind b) u)))))) H5)) H3 H4 H0 H1))))]) in (H0 (refl_equal PList PNil) (refl_equal C c) (refl_equal C e)))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p) (CHead c (Bind b) (lift1 p u)) (CHead e (Bind b) u))))))).(\lambda (c: C).(\lambda (u: T).(\lambda (H0: (drop1 (PCons n n0 p) c e)).(let H1 \def (match H0 return (\lambda (p0: PList).(\lambda (c0: C).(\lambda (c1: C).((eq PList p0 (PCons n n0 p)) \to ((eq C c0 c) \to ((eq C c1 e) \to (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u)))))))) with [(drop1_nil c0) \Rightarrow (\lambda (H1: (eq PList PNil (PCons n n0 p))).(\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq C c0 e)).((let H4 \def (eq_ind PList PNil (\lambda (e0: PList).(match e0 return (\lambda (_: ?).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H1) in (False_ind ((eq C c0 c) \to ((eq C c0 e) \to (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u)))) H4)) H2 H3)))) | (drop1_cons c1 c2 h d H1 c3 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds) (PCons n n0 p))).(\lambda (H4: (eq C c1 c)).(\lambda (H5: (eq C c3 e)).((let H6 \def (f_equal PList PList (\lambda (e0: PList).(match e0 return (\lambda (_: ?).PList) with [PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons n n0 p) H3) in ((let H7 \def (f_equal PList nat (\lambda (e0: PList).(match e0 return (\lambda (_: ?).nat) with [PNil \Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H3) in ((let H8 \def (f_equal PList nat (\lambda (e0: PList).(match e0 return (\lambda (_: ?).nat) with [PNil \Rightarrow h | (PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H3) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n1 d c1 c2) \to ((drop1 hds c2 c3) \to (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u))))))))) (\lambda (H9: (eq nat d n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n n1 c1 c2) \to ((drop1 hds c2 c3) \to (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u)))))))) (\lambda (H10: (eq PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c1 c) \to ((eq C c3 e) \to ((drop n n0 c1 c2) \to ((drop1 p0 c2 c3) \to (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u))))))) (\lambda (H11: (eq C c1 c)).(eq_ind C c (\lambda (c0: C).((eq C c3 e) \to ((drop n n0 c0 c2) \to ((drop1 p c2 c3) \to (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u)))))) (\lambda (H12: (eq C c3 e)).(eq_ind C e (\lambda (c0: C).((drop n n0 c c2) \to ((drop1 p c2 c0) \to (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u))))) (\lambda (H13: (drop n n0 c c2)).(\lambda (H14: (drop1 p c2 e)).(drop1_cons (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead c2 (Bind b) (lift1 p u)) n (S n0) (drop_skip_bind n n0 c c2 H13 b (lift1 p u)) (CHead e (Bind b) u) (Ss p) (H c2 u H14)))) c3 (sym_eq C c3 e H12))) c1 (sym_eq C c1 c H11))) hds (sym_eq PList hds p H10))) d (sym_eq nat d n0 H9))) h (sym_eq nat h n H8))) H7)) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList (PCons n n0 p)) (refl_equal C c) (refl_equal C e)))))))))) hds))).
+ \lambda (b: B).(\lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p) (CHead c (Bind b) (lift1 p u)) (CHead e (Bind b) u)))))) (\lambda (c: C).(\lambda (u: T).(\lambda (H: (drop1 PNil c e)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (p: PList).(\lambda (c0: C).(\lambda (c1: C).((eq PList p PNil) \to ((eq C c0 c) \to ((eq C c1 e) \to (drop1 PNil (CHead c (Bind b) u) (CHead e (Bind b) u))))))))) with [(drop1_nil c0) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq C c0 e)).(eq_ind C c (\lambda (c1: C).((eq C c1 e) \to (drop1 PNil (CHead c (Bind b) u) (CHead e (Bind b) u)))) (\lambda (H3: (eq C c e)).(eq_ind C e (\lambda (c: C).(drop1 PNil (CHead c (Bind b) u) (CHead e (Bind b) u))) (drop1_nil (CHead e (Bind b) u)) c (sym_eq C c e H3))) c0 (sym_eq C c0 c H1) H2)))) | (drop1_cons c1 c2 h d H0 c3 hds H1) \Rightarrow (\lambda (H2: (eq PList (PCons h d hds) PNil)).(\lambda (H3: (eq C c1 c)).(\lambda (H4: (eq C c3 e)).((let H5 \def (eq_ind PList (PCons h d hds) (\lambda (e0: PList).(match e0 return (\lambda (_: ?).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H2) in (False_ind ((eq C c1 c) \to ((eq C c3 e) \to ((drop h d c1 c2) \to ((drop1 hds c2 c3) \to (drop1 PNil (CHead c (Bind b) u) (CHead e (Bind b) u)))))) H5)) H3 H4 H0 H1))))]) in (H0 (refl_equal PList PNil) (refl_equal C c) (refl_equal C e)))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p) (CHead c (Bind b) (lift1 p u)) (CHead e (Bind b) u))))))).(\lambda (c: C).(\lambda (u: T).(\lambda (H0: (drop1 (PCons n n0 p) c e)).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (p0: PList).(\lambda (c0: C).(\lambda (c1: C).((eq PList p0 (PCons n n0 p)) \to ((eq C c0 c) \to ((eq C c1 e) \to (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u))))))))) with [(drop1_nil c0) \Rightarrow (\lambda (H1: (eq PList PNil (PCons n n0 p))).(\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq C c0 e)).((let H4 \def (eq_ind PList PNil (\lambda (e0: PList).(match e0 return (\lambda (_: ?).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H1) in (False_ind ((eq C c0 c) \to ((eq C c0 e) \to (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u)))) H4)) H2 H3)))) | (drop1_cons c1 c2 h d H1 c3 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds) (PCons n n0 p))).(\lambda (H4: (eq C c1 c)).(\lambda (H5: (eq C c3 e)).((let H6 \def (f_equal PList PList (\lambda (e0: PList).(match e0 return (\lambda (_: ?).PList) with [PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons n n0 p) H3) in ((let H7 \def (f_equal PList nat (\lambda (e0: PList).(match e0 return (\lambda (_: ?).nat) with [PNil \Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H3) in ((let H8 \def (f_equal PList nat (\lambda (e0: PList).(match e0 return (\lambda (_: ?).nat) with [PNil \Rightarrow h | (PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H3) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n1 d c1 c2) \to ((drop1 hds c2 c3) \to (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u))))))))) (\lambda (H9: (eq nat d n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n n1 c1 c2) \to ((drop1 hds c2 c3) \to (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u)))))))) (\lambda (H10: (eq PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c1 c) \to ((eq C c3 e) \to ((drop n n0 c1 c2) \to ((drop1 p0 c2 c3) \to (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u))))))) (\lambda (H11: (eq C c1 c)).(eq_ind C c (\lambda (c0: C).((eq C c3 e) \to ((drop n n0 c0 c2) \to ((drop1 p c2 c3) \to (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u)))))) (\lambda (H12: (eq C c3 e)).(eq_ind C e (\lambda (c0: C).((drop n n0 c c2) \to ((drop1 p c2 c0) \to (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u))))) (\lambda (H13: (drop n n0 c c2)).(\lambda (H14: (drop1 p c2 e)).(drop1_cons (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead c2 (Bind b) (lift1 p u)) n (S n0) (drop_skip_bind n n0 c c2 H13 b (lift1 p u)) (CHead e (Bind b) u) (Ss p) (H c2 u H14)))) c3 (sym_eq C c3 e H12))) c1 (sym_eq C c1 c H11))) hds (sym_eq PList hds p H10))) d (sym_eq nat d n0 H9))) h (sym_eq nat h n H8))) H7)) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList (PCons n n0 p)) (refl_equal C c) (refl_equal C e)))))))))) hds))).
 
 theorem drop1_cons_tail:
  \forall (c2: C).(\forall (c3: C).(\forall (h: nat).(\forall (d: nat).((drop h d c2 c3) \to (\forall (hds: PList).(\forall (c1: C).((drop1 hds c1 c2) \to (drop1 (PConsTail hds h d) c1 c3))))))))
 \def
- \lambda (c2: C).(\lambda (c3: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (drop h d c2 c3)).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1 c3)))) (\lambda (c1: C).(\lambda (H0: (drop1 PNil c1 c2)).(let H1 \def (match H0 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p PNil) \to ((eq C c c1) \to ((eq C c0 c2) \to (drop1 (PCons h d PNil) c1 c3))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c c1)).(\lambda (H3: (eq C c c2)).(eq_ind C c1 (\lambda (c0: C).((eq C c0 c2) \to (drop1 (PCons h d PNil) c1 c3))) (\lambda (H4: (eq C c1 c2)).(eq_ind C c2 (\lambda (c0: C).(drop1 (PCons h d PNil) c0 c3)) (drop1_cons c2 c3 h d H c3 PNil (drop1_nil c3)) c1 (sym_eq C c1 c2 H4))) c (sym_eq C c c1 H2) H3)))) | (drop1_cons c0 c4 h0 d0 H1 c5 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h0 d0 hds) PNil)).(\lambda (H4: (eq C c0 c1)).(\lambda (H5: (eq C c5 c2)).((let H6 \def (eq_ind PList (PCons h0 d0 hds) (\lambda (e: PList).(match e return (\lambda (_: ?).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c0 c1) \to ((eq C c5 c2) \to ((drop h0 d0 c0 c4) \to ((drop1 hds c4 c5) \to (drop1 (PCons h d PNil) c1 c3))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c1) (refl_equal C c2))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H0: ((\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1 c3))))).(\lambda (c1: C).(\lambda (H1: (drop1 (PCons n n0 p) c1 c2)).(let H2 \def (match H1 return (\lambda (p0: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p0 (PCons n n0 p)) \to ((eq C c c1) \to ((eq C c0 c2) \to (drop1 (PCons n n0 (PConsTail p h d)) c1 c3))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c c1)).(\lambda (H4: (eq C c c2)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e return (\lambda (_: ?).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c c1) \to ((eq C c c2) \to (drop1 (PCons n n0 (PConsTail p h d)) c1 c3))) H5)) H3 H4)))) | (drop1_cons c0 c4 h0 d0 H2 c5 hds H3) \Rightarrow (\lambda (H4: (eq PList (PCons h0 d0 hds) (PCons n n0 p))).(\lambda (H5: (eq C c0 c1)).(\lambda (H6: (eq C c5 c2)).((let H7 \def (f_equal PList PList (\lambda (e: PList).(match e return (\lambda (_: ?).PList) with [PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p])) (PCons h0 d0 hds) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_: ?).nat) with [PNil \Rightarrow d0 | (PCons _ n _) \Rightarrow n])) (PCons h0 d0 hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_: ?).nat) with [PNil \Rightarrow h0 | (PCons n _ _) \Rightarrow n])) (PCons h0 d0 hds) (PCons n n0 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d0 n0) \to ((eq PList hds p) \to ((eq C c0 c1) \to ((eq C c5 c2) \to ((drop n1 d0 c0 c4) \to ((drop1 hds c4 c5) \to (drop1 (PCons n n0 (PConsTail p h d)) c1 c3)))))))) (\lambda (H10: (eq nat d0 n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq C c0 c1) \to ((eq C c5 c2) \to ((drop n n1 c0 c4) \to ((drop1 hds c4 c5) \to (drop1 (PCons n n0 (PConsTail p h d)) c1 c3))))))) (\lambda (H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c0 c1) \to ((eq C c5 c2) \to ((drop n n0 c0 c4) \to ((drop1 p0 c4 c5) \to (drop1 (PCons n n0 (PConsTail p h d)) c1 c3)))))) (\lambda (H12: (eq C c0 c1)).(eq_ind C c1 (\lambda (c: C).((eq C c5 c2) \to ((drop n n0 c c4) \to ((drop1 p c4 c5) \to (drop1 (PCons n n0 (PConsTail p h d)) c1 c3))))) (\lambda (H13: (eq C c5 c2)).(eq_ind C c2 (\lambda (c: C).((drop n n0 c1 c4) \to ((drop1 p c4 c) \to (drop1 (PCons n n0 (PConsTail p h d)) c1 c3)))) (\lambda (H14: (drop n n0 c1 c4)).(\lambda (H15: (drop1 p c4 c2)).(drop1_cons c1 c4 n n0 H14 c3 (PConsTail p h d) (H0 c4 H15)))) c5 (sym_eq C c5 c2 H13))) c0 (sym_eq C c0 c1 H12))) hds (sym_eq PList hds p H11))) d0 (sym_eq nat d0 n0 H10))) h0 (sym_eq nat h0 n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p)) (refl_equal C c1) (refl_equal C c2))))))))) hds)))))).
+ \lambda (c2: C).(\lambda (c3: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (drop h d c2 c3)).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1 c3)))) (\lambda (c1: C).(\lambda (H0: (drop1 PNil c1 c2)).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p PNil) \to ((eq C c c1) \to ((eq C c0 c2) \to (drop1 (PCons h d PNil) c1 c3)))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c c1)).(\lambda (H3: (eq C c c2)).(eq_ind C c1 (\lambda (c0: C).((eq C c0 c2) \to (drop1 (PCons h d PNil) c1 c3))) (\lambda (H4: (eq C c1 c2)).(eq_ind C c2 (\lambda (c0: C).(drop1 (PCons h d PNil) c0 c3)) (drop1_cons c2 c3 h d H c3 PNil (drop1_nil c3)) c1 (sym_eq C c1 c2 H4))) c (sym_eq C c c1 H2) H3)))) | (drop1_cons c0 c4 h0 d0 H1 c5 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h0 d0 hds) PNil)).(\lambda (H4: (eq C c0 c1)).(\lambda (H5: (eq C c5 c2)).((let H6 \def (eq_ind PList (PCons h0 d0 hds) (\lambda (e: PList).(match e return (\lambda (_: ?).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c0 c1) \to ((eq C c5 c2) \to ((drop h0 d0 c0 c4) \to ((drop1 hds c4 c5) \to (drop1 (PCons h d PNil) c1 c3))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c1) (refl_equal C c2))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H0: ((\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1 c3))))).(\lambda (c1: C).(\lambda (H1: (drop1 (PCons n n0 p) c1 c2)).(let H2 \def (match H1 return (\lambda (_: ?).(\lambda (p0: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p0 (PCons n n0 p)) \to ((eq C c c1) \to ((eq C c0 c2) \to (drop1 (PCons n n0 (PConsTail p h d)) c1 c3)))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c c1)).(\lambda (H4: (eq C c c2)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e return (\lambda (_: ?).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c c1) \to ((eq C c c2) \to (drop1 (PCons n n0 (PConsTail p h d)) c1 c3))) H5)) H3 H4)))) | (drop1_cons c0 c4 h0 d0 H2 c5 hds H3) \Rightarrow (\lambda (H4: (eq PList (PCons h0 d0 hds) (PCons n n0 p))).(\lambda (H5: (eq C c0 c1)).(\lambda (H6: (eq C c5 c2)).((let H7 \def (f_equal PList PList (\lambda (e: PList).(match e return (\lambda (_: ?).PList) with [PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p])) (PCons h0 d0 hds) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_: ?).nat) with [PNil \Rightarrow d0 | (PCons _ n _) \Rightarrow n])) (PCons h0 d0 hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_: ?).nat) with [PNil \Rightarrow h0 | (PCons n _ _) \Rightarrow n])) (PCons h0 d0 hds) (PCons n n0 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d0 n0) \to ((eq PList hds p) \to ((eq C c0 c1) \to ((eq C c5 c2) \to ((drop n1 d0 c0 c4) \to ((drop1 hds c4 c5) \to (drop1 (PCons n n0 (PConsTail p h d)) c1 c3)))))))) (\lambda (H10: (eq nat d0 n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq C c0 c1) \to ((eq C c5 c2) \to ((drop n n1 c0 c4) \to ((drop1 hds c4 c5) \to (drop1 (PCons n n0 (PConsTail p h d)) c1 c3))))))) (\lambda (H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c0 c1) \to ((eq C c5 c2) \to ((drop n n0 c0 c4) \to ((drop1 p0 c4 c5) \to (drop1 (PCons n n0 (PConsTail p h d)) c1 c3)))))) (\lambda (H12: (eq C c0 c1)).(eq_ind C c1 (\lambda (c: C).((eq C c5 c2) \to ((drop n n0 c c4) \to ((drop1 p c4 c5) \to (drop1 (PCons n n0 (PConsTail p h d)) c1 c3))))) (\lambda (H13: (eq C c5 c2)).(eq_ind C c2 (\lambda (c: C).((drop n n0 c1 c4) \to ((drop1 p c4 c) \to (drop1 (PCons n n0 (PConsTail p h d)) c1 c3)))) (\lambda (H14: (drop n n0 c1 c4)).(\lambda (H15: (drop1 p c4 c2)).(drop1_cons c1 c4 n n0 H14 c3 (PConsTail p h d) (H0 c4 H15)))) c5 (sym_eq C c5 c2 H13))) c0 (sym_eq C c0 c1 H12))) hds (sym_eq PList hds p H11))) d0 (sym_eq nat d0 n0 H10))) h0 (sym_eq nat h0 n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p)) (refl_equal C c1) (refl_equal C c2))))))))) hds)))))).
 
 theorem lift1_free:
  \forall (hds: PList).(\forall (i: nat).(\forall (t: T).(eq T (lift1 hds (lift (S i) O t)) (lift (S (trans hds i)) O (ctrans hds i t)))))
@@ -761,17 +761,17 @@ definition cimp:
 theorem clear_gen_sort:
  \forall (x: C).(\forall (n: nat).((clear (CSort n) x) \to (\forall (P: Prop).P)))
 \def
- \lambda (x: C).(\lambda (n: nat).(\lambda (H: (clear (CSort n) x)).(\lambda (P: Prop).(let H0 \def (match H return (\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n)) \to ((eq C c0 x) \to P)))) with [(clear_bind b e u) \Rightarrow (\lambda (H0: (eq C (CHead e (Bind b) u) (CSort n))).(\lambda (H1: (eq C (CHead e (Bind b) u) x)).((let H2 \def (eq_ind C (CHead e (Bind b) u) (\lambda (e0: C).(match e0 return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H0) in (False_ind ((eq C (CHead e (Bind b) u) x) \to P) H2)) H1))) | (clear_flat e c H0 f u) \Rightarrow (\lambda (H1: (eq C (CHead e (Flat f) u) (CSort n))).(\lambda (H2: (eq C c x)).((let H3 \def (eq_ind C (CHead e (Flat f) u) (\lambda (e0: C).(match e0 return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H1) in (False_ind ((eq C c x) \to ((clear e c) \to P)) H3)) H2 H0)))]) in (H0 (refl_equal C (CSort n)) (refl_equal C x)))))).
+ \lambda (x: C).(\lambda (n: nat).(\lambda (H: (clear (CSort n) x)).(\lambda (P: Prop).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n)) \to ((eq C c0 x) \to P))))) with [(clear_bind b e u) \Rightarrow (\lambda (H0: (eq C (CHead e (Bind b) u) (CSort n))).(\lambda (H1: (eq C (CHead e (Bind b) u) x)).((let H2 \def (eq_ind C (CHead e (Bind b) u) (\lambda (e0: C).(match e0 return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H0) in (False_ind ((eq C (CHead e (Bind b) u) x) \to P) H2)) H1))) | (clear_flat e c H0 f u) \Rightarrow (\lambda (H1: (eq C (CHead e (Flat f) u) (CSort n))).(\lambda (H2: (eq C c x)).((let H3 \def (eq_ind C (CHead e (Flat f) u) (\lambda (e0: C).(match e0 return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H1) in (False_ind ((eq C c x) \to ((clear e c) \to P)) H3)) H2 H0)))]) in (H0 (refl_equal C (CSort n)) (refl_equal C x)))))).
 
 theorem clear_gen_bind:
  \forall (b: B).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear (CHead e (Bind b) u) x) \to (eq C x (CHead e (Bind b) u))))))
 \def
- \lambda (b: B).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H: (clear (CHead e (Bind b) u) x)).(let H0 \def (match H return (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e (Bind b) u)) \to ((eq C c0 x) \to (eq C x (CHead e (Bind b) u)))))) with [(clear_bind b0 e0 u0) \Rightarrow (\lambda (H0: (eq C (CHead e0 (Bind b0) u0) (CHead e (Bind b) u))).(\lambda (H1: (eq C (CHead e0 (Bind b0) u0) x)).((let H2 \def (f_equal C T (\lambda (e1: C).(match e1 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H0) in ((let H3 \def (f_equal C B (\lambda (e1: C).(match e1 return (\lambda (_: ?).B) with [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b0])])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H0) in ((let H4 \def (f_equal C C (\lambda (e1: C).(match e1 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow e0 | (CHead c _ _) \Rightarrow c])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H0) in (eq_ind C e (\lambda (c: C).((eq B b0 b) \to ((eq T u0 u) \to ((eq C (CHead c (Bind b0) u0) x) \to (eq C x (CHead e (Bind b) u)))))) (\lambda (H5: (eq B b0 b)).(eq_ind B b (\lambda (b1: B).((eq T u0 u) \to ((eq C (CHead e (Bind b1) u0) x) \to (eq C x (CHead e (Bind b) u))))) (\lambda (H6: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead e (Bind b) t) x) \to (eq C x (CHead e (Bind b) u)))) (\lambda (H7: (eq C (CHead e (Bind b) u) x)).(eq_ind C (CHead e (Bind b) u) (\lambda (c: C).(eq C c (CHead e (Bind b) u))) (refl_equal C (CHead e (Bind b) u)) x H7)) u0 (sym_eq T u0 u H6))) b0 (sym_eq B b0 b H5))) e0 (sym_eq C e0 e H4))) H3)) H2)) H1))) | (clear_flat e0 c H0 f u0) \Rightarrow (\lambda (H1: (eq C (CHead e0 (Flat f) u0) (CHead e (Bind b) u))).(\lambda (H2: (eq C c x)).((let H3 \def (eq_ind C (CHead e0 (Flat f) u0) (\lambda (e1: C).(match e1 return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (CHead e (Bind b) u) H1) in (False_ind ((eq C c x) \to ((clear e0 c) \to (eq C x (CHead e (Bind b) u)))) H3)) H2 H0)))]) in (H0 (refl_equal C (CHead e (Bind b) u)) (refl_equal C x))))))).
+ \lambda (b: B).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H: (clear (CHead e (Bind b) u) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e (Bind b) u)) \to ((eq C c0 x) \to (eq C x (CHead e (Bind b) u))))))) with [(clear_bind b0 e0 u0) \Rightarrow (\lambda (H0: (eq C (CHead e0 (Bind b0) u0) (CHead e (Bind b) u))).(\lambda (H1: (eq C (CHead e0 (Bind b0) u0) x)).((let H2 \def (f_equal C T (\lambda (e1: C).(match e1 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H0) in ((let H3 \def (f_equal C B (\lambda (e1: C).(match e1 return (\lambda (_: ?).B) with [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b0])])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H0) in ((let H4 \def (f_equal C C (\lambda (e1: C).(match e1 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow e0 | (CHead c _ _) \Rightarrow c])) (CHead e0 (Bind b0) u0) (CHead e (Bind b) u) H0) in (eq_ind C e (\lambda (c: C).((eq B b0 b) \to ((eq T u0 u) \to ((eq C (CHead c (Bind b0) u0) x) \to (eq C x (CHead e (Bind b) u)))))) (\lambda (H5: (eq B b0 b)).(eq_ind B b (\lambda (b1: B).((eq T u0 u) \to ((eq C (CHead e (Bind b1) u0) x) \to (eq C x (CHead e (Bind b) u))))) (\lambda (H6: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead e (Bind b) t) x) \to (eq C x (CHead e (Bind b) u)))) (\lambda (H7: (eq C (CHead e (Bind b) u) x)).(eq_ind C (CHead e (Bind b) u) (\lambda (c: C).(eq C c (CHead e (Bind b) u))) (refl_equal C (CHead e (Bind b) u)) x H7)) u0 (sym_eq T u0 u H6))) b0 (sym_eq B b0 b H5))) e0 (sym_eq C e0 e H4))) H3)) H2)) H1))) | (clear_flat e0 c H0 f u0) \Rightarrow (\lambda (H1: (eq C (CHead e0 (Flat f) u0) (CHead e (Bind b) u))).(\lambda (H2: (eq C c x)).((let H3 \def (eq_ind C (CHead e0 (Flat f) u0) (\lambda (e1: C).(match e1 return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (CHead e (Bind b) u) H1) in (False_ind ((eq C c x) \to ((clear e0 c) \to (eq C x (CHead e (Bind b) u)))) H3)) H2 H0)))]) in (H0 (refl_equal C (CHead e (Bind b) u)) (refl_equal C x))))))).
 
 theorem clear_gen_flat:
  \forall (f: F).(\forall (e: C).(\forall (x: C).(\forall (u: T).((clear (CHead e (Flat f) u) x) \to (clear e x)))))
 \def
- \lambda (f: F).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H: (clear (CHead e (Flat f) u) x)).(let H0 \def (match H return (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e (Flat f) u)) \to ((eq C c0 x) \to (clear e x))))) with [(clear_bind b e0 u0) \Rightarrow (\lambda (H0: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat f) u))).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) x)).((let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (e1: C).(match e1 return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e (Flat f) u) H0) in (False_ind ((eq C (CHead e0 (Bind b) u0) x) \to (clear e x)) H2)) H1))) | (clear_flat e0 c H0 f0 u0) \Rightarrow (\lambda (H1: (eq C (CHead e0 (Flat f0) u0) (CHead e (Flat f) u))).(\lambda (H2: (eq C c x)).((let H3 \def (f_equal C T (\lambda (e1: C).(match e1 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u) H1) in ((let H4 \def (f_equal C F (\lambda (e1: C).(match e1 return (\lambda (_: ?).F) with [(CSort _) \Rightarrow f0 | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).F) with [(Bind _) \Rightarrow f0 | (Flat f) \Rightarrow f])])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u) H1) in ((let H5 \def (f_equal C C (\lambda (e1: C).(match e1 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow e0 | (CHead c _ _) \Rightarrow c])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u) H1) in (eq_ind C e (\lambda (c0: C).((eq F f0 f) \to ((eq T u0 u) \to ((eq C c x) \to ((clear c0 c) \to (clear e x)))))) (\lambda (H6: (eq F f0 f)).(eq_ind F f (\lambda (_: F).((eq T u0 u) \to ((eq C c x) \to ((clear e c) \to (clear e x))))) (\lambda (H7: (eq T u0 u)).(eq_ind T u (\lambda (_: T).((eq C c x) \to ((clear e c) \to (clear e x)))) (\lambda (H8: (eq C c x)).(eq_ind C x (\lambda (c0: C).((clear e c0) \to (clear e x))) (\lambda (H9: (clear e x)).H9) c (sym_eq C c x H8))) u0 (sym_eq T u0 u H7))) f0 (sym_eq F f0 f H6))) e0 (sym_eq C e0 e H5))) H4)) H3)) H2 H0)))]) in (H0 (refl_equal C (CHead e (Flat f) u)) (refl_equal C x))))))).
+ \lambda (f: F).(\lambda (e: C).(\lambda (x: C).(\lambda (u: T).(\lambda (H: (clear (CHead e (Flat f) u) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e (Flat f) u)) \to ((eq C c0 x) \to (clear e x)))))) with [(clear_bind b e0 u0) \Rightarrow (\lambda (H0: (eq C (CHead e0 (Bind b) u0) (CHead e (Flat f) u))).(\lambda (H1: (eq C (CHead e0 (Bind b) u0) x)).((let H2 \def (eq_ind C (CHead e0 (Bind b) u0) (\lambda (e1: C).(match e1 return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead e (Flat f) u) H0) in (False_ind ((eq C (CHead e0 (Bind b) u0) x) \to (clear e x)) H2)) H1))) | (clear_flat e0 c H0 f0 u0) \Rightarrow (\lambda (H1: (eq C (CHead e0 (Flat f0) u0) (CHead e (Flat f) u))).(\lambda (H2: (eq C c x)).((let H3 \def (f_equal C T (\lambda (e1: C).(match e1 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u) H1) in ((let H4 \def (f_equal C F (\lambda (e1: C).(match e1 return (\lambda (_: ?).F) with [(CSort _) \Rightarrow f0 | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).F) with [(Bind _) \Rightarrow f0 | (Flat f) \Rightarrow f])])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u) H1) in ((let H5 \def (f_equal C C (\lambda (e1: C).(match e1 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow e0 | (CHead c _ _) \Rightarrow c])) (CHead e0 (Flat f0) u0) (CHead e (Flat f) u) H1) in (eq_ind C e (\lambda (c0: C).((eq F f0 f) \to ((eq T u0 u) \to ((eq C c x) \to ((clear c0 c) \to (clear e x)))))) (\lambda (H6: (eq F f0 f)).(eq_ind F f (\lambda (_: F).((eq T u0 u) \to ((eq C c x) \to ((clear e c) \to (clear e x))))) (\lambda (H7: (eq T u0 u)).(eq_ind T u (\lambda (_: T).((eq C c x) \to ((clear e c) \to (clear e x)))) (\lambda (H8: (eq C c x)).(eq_ind C x (\lambda (c0: C).((clear e c0) \to (clear e x))) (\lambda (H9: (clear e x)).H9) c (sym_eq C c x H8))) u0 (sym_eq T u0 u H7))) f0 (sym_eq F f0 f H6))) e0 (sym_eq C e0 e H5))) H4)) H3)) H2 H0)))]) in (H0 (refl_equal C (CHead e (Flat f) u)) (refl_equal C x))))))).
 
 theorem clear_gen_flat_r:
  \forall (f: F).(\forall (x: C).(\forall (e: C).(\forall (u: T).((clear x (CHead e (Flat f) u)) \to (\forall (P: Prop).P)))))
@@ -786,37 +786,37 @@ theorem clear_gen_all:
 theorem drop_clear:
  \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((drop (S i) O c1 c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear c1 (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))))))
 \def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (i: nat).((drop (S i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H: (drop (S i) O (CSort n) c2)).(and3_ind (eq C c2 (CSort n)) (eq nat (S i) O) (eq nat O O) (ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CSort n) (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) (\lambda (_: (eq C c2 (CSort n))).(\lambda (H1: (eq nat (S i) O)).(\lambda (_: (eq nat O O)).(let H3 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind (ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CSort n) (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) H3))))) (drop_gen_sort n (S i) O c2 H)))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (i: nat).((drop (S i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop (S i) O (CHead c k t) c2)).((match k return (\lambda (k0: K).((drop (r k0 i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c k0 t) (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))))) with [(Bind b) \Rightarrow (\lambda (H1: (drop (r (Bind b) i) O c c2)).(ex2_3_intro B C T (\lambda (b0: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c (Bind b) t) (CHead e (Bind b0) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) b c t (clear_bind b c t) H1)) | (Flat f) \Rightarrow (\lambda (H1: (drop (r (Flat f) i) O c c2)).(let H2 \def (H c2 i H1) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) (ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c (Flat f) t) (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H3: (clear c (CHead x1 (Bind x0) x2))).(\lambda (H4: (drop i O x1 c2)).(ex2_3_intro B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c (Flat f) t) (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) x0 x1 x2 (clear_flat c (CHead x1 (Bind x0) x2) H3 f t) H4)))))) H2)))]) (drop_gen_drop k c c2 t i H0))))))))) c1).
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (i: nat).((drop (S i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H: (drop (S i) O (CSort n) c2)).(and3_ind (eq C c2 (CSort n)) (eq nat (S i) O) (eq nat O O) (ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CSort n) (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) (\lambda (_: (eq C c2 (CSort n))).(\lambda (H1: (eq nat (S i) O)).(\lambda (_: (eq nat O O)).(let H3 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind (ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CSort n) (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) H3))))) (drop_gen_sort n (S i) O c2 H)))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (i: nat).((drop (S i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop (S i) O (CHead c k t) c2)).((match k return (\lambda (_: ?).(\lambda (k0: K).((drop (r k0 i) O c c2) \to (ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c k0 t) (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))))))) with [(Bind b) \Rightarrow (\lambda (H1: (drop (r (Bind b) i) O c c2)).(ex2_3_intro B C T (\lambda (b0: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c (Bind b) t) (CHead e (Bind b0) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) b c t (clear_bind b c t) H1)) | (Flat f) \Rightarrow (\lambda (H1: (drop (r (Flat f) i) O c c2)).(let H2 \def (H c2 i H1) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear c (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) (ex2_3 B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c (Flat f) t) (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2))))) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H3: (clear c (CHead x1 (Bind x0) x2))).(\lambda (H4: (drop i O x1 c2)).(ex2_3_intro B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear (CHead c (Flat f) t) (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop i O e c2)))) x0 x1 x2 (clear_flat c (CHead x1 (Bind x0) x2) H3 f t) H4)))))) H2)))]) (drop_gen_drop k c c2 t i H0))))))))) c1).
 
 theorem drop_clear_O:
  \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (u: T).((clear c (CHead e1 (Bind b) u)) \to (\forall (e2: C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c e2))))))))
 \def
- \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1: C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2: C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0 e2)))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (u: T).(\lambda (H: (clear (CSort n) (CHead e1 (Bind b) u))).(\lambda (e2: C).(\lambda (i: nat).(\lambda (_: (drop i O e1 e2)).(clear_gen_sort (CHead e1 (Bind b) u) n H (drop (S i) O (CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1: C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2: C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0 e2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (u: T).(\lambda (H0: (clear (CHead c0 k t) (CHead e1 (Bind b) u))).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H1: (drop i O e1 e2)).((match k return (\lambda (k0: K).((clear (CHead c0 k0 t) (CHead e1 (Bind b) u)) \to (drop (S i) O (CHead c0 k0 t) e2))) with [(Bind b0) \Rightarrow (\lambda (H2: (clear (CHead c0 (Bind b0) t) (CHead e1 (Bind b) u))).(let H3 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) u) t H2)) in ((let H4 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) u) t H2)) in ((let H5 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) u) t H2)) in (\lambda (H6: (eq B b b0)).(\lambda (H7: (eq C e1 c0)).(let H8 \def (eq_ind C e1 (\lambda (c: C).(drop i O c e2)) H1 c0 H7) in (eq_ind B b (\lambda (b1: B).(drop (S i) O (CHead c0 (Bind b1) t) e2)) (drop_drop (Bind b) i c0 e2 H8 t) b0 H6))))) H4)) H3))) | (Flat f) \Rightarrow (\lambda (H2: (clear (CHead c0 (Flat f) t) (CHead e1 (Bind b) u))).(drop_drop (Flat f) i c0 e2 (H e1 u (clear_gen_flat f c0 (CHead e1 (Bind b) u) t H2) e2 i H1) t))]) H0))))))))))) c)).
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1: C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2: C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0 e2)))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (u: T).(\lambda (H: (clear (CSort n) (CHead e1 (Bind b) u))).(\lambda (e2: C).(\lambda (i: nat).(\lambda (_: (drop i O e1 e2)).(clear_gen_sort (CHead e1 (Bind b) u) n H (drop (S i) O (CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1: C).(\forall (u: T).((clear c0 (CHead e1 (Bind b) u)) \to (\forall (e2: C).(\forall (i: nat).((drop i O e1 e2) \to (drop (S i) O c0 e2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (u: T).(\lambda (H0: (clear (CHead c0 k t) (CHead e1 (Bind b) u))).(\lambda (e2: C).(\lambda (i: nat).(\lambda (H1: (drop i O e1 e2)).((match k return (\lambda (_: ?).(\lambda (k0: K).((clear (CHead c0 k0 t) (CHead e1 (Bind b) u)) \to (drop (S i) O (CHead c0 k0 t) e2)))) with [(Bind b0) \Rightarrow (\lambda (H2: (clear (CHead c0 (Bind b0) t) (CHead e1 (Bind b) u))).(let H3 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) u) t H2)) in ((let H4 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) u) t H2)) in ((let H5 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead e1 (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) u) t H2)) in (\lambda (H6: (eq B b b0)).(\lambda (H7: (eq C e1 c0)).(let H8 \def (eq_ind C e1 (\lambda (c: C).(drop i O c e2)) H1 c0 H7) in (eq_ind B b (\lambda (b1: B).(drop (S i) O (CHead c0 (Bind b1) t) e2)) (drop_drop (Bind b) i c0 e2 H8 t) b0 H6))))) H4)) H3))) | (Flat f) \Rightarrow (\lambda (H2: (clear (CHead c0 (Flat f) t) (CHead e1 (Bind b) u))).(drop_drop (Flat f) i c0 e2 (H e1 u (clear_gen_flat f c0 (CHead e1 (Bind b) u) t H2) e2 i H1) t))]) H0))))))))))) c)).
 
 theorem drop_clear_S:
  \forall (x2: C).(\forall (x1: C).(\forall (h: nat).(\forall (d: nat).((drop h (S d) x1 x2) \to (\forall (b: B).(\forall (c2: C).(\forall (u: T).((clear x2 (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))))))))))
 \def
- \lambda (x2: C).(C_ind (\lambda (c: C).(\forall (x1: C).(\forall (h: nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2: C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))))))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_: (drop h (S d) x1 (CSort n))).(\lambda (b: B).(\lambda (c2: C).(\lambda (u: T).(\lambda (H0: (clear (CSort n) (CHead c2 (Bind b) u))).(clear_gen_sort (CHead c2 (Bind b) u) n H0 (ex2 C (\lambda (c1: C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))))))))))))) (\lambda (c: C).(\lambda (H: ((\forall (x1: C).(\forall (h: nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2: C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h (S d) x1 (CHead c k t))).(\lambda (b: B).(\lambda (c2: C).(\lambda (u: T).(\lambda (H1: (clear (CHead c k t) (CHead c2 (Bind b) u))).(ex2_ind C (\lambda (e: C).(eq C x1 (CHead e k (lift h (r k d) t)))) (\lambda (e: C).(drop h (r k d) e c)) (ex2 C (\lambda (c1: C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))) (\lambda (x: C).(\lambda (H2: (eq C x1 (CHead x k (lift h (r k d) t)))).(\lambda (H3: (drop h (r k d) x c)).(eq_ind_r C (CHead x k (lift h (r k d) t)) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear c0 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))) ((match k return (\lambda (k0: K).((clear (CHead c k0 t) (CHead c2 (Bind b) u)) \to ((drop h (r k0 d) x c) \to (ex2 C (\lambda (c1: C).(clear (CHead x k0 (lift h (r k0 d) t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))))) with [(Bind b0) \Rightarrow (\lambda (H4: (clear (CHead c (Bind b0) t) (CHead c2 (Bind b) u))).(\lambda (H5: (drop h (r (Bind b0) d) x c)).(let H6 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 (Bind b) u) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in ((let H7 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind b) u) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in (\lambda (H9: (eq B b b0)).(\lambda (H10: (eq C c2 c)).(eq_ind_r T t (\lambda (t0: T).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind b) (lift h d t0)))) (\lambda (c1: C).(drop h d c1 c2)))) (eq_ind_r C c (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind b) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c0)))) (eq_ind_r B b0 (\lambda (b1: B).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind b1) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)))) (ex_intro2 C (\lambda (c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind b0) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)) x (clear_bind b0 x (lift h d t)) H5) b H9) c2 H10) u H8)))) H7)) H6)))) | (Flat f) \Rightarrow (\lambda (H4: (clear (CHead c (Flat f) t) (CHead c2 (Bind b) u))).(\lambda (H5: (drop h (r (Flat f) d) x c)).(let H6 \def (H x h d H5 b c2 u (clear_gen_flat f c (CHead c2 (Bind b) u) t H4)) in (ex2_ind C (\lambda (c1: C).(clear x (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)) (ex2 C (\lambda (c1: C).(clear (CHead x (Flat f) (lift h (r (Flat f) d) t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))) (\lambda (x0: C).(\lambda (H7: (clear x (CHead x0 (Bind b) (lift h d u)))).(\lambda (H8: (drop h d x0 c2)).(ex_intro2 C (\lambda (c1: C).(clear (CHead x (Flat f) (lift h (r (Flat f) d) t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)) x0 (clear_flat x (CHead x0 (Bind b) (lift h d u)) H7 f (lift h (r (Flat f) d) t)) H8)))) H6))))]) H1 H3) x1 H2)))) (drop_gen_skip_r c x1 t h d k H0)))))))))))))) x2).
+ \lambda (x2: C).(C_ind (\lambda (c: C).(\forall (x1: C).(\forall (h: nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2: C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))))))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_: (drop h (S d) x1 (CSort n))).(\lambda (b: B).(\lambda (c2: C).(\lambda (u: T).(\lambda (H0: (clear (CSort n) (CHead c2 (Bind b) u))).(clear_gen_sort (CHead c2 (Bind b) u) n H0 (ex2 C (\lambda (c1: C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))))))))))))) (\lambda (c: C).(\lambda (H: ((\forall (x1: C).(\forall (h: nat).(\forall (d: nat).((drop h (S d) x1 c) \to (\forall (b: B).(\forall (c2: C).(\forall (u: T).((clear c (CHead c2 (Bind b) u)) \to (ex2 C (\lambda (c1: C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h (S d) x1 (CHead c k t))).(\lambda (b: B).(\lambda (c2: C).(\lambda (u: T).(\lambda (H1: (clear (CHead c k t) (CHead c2 (Bind b) u))).(ex2_ind C (\lambda (e: C).(eq C x1 (CHead e k (lift h (r k d) t)))) (\lambda (e: C).(drop h (r k d) e c)) (ex2 C (\lambda (c1: C).(clear x1 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))) (\lambda (x: C).(\lambda (H2: (eq C x1 (CHead x k (lift h (r k d) t)))).(\lambda (H3: (drop h (r k d) x c)).(eq_ind_r C (CHead x k (lift h (r k d) t)) (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear c0 (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)))) ((match k return (\lambda (_: ?).(\lambda (k0: K).((clear (CHead c k0 t) (CHead c2 (Bind b) u)) \to ((drop h (r k0 d) x c) \to (ex2 C (\lambda (c1: C).(clear (CHead x k0 (lift h (r k0 d) t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))))))) with [(Bind b0) \Rightarrow (\lambda (H4: (clear (CHead c (Bind b0) t) (CHead c2 (Bind b) u))).(\lambda (H5: (drop h (r (Bind b0) d) x c)).(let H6 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 (Bind b) u) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in ((let H7 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in ((let H8 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind b) u) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u) t H4)) in (\lambda (H9: (eq B b b0)).(\lambda (H10: (eq C c2 c)).(eq_ind_r T t (\lambda (t0: T).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind b) (lift h d t0)))) (\lambda (c1: C).(drop h d c1 c2)))) (eq_ind_r C c (\lambda (c0: C).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind b) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c0)))) (eq_ind_r B b0 (\lambda (b1: B).(ex2 C (\lambda (c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind b1) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)))) (ex_intro2 C (\lambda (c1: C).(clear (CHead x (Bind b0) (lift h (r (Bind b0) d) t)) (CHead c1 (Bind b0) (lift h d t)))) (\lambda (c1: C).(drop h d c1 c)) x (clear_bind b0 x (lift h d t)) H5) b H9) c2 H10) u H8)))) H7)) H6)))) | (Flat f) \Rightarrow (\lambda (H4: (clear (CHead c (Flat f) t) (CHead c2 (Bind b) u))).(\lambda (H5: (drop h (r (Flat f) d) x c)).(let H6 \def (H x h d H5 b c2 u (clear_gen_flat f c (CHead c2 (Bind b) u) t H4)) in (ex2_ind C (\lambda (c1: C).(clear x (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)) (ex2 C (\lambda (c1: C).(clear (CHead x (Flat f) (lift h (r (Flat f) d) t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2))) (\lambda (x0: C).(\lambda (H7: (clear x (CHead x0 (Bind b) (lift h d u)))).(\lambda (H8: (drop h d x0 c2)).(ex_intro2 C (\lambda (c1: C).(clear (CHead x (Flat f) (lift h (r (Flat f) d) t)) (CHead c1 (Bind b) (lift h d u)))) (\lambda (c1: C).(drop h d c1 c2)) x0 (clear_flat x (CHead x0 (Bind b) (lift h d u)) H7 f (lift h (r (Flat f) d) t)) H8)))) H6))))]) H1 H3) x1 H2)))) (drop_gen_skip_r c x1 t h d k H0)))))))))))))) x2).
 
 theorem clear_clear:
  \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (clear c2 c2)))
 \def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to (clear c2 c2)))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (H: (clear (CSort n) c2)).(clear_gen_sort c2 n H (clear c2 c2))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (clear c2 c2))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (H0: (clear (CHead c k t) c2)).((match k return (\lambda (k0: K).((clear (CHead c k0 t) c2) \to (clear c2 c2))) with [(Bind b) \Rightarrow (\lambda (H1: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: C).(clear c0 c0)) (clear_bind b c t) c2 (clear_gen_bind b c c2 t H1))) | (Flat f) \Rightarrow (\lambda (H1: (clear (CHead c (Flat f) t) c2)).(H c2 (clear_gen_flat f c c2 t H1)))]) H0))))))) c1).
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to (clear c2 c2)))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (H: (clear (CSort n) c2)).(clear_gen_sort c2 n H (clear c2 c2))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (clear c2 c2))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (H0: (clear (CHead c k t) c2)).((match k return (\lambda (_: ?).(\lambda (k0: K).((clear (CHead c k0 t) c2) \to (clear c2 c2)))) with [(Bind b) \Rightarrow (\lambda (H1: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: C).(clear c0 c0)) (clear_bind b c t) c2 (clear_gen_bind b c c2 t H1))) | (Flat f) \Rightarrow (\lambda (H1: (clear (CHead c (Flat f) t) c2)).(H c2 (clear_gen_flat f c c2 t H1)))]) H0))))))) c1).
 
 theorem clear_mono:
  \forall (c: C).(\forall (c1: C).((clear c c1) \to (\forall (c2: C).((clear c c2) \to (eq C c1 c2)))))
 \def
- \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1: C).((clear c0 c1) \to (\forall (c2: C).((clear c0 c2) \to (eq C c1 c2)))))) (\lambda (n: nat).(\lambda (c1: C).(\lambda (_: (clear (CSort n) c1)).(\lambda (c2: C).(\lambda (H0: (clear (CSort n) c2)).(clear_gen_sort c2 n H0 (eq C c1 c2))))))) (\lambda (c0: C).(\lambda (H: ((\forall (c1: C).((clear c0 c1) \to (\forall (c2: C).((clear c0 c2) \to (eq C c1 c2))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c1: C).(\lambda (H0: (clear (CHead c0 k t) c1)).(\lambda (c2: C).(\lambda (H1: (clear (CHead c0 k t) c2)).((match k return (\lambda (k0: K).((clear (CHead c0 k0 t) c1) \to ((clear (CHead c0 k0 t) c2) \to (eq C c1 c2)))) with [(Bind b) \Rightarrow (\lambda (H2: (clear (CHead c0 (Bind b) t) c1)).(\lambda (H3: (clear (CHead c0 (Bind b) t) c2)).(eq_ind_r C (CHead c0 (Bind b) t) (\lambda (c3: C).(eq C c1 c3)) (eq_ind_r C (CHead c0 (Bind b) t) (\lambda (c3: C).(eq C c3 (CHead c0 (Bind b) t))) (refl_equal C (CHead c0 (Bind b) t)) c1 (clear_gen_bind b c0 c1 t H2)) c2 (clear_gen_bind b c0 c2 t H3)))) | (Flat f) \Rightarrow (\lambda (H2: (clear (CHead c0 (Flat f) t) c1)).(\lambda (H3: (clear (CHead c0 (Flat f) t) c2)).(H c1 (clear_gen_flat f c0 c1 t H2) c2 (clear_gen_flat f c0 c2 t H3))))]) H0 H1))))))))) c).
+ \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1: C).((clear c0 c1) \to (\forall (c2: C).((clear c0 c2) \to (eq C c1 c2)))))) (\lambda (n: nat).(\lambda (c1: C).(\lambda (_: (clear (CSort n) c1)).(\lambda (c2: C).(\lambda (H0: (clear (CSort n) c2)).(clear_gen_sort c2 n H0 (eq C c1 c2))))))) (\lambda (c0: C).(\lambda (H: ((\forall (c1: C).((clear c0 c1) \to (\forall (c2: C).((clear c0 c2) \to (eq C c1 c2))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c1: C).(\lambda (H0: (clear (CHead c0 k t) c1)).(\lambda (c2: C).(\lambda (H1: (clear (CHead c0 k t) c2)).((match k return (\lambda (_: ?).(\lambda (k0: K).((clear (CHead c0 k0 t) c1) \to ((clear (CHead c0 k0 t) c2) \to (eq C c1 c2))))) with [(Bind b) \Rightarrow (\lambda (H2: (clear (CHead c0 (Bind b) t) c1)).(\lambda (H3: (clear (CHead c0 (Bind b) t) c2)).(eq_ind_r C (CHead c0 (Bind b) t) (\lambda (c3: C).(eq C c1 c3)) (eq_ind_r C (CHead c0 (Bind b) t) (\lambda (c3: C).(eq C c3 (CHead c0 (Bind b) t))) (refl_equal C (CHead c0 (Bind b) t)) c1 (clear_gen_bind b c0 c1 t H2)) c2 (clear_gen_bind b c0 c2 t H3)))) | (Flat f) \Rightarrow (\lambda (H2: (clear (CHead c0 (Flat f) t) c1)).(\lambda (H3: (clear (CHead c0 (Flat f) t) c2)).(H c1 (clear_gen_flat f c0 c1 t H2) c2 (clear_gen_flat f c0 c2 t H3))))]) H0 H1))))))))) c).
 
 theorem clear_trans:
  \forall (c1: C).(\forall (c: C).((clear c1 c) \to (\forall (c2: C).((clear c c2) \to (clear c1 c2)))))
 \def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c0: C).((clear c c0) \to (\forall (c2: C).((clear c0 c2) \to (clear c c2)))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (H: (clear (CSort n) c)).(\lambda (c2: C).(\lambda (_: (clear c c2)).(clear_gen_sort c n H (clear (CSort n) c2))))))) (\lambda (c: C).(\lambda (H: ((\forall (c0: C).((clear c c0) \to (\forall (c2: C).((clear c0 c2) \to (clear c c2))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c0: C).(\lambda (H0: (clear (CHead c k t) c0)).(\lambda (c2: C).(\lambda (H1: (clear c0 c2)).((match k return (\lambda (k0: K).((clear (CHead c k0 t) c0) \to (clear (CHead c k0 t) c2))) with [(Bind b) \Rightarrow (\lambda (H2: (clear (CHead c (Bind b) t) c0)).(let H3 \def (eq_ind C c0 (\lambda (c: C).(clear c c2)) H1 (CHead c (Bind b) t) (clear_gen_bind b c c0 t H2)) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(clear (CHead c (Bind b) t) c3)) (clear_bind b c t) c2 (clear_gen_bind b c c2 t H3)))) | (Flat f) \Rightarrow (\lambda (H2: (clear (CHead c (Flat f) t) c0)).(clear_flat c c2 (H c0 (clear_gen_flat f c c0 t H2) c2 H1) f t))]) H0))))))))) c1).
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c0: C).((clear c c0) \to (\forall (c2: C).((clear c0 c2) \to (clear c c2)))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (H: (clear (CSort n) c)).(\lambda (c2: C).(\lambda (_: (clear c c2)).(clear_gen_sort c n H (clear (CSort n) c2))))))) (\lambda (c: C).(\lambda (H: ((\forall (c0: C).((clear c c0) \to (\forall (c2: C).((clear c0 c2) \to (clear c c2))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c0: C).(\lambda (H0: (clear (CHead c k t) c0)).(\lambda (c2: C).(\lambda (H1: (clear c0 c2)).((match k return (\lambda (_: ?).(\lambda (k0: K).((clear (CHead c k0 t) c0) \to (clear (CHead c k0 t) c2)))) with [(Bind b) \Rightarrow (\lambda (H2: (clear (CHead c (Bind b) t) c0)).(let H3 \def (eq_ind C c0 (\lambda (c: C).(clear c c2)) H1 (CHead c (Bind b) t) (clear_gen_bind b c c0 t H2)) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(clear (CHead c (Bind b) t) c3)) (clear_bind b c t) c2 (clear_gen_bind b c c2 t H3)))) | (Flat f) \Rightarrow (\lambda (H2: (clear (CHead c (Flat f) t) c0)).(clear_flat c c2 (H c0 (clear_gen_flat f c c0 t H2) c2 H1) f t))]) H0))))))))) c1).
 
 theorem clear_ctail:
  \forall (b: B).(\forall (c1: C).(\forall (c2: C).(\forall (u2: T).((clear c1 (CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: T).(clear (CTail k u1 c1) (CHead (CTail k u1 c2) (Bind b) u2))))))))
 \def
- \lambda (b: B).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (u2: T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b) u2)))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H: (clear (CSort n) (CHead c2 (Bind b) u2))).(\lambda (k: K).(\lambda (u1: T).(match k return (\lambda (k0: K).(clear (CHead (CSort n) k0 u1) (CHead (CTail k0 u1 c2) (Bind b) u2))) with [(Bind b0) \Rightarrow (clear_gen_sort (CHead c2 (Bind b) u2) n H (clear (CHead (CSort n) (Bind b0) u1) (CHead (CTail (Bind b0) u1 c2) (Bind b) u2))) | (Flat f) \Rightarrow (clear_gen_sort (CHead c2 (Bind b) u2) n H (clear (CHead (CSort n) (Flat f) u1) (CHead (CTail (Flat f) u1 c2) (Bind b) u2)))]))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (u2: T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b) u2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H0: (clear (CHead c k t) (CHead c2 (Bind b) u2))).(\lambda (k0: K).(\lambda (u1: T).((match k return (\lambda (k1: K).((clear (CHead c k1 t) (CHead c2 (Bind b) u2)) \to (clear (CHead (CTail k0 u1 c) k1 t) (CHead (CTail k0 u1 c2) (Bind b) u2)))) with [(Bind b0) \Rightarrow (\lambda (H1: (clear (CHead c (Bind b0) t) (CHead c2 (Bind b) u2))).(let H2 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in ((let H3 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in ((let H4 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C c2 c)).(eq_ind_r T t (\lambda (t0: T).(clear (CHead (CTail k0 u1 c) (Bind b0) t) (CHead (CTail k0 u1 c2) (Bind b) t0))) (eq_ind_r C c (\lambda (c0: C).(clear (CHead (CTail k0 u1 c) (Bind b0) t) (CHead (CTail k0 u1 c0) (Bind b) t))) (eq_ind B b (\lambda (b1: B).(clear (CHead (CTail k0 u1 c) (Bind b1) t) (CHead (CTail k0 u1 c) (Bind b) t))) (clear_bind b (CTail k0 u1 c) t) b0 H5) c2 H6) u2 H4)))) H3)) H2))) | (Flat f) \Rightarrow (\lambda (H1: (clear (CHead c (Flat f) t) (CHead c2 (Bind b) u2))).(clear_flat (CTail k0 u1 c) (CHead (CTail k0 u1 c2) (Bind b) u2) (H c2 u2 (clear_gen_flat f c (CHead c2 (Bind b) u2) t H1) k0 u1) f t))]) H0)))))))))) c1)).
+ \lambda (b: B).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (u2: T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b) u2)))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H: (clear (CSort n) (CHead c2 (Bind b) u2))).(\lambda (k: K).(\lambda (u1: T).(match k return (\lambda (_: ?).(\lambda (k0: K).(clear (CHead (CSort n) k0 u1) (CHead (CTail k0 u1 c2) (Bind b) u2)))) with [(Bind b0) \Rightarrow (clear_gen_sort (CHead c2 (Bind b) u2) n H (clear (CHead (CSort n) (Bind b0) u1) (CHead (CTail (Bind b0) u1 c2) (Bind b) u2))) | (Flat f) \Rightarrow (clear_gen_sort (CHead c2 (Bind b) u2) n H (clear (CHead (CSort n) (Flat f) u1) (CHead (CTail (Flat f) u1 c2) (Bind b) u2)))]))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (u2: T).((clear c (CHead c2 (Bind b) u2)) \to (\forall (k: K).(\forall (u1: T).(clear (CTail k u1 c) (CHead (CTail k u1 c2) (Bind b) u2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (u2: T).(\lambda (H0: (clear (CHead c k t) (CHead c2 (Bind b) u2))).(\lambda (k0: K).(\lambda (u1: T).((match k return (\lambda (_: ?).(\lambda (k1: K).((clear (CHead c k1 t) (CHead c2 (Bind b) u2)) \to (clear (CHead (CTail k0 u1 c) k1 t) (CHead (CTail k0 u1 c2) (Bind b) u2))))) with [(Bind b0) \Rightarrow (\lambda (H1: (clear (CHead c (Bind b0) t) (CHead c2 (Bind b) u2))).(let H2 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in ((let H3 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in ((let H4 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind b) u2) (CHead c (Bind b0) t) (clear_gen_bind b0 c (CHead c2 (Bind b) u2) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C c2 c)).(eq_ind_r T t (\lambda (t0: T).(clear (CHead (CTail k0 u1 c) (Bind b0) t) (CHead (CTail k0 u1 c2) (Bind b) t0))) (eq_ind_r C c (\lambda (c0: C).(clear (CHead (CTail k0 u1 c) (Bind b0) t) (CHead (CTail k0 u1 c0) (Bind b) t))) (eq_ind B b (\lambda (b1: B).(clear (CHead (CTail k0 u1 c) (Bind b1) t) (CHead (CTail k0 u1 c) (Bind b) t))) (clear_bind b (CTail k0 u1 c) t) b0 H5) c2 H6) u2 H4)))) H3)) H2))) | (Flat f) \Rightarrow (\lambda (H1: (clear (CHead c (Flat f) t) (CHead c2 (Bind b) u2))).(clear_flat (CTail k0 u1 c) (CHead (CTail k0 u1 c2) (Bind b) u2) (H c2 u2 (clear_gen_flat f c (CHead c2 (Bind b) u2) t H1) k0 u1) f t))]) H0)))))))))) c1)).
 
 theorem getl_gen_all:
  \forall (c1: C).(\forall (c2: C).(\forall (i: nat).((getl i c1 c2) \to (ex2 C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e c2))))))
@@ -846,7 +846,7 @@ theorem getl_refl:
 theorem clear_getl_trans:
  \forall (i: nat).(\forall (c2: C).(\forall (c3: C).((getl i c2 c3) \to (\forall (c1: C).((clear c1 c2) \to (getl i c1 c3))))))
 \def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c2: C).(\forall (c3: C).((getl n c2 c3) \to (\forall (c1: C).((clear c1 c2) \to (getl n c1 c3))))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (H: (getl O c2 c3)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(getl_intro O c1 c3 c1 (drop_refl c1) (clear_trans c1 c2 H0 c3 (getl_gen_O c2 c3 H)))))))) (\lambda (n: nat).(\lambda (_: ((\forall (c2: C).(\forall (c3: C).((getl n c2 c3) \to (\forall (c1: C).((clear c1 c2) \to (getl n c1 c3)))))))).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (c3: C).((getl (S n) c c3) \to (\forall (c1: C).((clear c1 c) \to (getl (S n) c1 c3)))))) (\lambda (n0: nat).(\lambda (c3: C).(\lambda (H0: (getl (S n) (CSort n0) c3)).(\lambda (c1: C).(\lambda (_: (clear c1 (CSort n0))).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c1 c3))))))) (\lambda (c: C).(\lambda (_: ((\forall (c3: C).((getl (S n) c c3) \to (\forall (c1: C).((clear c1 c) \to (getl (S n) c1 c3))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t) c3)).(\lambda (c1: C).(\lambda (H2: (clear c1 (CHead c k t))).((match k return (\lambda (k0: K).((getl (S n) (CHead c k0 t) c3) \to ((clear c1 (CHead c k0 t)) \to (getl (S n) c1 c3)))) with [(Bind b) \Rightarrow (\lambda (H3: (getl (S n) (CHead c (Bind b) t) c3)).(\lambda (H4: (clear c1 (CHead c (Bind b) t))).(let H5 \def (getl_gen_all c c3 (r (Bind b) n) (getl_gen_S (Bind b) c c3 t n H3)) in (ex2_ind C (\lambda (e: C).(drop n O c e)) (\lambda (e: C).(clear e c3)) (getl (S n) c1 c3) (\lambda (x: C).(\lambda (H6: (drop n O c x)).(\lambda (H7: (clear x c3)).(getl_intro (S n) c1 c3 x (drop_clear_O b c1 c t H4 x n H6) H7)))) H5)))) | (Flat f) \Rightarrow (\lambda (_: (getl (S n) (CHead c (Flat f) t) c3)).(\lambda (H4: (clear c1 (CHead c (Flat f) t))).(clear_gen_flat_r f c1 c t H4 (getl (S n) c1 c3))))]) H1 H2))))))))) c2)))) i).
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c2: C).(\forall (c3: C).((getl n c2 c3) \to (\forall (c1: C).((clear c1 c2) \to (getl n c1 c3))))))) (\lambda (c2: C).(\lambda (c3: C).(\lambda (H: (getl O c2 c3)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(getl_intro O c1 c3 c1 (drop_refl c1) (clear_trans c1 c2 H0 c3 (getl_gen_O c2 c3 H)))))))) (\lambda (n: nat).(\lambda (_: ((\forall (c2: C).(\forall (c3: C).((getl n c2 c3) \to (\forall (c1: C).((clear c1 c2) \to (getl n c1 c3)))))))).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (c3: C).((getl (S n) c c3) \to (\forall (c1: C).((clear c1 c) \to (getl (S n) c1 c3)))))) (\lambda (n0: nat).(\lambda (c3: C).(\lambda (H0: (getl (S n) (CSort n0) c3)).(\lambda (c1: C).(\lambda (_: (clear c1 (CSort n0))).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c1 c3))))))) (\lambda (c: C).(\lambda (_: ((\forall (c3: C).((getl (S n) c c3) \to (\forall (c1: C).((clear c1 c) \to (getl (S n) c1 c3))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t) c3)).(\lambda (c1: C).(\lambda (H2: (clear c1 (CHead c k t))).((match k return (\lambda (_: ?).(\lambda (k0: K).((getl (S n) (CHead c k0 t) c3) \to ((clear c1 (CHead c k0 t)) \to (getl (S n) c1 c3))))) with [(Bind b) \Rightarrow (\lambda (H3: (getl (S n) (CHead c (Bind b) t) c3)).(\lambda (H4: (clear c1 (CHead c (Bind b) t))).(let H5 \def (getl_gen_all c c3 (r (Bind b) n) (getl_gen_S (Bind b) c c3 t n H3)) in (ex2_ind C (\lambda (e: C).(drop n O c e)) (\lambda (e: C).(clear e c3)) (getl (S n) c1 c3) (\lambda (x: C).(\lambda (H6: (drop n O c x)).(\lambda (H7: (clear x c3)).(getl_intro (S n) c1 c3 x (drop_clear_O b c1 c t H4 x n H6) H7)))) H5)))) | (Flat f) \Rightarrow (\lambda (_: (getl (S n) (CHead c (Flat f) t) c3)).(\lambda (H4: (clear c1 (CHead c (Flat f) t))).(clear_gen_flat_r f c1 c t H4 (getl (S n) c1 c3))))]) H1 H2))))))))) c2)))) i).
 
 theorem getl_clear_trans:
  \forall (i: nat).(\forall (c1: C).(\forall (c2: C).((getl i c1 c2) \to (\forall (c3: C).((clear c2 c3) \to (getl i c1 c3))))))
@@ -861,7 +861,7 @@ theorem getl_head:
 theorem getl_flat:
  \forall (c: C).(\forall (e: C).(\forall (h: nat).((getl h c e) \to (\forall (f: F).(\forall (u: T).(getl h (CHead c (Flat f) u) e))))))
 \def
- \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (H: (getl h c e)).(\lambda (f: F).(\lambda (u: T).(let H0 \def (getl_gen_all c e h H) in (ex2_ind C (\lambda (e0: C).(drop h O c e0)) (\lambda (e0: C).(clear e0 e)) (getl h (CHead c (Flat f) u) e) (\lambda (x: C).(\lambda (H1: (drop h O c x)).(\lambda (H2: (clear x e)).((match h return (\lambda (n: nat).((drop n O c x) \to (getl n (CHead c (Flat f) u) e))) with [O \Rightarrow (\lambda (H3: (drop O O c x)).(let H4 \def (eq_ind_r C x (\lambda (c: C).(clear c e)) H2 c (drop_gen_refl c x H3)) in (getl_intro O (CHead c (Flat f) u) e (CHead c (Flat f) u) (drop_refl (CHead c (Flat f) u)) (clear_flat c e H4 f u)))) | (S n) \Rightarrow (\lambda (H3: (drop (S n) O c x)).(getl_intro (S n) (CHead c (Flat f) u) e x (drop_drop (Flat f) n c x H3 u) H2))]) H1)))) H0))))))).
+ \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (H: (getl h c e)).(\lambda (f: F).(\lambda (u: T).(let H0 \def (getl_gen_all c e h H) in (ex2_ind C (\lambda (e0: C).(drop h O c e0)) (\lambda (e0: C).(clear e0 e)) (getl h (CHead c (Flat f) u) e) (\lambda (x: C).(\lambda (H1: (drop h O c x)).(\lambda (H2: (clear x e)).((match h return (\lambda (_: ?).(\lambda (n: nat).((drop n O c x) \to (getl n (CHead c (Flat f) u) e)))) with [O \Rightarrow (\lambda (H3: (drop O O c x)).(let H4 \def (eq_ind_r C x (\lambda (c: C).(clear c e)) H2 c (drop_gen_refl c x H3)) in (getl_intro O (CHead c (Flat f) u) e (CHead c (Flat f) u) (drop_refl (CHead c (Flat f) u)) (clear_flat c e H4 f u)))) | (S n) \Rightarrow (\lambda (H3: (drop (S n) O c x)).(getl_intro (S n) (CHead c (Flat f) u) e x (drop_drop (Flat f) n c x H3 u) H2))]) H1)))) H0))))))).
 
 theorem getl_drop:
  \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h: nat).((getl h c (CHead e (Bind b) u)) \to (drop (S h) O c e))))))
@@ -871,7 +871,7 @@ theorem getl_drop:
 theorem getl_clear_bind:
  \forall (b: B).(\forall (c: C).(\forall (e1: C).(\forall (v: T).((clear c (CHead e1 (Bind b) v)) \to (\forall (e2: C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c e2))))))))
 \def
- \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1: C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2: C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2)))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (v: T).(\lambda (H: (clear (CSort n) (CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n0: nat).(\lambda (_: (getl n0 e1 e2)).(clear_gen_sort (CHead e1 (Bind b) v) n H (getl (S n0) (CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1: C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2: C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (v: T).(\lambda (H0: (clear (CHead c0 k t) (CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n: nat).(\lambda (H1: (getl n e1 e2)).((match k return (\lambda (k0: K).((clear (CHead c0 k0 t) (CHead e1 (Bind b) v)) \to (getl (S n) (CHead c0 k0 t) e2))) with [(Bind b0) \Rightarrow (\lambda (H2: (clear (CHead c0 (Bind b0) t) (CHead e1 (Bind b) v))).(let H3 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 (Bind b) v) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) v) t H2)) in ((let H4 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (CHead e1 (Bind b) v) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) v) t H2)) in ((let H5 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead e1 (Bind b) v) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) v) t H2)) in (\lambda (H6: (eq B b b0)).(\lambda (H7: (eq C e1 c0)).(let H8 \def (eq_ind C e1 (\lambda (c: C).(getl n c e2)) H1 c0 H7) in (eq_ind B b (\lambda (b1: B).(getl (S n) (CHead c0 (Bind b1) t) e2)) (getl_head (Bind b) n c0 e2 H8 t) b0 H6))))) H4)) H3))) | (Flat f) \Rightarrow (\lambda (H2: (clear (CHead c0 (Flat f) t) (CHead e1 (Bind b) v))).(getl_flat c0 e2 (S n) (H e1 v (clear_gen_flat f c0 (CHead e1 (Bind b) v) t H2) e2 n H1) f t))]) H0))))))))))) c)).
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e1: C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2: C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2)))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (v: T).(\lambda (H: (clear (CSort n) (CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n0: nat).(\lambda (_: (getl n0 e1 e2)).(clear_gen_sort (CHead e1 (Bind b) v) n H (getl (S n0) (CSort n) e2))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e1: C).(\forall (v: T).((clear c0 (CHead e1 (Bind b) v)) \to (\forall (e2: C).(\forall (n: nat).((getl n e1 e2) \to (getl (S n) c0 e2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e1: C).(\lambda (v: T).(\lambda (H0: (clear (CHead c0 k t) (CHead e1 (Bind b) v))).(\lambda (e2: C).(\lambda (n: nat).(\lambda (H1: (getl n e1 e2)).((match k return (\lambda (_: ?).(\lambda (k0: K).((clear (CHead c0 k0 t) (CHead e1 (Bind b) v)) \to (getl (S n) (CHead c0 k0 t) e2)))) with [(Bind b0) \Rightarrow (\lambda (H2: (clear (CHead c0 (Bind b0) t) (CHead e1 (Bind b) v))).(let H3 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 (Bind b) v) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) v) t H2)) in ((let H4 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (CHead e1 (Bind b) v) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) v) t H2)) in ((let H5 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead e1 (Bind b) v) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e1 (Bind b) v) t H2)) in (\lambda (H6: (eq B b b0)).(\lambda (H7: (eq C e1 c0)).(let H8 \def (eq_ind C e1 (\lambda (c: C).(getl n c e2)) H1 c0 H7) in (eq_ind B b (\lambda (b1: B).(getl (S n) (CHead c0 (Bind b1) t) e2)) (getl_head (Bind b) n c0 e2 H8 t) b0 H6))))) H4)) H3))) | (Flat f) \Rightarrow (\lambda (H2: (clear (CHead c0 (Flat f) t) (CHead e1 (Bind b) v))).(getl_flat c0 e2 (S n) (H e1 v (clear_gen_flat f c0 (CHead e1 (Bind b) v) t H2) e2 n H1) f t))]) H0))))))))))) c)).
 
 theorem getl_ctail:
  \forall (b: B).(\forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d (Bind b) u)) \to (\forall (k: K).(\forall (v: T).(getl i (CTail k v c) (CHead (CTail k v d) (Bind b) u)))))))))
@@ -881,22 +881,22 @@ theorem getl_ctail:
 theorem getl_ctail_clen:
  \forall (b: B).(\forall (t: T).(\forall (c: C).(ex nat (\lambda (n: nat).(getl (clen c) (CTail (Bind b) t c) (CHead (CSort n) (Bind b) t))))))
 \def
- \lambda (b: B).(\lambda (t: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(ex nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))))) (\lambda (n: nat).(ex_intro nat (\lambda (n0: nat).(getl O (CHead (CSort n) (Bind b) t) (CHead (CSort n0) (Bind b) t))) n (getl_refl b (CSort n) t))) (\lambda (c0: C).(\lambda (H: (ex nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))))).(\lambda (k: K).(\lambda (t0: T).(let H0 \def H in (ex_ind nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))) (ex nat (\lambda (n: nat).(getl (s k (clen c0)) (CHead (CTail (Bind b) t c0) k t0) (CHead (CSort n) (Bind b) t)))) (\lambda (x: nat).(\lambda (H1: (getl (clen c0) (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t))).(match k return (\lambda (k0: K).(ex nat (\lambda (n: nat).(getl (s k0 (clen c0)) (CHead (CTail (Bind b) t c0) k0 t0) (CHead (CSort n) (Bind b) t))))) with [(Bind b0) \Rightarrow (ex_intro nat (\lambda (n: nat).(getl (S (clen c0)) (CHead (CTail (Bind b) t c0) (Bind b0) t0) (CHead (CSort n) (Bind b) t))) x (getl_head (Bind b0) (clen c0) (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t) H1 t0)) | (Flat f) \Rightarrow (ex_intro nat (\lambda (n: nat).(getl (clen c0) (CHead (CTail (Bind b) t c0) (Flat f) t0) (CHead (CSort n) (Bind b) t))) x (getl_flat (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t) (clen c0) H1 f t0))]))) H0)))))) c))).
+ \lambda (b: B).(\lambda (t: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(ex nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))))) (\lambda (n: nat).(ex_intro nat (\lambda (n0: nat).(getl O (CHead (CSort n) (Bind b) t) (CHead (CSort n0) (Bind b) t))) n (getl_refl b (CSort n) t))) (\lambda (c0: C).(\lambda (H: (ex nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))))).(\lambda (k: K).(\lambda (t0: T).(let H0 \def H in (ex_ind nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind b) t c0) (CHead (CSort n) (Bind b) t))) (ex nat (\lambda (n: nat).(getl (s k (clen c0)) (CHead (CTail (Bind b) t c0) k t0) (CHead (CSort n) (Bind b) t)))) (\lambda (x: nat).(\lambda (H1: (getl (clen c0) (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t))).(match k return (\lambda (_: ?).(\lambda (k0: K).(ex nat (\lambda (n: nat).(getl (s k0 (clen c0)) (CHead (CTail (Bind b) t c0) k0 t0) (CHead (CSort n) (Bind b) t)))))) with [(Bind b0) \Rightarrow (ex_intro nat (\lambda (n: nat).(getl (S (clen c0)) (CHead (CTail (Bind b) t c0) (Bind b0) t0) (CHead (CSort n) (Bind b) t))) x (getl_head (Bind b0) (clen c0) (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t) H1 t0)) | (Flat f) \Rightarrow (ex_intro nat (\lambda (n: nat).(getl (clen c0) (CHead (CTail (Bind b) t c0) (Flat f) t0) (CHead (CSort n) (Bind b) t))) x (getl_flat (CTail (Bind b) t c0) (CHead (CSort x) (Bind b) t) (clen c0) H1 f t0))]))) H0)))))) c))).
 
 theorem getl_dec:
  \forall (c: C).(\forall (i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl i c (CHead e (Bind b) v)))))) (\forall (d: C).((getl i c d) \to (\forall (P: Prop).P)))))
 \def
- \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl i c0 (CHead e (Bind b) v)))))) (\forall (d: C).((getl i c0 d) \to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda (i: nat).(or_intror (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl i (CSort n) (CHead e (Bind b) v)))))) (\forall (d: C).((getl i (CSort n) d) \to (\forall (P: Prop).P))) (\lambda (d: C).(\lambda (H: (getl i (CSort n) d)).(\lambda (P: Prop).(getl_gen_sort n i d H P))))))) (\lambda (c0: C).(\lambda (H: ((\forall (i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl i c0 (CHead e (Bind b) v)))))) (\forall (d: C).((getl i c0 d) \to (\forall (P: Prop).P))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (i: nat).(match i return (\lambda (n: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl n (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl n (CHead c0 k t) d) \to (\forall (P: Prop).P))))) with [O \Rightarrow (match k return (\lambda (k0: K).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 k0 t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O (CHead c0 k0 t) d) \to (\forall (P: Prop).P))))) with [(Bind b) \Rightarrow (or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead c0 (Bind b) t) (CHead e (Bind b0) v)))))) (\forall (d: C).((getl O (CHead c0 (Bind b) t) d) \to (\forall (P: Prop).P))) (ex_3_intro C B T (\lambda (e: C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead c0 (Bind b) t) (CHead e (Bind b0) v))))) c0 b t (getl_refl b c0 t))) | (Flat f) \Rightarrow (let H_x \def (H O) in (let H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v)))))) (\forall (d: C).((getl O c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P)))) (\lambda (H1: (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v))))))).(ex_3_ind C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P)))) (\lambda (x0: C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl O c0 (CHead x0 (Bind x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P))) (ex_3_intro C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_flat c0 (CHead x0 (Bind x1) x2) O H2 f t))))))) H1)) (\lambda (H1: ((\forall (d: C).((getl O c0 d) \to (\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P))) (\lambda (d: C).(\lambda (H2: (getl O (CHead c0 (Flat f) t) d)).(\lambda (P: Prop).(H1 d (getl_intro O c0 d c0 (drop_refl c0) (clear_gen_flat f c0 d t (getl_gen_O (CHead c0 (Flat f) t) d H2))) P)))))) H0)))]) | (S n) \Rightarrow (let H_x \def (H (r k n)) in (let H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v)))))) (\forall (d: C).((getl (r k n) c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P)))) (\lambda (H1: (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v))))))).(ex_3_ind C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P)))) (\lambda (x0: C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl (r k n) c0 (CHead x0 (Bind x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P))) (ex_3_intro C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_head k n c0 (CHead x0 (Bind x1) x2) H2 t))))))) H1)) (\lambda (H1: ((\forall (d: C).((getl (r k n) c0 d) \to (\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P))) (\lambda (d: C).(\lambda (H2: (getl (S n) (CHead c0 k t) d)).(\lambda (P: Prop).(H1 d (getl_gen_S k c0 d t n H2) P)))))) H0)))])))))) c).
+ \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl i c0 (CHead e (Bind b) v)))))) (\forall (d: C).((getl i c0 d) \to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda (i: nat).(or_intror (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl i (CSort n) (CHead e (Bind b) v)))))) (\forall (d: C).((getl i (CSort n) d) \to (\forall (P: Prop).P))) (\lambda (d: C).(\lambda (H: (getl i (CSort n) d)).(\lambda (P: Prop).(getl_gen_sort n i d H P))))))) (\lambda (c0: C).(\lambda (H: ((\forall (i: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl i c0 (CHead e (Bind b) v)))))) (\forall (d: C).((getl i c0 d) \to (\forall (P: Prop).P))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (i: nat).(match i return (\lambda (_: ?).(\lambda (n: nat).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl n (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl n (CHead c0 k t) d) \to (\forall (P: Prop).P)))))) with [O \Rightarrow (match k return (\lambda (_: ?).(\lambda (k0: K).(or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 k0 t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O (CHead c0 k0 t) d) \to (\forall (P: Prop).P)))))) with [(Bind b) \Rightarrow (or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead c0 (Bind b) t) (CHead e (Bind b0) v)))))) (\forall (d: C).((getl O (CHead c0 (Bind b) t) d) \to (\forall (P: Prop).P))) (ex_3_intro C B T (\lambda (e: C).(\lambda (b0: B).(\lambda (v: T).(getl O (CHead c0 (Bind b) t) (CHead e (Bind b0) v))))) c0 b t (getl_refl b c0 t))) | (Flat f) \Rightarrow (let H_x \def (H O) in (let H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v)))))) (\forall (d: C).((getl O c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P)))) (\lambda (H1: (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v))))))).(ex_3_ind C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O c0 (CHead e (Bind b) v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P)))) (\lambda (x0: C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl O c0 (CHead x0 (Bind x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P))) (ex_3_intro C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_flat c0 (CHead x0 (Bind x1) x2) O H2 f t))))))) H1)) (\lambda (H1: ((\forall (d: C).((getl O c0 d) \to (\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl O (CHead c0 (Flat f) t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl O (CHead c0 (Flat f) t) d) \to (\forall (P: Prop).P))) (\lambda (d: C).(\lambda (H2: (getl O (CHead c0 (Flat f) t) d)).(\lambda (P: Prop).(H1 d (getl_intro O c0 d c0 (drop_refl c0) (clear_gen_flat f c0 d t (getl_gen_O (CHead c0 (Flat f) t) d H2))) P)))))) H0)))]) | (S n) \Rightarrow (let H_x \def (H (r k n)) in (let H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v)))))) (\forall (d: C).((getl (r k n) c0 d) \to (\forall (P: Prop).P))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P)))) (\lambda (H1: (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v))))))).(ex_3_ind C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (r k n) c0 (CHead e (Bind b) v))))) (or (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P)))) (\lambda (x0: C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl (r k n) c0 (CHead x0 (Bind x1) x2))).(or_introl (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P))) (ex_3_intro C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v))))) x0 x1 x2 (getl_head k n c0 (CHead x0 (Bind x1) x2) H2 t))))))) H1)) (\lambda (H1: ((\forall (d: C).((getl (r k n) c0 d) \to (\forall (P: Prop).P))))).(or_intror (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl (S n) (CHead c0 k t) (CHead e (Bind b) v)))))) (\forall (d: C).((getl (S n) (CHead c0 k t) d) \to (\forall (P: Prop).P))) (\lambda (d: C).(\lambda (H2: (getl (S n) (CHead c0 k t) d)).(\lambda (P: Prop).(H1 d (getl_gen_S k c0 d t n H2) P)))))) H0)))])))))) c).
 
 theorem clear_cle:
  \forall (c1: C).(\forall (c2: C).((clear c1 c2) \to (cle c2 c1)))
 \def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to (le (cweight c2) (cweight c))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (H: (clear (CSort n) c2)).(clear_gen_sort c2 n H (le (cweight c2) O))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (le (cweight c2) (cweight c)))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (H0: (clear (CHead c k t) c2)).((match k return (\lambda (k0: K).((clear (CHead c k0 t) c2) \to (le (cweight c2) (plus (cweight c) (tweight t))))) with [(Bind b) \Rightarrow (\lambda (H1: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: C).(le (cweight c0) (plus (cweight c) (tweight t)))) (le_n (plus (cweight c) (tweight t))) c2 (clear_gen_bind b c c2 t H1))) | (Flat f) \Rightarrow (\lambda (H1: (clear (CHead c (Flat f) t) c2)).(le_S_n (cweight c2) (plus (cweight c) (tweight t)) (le_n_S (cweight c2) (plus (cweight c) (tweight t)) (le_plus_trans (cweight c2) (cweight c) (tweight t) (H c2 (clear_gen_flat f c c2 t H1))))))]) H0))))))) c1).
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).((clear c c2) \to (le (cweight c2) (cweight c))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (H: (clear (CSort n) c2)).(clear_gen_sort c2 n H (le (cweight c2) O))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).((clear c c2) \to (le (cweight c2) (cweight c)))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (H0: (clear (CHead c k t) c2)).((match k return (\lambda (_: ?).(\lambda (k0: K).((clear (CHead c k0 t) c2) \to (le (cweight c2) (plus (cweight c) (tweight t)))))) with [(Bind b) \Rightarrow (\lambda (H1: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: C).(le (cweight c0) (plus (cweight c) (tweight t)))) (le_n (plus (cweight c) (tweight t))) c2 (clear_gen_bind b c c2 t H1))) | (Flat f) \Rightarrow (\lambda (H1: (clear (CHead c (Flat f) t) c2)).(le_S_n (cweight c2) (plus (cweight c) (tweight t)) (le_n_S (cweight c2) (plus (cweight c) (tweight t)) (le_plus_trans (cweight c2) (cweight c) (tweight t) (H c2 (clear_gen_flat f c c2 t H1))))))]) H0))))))) c1).
 
 theorem getl_flt:
  \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead e (Bind b) u)) \to (flt e u c (TLRef i)))))))
 \def
- \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to (flt e u c0 (TLRef i))))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (getl i (CSort n) (CHead e (Bind b) u))).(getl_gen_sort n i (CHead e (Bind b) u) H (flt e u (CSort n) (TLRef i)))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to (flt e u c0 (TLRef i)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(match i return (\lambda (n: nat).((getl n (CHead c0 k t) (CHead e (Bind b) u)) \to (flt e u (CHead c0 k t) (TLRef n)))) with [O \Rightarrow (\lambda (H0: (getl O (CHead c0 k t) (CHead e (Bind b) u))).((match k return (\lambda (k0: K).((clear (CHead c0 k0 t) (CHead e (Bind b) u)) \to (flt e u (CHead c0 k0 t) (TLRef O)))) with [(Bind b0) \Rightarrow (\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e (Bind b) u))).(let H2 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow e | (CHead c _ _) \Rightarrow c])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in ((let H3 \def (f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: ?).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in ((let H4 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C e c0)).(eq_ind_r T t (\lambda (t0: T).(flt e t0 (CHead c0 (Bind b0) t) (TLRef O))) (eq_ind_r C c0 (\lambda (c1: C).(flt c1 t (CHead c0 (Bind b0) t) (TLRef O))) (eq_ind B b (\lambda (b1: B).(flt c0 t (CHead c0 (Bind b1) t) (TLRef O))) (flt_arith0 (Bind b) c0 t O) b0 H5) e H6) u H4)))) H3)) H2))) | (Flat f) \Rightarrow (\lambda (H1: (clear (CHead c0 (Flat f) t) (CHead e (Bind b) u))).(flt_arith1 (Bind b) e c0 u (clear_cle c0 (CHead e (Bind b) u) (clear_gen_flat f c0 (CHead e (Bind b) u) t H1)) (Flat f) t O))]) (getl_gen_O (CHead c0 k t) (CHead e (Bind b) u) H0))) | (S n) \Rightarrow (\lambda (H0: (getl (S n) (CHead c0 k t) (CHead e (Bind b) u))).(let H_y \def (H e u (r k n) (getl_gen_S k c0 (CHead e (Bind b) u) t n H0)) in (flt_arith2 e c0 u (r k n) H_y k t (S n))))])))))))) c)).
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to (flt e u c0 (TLRef i))))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (getl i (CSort n) (CHead e (Bind b) u))).(getl_gen_sort n i (CHead e (Bind b) u) H (flt e u (CSort n) (TLRef i)))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead e (Bind b) u)) \to (flt e u c0 (TLRef i)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (i: nat).(match i return (\lambda (_: ?).(\lambda (n: nat).((getl n (CHead c0 k t) (CHead e (Bind b) u)) \to (flt e u (CHead c0 k t) (TLRef n))))) with [O \Rightarrow (\lambda (H0: (getl O (CHead c0 k t) (CHead e (Bind b) u))).((match k return (\lambda (_: ?).(\lambda (k0: K).((clear (CHead c0 k0 t) (CHead e (Bind b) u)) \to (flt e u (CHead c0 k0 t) (TLRef O))))) with [(Bind b0) \Rightarrow (\lambda (H1: (clear (CHead c0 (Bind b0) t) (CHead e (Bind b) u))).(let H2 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow e | (CHead c _ _) \Rightarrow c])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in ((let H3 \def (f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: ?).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in ((let H4 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead e (Bind b) u) (CHead c0 (Bind b0) t) (clear_gen_bind b0 c0 (CHead e (Bind b) u) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C e c0)).(eq_ind_r T t (\lambda (t0: T).(flt e t0 (CHead c0 (Bind b0) t) (TLRef O))) (eq_ind_r C c0 (\lambda (c1: C).(flt c1 t (CHead c0 (Bind b0) t) (TLRef O))) (eq_ind B b (\lambda (b1: B).(flt c0 t (CHead c0 (Bind b1) t) (TLRef O))) (flt_arith0 (Bind b) c0 t O) b0 H5) e H6) u H4)))) H3)) H2))) | (Flat f) \Rightarrow (\lambda (H1: (clear (CHead c0 (Flat f) t) (CHead e (Bind b) u))).(flt_arith1 (Bind b) e c0 u (clear_cle c0 (CHead e (Bind b) u) (clear_gen_flat f c0 (CHead e (Bind b) u) t H1)) (Flat f) t O))]) (getl_gen_O (CHead c0 k t) (CHead e (Bind b) u) H0))) | (S n) \Rightarrow (\lambda (H0: (getl (S n) (CHead c0 k t) (CHead e (Bind b) u))).(let H_y \def (H e u (r k n) (getl_gen_S k c0 (CHead e (Bind b) u) t n H0)) in (flt_arith2 e c0 u (r k n) H_y k t (S n))))])))))))) c)).
 
 theorem getl_gen_flat:
  \forall (f: F).(\forall (e: C).(\forall (d: C).(\forall (v: T).(\forall (i: nat).((getl i (CHead e (Flat f) v) d) \to (getl i e d))))))
@@ -911,12 +911,12 @@ theorem getl_gen_bind:
 theorem getl_gen_tail:
  \forall (k: K).(\forall (b: B).(\forall (u1: T).(\forall (u2: T).(\forall (c2: C).(\forall (c1: C).(\forall (i: nat).((getl i (CTail k u1 c1) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl i c1 (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c1))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))))))))))
 \def
- \lambda (k: K).(\lambda (b: B).(\lambda (u1: T).(\lambda (u2: T).(\lambda (c2: C).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (i: nat).((getl i (CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))))) (\lambda (n: nat).(\lambda (i: nat).(match i return (\lambda (n0: nat).((getl n0 (CTail k u1 (CSort n)) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl n0 (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n0 (clen (CSort n)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n1: nat).(eq C c2 (CSort n1))))))) with [O \Rightarrow (\lambda (H: (getl O (CHead (CSort n) k u1) (CHead c2 (Bind b) u2))).((match k return (\lambda (k0: K).((clear (CHead (CSort n) k0 u1) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k0 u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K k0 (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))))) with [(Bind b0) \Rightarrow (\lambda (H0: (clear (CHead (CSort n) (Bind b0) u1) (CHead c2 (Bind b) u2))).(let H1 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) u1 H0)) in ((let H2 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) u1 H0)) in ((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) u1 H0)) in (\lambda (H4: (eq B b b0)).(\lambda (H5: (eq C c2 (CSort n))).(eq_ind_r C (CSort n) (\lambda (c: C).(or (ex2 C (\lambda (e: C).(eq C c (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c (CSort n0)))))) (eq_ind_r T u1 (\lambda (t: T).(or (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b) t)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b))) (\lambda (_: nat).(eq T u1 t)) (\lambda (n0: nat).(eq C (CSort n) (CSort n0)))))) (eq_ind_r B b0 (\lambda (b1: B).(or (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b1) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b1))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort n0)))))) (or_intror (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b0) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b0))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b0))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort n0))) n (refl_equal nat O) (refl_equal K (Bind b0)) (refl_equal T u1) (refl_equal C (CSort n)))) b H4) u2 H3) c2 H5)))) H2)) H1))) | (Flat f) \Rightarrow (\lambda (H0: (clear (CHead (CSort n) (Flat f) u1) (CHead c2 (Bind b) u2))).(clear_gen_sort (CHead c2 (Bind b) u2) n (clear_gen_flat f (CSort n) (CHead c2 (Bind b) u2) u1 H0) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail (Flat f) u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Flat f) (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0)))))))]) (getl_gen_O (CHead (CSort n) k u1) (CHead c2 (Bind b) u2) H))) | (S n0) \Rightarrow (\lambda (H: (getl (S n0) (CHead (CSort n) k u1) (CHead c2 (Bind b) u2))).(getl_gen_sort n (r k n0) (CHead c2 (Bind b) u2) (getl_gen_S k (CSort n) (CHead c2 (Bind b) u2) u1 n0 H) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n0) (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n0) O)) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n1: nat).(eq C c2 (CSort n1)))))))]))) (\lambda (c: C).(\lambda (H: ((\forall (i: nat).((getl i (CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))))))).(\lambda (k0: K).(\lambda (t: T).(\lambda (i: nat).(match i return (\lambda (n: nat).((getl n (CTail k u1 (CHead c k0 t)) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n (clen (CHead c k0 t)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))))) with [O \Rightarrow (\lambda (H0: (getl O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b) u2))).((match k0 return (\lambda (k1: K).((clear (CHead (CTail k u1 c) k1 t) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c k1 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s k1 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))))) with [(Bind b0) \Rightarrow (\lambda (H1: (clear (CHead (CTail k u1 c) (Bind b0) t) (CHead c2 (Bind b) u2))).(let H2 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t) (clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) in ((let H3 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t) (clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) in ((let H4 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t) (clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C c2 (CTail k u1 c))).(eq_ind T u2 (\lambda (t0: T).(or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b0) t0) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b0) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))) (eq_ind B b (\lambda (b1: B).(or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b1) u2) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b1) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))) (let H7 \def (eq_ind C c2 (\lambda (c0: C).(\forall (i: nat).((getl i (CTail k u1 c) (CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 (CSort n)))))))) H (CTail k u1 c) H6) in (eq_ind_r C (CTail k u1 c) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 (CSort n)))))) (or_introl (ex2 C (\lambda (e: C).(eq C (CTail k u1 c) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C (CTail k u1 c) (CSort n)))) (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 c) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e (Bind b) u2))) c (refl_equal C (CTail k u1 c)) (getl_refl b c u2))) c2 H6)) b0 H5) t H4)))) H3)) H2))) | (Flat f) \Rightarrow (\lambda (H1: (clear (CHead (CTail k u1 c) (Flat f) t) (CHead c2 (Bind b) u2))).(let H2 \def (H O (getl_intro O (CTail k u1 c) (CHead c2 (Bind b) u2) (CTail k u1 c) (drop_refl (CTail k u1 c)) (clear_gen_flat f (CTail k u1 c) (CHead c2 (Bind b) u2) t H1))) in (or_ind (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))) (\lambda (H3: (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O c (CHead e (Bind b) u2))))).(ex2_ind C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O c (CHead e (Bind b) u2))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CTail k u1 x))).(\lambda (H5: (getl O c (CHead x (Bind b) u2))).(eq_ind_r C (CTail k u1 x) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 (CSort n)))))) (or_introl (ex2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C (CTail k u1 x) (CSort n)))) (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2))) x (refl_equal C (CTail k u1 x)) (getl_flat c (CHead x (Bind b) u2) O H5 f t))) c2 H4)))) H3)) (\lambda (H3: (ex4 nat (\lambda (_: nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))).(ex4_ind nat (\lambda (_: nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))) (\lambda (x0: nat).(\lambda (H4: (eq nat O (clen c))).(\lambda (H5: (eq K k (Bind b))).(\lambda (H6: (eq T u1 u2)).(\lambda (H7: (eq C c2 (CSort x0))).(eq_ind_r C (CSort x0) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 (CSort n)))))) (eq_ind T u1 (\lambda (t0: T).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 t0)) (\lambda (n: nat).(eq C (CSort x0) (CSort n)))))) (eq_ind_r K (Bind b) (\lambda (k1: K).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k1 u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k1 (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n: nat).(eq C (CSort x0) (CSort n)))))) (or_intror (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n: nat).(eq C (CSort x0) (CSort n)))) (ex4_intro nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n: nat).(eq C (CSort x0) (CSort n))) x0 H4 (refl_equal K (Bind b)) (refl_equal T u1) (refl_equal C (CSort x0)))) k H5) u2 H6) c2 H7)))))) H3)) H2)))]) (getl_gen_O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b) u2) H0))) | (S n) \Rightarrow (\lambda (H0: (getl (S n) (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b) u2))).(let H_x \def (H (r k0 n) (getl_gen_S k0 (CTail k u1 c) (CHead c2 (Bind b) u2) t n H0)) in (let H1 \def H_x in (or_ind (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 n) c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (r k0 n) (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0)))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))) (\lambda (H2: (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 n) c (CHead e (Bind b) u2))))).(ex2_ind C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 n) c (CHead e (Bind b) u2))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CTail k u1 x))).(\lambda (H4: (getl (r k0 n) c (CHead x (Bind b) u2))).(let H5 \def (eq_ind C c2 (\lambda (c0: C).(getl (r k0 n) (CTail k u1 c) (CHead c0 (Bind b) u2))) (getl_gen_S k0 (CTail k u1 c) (CHead c2 (Bind b) u2) t n H0) (CTail k u1 x) H3) in (eq_ind_r C (CTail k u1 x) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort n0)))))) (or_introl (ex2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C (CTail k u1 x) (CSort n0)))) (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2))) x (refl_equal C (CTail k u1 x)) (getl_head k0 n c (CHead x (Bind b) u2) H4 t))) c2 H3))))) H2)) (\lambda (H2: (ex4 nat (\lambda (_: nat).(eq nat (r k0 n) (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))).(ex4_ind nat (\lambda (_: nat).(eq nat (r k0 n) (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))) (\lambda (x0: nat).(\lambda (H3: (eq nat (r k0 n) (clen c))).(\lambda (H4: (eq K k (Bind b))).(\lambda (H5: (eq T u1 u2)).(\lambda (H6: (eq C c2 (CSort x0))).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(getl (r k0 n) (CTail k u1 c) (CHead c0 (Bind b) u2))) (getl_gen_S k0 (CTail k u1 c) (CHead c2 (Bind b) u2) t n H0) (CSort x0) H6) in (eq_ind_r C (CSort x0) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort n0)))))) (let H8 \def (eq_ind_r T u2 (\lambda (t: T).(getl (r k0 n) (CTail k u1 c) (CHead (CSort x0) (Bind b) t))) H7 u1 H5) in (eq_ind T u1 (\lambda (t0: T).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 t0)) (\lambda (n0: nat).(eq C (CSort x0) (CSort n0)))))) (let H9 \def (eq_ind K k (\lambda (k: K).(getl (r k0 n) (CTail k u1 c) (CHead (CSort x0) (Bind b) u1))) H8 (Bind b) H4) in (eq_ind_r K (Bind b) (\lambda (k1: K).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k1 u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k1 (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort n0)))))) (eq_ind nat (r k0 n) (\lambda (n0: nat).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 n0))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n1: nat).(eq C (CSort x0) (CSort n1)))))) (eq_ind_r nat (S n) (\lambda (n0: nat).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) n0)) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n1: nat).(eq C (CSort x0) (CSort n1)))))) (or_intror (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (S n))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat (S n) (S n))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort n0))) x0 (refl_equal nat (S n)) (refl_equal K (Bind b)) (refl_equal T u1) (refl_equal C (CSort x0)))) (s k0 (r k0 n)) (s_r k0 n)) (clen c) H3) k H4)) u2 H5)) c2 H6))))))) H2)) H1))))])))))) c1)))))).
+ \lambda (k: K).(\lambda (b: B).(\lambda (u1: T).(\lambda (u2: T).(\lambda (c2: C).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (i: nat).((getl i (CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))))) (\lambda (n: nat).(\lambda (i: nat).(match i return (\lambda (_: ?).(\lambda (n0: nat).((getl n0 (CTail k u1 (CSort n)) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl n0 (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n0 (clen (CSort n)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n1: nat).(eq C c2 (CSort n1)))))))) with [O \Rightarrow (\lambda (H: (getl O (CHead (CSort n) k u1) (CHead c2 (Bind b) u2))).((match k return (\lambda (_: ?).(\lambda (k0: K).((clear (CHead (CSort n) k0 u1) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k0 u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K k0 (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0)))))))) with [(Bind b0) \Rightarrow (\lambda (H0: (clear (CHead (CSort n) (Bind b0) u1) (CHead c2 (Bind b) u2))).(let H1 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) u1 H0)) in ((let H2 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) u1 H0)) in ((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind b) u2) (CHead (CSort n) (Bind b0) u1) (clear_gen_bind b0 (CSort n) (CHead c2 (Bind b) u2) u1 H0)) in (\lambda (H4: (eq B b b0)).(\lambda (H5: (eq C c2 (CSort n))).(eq_ind_r C (CSort n) (\lambda (c: C).(or (ex2 C (\lambda (e: C).(eq C c (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c (CSort n0)))))) (eq_ind_r T u1 (\lambda (t: T).(or (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b) t)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b))) (\lambda (_: nat).(eq T u1 t)) (\lambda (n0: nat).(eq C (CSort n) (CSort n0)))))) (eq_ind_r B b0 (\lambda (b1: B).(or (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b1) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b1))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort n0)))))) (or_intror (ex2 C (\lambda (e: C).(eq C (CSort n) (CTail (Bind b0) u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b0) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b0))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Bind b0) (Bind b0))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort n) (CSort n0))) n (refl_equal nat O) (refl_equal K (Bind b0)) (refl_equal T u1) (refl_equal C (CSort n)))) b H4) u2 H3) c2 H5)))) H2)) H1))) | (Flat f) \Rightarrow (\lambda (H0: (clear (CHead (CSort n) (Flat f) u1) (CHead c2 (Bind b) u2))).(clear_gen_sort (CHead c2 (Bind b) u2) n (clear_gen_flat f (CSort n) (CHead c2 (Bind b) u2) u1 H0) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail (Flat f) u1 e))) (\lambda (e: C).(getl O (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O O)) (\lambda (_: nat).(eq K (Flat f) (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0)))))))]) (getl_gen_O (CHead (CSort n) k u1) (CHead c2 (Bind b) u2) H))) | (S n0) \Rightarrow (\lambda (H: (getl (S n0) (CHead (CSort n) k u1) (CHead c2 (Bind b) u2))).(getl_gen_sort n (r k n0) (CHead c2 (Bind b) u2) (getl_gen_S k (CSort n) (CHead c2 (Bind b) u2) u1 n0 H) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n0) (CSort n) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n0) O)) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n1: nat).(eq C c2 (CSort n1)))))))]))) (\lambda (c: C).(\lambda (H: ((\forall (i: nat).((getl i (CTail k u1 c) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))))))).(\lambda (k0: K).(\lambda (t: T).(\lambda (i: nat).(match i return (\lambda (_: ?).(\lambda (n: nat).((getl n (CTail k u1 (CHead c k0 t)) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl n (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat n (clen (CHead c k0 t)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0)))))))) with [O \Rightarrow (\lambda (H0: (getl O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b) u2))).((match k0 return (\lambda (_: ?).(\lambda (k1: K).((clear (CHead (CTail k u1 c) k1 t) (CHead c2 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c k1 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s k1 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))))) with [(Bind b0) \Rightarrow (\lambda (H1: (clear (CHead (CTail k u1 c) (Bind b0) t) (CHead c2 (Bind b) u2))).(let H2 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t) (clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) in ((let H3 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t) (clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) in ((let H4 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind b) u2) (CHead (CTail k u1 c) (Bind b0) t) (clear_gen_bind b0 (CTail k u1 c) (CHead c2 (Bind b) u2) t H1)) in (\lambda (H5: (eq B b b0)).(\lambda (H6: (eq C c2 (CTail k u1 c))).(eq_ind T u2 (\lambda (t0: T).(or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b0) t0) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b0) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))) (eq_ind B b (\lambda (b1: B).(or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b1) u2) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b1) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))))) (let H7 \def (eq_ind C c2 (\lambda (c0: C).(\forall (i: nat).((getl i (CTail k u1 c) (CHead c0 (Bind b) u2)) \to (or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl i c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat i (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 (CSort n)))))))) H (CTail k u1 c) H6) in (eq_ind_r C (CTail k u1 c) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 (CSort n)))))) (or_introl (ex2 C (\lambda (e: C).(eq C (CTail k u1 c) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Bind b) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C (CTail k u1 c) (CSort n)))) (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 c) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Bind b) u2) (CHead e (Bind b) u2))) c (refl_equal C (CTail k u1 c)) (getl_refl b c u2))) c2 H6)) b0 H5) t H4)))) H3)) H2))) | (Flat f) \Rightarrow (\lambda (H1: (clear (CHead (CTail k u1 c) (Flat f) t) (CHead c2 (Bind b) u2))).(let H2 \def (H O (getl_intro O (CTail k u1 c) (CHead c2 (Bind b) u2) (CTail k u1 c) (drop_refl (CTail k u1 c)) (clear_gen_flat f (CTail k u1 c) (CHead c2 (Bind b) u2) t H1))) in (or_ind (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n)))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))) (\lambda (H3: (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O c (CHead e (Bind b) u2))))).(ex2_ind C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O c (CHead e (Bind b) u2))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CTail k u1 x))).(\lambda (H5: (getl O c (CHead x (Bind b) u2))).(eq_ind_r C (CTail k u1 x) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 (CSort n)))))) (or_introl (ex2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C (CTail k u1 x) (CSort n)))) (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2))) x (refl_equal C (CTail k u1 x)) (getl_flat c (CHead x (Bind b) u2) O H5 f t))) c2 H4)))) H3)) (\lambda (H3: (ex4 nat (\lambda (_: nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))).(ex4_ind nat (\lambda (_: nat).(eq nat O (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))) (\lambda (x0: nat).(\lambda (H4: (eq nat O (clen c))).(\lambda (H5: (eq K k (Bind b))).(\lambda (H6: (eq T u1 u2)).(\lambda (H7: (eq C c2 (CSort x0))).(eq_ind_r C (CSort x0) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c0 (CSort n)))))) (eq_ind T u1 (\lambda (t0: T).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 t0)) (\lambda (n: nat).(eq C (CSort x0) (CSort n)))))) (eq_ind_r K (Bind b) (\lambda (k1: K).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k1 u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K k1 (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n: nat).(eq C (CSort x0) (CSort n)))))) (or_intror (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl O (CHead c (Flat f) t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n: nat).(eq C (CSort x0) (CSort n)))) (ex4_intro nat (\lambda (_: nat).(eq nat O (s (Flat f) (clen c)))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n: nat).(eq C (CSort x0) (CSort n))) x0 H4 (refl_equal K (Bind b)) (refl_equal T u1) (refl_equal C (CSort x0)))) k H5) u2 H6) c2 H7)))))) H3)) H2)))]) (getl_gen_O (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b) u2) H0))) | (S n) \Rightarrow (\lambda (H0: (getl (S n) (CHead (CTail k u1 c) k0 t) (CHead c2 (Bind b) u2))).(let H_x \def (H (r k0 n) (getl_gen_S k0 (CTail k u1 c) (CHead c2 (Bind b) u2) t n H0)) in (let H1 \def H_x in (or_ind (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 n) c (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (r k0 n) (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0)))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))) (\lambda (H2: (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 n) c (CHead e (Bind b) u2))))).(ex2_ind C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (r k0 n) c (CHead e (Bind b) u2))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CTail k u1 x))).(\lambda (H4: (getl (r k0 n) c (CHead x (Bind b) u2))).(let H5 \def (eq_ind C c2 (\lambda (c0: C).(getl (r k0 n) (CTail k u1 c) (CHead c0 (Bind b) u2))) (getl_gen_S k0 (CTail k u1 c) (CHead c2 (Bind b) u2) t n H0) (CTail k u1 x) H3) in (eq_ind_r C (CTail k u1 x) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort n0)))))) (or_introl (ex2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C (CTail k u1 x) (CSort n0)))) (ex_intro2 C (\lambda (e: C).(eq C (CTail k u1 x) (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2))) x (refl_equal C (CTail k u1 x)) (getl_head k0 n c (CHead x (Bind b) u2) H4 t))) c2 H3))))) H2)) (\lambda (H2: (ex4 nat (\lambda (_: nat).(eq nat (r k0 n) (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n: nat).(eq C c2 (CSort n))))).(ex4_ind nat (\lambda (_: nat).(eq nat (r k0 n) (clen c))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))) (or (ex2 C (\lambda (e: C).(eq C c2 (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c2 (CSort n0))))) (\lambda (x0: nat).(\lambda (H3: (eq nat (r k0 n) (clen c))).(\lambda (H4: (eq K k (Bind b))).(\lambda (H5: (eq T u1 u2)).(\lambda (H6: (eq C c2 (CSort x0))).(let H7 \def (eq_ind C c2 (\lambda (c0: C).(getl (r k0 n) (CTail k u1 c) (CHead c0 (Bind b) u2))) (getl_gen_S k0 (CTail k u1 c) (CHead c2 (Bind b) u2) t n H0) (CSort x0) H6) in (eq_ind_r C (CSort x0) (\lambda (c0: C).(or (ex2 C (\lambda (e: C).(eq C c0 (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u2)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 u2)) (\lambda (n0: nat).(eq C c0 (CSort n0)))))) (let H8 \def (eq_ind_r T u2 (\lambda (t: T).(getl (r k0 n) (CTail k u1 c) (CHead (CSort x0) (Bind b) t))) H7 u1 H5) in (eq_ind T u1 (\lambda (t0: T).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) t0)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k (Bind b))) (\lambda (_: nat).(eq T u1 t0)) (\lambda (n0: nat).(eq C (CSort x0) (CSort n0)))))) (let H9 \def (eq_ind K k (\lambda (k: K).(getl (r k0 n) (CTail k u1 c) (CHead (CSort x0) (Bind b) u1))) H8 (Bind b) H4) in (eq_ind_r K (Bind b) (\lambda (k1: K).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail k1 u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 (clen c)))) (\lambda (_: nat).(eq K k1 (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort n0)))))) (eq_ind nat (r k0 n) (\lambda (n0: nat).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (s k0 n0))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n1: nat).(eq C (CSort x0) (CSort n1)))))) (eq_ind_r nat (S n) (\lambda (n0: nat).(or (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) n0)) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n1: nat).(eq C (CSort x0) (CSort n1)))))) (or_intror (ex2 C (\lambda (e: C).(eq C (CSort x0) (CTail (Bind b) u1 e))) (\lambda (e: C).(getl (S n) (CHead c k0 t) (CHead e (Bind b) u1)))) (ex4 nat (\lambda (_: nat).(eq nat (S n) (S n))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort n0)))) (ex4_intro nat (\lambda (_: nat).(eq nat (S n) (S n))) (\lambda (_: nat).(eq K (Bind b) (Bind b))) (\lambda (_: nat).(eq T u1 u1)) (\lambda (n0: nat).(eq C (CSort x0) (CSort n0))) x0 (refl_equal nat (S n)) (refl_equal K (Bind b)) (refl_equal T u1) (refl_equal C (CSort x0)))) (s k0 (r k0 n)) (s_r k0 n)) (clen c) H3) k H4)) u2 H5)) c2 H6))))))) H2)) H1))))])))))) c1)))))).
 
 theorem cimp_flat_sx:
  \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp (CHead c (Flat f) v) c)))
 \def
- \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1: C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h (CHead c (Flat f) v) (CHead d1 (Bind b) w))).((match h return (\lambda (n: nat).((getl n (CHead c (Flat f) v) (CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl n c (CHead d2 (Bind b) w)))))) with [O \Rightarrow (\lambda (H0: (getl O (CHead c (Flat f) v) (CHead d1 (Bind b) w))).(ex_intro C (\lambda (d2: C).(getl O c (CHead d2 (Bind b) w))) d1 (getl_intro O c (CHead d1 (Bind b) w) c (drop_refl c) (clear_gen_flat f c (CHead d1 (Bind b) w) v (getl_gen_O (CHead c (Flat f) v) (CHead d1 (Bind b) w) H0))))) | (S n) \Rightarrow (\lambda (H0: (getl (S n) (CHead c (Flat f) v) (CHead d1 (Bind b) w))).(ex_intro C (\lambda (d2: C).(getl (S n) c (CHead d2 (Bind b) w))) d1 (getl_gen_S (Flat f) c (CHead d1 (Bind b) w) v n H0)))]) H)))))))).
+ \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (b: B).(\lambda (d1: C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H: (getl h (CHead c (Flat f) v) (CHead d1 (Bind b) w))).((match h return (\lambda (_: ?).(\lambda (n: nat).((getl n (CHead c (Flat f) v) (CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl n c (CHead d2 (Bind b) w))))))) with [O \Rightarrow (\lambda (H0: (getl O (CHead c (Flat f) v) (CHead d1 (Bind b) w))).(ex_intro C (\lambda (d2: C).(getl O c (CHead d2 (Bind b) w))) d1 (getl_intro O c (CHead d1 (Bind b) w) c (drop_refl c) (clear_gen_flat f c (CHead d1 (Bind b) w) v (getl_gen_O (CHead c (Flat f) v) (CHead d1 (Bind b) w) H0))))) | (S n) \Rightarrow (\lambda (H0: (getl (S n) (CHead c (Flat f) v) (CHead d1 (Bind b) w))).(ex_intro C (\lambda (d2: C).(getl (S n) c (CHead d2 (Bind b) w))) d1 (getl_gen_S (Flat f) c (CHead d1 (Bind b) w) v n H0)))]) H)))))))).
 
 theorem cimp_flat_dx:
  \forall (f: F).(\forall (c: C).(\forall (v: T).(cimp c (CHead c (Flat f) v))))
@@ -926,7 +926,7 @@ theorem cimp_flat_dx:
 theorem cimp_bind:
  \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall (v: T).(cimp (CHead c1 (Bind b) v) (CHead c2 (Bind b) v))))))
 \def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1: C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda (b: B).(\lambda (v: T).(\lambda (b0: B).(\lambda (d1: C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H0: (getl h (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w))).((match h return (\lambda (n: nat).((getl n (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w)) \to (ex C (\lambda (d2: C).(getl n (CHead c2 (Bind b) v) (CHead d2 (Bind b0) w)))))) with [O \Rightarrow (\lambda (H1: (getl O (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w))).(let H2 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind b0) w) (CHead c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in ((let H3 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b0])])) (CHead d1 (Bind b0) w) (CHead c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in ((let H4 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind b0) w) (CHead c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in (\lambda (H5: (eq B b0 b)).(\lambda (_: (eq C d1 c1)).(eq_ind_r T v (\lambda (t: T).(ex C (\lambda (d2: C).(getl O (CHead c2 (Bind b) v) (CHead d2 (Bind b0) t))))) (eq_ind_r B b (\lambda (b1: B).(ex C (\lambda (d2: C).(getl O (CHead c2 (Bind b) v) (CHead d2 (Bind b1) v))))) (ex_intro C (\lambda (d2: C).(getl O (CHead c2 (Bind b) v) (CHead d2 (Bind b) v))) c2 (getl_refl b c2 v)) b0 H5) w H4)))) H3)) H2))) | (S n) \Rightarrow (\lambda (H1: (getl (S n) (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w))).(let H_x \def (H b0 d1 w (r (Bind b) n) (getl_gen_S (Bind b) c1 (CHead d1 (Bind b0) w) v n H1)) in (let H2 \def H_x in (ex_ind C (\lambda (d2: C).(getl (r (Bind b) n) c2 (CHead d2 (Bind b0) w))) (ex C (\lambda (d2: C).(getl (S n) (CHead c2 (Bind b) v) (CHead d2 (Bind b0) w)))) (\lambda (x: C).(\lambda (H3: (getl (r (Bind b) n) c2 (CHead x (Bind b0) w))).(ex_intro C (\lambda (d2: C).(getl (S n) (CHead c2 (Bind b) v) (CHead d2 (Bind b0) w))) x (getl_head (Bind b) n c2 (CHead x (Bind b0) w) H3 v)))) H2))))]) H0)))))))))).
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: ((\forall (b: B).(\forall (d1: C).(\forall (w: T).(\forall (h: nat).((getl h c1 (CHead d1 (Bind b) w)) \to (ex C (\lambda (d2: C).(getl h c2 (CHead d2 (Bind b) w))))))))))).(\lambda (b: B).(\lambda (v: T).(\lambda (b0: B).(\lambda (d1: C).(\lambda (w: T).(\lambda (h: nat).(\lambda (H0: (getl h (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w))).((match h return (\lambda (_: ?).(\lambda (n: nat).((getl n (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w)) \to (ex C (\lambda (d2: C).(getl n (CHead c2 (Bind b) v) (CHead d2 (Bind b0) w))))))) with [O \Rightarrow (\lambda (H1: (getl O (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w))).(let H2 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind b0) w) (CHead c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in ((let H3 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow b0 | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b0])])) (CHead d1 (Bind b0) w) (CHead c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in ((let H4 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind b0) w) (CHead c1 (Bind b) v) (clear_gen_bind b c1 (CHead d1 (Bind b0) w) v (getl_gen_O (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w) H1))) in (\lambda (H5: (eq B b0 b)).(\lambda (_: (eq C d1 c1)).(eq_ind_r T v (\lambda (t: T).(ex C (\lambda (d2: C).(getl O (CHead c2 (Bind b) v) (CHead d2 (Bind b0) t))))) (eq_ind_r B b (\lambda (b1: B).(ex C (\lambda (d2: C).(getl O (CHead c2 (Bind b) v) (CHead d2 (Bind b1) v))))) (ex_intro C (\lambda (d2: C).(getl O (CHead c2 (Bind b) v) (CHead d2 (Bind b) v))) c2 (getl_refl b c2 v)) b0 H5) w H4)))) H3)) H2))) | (S n) \Rightarrow (\lambda (H1: (getl (S n) (CHead c1 (Bind b) v) (CHead d1 (Bind b0) w))).(let H_x \def (H b0 d1 w (r (Bind b) n) (getl_gen_S (Bind b) c1 (CHead d1 (Bind b0) w) v n H1)) in (let H2 \def H_x in (ex_ind C (\lambda (d2: C).(getl (r (Bind b) n) c2 (CHead d2 (Bind b0) w))) (ex C (\lambda (d2: C).(getl (S n) (CHead c2 (Bind b) v) (CHead d2 (Bind b0) w)))) (\lambda (x: C).(\lambda (H3: (getl (r (Bind b) n) c2 (CHead x (Bind b0) w))).(ex_intro C (\lambda (d2: C).(getl (S n) (CHead c2 (Bind b) v) (CHead d2 (Bind b0) w))) x (getl_head (Bind b) n c2 (CHead x (Bind b0) w) H3 v)))) H2))))]) H0)))))))))).
 
 theorem getl_mono:
  \forall (c: C).(\forall (x1: C).(\forall (h: nat).((getl h c x1) \to (\forall (x2: C).((getl h c x2) \to (eq C x1 x2))))))
@@ -936,12 +936,12 @@ theorem getl_mono:
 theorem getl_clear_conf:
  \forall (i: nat).(\forall (c1: C).(\forall (c3: C).((getl i c1 c3) \to (\forall (c2: C).((clear c1 c2) \to (getl i c2 c3))))))
 \def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c3: C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2) \to (getl n c2 c3))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H: (getl O c1 c3)).(\lambda (c2: C).(\lambda (H0: (clear c1 c2)).(eq_ind C c3 (\lambda (c: C).(getl O c c3)) (let H1 \def (clear_gen_all c1 c3 (getl_gen_O c1 c3 H)) in (ex_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c3 (CHead e (Bind b) u))))) (getl O c3 c3) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H2: (eq C c3 (CHead x1 (Bind x0) x2))).(let H3 \def (eq_ind C c3 (\lambda (c: C).(clear c1 c)) (getl_gen_O c1 c3 H) (CHead x1 (Bind x0) x2) H2) in (eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c: C).(getl O c c)) (getl_refl x0 x1 x2) c3 H2)))))) H1)) c2 (clear_mono c1 c3 (getl_gen_O c1 c3 H) c2 H0))))))) (\lambda (n: nat).(\lambda (_: ((\forall (c1: C).(\forall (c3: C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2) \to (getl n c2 c3)))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c3: C).((getl (S n) c c3) \to (\forall (c2: C).((clear c c2) \to (getl (S n) c2 c3)))))) (\lambda (n0: nat).(\lambda (c3: C).(\lambda (H0: (getl (S n) (CSort n0) c3)).(\lambda (c2: C).(\lambda (_: (clear (CSort n0) c2)).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c2 c3))))))) (\lambda (c: C).(\lambda (H0: ((\forall (c3: C).((getl (S n) c c3) \to (\forall (c2: C).((clear c c2) \to (getl (S n) c2 c3))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t) c3)).(\lambda (c2: C).(\lambda (H2: (clear (CHead c k t) c2)).((match k return (\lambda (k0: K).((getl (S n) (CHead c k0 t) c3) \to ((clear (CHead c k0 t) c2) \to (getl (S n) c2 c3)))) with [(Bind b) \Rightarrow (\lambda (H3: (getl (S n) (CHead c (Bind b) t) c3)).(\lambda (H4: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: C).(getl (S n) c0 c3)) (getl_head (Bind b) n c c3 (getl_gen_S (Bind b) c c3 t n H3) t) c2 (clear_gen_bind b c c2 t H4)))) | (Flat f) \Rightarrow (\lambda (H3: (getl (S n) (CHead c (Flat f) t) c3)).(\lambda (H4: (clear (CHead c (Flat f) t) c2)).(H0 c3 (getl_gen_S (Flat f) c c3 t n H3) c2 (clear_gen_flat f c c2 t H4))))]) H1 H2))))))))) c1)))) i).
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c3: C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2) \to (getl n c2 c3))))))) (\lambda (c1: C).(\lambda (c3: C).(\lambda (H: (getl O c1 c3)).(\lambda (c2: C).(\lambda (H0: (clear c1 c2)).(eq_ind C c3 (\lambda (c: C).(getl O c c3)) (let H1 \def (clear_gen_all c1 c3 (getl_gen_O c1 c3 H)) in (ex_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (u: T).(eq C c3 (CHead e (Bind b) u))))) (getl O c3 c3) (\lambda (x0: B).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H2: (eq C c3 (CHead x1 (Bind x0) x2))).(let H3 \def (eq_ind C c3 (\lambda (c: C).(clear c1 c)) (getl_gen_O c1 c3 H) (CHead x1 (Bind x0) x2) H2) in (eq_ind_r C (CHead x1 (Bind x0) x2) (\lambda (c: C).(getl O c c)) (getl_refl x0 x1 x2) c3 H2)))))) H1)) c2 (clear_mono c1 c3 (getl_gen_O c1 c3 H) c2 H0))))))) (\lambda (n: nat).(\lambda (_: ((\forall (c1: C).(\forall (c3: C).((getl n c1 c3) \to (\forall (c2: C).((clear c1 c2) \to (getl n c2 c3)))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c3: C).((getl (S n) c c3) \to (\forall (c2: C).((clear c c2) \to (getl (S n) c2 c3)))))) (\lambda (n0: nat).(\lambda (c3: C).(\lambda (H0: (getl (S n) (CSort n0) c3)).(\lambda (c2: C).(\lambda (_: (clear (CSort n0) c2)).(getl_gen_sort n0 (S n) c3 H0 (getl (S n) c2 c3))))))) (\lambda (c: C).(\lambda (H0: ((\forall (c3: C).((getl (S n) c c3) \to (\forall (c2: C).((clear c c2) \to (getl (S n) c2 c3))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3: C).(\lambda (H1: (getl (S n) (CHead c k t) c3)).(\lambda (c2: C).(\lambda (H2: (clear (CHead c k t) c2)).((match k return (\lambda (_: ?).(\lambda (k0: K).((getl (S n) (CHead c k0 t) c3) \to ((clear (CHead c k0 t) c2) \to (getl (S n) c2 c3))))) with [(Bind b) \Rightarrow (\lambda (H3: (getl (S n) (CHead c (Bind b) t) c3)).(\lambda (H4: (clear (CHead c (Bind b) t) c2)).(eq_ind_r C (CHead c (Bind b) t) (\lambda (c0: C).(getl (S n) c0 c3)) (getl_head (Bind b) n c c3 (getl_gen_S (Bind b) c c3 t n H3) t) c2 (clear_gen_bind b c c2 t H4)))) | (Flat f) \Rightarrow (\lambda (H3: (getl (S n) (CHead c (Flat f) t) c3)).(\lambda (H4: (clear (CHead c (Flat f) t) c2)).(H0 c3 (getl_gen_S (Flat f) c c3 t n H3) c2 (clear_gen_flat f c c2 t H4))))]) H1 H2))))))))) c1)))) i).
 
 theorem getl_drop_conf_lt:
  \forall (b: B).(\forall (c: C).(\forall (c0: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead c0 (Bind b) u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c0 e0)))))))))))))
 \def
- \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1: C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead c1 (Bind b) u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i d)) c0 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (getl i (CSort n) (CHead c0 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_: (drop h (S (plus i d)) (CSort n) e)).(getl_gen_sort n i (CHead c0 (Bind b) u) H (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c0 e0)))))))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (c1: C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead c1 (Bind b) u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i d)) c0 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i (CHead c0 k t) (CHead c1 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (drop h (S (plus i d)) (CHead c0 k t) e)).(let H2 \def (getl_gen_all (CHead c0 k t) (CHead c1 (Bind b) u) i H0) in (ex2_ind C (\lambda (e0: C).(drop i O (CHead c0 k t) e0)) (\lambda (e0: C).(clear e0 (CHead c1 (Bind b) u))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x: C).(\lambda (H3: (drop i O (CHead c0 k t) x)).(\lambda (H4: (clear x (CHead c1 (Bind b) u))).((match x return (\lambda (c2: C).((drop i O (CHead c0 k t) c2) \to ((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) with [(CSort n) \Rightarrow (\lambda (_: (drop i O (CHead c0 k t) (CSort n))).(\lambda (H6: (clear (CSort n) (CHead c1 (Bind b) u))).(clear_gen_sort (CHead c1 (Bind b) u) n H6 (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) | (CHead c2 k0 t0) \Rightarrow (\lambda (H5: (drop i O (CHead c0 k t) (CHead c2 k0 t0))).(\lambda (H6: (clear (CHead c2 k0 t0) (CHead c1 (Bind b) u))).((match k0 return (\lambda (k1: K).((drop i O (CHead c0 k t) (CHead c2 k1 t0)) \to ((clear (CHead c2 k1 t0) (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) with [(Bind b0) \Rightarrow (\lambda (H7: (drop i O (CHead c0 k t) (CHead c2 (Bind b0) t0))).(\lambda (H8: (clear (CHead c2 (Bind b0) t0) (CHead c1 (Bind b) u))).(let H9 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind b) u) (CHead c2 (Bind b0) t0) (clear_gen_bind b0 c2 (CHead c1 (Bind b) u) t0 H8)) in ((let H10 \def (f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: ?).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (CHead c1 (Bind b) u) (CHead c2 (Bind b0) t0) (clear_gen_bind b0 c2 (CHead c1 (Bind b) u) t0 H8)) in ((let H11 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 (Bind b) u) (CHead c2 (Bind b0) t0) (clear_gen_bind b0 c2 (CHead c1 (Bind b) u) t0 H8)) in (\lambda (H12: (eq B b b0)).(\lambda (H13: (eq C c1 c2)).(let H14 \def (eq_ind_r T t0 (\lambda (t0: T).(drop i O (CHead c0 k t) (CHead c2 (Bind b0) t0))) H7 u H11) in (let H15 \def (eq_ind_r B b0 (\lambda (b: B).(drop i O (CHead c0 k t) (CHead c2 (Bind b) u))) H14 b H12) in (let H16 \def (eq_ind_r C c2 (\lambda (c: C).(drop i O (CHead c0 k t) (CHead c (Bind b) u))) H15 c1 H13) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r (Bind b) d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop i O e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r (Bind b) d) c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (H17: (eq T u (lift h (r (Bind b) d) x0))).(\lambda (H18: (drop i O e (CHead x1 (Bind b) x0))).(\lambda (H19: (drop h (r (Bind b) d) c1 x1)).(eq_ind_r T (lift h (r (Bind b) d) x0) (\lambda (t1: T).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T (lift h (r (Bind b) d) x0) (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x0 x1 (refl_equal T (lift h d x0)) (getl_intro i e (CHead x1 (Bind b) x0) (CHead x1 (Bind b) x0) H18 (clear_bind b x1 x0)) H19) u H17)))))) (drop_conf_lt (Bind b) i u c1 (CHead c0 k t) H16 e h d H1)))))))) H10)) H9)))) | (Flat f) \Rightarrow (\lambda (H7: (drop i O (CHead c0 k t) (CHead c2 (Flat f) t0))).(\lambda (H8: (clear (CHead c2 (Flat f) t0) (CHead c1 (Bind b) u))).((match i return (\lambda (n: nat).((drop h (S (plus n d)) (CHead c0 k t) e) \to ((drop n O (CHead c0 k t) (CHead c2 (Flat f) t0)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) with [O \Rightarrow (\lambda (H9: (drop h (S (plus O d)) (CHead c0 k t) e)).(\lambda (H10: (drop O O (CHead c0 k t) (CHead c2 (Flat f) t0))).(let H11 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k t) (CHead c2 (Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead c2 (Flat f) t0) H10)) in ((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c0 k t) (CHead c2 (Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead c2 (Flat f) t0) H10)) in ((let H13 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow t | (CHead _ _ t) \Rightarrow t])) (CHead c0 k t) (CHead c2 (Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead c2 (Flat f) t0) H10)) in (\lambda (H14: (eq K k (Flat f))).(\lambda (H15: (eq C c0 c2)).(let H16 \def (eq_ind_r C c2 (\lambda (c: C).(clear c (CHead c1 (Bind b) u))) (clear_gen_flat f c2 (CHead c1 (Bind b) u) t0 H8) c0 H15) in (let H17 \def (eq_ind K k (\lambda (k: K).(drop h (S (plus O d)) (CHead c0 k t) e)) H9 (Flat f) H14) in (ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Flat f) (plus O d)) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Flat f) (plus O d)) c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H18: (eq C e (CHead x0 (Flat f) x1))).(\lambda (H19: (eq T t (lift h (r (Flat f) (plus O d)) x1))).(\lambda (H20: (drop h (r (Flat f) (plus O d)) c0 x0)).(let H21 \def (f_equal T T (\lambda (e0: T).e0) t (lift h (r (Flat f) (plus O d)) x1) H19) in (eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c3: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O c3 (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H22 \def (H c1 u O (getl_intro O c0 (CHead c1 (Bind b) u) c0 (drop_refl c0) H16) x0 h d H20) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O x0 (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x0 (Flat f) x1) (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H23: (eq T u (lift h d x2))).(\lambda (H24: (getl O x0 (CHead x3 (Bind b) x2))).(\lambda (H25: (drop h d c1 x3)).(let H26 \def (eq_ind T u (\lambda (t: T).(clear c0 (CHead c1 (Bind b) t))) H16 (lift h d x2) H23) in (eq_ind_r T (lift h d x2) (\lambda (t1: T).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x0 (Flat f) x1) (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T (lift h d x2) (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x0 (Flat f) x1) (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x2 x3 (refl_equal T (lift h d x2)) (getl_flat x0 (CHead x3 (Bind b) x2) O H24 f x1) H25) u H23))))))) H22)) e H18))))))) (drop_gen_skip_l c0 e t h (plus O d) (Flat f) H17))))))) H12)) H11)))) | (S n) \Rightarrow (\lambda (H9: (drop h (S (plus (S n) d)) (CHead c0 k t) e)).(\lambda (H10: (drop (S n) O (CHead c0 k t) (CHead c2 (Flat f) t0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k (plus (S n) d)) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus (S n) d)) c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S n) e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H11: (eq C e (CHead x0 k x1))).(\lambda (H12: (eq T t (lift h (r k (plus (S n) d)) x1))).(\lambda (H13: (drop h (r k (plus (S n) d)) c0 x0)).(let H14 \def (f_equal T T (\lambda (e0: T).e0) t (lift h (r k (plus (S n) d)) x1) H12) in (eq_ind_r C (CHead x0 k x1) (\lambda (c3: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S n) c3 (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H15 \def (eq_ind nat (r k (plus (S n) d)) (\lambda (n: nat).(drop h n c0 x0)) H13 (plus (r k (S n)) d) (r_plus k (S n) d)) in (let H16 \def (eq_ind nat (r k (S n)) (\lambda (n: nat).(drop h (plus n d) c0 x0)) H15 (S (r k n)) (r_S k n)) in (let H17 \def (H c1 u (r k n) (getl_intro (r k n) c0 (CHead c1 (Bind b) u) (CHead c2 (Flat f) t0) (drop_gen_drop k c0 (CHead c2 (Flat f) t0) t n H10) (clear_flat c2 (CHead c1 (Bind b) u) (clear_gen_flat f c2 (CHead c1 (Bind b) u) t0 H8) f t0)) x0 h d H16) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (r k n) x0 (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S n) (CHead x0 k x1) (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H18: (eq T u (lift h d x2))).(\lambda (H19: (getl (r k n) x0 (CHead x3 (Bind b) x2))).(\lambda (H20: (drop h d c1 x3)).(let H21 \def (eq_ind T u (\lambda (t: T).(clear c2 (CHead c1 (Bind b) t))) (clear_gen_flat f c2 (CHead c1 (Bind b) u) t0 H8) (lift h d x2) H18) in (eq_ind_r T (lift h d x2) (\lambda (t1: T).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S n) (CHead x0 k x1) (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T (lift h d x2) (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S n) (CHead x0 k x1) (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x2 x3 (refl_equal T (lift h d x2)) (getl_head k n x0 (CHead x3 (Bind b) x2) H19 x1) H20) u H18))))))) H17)))) e H11))))))) (drop_gen_skip_l c0 e t h (plus (S n) d) k H9))))]) H1 H7)))]) H5 H6)))]) H3 H4)))) H2)))))))))))))) c)).
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (c1: C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead c1 (Bind b) u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i d)) c0 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (getl i (CSort n) (CHead c0 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (_: (drop h (S (plus i d)) (CSort n) e)).(getl_gen_sort n i (CHead c0 (Bind b) u) H (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c0 e0)))))))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (c1: C).(\forall (u: T).(\forall (i: nat).((getl i c0 (CHead c1 (Bind b) u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i d)) c0 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i (CHead c0 k t) (CHead c1 (Bind b) u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (drop h (S (plus i d)) (CHead c0 k t) e)).(let H2 \def (getl_gen_all (CHead c0 k t) (CHead c1 (Bind b) u) i H0) in (ex2_ind C (\lambda (e0: C).(drop i O (CHead c0 k t) e0)) (\lambda (e0: C).(clear e0 (CHead c1 (Bind b) u))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x: C).(\lambda (H3: (drop i O (CHead c0 k t) x)).(\lambda (H4: (clear x (CHead c1 (Bind b) u))).((match x return (\lambda (_: ?).(\lambda (c2: C).((drop i O (CHead c0 k t) c2) \to ((clear c2 (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))))))) with [(CSort n) \Rightarrow (\lambda (_: (drop i O (CHead c0 k t) (CSort n))).(\lambda (H6: (clear (CSort n) (CHead c1 (Bind b) u))).(clear_gen_sort (CHead c1 (Bind b) u) n H6 (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))))) | (CHead c2 k0 t0) \Rightarrow (\lambda (H5: (drop i O (CHead c0 k t) (CHead c2 k0 t0))).(\lambda (H6: (clear (CHead c2 k0 t0) (CHead c1 (Bind b) u))).((match k0 return (\lambda (_: ?).(\lambda (k1: K).((drop i O (CHead c0 k t) (CHead c2 k1 t0)) \to ((clear (CHead c2 k1 t0) (CHead c1 (Bind b) u)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))))))) with [(Bind b0) \Rightarrow (\lambda (H7: (drop i O (CHead c0 k t) (CHead c2 (Bind b0) t0))).(\lambda (H8: (clear (CHead c2 (Bind b0) t0) (CHead c1 (Bind b) u))).(let H9 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind b) u) (CHead c2 (Bind b0) t0) (clear_gen_bind b0 c2 (CHead c1 (Bind b) u) t0 H8)) in ((let H10 \def (f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: ?).B) with [(CSort _) \Rightarrow b | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (CHead c1 (Bind b) u) (CHead c2 (Bind b0) t0) (clear_gen_bind b0 c2 (CHead c1 (Bind b) u) t0 H8)) in ((let H11 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 (Bind b) u) (CHead c2 (Bind b0) t0) (clear_gen_bind b0 c2 (CHead c1 (Bind b) u) t0 H8)) in (\lambda (H12: (eq B b b0)).(\lambda (H13: (eq C c1 c2)).(let H14 \def (eq_ind_r T t0 (\lambda (t0: T).(drop i O (CHead c0 k t) (CHead c2 (Bind b0) t0))) H7 u H11) in (let H15 \def (eq_ind_r B b0 (\lambda (b: B).(drop i O (CHead c0 k t) (CHead c2 (Bind b) u))) H14 b H12) in (let H16 \def (eq_ind_r C c2 (\lambda (c: C).(drop i O (CHead c0 k t) (CHead c (Bind b) u))) H15 c1 H13) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r (Bind b) d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop i O e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r (Bind b) d) c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (H17: (eq T u (lift h (r (Bind b) d) x0))).(\lambda (H18: (drop i O e (CHead x1 (Bind b) x0))).(\lambda (H19: (drop h (r (Bind b) d) c1 x1)).(eq_ind_r T (lift h (r (Bind b) d) x0) (\lambda (t1: T).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T (lift h (r (Bind b) d) x0) (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x0 x1 (refl_equal T (lift h d x0)) (getl_intro i e (CHead x1 (Bind b) x0) (CHead x1 (Bind b) x0) H18 (clear_bind b x1 x0)) H19) u H17)))))) (drop_conf_lt (Bind b) i u c1 (CHead c0 k t) H16 e h d H1)))))))) H10)) H9)))) | (Flat f) \Rightarrow (\lambda (H7: (drop i O (CHead c0 k t) (CHead c2 (Flat f) t0))).(\lambda (H8: (clear (CHead c2 (Flat f) t0) (CHead c1 (Bind b) u))).((match i return (\lambda (_: ?).(\lambda (n: nat).((drop h (S (plus n d)) (CHead c0 k t) e) \to ((drop n O (CHead c0 k t) (CHead c2 (Flat f) t0)) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl n e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))))))) with [O \Rightarrow (\lambda (H9: (drop h (S (plus O d)) (CHead c0 k t) e)).(\lambda (H10: (drop O O (CHead c0 k t) (CHead c2 (Flat f) t0))).(let H11 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k t) (CHead c2 (Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead c2 (Flat f) t0) H10)) in ((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c0 k t) (CHead c2 (Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead c2 (Flat f) t0) H10)) in ((let H13 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow t | (CHead _ _ t) \Rightarrow t])) (CHead c0 k t) (CHead c2 (Flat f) t0) (drop_gen_refl (CHead c0 k t) (CHead c2 (Flat f) t0) H10)) in (\lambda (H14: (eq K k (Flat f))).(\lambda (H15: (eq C c0 c2)).(let H16 \def (eq_ind_r C c2 (\lambda (c: C).(clear c (CHead c1 (Bind b) u))) (clear_gen_flat f c2 (CHead c1 (Bind b) u) t0 H8) c0 H15) in (let H17 \def (eq_ind K k (\lambda (k: K).(drop h (S (plus O d)) (CHead c0 k t) e)) H9 (Flat f) H14) in (ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Flat f) (plus O d)) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Flat f) (plus O d)) c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H18: (eq C e (CHead x0 (Flat f) x1))).(\lambda (H19: (eq T t (lift h (r (Flat f) (plus O d)) x1))).(\lambda (H20: (drop h (r (Flat f) (plus O d)) c0 x0)).(let H21 \def (f_equal T T (\lambda (e0: T).e0) t (lift h (r (Flat f) (plus O d)) x1) H19) in (eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c3: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O c3 (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H22 \def (H c1 u O (getl_intro O c0 (CHead c1 (Bind b) u) c0 (drop_refl c0) H16) x0 h d H20) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O x0 (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x0 (Flat f) x1) (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H23: (eq T u (lift h d x2))).(\lambda (H24: (getl O x0 (CHead x3 (Bind b) x2))).(\lambda (H25: (drop h d c1 x3)).(let H26 \def (eq_ind T u (\lambda (t: T).(clear c0 (CHead c1 (Bind b) t))) H16 (lift h d x2) H23) in (eq_ind_r T (lift h d x2) (\lambda (t1: T).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x0 (Flat f) x1) (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T (lift h d x2) (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl O (CHead x0 (Flat f) x1) (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x2 x3 (refl_equal T (lift h d x2)) (getl_flat x0 (CHead x3 (Bind b) x2) O H24 f x1) H25) u H23))))))) H22)) e H18))))))) (drop_gen_skip_l c0 e t h (plus O d) (Flat f) H17))))))) H12)) H11)))) | (S n) \Rightarrow (\lambda (H9: (drop h (S (plus (S n) d)) (CHead c0 k t) e)).(\lambda (H10: (drop (S n) O (CHead c0 k t) (CHead c2 (Flat f) t0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k (plus (S n) d)) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus (S n) d)) c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S n) e (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H11: (eq C e (CHead x0 k x1))).(\lambda (H12: (eq T t (lift h (r k (plus (S n) d)) x1))).(\lambda (H13: (drop h (r k (plus (S n) d)) c0 x0)).(let H14 \def (f_equal T T (\lambda (e0: T).e0) t (lift h (r k (plus (S n) d)) x1) H12) in (eq_ind_r C (CHead x0 k x1) (\lambda (c3: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S n) c3 (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (let H15 \def (eq_ind nat (r k (plus (S n) d)) (\lambda (n: nat).(drop h n c0 x0)) H13 (plus (r k (S n)) d) (r_plus k (S n) d)) in (let H16 \def (eq_ind nat (r k (S n)) (\lambda (n: nat).(drop h (plus n d) c0 x0)) H15 (S (r k n)) (r_S k n)) in (let H17 \def (H c1 u (r k n) (getl_intro (r k n) c0 (CHead c1 (Bind b) u) (CHead c2 (Flat f) t0) (drop_gen_drop k c0 (CHead c2 (Flat f) t0) t n H10) (clear_flat c2 (CHead c1 (Bind b) u) (clear_gen_flat f c2 (CHead c1 (Bind b) u) t0 H8) f t0)) x0 h d H16) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (r k n) x0 (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S n) (CHead x0 k x1) (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0)))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H18: (eq T u (lift h d x2))).(\lambda (H19: (getl (r k n) x0 (CHead x3 (Bind b) x2))).(\lambda (H20: (drop h d c1 x3)).(let H21 \def (eq_ind T u (\lambda (t: T).(clear c2 (CHead c1 (Bind b) t))) (clear_gen_flat f c2 (CHead c1 (Bind b) u) t0 H8) (lift h d x2) H18) in (eq_ind_r T (lift h d x2) (\lambda (t1: T).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T t1 (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S n) (CHead x0 k x1) (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))))) (ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T (lift h d x2) (lift h d v)))) (\lambda (v: T).(\lambda (e0: C).(getl (S n) (CHead x0 k x1) (CHead e0 (Bind b) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h d c1 e0))) x2 x3 (refl_equal T (lift h d x2)) (getl_head k n x0 (CHead x3 (Bind b) x2) H19 x1) H20) u H18))))))) H17)))) e H11))))))) (drop_gen_skip_l c0 e t h (plus (S n) d) k H9))))]) H1 H7)))]) H5 H6)))]) H3 H4)))) H2)))))))))))))) c)).
 
 theorem getl_drop_conf_ge:
  \forall (i: nat).(\forall (a: C).(\forall (c: C).((getl i c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le (plus d h) i) \to (getl (minus i h) e a)))))))))
@@ -986,12 +986,12 @@ theorem getl_drop_trans:
 theorem getl_trans:
  \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((getl h c1 c2) \to (\forall (e2: C).((getl i c2 e2) \to (getl (plus i h) c1 e2)))))))
 \def
- \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H: (getl h c1 c2)).(\lambda (e2: C).(\lambda (H0: (getl i c2 e2)).(let H1 \def (getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) (\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2) (\lambda (x: C).(\lambda (H2: (drop i O c2 x)).(\lambda (H3: (clear x e2)).((match i return (\lambda (n: nat).((drop n O c2 x) \to (getl (plus n h) c1 e2))) with [O \Rightarrow (\lambda (H4: (drop O O c2 x)).(let H5 \def (eq_ind_r C x (\lambda (c: C).(clear c e2)) H3 c2 (drop_gen_refl c2 x H4)) in (getl_clear_trans (plus O h) c1 c2 H e2 H5))) | (S n) \Rightarrow (\lambda (H4: (drop (S n) O c2 x)).(let H_y \def (getl_drop_trans c1 c2 h H x n H4) in (getl_intro (plus (S n) h) c1 e2 x H_y H3)))]) H2)))) H1)))))))).
+ \lambda (i: nat).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (H: (getl h c1 c2)).(\lambda (e2: C).(\lambda (H0: (getl i c2 e2)).(let H1 \def (getl_gen_all c2 e2 i H0) in (ex2_ind C (\lambda (e: C).(drop i O c2 e)) (\lambda (e: C).(clear e e2)) (getl (plus i h) c1 e2) (\lambda (x: C).(\lambda (H2: (drop i O c2 x)).(\lambda (H3: (clear x e2)).((match i return (\lambda (_: ?).(\lambda (n: nat).((drop n O c2 x) \to (getl (plus n h) c1 e2)))) with [O \Rightarrow (\lambda (H4: (drop O O c2 x)).(let H5 \def (eq_ind_r C x (\lambda (c: C).(clear c e2)) H3 c2 (drop_gen_refl c2 x H4)) in (getl_clear_trans (plus O h) c1 c2 H e2 H5))) | (S n) \Rightarrow (\lambda (H4: (drop (S n) O c2 x)).(let H_y \def (getl_drop_trans c1 c2 h H x n H4) in (getl_intro (plus (S n) h) c1 e2 x H_y H3)))]) H2)))) H1)))))))).
 
 theorem drop1_getl_trans:
  \forall (hds: PList).(\forall (c1: C).(\forall (c2: C).((drop1 hds c2 c1) \to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to (ex C (\lambda (e2: C).(getl (trans hds i) c2 (CHead e2 (Bind b) (ctrans hds i v)))))))))))))
 \def
- \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c1: C).(\forall (c2: C).((drop1 p c2 c1) \to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to (ex C (\lambda (e2: C).(getl (trans p i) c2 (CHead e2 (Bind b) (ctrans p i v)))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c2 c1)).(\lambda (b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c1 (CHead e1 (Bind b) v))).(let H1 \def (match H return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 c1) \to (ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c c2)).(\lambda (H3: (eq C c c1)).(eq_ind C c2 (\lambda (c0: C).((eq C c0 c1) \to (ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v)))))) (\lambda (H4: (eq C c2 c1)).(eq_ind C c1 (\lambda (c0: C).(ex C (\lambda (e2: C).(getl i c0 (CHead e2 (Bind b) v))))) (ex_intro C (\lambda (e2: C).(getl i c1 (CHead e2 (Bind b) v))) e1 H0) c2 (sym_eq C c2 c1 H4))) c (sym_eq C c c2 H2) H3)))) | (drop1_cons c0 c3 h d H1 c4 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c0 c2)).(\lambda (H5: (eq C c4 c1)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e: PList).(match e return (\lambda (_: ?).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c0 c2) \to ((eq C c4 c1) \to ((drop h d c0 c3) \to ((drop1 hds c3 c4) \to (ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v)))))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c2) (refl_equal C c1))))))))))) (\lambda (h: nat).(\lambda (d: nat).(\lambda (hds0: PList).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((drop1 hds0 c2 c1) \to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to (ex C (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (ctrans hds0 i v))))))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H0: (drop1 (PCons h d hds0) c2 c1)).(\lambda (b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl i c1 (CHead e1 (Bind b) v))).(let H2 \def (match H0 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p (PCons h d hds0)) \to ((eq C c c2) \to ((eq C c0 c1) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)]))))))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil (PCons h d hds0))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c c1)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e return (\lambda (_: ?).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons h d hds0) H2) in (False_ind ((eq C c c2) \to ((eq C c c1) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)]))))))) H5)) H3 H4)))) | (drop1_cons c0 c3 h0 d0 H2 c4 hds0 H3) \Rightarrow (\lambda (H4: (eq PList (PCons h0 d0 hds0) (PCons h d hds0))).(\lambda (H5: (eq C c0 c2)).(\lambda (H6: (eq C c4 c1)).((let H7 \def (f_equal PList PList (\lambda (e: PList).(match e return (\lambda (_: ?).PList) with [PNil \Rightarrow hds0 | (PCons _ _ p) \Rightarrow p])) (PCons h0 d0 hds0) (PCons h d hds0) H4) in ((let H8 \def (f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_: ?).nat) with [PNil \Rightarrow d0 | (PCons _ n _) \Rightarrow n])) (PCons h0 d0 hds0) (PCons h d hds0) H4) in ((let H9 \def (f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_: ?).nat) with [PNil \Rightarrow h0 | (PCons n _ _) \Rightarrow n])) (PCons h0 d0 hds0) (PCons h d hds0) H4) in (eq_ind nat h (\lambda (n: nat).((eq nat d0 d) \to ((eq PList hds0 hds0) \to ((eq C c0 c2) \to ((eq C c4 c1) \to ((drop n d0 c0 c3) \to ((drop1 hds0 c3 c4) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)])))))))))))) (\lambda (H10: (eq nat d0 d)).(eq_ind nat d (\lambda (n: nat).((eq PList hds0 hds0) \to ((eq C c0 c2) \to ((eq C c4 c1) \to ((drop h n c0 c3) \to ((drop1 hds0 c3 c4) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)]))))))))))) (\lambda (H11: (eq PList hds0 hds0)).(eq_ind PList hds0 (\lambda (p: PList).((eq C c0 c2) \to ((eq C c4 c1) \to ((drop h d c0 c3) \to ((drop1 p c3 c4) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)])))))))))) (\lambda (H12: (eq C c0 c2)).(eq_ind C c2 (\lambda (c: C).((eq C c4 c1) \to ((drop h d c c3) \to ((drop1 hds0 c3 c4) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)]))))))))) (\lambda (H13: (eq C c4 c1)).(eq_ind C c1 (\lambda (c: C).((drop h d c2 c3) \to ((drop1 hds0 c3 c) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)])))))))) (\lambda (H14: (drop h d c2 c3)).(\lambda (H15: (drop1 hds0 c3 c1)).(xinduction bool (blt (trans hds0 i) d) (\lambda (b0: bool).(ex C (\lambda (e2: C).(getl (match b0 with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match b0 with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)])))))) (\lambda (x_x: bool).(bool_ind (\lambda (b0: bool).((eq bool (blt (trans hds0 i) d) b0) \to (ex C (\lambda (e2: C).(getl (match b0 with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match b0 with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)]))))))) (\lambda (H0: (eq bool (blt (trans hds0 i) d) true)).(let H_x \def (H c1 c3 H15 b e1 v i H1) in (let H16 \def H_x in (ex_ind C (\lambda (e2: C).(getl (trans hds0 i) c3 (CHead e2 (Bind b) (ctrans hds0 i v)))) (ex C (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)))))) (\lambda (x: C).(\lambda (H17: (getl (trans hds0 i) c3 (CHead x (Bind b) (ctrans hds0 i v)))).(let H_x0 \def (drop_getl_trans_lt (trans hds0 i) d (le_S_n (S (trans hds0 i)) d (lt_le_S (S (trans hds0 i)) (S d) (blt_lt (S d) (S (trans hds0 i)) H0))) c2 c3 h H14 b x (ctrans hds0 i v) H17) in (let H \def H_x0 in (ex2_ind C (\lambda (e1: C).(getl (trans hds0 i) c2 (CHead e1 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v))))) (\lambda (e1: C).(drop h (minus d (S (trans hds0 i))) e1 x)) (ex C (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)))))) (\lambda (x0: C).(\lambda (H1: (getl (trans hds0 i) c2 (CHead x0 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v))))).(\lambda (_: (drop h (minus d (S (trans hds0 i))) x0 x)).(ex_intro C (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v))))) x0 H1)))) H))))) H16)))) (\lambda (H0: (eq bool (blt (trans hds0 i) d) false)).(let H_x \def (H c1 c3 H15 b e1 v i H1) in (let H16 \def H_x in (ex_ind C (\lambda (e2: C).(getl (trans hds0 i) c3 (CHead e2 (Bind b) (ctrans hds0 i v)))) (ex C (\lambda (e2: C).(getl (plus (trans hds0 i) h) c2 (CHead e2 (Bind b) (ctrans hds0 i v))))) (\lambda (x: C).(\lambda (H17: (getl (trans hds0 i) c3 (CHead x (Bind b) (ctrans hds0 i v)))).(let H \def (drop_getl_trans_ge (trans hds0 i) c2 c3 d h H14 (CHead x (Bind b) (ctrans hds0 i v)) H17) in (ex_intro C (\lambda (e2: C).(getl (plus (trans hds0 i) h) c2 (CHead e2 (Bind b) (ctrans hds0 i v)))) x (H (bge_le d (trans hds0 i) H0)))))) H16)))) x_x))))) c4 (sym_eq C c4 c1 H13))) c0 (sym_eq C c0 c2 H12))) hds0 (sym_eq PList hds0 hds0 H11))) d0 (sym_eq nat d0 d H10))) h0 (sym_eq nat h0 h H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons h d hds0)) (refl_equal C c2) (refl_equal C c1))))))))))))))) hds).
+ \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c1: C).(\forall (c2: C).((drop1 p c2 c1) \to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to (ex C (\lambda (e2: C).(getl (trans p i) c2 (CHead e2 (Bind b) (ctrans p i v)))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c2 c1)).(\lambda (b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl i c1 (CHead e1 (Bind b) v))).(let H1 \def (match H return (\lambda (_: ?).(\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 c1) \to (ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v))))))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c c2)).(\lambda (H3: (eq C c c1)).(eq_ind C c2 (\lambda (c0: C).((eq C c0 c1) \to (ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v)))))) (\lambda (H4: (eq C c2 c1)).(eq_ind C c1 (\lambda (c0: C).(ex C (\lambda (e2: C).(getl i c0 (CHead e2 (Bind b) v))))) (ex_intro C (\lambda (e2: C).(getl i c1 (CHead e2 (Bind b) v))) e1 H0) c2 (sym_eq C c2 c1 H4))) c (sym_eq C c c2 H2) H3)))) | (drop1_cons c0 c3 h d H1 c4 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c0 c2)).(\lambda (H5: (eq C c4 c1)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e: PList).(match e return (\lambda (_: ?).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c0 c2) \to ((eq C c4 c1) \to ((drop h d c0 c3) \to ((drop1 hds c3 c4) \to (ex C (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v)))))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c2) (refl_equal C c1))))))))))) (\lambda (h: nat).(\lambda (d: nat).(\lambda (hds0: PList).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((drop1 hds0 c2 c1) \to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to (ex C (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (ctrans hds0 i v))))))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H0: (drop1 (PCons h d hds0) c2 c1)).(\lambda (b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H1: (getl i c1 (CHead e1 (Bind b) v))).(let H2 \def (match H0 return (\lambda (_: ?).(\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p (PCons h d hds0)) \to ((eq C c c2) \to ((eq C c0 c1) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)])))))))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil (PCons h d hds0))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c c1)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e return (\lambda (_: ?).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons h d hds0) H2) in (False_ind ((eq C c c2) \to ((eq C c c1) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)]))))))) H5)) H3 H4)))) | (drop1_cons c0 c3 h0 d0 H2 c4 hds0 H3) \Rightarrow (\lambda (H4: (eq PList (PCons h0 d0 hds0) (PCons h d hds0))).(\lambda (H5: (eq C c0 c2)).(\lambda (H6: (eq C c4 c1)).((let H7 \def (f_equal PList PList (\lambda (e: PList).(match e return (\lambda (_: ?).PList) with [PNil \Rightarrow hds0 | (PCons _ _ p) \Rightarrow p])) (PCons h0 d0 hds0) (PCons h d hds0) H4) in ((let H8 \def (f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_: ?).nat) with [PNil \Rightarrow d0 | (PCons _ n _) \Rightarrow n])) (PCons h0 d0 hds0) (PCons h d hds0) H4) in ((let H9 \def (f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_: ?).nat) with [PNil \Rightarrow h0 | (PCons n _ _) \Rightarrow n])) (PCons h0 d0 hds0) (PCons h d hds0) H4) in (eq_ind nat h (\lambda (n: nat).((eq nat d0 d) \to ((eq PList hds0 hds0) \to ((eq C c0 c2) \to ((eq C c4 c1) \to ((drop n d0 c0 c3) \to ((drop1 hds0 c3 c4) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)])))))))))))) (\lambda (H10: (eq nat d0 d)).(eq_ind nat d (\lambda (n: nat).((eq PList hds0 hds0) \to ((eq C c0 c2) \to ((eq C c4 c1) \to ((drop h n c0 c3) \to ((drop1 hds0 c3 c4) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)]))))))))))) (\lambda (H11: (eq PList hds0 hds0)).(eq_ind PList hds0 (\lambda (p: PList).((eq C c0 c2) \to ((eq C c4 c1) \to ((drop h d c0 c3) \to ((drop1 p c3 c4) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)])))))))))) (\lambda (H12: (eq C c0 c2)).(eq_ind C c2 (\lambda (c: C).((eq C c4 c1) \to ((drop h d c c3) \to ((drop1 hds0 c3 c4) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)]))))))))) (\lambda (H13: (eq C c4 c1)).(eq_ind C c1 (\lambda (c: C).((drop h d c2 c3) \to ((drop1 hds0 c3 c) \to (ex C (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match (blt (trans hds0 i) d) with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)])))))))) (\lambda (H14: (drop h d c2 c3)).(\lambda (H15: (drop1 hds0 c3 c1)).(xinduction bool (blt (trans hds0 i) d) (\lambda (b0: bool).(ex C (\lambda (e2: C).(getl (match b0 with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match b0 with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)])))))) (\lambda (x_x: bool).(bool_ind (\lambda (b0: bool).((eq bool (blt (trans hds0 i) d) b0) \to (ex C (\lambda (e2: C).(getl (match b0 with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (match b0 with [true \Rightarrow (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)) | false \Rightarrow (ctrans hds0 i v)]))))))) (\lambda (H0: (eq bool (blt (trans hds0 i) d) true)).(let H_x \def (H c1 c3 H15 b e1 v i H1) in (let H16 \def H_x in (ex_ind C (\lambda (e2: C).(getl (trans hds0 i) c3 (CHead e2 (Bind b) (ctrans hds0 i v)))) (ex C (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)))))) (\lambda (x: C).(\lambda (H17: (getl (trans hds0 i) c3 (CHead x (Bind b) (ctrans hds0 i v)))).(let H_x0 \def (drop_getl_trans_lt (trans hds0 i) d (le_S_n (S (trans hds0 i)) d (lt_le_S (S (trans hds0 i)) (S d) (blt_lt (S d) (S (trans hds0 i)) H0))) c2 c3 h H14 b x (ctrans hds0 i v) H17) in (let H \def H_x0 in (ex2_ind C (\lambda (e1: C).(getl (trans hds0 i) c2 (CHead e1 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v))))) (\lambda (e1: C).(drop h (minus d (S (trans hds0 i))) e1 x)) (ex C (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v)))))) (\lambda (x0: C).(\lambda (H1: (getl (trans hds0 i) c2 (CHead x0 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v))))).(\lambda (_: (drop h (minus d (S (trans hds0 i))) x0 x)).(ex_intro C (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans hds0 i))) (ctrans hds0 i v))))) x0 H1)))) H))))) H16)))) (\lambda (H0: (eq bool (blt (trans hds0 i) d) false)).(let H_x \def (H c1 c3 H15 b e1 v i H1) in (let H16 \def H_x in (ex_ind C (\lambda (e2: C).(getl (trans hds0 i) c3 (CHead e2 (Bind b) (ctrans hds0 i v)))) (ex C (\lambda (e2: C).(getl (plus (trans hds0 i) h) c2 (CHead e2 (Bind b) (ctrans hds0 i v))))) (\lambda (x: C).(\lambda (H17: (getl (trans hds0 i) c3 (CHead x (Bind b) (ctrans hds0 i v)))).(let H \def (drop_getl_trans_ge (trans hds0 i) c2 c3 d h H14 (CHead x (Bind b) (ctrans hds0 i v)) H17) in (ex_intro C (\lambda (e2: C).(getl (plus (trans hds0 i) h) c2 (CHead e2 (Bind b) (ctrans hds0 i v)))) x (H (bge_le d (trans hds0 i) H0)))))) H16)))) x_x))))) c4 (sym_eq C c4 c1 H13))) c0 (sym_eq C c0 c2 H12))) hds0 (sym_eq PList hds0 hds0 H11))) d0 (sym_eq nat d0 d H10))) h0 (sym_eq nat h0 h H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons h d hds0)) (refl_equal C c2) (refl_equal C c1))))))))))))))) hds).
 
 theorem cimp_getl_conf:
  \forall (c1: C).(\forall (c2: C).((cimp c1 c2) \to (\forall (b: B).(\forall (d1: C).(\forall (w: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind b) w)) \to (ex2 C (\lambda (d2: C).(cimp d1 d2)) (\lambda (d2: C).(getl i c2 (CHead d2 (Bind b) w)))))))))))
@@ -1007,17 +1007,17 @@ inductive subst0: nat \to (T \to (T \to (T \to Prop))) \def
 theorem subst0_gen_sort:
  \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 i v (TSort n) x) \to (\forall (P: Prop).P)))))
 \def
- \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda (H: (subst0 i v (TSort n) x)).(\lambda (P: Prop).(let H0 \def (match H return (\lambda (n0: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).((eq nat n0 i) \to ((eq T t v) \to ((eq T t0 (TSort n)) \to ((eq T t1 x) \to P)))))))) with [(subst0_lref v0 i0) \Rightarrow (\lambda (H0: (eq nat i0 i)).(\lambda (H1: (eq T v0 v)).(\lambda (H2: (eq T (TLRef i0) (TSort n))).(\lambda (H3: (eq T (lift (S i0) O v0) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (TLRef n0) (TSort n)) \to ((eq T (lift (S n0) O v0) x) \to P)))) (\lambda (H4: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T (TLRef i) (TSort n)) \to ((eq T (lift (S i) O t) x) \to P))) (\lambda (H5: (eq T (TLRef i) (TSort n))).(let H6 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) H5) in (False_ind ((eq T (lift (S i) O v) x) \to P) H6))) v0 (sym_eq T v0 v H4))) i0 (sym_eq nat i0 i H0) H1 H2 H3))))) | (subst0_fst v0 u2 u1 i0 H0 t k) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u1 t) (TSort n))).(\lambda (H4: (eq T (THead k u2 t) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (THead k u1 t) (TSort n)) \to ((eq T (THead k u2 t) x) \to ((subst0 n0 v0 u1 u2) \to P))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq T (THead k u1 t) (TSort n)) \to ((eq T (THead k u2 t) x) \to ((subst0 i t0 u1 u2) \to P)))) (\lambda (H6: (eq T (THead k u1 t) (TSort n))).(let H7 \def (eq_ind T (THead k u1 t) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H6) in (False_ind ((eq T (THead k u2 t) x) \to ((subst0 i v u1 u2) \to P)) H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_snd k v0 t2 t1 i0 H0 u) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u t1) (TSort n))).(\lambda (H4: (eq T (THead k u t2) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (THead k u t1) (TSort n)) \to ((eq T (THead k u t2) x) \to ((subst0 (s k n0) v0 t1 t2) \to P))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T (THead k u t1) (TSort n)) \to ((eq T (THead k u t2) x) \to ((subst0 (s k i) t t1 t2) \to P)))) (\lambda (H6: (eq T (THead k u t1) (TSort n))).(let H7 \def (eq_ind T (THead k u t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H6) in (False_ind ((eq T (THead k u t2) x) \to ((subst0 (s k i) v t1 t2) \to P)) H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_both v0 u1 u2 i0 H0 k t1 t2 H1) \Rightarrow (\lambda (H2: (eq nat i0 i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq T (THead k u1 t1) (TSort n))).(\lambda (H5: (eq T (THead k u2 t2) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (THead k u1 t1) (TSort n)) \to ((eq T (THead k u2 t2) x) \to ((subst0 n0 v0 u1 u2) \to ((subst0 (s k n0) v0 t1 t2) \to P)))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T (THead k u1 t1) (TSort n)) \to ((eq T (THead k u2 t2) x) \to ((subst0 i t u1 u2) \to ((subst0 (s k i) t t1 t2) \to P))))) (\lambda (H7: (eq T (THead k u1 t1) (TSort n))).(let H8 \def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H7) in (False_ind ((eq T (THead k u2 t2) x) \to ((subst0 i v u1 u2) \to ((subst0 (s k i) v t1 t2) \to P))) H8))) v0 (sym_eq T v0 v H6))) i0 (sym_eq nat i0 i H2) H3 H4 H5 H0 H1)))))]) in (H0 (refl_equal nat i) (refl_equal T v) (refl_equal T (TSort n)) (refl_equal T x)))))))).
+ \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda (H: (subst0 i v (TSort n) x)).(\lambda (P: Prop).(let H0 \def (match H return (\lambda (_: ?).(\lambda (n0: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).((eq nat n0 i) \to ((eq T t v) \to ((eq T t0 (TSort n)) \to ((eq T t1 x) \to P))))))))) with [(subst0_lref v0 i0) \Rightarrow (\lambda (H0: (eq nat i0 i)).(\lambda (H1: (eq T v0 v)).(\lambda (H2: (eq T (TLRef i0) (TSort n))).(\lambda (H3: (eq T (lift (S i0) O v0) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (TLRef n0) (TSort n)) \to ((eq T (lift (S n0) O v0) x) \to P)))) (\lambda (H4: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T (TLRef i) (TSort n)) \to ((eq T (lift (S i) O t) x) \to P))) (\lambda (H5: (eq T (TLRef i) (TSort n))).(let H6 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort n) H5) in (False_ind ((eq T (lift (S i) O v) x) \to P) H6))) v0 (sym_eq T v0 v H4))) i0 (sym_eq nat i0 i H0) H1 H2 H3))))) | (subst0_fst v0 u2 u1 i0 H0 t k) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u1 t) (TSort n))).(\lambda (H4: (eq T (THead k u2 t) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (THead k u1 t) (TSort n)) \to ((eq T (THead k u2 t) x) \to ((subst0 n0 v0 u1 u2) \to P))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq T (THead k u1 t) (TSort n)) \to ((eq T (THead k u2 t) x) \to ((subst0 i t0 u1 u2) \to P)))) (\lambda (H6: (eq T (THead k u1 t) (TSort n))).(let H7 \def (eq_ind T (THead k u1 t) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H6) in (False_ind ((eq T (THead k u2 t) x) \to ((subst0 i v u1 u2) \to P)) H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_snd k v0 t2 t1 i0 H0 u) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u t1) (TSort n))).(\lambda (H4: (eq T (THead k u t2) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (THead k u t1) (TSort n)) \to ((eq T (THead k u t2) x) \to ((subst0 (s k n0) v0 t1 t2) \to P))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T (THead k u t1) (TSort n)) \to ((eq T (THead k u t2) x) \to ((subst0 (s k i) t t1 t2) \to P)))) (\lambda (H6: (eq T (THead k u t1) (TSort n))).(let H7 \def (eq_ind T (THead k u t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H6) in (False_ind ((eq T (THead k u t2) x) \to ((subst0 (s k i) v t1 t2) \to P)) H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_both v0 u1 u2 i0 H0 k t1 t2 H1) \Rightarrow (\lambda (H2: (eq nat i0 i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq T (THead k u1 t1) (TSort n))).(\lambda (H5: (eq T (THead k u2 t2) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (THead k u1 t1) (TSort n)) \to ((eq T (THead k u2 t2) x) \to ((subst0 n0 v0 u1 u2) \to ((subst0 (s k n0) v0 t1 t2) \to P)))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T (THead k u1 t1) (TSort n)) \to ((eq T (THead k u2 t2) x) \to ((subst0 i t u1 u2) \to ((subst0 (s k i) t t1 t2) \to P))))) (\lambda (H7: (eq T (THead k u1 t1) (TSort n))).(let H8 \def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H7) in (False_ind ((eq T (THead k u2 t2) x) \to ((subst0 i v u1 u2) \to ((subst0 (s k i) v t1 t2) \to P))) H8))) v0 (sym_eq T v0 v H6))) i0 (sym_eq nat i0 i H2) H3 H4 H5 H0 H1)))))]) in (H0 (refl_equal nat i) (refl_equal T v) (refl_equal T (TSort n)) (refl_equal T x)))))))).
 
 theorem subst0_gen_lref:
  \forall (v: T).(\forall (x: T).(\forall (i: nat).(\forall (n: nat).((subst0 i v (TLRef n) x) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))))
 \def
- \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda (H: (subst0 i v (TLRef n) x)).(let H0 \def (match H return (\lambda (n0: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).((eq nat n0 i) \to ((eq T t v) \to ((eq T t0 (TLRef n)) \to ((eq T t1 x) \to (land (eq nat n i) (eq T x (lift (S n) O v))))))))))) with [(subst0_lref v0 i0) \Rightarrow (\lambda (H0: (eq nat i0 i)).(\lambda (H1: (eq T v0 v)).(\lambda (H2: (eq T (TLRef i0) (TLRef n))).(\lambda (H3: (eq T (lift (S i0) O v0) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (TLRef n0) (TLRef n)) \to ((eq T (lift (S n0) O v0) x) \to (land (eq nat n i) (eq T x (lift (S n) O v))))))) (\lambda (H4: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O t) x) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))) (\lambda (H5: (eq T (TLRef i) (TLRef n))).(let H6 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow i | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H5) in (eq_ind nat n (\lambda (n0: nat).((eq T (lift (S n0) O v) x) \to (land (eq nat n n0) (eq T x (lift (S n) O v))))) (\lambda (H7: (eq T (lift (S n) O v) x)).(eq_ind T (lift (S n) O v) (\lambda (t: T).(land (eq nat n n) (eq T t (lift (S n) O v)))) (conj (eq nat n n) (eq T (lift (S n) O v) (lift (S n) O v)) (refl_equal nat n) (refl_equal T (lift (S n) O v))) x H7)) i (sym_eq nat i n H6)))) v0 (sym_eq T v0 v H4))) i0 (sym_eq nat i0 i H0) H1 H2 H3))))) | (subst0_fst v0 u2 u1 i0 H0 t k) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u1 t) (TLRef n))).(\lambda (H4: (eq T (THead k u2 t) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (THead k u1 t) (TLRef n)) \to ((eq T (THead k u2 t) x) \to ((subst0 n0 v0 u1 u2) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq T (THead k u1 t) (TLRef n)) \to ((eq T (THead k u2 t) x) \to ((subst0 i t0 u1 u2) \to (land (eq nat n i) (eq T x (lift (S n) O v))))))) (\lambda (H6: (eq T (THead k u1 t) (TLRef n))).(let H7 \def (eq_ind T (THead k u1 t) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H6) in (False_ind ((eq T (THead k u2 t) x) \to ((subst0 i v u1 u2) \to (land (eq nat n i) (eq T x (lift (S n) O v))))) H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_snd k v0 t2 t1 i0 H0 u) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u t1) (TLRef n))).(\lambda (H4: (eq T (THead k u t2) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (THead k u t1) (TLRef n)) \to ((eq T (THead k u t2) x) \to ((subst0 (s k n0) v0 t1 t2) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T (THead k u t1) (TLRef n)) \to ((eq T (THead k u t2) x) \to ((subst0 (s k i) t t1 t2) \to (land (eq nat n i) (eq T x (lift (S n) O v))))))) (\lambda (H6: (eq T (THead k u t1) (TLRef n))).(let H7 \def (eq_ind T (THead k u t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H6) in (False_ind ((eq T (THead k u t2) x) \to ((subst0 (s k i) v t1 t2) \to (land (eq nat n i) (eq T x (lift (S n) O v))))) H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_both v0 u1 u2 i0 H0 k t1 t2 H1) \Rightarrow (\lambda (H2: (eq nat i0 i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq T (THead k u1 t1) (TLRef n))).(\lambda (H5: (eq T (THead k u2 t2) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (THead k u1 t1) (TLRef n)) \to ((eq T (THead k u2 t2) x) \to ((subst0 n0 v0 u1 u2) \to ((subst0 (s k n0) v0 t1 t2) \to (land (eq nat n i) (eq T x (lift (S n) O v))))))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T (THead k u1 t1) (TLRef n)) \to ((eq T (THead k u2 t2) x) \to ((subst0 i t u1 u2) \to ((subst0 (s k i) t t1 t2) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))))) (\lambda (H7: (eq T (THead k u1 t1) (TLRef n))).(let H8 \def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H7) in (False_ind ((eq T (THead k u2 t2) x) \to ((subst0 i v u1 u2) \to ((subst0 (s k i) v t1 t2) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))) H8))) v0 (sym_eq T v0 v H6))) i0 (sym_eq nat i0 i H2) H3 H4 H5 H0 H1)))))]) in (H0 (refl_equal nat i) (refl_equal T v) (refl_equal T (TLRef n)) (refl_equal T x))))))).
+ \lambda (v: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda (H: (subst0 i v (TLRef n) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (n0: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t1: T).((eq nat n0 i) \to ((eq T t v) \to ((eq T t0 (TLRef n)) \to ((eq T t1 x) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))))))))) with [(subst0_lref v0 i0) \Rightarrow (\lambda (H0: (eq nat i0 i)).(\lambda (H1: (eq T v0 v)).(\lambda (H2: (eq T (TLRef i0) (TLRef n))).(\lambda (H3: (eq T (lift (S i0) O v0) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (TLRef n0) (TLRef n)) \to ((eq T (lift (S n0) O v0) x) \to (land (eq nat n i) (eq T x (lift (S n) O v))))))) (\lambda (H4: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O t) x) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))) (\lambda (H5: (eq T (TLRef i) (TLRef n))).(let H6 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow i | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H5) in (eq_ind nat n (\lambda (n0: nat).((eq T (lift (S n0) O v) x) \to (land (eq nat n n0) (eq T x (lift (S n) O v))))) (\lambda (H7: (eq T (lift (S n) O v) x)).(eq_ind T (lift (S n) O v) (\lambda (t: T).(land (eq nat n n) (eq T t (lift (S n) O v)))) (conj (eq nat n n) (eq T (lift (S n) O v) (lift (S n) O v)) (refl_equal nat n) (refl_equal T (lift (S n) O v))) x H7)) i (sym_eq nat i n H6)))) v0 (sym_eq T v0 v H4))) i0 (sym_eq nat i0 i H0) H1 H2 H3))))) | (subst0_fst v0 u2 u1 i0 H0 t k) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u1 t) (TLRef n))).(\lambda (H4: (eq T (THead k u2 t) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (THead k u1 t) (TLRef n)) \to ((eq T (THead k u2 t) x) \to ((subst0 n0 v0 u1 u2) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq T (THead k u1 t) (TLRef n)) \to ((eq T (THead k u2 t) x) \to ((subst0 i t0 u1 u2) \to (land (eq nat n i) (eq T x (lift (S n) O v))))))) (\lambda (H6: (eq T (THead k u1 t) (TLRef n))).(let H7 \def (eq_ind T (THead k u1 t) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H6) in (False_ind ((eq T (THead k u2 t) x) \to ((subst0 i v u1 u2) \to (land (eq nat n i) (eq T x (lift (S n) O v))))) H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_snd k v0 t2 t1 i0 H0 u) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k u t1) (TLRef n))).(\lambda (H4: (eq T (THead k u t2) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (THead k u t1) (TLRef n)) \to ((eq T (THead k u t2) x) \to ((subst0 (s k n0) v0 t1 t2) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T (THead k u t1) (TLRef n)) \to ((eq T (THead k u t2) x) \to ((subst0 (s k i) t t1 t2) \to (land (eq nat n i) (eq T x (lift (S n) O v))))))) (\lambda (H6: (eq T (THead k u t1) (TLRef n))).(let H7 \def (eq_ind T (THead k u t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H6) in (False_ind ((eq T (THead k u t2) x) \to ((subst0 (s k i) v t1 t2) \to (land (eq nat n i) (eq T x (lift (S n) O v))))) H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_both v0 u1 u2 i0 H0 k t1 t2 H1) \Rightarrow (\lambda (H2: (eq nat i0 i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq T (THead k u1 t1) (TLRef n))).(\lambda (H5: (eq T (THead k u2 t2) x)).(eq_ind nat i (\lambda (n0: nat).((eq T v0 v) \to ((eq T (THead k u1 t1) (TLRef n)) \to ((eq T (THead k u2 t2) x) \to ((subst0 n0 v0 u1 u2) \to ((subst0 (s k n0) v0 t1 t2) \to (land (eq nat n i) (eq T x (lift (S n) O v))))))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T (THead k u1 t1) (TLRef n)) \to ((eq T (THead k u2 t2) x) \to ((subst0 i t u1 u2) \to ((subst0 (s k i) t t1 t2) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))))) (\lambda (H7: (eq T (THead k u1 t1) (TLRef n))).(let H8 \def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H7) in (False_ind ((eq T (THead k u2 t2) x) \to ((subst0 i v u1 u2) \to ((subst0 (s k i) v t1 t2) \to (land (eq nat n i) (eq T x (lift (S n) O v)))))) H8))) v0 (sym_eq T v0 v H6))) i0 (sym_eq nat i0 i H2) H3 H4 H5 H0 H1)))))]) in (H0 (refl_equal nat i) (refl_equal T v) (refl_equal T (TLRef n)) (refl_equal T x))))))).
 
 theorem subst0_gen_head:
  \forall (k: K).(\forall (v: T).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).(\forall (i: nat).((subst0 i v (THead k u1 t1) x) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))))))
 \def
- \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (H: (subst0 i v (THead k u1 t1) x)).(let H0 \def (match H return (\lambda (n: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t2: T).((eq nat n i) \to ((eq T t v) \to ((eq T t0 (THead k u1 t1)) \to ((eq T t2 x) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))))))))))))) with [(subst0_lref v0 i0) \Rightarrow (\lambda (H0: (eq nat i0 i)).(\lambda (H1: (eq T v0 v)).(\lambda (H2: (eq T (TLRef i0) (THead k u1 t1))).(\lambda (H3: (eq T (lift (S i0) O v0) x)).(eq_ind nat i (\lambda (n: nat).((eq T v0 v) \to ((eq T (TLRef n) (THead k u1 t1)) \to ((eq T (lift (S n) O v0) x) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))))) (\lambda (H4: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T (TLRef i) (THead k u1 t1)) \to ((eq T (lift (S i) O t) x) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))) (\lambda (H5: (eq T (TLRef i) (THead k u1 t1))).(let H6 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k u1 t1) H5) in (False_ind ((eq T (lift (S i) O v) x) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))) H6))) v0 (sym_eq T v0 v H4))) i0 (sym_eq nat i0 i H0) H1 H2 H3))))) | (subst0_fst v0 u2 u0 i0 H0 t k0) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k0 u0 t) (THead k u1 t1))).(\lambda (H4: (eq T (THead k0 u2 t) x)).(eq_ind nat i (\lambda (n: nat).((eq T v0 v) \to ((eq T (THead k0 u0 t) (THead k u1 t1)) \to ((eq T (THead k0 u2 t) x) \to ((subst0 n v0 u0 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T x (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq T (THead k0 u0 t) (THead k u1 t1)) \to ((eq T (THead k0 u2 t) x) \to ((subst0 i t0 u0 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T x (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))))) (\lambda (H6: (eq T (THead k0 u0 t) (THead k u1 t1))).(let H7 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t) \Rightarrow t])) (THead k0 u0 t) (THead k u1 t1) H6) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k0 u0 t) (THead k u1 t1) H6) in ((let H9 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k _ _) \Rightarrow k])) (THead k0 u0 t) (THead k u1 t1) H6) in (eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to ((eq T t t1) \to ((eq T (THead k1 u2 t) x) \to ((subst0 i v u0 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T x (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))))) (\lambda (H10: (eq T u0 u1)).(eq_ind T u1 (\lambda (t0: T).((eq T t t1) \to ((eq T (THead k u2 t) x) \to ((subst0 i v t0 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T x (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))))) (\lambda (H11: (eq T t t1)).(eq_ind T t1 (\lambda (t0: T).((eq T (THead k u2 t0) x) \to ((subst0 i v u1 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T x (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))) (\lambda (H12: (eq T (THead k u2 t1) x)).(eq_ind T (THead k u2 t1) (\lambda (t0: T).((subst0 i v u1 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T t0 (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T t0 (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t0 (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))) (\lambda (H13: (subst0 i v u1 u2)).(or3_intro0 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k u2 t1) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t1) (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))) (ex_intro2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3)) u2 (refl_equal T (THead k u2 t1)) H13))) x H12)) t (sym_eq T t t1 H11))) u0 (sym_eq T u0 u1 H10))) k0 (sym_eq K k0 k H9))) H8)) H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_snd k0 v0 t2 t0 i0 H0 u) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k0 u t0) (THead k u1 t1))).(\lambda (H4: (eq T (THead k0 u t2) x)).(eq_ind nat i (\lambda (n: nat).((eq T v0 v) \to ((eq T (THead k0 u t0) (THead k u1 t1)) \to ((eq T (THead k0 u t2) x) \to ((subst0 (s k0 n) v0 t0 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T (THead k0 u t0) (THead k u1 t1)) \to ((eq T (THead k0 u t2) x) \to ((subst0 (s k0 i) t t0 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))))))))) (\lambda (H6: (eq T (THead k0 u t0) (THead k u1 t1))).(let H7 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H6) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H6) in ((let H9 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k _ _) \Rightarrow k])) (THead k0 u t0) (THead k u1 t1) H6) in (eq_ind K k (\lambda (k1: K).((eq T u u1) \to ((eq T t0 t1) \to ((eq T (THead k1 u t2) x) \to ((subst0 (s k1 i) v t0 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))))))))) (\lambda (H10: (eq T u u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead k t t2) x) \to ((subst0 (s k i) v t0 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))))))))) (\lambda (H11: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead k u1 t2) x) \to ((subst0 (s k i) v t t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))))))) (\lambda (H12: (eq T (THead k u1 t2) x)).(eq_ind T (THead k u1 t2) (\lambda (t: T).((subst0 (s k i) v t1 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T t (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))))))) (\lambda (H13: (subst0 (s k i) v t1 t2)).(or3_intro1 (ex2 T (\lambda (u2: T).(eq T (THead k u1 t2) (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k u1 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead k u1 t2) (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))) (ex_intro2 T (\lambda (t3: T).(eq T (THead k u1 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3)) t2 (refl_equal T (THead k u1 t2)) H13))) x H12)) t0 (sym_eq T t0 t1 H11))) u (sym_eq T u u1 H10))) k0 (sym_eq K k0 k H9))) H8)) H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_both v0 u0 u2 i0 H0 k0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq nat i0 i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq T (THead k0 u0 t0) (THead k u1 t1))).(\lambda (H5: (eq T (THead k0 u2 t2) x)).(eq_ind nat i (\lambda (n: nat).((eq T v0 v) \to ((eq T (THead k0 u0 t0) (THead k u1 t1)) \to ((eq T (THead k0 u2 t2) x) \to ((subst0 n v0 u0 u2) \to ((subst0 (s k0 n) v0 t0 t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))))))))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T (THead k0 u0 t0) (THead k u1 t1)) \to ((eq T (THead k0 u2 t2) x) \to ((subst0 i t u0 u2) \to ((subst0 (s k0 i) t t0 t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))))))))) (\lambda (H7: (eq T (THead k0 u0 t0) (THead k u1 t1))).(let H8 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H7) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H7) in ((let H10 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k _ _) \Rightarrow k])) (THead k0 u0 t0) (THead k u1 t1) H7) in (eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k1 u2 t2) x) \to ((subst0 i v u0 u2) \to ((subst0 (s k1 i) v t0 t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))))))))))) (\lambda (H11: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead k u2 t2) x) \to ((subst0 i v t u2) \to ((subst0 (s k i) v t0 t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))))))))) (\lambda (H12: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead k u2 t2) x) \to ((subst0 i v u1 u2) \to ((subst0 (s k i) v t t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))))))))) (\lambda (H13: (eq T (THead k u2 t2) x)).(eq_ind T (THead k u2 t2) (\lambda (t: T).((subst0 i v u1 u2) \to ((subst0 (s k i) v t1 t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T t (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T t (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))))))) (\lambda (H14: (subst0 i v u1 u2)).(\lambda (H15: (subst0 (s k i) v t1 t2)).(or3_intro2 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T (THead k u2 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))) u2 t2 (refl_equal T (THead k u2 t2)) H14 H15)))) x H13)) t0 (sym_eq T t0 t1 H12))) u0 (sym_eq T u0 u1 H11))) k0 (sym_eq K k0 k H10))) H9)) H8))) v0 (sym_eq T v0 v H6))) i0 (sym_eq nat i0 i H2) H3 H4 H5 H0 H1)))))]) in (H0 (refl_equal nat i) (refl_equal T v) (refl_equal T (THead k u1 t1)) (refl_equal T x))))))))).
+ \lambda (k: K).(\lambda (v: T).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (i: nat).(\lambda (H: (subst0 i v (THead k u1 t1) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (n: nat).(\lambda (t: T).(\lambda (t0: T).(\lambda (t2: T).((eq nat n i) \to ((eq T t v) \to ((eq T t0 (THead k u1 t1)) \to ((eq T t2 x) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))))))))))))) with [(subst0_lref v0 i0) \Rightarrow (\lambda (H0: (eq nat i0 i)).(\lambda (H1: (eq T v0 v)).(\lambda (H2: (eq T (TLRef i0) (THead k u1 t1))).(\lambda (H3: (eq T (lift (S i0) O v0) x)).(eq_ind nat i (\lambda (n: nat).((eq T v0 v) \to ((eq T (TLRef n) (THead k u1 t1)) \to ((eq T (lift (S n) O v0) x) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))))) (\lambda (H4: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T (TLRef i) (THead k u1 t1)) \to ((eq T (lift (S i) O t) x) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))) (\lambda (H5: (eq T (TLRef i) (THead k u1 t1))).(let H6 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k u1 t1) H5) in (False_ind ((eq T (lift (S i) O v) x) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))) H6))) v0 (sym_eq T v0 v H4))) i0 (sym_eq nat i0 i H0) H1 H2 H3))))) | (subst0_fst v0 u2 u0 i0 H0 t k0) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k0 u0 t) (THead k u1 t1))).(\lambda (H4: (eq T (THead k0 u2 t) x)).(eq_ind nat i (\lambda (n: nat).((eq T v0 v) \to ((eq T (THead k0 u0 t) (THead k u1 t1)) \to ((eq T (THead k0 u2 t) x) \to ((subst0 n v0 u0 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T x (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq T (THead k0 u0 t) (THead k u1 t1)) \to ((eq T (THead k0 u2 t) x) \to ((subst0 i t0 u0 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T x (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))))) (\lambda (H6: (eq T (THead k0 u0 t) (THead k u1 t1))).(let H7 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t) \Rightarrow t])) (THead k0 u0 t) (THead k u1 t1) H6) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k0 u0 t) (THead k u1 t1) H6) in ((let H9 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k _ _) \Rightarrow k])) (THead k0 u0 t) (THead k u1 t1) H6) in (eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to ((eq T t t1) \to ((eq T (THead k1 u2 t) x) \to ((subst0 i v u0 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T x (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))))) (\lambda (H10: (eq T u0 u1)).(eq_ind T u1 (\lambda (t0: T).((eq T t t1) \to ((eq T (THead k u2 t) x) \to ((subst0 i v t0 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T x (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))))) (\lambda (H11: (eq T t t1)).(eq_ind T t1 (\lambda (t0: T).((eq T (THead k u2 t0) x) \to ((subst0 i v u1 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T x (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T x (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))))))) (\lambda (H12: (eq T (THead k u2 t1) x)).(eq_ind T (THead k u2 t1) (\lambda (t0: T).((subst0 i v u1 u2) \to (or3 (ex2 T (\lambda (u3: T).(eq T t0 (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T t0 (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t0 (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2))))))) (\lambda (H13: (subst0 i v u1 u2)).(or3_intro0 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t2: T).(eq T (THead k u2 t1) (THead k u1 t2))) (\lambda (t2: T).(subst0 (s k i) v t1 t2))) (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k u2 t1) (THead k u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i) v t1 t2)))) (ex_intro2 T (\lambda (u3: T).(eq T (THead k u2 t1) (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3)) u2 (refl_equal T (THead k u2 t1)) H13))) x H12)) t (sym_eq T t t1 H11))) u0 (sym_eq T u0 u1 H10))) k0 (sym_eq K k0 k H9))) H8)) H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_snd k0 v0 t2 t0 i0 H0 u) \Rightarrow (\lambda (H1: (eq nat i0 i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq T (THead k0 u t0) (THead k u1 t1))).(\lambda (H4: (eq T (THead k0 u t2) x)).(eq_ind nat i (\lambda (n: nat).((eq T v0 v) \to ((eq T (THead k0 u t0) (THead k u1 t1)) \to ((eq T (THead k0 u t2) x) \to ((subst0 (s k0 n) v0 t0 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T (THead k0 u t0) (THead k u1 t1)) \to ((eq T (THead k0 u t2) x) \to ((subst0 (s k0 i) t t0 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))))))))) (\lambda (H6: (eq T (THead k0 u t0) (THead k u1 t1))).(let H7 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H6) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead k0 u t0) (THead k u1 t1) H6) in ((let H9 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k _ _) \Rightarrow k])) (THead k0 u t0) (THead k u1 t1) H6) in (eq_ind K k (\lambda (k1: K).((eq T u u1) \to ((eq T t0 t1) \to ((eq T (THead k1 u t2) x) \to ((subst0 (s k1 i) v t0 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))))))))) (\lambda (H10: (eq T u u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead k t t2) x) \to ((subst0 (s k i) v t0 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))))))))) (\lambda (H11: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead k u1 t2) x) \to ((subst0 (s k i) v t t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T x (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))))))) (\lambda (H12: (eq T (THead k u1 t2) x)).(eq_ind T (THead k u1 t2) (\lambda (t: T).((subst0 (s k i) v t1 t2) \to (or3 (ex2 T (\lambda (u2: T).(eq T t (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T t (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))))))) (\lambda (H13: (subst0 (s k i) v t1 t2)).(or3_intro1 (ex2 T (\lambda (u2: T).(eq T (THead k u1 t2) (THead k u2 t1))) (\lambda (u2: T).(subst0 i v u1 u2))) (ex2 T (\lambda (t3: T).(eq T (THead k u1 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead k u1 t2) (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i v u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))) (ex_intro2 T (\lambda (t3: T).(eq T (THead k u1 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3)) t2 (refl_equal T (THead k u1 t2)) H13))) x H12)) t0 (sym_eq T t0 t1 H11))) u (sym_eq T u u1 H10))) k0 (sym_eq K k0 k H9))) H8)) H7))) v0 (sym_eq T v0 v H5))) i0 (sym_eq nat i0 i H1) H2 H3 H4 H0))))) | (subst0_both v0 u0 u2 i0 H0 k0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq nat i0 i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq T (THead k0 u0 t0) (THead k u1 t1))).(\lambda (H5: (eq T (THead k0 u2 t2) x)).(eq_ind nat i (\lambda (n: nat).((eq T v0 v) \to ((eq T (THead k0 u0 t0) (THead k u1 t1)) \to ((eq T (THead k0 u2 t2) x) \to ((subst0 n v0 u0 u2) \to ((subst0 (s k0 n) v0 t0 t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))))))))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq T (THead k0 u0 t0) (THead k u1 t1)) \to ((eq T (THead k0 u2 t2) x) \to ((subst0 i t u0 u2) \to ((subst0 (s k0 i) t t0 t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))))))))) (\lambda (H7: (eq T (THead k0 u0 t0) (THead k u1 t1))).(let H8 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H7) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k0 u0 t0) (THead k u1 t1) H7) in ((let H10 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k _ _) \Rightarrow k])) (THead k0 u0 t0) (THead k u1 t1) H7) in (eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k1 u2 t2) x) \to ((subst0 i v u0 u2) \to ((subst0 (s k1 i) v t0 t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))))))))))) (\lambda (H11: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead k u2 t2) x) \to ((subst0 i v t u2) \to ((subst0 (s k i) v t0 t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))))))))) (\lambda (H12: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead k u2 t2) x) \to ((subst0 i v u1 u2) \to ((subst0 (s k i) v t t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T x (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T x (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))))))))) (\lambda (H13: (eq T (THead k u2 t2) x)).(eq_ind T (THead k u2 t2) (\lambda (t: T).((subst0 i v u1 u2) \to ((subst0 (s k i) v t1 t2) \to (or3 (ex2 T (\lambda (u3: T).(eq T t (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T t (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))))))) (\lambda (H14: (subst0 i v u1 u2)).(\lambda (H15: (subst0 (s k i) v t1 t2)).(or3_intro2 (ex2 T (\lambda (u3: T).(eq T (THead k u2 t2) (THead k u3 t1))) (\lambda (u3: T).(subst0 i v u1 u3))) (ex2 T (\lambda (t3: T).(eq T (THead k u2 t2) (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i) v t1 t3))) (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3)))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead k u2 t2) (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i) v t1 t3))) u2 t2 (refl_equal T (THead k u2 t2)) H14 H15)))) x H13)) t0 (sym_eq T t0 t1 H12))) u0 (sym_eq T u0 u1 H11))) k0 (sym_eq K k0 k H10))) H9)) H8))) v0 (sym_eq T v0 v H6))) i0 (sym_eq nat i0 i H2) H3 H4 H5 H0 H1)))))]) in (H0 (refl_equal nat i) (refl_equal T v) (refl_equal T (THead k u1 t1)) (refl_equal T x))))))))).
 
 theorem subst0_refl:
  \forall (u: T).(\forall (t: T).(\forall (d: nat).((subst0 d u t t) \to (\forall (P: Prop).P))))
@@ -1221,52 +1221,52 @@ theorem csubst0_both_bind:
 theorem csubst0_gen_sort:
  \forall (x: C).(\forall (v: T).(\forall (i: nat).(\forall (n: nat).((csubst0 i v (CSort n) x) \to (\forall (P: Prop).P)))))
 \def
- \lambda (x: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda (H: (csubst0 i v (CSort n) x)).(\lambda (P: Prop).(let H0 \def (match H return (\lambda (n0: nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 i) \to ((eq T t v) \to ((eq C c (CSort n)) \to ((eq C c0 x) \to P)))))))) with [(csubst0_snd k i0 v0 u1 u2 H0 c) \Rightarrow (\lambda (H1: (eq nat (s k i0) i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq C (CHead c k u1) (CSort n))).(\lambda (H4: (eq C (CHead c k u2) x)).(eq_ind nat (s k i0) (\lambda (_: nat).((eq T v0 v) \to ((eq C (CHead c k u1) (CSort n)) \to ((eq C (CHead c k u2) x) \to ((subst0 i0 v0 u1 u2) \to P))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq C (CHead c k u1) (CSort n)) \to ((eq C (CHead c k u2) x) \to ((subst0 i0 t u1 u2) \to P)))) (\lambda (H6: (eq C (CHead c k u1) (CSort n))).(let H7 \def (eq_ind C (CHead c k u1) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H6) in (False_ind ((eq C (CHead c k u2) x) \to ((subst0 i0 v u1 u2) \to P)) H7))) v0 (sym_eq T v0 v H5))) i H1 H2 H3 H4 H0))))) | (csubst0_fst k i0 c1 c2 v0 H0 u) \Rightarrow (\lambda (H1: (eq nat (s k i0) i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq C (CHead c1 k u) (CSort n))).(\lambda (H4: (eq C (CHead c2 k u) x)).(eq_ind nat (s k i0) (\lambda (_: nat).((eq T v0 v) \to ((eq C (CHead c1 k u) (CSort n)) \to ((eq C (CHead c2 k u) x) \to ((csubst0 i0 v0 c1 c2) \to P))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq C (CHead c1 k u) (CSort n)) \to ((eq C (CHead c2 k u) x) \to ((csubst0 i0 t c1 c2) \to P)))) (\lambda (H6: (eq C (CHead c1 k u) (CSort n))).(let H7 \def (eq_ind C (CHead c1 k u) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H6) in (False_ind ((eq C (CHead c2 k u) x) \to ((csubst0 i0 v c1 c2) \to P)) H7))) v0 (sym_eq T v0 v H5))) i H1 H2 H3 H4 H0))))) | (csubst0_both k i0 v0 u1 u2 H0 c1 c2 H1) \Rightarrow (\lambda (H2: (eq nat (s k i0) i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq C (CHead c1 k u1) (CSort n))).(\lambda (H5: (eq C (CHead c2 k u2) x)).(eq_ind nat (s k i0) (\lambda (_: nat).((eq T v0 v) \to ((eq C (CHead c1 k u1) (CSort n)) \to ((eq C (CHead c2 k u2) x) \to ((subst0 i0 v0 u1 u2) \to ((csubst0 i0 v0 c1 c2) \to P)))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq C (CHead c1 k u1) (CSort n)) \to ((eq C (CHead c2 k u2) x) \to ((subst0 i0 t u1 u2) \to ((csubst0 i0 t c1 c2) \to P))))) (\lambda (H7: (eq C (CHead c1 k u1) (CSort n))).(let H8 \def (eq_ind C (CHead c1 k u1) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H7) in (False_ind ((eq C (CHead c2 k u2) x) \to ((subst0 i0 v u1 u2) \to ((csubst0 i0 v c1 c2) \to P))) H8))) v0 (sym_eq T v0 v H6))) i H2 H3 H4 H5 H0 H1)))))]) in (H0 (refl_equal nat i) (refl_equal T v) (refl_equal C (CSort n)) (refl_equal C x)))))))).
+ \lambda (x: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (n: nat).(\lambda (H: (csubst0 i v (CSort n) x)).(\lambda (P: Prop).(let H0 \def (match H return (\lambda (_: ?).(\lambda (n0: nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 i) \to ((eq T t v) \to ((eq C c (CSort n)) \to ((eq C c0 x) \to P))))))))) with [(csubst0_snd k i0 v0 u1 u2 H0 c) \Rightarrow (\lambda (H1: (eq nat (s k i0) i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq C (CHead c k u1) (CSort n))).(\lambda (H4: (eq C (CHead c k u2) x)).(eq_ind nat (s k i0) (\lambda (_: nat).((eq T v0 v) \to ((eq C (CHead c k u1) (CSort n)) \to ((eq C (CHead c k u2) x) \to ((subst0 i0 v0 u1 u2) \to P))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq C (CHead c k u1) (CSort n)) \to ((eq C (CHead c k u2) x) \to ((subst0 i0 t u1 u2) \to P)))) (\lambda (H6: (eq C (CHead c k u1) (CSort n))).(let H7 \def (eq_ind C (CHead c k u1) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H6) in (False_ind ((eq C (CHead c k u2) x) \to ((subst0 i0 v u1 u2) \to P)) H7))) v0 (sym_eq T v0 v H5))) i H1 H2 H3 H4 H0))))) | (csubst0_fst k i0 c1 c2 v0 H0 u) \Rightarrow (\lambda (H1: (eq nat (s k i0) i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq C (CHead c1 k u) (CSort n))).(\lambda (H4: (eq C (CHead c2 k u) x)).(eq_ind nat (s k i0) (\lambda (_: nat).((eq T v0 v) \to ((eq C (CHead c1 k u) (CSort n)) \to ((eq C (CHead c2 k u) x) \to ((csubst0 i0 v0 c1 c2) \to P))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq C (CHead c1 k u) (CSort n)) \to ((eq C (CHead c2 k u) x) \to ((csubst0 i0 t c1 c2) \to P)))) (\lambda (H6: (eq C (CHead c1 k u) (CSort n))).(let H7 \def (eq_ind C (CHead c1 k u) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H6) in (False_ind ((eq C (CHead c2 k u) x) \to ((csubst0 i0 v c1 c2) \to P)) H7))) v0 (sym_eq T v0 v H5))) i H1 H2 H3 H4 H0))))) | (csubst0_both k i0 v0 u1 u2 H0 c1 c2 H1) \Rightarrow (\lambda (H2: (eq nat (s k i0) i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq C (CHead c1 k u1) (CSort n))).(\lambda (H5: (eq C (CHead c2 k u2) x)).(eq_ind nat (s k i0) (\lambda (_: nat).((eq T v0 v) \to ((eq C (CHead c1 k u1) (CSort n)) \to ((eq C (CHead c2 k u2) x) \to ((subst0 i0 v0 u1 u2) \to ((csubst0 i0 v0 c1 c2) \to P)))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq C (CHead c1 k u1) (CSort n)) \to ((eq C (CHead c2 k u2) x) \to ((subst0 i0 t u1 u2) \to ((csubst0 i0 t c1 c2) \to P))))) (\lambda (H7: (eq C (CHead c1 k u1) (CSort n))).(let H8 \def (eq_ind C (CHead c1 k u1) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H7) in (False_ind ((eq C (CHead c2 k u2) x) \to ((subst0 i0 v u1 u2) \to ((csubst0 i0 v c1 c2) \to P))) H8))) v0 (sym_eq T v0 v H6))) i H2 H3 H4 H5 H0 H1)))))]) in (H0 (refl_equal nat i) (refl_equal T v) (refl_equal C (CSort n)) (refl_equal C x)))))))).
 
 theorem csubst0_gen_head:
  \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall (v: T).(\forall (i: nat).((csubst0 i v (CHead c1 k u1) x) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2))))))))))))
 \def
- \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda (H: (csubst0 i v (CHead c1 k u1) x)).(let H0 \def (match H return (\lambda (n: nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).((eq nat n i) \to ((eq T t v) \to ((eq C c (CHead c1 k u1)) \to ((eq C c0 x) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))))))))))))) with [(csubst0_snd k0 i0 v0 u0 u2 H0 c) \Rightarrow (\lambda (H1: (eq nat (s k0 i0) i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq C (CHead c k0 u0) (CHead c1 k u1))).(\lambda (H4: (eq C (CHead c k0 u2) x)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c k0 u0) (CHead c1 k u1)) \to ((eq C (CHead c k0 u2) x) \to ((subst0 i0 v0 u0 u2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat n (s k j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2))))))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq C (CHead c k0 u0) (CHead c1 k u1)) \to ((eq C (CHead c k0 u2) x) \to ((subst0 i0 t u0 u2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))))))))) (\lambda (H6: (eq C (CHead c k0 u0) (CHead c1 k u1))).(let H7 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c k0 u0) (CHead c1 k u1) H6) in ((let H8 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c k0 u0) (CHead c1 k u1) H6) in ((let H9 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c | (CHead c _ _) \Rightarrow c])) (CHead c k0 u0) (CHead c1 k u1) H6) in (eq_ind C c1 (\lambda (c0: C).((eq K k0 k) \to ((eq T u0 u1) \to ((eq C (CHead c0 k0 u2) x) \to ((subst0 i0 v u0 u2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2))))))))))) (\lambda (H10: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to ((eq C (CHead c1 k1 u2) x) \to ((subst0 i0 v u0 u2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))))))))) (\lambda (H11: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c1 k u2) x) \to ((subst0 i0 v t u2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2))))))))) (\lambda (H12: (eq C (CHead c1 k u2) x)).(eq_ind C (CHead c1 k u2) (\lambda (c0: C).((subst0 i0 v u1 u2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C c0 (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))))))) (\lambda (H13: (subst0 i0 v u1 u2)).(let H \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i0) i)) H1 k H10) in (or3_intro0 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2))))) (ex3_2_intro T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3))) u2 i0 (refl_equal nat (s k i0)) (refl_equal C (CHead c1 k u2)) H13)))) x H12)) u0 (sym_eq T u0 u1 H11))) k0 (sym_eq K k0 k H10))) c (sym_eq C c c1 H9))) H8)) H7))) v0 (sym_eq T v0 v H5))) i H1 H2 H3 H4 H0))))) | (csubst0_fst k0 i0 c0 c2 v0 H0 u) \Rightarrow (\lambda (H1: (eq nat (s k0 i0) i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq C (CHead c0 k0 u) (CHead c1 k u1))).(\lambda (H4: (eq C (CHead c2 k0 u) x)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c0 k0 u) (CHead c1 k u1)) \to ((eq C (CHead c2 k0 u) x) \to ((csubst0 i0 v0 c0 c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat n (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq C (CHead c0 k0 u) (CHead c1 k u1)) \to ((eq C (CHead c2 k0 u) x) \to ((csubst0 i0 t c0 c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))))))) (\lambda (H6: (eq C (CHead c0 k0 u) (CHead c1 k u1))).(let H7 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u) (CHead c1 k u1) H6) in ((let H8 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c0 k0 u) (CHead c1 k u1) H6) in ((let H9 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u) (CHead c1 k u1) H6) in (eq_ind C c1 (\lambda (c: C).((eq K k0 k) \to ((eq T u u1) \to ((eq C (CHead c2 k0 u) x) \to ((csubst0 i0 v c c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))))))))) (\lambda (H10: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T u u1) \to ((eq C (CHead c2 k1 u) x) \to ((csubst0 i0 v c1 c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))))))) (\lambda (H11: (eq T u u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 k t) x) \to ((csubst0 i0 v c1 c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))))))) (\lambda (H12: (eq C (CHead c2 k u1) x)).(eq_ind C (CHead c2 k u1) (\lambda (c: C).((csubst0 i0 v c1 c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))))) (\lambda (H13: (csubst0 i0 v c1 c2)).(let H \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i0) i)) H1 k H10) in (or3_intro1 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))) (ex3_2_intro C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))) c2 i0 (refl_equal nat (s k i0)) (refl_equal C (CHead c2 k u1)) H13)))) x H12)) u (sym_eq T u u1 H11))) k0 (sym_eq K k0 k H10))) c0 (sym_eq C c0 c1 H9))) H8)) H7))) v0 (sym_eq T v0 v H5))) i H1 H2 H3 H4 H0))))) | (csubst0_both k0 i0 v0 u0 u2 H0 c0 c2 H1) \Rightarrow (\lambda (H2: (eq nat (s k0 i0) i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(\lambda (H5: (eq C (CHead c2 k0 u2) x)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c0 k0 u0) (CHead c1 k u1)) \to ((eq C (CHead c2 k0 u2) x) \to ((subst0 i0 v0 u0 u2) \to ((csubst0 i0 v0 c0 c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat n (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))))))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq C (CHead c0 k0 u0) (CHead c1 k u1)) \to ((eq C (CHead c2 k0 u2) x) \to ((subst0 i0 t u0 u2) \to ((csubst0 i0 t c0 c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))))))))) (\lambda (H7: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(let H8 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c1 k u1) H7) in ((let H9 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c0 k0 u0) (CHead c1 k u1) H7) in ((let H10 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k u1) H7) in (eq_ind C c1 (\lambda (c: C).((eq K k0 k) \to ((eq T u0 u1) \to ((eq C (CHead c2 k0 u2) x) \to ((subst0 i0 v u0 u2) \to ((csubst0 i0 v c c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))))))))) (\lambda (H11: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to ((eq C (CHead c2 k1 u2) x) \to ((subst0 i0 v u0 u2) \to ((csubst0 i0 v c1 c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))))))))) (\lambda (H12: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 k u2) x) \to ((subst0 i0 v t u2) \to ((csubst0 i0 v c1 c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))))))) (\lambda (H13: (eq C (CHead c2 k u2) x)).(eq_ind C (CHead c2 k u2) (\lambda (c: C).((subst0 i0 v u1 u2) \to ((csubst0 i0 v c1 c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C c (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))))))) (\lambda (H14: (subst0 i0 v u1 u2)).(\lambda (H15: (csubst0 i0 v c1 c2)).(let H \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i0) i)) H2 k H11) in (or3_intro2 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))) (ex4_3_intro T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) u2 c2 i0 (refl_equal nat (s k i0)) (refl_equal C (CHead c2 k u2)) H14 H15))))) x H13)) u0 (sym_eq T u0 u1 H12))) k0 (sym_eq K k0 k H11))) c0 (sym_eq C c0 c1 H10))) H9)) H8))) v0 (sym_eq T v0 v H6))) i H2 H3 H4 H5 H0 H1)))))]) in (H0 (refl_equal nat i) (refl_equal T v) (refl_equal C (CHead c1 k u1)) (refl_equal C x))))))))).
+ \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda (H: (csubst0 i v (CHead c1 k u1) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (n: nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).((eq nat n i) \to ((eq T t v) \to ((eq C c (CHead c1 k u1)) \to ((eq C c0 x) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2))))))))))))))) with [(csubst0_snd k0 i0 v0 u0 u2 H0 c) \Rightarrow (\lambda (H1: (eq nat (s k0 i0) i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq C (CHead c k0 u0) (CHead c1 k u1))).(\lambda (H4: (eq C (CHead c k0 u2) x)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c k0 u0) (CHead c1 k u1)) \to ((eq C (CHead c k0 u2) x) \to ((subst0 i0 v0 u0 u2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat n (s k j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2))))))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq C (CHead c k0 u0) (CHead c1 k u1)) \to ((eq C (CHead c k0 u2) x) \to ((subst0 i0 t u0 u2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))))))))) (\lambda (H6: (eq C (CHead c k0 u0) (CHead c1 k u1))).(let H7 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c k0 u0) (CHead c1 k u1) H6) in ((let H8 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c k0 u0) (CHead c1 k u1) H6) in ((let H9 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c | (CHead c _ _) \Rightarrow c])) (CHead c k0 u0) (CHead c1 k u1) H6) in (eq_ind C c1 (\lambda (c0: C).((eq K k0 k) \to ((eq T u0 u1) \to ((eq C (CHead c0 k0 u2) x) \to ((subst0 i0 v u0 u2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2))))))))))) (\lambda (H10: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to ((eq C (CHead c1 k1 u2) x) \to ((subst0 i0 v u0 u2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))))))))) (\lambda (H11: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c1 k u2) x) \to ((subst0 i0 v t u2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2))))))))) (\lambda (H12: (eq C (CHead c1 k u2) x)).(eq_ind C (CHead c1 k u2) (\lambda (c0: C).((subst0 i0 v u1 u2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C c0 (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C c0 (CHead c2 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))))))) (\lambda (H13: (subst0 i0 v u1 u2)).(let H \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i0) i)) H1 k H10) in (or3_intro0 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: T).(\lambda (c2: C).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c2 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2))))) (ex3_2_intro T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c1 k u2) (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3))) u2 i0 (refl_equal nat (s k i0)) (refl_equal C (CHead c1 k u2)) H13)))) x H12)) u0 (sym_eq T u0 u1 H11))) k0 (sym_eq K k0 k H10))) c (sym_eq C c c1 H9))) H8)) H7))) v0 (sym_eq T v0 v H5))) i H1 H2 H3 H4 H0))))) | (csubst0_fst k0 i0 c0 c2 v0 H0 u) \Rightarrow (\lambda (H1: (eq nat (s k0 i0) i)).(\lambda (H2: (eq T v0 v)).(\lambda (H3: (eq C (CHead c0 k0 u) (CHead c1 k u1))).(\lambda (H4: (eq C (CHead c2 k0 u) x)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c0 k0 u) (CHead c1 k u1)) \to ((eq C (CHead c2 k0 u) x) \to ((csubst0 i0 v0 c0 c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat n (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))))))))) (\lambda (H5: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq C (CHead c0 k0 u) (CHead c1 k u1)) \to ((eq C (CHead c2 k0 u) x) \to ((csubst0 i0 t c0 c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))))))) (\lambda (H6: (eq C (CHead c0 k0 u) (CHead c1 k u1))).(let H7 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u) (CHead c1 k u1) H6) in ((let H8 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c0 k0 u) (CHead c1 k u1) H6) in ((let H9 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u) (CHead c1 k u1) H6) in (eq_ind C c1 (\lambda (c: C).((eq K k0 k) \to ((eq T u u1) \to ((eq C (CHead c2 k0 u) x) \to ((csubst0 i0 v c c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))))))))) (\lambda (H10: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T u u1) \to ((eq C (CHead c2 k1 u) x) \to ((csubst0 i0 v c1 c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))))))) (\lambda (H11: (eq T u u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 k t) x) \to ((csubst0 i0 v c1 c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))))))) (\lambda (H12: (eq C (CHead c2 k u1) x)).(eq_ind C (CHead c2 k u1) (\lambda (c: C).((csubst0 i0 v c1 c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))))) (\lambda (H13: (csubst0 i0 v c1 c2)).(let H \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i0) i)) H1 k H10) in (or3_intro1 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))) (ex3_2_intro C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u1) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))) c2 i0 (refl_equal nat (s k i0)) (refl_equal C (CHead c2 k u1)) H13)))) x H12)) u (sym_eq T u u1 H11))) k0 (sym_eq K k0 k H10))) c0 (sym_eq C c0 c1 H9))) H8)) H7))) v0 (sym_eq T v0 v H5))) i H1 H2 H3 H4 H0))))) | (csubst0_both k0 i0 v0 u0 u2 H0 c0 c2 H1) \Rightarrow (\lambda (H2: (eq nat (s k0 i0) i)).(\lambda (H3: (eq T v0 v)).(\lambda (H4: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(\lambda (H5: (eq C (CHead c2 k0 u2) x)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c0 k0 u0) (CHead c1 k u1)) \to ((eq C (CHead c2 k0 u2) x) \to ((subst0 i0 v0 u0 u2) \to ((csubst0 i0 v0 c0 c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat n (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat n (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))))))))) (\lambda (H6: (eq T v0 v)).(eq_ind T v (\lambda (t: T).((eq C (CHead c0 k0 u0) (CHead c1 k u1)) \to ((eq C (CHead c2 k0 u2) x) \to ((subst0 i0 t u0 u2) \to ((csubst0 i0 t c0 c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))))))))) (\lambda (H7: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(let H8 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c1 k u1) H7) in ((let H9 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c0 k0 u0) (CHead c1 k u1) H7) in ((let H10 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k u1) H7) in (eq_ind C c1 (\lambda (c: C).((eq K k0 k) \to ((eq T u0 u1) \to ((eq C (CHead c2 k0 u2) x) \to ((subst0 i0 v u0 u2) \to ((csubst0 i0 v c c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k0 i0) (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))))))))) (\lambda (H11: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to ((eq C (CHead c2 k1 u2) x) \to ((subst0 i0 v u0 u2) \to ((csubst0 i0 v c1 c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k1 i0) (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))))))))) (\lambda (H12: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 k u2) x) \to ((subst0 i0 v t u2) \to ((csubst0 i0 v c1 c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C x (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))))))))) (\lambda (H13: (eq C (CHead c2 k u2) x)).(eq_ind C (CHead c2 k u2) (\lambda (c: C).((subst0 i0 v u1 u2) \to ((csubst0 i0 v c1 c2) \to (or3 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C c (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))))))) (\lambda (H14: (subst0 i0 v u1 u2)).(\lambda (H15: (csubst0 i0 v c1 c2)).(let H \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i0) i)) H2 k H11) in (or3_intro2 (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (u3: T).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c1 k u3)))) (\lambda (u3: T).(\lambda (j: nat).(subst0 j v u1 u3)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))) (ex4_3_intro T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i0) (s k j))))) (\lambda (u3: T).(\lambda (c3: C).(\lambda (_: nat).(eq C (CHead c2 k u2) (CHead c3 k u3))))) (\lambda (u3: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u3)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) u2 c2 i0 (refl_equal nat (s k i0)) (refl_equal C (CHead c2 k u2)) H14 H15))))) x H13)) u0 (sym_eq T u0 u1 H12))) k0 (sym_eq K k0 k H11))) c0 (sym_eq C c0 c1 H10))) H9)) H8))) v0 (sym_eq T v0 v H6))) i H2 H3 H4 H5 H0 H1)))))]) in (H0 (refl_equal nat i) (refl_equal T v) (refl_equal C (CHead c1 k u1)) (refl_equal C x))))))))).
 
 theorem csubst0_drop_gt:
  \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O c1 e) \to (drop n O c2 e)))))))))
 \def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c1 e) \to (drop n0 O c2 e)))))))))) (\lambda (i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O O c1 e)).(let H2 \def (match H return (\lambda (n: nat).((eq nat n O) \to (drop O O c2 e))) with [le_n \Rightarrow (\lambda (H2: (eq nat (S i) O)).(let H3 \def (eq_ind nat (S i) (\lambda (e0: nat).(match e0 return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind (drop O O c2 e) H3))) | (le_S m H2) \Rightarrow (\lambda (H3: (eq nat (S m) O)).((let H4 \def (eq_ind nat (S m) (\lambda (e0: nat).(match e0 return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind ((le (S i) m) \to (drop O O c2 e)) H4)) H2))]) in (H2 (refl_equal nat O))))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c1 e) \to (drop n0 O c2 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v (CSort n1) c2)).(\lambda (e: C).(\lambda (H2: (drop (S n0) O (CSort n1) e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (drop (S n0) O c2 e) (\lambda (H3: (eq C e (CSort n1))).(\lambda (H4: (eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(drop (S n0) O c2 c)) (let H6 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (drop (S n0) O c2 (CSort n1)) H6)) e H3)))) (drop_gen_sort n1 (S n0) O e H2)))))))) (\lambda (c: C).(\lambda (H1: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H2: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda (H3: (drop (S n0) O (CHead c k t) e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (drop (S n0) O c2 e) (\lambda (H4: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (drop (S n0) O c2 e) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead c k x0))).(\lambda (_: (subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(drop (S n0) O c0 e)) (let H8 \def (eq_ind nat i (\lambda (n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) H1 (s k x1) H5) in (let H9 \def (eq_ind nat i (\lambda (n: nat).(lt n (S n0))) H0 (s k x1) H5) in ((match k return (\lambda (k0: K).((drop (r k0 n0) O c e) \to (((\forall (c2: C).(\forall (v: T).((csubst0 (s k0 x1) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead c k0 x0) e))))) with [(Bind b) \Rightarrow (\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 c e H10 x0)))) | (Flat f) \Rightarrow (\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat f) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead c (Flat f) x0) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 c e H10 x0)) (\lambda (H13: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead c (Flat f) x0) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e H10 x0)))) H13)) (lt_gen_xS x1 n0 H12)))))]) (drop_gen_drop k c e t n0 H3) H8 H9))) c2 H6)))))) H4)) (\lambda (H4: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S n0) O c2 e) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead x0 k t))).(\lambda (H7: (csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c0: C).(drop (S n0) O c0 e)) (let H8 \def (eq_ind nat i (\lambda (n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) H1 (s k x1) H5) in (let H9 \def (eq_ind nat i (\lambda (n: nat).(lt n (S n0))) H0 (s k x1) H5) in ((match k return (\lambda (k0: K).((drop (r k0 n0) O c e) \to (((\forall (c2: C).(\forall (v: T).((csubst0 (s k0 x1) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead x0 k0 t) e))))) with [(Bind b) \Rightarrow (\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (H12: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 x0 e (H x1 (lt_S_n x1 n0 H12) c x0 v H7 e H10) t)))) | (Flat f) \Rightarrow (\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (H11: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat f) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead x0 (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 x0 e (H11 x0 v H7 e H10) t)) (\lambda (H13: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead x0 (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 x0 e (H11 x0 v H7 e H10) t)))) H13)) (lt_gen_xS x1 n0 H12)))))]) (drop_gen_drop k c e t n0 H3) H8 H9))) c2 H6)))))) H4)) (\lambda (H4: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O c2 e) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H5: (eq nat i (s k x2))).(\lambda (H6: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H8: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c0: C).(drop (S n0) O c0 e)) (let H9 \def (eq_ind nat i (\lambda (n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) H1 (s k x2) H5) in (let H10 \def (eq_ind nat i (\lambda (n: nat).(lt n (S n0))) H0 (s k x2) H5) in ((match k return (\lambda (k0: K).((drop (r k0 n0) O c e) \to (((\forall (c2: C).(\forall (v: T).((csubst0 (s k0 x2) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) \to ((lt (s k0 x2) (S n0)) \to (drop (S n0) O (CHead x1 k0 x0) e))))) with [(Bind b) \Rightarrow (\lambda (H11: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) x2) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (H13: (lt (s (Bind b) x2) (S n0))).(drop_drop (Bind b) n0 x1 e (H x2 (lt_S_n x2 n0 H13) c x1 v H8 e H11) x0)))) | (Flat f) \Rightarrow (\lambda (H11: (drop (r (Flat f) n0) O c e)).(\lambda (H12: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat f) x2) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S n0))).(or_ind (eq nat x2 O) (ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (_: (eq nat x2 O)).(drop_drop (Flat f) n0 x1 e (H12 x1 v H8 e H11) x0)) (\lambda (H14: (ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (x: nat).(\lambda (_: (eq nat x2 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 x1 e (H12 x1 v H8 e H11) x0)))) H14)) (lt_gen_xS x2 n0 H13)))))]) (drop_gen_drop k c e t n0 H3) H9 H10))) c2 H6)))))))) H4)) (csubst0_gen_head k c c2 t v i H2))))))))))) c1)))))) n).
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c1 e) \to (drop n0 O c2 e)))))))))) (\lambda (i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O O c1 e)).(let H2 \def (match H return (\lambda (_: ?).(\lambda (n: nat).((eq nat n O) \to (drop O O c2 e)))) with [le_n \Rightarrow (\lambda (H2: (eq nat (S i) O)).(let H3 \def (eq_ind nat (S i) (\lambda (e0: nat).(match e0 return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind (drop O O c2 e) H3))) | (le_S m H2) \Rightarrow (\lambda (H3: (eq nat (S m) O)).((let H4 \def (eq_ind nat (S m) (\lambda (e0: nat).(match e0 return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind ((le (S i) m) \to (drop O O c2 e)) H4)) H2))]) in (H2 (refl_equal nat O))))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c1 e) \to (drop n0 O c2 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v (CSort n1) c2)).(\lambda (e: C).(\lambda (H2: (drop (S n0) O (CSort n1) e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (drop (S n0) O c2 e) (\lambda (H3: (eq C e (CSort n1))).(\lambda (H4: (eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(drop (S n0) O c2 c)) (let H6 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (drop (S n0) O c2 (CSort n1)) H6)) e H3)))) (drop_gen_sort n1 (S n0) O e H2)))))))) (\lambda (c: C).(\lambda (H1: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H2: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda (H3: (drop (S n0) O (CHead c k t) e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (drop (S n0) O c2 e) (\lambda (H4: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (drop (S n0) O c2 e) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead c k x0))).(\lambda (_: (subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c0: C).(drop (S n0) O c0 e)) (let H8 \def (eq_ind nat i (\lambda (n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) H1 (s k x1) H5) in (let H9 \def (eq_ind nat i (\lambda (n: nat).(lt n (S n0))) H0 (s k x1) H5) in ((match k return (\lambda (_: ?).(\lambda (k0: K).((drop (r k0 n0) O c e) \to (((\forall (c2: C).(\forall (v: T).((csubst0 (s k0 x1) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead c k0 x0) e)))))) with [(Bind b) \Rightarrow (\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 c e H10 x0)))) | (Flat f) \Rightarrow (\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat f) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead c (Flat f) x0) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 c e H10 x0)) (\lambda (H13: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead c (Flat f) x0) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e H10 x0)))) H13)) (lt_gen_xS x1 n0 H12)))))]) (drop_gen_drop k c e t n0 H3) H8 H9))) c2 H6)))))) H4)) (\lambda (H4: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S n0) O c2 e) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead x0 k t))).(\lambda (H7: (csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c0: C).(drop (S n0) O c0 e)) (let H8 \def (eq_ind nat i (\lambda (n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) H1 (s k x1) H5) in (let H9 \def (eq_ind nat i (\lambda (n: nat).(lt n (S n0))) H0 (s k x1) H5) in ((match k return (\lambda (_: ?).(\lambda (k0: K).((drop (r k0 n0) O c e) \to (((\forall (c2: C).(\forall (v: T).((csubst0 (s k0 x1) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) \to ((lt (s k0 x1) (S n0)) \to (drop (S n0) O (CHead x0 k0 t) e)))))) with [(Bind b) \Rightarrow (\lambda (H10: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (H12: (lt (s (Bind b) x1) (S n0))).(drop_drop (Bind b) n0 x0 e (H x1 (lt_S_n x1 n0 H12) c x0 v H7 e H10) t)))) | (Flat f) \Rightarrow (\lambda (H10: (drop (r (Flat f) n0) O c e)).(\lambda (H11: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat f) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(or_ind (eq nat x1 O) (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead x0 (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 x0 e (H11 x0 v H7 e H10) t)) (\lambda (H13: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead x0 (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 x0 e (H11 x0 v H7 e H10) t)))) H13)) (lt_gen_xS x1 n0 H12)))))]) (drop_gen_drop k c e t n0 H3) H8 H9))) c2 H6)))))) H4)) (\lambda (H4: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O c2 e) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H5: (eq nat i (s k x2))).(\lambda (H6: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H8: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c0: C).(drop (S n0) O c0 e)) (let H9 \def (eq_ind nat i (\lambda (n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) H1 (s k x2) H5) in (let H10 \def (eq_ind nat i (\lambda (n: nat).(lt n (S n0))) H0 (s k x2) H5) in ((match k return (\lambda (_: ?).(\lambda (k0: K).((drop (r k0 n0) O c e) \to (((\forall (c2: C).(\forall (v: T).((csubst0 (s k0 x2) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e))))))) \to ((lt (s k0 x2) (S n0)) \to (drop (S n0) O (CHead x1 k0 x0) e)))))) with [(Bind b) \Rightarrow (\lambda (H11: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) x2) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (H13: (lt (s (Bind b) x2) (S n0))).(drop_drop (Bind b) n0 x1 e (H x2 (lt_S_n x2 n0 H13) c x1 v H8 e H11) x0)))) | (Flat f) \Rightarrow (\lambda (H11: (drop (r (Flat f) n0) O c e)).(\lambda (H12: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat f) x2) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (drop (S n0) O c2 e)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S n0))).(or_ind (eq nat x2 O) (ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (_: (eq nat x2 O)).(drop_drop (Flat f) n0 x1 e (H12 x1 v H8 e H11) x0)) (\lambda (H14: (ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead x1 (Flat f) x0) e) (\lambda (x: nat).(\lambda (_: (eq nat x2 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 x1 e (H12 x1 v H8 e H11) x0)))) H14)) (lt_gen_xS x2 n0 H13)))))]) (drop_gen_drop k c e t n0 H3) H9 H10))) c2 H6)))))))) H4)) (csubst0_gen_head k c c2 t v i H2))))))))))) c1)))))) n).
 
 theorem csubst0_drop_gt_back:
  \forall (n: nat).(\forall (i: nat).((lt i n) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O c2 e) \to (drop n O c1 e)))))))))
 \def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c2 e) \to (drop n0 O c1 e)))))))))) (\lambda (i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O O c2 e)).(let H2 \def (match H return (\lambda (n: nat).((eq nat n O) \to (drop O O c1 e))) with [le_n \Rightarrow (\lambda (H2: (eq nat (S i) O)).(let H3 \def (eq_ind nat (S i) (\lambda (e0: nat).(match e0 return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind (drop O O c1 e) H3))) | (le_S m H2) \Rightarrow (\lambda (H3: (eq nat (S m) O)).((let H4 \def (eq_ind nat (S m) (\lambda (e0: nat).(match e0 return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind ((le (S i) m) \to (drop O O c1 e)) H4)) H2))]) in (H2 (refl_equal nat O))))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c2 e) \to (drop n0 O c1 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H1: (csubst0 i v (CSort n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2 e)).(csubst0_gen_sort c2 v i n1 H1 (drop (S n0) O (CSort n1) e)))))))) (\lambda (c: C).(\lambda (H1: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H2: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda (H3: (drop (S n0) O c2 e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (drop (S n0) O (CHead c k t) e) (\lambda (H4: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (drop (S n0) O (CHead c k t) e) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead c k x0))).(\lambda (_: (subst0 x1 v t x0)).(let H8 \def (eq_ind C c2 (\lambda (c: C).(drop (S n0) O c e)) H3 (CHead c k x0) H6) in (let H9 \def (eq_ind nat i (\lambda (n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) H1 (s k x1) H5) in (let H10 \def (eq_ind nat i (\lambda (n: nat).(lt n (S n0))) H0 (s k x1) H5) in ((match k return (\lambda (k0: K).(((\forall (c2: C).(\forall (v: T).((csubst0 (s k0 x1) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) \to ((lt (s k0 x1) (S n0)) \to ((drop (r k0 n0) O c e) \to (drop (S n0) O (CHead c k0 t) e))))) with [(Bind b) \Rightarrow (\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(\lambda (H13: (drop (r (Bind b) n0) O c e)).(drop_drop (Bind b) n0 c e H13 t)))) | (Flat f) \Rightarrow (\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat f) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(\lambda (H13: (drop (r (Flat f) n0) O c e)).(or_ind (eq nat x1 O) (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 c e H13 t)) (\lambda (H14: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e H13 t)))) H14)) (lt_gen_xS x1 n0 H12)))))]) H9 H10 (drop_gen_drop k c e x0 n0 H8)))))))))) H4)) (\lambda (H4: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S n0) O (CHead c k t) e) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead x0 k t))).(\lambda (H7: (csubst0 x1 v c x0)).(let H8 \def (eq_ind C c2 (\lambda (c: C).(drop (S n0) O c e)) H3 (CHead x0 k t) H6) in (let H9 \def (eq_ind nat i (\lambda (n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) H1 (s k x1) H5) in (let H10 \def (eq_ind nat i (\lambda (n: nat).(lt n (S n0))) H0 (s k x1) H5) in ((match k return (\lambda (k0: K).(((\forall (c2: C).(\forall (v: T).((csubst0 (s k0 x1) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) \to ((lt (s k0 x1) (S n0)) \to ((drop (r k0 n0) O x0 e) \to (drop (S n0) O (CHead c k0 t) e))))) with [(Bind b) \Rightarrow (\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))).(\lambda (H12: (lt (s (Bind b) x1) (S n0))).(\lambda (H13: (drop (r (Bind b) n0) O x0 e)).(drop_drop (Bind b) n0 c e (H x1 (lt_S_n x1 n0 H12) c x0 v H7 e H13) t)))) | (Flat f) \Rightarrow (\lambda (H11: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat f) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(\lambda (H13: (drop (r (Flat f) n0) O x0 e)).(or_ind (eq nat x1 O) (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 c e (H11 x0 v H7 e H13) t)) (\lambda (H14: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H11 x0 v H7 e H13) t)))) H14)) (lt_gen_xS x1 n0 H12)))))]) H9 H10 (drop_gen_drop k x0 e t n0 H8)))))))))) H4)) (\lambda (H4: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O (CHead c k t) e) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H5: (eq nat i (s k x2))).(\lambda (H6: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H8: (csubst0 x2 v c x1)).(let H9 \def (eq_ind C c2 (\lambda (c: C).(drop (S n0) O c e)) H3 (CHead x1 k x0) H6) in (let H10 \def (eq_ind nat i (\lambda (n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) H1 (s k x2) H5) in (let H11 \def (eq_ind nat i (\lambda (n: nat).(lt n (S n0))) H0 (s k x2) H5) in ((match k return (\lambda (k0: K).(((\forall (c2: C).(\forall (v: T).((csubst0 (s k0 x2) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) \to ((lt (s k0 x2) (S n0)) \to ((drop (r k0 n0) O x1 e) \to (drop (S n0) O (CHead c k0 t) e))))) with [(Bind b) \Rightarrow (\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) x2) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))).(\lambda (H13: (lt (s (Bind b) x2) (S n0))).(\lambda (H14: (drop (r (Bind b) n0) O x1 e)).(drop_drop (Bind b) n0 c e (H x2 (lt_S_n x2 n0 H13) c x1 v H8 e H14) t)))) | (Flat f) \Rightarrow (\lambda (H12: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat f) x2) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S n0))).(\lambda (H14: (drop (r (Flat f) n0) O x1 e)).(or_ind (eq nat x2 O) (ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x2 O)).(drop_drop (Flat f) n0 c e (H12 x1 v H8 e H14) t)) (\lambda (H15: (ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x2 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H12 x1 v H8 e H14) t)))) H15)) (lt_gen_xS x2 n0 H13)))))]) H10 H11 (drop_gen_drop k x1 e x0 n0 H9)))))))))))) H4)) (csubst0_gen_head k c c2 t v i H2))))))))))) c1)))))) n).
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c2 e) \to (drop n0 O c1 e)))))))))) (\lambda (i: nat).(\lambda (H: (lt i O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (_: (drop O O c2 e)).(let H2 \def (match H return (\lambda (_: ?).(\lambda (n: nat).((eq nat n O) \to (drop O O c1 e)))) with [le_n \Rightarrow (\lambda (H2: (eq nat (S i) O)).(let H3 \def (eq_ind nat (S i) (\lambda (e0: nat).(match e0 return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind (drop O O c1 e) H3))) | (le_S m H2) \Rightarrow (\lambda (H3: (eq nat (S m) O)).((let H4 \def (eq_ind nat (S m) (\lambda (e0: nat).(match e0 return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind ((le (S i) m) \to (drop O O c1 e)) H4)) H2))]) in (H2 (refl_equal nat O))))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (i: nat).((lt i n0) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c2 e) \to (drop n0 O c1 e))))))))))).(\lambda (i: nat).(\lambda (H0: (lt i (S n0))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H1: (csubst0 i v (CSort n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2 e)).(csubst0_gen_sort c2 v i n1 H1 (drop (S n0) O (CSort n1) e)))))))) (\lambda (c: C).(\lambda (H1: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H2: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda (H3: (drop (S n0) O c2 e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (drop (S n0) O (CHead c k t) e) (\lambda (H4: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (drop (S n0) O (CHead c k t) e) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead c k x0))).(\lambda (_: (subst0 x1 v t x0)).(let H8 \def (eq_ind C c2 (\lambda (c: C).(drop (S n0) O c e)) H3 (CHead c k x0) H6) in (let H9 \def (eq_ind nat i (\lambda (n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) H1 (s k x1) H5) in (let H10 \def (eq_ind nat i (\lambda (n: nat).(lt n (S n0))) H0 (s k x1) H5) in ((match k return (\lambda (_: ?).(\lambda (k0: K).(((\forall (c2: C).(\forall (v: T).((csubst0 (s k0 x1) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) \to ((lt (s k0 x1) (S n0)) \to ((drop (r k0 n0) O c e) \to (drop (S n0) O (CHead c k0 t) e)))))) with [(Bind b) \Rightarrow (\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))).(\lambda (_: (lt (s (Bind b) x1) (S n0))).(\lambda (H13: (drop (r (Bind b) n0) O c e)).(drop_drop (Bind b) n0 c e H13 t)))) | (Flat f) \Rightarrow (\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat f) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(\lambda (H13: (drop (r (Flat f) n0) O c e)).(or_ind (eq nat x1 O) (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 c e H13 t)) (\lambda (H14: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e H13 t)))) H14)) (lt_gen_xS x1 n0 H12)))))]) H9 H10 (drop_gen_drop k c e x0 n0 H8)))))))))) H4)) (\lambda (H4: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (drop (S n0) O (CHead c k t) e) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H5: (eq nat i (s k x1))).(\lambda (H6: (eq C c2 (CHead x0 k t))).(\lambda (H7: (csubst0 x1 v c x0)).(let H8 \def (eq_ind C c2 (\lambda (c: C).(drop (S n0) O c e)) H3 (CHead x0 k t) H6) in (let H9 \def (eq_ind nat i (\lambda (n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) H1 (s k x1) H5) in (let H10 \def (eq_ind nat i (\lambda (n: nat).(lt n (S n0))) H0 (s k x1) H5) in ((match k return (\lambda (_: ?).(\lambda (k0: K).(((\forall (c2: C).(\forall (v: T).((csubst0 (s k0 x1) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) \to ((lt (s k0 x1) (S n0)) \to ((drop (r k0 n0) O x0 e) \to (drop (S n0) O (CHead c k0 t) e)))))) with [(Bind b) \Rightarrow (\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))).(\lambda (H12: (lt (s (Bind b) x1) (S n0))).(\lambda (H13: (drop (r (Bind b) n0) O x0 e)).(drop_drop (Bind b) n0 c e (H x1 (lt_S_n x1 n0 H12) c x0 v H7 e H13) t)))) | (Flat f) \Rightarrow (\lambda (H11: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat f) x1) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))).(\lambda (H12: (lt (s (Flat f) x1) (S n0))).(\lambda (H13: (drop (r (Flat f) n0) O x0 e)).(or_ind (eq nat x1 O) (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x1 O)).(drop_drop (Flat f) n0 c e (H11 x0 v H7 e H13) t)) (\lambda (H14: (ex2 nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x1 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x1 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H11 x0 v H7 e H13) t)))) H14)) (lt_gen_xS x1 n0 H12)))))]) H9 H10 (drop_gen_drop k x0 e t n0 H8)))))))))) H4)) (\lambda (H4: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat i (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (drop (S n0) O (CHead c k t) e) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H5: (eq nat i (s k x2))).(\lambda (H6: (eq C c2 (CHead x1 k x0))).(\lambda (_: (subst0 x2 v t x0)).(\lambda (H8: (csubst0 x2 v c x1)).(let H9 \def (eq_ind C c2 (\lambda (c: C).(drop (S n0) O c e)) H3 (CHead x1 k x0) H6) in (let H10 \def (eq_ind nat i (\lambda (n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) H1 (s k x2) H5) in (let H11 \def (eq_ind nat i (\lambda (n: nat).(lt n (S n0))) H0 (s k x2) H5) in ((match k return (\lambda (_: ?).(\lambda (k0: K).(((\forall (c2: C).(\forall (v: T).((csubst0 (s k0 x2) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e))))))) \to ((lt (s k0 x2) (S n0)) \to ((drop (r k0 n0) O x1 e) \to (drop (S n0) O (CHead c k0 t) e)))))) with [(Bind b) \Rightarrow (\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) x2) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))).(\lambda (H13: (lt (s (Bind b) x2) (S n0))).(\lambda (H14: (drop (r (Bind b) n0) O x1 e)).(drop_drop (Bind b) n0 c e (H x2 (lt_S_n x2 n0 H13) c x1 v H8 e H14) t)))) | (Flat f) \Rightarrow (\lambda (H12: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat f) x2) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (drop (S n0) O c e)))))))).(\lambda (H13: (lt (s (Flat f) x2) (S n0))).(\lambda (H14: (drop (r (Flat f) n0) O x1 e)).(or_ind (eq nat x2 O) (ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0))) (drop (S n0) O (CHead c (Flat f) t) e) (\lambda (_: (eq nat x2 O)).(drop_drop (Flat f) n0 c e (H12 x1 v H8 e H14) t)) (\lambda (H15: (ex2 nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)))).(ex2_ind nat (\lambda (m: nat).(eq nat x2 (S m))) (\lambda (m: nat).(lt m n0)) (drop (S n0) O (CHead c (Flat f) t) e) (\lambda (x: nat).(\lambda (_: (eq nat x2 (S x))).(\lambda (_: (lt x n0)).(drop_drop (Flat f) n0 c e (H12 x1 v H8 e H14) t)))) H15)) (lt_gen_xS x2 n0 H13)))))]) H10 H11 (drop_gen_drop k x1 e x0 n0 H9)))))))))))) H4)) (csubst0_gen_head k c c2 t v i H2))))))))))) c1)))))) n).
 
 theorem csubst0_drop_lt:
  \forall (n: nat).(\forall (i: nat).((lt n i) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n O c1 e) \to (or4 (drop n O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2))))))))))))))))
 \def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 e2))))))))))))))))) (\lambda (i: nat).(\lambda (_: (lt O i)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (H1: (drop O O c1 e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k O)) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k O)) v e1 e2))))))))) (csubst0_ind (\lambda (n0: nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).(or4 (drop O O c0 c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n0 (s k O)) t e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n0 (s k O)) t e1 e2)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(let H3 \def (eq_ind_r nat i0 (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) in (or4_intro1 (drop O O (CHead c k u2) (CHead c k u1)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k u1) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c k u2) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k u1) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w))))) k c u1 u2 (refl_equal C (CHead c k u1)) (drop_refl (CHead c k u2)) H3)))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2: (csubst0 i0 v0 c3 c4)).(\lambda (H3: (or4 (drop O O c4 c3) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k O)) v0 u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k O)) v0 u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k O)) v0 e1 e2))))))))).(\lambda (u: T).(let H4 \def (eq_ind_r nat i0 (\lambda (n: nat).(csubst0 n v0 c3 c4)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) in (let H5 \def (eq_ind_r nat i0 (\lambda (n: nat).(or4 (drop O O c4 c3) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k O)) v0 u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k O)) v0 u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n (s k O)) v0 e1 e2))))))))) H3 (minus (s k i0) (s k O)) (s_arith0 k i0)) in (or4_intro2 (drop O O (CHead c4 k u) (CHead c3 k u)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k u) (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0 u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k u) (CHead e1 k0 u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0 u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))) k c3 c4 u (refl_equal C (CHead c3 k u)) (drop_refl (CHead c4 k u)) H4)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i0 v0 c3 c4)).(\lambda (_: (or4 (drop O O c4 c3) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k O)) v0 u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k O)) v0 u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k O)) v0 e1 e2))))))))).(let H5 \def (eq_ind_r nat i0 (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) in (let H6 \def (eq_ind_r nat i0 (\lambda (n: nat).(csubst0 n v0 c3 c4)) H3 (minus (s k i0) (s k O)) (s_arith0 k i0)) in (or4_intro3 (drop O O (CHead c4 k u2) (CHead c3 k u1)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 k u1) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c4 k u2) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k u1) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k u1) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2)))))) k c3 c4 u1 u2 (refl_equal C (CHead c3 k u1)) (drop_refl (CHead c4 k u2)) H5 H6)))))))))))))) i v c1 c2 H0) e (drop_gen_refl c1 e H1)))))))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt n0 i) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 e2)))))))))))))))))).(\lambda (i: nat).(\lambda (H: (lt (S n0) i)).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))))))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v (CSort n1) c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CSort n1) e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))))) (\lambda (H2: (eq C e (CSort n1))).(\lambda (H3: (eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(or4 (drop (S n0) O c2 c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2))))))))) (let H5 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or4 (drop (S n0) O c2 (CSort n1)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort n1) (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))))) H5)) e H2)))) (drop_gen_sort n1 (S n0) O e H1)))))))) (\lambda (c: C).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H1: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda (H2: (drop (S n0) O (CHead c k t) e)).(let H3 \def (match H1 return (\lambda (n: nat).(\lambda (t0: T).(\lambda (c0: C).(\lambda (c1: C).((eq nat n i) \to ((eq T t0 v) \to ((eq C c0 (CHead c k t)) \to ((eq C c1 c2) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))))))))))))) with [(csubst0_snd k0 i0 v0 u1 u2 H3 c0) \Rightarrow (\lambda (H4: (eq nat (s k0 i0) i)).(\lambda (H5: (eq T v0 v)).(\lambda (H6: (eq C (CHead c0 k0 u1) (CHead c k t))).(\lambda (H7: (eq C (CHead c0 k0 u2) c2)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c0 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i0 v0 u1 u2) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2))))))))))))) (\lambda (H8: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq C (CHead c0 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i0 t0 u1 u2) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))))))))) (\lambda (H9: (eq C (CHead c0 k0 u1) (CHead c k t))).(let H10 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u1) (CHead c k t) H9) in ((let H11 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c0 k0 u1) (CHead c k t) H9) in ((let H12 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u1) (CHead c k t) H9) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to ((eq T u1 t) \to ((eq C (CHead c k0 u2) c2) \to ((subst0 i0 v u1 u2) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2))))))))))))) (\lambda (H13: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq T u1 t) \to ((eq C (CHead c k u2) c2) \to ((subst0 i0 v u1 u2) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda (H14: (eq T u1 t)).(eq_ind T t (\lambda (t: T).((eq C (CHead c k u2) c2) \to ((subst0 i0 v t u2) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2))))))))))) (\lambda (H15: (eq C (CHead c k u2) c2)).(eq_ind C (CHead c k u2) (\lambda (c: C).((subst0 i0 v t u2) \to (or4 (drop (S n0) O c e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c (CHead e0 k w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c (CHead e2 k u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c (CHead e2 k w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))) (\lambda (_: (subst0 i0 v t u2)).(let H1 \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i0) i)) H4 k H13) in (let H17 \def (eq_ind_r nat i (\lambda (n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))))))))))) H0 (s k i0) H1) in (let H18 \def (eq_ind_r nat i (\lambda (n: nat).(lt (S n0) n)) H (s k i0) H1) in (K_ind (\lambda (k: K).((drop (r k n0) O c e) \to (((\forall (c2: C).(\forall (v: T).((csubst0 (s k i0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 (S n0))) v e1 e2)))))))))))))) \to ((lt (S n0) (s k i0)) \to (or4 (drop (S n0) O (CHead c k u2) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k u2) (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c k u2) (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k u2) (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda (b: B).(\lambda (H2: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) i0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Bind b) i0))).(or4_intro0 (drop (S n0) O (CHead c (Bind b) u2) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))) (drop_drop (Bind b) n0 c e H2 u2)))))) (\lambda (f: F).(\lambda (H2: (drop (r (Flat f) n0) O c e)).(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat f) i0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) i0))).(or4_intro0 (drop (S n0) O (CHead c (Flat f) u2) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (drop_drop (Flat f) n0 c e H2 u2)))))) k (drop_gen_drop k c e t n0 H2) H17 H18))))) c2 H15)) u1 (sym_eq T u1 t H14))) k0 (sym_eq K k0 k H13))) c0 (sym_eq C c0 c H12))) H11)) H10))) v0 (sym_eq T v0 v H8))) i H4 H5 H6 H7 H3))))) | (csubst0_fst k0 i0 c1 c0 v0 H3 u) \Rightarrow (\lambda (H4: (eq nat (s k0 i0) i)).(\lambda (H5: (eq T v0 v)).(\lambda (H6: (eq C (CHead c1 k0 u) (CHead c k t))).(\lambda (H7: (eq C (CHead c0 k0 u) c2)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c1 k0 u) (CHead c k t)) \to ((eq C (CHead c0 k0 u) c2) \to ((csubst0 i0 v0 c1 c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2))))))))))))) (\lambda (H8: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq C (CHead c1 k0 u) (CHead c k t)) \to ((eq C (CHead c0 k0 u) c2) \to ((csubst0 i0 t0 c1 c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))))))))) (\lambda (H9: (eq C (CHead c1 k0 u) (CHead c k t))).(let H10 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k0 u) (CHead c k t) H9) in ((let H11 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c1 k0 u) (CHead c k t) H9) in ((let H12 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k0 u) (CHead c k t) H9) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to ((eq T u t) \to ((eq C (CHead c0 k0 u) c2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2))))))))))))) (\lambda (H13: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq T u t) \to ((eq C (CHead c0 k u) c2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k1 u0))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda (H14: (eq T u t)).(eq_ind T t (\lambda (t: T).((eq C (CHead c0 k t) c2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead e2 k u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2))))))))))) (\lambda (H15: (eq C (CHead c0 k t) c2)).(eq_ind C (CHead c0 k t) (\lambda (c2: C).((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead e2 k u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))) (\lambda (H16: (csubst0 i0 v c c0)).(let H1 \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i0) i)) H4 k H13) in (let H17 \def (eq_ind_r nat i (\lambda (n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))))))))))) H0 (s k i0) H1) in (let H18 \def (eq_ind_r nat i (\lambda (n: nat).(lt (S n0) n)) H (s k i0) H1) in (K_ind (\lambda (k: K).((drop (r k n0) O c e) \to (((\forall (c2: C).(\forall (v: T).((csubst0 (s k i0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 (S n0))) v e1 e2)))))))))))))) \to ((lt (S n0) (s k i0)) \to (or4 (drop (S n0) O (CHead c0 k t) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 k t) (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 k t) (CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k1 u0))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 k t) (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda (b: B).(\lambda (H2: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) i0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))))))))))).(\lambda (H: (lt (S n0) (s (Bind b) i0))).(let H19 \def (IHn i0 (le_S_n (S n0) i0 H) c c0 v H16 e H2) in (or4_ind (drop n0 O c0 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus i0 (s k n0)) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n0 O c0 (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k n0)) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus i0 (s k n0)) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k n0)) v e1 e2))))))) (or4 (drop (S n0) O (CHead c0 (Bind b) t) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (H20: (drop n0 O c0 e)).(or4_intro0 (drop (S n0) O (CHead c0 (Bind b) t) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))) (drop_drop (Bind b) n0 c0 e H20 t))) (\lambda (H20: (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k n0)) v u w))))))).(ex3_4_ind K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus i0 (s k n0)) v u0 w))))) (or4 (drop (S n0) O (CHead c0 (Bind b) t) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H21: (eq C e (CHead x1 x0 x2))).(\lambda (H22: (drop n0 O c0 (CHead x1 x0 x3))).(\lambda (H23: (subst0 (minus i0 (s x0 n0)) v x2 x3)).(eq_ind_r C (CHead x1 x0 x2) (\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Bind b) t) c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c0 (Bind b) t) (CHead x1 x0 x2)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 x0 x2) (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))) (ex3_4_intro K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w))))) x0 x1 x2 x3 (refl_equal C (CHead x1 x0 x2)) (drop_drop (Bind b) n0 c0 (CHead x1 x0 x3) H22 t) (eq_ind_r nat (S (s x0 n0)) (\lambda (n: nat).(subst0 (minus (s (Bind b) i0) n) v x2 x3)) H23 (s x0 (S n0)) (s_S x0 n0)))) e H21)))))))) H20)) (\lambda (H20: (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k n0)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n0 O c0 (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k n0)) v e1 e2))))) (or4 (drop (S n0) O (CHead c0 (Bind b) t) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H21: (eq C e (CHead x1 x0 x3))).(\lambda (H22: (drop n0 O c0 (CHead x2 x0 x3))).(\lambda (H23: (csubst0 (minus i0 (s x0 n0)) v x1 x2)).(eq_ind_r C (CHead x1 x0 x3) (\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Bind b) t) c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda 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(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c0 (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus i0 (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1 e2))))))) (or4 (drop (S n0) O (CHead c0 (Flat f) t) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda 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C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H21: (eq C e (CHead x1 x0 x2))).(\lambda (H22: (drop (S n0) O c0 (CHead x1 x0 x3))).(\lambda (H23: (subst0 (minus i0 (s x0 (S n0))) v x2 x3)).(eq_ind_r C (CHead x1 x0 x2) (\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Flat f) t) c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S 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T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c0 (Flat f) t) (CHead x1 x0 x2)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 x0 x2) (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (ex3_4_intro K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w))))) x0 x1 x2 x3 (refl_equal C (CHead x1 x0 x2)) (drop_drop (Flat f) n0 c0 (CHead x1 x0 x3) H22 t) H23)) e H21)))))))) H20)) (\lambda (H20: (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c0 (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1 e2))))) (or4 (drop (S n0) O (CHead c0 (Flat f) t) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H21: (eq C e (CHead x1 x0 x3))).(\lambda (H22: (drop (S n0) O c0 (CHead x2 x0 x3))).(\lambda (H23: (csubst0 (minus i0 (s x0 (S n0))) v x1 x2)).(eq_ind_r C (CHead x1 x0 x3) (\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Flat f) t) c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c0 (Flat f) t) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 x0 x3) (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 x0 x3) (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))) x0 x1 x2 x3 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Flat f) n0 c0 (CHead x2 x0 x3) H22 t) H23)) e H21)))))))) H20)) (\lambda (H20: (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus i0 (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1 e2)))))) (or4 (drop (S n0) O (CHead c0 (Flat f) t) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H21: (eq C e (CHead x1 x0 x3))).(\lambda (H22: (drop (S n0) O c0 (CHead x2 x0 x4))).(\lambda (H23: (subst0 (minus i0 (s x0 (S n0))) v x3 x4)).(\lambda (H24: (csubst0 (minus i0 (s x0 (S n0))) v x1 x2)).(eq_ind_r C (CHead x1 x0 x3) (\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Flat f) t) c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c0 (Flat f) t) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 x0 x3) (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (ex4_5_intro K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Flat f) n0 c0 (CHead x2 x0 x4) H22 t) H23 H24)) e H21)))))))))) H20)) H19)))))) k (drop_gen_drop k c e t n0 H2) H17 H18))))) c2 H15)) u (sym_eq T u t H14))) k0 (sym_eq K k0 k H13))) c1 (sym_eq C c1 c H12))) H11)) H10))) v0 (sym_eq T v0 v H8))) i H4 H5 H6 H7 H3))))) | (csubst0_both k0 i0 v0 u1 u2 H3 c1 c0 H4) \Rightarrow (\lambda (H5: (eq nat (s k0 i0) i)).(\lambda (H6: (eq T v0 v)).(\lambda (H7: (eq C (CHead c1 k0 u1) (CHead c k t))).(\lambda (H8: (eq C (CHead c0 k0 u2) c2)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c1 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i0 v0 u1 u2) \to ((csubst0 i0 v0 c1 c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))))))))))) (\lambda (H9: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq C (CHead c1 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i0 t0 u1 u2) \to ((csubst0 i0 t0 c1 c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2))))))))))))) (\lambda (H10: (eq C (CHead c1 k0 u1) (CHead c k t))).(let H11 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k0 u1) (CHead c k t) H10) in ((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c1 k0 u1) (CHead c k t) H10) in ((let H13 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k0 u1) (CHead c k t) H10) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to ((eq T u1 t) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i0 v u1 u2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))))))))))) (\lambda (H14: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq T u1 t) \to ((eq C (CHead c0 k u2) c2) \to ((subst0 i0 v u1 u2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2))))))))))))) (\lambda (H15: (eq T u1 t)).(eq_ind T t (\lambda (t: T).((eq C (CHead c0 k u2) c2) \to ((subst0 i0 v t u2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda (H16: (eq C (CHead c0 k u2) c2)).(eq_ind C (CHead c0 k u2) (\lambda (c2: C).((subst0 i0 v t u2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2))))))))))) (\lambda (_: (subst0 i0 v t u2)).(\lambda (H18: (csubst0 i0 v c c0)).(let H1 \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i0) i)) H5 k H14) in (let H19 \def (eq_ind_r nat i (\lambda (n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))))))))))) H0 (s k i0) H1) in (let H20 \def (eq_ind_r nat i (\lambda (n: nat).(lt (S n0) n)) H (s k i0) H1) in (K_ind (\lambda (k: K).((drop (r k n0) O c e) \to (((\forall (c2: C).(\forall (v: T).((csubst0 (s k i0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 (S n0))) v e1 e2)))))))))))))) \to ((lt (S n0) (s k i0)) \to (or4 (drop (S n0) O (CHead c0 k u2) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 k u2) (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 k u2) (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 k u2) (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda (b: B).(\lambda (H2: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) i0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: 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i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Bind b) n0 c0 (CHead x2 x0 x4) H24 u2) (eq_ind_r nat (S (s x0 n0)) (\lambda (n: nat).(subst0 (minus (s (Bind b) i0) n) v x3 x4)) H25 (s x0 (S n0)) (s_S x0 n0)) (eq_ind_r nat (S (s x0 n0)) (\lambda (n: nat).(csubst0 (minus (s (Bind b) i0) n) v x1 x2)) H26 (s x0 (S n0)) (s_S x0 n0)))) e H23)))))))))) H22)) H21)))))) (\lambda (f: F).(\lambda (H2: (drop (r (Flat f) n0) O c e)).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat f) i0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: 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(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (H22: (drop (S n0) O c0 e)).(or4_intro0 (drop (S n0) O (CHead c0 (Flat f) u2) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: 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(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H23: (eq C e (CHead x1 x0 x2))).(\lambda (H24: (drop (S n0) O c0 (CHead x1 x0 x3))).(\lambda (H25: (subst0 (minus i0 (s x0 (S n0))) v x2 x3)).(eq_ind_r C (CHead x1 x0 x2) (\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Flat f) u2) c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c0 (Flat f) u2) (CHead x1 x0 x2)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 x0 x2) (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (ex3_4_intro K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x2) (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w))))) x0 x1 x2 x3 (refl_equal C (CHead x1 x0 x2)) (drop_drop (Flat f) n0 c0 (CHead x1 x0 x3) H24 u2) H25)) e H23)))))))) H22)) (\lambda (H22: (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1 e2))))) (or4 (drop (S n0) O (CHead c0 (Flat f) u2) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H23: (eq C e (CHead x1 x0 x3))).(\lambda (H24: (drop (S n0) O c0 (CHead x2 x0 x3))).(\lambda (H25: (csubst0 (minus i0 (s x0 (S n0))) v x1 x2)).(eq_ind_r C (CHead x1 x0 x3) (\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Flat f) u2) c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c0 (Flat f) u2) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 x0 x3) (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 x0 x3) (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))) x0 x1 x2 x3 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Flat f) n0 c0 (CHead x2 x0 x3) H24 u2) H25)) e H23)))))))) H22)) (\lambda (H22: (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1 e2)))))) (or4 (drop (S n0) O (CHead c0 (Flat f) u2) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H23: (eq C e (CHead x1 x0 x3))).(\lambda (H24: (drop (S n0) O c0 (CHead x2 x0 x4))).(\lambda (H25: (subst0 (minus i0 (s x0 (S n0))) v x3 x4)).(\lambda (H26: (csubst0 (minus i0 (s x0 (S n0))) v x1 x2)).(eq_ind_r C (CHead x1 x0 x3) (\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Flat f) u2) c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c0 (Flat f) u2) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 x0 x3) (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (ex4_5_intro K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Flat f) n0 c0 (CHead x2 x0 x4) H24 u2) H25 H26)) e H23)))))))))) H22)) H21)))))) k (drop_gen_drop k c e t n0 H2) H19 H20)))))) c2 H16)) u1 (sym_eq T u1 t H15))) k0 (sym_eq K k0 k H14))) c1 (sym_eq C c1 c H13))) H12)) H11))) v0 (sym_eq T v0 v H9))) i H5 H6 H7 H8 H3 H4)))))]) in (H3 (refl_equal nat i) (refl_equal T v) (refl_equal C (CHead c k t)) (refl_equal C c2)))))))))))) c1)))))) n).
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (i: nat).((lt n0 i) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 e2))))))))))))))))) (\lambda (i: nat).(\lambda (_: (lt O i)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (H1: (drop O O c1 e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k O)) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k O)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k O)) v e1 e2))))))))) (csubst0_ind (\lambda (n0: nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).(or4 (drop O O c0 c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n0 (s k O)) t e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n0 (s k O)) t u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n0 (s k O)) t e1 e2)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c: C).(let H3 \def (eq_ind_r nat i0 (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) in (or4_intro1 (drop O O (CHead c k u2) (CHead c k u1)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k u1) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c k u2) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k u1) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k u2) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w))))) k c u1 u2 (refl_equal C (CHead c k u1)) (drop_refl (CHead c k u2)) H3)))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2: (csubst0 i0 v0 c3 c4)).(\lambda (H3: (or4 (drop O O c4 c3) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k O)) v0 u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k O)) v0 u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k O)) v0 e1 e2))))))))).(\lambda (u: T).(let H4 \def (eq_ind_r nat i0 (\lambda (n: nat).(csubst0 n v0 c3 c4)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) in (let H5 \def (eq_ind_r nat i0 (\lambda (n: nat).(or4 (drop O O c4 c3) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k O)) v0 u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k O)) v0 u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n (s k O)) v0 e1 e2))))))))) H3 (minus (s k i0) (s k O)) (s_arith0 k i0)) in (or4_intro2 (drop O O (CHead c4 k u) (CHead c3 k u)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k u) (CHead e0 k0 u0)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u0 w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0 u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k u) (CHead e1 k0 u0))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u0 w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex3_4_intro K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k u) (CHead e1 k0 u0)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 k u) (CHead e2 k0 u0)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))) k c3 c4 u (refl_equal C (CHead c3 k u)) (drop_refl (CHead c4 k u)) H4)))))))))))) (\lambda (k: K).(\lambda (i0: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i0 v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i0 v0 c3 c4)).(\lambda (_: (or4 (drop O O c4 c3) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k O)) v0 u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k O)) v0 u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k O)) v0 e1 e2))))))))).(let H5 \def (eq_ind_r nat i0 (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 (minus (s k i0) (s k O)) (s_arith0 k i0)) in (let H6 \def (eq_ind_r nat i0 (\lambda (n: nat).(csubst0 n v0 c3 c4)) H3 (minus (s k i0) (s k O)) (s_arith0 k i0)) in (or4_intro3 (drop O O (CHead c4 k u2) (CHead c3 k u1)) (ex3_4 K C T T (\lambda (k0: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k u1) (CHead e0 k0 u)))))) (\lambda (k0: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e0 k0 w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (ex3_4 K C C T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 k u1) (CHead e1 k0 u)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c4 k u2) (CHead e2 k0 u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2)))))) (ex4_5 K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k u1) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2))))))) (ex4_5_intro K C C T T (\lambda (k0: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k u1) (CHead e1 k0 u))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k u2) (CHead e2 k0 w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 O)) v0 u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 O)) v0 e1 e2)))))) k c3 c4 u1 u2 (refl_equal C (CHead c3 k u1)) (drop_refl (CHead c4 k u2)) H5 H6)))))))))))))) i v c1 c2 H0) e (drop_gen_refl c1 e H1)))))))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (i: nat).((lt n0 i) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n0)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k n0)) v e1 e2)))))))))))))))))).(\lambda (i: nat).(\lambda (H: (lt (S n0) i)).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))))))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 i v (CSort n1) c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CSort n1) e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))))) (\lambda (H2: (eq C e (CSort n1))).(\lambda (H3: (eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(or4 (drop (S n0) O c2 c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2))))))))) (let H5 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or4 (drop (S n0) O c2 (CSort n1)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort n1) (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))))) H5)) e H2)))) (drop_gen_sort n1 (S n0) O e H1)))))))) (\lambda (c: C).(\lambda (H0: ((\forall (c2: C).(\forall (v: T).((csubst0 i v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H1: (csubst0 i v (CHead c k t) c2)).(\lambda (e: C).(\lambda (H2: (drop (S n0) O (CHead c k t) e)).(let H3 \def (match H1 return (\lambda (_: ?).(\lambda (n: nat).(\lambda (t0: T).(\lambda (c0: C).(\lambda (c1: C).((eq nat n i) \to ((eq T t0 v) \to ((eq C c0 (CHead c k t)) \to ((eq C c1 c2) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k (S n0))) v e1 e2))))))))))))))))) with [(csubst0_snd k0 i0 v0 u1 u2 H3 c0) \Rightarrow (\lambda (H4: (eq nat (s k0 i0) i)).(\lambda (H5: (eq T v0 v)).(\lambda (H6: (eq C (CHead c0 k0 u1) (CHead c k t))).(\lambda (H7: (eq C (CHead c0 k0 u2) c2)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c0 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i0 v0 u1 u2) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2))))))))))))) (\lambda (H8: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq C (CHead c0 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i0 t0 u1 u2) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))))))))) (\lambda (H9: (eq C (CHead c0 k0 u1) (CHead c k t))).(let H10 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u1) (CHead c k t) H9) in ((let H11 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c0 k0 u1) (CHead c k t) H9) in ((let H12 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u1) (CHead c k t) H9) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to ((eq T u1 t) \to ((eq C (CHead c k0 u2) c2) \to ((subst0 i0 v u1 u2) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2))))))))))))) (\lambda (H13: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq T u1 t) \to ((eq C (CHead c k u2) c2) \to ((subst0 i0 v u1 u2) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda (H14: (eq T u1 t)).(eq_ind T t (\lambda (t: T).((eq C (CHead c k u2) c2) \to ((subst0 i0 v t u2) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2))))))))))) (\lambda (H15: (eq C (CHead c k u2) c2)).(eq_ind C (CHead c k u2) (\lambda (c: C).((subst0 i0 v t u2) \to (or4 (drop (S n0) O c e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c (CHead e0 k w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c (CHead e2 k u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c (CHead e2 k w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))) (\lambda (_: (subst0 i0 v t u2)).(let H1 \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i0) i)) H4 k H13) in (let H17 \def (eq_ind_r nat i (\lambda (n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))))))))))) H0 (s k i0) H1) in (let H18 \def (eq_ind_r nat i (\lambda (n: nat).(lt (S n0) n)) H (s k i0) H1) in (K_ind (\lambda (k: K).((drop (r k n0) O c e) \to (((\forall (c2: C).(\forall (v: T).((csubst0 (s k i0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 (S n0))) v e1 e2)))))))))))))) \to ((lt (S n0) (s k i0)) \to (or4 (drop (S n0) O (CHead c k u2) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k u2) (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c k u2) (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c k u2) (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda (b: B).(\lambda (H2: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) i0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Bind b) i0))).(or4_intro0 (drop (S n0) O (CHead c (Bind b) u2) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))) (drop_drop (Bind b) n0 c e H2 u2)))))) (\lambda (f: F).(\lambda (H2: (drop (r (Flat f) n0) O c e)).(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Flat f) i0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))))))))))).(\lambda (_: (lt (S n0) (s (Flat f) i0))).(or4_intro0 (drop (S n0) O (CHead c (Flat f) u2) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (drop_drop (Flat f) n0 c e H2 u2)))))) k (drop_gen_drop k c e t n0 H2) H17 H18))))) c2 H15)) u1 (sym_eq T u1 t H14))) k0 (sym_eq K k0 k H13))) c0 (sym_eq C c0 c H12))) H11)) H10))) v0 (sym_eq T v0 v H8))) i H4 H5 H6 H7 H3))))) | (csubst0_fst k0 i0 c1 c0 v0 H3 u) \Rightarrow (\lambda (H4: (eq nat (s k0 i0) i)).(\lambda (H5: (eq T v0 v)).(\lambda (H6: (eq C (CHead c1 k0 u) (CHead c k t))).(\lambda (H7: (eq C (CHead c0 k0 u) c2)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c1 k0 u) (CHead c k t)) \to ((eq C (CHead c0 k0 u) c2) \to ((csubst0 i0 v0 c1 c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2))))))))))))) (\lambda (H8: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq C (CHead c1 k0 u) (CHead c k t)) \to ((eq C (CHead c0 k0 u) c2) \to ((csubst0 i0 t0 c1 c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))))))))) (\lambda (H9: (eq C (CHead c1 k0 u) (CHead c k t))).(let H10 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k0 u) (CHead c k t) H9) in ((let H11 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c1 k0 u) (CHead c k t) H9) in ((let H12 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k0 u) (CHead c k t) H9) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to ((eq T u t) \to ((eq C (CHead c0 k0 u) c2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2))))))))))))) (\lambda (H13: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq T u t) \to ((eq C (CHead c0 k u) c2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k1 u0))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda (H14: (eq T u t)).(eq_ind T t (\lambda (t: T).((eq C (CHead c0 k t) c2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead e2 k u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2))))))))))) (\lambda (H15: (eq C (CHead c0 k t) c2)).(eq_ind C (CHead c0 k t) (\lambda (c2: C).((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c2 (CHead e2 k u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))) (\lambda (H16: (csubst0 i0 v c c0)).(let H1 \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i0) i)) H4 k H13) in (let H17 \def (eq_ind_r nat i (\lambda (n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))))))))))) H0 (s k i0) H1) in (let H18 \def (eq_ind_r nat i (\lambda (n: nat).(lt (S n0) n)) H (s k i0) H1) in (K_ind (\lambda (k: K).((drop (r k n0) O c e) \to (((\forall (c2: C).(\forall (v: T).((csubst0 (s k i0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 (S n0))) v e1 e2)))))))))))))) \to ((lt (S n0) (s k i0)) \to (or4 (drop (S n0) O (CHead c0 k t) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k1 u0)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 k t) (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k1 u0)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 k t) (CHead e2 k1 u0)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k1 u0))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 k t) (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u0 w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda (b: B).(\lambda (H2: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) i0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))))))))))).(\lambda (H: (lt (S n0) (s (Bind b) i0))).(let H19 \def (IHn i0 (le_S_n (S n0) i0 H) c c0 v H16 e H2) in (or4_ind (drop n0 O c0 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus i0 (s k n0)) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n0 O c0 (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k n0)) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus i0 (s k n0)) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k n0)) v e1 e2))))))) (or4 (drop (S n0) O (CHead c0 (Bind b) t) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (H20: (drop n0 O c0 e)).(or4_intro0 (drop (S n0) O (CHead c0 (Bind b) t) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))) (drop_drop (Bind b) n0 c0 e H20 t))) (\lambda (H20: (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k n0)) v u w))))))).(ex3_4_ind K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus i0 (s k n0)) v u0 w))))) (or4 (drop (S n0) O (CHead c0 (Bind b) t) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k 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K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c0 (Bind b) t) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: 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C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H21: (eq C e (CHead x1 x0 x3))).(\lambda (H22: (drop (S n0) O c0 (CHead x2 x0 x3))).(\lambda (H23: (csubst0 (minus i0 (s x0 (S n0))) v x1 x2)).(eq_ind_r C (CHead x1 x0 x3) (\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Flat f) t) c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c0 (Flat f) t) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 x0 x3) (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 x0 x3) (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))) x0 x1 x2 x3 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Flat f) n0 c0 (CHead x2 x0 x3) H22 t) H23)) e H21)))))))) H20)) (\lambda (H20: (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus i0 (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1 e2)))))) (or4 (drop (S n0) O (CHead c0 (Flat f) t) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H21: (eq C e (CHead x1 x0 x3))).(\lambda (H22: (drop (S n0) O c0 (CHead x2 x0 x4))).(\lambda (H23: (subst0 (minus i0 (s x0 (S n0))) v x3 x4)).(\lambda (H24: (csubst0 (minus i0 (s x0 (S n0))) v x1 x2)).(eq_ind_r C (CHead x1 x0 x3) (\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Flat f) t) c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c0 (Flat f) t) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k u0)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 x0 x3) (CHead e1 k u0)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k u0)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (ex4_5_intro K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k u0))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) t) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u0 w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Flat f) n0 c0 (CHead x2 x0 x4) H22 t) H23 H24)) e H21)))))))))) H20)) H19)))))) k (drop_gen_drop k c e t n0 H2) H17 H18))))) c2 H15)) u (sym_eq T u t H14))) k0 (sym_eq K k0 k H13))) c1 (sym_eq C c1 c H12))) H11)) H10))) v0 (sym_eq T v0 v H8))) i H4 H5 H6 H7 H3))))) | (csubst0_both k0 i0 v0 u1 u2 H3 c1 c0 H4) \Rightarrow (\lambda (H5: (eq nat (s k0 i0) i)).(\lambda (H6: (eq T v0 v)).(\lambda (H7: (eq C (CHead c1 k0 u1) (CHead c k t))).(\lambda (H8: (eq C (CHead c0 k0 u2) c2)).(eq_ind nat (s k0 i0) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c1 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i0 v0 u1 u2) \to ((csubst0 i0 v0 c1 c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))))))))))) (\lambda (H9: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq C (CHead c1 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i0 t0 u1 u2) \to ((csubst0 i0 t0 c1 c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2))))))))))))) (\lambda (H10: (eq C (CHead c1 k0 u1) (CHead c k t))).(let H11 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k0 u1) (CHead c k t) H10) in ((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c1 k0 u1) (CHead c k t) H10) in ((let H13 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k0 u1) (CHead c k t) H10) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to ((eq T u1 t) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i0 v u1 u2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k0 i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k0 i0) (s k (S n0))) v e1 e2)))))))))))))) (\lambda (H14: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq T u1 t) \to ((eq C (CHead c0 k u2) c2) \to ((subst0 i0 v u1 u2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2))))))))))))) (\lambda (H15: (eq T u1 t)).(eq_ind T t (\lambda (t: T).((eq C (CHead c0 k u2) c2) \to ((subst0 i0 v t u2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda (H16: (eq C (CHead c0 k u2) c2)).(eq_ind C (CHead c0 k u2) (\lambda (c2: C).((subst0 i0 v t u2) \to ((csubst0 i0 v c c0) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2))))))))))) (\lambda (_: (subst0 i0 v t u2)).(\lambda (H18: (csubst0 i0 v c c0)).(let H1 \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i0) i)) H5 k H14) in (let H19 \def (eq_ind_r nat i (\lambda (n: nat).(\forall (c2: C).(\forall (v: T).((csubst0 n v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus n (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus n (s k (S n0))) v e1 e2)))))))))))))) H0 (s k i0) H1) in (let H20 \def (eq_ind_r nat i (\lambda (n: nat).(lt (S n0) n)) H (s k i0) H1) in (K_ind (\lambda (k: K).((drop (r k n0) O c e) \to (((\forall (c2: C).(\forall (v: T).((csubst0 (s k i0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k0: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k0 (S n0))) v u w)))))) (\lambda (k0: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k0 (S n0))) v e1 e2)))))))))))))) \to ((lt (S n0) (s k i0)) \to (or4 (drop (S n0) O (CHead c0 k u2) e) (ex3_4 K C T T (\lambda (k1: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k1 u)))))) (\lambda (k1: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 k u2) (CHead e0 k1 w)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k1 u)))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 k u2) (CHead e2 k1 u)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k1: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k1 u))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 k u2) (CHead e2 k1 w))))))) (\lambda (k1: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s k i0) (s k1 (S n0))) v u w)))))) (\lambda (k1: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s k i0) (s k1 (S n0))) v e1 e2)))))))))))) (\lambda (b: B).(\lambda (H2: (drop (r (Bind b) n0) O c e)).(\lambda (_: ((\forall (c2: C).(\forall (v: T).((csubst0 (s (Bind b) i0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))))))))))).(\lambda (H: (lt (S n0) (s (Bind b) i0))).(let H21 \def (IHn i0 (le_S_n (S n0) i0 H) c c0 v H18 e H2) in (or4_ind (drop n0 O c0 e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k n0)) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c0 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i0 (s k n0)) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k n0)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k n0)) v e1 e2))))))) (or4 (drop (S n0) O (CHead c0 (Bind b) u2) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (H22: (drop n0 O c0 e)).(or4_intro0 (drop (S n0) O (CHead c0 (Bind b) u2) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))) (drop_drop (Bind b) n0 c0 e H22 u2))) (\lambda (H22: (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k n0)) v u w))))))).(ex3_4_ind K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k n0)) v u w))))) (or4 (drop (S n0) O (CHead c0 (Bind b) u2) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O 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(\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c0 (Bind b) u2) (CHead x1 x0 x2)) (ex3_4 K C T T (\lambda (k: K).(\lambda 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(CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Bind b) i0) (s k (S n0))) v e1 e2))))) x0 x1 x2 x3 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Bind b) n0 c0 (CHead x2 x0 x3) H24 u2) (eq_ind_r nat (S (s x0 n0)) (\lambda (n: nat).(csubst0 (minus (s (Bind b) i0) n) v x1 x2)) H25 (s x0 (S n0)) (s_S x0 n0)))) e H23)))))))) H22)) (\lambda (H22: (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k n0)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k n0)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k n0)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k n0)) v e1 e2)))))) (or4 (drop (S n0) O (CHead c0 (Bind b) u2) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Bind b) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Bind b) u2) (CHead e2 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(Flat f) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c0 (Flat f) u2) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 x0 x3) (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (ex3_4_intro K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 x0 x3) (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))) x0 x1 x2 x3 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Flat f) n0 c0 (CHead x2 x0 x3) H24 u2) H25)) e H23)))))))) H22)) (\lambda (H22: (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i0 (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i0 (s k (S n0))) v e1 e2)))))) (or4 (drop (S n0) O (CHead c0 (Flat f) u2) e) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H23: (eq C e (CHead x1 x0 x3))).(\lambda (H24: (drop (S n0) O c0 (CHead x2 x0 x4))).(\lambda (H25: (subst0 (minus i0 (s x0 (S n0))) v x3 x4)).(\lambda (H26: (csubst0 (minus i0 (s x0 (S n0))) v x1 x2)).(eq_ind_r C (CHead x1 x0 x3) (\lambda (c: C).(or4 (drop (S n0) O (CHead c0 (Flat f) u2) c) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c0 (Flat f) u2) (CHead x1 x0 x3)) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 x0 x3) (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2))))))) (ex4_5_intro K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 x0 x3) (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O (CHead c0 (Flat f) u2) (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus (s (Flat f) i0) (s k (S n0))) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus (s (Flat f) i0) (s k (S n0))) v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 x0 x3)) (drop_drop (Flat f) n0 c0 (CHead x2 x0 x4) H24 u2) H25 H26)) e H23)))))))))) H22)) H21)))))) k (drop_gen_drop k c e t n0 H2) H19 H20)))))) c2 H16)) u1 (sym_eq T u1 t H15))) k0 (sym_eq K k0 k H14))) c1 (sym_eq C c1 c H13))) H12)) H11))) v0 (sym_eq T v0 v H9))) i H5 H6 H7 H8 H3 H4)))))]) in (H3 (refl_equal nat i) (refl_equal T v) (refl_equal C (CHead c k t)) (refl_equal C c2)))))))))))) c1)))))) n).
 
 theorem csubst0_drop_eq:
  \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n v c1 c2) \to (\forall (e: C).((drop n O c1 e) \to (or4 (drop n O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))
 \def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c1 e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2)) (or4 (drop O O c2 c1) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c1 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (y: nat).(\lambda (H1: (csubst0 y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop O O c0 c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O t u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O t u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O t e1 e2))))))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s k0 i) O) \to (or4 (drop O O (CHead c k0 u2) (CHead c k0 u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k0 u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k0 u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k0 u1) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c k0 u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k0 u1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k0 u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or4 (drop O O (CHead c (Bind b) u2) (CHead c (Bind b) u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Bind b) u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))) H4)))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i O)).(let H4 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 O H3) in (or4_intro1 (drop O O (CHead c (Flat f) u2) (CHead c (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u1)) (drop_refl (CHead c (Flat f) u2)) H4))))))))))) k)) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4: C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))) \to (\forall (u: T).((eq nat (s k0 i) O) \to (or4 (drop O O (CHead c4 k0 u) (CHead c3 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k0 u) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 k0 u) (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k0 u) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3 c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat (S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (or4 (drop O O (CHead c4 (Bind b) u) (CHead c3 (Bind b) u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u) (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Bind b) u) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 (Bind b) u) (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Bind b) u) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2: (csubst0 i v0 c3 c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i O)).(let H5 \def (eq_ind nat i (\lambda (n: nat).((eq nat n O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v0 c3 c4)) H2 O H4) in (or4_intro2 (drop O O (CHead c4 (Flat f) u) (CHead c3 (Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c3 (Flat f) u)) (drop_refl (CHead c4 (Flat f) u)) H6))))))))))))) k)) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4: C).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop O O (CHead c4 k0 u2) (CHead c3 k0 u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0 u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k0 u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 k0 u1) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c4 k0 u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0 u1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k0 u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3 c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5) in (False_ind (or4 (drop O O (CHead c4 (Bind b) u2) (CHead c3 (Bind b) u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c4 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))) H6))))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3 c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (H5: (eq nat i O)).(let H6 \def (eq_ind nat i (\lambda (n: nat).((eq nat n O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))) H4 O H5) in (let H7 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v0 c3 c4)) H3 O H5) in (let H8 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 O H5) in (or4_intro3 (drop O O (CHead c4 (Flat f) u2) (CHead c3 (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c3 (Flat f) u1)) (drop_refl (CHead c4 (Flat f) u2)) H8 H7)))))))))))))))) k)) y v c1 c2 H1))) H) e (drop_gen_refl c1 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 (S n0) v (CSort n1) c2)).(\lambda (e: C).(\lambda (H0: (drop (S n0) O (CSort n1) e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H1: (eq C e (CSort n1))).(\lambda (H2: (eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(or4 (drop (S n0) O c2 c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (let H4 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind (or4 (drop (S n0) O c2 (CSort n1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort n1) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) H4)) e H1)))) (drop_gen_sort n1 (S n0) O e H0)))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 (S n0) v (CHead c k t) c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CHead c k t) e)).(let H2 \def (match H0 return (\lambda (n: nat).(\lambda (t0: T).(\lambda (c0: C).(\lambda (c1: C).((eq nat n (S n0)) \to ((eq T t0 v) \to ((eq C c0 (CHead c k t)) \to ((eq C c1 c2) \to (or4 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))))) with [(csubst0_snd k0 i v0 u1 u2 H2 c0) \Rightarrow (\lambda (H3: (eq nat (s k0 i) (S n0))).(\lambda (H4: (eq T v0 v)).(\lambda (H5: (eq C (CHead c0 k0 u1) (CHead c k t))).(\lambda (H6: (eq C (CHead c0 k0 u2) c2)).((let H7 \def (f_equal nat nat (\lambda (e0: nat).e0) (s k0 i) (S n0) H3) in (eq_ind nat (s k0 i) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c0 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i v0 u1 u2) \to (or4 (drop n O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))) (\lambda (H8: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq C (CHead c0 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i t0 u1 u2) \to (or4 (drop (s k0 i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))) (\lambda (H9: (eq C (CHead c0 k0 u1) (CHead c k t))).(let H10 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u1) (CHead c k t) H9) in ((let H11 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c0 k0 u1) (CHead c k t) H9) in ((let H12 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u1) (CHead c k t) H9) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to ((eq T u1 t) \to ((eq C (CHead c k0 u2) c2) \to ((subst0 i v u1 u2) \to (or4 (drop (s k0 i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))) (\lambda (H13: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq T u1 t) \to ((eq C (CHead c k u2) c2) \to ((subst0 i v u1 u2) \to (or4 (drop (s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))) (\lambda (H14: (eq T u1 t)).(eq_ind T t (\lambda (t: T).((eq C (CHead c k u2) c2) \to ((subst0 i v t u2) \to (or4 (drop (s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (H15: (eq C (CHead c k u2) c2)).(eq_ind C (CHead c k u2) (\lambda (c: C).((subst0 i v t u2) \to (or4 (drop (s k i) O c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k i) O c (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))) (\lambda (H16: (subst0 i v t u2)).(let H0 \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i) (S n0))) H7 k H13) in (K_ind (\lambda (k: K).((drop (r k n0) O c e) \to ((eq nat (s k i) (S n0)) \to (or4 (drop (s k i) O (CHead c k u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O (CHead c k u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k i) O (CHead c k u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O (CHead c k u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H1: (drop (r (Bind b) n0) O c e)).(\lambda (H17: (eq nat (s (Bind b) i) (S n0))).(let H18 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S n0) H17) in (let H19 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v t u2)) H16 n0 H18) in (eq_ind_r nat n0 (\lambda (n: nat).(or4 (drop (s (Bind b) n) O (CHead c (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n) O (CHead c (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n) O (CHead c (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n) O (CHead c (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro0 (drop (s (Bind b) n0) O (CHead c (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e H1 u2)) i H18)))))) (\lambda (f: F).(\lambda (H1: (drop (r (Flat f) n0) O c e)).(\lambda (H17: (eq nat (s (Flat f) i) (S n0))).(let H18 \def (f_equal nat nat (\lambda (e0: nat).e0) i (S n0) H17) in (let H19 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v t u2)) H16 (S n0) H18) in (eq_ind_r nat (S n0) (\lambda (n: nat).(or4 (drop (s (Flat f) n) O (CHead c (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) n) O (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) n) O (CHead c (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) n) O (CHead c (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro0 (drop (s (Flat f) (S n0)) O (CHead c (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e H1 u2)) i H18)))))) k (drop_gen_drop k c e t n0 H1) H0))) c2 H15)) u1 (sym_eq T u1 t H14))) k0 (sym_eq K k0 k H13))) c0 (sym_eq C c0 c H12))) H11)) H10))) v0 (sym_eq T v0 v H8))) (S n0) H7)) H4 H5 H6 H2))))) | (csubst0_fst k0 i c1 c0 v0 H2 u) \Rightarrow (\lambda (H3: (eq nat (s k0 i) (S n0))).(\lambda (H4: (eq T v0 v)).(\lambda (H5: (eq C (CHead c1 k0 u) (CHead c k t))).(\lambda (H6: (eq C (CHead c0 k0 u) c2)).((let H7 \def (f_equal nat nat (\lambda (e0: nat).e0) (s k0 i) (S n0) H3) in (eq_ind nat (s k0 i) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c1 k0 u) (CHead c k t)) \to ((eq C (CHead c0 k0 u) c2) \to ((csubst0 i v0 c1 c0) \to (or4 (drop n O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n O c2 (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))) (\lambda (H8: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq C (CHead c1 k0 u) (CHead c k t)) \to ((eq C (CHead c0 k0 u) c2) \to ((csubst0 i t0 c1 c0) \to (or4 (drop (s k0 i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))) (\lambda (H9: (eq C (CHead c1 k0 u) (CHead c k t))).(let H10 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k0 u) (CHead c k t) H9) in ((let H11 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c1 k0 u) (CHead c k t) H9) in ((let H12 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k0 u) (CHead c k t) H9) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to ((eq T u t) \to ((eq C (CHead c0 k0 u) c2) \to ((csubst0 i v c c0) \to (or4 (drop (s k0 i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))) (\lambda (H13: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq T u t) \to ((eq C (CHead c0 k u) c2) \to ((csubst0 i v c c0) \to (or4 (drop (s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k i) O c2 (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))) (\lambda (H14: (eq T u t)).(eq_ind T t (\lambda (t: T).((eq C (CHead c0 k t) c2) \to ((csubst0 i v c c0) \to (or4 (drop (s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k i) O c2 (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (H15: (eq C (CHead c0 k t) c2)).(eq_ind C (CHead c0 k t) (\lambda (c2: C).((csubst0 i v c c0) \to (or4 (drop (s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k i) O c2 (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))) (\lambda (H16: (csubst0 i v c c0)).(let H0 \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i) (S n0))) H7 k H13) in (K_ind (\lambda (k: K).((drop (r k n0) O c e) \to ((eq nat (s k i) (S n0)) \to (or4 (drop (s k i) O (CHead c0 k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O (CHead c0 k t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k i) O (CHead c0 k t) (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O (CHead c0 k t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H1: (drop (r (Bind b) n0) O c e)).(\lambda (H17: (eq nat (s (Bind b) i) (S n0))).(let H18 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S n0) H17) in (let H19 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v c c0)) H16 n0 H18) in (eq_ind_r nat n0 (\lambda (n: nat).(or4 (drop (s (Bind b) n) O (CHead c0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Bind b) n) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (let H \def (IHn c c0 v H19 e H1) in (or4_ind (drop n0 O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n0 O c0 (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: 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(f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x2)) (drop_drop (Bind b) n0 c0 (CHead x1 (Flat x0) x3) H22 t) H23)) e H21)))))))) H20)) (\lambda (H20: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n0 O 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u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H21: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H22: (drop n0 O c0 (CHead x2 (Flat x0) x3))).(\lambda (H23: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) t) c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: 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(CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H21: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H22: (drop n0 O c0 (CHead x2 (Flat x0) x4))).(\lambda (H23: (subst0 O v x3 x4)).(\lambda (H24: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) t) c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Bind b) n0 c0 (CHead x2 (Flat x0) x4) H22 t) H23 H24)) e H21)))))))))) H20)) H)) i H18)))))) (\lambda (f: F).(\lambda (H1: (drop (r (Flat f) n0) O c e)).(\lambda (H17: (eq nat (s (Flat f) i) (S n0))).(let H18 \def (f_equal nat nat (\lambda (e0: nat).e0) i (S n0) H17) in (let H19 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v c c0)) H16 (S n0) H18) in (eq_ind_r nat (S n0) (\lambda (n: nat).(or4 (drop (s (Flat f) n) O (CHead c0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) n) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) n) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) n) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (let H20 \def (H c0 v H19 e H1) in (or4_ind (drop (S n0) O c0 e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c0 (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H21: (drop (S n0) O c0 e)).(or4_intro0 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c0 e H21 t))) (\lambda (H21: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H22: (eq C e (CHead x1 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C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead x1 (Flat x0) x2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x2)) (drop_drop (Flat f) n0 c0 (CHead x1 (Flat x0) x3) H23 t) H24)) e H22)))))))) H21)) (\lambda (H21: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c0 (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H22: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H23: (drop (S n0) O c0 (CHead x2 (Flat x0) x3))).(\lambda (H24: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) c) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Flat f) n0 c0 (CHead x2 (Flat x0) x3) H23 t) H24)) e H22)))))))) H21)) (\lambda (H21: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H22: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H23: (drop (S n0) O c0 (CHead x2 (Flat x0) x4))).(\lambda (H24: (subst0 O v x3 x4)).(\lambda (H25: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) c) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Flat f) n0 c0 (CHead x2 (Flat x0) x4) H23 t) H24 H25)) e H22)))))))))) H21)) H20)) i H18)))))) k (drop_gen_drop k c e t n0 H1) H0))) c2 H15)) u (sym_eq T u t H14))) k0 (sym_eq K k0 k H13))) c1 (sym_eq C c1 c H12))) H11)) H10))) v0 (sym_eq T v0 v H8))) (S n0) H7)) H4 H5 H6 H2))))) | (csubst0_both k0 i v0 u1 u2 H2 c1 c0 H3) \Rightarrow (\lambda (H4: (eq nat (s k0 i) (S n0))).(\lambda (H5: (eq T v0 v)).(\lambda (H6: (eq C (CHead c1 k0 u1) (CHead c k t))).(\lambda (H7: (eq C (CHead c0 k0 u2) c2)).((let H8 \def (f_equal nat nat (\lambda (e0: nat).e0) (s k0 i) (S n0) H4) in (eq_ind nat (s k0 i) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c1 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i v0 u1 u2) \to ((csubst0 i v0 c1 c0) \to (or4 (drop n O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (H9: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq C (CHead c1 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i t0 u1 u2) \to ((csubst0 i t0 c1 c0) \to (or4 (drop (s k0 i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))) (\lambda (H10: (eq C (CHead c1 k0 u1) (CHead c k t))).(let H11 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k0 u1) (CHead c k t) H10) in ((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c1 k0 u1) (CHead c k t) H10) in ((let H13 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k0 u1) (CHead c k t) H10) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to ((eq T u1 t) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i v u1 u2) \to ((csubst0 i v c c0) \to (or4 (drop (s k0 i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (H14: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq T u1 t) \to ((eq C (CHead c0 k u2) c2) \to ((subst0 i v u1 u2) \to ((csubst0 i v c c0) \to (or4 (drop (s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))) (\lambda (H15: (eq T u1 t)).(eq_ind T t (\lambda (t: T).((eq C (CHead c0 k u2) c2) \to ((subst0 i v t u2) \to ((csubst0 i v c c0) \to (or4 (drop (s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))) (\lambda (H16: (eq C (CHead c0 k u2) c2)).(eq_ind C (CHead c0 k u2) (\lambda (c2: C).((subst0 i v t u2) \to ((csubst0 i v c c0) \to (or4 (drop (s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (H17: (subst0 i v t u2)).(\lambda (H18: (csubst0 i v c c0)).(let H0 \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i) (S n0))) H8 k H14) in (K_ind (\lambda (k: K).((drop (r k n0) O c e) \to ((eq nat (s k i) (S n0)) \to (or4 (drop (s k i) O (CHead c0 k u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O (CHead c0 k u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k i) O (CHead c0 k u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O (CHead c0 k u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H1: (drop (r (Bind b) n0) O c e)).(\lambda (H19: (eq nat (s (Bind b) i) (S n0))).(let H20 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S n0) H19) in (let H21 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v c c0)) H18 n0 H20) in (let H22 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v t u2)) H17 n0 H20) in (eq_ind_r nat n0 (\lambda (n: nat).(or4 (drop (s (Bind b) n) O (CHead c0 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (let H \def (IHn c c0 v H21 e H1) in (or4_ind (drop n0 O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H23: (drop n0 O c0 e)).(or4_intro0 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c0 e H23 u2))) (\lambda (H23: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H24: (eq C e (CHead x1 (Flat x0) x2))).(\lambda (H25: (drop n0 O c0 (CHead x1 (Flat x0) x3))).(\lambda (H26: (subst0 O v x2 x3)).(eq_ind_r C (CHead x1 (Flat x0) x2) (\lambda (c: C).(or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead x1 (Flat x0) x2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x2)) (drop_drop (Bind b) n0 c0 (CHead x1 (Flat x0) x3) H25 u2) H26)) e H24)))))))) H23)) (\lambda (H23: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H24: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H25: (drop n0 O c0 (CHead x2 (Flat x0) x3))).(\lambda (H26: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Bind b) n0 c0 (CHead x2 (Flat x0) x3) H25 u2) H26)) e H24)))))))) H23)) (\lambda (H23: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H24: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H25: (drop n0 O c0 (CHead x2 (Flat x0) x4))).(\lambda (H26: (subst0 O v x3 x4)).(\lambda (H27: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Bind b) n0 c0 (CHead x2 (Flat x0) x4) H25 u2) H26 H27)) e H24)))))))))) H23)) H)) i H20))))))) (\lambda (f: F).(\lambda (H1: (drop (r (Flat f) n0) O c e)).(\lambda (H19: (eq nat (s (Flat f) i) (S n0))).(let H20 \def (f_equal nat nat (\lambda (e0: nat).e0) i (S n0) H19) in (let H21 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v c c0)) H18 (S n0) H20) in (let H22 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v t u2)) H17 (S n0) H20) in (eq_ind_r nat (S n0) (\lambda (n: nat).(or4 (drop (s (Flat f) n) O (CHead c0 (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) n) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) n) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) n) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (let H23 \def (H c0 v H21 e H1) in (or4_ind (drop (S n0) O c0 e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H24: (drop (S n0) O c0 e)).(or4_intro0 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c0 e H24 u2))) (\lambda (H24: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H25: (eq C e (CHead x1 (Flat x0) x2))).(\lambda (H26: (drop (S n0) O c0 (CHead x1 (Flat x0) x3))).(\lambda (H27: (subst0 O v x2 x3)).(eq_ind_r C (CHead x1 (Flat x0) x2) (\lambda (c: C).(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) c) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead x1 (Flat x0) x2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x2)) (drop_drop (Flat f) n0 c0 (CHead x1 (Flat x0) x3) H26 u2) H27)) e H25)))))))) H24)) (\lambda (H24: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H25: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H26: (drop (S n0) O c0 (CHead x2 (Flat x0) x3))).(\lambda (H27: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) c) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda 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w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Flat f) n0 c0 (CHead x2 (Flat x0) x3) H26 u2) H27)) e H25)))))))) H24)) (\lambda (H24: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H25: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H26: (drop (S n0) O c0 (CHead x2 (Flat x0) x4))).(\lambda (H27: (subst0 O v x3 x4)).(\lambda (H28: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) c) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Flat f) n0 c0 (CHead x2 (Flat x0) x4) H26 u2) H27 H28)) e H25)))))))))) H24)) H23)) i H20))))))) k (drop_gen_drop k c e t n0 H1) H0)))) c2 H16)) u1 (sym_eq T u1 t H15))) k0 (sym_eq K k0 k H14))) c1 (sym_eq C c1 c H13))) H12)) H11))) v0 (sym_eq T v0 v H9))) (S n0) H8)) H5 H6 H7 H2 H3)))))]) in (H2 (refl_equal nat (S n0)) (refl_equal T v) (refl_equal C (CHead c k t)) (refl_equal C c2)))))))))))) c1)))) n).
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c1 e)).(eq_ind C c1 (\lambda (c: C).(or4 (drop O O c2 c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2)) (or4 (drop O O c2 c1) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c1 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c1 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (y: nat).(\lambda (H1: (csubst0 y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop O O c0 c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O t u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O t u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O t e1 e2))))))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s k0 i) O) \to (or4 (drop O O (CHead c k0 u2) (CHead c k0 u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k0 u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k0 u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c k0 u1) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c k0 u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c k0 u1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c k0 u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or4 (drop O O (CHead c (Bind b) u2) (CHead c (Bind b) u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Bind b) u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Bind b) u1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))) H4)))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i O)).(let H4 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 O H3) in (or4_intro1 (drop O O (CHead c (Flat f) u2) (CHead c (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c (Flat f) u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u1)) (drop_refl (CHead c (Flat f) u2)) H4))))))))))) k)) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4: C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))) \to (\forall (u: T).((eq nat (s k0 i) O) \to (or4 (drop O O (CHead c4 k0 u) (CHead c3 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k0 u) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 k0 u) (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 k0 u) (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k0 u) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3 c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat (S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (or4 (drop O O (CHead c4 (Bind b) u) (CHead c3 (Bind b) u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u) (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Bind b) u) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 (Bind b) u) (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Bind b) u) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2: (csubst0 i v0 c3 c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i O)).(let H5 \def (eq_ind nat i (\lambda (n: nat).((eq nat n O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v0 c3 c4)) H2 O H4) in (or4_intro2 (drop O O (CHead c4 (Flat f) u) (CHead c3 (Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v0 u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop O O (CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c3 (Flat f) u)) (drop_refl (CHead c4 (Flat f) u)) H6))))))))))))) k)) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4: C).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop O O (CHead c4 k0 u2) (CHead c3 k0 u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0 u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k0 u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 k0 u1) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c4 k0 u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 k0 u1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 k0 u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3 c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5) in (False_ind (or4 (drop O O (CHead c4 (Bind b) u2) (CHead c3 (Bind b) u1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c4 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))) H6))))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3 c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (H5: (eq nat i O)).(let H6 \def (eq_ind nat i (\lambda (n: nat).((eq nat n O) \to (or4 (drop O O c4 c3) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O c4 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c3 (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O c4 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))) H4 O H5) in (let H7 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v0 c3 c4)) H3 O H5) in (let H8 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 O H5) in (or4_intro3 (drop O O (CHead c4 (Flat f) u2) (CHead c3 (Flat f) u1)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop O O (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop O O (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v0 u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c3 (Flat f) u1)) (drop_refl (CHead c4 (Flat f) u2)) H8 H7)))))))))))))))) k)) y v c1 c2 H1))) H) e (drop_gen_refl c1 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c1 e) \to (or4 (drop n0 O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (_: (csubst0 (S n0) v (CSort n1) c2)).(\lambda (e: C).(\lambda (H0: (drop (S n0) O (CSort n1) e)).(and3_ind (eq C e (CSort n1)) (eq nat (S n0) O) (eq nat O O) (or4 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H1: (eq C e (CSort n1))).(\lambda (H2: (eq nat (S n0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n1) (\lambda (c: C).(or4 (drop (S n0) O c2 c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (let H4 \def (eq_ind nat (S n0) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind (or4 (drop (S n0) O c2 (CSort n1)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CSort n1) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CSort n1) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) H4)) e H1)))) (drop_gen_sort n1 (S n0) O e H0)))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to (\forall (e: C).((drop (S n0) O c e) \to (or4 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 (S n0) v (CHead c k t) c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O (CHead c k t) e)).(let H2 \def (match H0 return (\lambda (_: ?).(\lambda (n: nat).(\lambda (t0: T).(\lambda (c0: C).(\lambda (c1: C).((eq nat n (S n0)) \to ((eq T t0 v) \to ((eq C c0 (CHead c k t)) \to ((eq C c1 c2) \to (or4 (drop (S n0) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))))) with [(csubst0_snd k0 i v0 u1 u2 H2 c0) \Rightarrow (\lambda (H3: (eq nat (s k0 i) (S n0))).(\lambda (H4: (eq T v0 v)).(\lambda (H5: (eq C (CHead c0 k0 u1) (CHead c k t))).(\lambda (H6: (eq C (CHead c0 k0 u2) c2)).((let H7 \def (f_equal nat nat (\lambda (e0: nat).e0) (s k0 i) (S n0) H3) in (eq_ind nat (s k0 i) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c0 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i v0 u1 u2) \to (or4 (drop n O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))) (\lambda (H8: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq C (CHead c0 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i t0 u1 u2) \to (or4 (drop (s k0 i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))) (\lambda (H9: (eq C (CHead c0 k0 u1) (CHead c k t))).(let H10 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u1) (CHead c k t) H9) in ((let H11 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c0 k0 u1) (CHead c k t) H9) in ((let H12 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u1) (CHead c k t) H9) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to ((eq T u1 t) \to ((eq C (CHead c k0 u2) c2) \to ((subst0 i v u1 u2) \to (or4 (drop (s k0 i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))) (\lambda (H13: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq T u1 t) \to ((eq C (CHead c k u2) c2) \to ((subst0 i v u1 u2) \to (or4 (drop (s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))) (\lambda (H14: (eq T u1 t)).(eq_ind T t (\lambda (t: T).((eq C (CHead c k u2) c2) \to ((subst0 i v t u2) \to (or4 (drop (s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (H15: (eq C (CHead c k u2) c2)).(eq_ind C (CHead c k u2) (\lambda (c: C).((subst0 i v t u2) \to (or4 (drop (s k i) O c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k i) O c (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))) (\lambda (H16: (subst0 i v t u2)).(let H0 \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i) (S n0))) H7 k H13) in (K_ind (\lambda (k: K).((drop (r k n0) O c e) \to ((eq nat (s k i) (S n0)) \to (or4 (drop (s k i) O (CHead c k u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O (CHead c k u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k i) O (CHead c k u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O (CHead c k u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H1: (drop (r (Bind b) n0) O c e)).(\lambda (H17: (eq nat (s (Bind b) i) (S n0))).(let H18 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S n0) H17) in (let H19 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v t u2)) H16 n0 H18) in (eq_ind_r nat n0 (\lambda (n: nat).(or4 (drop (s (Bind b) n) O (CHead c (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n) O (CHead c (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n) O (CHead c (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n) O (CHead c (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro0 (drop (s (Bind b) n0) O (CHead c (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e H1 u2)) i H18)))))) (\lambda (f: F).(\lambda (H1: (drop (r (Flat f) n0) O c e)).(\lambda (H17: (eq nat (s (Flat f) i) (S n0))).(let H18 \def (f_equal nat nat (\lambda (e0: nat).e0) i (S n0) H17) in (let H19 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v t u2)) H16 (S n0) H18) in (eq_ind_r nat (S n0) (\lambda (n: nat).(or4 (drop (s (Flat f) n) O (CHead c (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) n) O (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) n) O (CHead c (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) n) O (CHead c (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro0 (drop (s (Flat f) (S n0)) O (CHead c (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e H1 u2)) i H18)))))) k (drop_gen_drop k c e t n0 H1) H0))) c2 H15)) u1 (sym_eq T u1 t H14))) k0 (sym_eq K k0 k H13))) c0 (sym_eq C c0 c H12))) H11)) H10))) v0 (sym_eq T v0 v H8))) (S n0) H7)) H4 H5 H6 H2))))) | (csubst0_fst k0 i c1 c0 v0 H2 u) \Rightarrow (\lambda (H3: (eq nat (s k0 i) (S n0))).(\lambda (H4: (eq T v0 v)).(\lambda (H5: (eq C (CHead c1 k0 u) (CHead c k t))).(\lambda (H6: (eq C (CHead c0 k0 u) c2)).((let H7 \def (f_equal nat nat (\lambda (e0: nat).e0) (s k0 i) (S n0) H3) in (eq_ind nat (s k0 i) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c1 k0 u) (CHead c k t)) \to ((eq C (CHead c0 k0 u) c2) \to ((csubst0 i v0 c1 c0) \to (or4 (drop n O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n O c2 (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))) (\lambda (H8: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq C (CHead c1 k0 u) (CHead c k t)) \to ((eq C (CHead c0 k0 u) c2) \to ((csubst0 i t0 c1 c0) \to (or4 (drop (s k0 i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))) (\lambda (H9: (eq C (CHead c1 k0 u) (CHead c k t))).(let H10 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k0 u) (CHead c k t) H9) in ((let H11 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c1 k0 u) (CHead c k t) H9) in ((let H12 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k0 u) (CHead c k t) H9) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to ((eq T u t) \to ((eq C (CHead c0 k0 u) c2) \to ((csubst0 i v c c0) \to (or4 (drop (s k0 i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))) (\lambda (H13: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq T u t) \to ((eq C (CHead c0 k u) c2) \to ((csubst0 i v c c0) \to (or4 (drop (s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k i) O c2 (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))) (\lambda (H14: (eq T u t)).(eq_ind T t (\lambda (t: T).((eq C (CHead c0 k t) c2) \to ((csubst0 i v c c0) \to (or4 (drop (s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k i) O c2 (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (H15: (eq C (CHead c0 k t) c2)).(eq_ind C (CHead c0 k t) (\lambda (c2: C).((csubst0 i v c c0) \to (or4 (drop (s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k i) O c2 (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))) (\lambda (H16: (csubst0 i v c c0)).(let H0 \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i) (S n0))) H7 k H13) in (K_ind (\lambda (k: K).((drop (r k n0) O c e) \to ((eq nat (s k i) (S n0)) \to (or4 (drop (s k i) O (CHead c0 k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O (CHead c0 k t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s k i) O (CHead c0 k t) (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O (CHead c0 k t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H1: (drop (r (Bind b) n0) O c e)).(\lambda (H17: (eq nat (s (Bind b) i) (S n0))).(let H18 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S n0) H17) in (let H19 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v c c0)) H16 n0 H18) in (eq_ind_r nat n0 (\lambda (n: nat).(or4 (drop (s (Bind b) n) O (CHead c0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Bind b) n) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (let H \def (IHn c c0 v H19 e H1) in (or4_ind (drop n0 O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop n0 O c0 (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H20: (drop n0 O c0 e)).(or4_intro0 (drop (s (Bind b) n0) O (CHead c0 (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) 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e2))))))) (ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x2)) (drop_drop (Bind b) n0 c0 (CHead x1 (Flat x0) x3) H22 t) H23)) e H21)))))))) H20)) (\lambda (H20: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: 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v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H21: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H22: (drop n0 O c0 (CHead x2 (Flat x0) x4))).(\lambda (H23: (subst0 O v x3 x4)).(\lambda (H24: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) t) c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: 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(w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u0)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u0))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) t) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Bind b) n0 c0 (CHead x2 (Flat x0) x4) H22 t) H23 H24)) e H21)))))))))) H20)) H)) i H18)))))) (\lambda (f: F).(\lambda (H1: (drop (r (Flat f) n0) O c e)).(\lambda (H17: (eq nat (s (Flat f) i) (S n0))).(let H18 \def (f_equal nat nat (\lambda (e0: nat).e0) i (S n0) H17) in (let H19 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v c c0)) H16 (S n0) H18) in (eq_ind_r nat (S n0) (\lambda (n: nat).(or4 (drop (s (Flat f) n) O (CHead c0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) n) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) n) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) n) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (let H20 \def (H c0 v H19 e H1) in (or4_ind (drop (S n0) O c0 e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c0 (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead 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f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c0 e H21 t))) (\lambda (H21: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H22: (eq C e (CHead x1 (Flat x0) x2))).(\lambda (H23: (drop (S n0) O c0 (CHead x1 (Flat x0) x3))).(\lambda (H24: (subst0 O v x2 x3)).(eq_ind_r C (CHead x1 (Flat x0) x2) (\lambda (c: C).(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) c) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead x1 (Flat x0) x2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x2)) (drop_drop (Flat f) n0 c0 (CHead x1 (Flat x0) x3) H23 t) H24)) e H22)))))))) H21)) (\lambda (H21: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (S n0) O c0 (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H22: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H23: (drop (S n0) O c0 (CHead x2 (Flat x0) x3))).(\lambda (H24: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) c) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Flat f) n0 c0 (CHead x2 (Flat x0) x3) H23 t) H24)) e H22)))))))) H21)) (\lambda (H21: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C e (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H22: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H23: (drop (S n0) O c0 (CHead x2 (Flat x0) x4))).(\lambda (H24: (subst0 O v x3 x4)).(\lambda (H25: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) c) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C c (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u0))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) t) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u0: T).(\lambda (w: T).(subst0 O v u0 w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Flat f) n0 c0 (CHead x2 (Flat x0) x4) H23 t) H24 H25)) e H22)))))))))) H21)) H20)) i H18)))))) k (drop_gen_drop k c e t n0 H1) H0))) c2 H15)) u (sym_eq T u t H14))) k0 (sym_eq K k0 k H13))) c1 (sym_eq C c1 c H12))) H11)) H10))) v0 (sym_eq T v0 v H8))) (S n0) H7)) H4 H5 H6 H2))))) | (csubst0_both k0 i v0 u1 u2 H2 c1 c0 H3) \Rightarrow (\lambda (H4: (eq nat (s k0 i) (S n0))).(\lambda (H5: (eq T v0 v)).(\lambda (H6: (eq C (CHead c1 k0 u1) (CHead c k t))).(\lambda (H7: (eq C (CHead c0 k0 u2) c2)).((let H8 \def (f_equal nat nat (\lambda (e0: nat).e0) (s k0 i) (S n0) H4) in (eq_ind nat (s k0 i) (\lambda (n: nat).((eq T v0 v) \to ((eq C (CHead c1 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i v0 u1 u2) \to ((csubst0 i v0 c1 c0) \to (or4 (drop n O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (H9: (eq T v0 v)).(eq_ind T v (\lambda (t0: T).((eq C (CHead c1 k0 u1) (CHead c k t)) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i t0 u1 u2) \to ((csubst0 i t0 c1 c0) \to (or4 (drop (s k0 i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))) (\lambda (H10: (eq C (CHead c1 k0 u1) (CHead c k t))).(let H11 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k0 u1) (CHead c k t) H10) in ((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c1 k0 u1) (CHead c k t) H10) in ((let H13 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k0 u1) (CHead c k t) H10) in (eq_ind C c (\lambda (c: C).((eq K k0 k) \to ((eq T u1 t) \to ((eq C (CHead c0 k0 u2) c2) \to ((subst0 i v u1 u2) \to ((csubst0 i v c c0) \to (or4 (drop (s k0 i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k0 i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (H14: (eq K k0 k)).(eq_ind K k (\lambda (k: K).((eq T u1 t) \to ((eq C (CHead c0 k u2) c2) \to ((subst0 i v u1 u2) \to ((csubst0 i v c c0) \to (or4 (drop (s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))) (\lambda (H15: (eq T u1 t)).(eq_ind T t (\lambda (t: T).((eq C (CHead c0 k u2) c2) \to ((subst0 i v t u2) \to ((csubst0 i v c c0) \to (or4 (drop (s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))) (\lambda (H16: (eq C (CHead c0 k u2) c2)).(eq_ind C (CHead c0 k u2) (\lambda (c2: C).((subst0 i v t u2) \to ((csubst0 i v c c0) \to (or4 (drop (s k i) O c2 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k i) O c2 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O c2 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (H17: (subst0 i v t u2)).(\lambda (H18: (csubst0 i v c c0)).(let H0 \def (eq_ind K k0 (\lambda (k: K).(eq nat (s k i) (S n0))) H8 k H14) in (K_ind (\lambda (k: K).((drop (r k n0) O c e) \to ((eq nat (s k i) (S n0)) \to (or4 (drop (s k i) O (CHead c0 k u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O (CHead c0 k u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s k i) O (CHead c0 k u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s k i) O (CHead c0 k u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) (\lambda (b: B).(\lambda (H1: (drop (r (Bind b) n0) O c e)).(\lambda (H19: (eq nat (s (Bind b) i) (S n0))).(let H20 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S n0) H19) in (let H21 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v c c0)) H18 n0 H20) in (let H22 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v t u2)) H17 n0 H20) in (eq_ind_r nat n0 (\lambda (n: nat).(or4 (drop (s (Bind b) n) O (CHead c0 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (let H \def (IHn c c0 v H21 e H1) in (or4_ind (drop n0 O c0 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind 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(ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H24: (eq C e (CHead x1 (Flat x0) x2))).(\lambda (H25: (drop n0 O c0 (CHead x1 (Flat x0) x3))).(\lambda (H26: (subst0 O v x2 x3)).(eq_ind_r C (CHead x1 (Flat x0) x2) (\lambda (c: C).(or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: 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(CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x2)) (drop_drop (Bind b) n0 c0 (CHead x1 (Flat x0) x3) H25 u2) H26)) e H24)))))))) H23)) (\lambda (H23: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n0 O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H24: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H25: (drop n0 O c0 (CHead x2 (Flat x0) x3))).(\lambda (H26: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Bind b) n0 c0 (CHead x2 (Flat x0) x3) H25 u2) H26)) e H24)))))))) H23)) (\lambda (H23: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n0 O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H24: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H25: (drop n0 O c0 (CHead x2 (Flat x0) x4))).(\lambda (H26: (subst0 O v x3 x4)).(\lambda (H27: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Bind b) n0) O (CHead c0 (Bind b) u2) (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Bind b) n0 c0 (CHead x2 (Flat x0) x4) H25 u2) H26 H27)) e H24)))))))))) H23)) H)) i H20))))))) (\lambda (f: F).(\lambda (H1: (drop (r (Flat f) n0) O c e)).(\lambda (H19: (eq nat (s (Flat f) i) (S n0))).(let H20 \def (f_equal nat nat (\lambda (e0: nat).e0) i (S n0) H19) in (let H21 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v c c0)) H18 (S n0) H20) in (let H22 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v t u2)) H17 (S n0) H20) in (eq_ind_r nat (S n0) (\lambda (n: nat).(or4 (drop (s (Flat f) n) O (CHead c0 (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) n) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) n) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) n) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (let H23 \def (H c0 v H21 e H1) in (or4_ind (drop (S n0) O c0 e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H24: (drop (S n0) O c0 e)).(or4_intro0 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c0 e H24 u2))) (\lambda (H24: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f) u)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))))).(ex3_4_ind F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H25: (eq C e (CHead x1 (Flat x0) x2))).(\lambda (H26: (drop (S n0) O c0 (CHead x1 (Flat x0) x3))).(\lambda (H27: (subst0 O v x2 x3)).(eq_ind_r C (CHead x1 (Flat x0) x2) (\lambda (c: C).(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) c) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead x1 (Flat x0) x2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x2) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x2)) (drop_drop (Flat f) n0 c0 (CHead x1 (Flat x0) x3) H26 u2) H27)) e H25)))))))) H24)) (\lambda (H24: (ex3_4 F C C T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f) u)))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H25: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H26: (drop (S n0) O c0 (CHead x2 (Flat x0) x3))).(\lambda (H27: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) c) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x0 x1 x2 x3 (refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Flat f) n0 c0 (CHead x2 (Flat x0) x3) H26 u2) H27)) e H25)))))))) H24)) (\lambda (H24: (ex4_5 F C C T T (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f) u))))))) (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (S n0) O c0 (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: F).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H25: (eq C e (CHead x1 (Flat x0) x3))).(\lambda (H26: (drop (S n0) O c0 (CHead x2 (Flat x0) x4))).(\lambda (H27: (subst0 O v x3 x4)).(\lambda (H28: (csubst0 O v x1 x2)).(eq_ind_r C (CHead x1 (Flat x0) x3) (\lambda (c: C).(or4 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) c) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead x1 (Flat x0) x3)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e0 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e0 (Flat f0) w)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Flat x0) x3) (CHead e1 (Flat f0) u))))))) (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop (s (Flat f) (S n0)) O (CHead c0 (Flat f) u2) (CHead e2 (Flat f0) w))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 O v u w)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x0 x1 x2 x3 x4 (refl_equal C (CHead x1 (Flat x0) x3)) (drop_drop (Flat f) n0 c0 (CHead x2 (Flat x0) x4) H26 u2) H27 H28)) e H25)))))))))) H24)) H23)) i H20))))))) k (drop_gen_drop k c e t n0 H1) H0)))) c2 H16)) u1 (sym_eq T u1 t H15))) k0 (sym_eq K k0 k H14))) c1 (sym_eq C c1 c H13))) H12)) H11))) v0 (sym_eq T v0 v H9))) (S n0) H8)) H5 H6 H7 H2 H3)))))]) in (H2 (refl_equal nat (S n0)) (refl_equal T v) (refl_equal C (CHead c k t)) (refl_equal C c2)))))))))))) c1)))) n).
 
 theorem csubst0_drop_eq_back:
  \forall (n: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n v c1 c2) \to (\forall (e: C).((drop n O c2 e) \to (or4 (drop n O c1 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))
 \def
- \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c2 e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c2 e)).(eq_ind C c2 (\lambda (c: C).(or4 (drop O O c1 c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2)) (or4 (drop O O c1 c2) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c2 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (y: nat).(\lambda (H1: (csubst0 y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop O O c c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O t u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O t u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O t e1 e2))))))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s k0 i) O) \to (or4 (drop O O (CHead c k0 u1) (CHead c k0 u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c k0 u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c k0 u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c k0 u2) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c k0 u1) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c k0 u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c k0 u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or4 (drop O O (CHead c (Bind b) u1) (CHead c (Bind b) u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Bind b) u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c (Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Bind b) u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c (Bind b) u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))) H4)))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i O)).(let H4 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 O H3) in (or4_intro1 (drop O O (CHead c (Flat f) u1) (CHead c (Flat f) u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u2)) (drop_refl (CHead c (Flat f) u1)) H4))))))))))) k)) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4: C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))) \to (\forall (u: T).((eq nat (s k0 i) O) \to (or4 (drop O O (CHead c3 k0 u) (CHead c4 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 k0 u) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop O O (CHead c3 k0 u) (CHead e1 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 k0 u) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3 c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat (S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (or4 (drop O O (CHead c3 (Bind b) u) (CHead c4 (Bind b) u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Bind b) u) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 (Bind b) u) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Bind b) u) (CHead e2 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Bind b) u) (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 (Bind b) u) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2: (csubst0 i v0 c3 c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i O)).(let H5 \def (eq_ind nat i (\lambda (n: nat).((eq nat n O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v0 c3 c4)) H2 O H4) in (or4_intro2 (drop O O (CHead c3 (Flat f) u) (CHead c4 (Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Flat f) u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c4 (Flat f) u)) (drop_refl (CHead c3 (Flat f) u)) H6))))))))))))) k)) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4: C).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop O O (CHead c3 k0 u1) (CHead c4 k0 u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0 u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 k0 u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 k0 u2) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c3 k0 u1) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0 u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 k0 u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3 c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5) in (False_ind (or4 (drop O O (CHead c3 (Bind b) u1) (CHead c4 (Bind b) u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))) H6))))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3 c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (H5: (eq nat i O)).(let H6 \def (eq_ind nat i (\lambda (n: nat).((eq nat n O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))) H4 O H5) in (let H7 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v0 c3 c4)) H3 O H5) in (let H8 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 O H5) in (or4_intro3 (drop O O (CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c4 (Flat f) u2)) (drop_refl (CHead c3 (Flat f) u1)) H8 H7)))))))))))))))) k)) y v c1 c2 H1))) H) e (drop_gen_refl c2 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c2 e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: (csubst0 (S n0) v (CSort n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2 e)).(csubst0_gen_sort c2 v (S n0) n1 H (or4 (drop (S n0) O (CSort n1) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CSort n1) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 (S n0) v (CHead c k t) c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O c2 e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat (S n0) (s k x1))).(\lambda (H4: (eq C c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(let H6 \def (eq_ind C c2 (\lambda (c: C).(drop (S n0) O c e)) H1 (CHead c k x0) H4) in ((match k return (\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to ((drop (r k0 n0) O c e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) with [(Bind b) \Rightarrow (\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(\lambda (H8: (drop (r (Bind b) n0) O c e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow n0 | (S n) \Rightarrow n])) (S n0) (S x1) H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5 n0 H9) in (or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e H8 t)))))) | (Flat f) \Rightarrow (\lambda (H7: (eq nat (S n0) (s (Flat f) x1))).(\lambda (H8: (drop (r (Flat f) n0) O c e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).e0) (S n0) x1 H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5 (S n0) H9) in (or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e H8 t))))))]) H3 (drop_gen_drop k c e x0 n0 H6)))))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S n0) (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 v c x0)).(let H6 \def (eq_ind C c2 (\lambda (c: C).(drop (S n0) O c e)) H1 (CHead x0 k t) H4) in ((match k return (\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to ((drop (r k0 n0) O x0 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) with [(Bind b) \Rightarrow (\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(\lambda (H8: (drop (r (Bind b) n0) O x0 e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow n0 | (S n) \Rightarrow n])) (S n0) (S x1) H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H5 n0 H9) in (let H11 \def (IHn c x0 v H10 e H8) in (or4_ind (drop n0 O c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H12: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H14: (drop n0 O c (CHead x3 (Flat x2) x4))).(\lambda (H15: (subst0 O v x4 x5)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Bind b) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0 c (CHead x3 (Flat x2) x4) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (H13: (eq C e 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(CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Bind b) t) (CHead x4 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: 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F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop (Bind b) n0 c (CHead x3 (Flat x2) x5) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: 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T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H13: (eq C e (CHead x4 (Flat x2) x6))).(\lambda (H14: (drop n0 O c (CHead x3 (Flat x2) x5))).(\lambda (H15: (subst0 O v x5 x6)).(\lambda (H16: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x4 (Flat x2) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Bind b) t) (CHead x4 (Flat x2) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6)) (drop_drop (Bind b) n0 c (CHead x3 (Flat x2) x5) H14 t) H15 H16)) e H13)))))))))) H12)) H11)))))) | (Flat f) \Rightarrow (\lambda (H7: (eq nat (S n0) (s (Flat f) x1))).(\lambda 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C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H12: (drop (S n0) O c e)).(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H14: (drop (S n0) O c (CHead x3 (Flat x2) x4))).(\lambda (H15: (subst0 O v x4 x5)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Flat f) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x4) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (H13: (eq C e (CHead x4 (Flat x2) x5))).(\lambda (H14: (drop (S n0) O c (CHead x3 (Flat x2) x5))).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x4 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat f) t) (CHead x4 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x5) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H13: (eq C e (CHead x4 (Flat x2) x6))).(\lambda (H14: (drop (S n0) O c (CHead x3 (Flat x2) x5))).(\lambda (H15: (subst0 O v x5 x6)).(\lambda (H16: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x4 (Flat x2) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat f) t) (CHead x4 (Flat x2) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6)) (drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x5) H14 t) H15 H16)) e H13)))))))))) H12)) H11))))))]) H3 (drop_gen_drop k x0 e t n0 H6)))))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H3: (eq nat (S n0) (s k x2))).(\lambda (H4: (eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 v c x1)).(let H7 \def (eq_ind C c2 (\lambda (c: C).(drop (S n0) O c e)) H1 (CHead x1 k x0) H4) in ((match k return (\lambda (k0: K).((eq nat (S n0) (s k0 x2)) \to ((drop (r k0 n0) O x1 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))) with [(Bind b) \Rightarrow (\lambda (H8: (eq nat (S n0) (s (Bind b) x2))).(\lambda (H9: (drop (r (Bind b) n0) O x1 e)).(let H10 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow n0 | (S n) \Rightarrow n])) (S n0) (S x2) H8) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c x1)) H6 n0 H10) in (let H12 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v t x0)) H5 n0 H10) in (let H13 \def (IHn c x1 v H11 e H9) in (or4_ind (drop n0 O c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H14: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e H14 t))) (\lambda (H14: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) x6))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3) x5))).(\lambda (H17: (subst0 O v x5 x6)).(eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: 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T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Bind b) t) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Bind b) n0 c (CHead x4 (Flat x3) x5) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) x6))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3) x6))).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Bind b) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop (Bind b) n0 c (CHead x4 (Flat x3) x6) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14: (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x6 x7)).(\lambda (H18: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x7) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Bind b) t) (CHead x5 (Flat x3) x7)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat x3) x7)) (drop_drop (Bind b) n0 c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e H15)))))))))) H14)) H13))))))) | (Flat f) \Rightarrow (\lambda (H8: (eq nat (S n0) (s (Flat f) x2))).(\lambda (H9: (drop (r (Flat f) n0) O x1 e)).(let H10 \def (f_equal nat nat (\lambda (e0: nat).e0) (S n0) x2 H8) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c x1)) H6 (S n0) H10) in (let H12 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v t x0)) H5 (S n0) H10) in (let H13 \def (H x1 v H11 e H9) in (or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H14: (drop (S n0) O c e)).(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e H14 t))) (\lambda (H14: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) x6))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3) x5))).(\lambda (H17: (subst0 O v x5 x6)).(eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Flat f) t) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x5) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) x6))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3) x6))).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat f) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14: (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x6 x7)).(\lambda (H18: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x7) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat f) t) (CHead x5 (Flat x3) x7)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat x3) x7)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e H15)))))))))) H14)) H13)))))))]) H3 (drop_gen_drop k x1 e x0 n0 H7)))))))))) H2)) (csubst0_gen_head k c c2 t v (S n0) H0))))))))))) c1)))) n).
+ \lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c2 e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: (csubst0 O v c1 c2)).(\lambda (e: C).(\lambda (H0: (drop O O c2 e)).(eq_ind C c2 (\lambda (c: C).(or4 (drop O O c1 c) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (insert_eq nat O (\lambda (n0: nat).(csubst0 n0 v c1 c2)) (or4 (drop O O c1 c2) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c2 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c2 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (y: nat).(\lambda (H1: (csubst0 y v c1 c2)).(csubst0_ind (\lambda (n0: nat).(\lambda (t: T).(\lambda (c: C).(\lambda (c0: C).((eq nat n0 O) \to (or4 (drop O O c c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O t u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O t e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O t u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O t e1 e2))))))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c: C).((eq nat (s k0 i) O) \to (or4 (drop O O (CHead c k0 u1) (CHead c k0 u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c k0 u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c k0 u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c k0 u2) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c k0 u1) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c k0 u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c k0 u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat (S i) O)).(let H4 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind (or4 (drop O O (CHead c (Bind b) u1) (CHead c (Bind b) u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Bind b) u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c (Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Bind b) u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c (Bind b) u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))) H4)))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c: C).(\lambda (H3: (eq nat i O)).(let H4 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 O H3) in (or4_intro1 (drop O O (CHead c (Flat f) u1) (CHead c (Flat f) u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c (Flat f) u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c (Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4))))) f c u1 u2 (refl_equal C (CHead c (Flat f) u2)) (drop_refl (CHead c (Flat f) u1)) H4))))))))))) k)) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (c3: C).(\forall (c4: C).(\forall (v0: T).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))) \to (\forall (u: T).((eq nat (s k0 i) O) \to (or4 (drop O O (CHead c3 k0 u) (CHead c4 k0 u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 k0 u) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop O O (CHead c3 k0 u) (CHead e1 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 k0 u) (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 k0 u) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (_: (csubst0 i v0 c3 c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat (S i) O)).(let H5 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (or4 (drop O O (CHead c3 (Bind b) u) (CHead c4 (Bind b) u)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Bind b) u) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 (Bind b) u) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Bind b) u) (CHead e2 (Flat f) u0)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind b) u) (CHead e1 (Flat f) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Bind b) u) (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 (Bind b) u) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))) H5))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (c3: C).(\lambda (c4: C).(\lambda (v0: T).(\lambda (H2: (csubst0 i v0 c3 c4)).(\lambda (H3: (((eq nat i O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (u: T).(\lambda (H4: (eq nat i O)).(let H5 \def (eq_ind nat i (\lambda (n: nat).((eq nat n O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))) H3 O H4) in (let H6 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v0 c3 c4)) H2 O H4) in (or4_intro2 (drop O O (CHead c3 (Flat f) u) (CHead c4 (Flat f) u)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Flat f) u) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead c4 (Flat f) u) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u0: T).(eq C (CHead c4 (Flat f) u) (CHead e2 (Flat f0) u0)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u0: T).(drop O O (CHead c3 (Flat f) u) (CHead e1 (Flat f0) u0)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2))))) f c3 c4 u (refl_equal C (CHead c4 (Flat f) u)) (drop_refl (CHead c3 (Flat f) u)) H6))))))))))))) k)) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (v0: T).(\forall (u1: T).(\forall (u2: T).((subst0 i v0 u1 u2) \to (\forall (c3: C).(\forall (c4: C).((csubst0 i v0 c3 c4) \to ((((eq nat i O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))) \to ((eq nat (s k0 i) O) \to (or4 (drop O O (CHead c3 k0 u1) (CHead c4 k0 u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0 u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 k0 u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 k0 u2) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c3 k0 u1) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 k0 u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 k0 u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))))))))))) (\lambda (b: B).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (subst0 i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (_: (csubst0 i v0 c3 c4)).(\lambda (_: (((eq nat i O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (H5: (eq nat (S i) O)).(let H6 \def (eq_ind nat (S i) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5) in (False_ind (or4 (drop O O (CHead c3 (Bind b) u1) (CHead c4 (Bind b) u2)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) u2) (CHead e0 (Flat f) u4)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Bind b) u1) (CHead e0 (Flat f) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 (Bind b) u2) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Bind b) u2) (CHead e2 (Flat f) u4))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Bind b) u1) (CHead e1 (Flat f) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))) H6))))))))))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (v0: T).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (subst0 i v0 u1 u2)).(\lambda (c3: C).(\lambda (c4: C).(\lambda (H3: (csubst0 i v0 c3 c4)).(\lambda (H4: (((eq nat i O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))))))).(\lambda (H5: (eq nat i O)).(let H6 \def (eq_ind nat i (\lambda (n: nat).((eq nat n O) \to (or4 (drop O O c3 c4) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c4 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O c3 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c4 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop O O c3 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v0 u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))))))) H4 O H5) in (let H7 \def (eq_ind nat i (\lambda (n: nat).(csubst0 n v0 c3 c4)) H3 O H5) in (let H8 \def (eq_ind nat i (\lambda (n: nat).(subst0 n v0 u1 u2)) H2 O H5) in (or4_intro3 (drop O O (CHead c3 (Flat f) u1) (CHead c4 (Flat f) u2)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) u2) (CHead e0 (Flat f0) u4)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead e0 (Flat f0) u3)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop O O (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u4: T).(eq C (CHead c4 (Flat f) u2) (CHead e2 (Flat f0) u4))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (_: T).(drop O O (CHead c3 (Flat f) u1) (CHead e1 (Flat f0) u3))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u3: T).(\lambda (u4: T).(subst0 O v0 u3 u4)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v0 e1 e2)))))) f c3 c4 u1 u2 (refl_equal C (CHead c4 (Flat f) u2)) (drop_refl (CHead c3 (Flat f) u1)) H8 H7)))))))))))))))) k)) y v c1 c2 H1))) H) e (drop_gen_refl c2 e H0)))))))) (\lambda (n0: nat).(\lambda (IHn: ((\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 n0 v c1 c2) \to (\forall (e: C).((drop n0 O c2 e) \to (or4 (drop n0 O c1 e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c1 (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c1 (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c1 (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))))) (\lambda (n1: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: (csubst0 (S n0) v (CSort n1) c2)).(\lambda (e: C).(\lambda (_: (drop (S n0) O c2 e)).(csubst0_gen_sort c2 v (S n0) n1 H (or4 (drop (S n0) O (CSort n1) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CSort n1) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CSort n1) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 (S n0) v c c2) \to (\forall (e: C).((drop (S n0) O c2 e) \to (or4 (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 (S n0) v (CHead c k t) c2)).(\lambda (e: C).(\lambda (H1: (drop (S n0) O c2 e)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat (S n0) (s k x1))).(\lambda (H4: (eq C c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(let H6 \def (eq_ind C c2 (\lambda (c: C).(drop (S n0) O c e)) H1 (CHead c k x0) H4) in ((match k return (\lambda (_: ?).(\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to ((drop (r k0 n0) O c e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))) with [(Bind b) \Rightarrow (\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(\lambda (H8: (drop (r (Bind b) n0) O c e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow n0 | (S n) \Rightarrow n])) (S n0) (S x1) H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5 n0 H9) in (or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e H8 t)))))) | (Flat f) \Rightarrow (\lambda (H7: (eq nat (S n0) (s (Flat f) x1))).(\lambda (H8: (drop (r (Flat f) n0) O c e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).e0) (S n0) x1 H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5 (S n0) H9) in (or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e H8 t))))))]) H3 (drop_gen_drop k c e x0 n0 H6)))))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S n0) (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 v c x0)).(let H6 \def (eq_ind C c2 (\lambda (c: C).(drop (S n0) O c e)) H1 (CHead x0 k t) H4) in ((match k return (\lambda (_: ?).(\lambda (k0: K).((eq nat (S n0) (s k0 x1)) \to ((drop (r k0 n0) O x0 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))) with [(Bind b) \Rightarrow (\lambda (H7: (eq nat (S n0) (s (Bind b) x1))).(\lambda (H8: (drop (r (Bind b) n0) O x0 e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow n0 | (S n) \Rightarrow n])) (S n0) (S x1) H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H5 n0 H9) in (let H11 \def (IHn c x0 v H10 e H8) in (or4_ind (drop n0 O c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H12: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H14: (drop n0 O c (CHead x3 (Flat x2) x4))).(\lambda (H15: (subst0 O 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F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Bind b) n0 c (CHead x3 (Flat x2) x4) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) 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C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Bind b) t) (CHead x4 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop (Bind b) n0 c (CHead x3 (Flat x2) x5) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H13: (eq C e (CHead x4 (Flat x2) x6))).(\lambda (H14: (drop n0 O c (CHead x3 (Flat x2) x5))).(\lambda (H15: (subst0 O v x5 x6)).(\lambda (H16: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x4 (Flat x2) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Bind b) t) (CHead x4 (Flat x2) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6)) (drop_drop (Bind b) n0 c (CHead x3 (Flat x2) x5) H14 t) H15 H16)) e H13)))))))))) H12)) H11)))))) | (Flat f) \Rightarrow (\lambda (H7: (eq nat (S n0) (s (Flat f) x1))).(\lambda (H8: (drop (r (Flat f) n0) O x0 e)).(let H9 \def (f_equal nat nat (\lambda (e0: nat).e0) (S n0) x1 H7) in (let H10 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H5 (S n0) H9) in (let H11 \def (H x0 v H10 e H8) in (or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H12: (drop (S n0) O c e)).(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e H12 t))) (\lambda (H12: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H13: (eq C e (CHead x3 (Flat x2) x5))).(\lambda (H14: (drop (S n0) O c (CHead x3 (Flat x2) x4))).(\lambda (H15: (subst0 O v x4 x5)).(eq_ind_r C (CHead x3 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Flat f) t) (CHead x3 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x3 (Flat x2) x5) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) x2 x3 x4 x5 (refl_equal C (CHead x3 (Flat x2) x5)) (drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x4) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (H13: (eq C e (CHead x4 (Flat x2) x5))).(\lambda (H14: (drop (S n0) O c (CHead x3 (Flat x2) x5))).(\lambda (H15: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x4 (Flat x2) x5) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat f) t) (CHead x4 (Flat x2) x5)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x5) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x2 x3 x4 x5 (refl_equal C (CHead x4 (Flat x2) x5)) (drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x5) H14 t) H15)) e H13)))))))) H12)) (\lambda (H12: (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x2: F).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H13: (eq C e (CHead x4 (Flat x2) x6))).(\lambda (H14: (drop (S n0) O c (CHead x3 (Flat x2) x5))).(\lambda (H15: (subst0 O v x5 x6)).(\lambda (H16: (csubst0 O v x3 x4)).(eq_ind_r C (CHead x4 (Flat x2) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat f) t) (CHead x4 (Flat x2) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x2) x6) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x2 x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x2) x6)) (drop_drop (Flat f) n0 c (CHead x3 (Flat x2) x5) H14 t) H15 H16)) e H13)))))))))) H12)) H11))))))]) H3 (drop_gen_drop k x0 e t n0 H6)))))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S n0) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (drop (S n0) O (CHead c k t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H3: (eq nat (S n0) (s k x2))).(\lambda (H4: (eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 v c x1)).(let H7 \def (eq_ind C c2 (\lambda (c: C).(drop (S n0) O c e)) H1 (CHead x1 k x0) H4) in ((match k return (\lambda (_: ?).(\lambda (k0: K).((eq nat (S n0) (s k0 x2)) \to ((drop (r k0 n0) O x1 e) \to (or4 (drop (S n0) O (CHead c k0 t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c k0 t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))))))) with [(Bind b) \Rightarrow (\lambda (H8: (eq nat (S n0) (s (Bind b) x2))).(\lambda (H9: (drop (r (Bind b) n0) O x1 e)).(let H10 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow n0 | (S n) \Rightarrow n])) (S n0) (S x2) H8) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c x1)) H6 n0 H10) in (let H12 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v t x0)) H5 n0 H10) in (let H13 \def (IHn c x1 v H11 e H9) in (or4_ind (drop n0 O c e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H14: (drop n0 O c e)).(or4_intro0 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Bind b) n0 c e H14 t))) (\lambda (H14: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: 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u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) x6))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3) x5))).(\lambda (H17: (subst0 O v x5 x6)).(eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 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C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Bind b) t) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Bind b) n0 c (CHead x4 (Flat x3) x5) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop n0 O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) x6))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3) x6))).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x6) (\lambda (c0: 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v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop (Bind b) n0 c (CHead x4 (Flat x3) x6) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14: (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop n0 O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Bind b) t) e) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H16: (drop n0 O c (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x6 x7)).(\lambda (H18: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x7) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Bind b) t) c0) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Bind b) t) (CHead x5 (Flat x3) x7)) (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Bind b) t) (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat x3) x7)) (drop_drop (Bind b) n0 c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e H15)))))))))) H14)) H13))))))) | (Flat f) \Rightarrow (\lambda (H8: (eq nat (S n0) (s (Flat f) x2))).(\lambda (H9: (drop (r (Flat f) n0) O x1 e)).(let H10 \def (f_equal nat nat (\lambda (e0: nat).e0) (S n0) x2 H8) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c x1)) H6 (S n0) H10) in (let H12 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v t x0)) H5 (S n0) H10) in (let H13 \def (H x1 v H11 e H9) in (or4_ind (drop (S n0) O c e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (H14: (drop (S n0) O c e)).(or4_intro0 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (drop_drop (Flat f) n0 c e H14 t))) (\lambda (H14: (ex3_4 F C T T (\lambda (f: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f) u2)))))) (\lambda (f: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))))).(ex3_4_ind F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x4 (Flat x3) x6))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3) x5))).(\lambda (H17: (subst0 O v x5 x6)).(eq_ind_r C (CHead x4 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro1 (drop (S n0) O (CHead c (Flat f) t) (CHead x4 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x4 (Flat x3) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2))))) x3 x4 x5 x6 (refl_equal C (CHead x4 (Flat x3) x6)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x5) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14: (ex3_4 F C C T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f) u)))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))))).(ex3_4_ind F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O c (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) x6))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3) x6))).(\lambda (H17: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x6) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro2 (drop (S n0) O (CHead c (Flat f) t) (CHead x5 (Flat x3) x6)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex3_4_intro F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x6) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2))))) x3 x4 x5 x6 (refl_equal C (CHead x5 (Flat x3) x6)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17)) e H15)))))))) H14)) (\lambda (H14: (ex4_5 F C C T T (\lambda (f: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f) u2))))))) (\lambda (f: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))).(ex4_5_ind F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O c (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) (or4 (drop (S n0) O (CHead c (Flat f) t) e) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C e (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C e (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))))) (\lambda (x3: F).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H15: (eq C e (CHead x5 (Flat x3) x7))).(\lambda (H16: (drop (S n0) O c (CHead x4 (Flat x3) x6))).(\lambda (H17: (subst0 O v x6 x7)).(\lambda (H18: (csubst0 O v x4 x5)).(eq_ind_r C (CHead x5 (Flat x3) x7) (\lambda (c0: C).(or4 (drop (S n0) O (CHead c (Flat f) t) c0) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C c0 (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C c0 (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))))) (or4_intro3 (drop (S n0) O (CHead c (Flat f) t) (CHead x5 (Flat x3) x7)) (ex3_4 F C T T (\lambda (f0: F).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e0 (Flat f0) u2)))))) (\lambda (f0: F).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e0 (Flat f0) u1)))))) (\lambda (_: F).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (ex3_4 F C C T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f0) u)))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 O v e1 e2)))))) (ex4_5 F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2))))))) (ex4_5_intro F C C T T (\lambda (f0: F).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(eq C (CHead x5 (Flat x3) x7) (CHead e2 (Flat f0) u2))))))) (\lambda (f0: F).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(drop (S n0) O (CHead c (Flat f) t) (CHead e1 (Flat f0) u1))))))) (\lambda (_: F).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 O v u1 u2)))))) (\lambda (_: F).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 O v e1 e2)))))) x3 x4 x5 x6 x7 (refl_equal C (CHead x5 (Flat x3) x7)) (drop_drop (Flat f) n0 c (CHead x4 (Flat x3) x6) H16 t) H17 H18)) e H15)))))))))) H14)) H13)))))))]) H3 (drop_gen_drop k x1 e x0 n0 H7)))))))))) H2)) (csubst0_gen_head k c c2 t v (S n0) H0))))))))))) c1)))) n).
 
 theorem csubst0_clear_O:
  \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to (\forall (c: C).((clear c1 c) \to (clear c2 c))))))
 \def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c c0) \to (clear c2 c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: (csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear (CSort n) c)).(csubst0_gen_sort c2 v O n H (clear c2 c)))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c c0) \to (clear c2 c0)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O v (CHead c k t) c2)).(\lambda (c0: C).(\lambda (H1: (clear (CHead c k t) c0)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (clear c2 c0) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (clear c2 c0) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat O (s k x1))).(\lambda (H4: (eq C c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c3: C).(clear c3 c0)) ((match k return (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 x1)) \to (clear (CHead c k0 x0) c0)))) with [(Bind b) \Rightarrow (\lambda (_: (clear (CHead c (Bind b) t) c0)).(\lambda (H7: (eq nat O (s (Bind b) x1))).(let H8 \def (eq_ind nat O (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7) in (False_ind (clear (CHead c (Bind b) x0) c0) H8)))) | (Flat f) \Rightarrow (\lambda (H6: (clear (CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat O (s (Flat f) x1))).(let H8 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5 O H7) in (clear_flat c c0 (clear_gen_flat f c c0 t H6) f x0))))]) H1 H3) c2 H4)))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (clear c2 c0) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat O (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c3: C).(clear c3 c0)) ((match k return (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 x1)) \to (clear (CHead x0 k0 t) c0)))) with [(Bind b) \Rightarrow (\lambda (_: (clear (CHead c (Bind b) t) c0)).(\lambda (H7: (eq nat O (s (Bind b) x1))).(let H8 \def (eq_ind nat O (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7) in (False_ind (clear (CHead x0 (Bind b) t) c0) H8)))) | (Flat f) \Rightarrow (\lambda (H6: (clear (CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat O (s (Flat f) x1))).(let H8 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H5 O H7) in (clear_flat x0 c0 (H x0 v H8 c0 (clear_gen_flat f c c0 t H6)) f t))))]) H1 H3) c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (clear c2 c0) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H3: (eq nat O (s k x2))).(\lambda (H4: (eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c3: C).(clear c3 c0)) ((match k return (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 x2)) \to (clear (CHead x1 k0 x0) c0)))) with [(Bind b) \Rightarrow (\lambda (_: (clear (CHead c (Bind b) t) c0)).(\lambda (H8: (eq nat O (s (Bind b) x2))).(let H9 \def (eq_ind nat O (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x2) H8) in (False_ind (clear (CHead x1 (Bind b) x0) c0) H9)))) | (Flat f) \Rightarrow (\lambda (H7: (clear (CHead c (Flat f) t) c0)).(\lambda (H8: (eq nat O (s (Flat f) x2))).(let H9 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c x1)) H6 O H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v t x0)) H5 O H8) in (clear_flat x1 c0 (H x1 v H9 c0 (clear_gen_flat f c c0 t H7)) f x0)))))]) H1 H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 t v O H0))))))))))) c1).
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c c0) \to (clear c2 c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: (csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear (CSort n) c)).(csubst0_gen_sort c2 v O n H (clear c2 c)))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c c0) \to (clear c2 c0)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O v (CHead c k t) c2)).(\lambda (c0: C).(\lambda (H1: (clear (CHead c k t) c0)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (clear c2 c0) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (clear c2 c0) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat O (s k x1))).(\lambda (H4: (eq C c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c3: C).(clear c3 c0)) ((match k return (\lambda (_: ?).(\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 x1)) \to (clear (CHead c k0 x0) c0))))) with [(Bind b) \Rightarrow (\lambda (_: (clear (CHead c (Bind b) t) c0)).(\lambda (H7: (eq nat O (s (Bind b) x1))).(let H8 \def (eq_ind nat O (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7) in (False_ind (clear (CHead c (Bind b) x0) c0) H8)))) | (Flat f) \Rightarrow (\lambda (H6: (clear (CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat O (s (Flat f) x1))).(let H8 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5 O H7) in (clear_flat c c0 (clear_gen_flat f c c0 t H6) f x0))))]) H1 H3) c2 H4)))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (clear c2 c0) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat O (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c3: C).(clear c3 c0)) ((match k return (\lambda (_: ?).(\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 x1)) \to (clear (CHead x0 k0 t) c0))))) with [(Bind b) \Rightarrow (\lambda (_: (clear (CHead c (Bind b) t) c0)).(\lambda (H7: (eq nat O (s (Bind b) x1))).(let H8 \def (eq_ind nat O (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7) in (False_ind (clear (CHead x0 (Bind b) t) c0) H8)))) | (Flat f) \Rightarrow (\lambda (H6: (clear (CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat O (s (Flat f) x1))).(let H8 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H5 O H7) in (clear_flat x0 c0 (H x0 v H8 c0 (clear_gen_flat f c c0 t H6)) f t))))]) H1 H3) c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (clear c2 c0) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H3: (eq nat O (s k x2))).(\lambda (H4: (eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c3: C).(clear c3 c0)) ((match k return (\lambda (_: ?).(\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat O (s k0 x2)) \to (clear (CHead x1 k0 x0) c0))))) with [(Bind b) \Rightarrow (\lambda (_: (clear (CHead c (Bind b) t) c0)).(\lambda (H8: (eq nat O (s (Bind b) x2))).(let H9 \def (eq_ind nat O (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x2) H8) in (False_ind (clear (CHead x1 (Bind b) x0) c0) H9)))) | (Flat f) \Rightarrow (\lambda (H7: (clear (CHead c (Flat f) t) c0)).(\lambda (H8: (eq nat O (s (Flat f) x2))).(let H9 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c x1)) H6 O H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v t x0)) H5 O H8) in (clear_flat x1 c0 (H x1 v H9 c0 (clear_gen_flat f c c0 t H7)) f x0)))))]) H1 H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 t v O H0))))))))))) c1).
 
 theorem csubst0_clear_O_back:
  \forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c1 c2) \to (\forall (c: C).((clear c2 c) \to (clear c1 c))))))
 \def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c2 c0) \to (clear c c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: (csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear c2 c)).(csubst0_gen_sort c2 v O n H (clear (CSort n) c)))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c2 c0) \to (clear c c0)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O v (CHead c k t) c2)).(\lambda (c0: C).(\lambda (H1: (clear c2 c0)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (clear (CHead c k t) c0) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (clear (CHead c k t) c0) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat O (s k x1))).(\lambda (H4: (eq C c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(let H6 \def (eq_ind C c2 (\lambda (c: C).(clear c c0)) H1 (CHead c k x0) H4) in ((match k return (\lambda (k0: K).((eq nat O (s k0 x1)) \to ((clear (CHead c k0 x0) c0) \to (clear (CHead c k0 t) c0)))) with [(Bind b) \Rightarrow (\lambda (H7: (eq nat O (s (Bind b) x1))).(\lambda (_: (clear (CHead c (Bind b) x0) c0)).(let H9 \def (eq_ind nat O (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7) in (False_ind (clear (CHead c (Bind b) t) c0) H9)))) | (Flat f) \Rightarrow (\lambda (H7: (eq nat O (s (Flat f) x1))).(\lambda (H8: (clear (CHead c (Flat f) x0) c0)).(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5 O H7) in (clear_flat c c0 (clear_gen_flat f c c0 x0 H8) f t))))]) H3 H6))))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (clear (CHead c k t) c0) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat O (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 v c x0)).(let H6 \def (eq_ind C c2 (\lambda (c: C).(clear c c0)) H1 (CHead x0 k t) H4) in ((match k return (\lambda (k0: K).((eq nat O (s k0 x1)) \to ((clear (CHead x0 k0 t) c0) \to (clear (CHead c k0 t) c0)))) with [(Bind b) \Rightarrow (\lambda (H7: (eq nat O (s (Bind b) x1))).(\lambda (_: (clear (CHead x0 (Bind b) t) c0)).(let H9 \def (eq_ind nat O (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7) in (False_ind (clear (CHead c (Bind b) t) c0) H9)))) | (Flat f) \Rightarrow (\lambda (H7: (eq nat O (s (Flat f) x1))).(\lambda (H8: (clear (CHead x0 (Flat f) t) c0)).(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H5 O H7) in (clear_flat c c0 (H x0 v H9 c0 (clear_gen_flat f x0 c0 t H8)) f t))))]) H3 H6))))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (clear (CHead c k t) c0) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H3: (eq nat O (s k x2))).(\lambda (H4: (eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 v c x1)).(let H7 \def (eq_ind C c2 (\lambda (c: C).(clear c c0)) H1 (CHead x1 k x0) H4) in ((match k return (\lambda (k0: K).((eq nat O (s k0 x2)) \to ((clear (CHead x1 k0 x0) c0) \to (clear (CHead c k0 t) c0)))) with [(Bind b) \Rightarrow (\lambda (H8: (eq nat O (s (Bind b) x2))).(\lambda (_: (clear (CHead x1 (Bind b) x0) c0)).(let H10 \def (eq_ind nat O (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x2) H8) in (False_ind (clear (CHead c (Bind b) t) c0) H10)))) | (Flat f) \Rightarrow (\lambda (H8: (eq nat O (s (Flat f) x2))).(\lambda (H9: (clear (CHead x1 (Flat f) x0) c0)).(let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c x1)) H6 O H8) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v t x0)) H5 O H8) in (clear_flat c c0 (H x1 v H10 c0 (clear_gen_flat f x1 c0 x0 H9)) f t)))))]) H3 H7))))))))) H2)) (csubst0_gen_head k c c2 t v O H0))))))))))) c1).
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c2 c0) \to (clear c c0))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: (csubst0 O v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear c2 c)).(csubst0_gen_sort c2 v O n H (clear (CSort n) c)))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).((csubst0 O v c c2) \to (\forall (c0: C).((clear c2 c0) \to (clear c c0)))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 O v (CHead c k t) c2)).(\lambda (c0: C).(\lambda (H1: (clear c2 c0)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (clear (CHead c k t) c0) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (clear (CHead c k t) c0) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat O (s k x1))).(\lambda (H4: (eq C c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(let H6 \def (eq_ind C c2 (\lambda (c: C).(clear c c0)) H1 (CHead c k x0) H4) in ((match k return (\lambda (_: ?).(\lambda (k0: K).((eq nat O (s k0 x1)) \to ((clear (CHead c k0 x0) c0) \to (clear (CHead c k0 t) c0))))) with [(Bind b) \Rightarrow (\lambda (H7: (eq nat O (s (Bind b) x1))).(\lambda (_: (clear (CHead c (Bind b) x0) c0)).(let H9 \def (eq_ind nat O (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7) in (False_ind (clear (CHead c (Bind b) t) c0) H9)))) | (Flat f) \Rightarrow (\lambda (H7: (eq nat O (s (Flat f) x1))).(\lambda (H8: (clear (CHead c (Flat f) x0) c0)).(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5 O H7) in (clear_flat c c0 (clear_gen_flat f c c0 x0 H8) f t))))]) H3 H6))))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (clear (CHead c k t) c0) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat O (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 v c x0)).(let H6 \def (eq_ind C c2 (\lambda (c: C).(clear c c0)) H1 (CHead x0 k t) H4) in ((match k return (\lambda (_: ?).(\lambda (k0: K).((eq nat O (s k0 x1)) \to ((clear (CHead x0 k0 t) c0) \to (clear (CHead c k0 t) c0))))) with [(Bind b) \Rightarrow (\lambda (H7: (eq nat O (s (Bind b) x1))).(\lambda (_: (clear (CHead x0 (Bind b) t) c0)).(let H9 \def (eq_ind nat O (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x1) H7) in (False_ind (clear (CHead c (Bind b) t) c0) H9)))) | (Flat f) \Rightarrow (\lambda (H7: (eq nat O (s (Flat f) x1))).(\lambda (H8: (clear (CHead x0 (Flat f) t) c0)).(let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H5 O H7) in (clear_flat c c0 (H x0 v H9 c0 (clear_gen_flat f x0 c0 t H8)) f t))))]) H3 H6))))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat O (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (clear (CHead c k t) c0) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H3: (eq nat O (s k x2))).(\lambda (H4: (eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 v c x1)).(let H7 \def (eq_ind C c2 (\lambda (c: C).(clear c c0)) H1 (CHead x1 k x0) H4) in ((match k return (\lambda (_: ?).(\lambda (k0: K).((eq nat O (s k0 x2)) \to ((clear (CHead x1 k0 x0) c0) \to (clear (CHead c k0 t) c0))))) with [(Bind b) \Rightarrow (\lambda (H8: (eq nat O (s (Bind b) x2))).(\lambda (_: (clear (CHead x1 (Bind b) x0) c0)).(let H10 \def (eq_ind nat O (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S x2) H8) in (False_ind (clear (CHead c (Bind b) t) c0) H10)))) | (Flat f) \Rightarrow (\lambda (H8: (eq nat O (s (Flat f) x2))).(\lambda (H9: (clear (CHead x1 (Flat f) x0) c0)).(let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c x1)) H6 O H8) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v t x0)) H5 O H8) in (clear_flat c c0 (H x1 v H10 c0 (clear_gen_flat f x1 c0 x0 H9)) f t)))))]) H3 H7))))))))) H2)) (csubst0_gen_head k c c2 t v O H0))))))))))) c1).
 
 theorem csubst0_clear_S:
  \forall (c1: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0 (S i) v c1 c2) \to (\forall (c: C).((clear c1 c) \to (or4 (clear c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))))))
 \def
- \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0 (S i) v c c2) \to (\forall (c0: C).((clear c c0) \to (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H: (csubst0 (S i) v (CSort n) c2)).(\lambda (c: C).(\lambda (_: (clear (CSort n) c)).(csubst0_gen_sort c2 v (S i) n H (or4 (clear c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))))))))))) (\lambda (c: C).(\lambda (H: ((\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0 (S i) v c c2) \to (\forall (c0: C).((clear c c0) \to (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c2: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (csubst0 (S i) v (CHead c k t) c2)).(\lambda (c0: C).(\lambda (H1: (clear (CHead c k t) c0)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))))) (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (H2: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat (S i) (s k x1))).(\lambda (H4: (eq C c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) ((match k return (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat (S i) (s k0 x1)) \to (or4 (clear (CHead c k0 x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c k0 x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c k0 x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c k0 x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))) with [(Bind b) \Rightarrow (\lambda (H6: (clear (CHead c (Bind b) t) c0)).(\lambda (H7: (eq nat (S i) (s (Bind b) x1))).(let H8 \def (f_equal nat nat (\lambda (e: nat).(match e return (\lambda (_: ?).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5 i H8) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(or4 (clear (CHead c (Bind b) x0) c3) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (or4_intro1 (clear (CHead c (Bind b) x0) (CHead c (Bind b) t)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))) b c t x0 (refl_equal C (CHead c (Bind b) t)) (clear_bind b c x0) H9)) c0 (clear_gen_bind b c c0 t H6)))))) | (Flat f) \Rightarrow (\lambda (H6: (clear (CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat (S i) (s (Flat f) x1))).(let H8 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5 (S i) H8) in (or4_intro0 (clear (CHead c (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat c c0 (clear_gen_flat f c c0 t H6) f x0))))))]) H1 H3) c2 H4)))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S i) (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) ((match k return (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat (S i) (s k0 x1)) \to (or4 (clear (CHead x0 k0 t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 k0 t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 k0 t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 k0 t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))) with [(Bind b) \Rightarrow (\lambda (H6: (clear (CHead c (Bind b) t) c0)).(\lambda (H7: (eq nat (S i) (s (Bind b) x1))).(let H8 \def (f_equal nat nat (\lambda (e: nat).(match e return (\lambda (_: ?).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H5 i H8) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(or4 (clear (CHead x0 (Bind b) t) c3) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Bind b) t) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (or4_intro2 (clear (CHead x0 (Bind b) t) (CHead c (Bind b) t)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Bind b) t) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Bind b) t) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))) b c x0 t (refl_equal C (CHead c (Bind b) t)) (clear_bind b x0 t) H9)) c0 (clear_gen_bind b c c0 t H6)))))) | (Flat f) \Rightarrow (\lambda (H6: (clear (CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat (S i) (s (Flat f) x1))).(let H8 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H5 (S i) H8) in (let H10 \def (H x0 v i H9 c0 (clear_gen_flat f c c0 t H6)) in (or4_ind (clear x0 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x0 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (H11: (clear x0 c0)).(or4_intro0 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat x0 c0 H11 f t))) (\lambda (H11: (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: 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u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))) x2 x3 x4 x5 H12 (clear_flat x0 (CHead x3 (Bind x2) x5) H13 f t) H14))))))))) H11)) (\lambda (H11: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x0 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x0 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (H12: (eq C c0 (CHead x3 (Bind x2) x5))).(\lambda (H13: (clear x0 (CHead x4 (Bind x2) x5))).(\lambda (H14: (csubst0 i v x3 x4)).(or4_intro2 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: 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(ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))) x2 x3 x4 x5 H12 (clear_flat x0 (CHead x4 (Bind x2) x5) H13 f t) H14))))))))) H11)) (\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H12: (eq C c0 (CHead x3 (Bind x2) x5))).(\lambda (H13: (clear x0 (CHead x4 (Bind x2) x6))).(\lambda (H14: (subst0 i v x5 x6)).(\lambda (H15: (csubst0 i v x3 x4)).(or4_intro3 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x2 x3 x4 x5 x6 H12 (clear_flat x0 (CHead x4 (Bind x2) x6) H13 f t) H14 H15))))))))))) H11)) H10))))))]) H1 H3) c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H3: (eq nat (S i) (s k x2))).(\lambda (H4: (eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) ((match k return (\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat (S i) (s k0 x2)) \to (or4 (clear (CHead x1 k0 x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 k0 x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))))) with [(Bind b) \Rightarrow (\lambda (H7: (clear (CHead c (Bind b) t) c0)).(\lambda (H8: (eq nat (S i) (s (Bind b) x2))).(let H9 \def (f_equal nat nat (\lambda (e: nat).(match e return (\lambda (_: ?).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S x2) H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c x1)) H6 i H9) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v t x0)) H5 i H9) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(or4 (clear (CHead x1 (Bind b) x0) c3) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (or4_intro3 (clear (CHead x1 (Bind b) x0) (CHead c (Bind b) t)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))) b c x1 t x0 (refl_equal C (CHead c (Bind b) t)) (clear_bind b x1 x0) H11 H10)) c0 (clear_gen_bind b c c0 t H7))))))) | (Flat f) \Rightarrow (\lambda (H7: (clear (CHead c (Flat f) t) c0)).(\lambda (H8: (eq nat (S i) (s (Flat f) x2))).(let H9 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x2) H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c x1)) H6 (S i) H9) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v t x0)) H5 (S i) H9) in (let H12 \def (H x1 v i H10 c0 (clear_gen_flat f c c0 t H7)) in (or4_ind (clear x1 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (H13: (clear x1 c0)).(or4_intro0 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat x1 c0 H13 f x0))) (\lambda (H13: (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H14: (eq C c0 (CHead x4 (Bind x3) x5))).(\lambda (H15: (clear x1 (CHead x4 (Bind x3) x6))).(\lambda (H16: (subst0 i v x5 x6)).(or4_intro1 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))) x3 x4 x5 x6 H14 (clear_flat x1 (CHead x4 (Bind x3) x6) H15 f x0) H16))))))))) H13)) (\lambda (H13: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (H14: (eq C c0 (CHead x4 (Bind x3) x6))).(\lambda (H15: (clear x1 (CHead x5 (Bind x3) x6))).(\lambda (H16: (csubst0 i v x4 x5)).(or4_intro2 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))) x3 x4 x5 x6 H14 (clear_flat x1 (CHead x5 (Bind x3) x6) H15 f x0) H16))))))))) H13)) (\lambda (H13: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))) (or4 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H14: (eq C c0 (CHead x4 (Bind x3) x6))).(\lambda (H15: (clear x1 (CHead x5 (Bind x3) x7))).(\lambda (H16: (subst0 i v x6 x7)).(\lambda (H17: (csubst0 i v x4 x5)).(or4_intro3 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda 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+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (v: T).(\forall (i: nat).((csubst0 (S i) v c c2) \to (\forall (c0: C).((clear c c0) \to (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 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u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C c2 (CHead c k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v t u2))) (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq nat (S i) (s k x1))).(\lambda (H4: (eq C c2 (CHead c k x0))).(\lambda (H5: (subst0 x1 v t x0)).(eq_ind_r C (CHead c k x0) (\lambda (c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) ((match k return (\lambda (_: ?).(\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat (S i) (s k0 x1)) \to (or4 (clear (CHead c k0 x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear 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c (Bind b) t) c0)).(\lambda (H7: (eq nat (S i) (s (Bind b) x1))).(let H8 \def (f_equal nat nat (\lambda (e: nat).(match e return (\lambda (_: ?).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5 i H8) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(or4 (clear (CHead c (Bind b) x0) c3) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (or4_intro1 (clear (CHead c (Bind b) x0) (CHead c (Bind b) t)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))) b c t x0 (refl_equal C (CHead c (Bind b) t)) (clear_bind b c x0) H9)) c0 (clear_gen_bind b c c0 t H6)))))) | (Flat f) \Rightarrow (\lambda (H6: (clear (CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat (S i) (s (Flat f) x1))).(let H8 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v t x0)) H5 (S i) H8) in (or4_intro0 (clear (CHead c (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead c (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead c (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat c c0 (clear_gen_flat f c c0 t H6) f x0))))))]) H1 H3) c2 H4)))))) H2)) (\lambda (H2: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k t)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3))) (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H3: (eq nat (S i) (s k x1))).(\lambda (H4: (eq C c2 (CHead x0 k t))).(\lambda (H5: (csubst0 x1 v c x0)).(eq_ind_r C (CHead x0 k t) (\lambda (c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) ((match k return (\lambda (_: ?).(\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat (S i) (s k0 x1)) \to (or4 (clear (CHead x0 k0 t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 k0 t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 k0 t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 k0 t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))))))) with [(Bind b) \Rightarrow (\lambda (H6: (clear (CHead c (Bind b) t) c0)).(\lambda (H7: (eq nat (S i) (s (Bind b) x1))).(let H8 \def (f_equal nat nat (\lambda (e: nat).(match e return (\lambda (_: ?).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H5 i H8) in (eq_ind_r C (CHead c (Bind b) t) (\lambda (c3: C).(or4 (clear (CHead x0 (Bind b) t) c3) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Bind b) t) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (or4_intro2 (clear (CHead x0 (Bind b) t) (CHead c (Bind b) t)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Bind b) t) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Bind b) t) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Bind b) t) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))) b c x0 t (refl_equal C (CHead c (Bind b) t)) (clear_bind b x0 t) H9)) c0 (clear_gen_bind b c c0 t H6)))))) | (Flat f) \Rightarrow (\lambda (H6: (clear (CHead c (Flat f) t) c0)).(\lambda (H7: (eq nat (S i) (s (Flat f) x1))).(let H8 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x1) H7) in (let H9 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c x0)) H5 (S i) H8) in (let H10 \def (H x0 v i H9 c0 (clear_gen_flat f c c0 t H6)) in (or4_ind (clear x0 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x0 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (H11: (clear x0 c0)).(or4_intro0 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat x0 c0 H11 f t))) (\lambda (H11: (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H12: (eq C c0 (CHead x3 (Bind x2) x4))).(\lambda (H13: (clear x0 (CHead x3 (Bind x2) x5))).(\lambda (H14: (subst0 i v x4 x5)).(or4_intro1 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))) x2 x3 x4 x5 H12 (clear_flat x0 (CHead x3 (Bind x2) x5) H13 f t) H14))))))))) H11)) (\lambda (H11: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda 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C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))) x2 x3 x4 x5 H12 (clear_flat x0 (CHead x4 (Bind x2) x5) H13 f t) H14))))))))) H11)) (\lambda (H11: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x0 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))) (or4 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x2: B).(\lambda (x3: C).(\lambda (x4: C).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H12: (eq C c0 (CHead x3 (Bind x2) x5))).(\lambda (H13: (clear x0 (CHead x4 (Bind x2) x6))).(\lambda (H14: (subst0 i v x5 x6)).(\lambda (H15: (csubst0 i v x3 x4)).(or4_intro3 (clear (CHead x0 (Flat f) t) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x0 (Flat f) t) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x2 x3 x4 x5 x6 H12 (clear_flat x0 (CHead x4 (Bind x2) x6) H13 f t) H14 H15))))))))))) H11)) H10))))))]) H1 H3) c2 H4)))))) H2)) (\lambda (H2: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c c2)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (S i) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C c2 (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v t u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c c3)))) (or4 (clear c2 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H3: (eq nat (S i) (s k x2))).(\lambda (H4: (eq C c2 (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v t x0)).(\lambda (H6: (csubst0 x2 v c x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c3: C).(or4 (clear c3 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear c3 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear c3 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) ((match k return (\lambda (_: ?).(\lambda (k0: K).((clear (CHead c k0 t) c0) \to ((eq nat (S i) (s k0 x2)) \to (or4 (clear (CHead x1 k0 x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 k0 x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 k0 x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 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(b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c3 (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c3 (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))))) (or4_intro3 (clear (CHead x1 (Bind b) x0) (CHead c (Bind b) t)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e (Bind b0) u1)))))) (\lambda (b0: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e (Bind b0) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C (CHead c (Bind b) t) (CHead e1 (Bind b0) u1))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Bind b) x0) (CHead e2 (Bind b0) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))) b c x1 t x0 (refl_equal C (CHead c (Bind b) t)) (clear_bind b x1 x0) H11 H10)) c0 (clear_gen_bind b c c0 t H7))))))) | (Flat f) \Rightarrow (\lambda (H7: (clear (CHead c (Flat f) t) c0)).(\lambda (H8: (eq nat (S i) (s (Flat f) x2))).(let H9 \def (f_equal nat nat (\lambda (e: nat).e) (S i) (s (Flat f) x2) H8) in (let H10 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c x1)) H6 (S i) H9) in (let H11 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v t x0)) H5 (S i) H9) in (let H12 \def (H x1 v i H10 c0 (clear_gen_flat f c c0 t H7)) in (or4_ind (clear x1 c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: 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B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (H13: (clear x1 c0)).(or4_intro0 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (clear_flat x1 c0 H13 f x0))) (\lambda (H13: (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: 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(e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2))))) x3 x4 x5 x6 H14 (clear_flat x1 (CHead x4 (Bind x3) x6) H15 f x0) H16))))))))) H13)) (\lambda (H13: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x1 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 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x0) H16))))))))) H13)) (\lambda (H13: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x1 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))) (or4 (clear 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C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))))) (\lambda (x3: B).(\lambda (x4: C).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H14: (eq C c0 (CHead x4 (Bind x3) x6))).(\lambda (H15: (clear x1 (CHead x5 (Bind x3) x7))).(\lambda (H16: (subst0 i v x6 x7)).(\lambda (H17: (csubst0 i v x4 x5)).(or4_intro3 (clear (CHead x1 (Flat f) x0) c0) (ex3_4 B C T T (\lambda (b: B).(\lambda (e: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c0 (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 i v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C c0 (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear (CHead x1 (Flat f) x0) (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 i v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 i v e1 e2)))))) x3 x4 x5 x6 x7 H14 (clear_flat x1 (CHead x5 (Bind x3) x7) H15 f x0) H16 H17))))))))))) H13)) H12)))))))]) H1 H3) c2 H4)))))))) H2)) (csubst0_gen_head k c c2 t v (S i) H0)))))))))))) c1).
 
 theorem csubst0_getl_ge:
  \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1 e) \to (getl n c2 e)))))))))
@@ -1276,7 +1276,7 @@ theorem csubst0_getl_ge:
 theorem csubst0_getl_lt:
  \forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c1 e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))))))))))
 \def
- \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (H1: (getl n c1 e)).(let H2 \def (getl_gen_all c1 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c1 e0)) (\lambda (e0: C).(clear e0 e)) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x: C).(\lambda (H3: (drop n O c1 x)).(\lambda (H4: (clear x e)).(let H5 \def (csubst0_drop_lt n i H c1 c2 v H0 x H3) in (or4_ind (drop n O c2 x) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (H6: (drop n O c2 x)).(or4_intro0 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e x H6 H4))) (\lambda (H6: (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w))))))).(ex3_4_ind K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq C x (CHead x1 x0 x2))).(\lambda (H8: (drop n O c2 (CHead x1 x0 x3))).(\lambda (H9: (subst0 (minus i (s x0 n)) v x2 x3)).(let H10 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 (CHead x1 x0 x2) H7) in ((match x0 return (\lambda (k: K).((drop n O c2 (CHead x1 k x3)) \to ((subst0 (minus i (s k n)) v x2 x3) \to ((clear (CHead x1 k x2) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))) with [(Bind b) \Rightarrow (\lambda (H11: (drop n O c2 (CHead x1 (Bind b) x3))).(\lambda (H12: (subst0 (minus i (s (Bind b) n)) v x2 x3)).(\lambda (H13: (clear (CHead x1 (Bind b) x2) e)).(eq_ind_r C (CHead x1 (Bind b) x2) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1 (getl n c2 (CHead x1 (Bind b) x2)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x2) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))) b x1 x2 x3 (refl_equal C (CHead x1 (Bind b) x2)) (getl_intro n c2 (CHead x1 (Bind b) x3) (CHead x1 (Bind b) x3) H11 (clear_bind b x1 x3)) H12)) e (clear_gen_bind b x1 e x2 H13))))) | (Flat f) \Rightarrow (\lambda (H11: (drop n O c2 (CHead x1 (Flat f) x3))).(\lambda (_: (subst0 (minus i (s (Flat f) n)) v x2 x3)).(\lambda (H13: (clear (CHead x1 (Flat f) x2) e)).(or4_intro0 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e (CHead x1 (Flat f) x3) H11 (clear_flat x1 e (clear_gen_flat f x1 e x2 H13) f x3))))))]) H8 H9 H10))))))))) H6)) (\lambda (H6: (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H7: (eq C x (CHead x1 x0 x3))).(\lambda (H8: (drop n O c2 (CHead x2 x0 x3))).(\lambda (H9: (csubst0 (minus i (s x0 n)) v x1 x2)).(let H10 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 (CHead x1 x0 x3) H7) in ((match x0 return (\lambda (k: K).((drop n O c2 (CHead x2 k x3)) \to ((csubst0 (minus i (s k n)) v x1 x2) \to ((clear (CHead x1 k x3) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))) with [(Bind b) \Rightarrow (\lambda (H11: (drop n O c2 (CHead x2 (Bind b) x3))).(\lambda (H12: (csubst0 (minus i (s (Bind b) n)) v x1 x2)).(\lambda (H13: (clear (CHead x1 (Bind b) x3) e)).(eq_ind_r C (CHead x1 (Bind b) x3) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 (CHead x1 (Bind b) x3)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex3_4_intro B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) b x1 x2 x3 (refl_equal C (CHead x1 (Bind b) x3)) (getl_intro n c2 (CHead x2 (Bind b) x3) (CHead x2 (Bind b) x3) H11 (clear_bind b x2 x3)) H12)) e (clear_gen_bind b x1 e x3 H13))))) | (Flat f) \Rightarrow (\lambda (H11: (drop n O c2 (CHead x2 (Flat f) x3))).(\lambda (H12: (csubst0 (minus i (s (Flat f) n)) v x1 x2)).(\lambda (H13: (clear (CHead x1 (Flat f) x3) e)).(let H14 \def (eq_ind nat (minus i n) (\lambda (n: nat).(csubst0 n v x1 x2)) H12 (S (minus i (S n))) (minus_x_Sy i n H)) in (let H15 \def (csubst0_clear_S x1 x2 v (minus i (S n)) H14 e (clear_gen_flat f x1 e x3 H13)) in (or4_ind (clear x2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (H16: (clear x2 e)).(or4_intro0 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e (CHead x2 (Flat f) x3) H11 (clear_flat x2 e H16 f x3)))) (\lambda (H16: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: 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(CHead x5 (Bind x4) x6)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x6) (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x5 (Bind x4) x6) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x6) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x6) (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))) x4 x5 x6 x7 (refl_equal C (CHead x5 (Bind x4) x6)) (getl_intro n c2 (CHead x5 (Bind x4) x7) (CHead x2 (Flat f) x3) H11 (clear_flat x2 (CHead x5 (Bind x4) x7) H18 f x3)) H19)) e H17)))))))) H16)) (\lambda (H16: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x4: B).(\lambda (x5: C).(\lambda (x6: C).(\lambda (x7: T).(\lambda (H17: (eq C e (CHead x5 (Bind x4) x7))).(\lambda (H18: (clear x2 (CHead x6 (Bind x4) x7))).(\lambda (H19: (csubst0 (minus i (S n)) v x5 x6)).(eq_ind_r C (CHead x5 (Bind x4) x7) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 (CHead x5 (Bind x4) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) x4 x5 x6 x7 (refl_equal C (CHead x5 (Bind x4) x7)) (getl_intro n c2 (CHead x6 (Bind x4) x7) (CHead x2 (Flat f) x3) H11 (clear_flat x2 (CHead x6 (Bind x4) x7) H18 f x3)) H19)) e H17)))))))) H16)) (\lambda (H16: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x4: B).(\lambda (x5: C).(\lambda (x6: C).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H17: (eq C e (CHead x5 (Bind x4) x7))).(\lambda (H18: (clear x2 (CHead x6 (Bind x4) x8))).(\lambda (H19: (subst0 (minus i (S n)) v x7 x8)).(\lambda (H20: (csubst0 (minus i (S n)) v x5 x6)).(eq_ind_r C (CHead x5 (Bind x4) x7) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 (getl n c2 (CHead x5 (Bind x4) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) x4 x5 x6 x7 x8 (refl_equal C (CHead x5 (Bind x4) x7)) (getl_intro n c2 (CHead x6 (Bind x4) x8) (CHead x2 (Flat f) x3) H11 (clear_flat x2 (CHead x6 (Bind x4) x8) H18 f x3)) H19 H20)) e H17)))))))))) H16)) H15))))))]) H8 H9 H10))))))))) H6)) (\lambda (H6: (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H7: (eq C x (CHead x1 x0 x3))).(\lambda (H8: (drop n O c2 (CHead x2 x0 x4))).(\lambda (H9: (subst0 (minus i (s x0 n)) v x3 x4)).(\lambda (H10: (csubst0 (minus i (s x0 n)) v x1 x2)).(let H11 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 (CHead x1 x0 x3) H7) in ((match x0 return (\lambda (k: K).((drop n O c2 (CHead x2 k x4)) \to ((subst0 (minus i (s k n)) v x3 x4) \to ((csubst0 (minus i (s k n)) v x1 x2) \to ((clear (CHead x1 k x3) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))))))) with [(Bind b) \Rightarrow (\lambda (H12: (drop n O c2 (CHead x2 (Bind b) x4))).(\lambda (H13: (subst0 (minus i (s (Bind b) n)) v x3 x4)).(\lambda (H14: (csubst0 (minus i (s (Bind b) n)) v x1 x2)).(\lambda (H15: (clear (CHead x1 (Bind b) x3) e)).(eq_ind_r C (CHead x1 (Bind b) x3) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 (getl n c2 (CHead x1 (Bind b) x3)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) b x1 x2 x3 x4 (refl_equal C (CHead x1 (Bind b) x3)) (getl_intro n c2 (CHead x2 (Bind b) x4) (CHead x2 (Bind b) x4) H12 (clear_bind b x2 x4)) H13 H14)) e (clear_gen_bind b x1 e x3 H15)))))) | (Flat f) \Rightarrow (\lambda (H12: (drop n O c2 (CHead x2 (Flat f) x4))).(\lambda (_: (subst0 (minus i (s (Flat f) n)) v x3 x4)).(\lambda (H14: (csubst0 (minus i (s (Flat f) n)) v x1 x2)).(\lambda (H15: (clear (CHead x1 (Flat f) x3) e)).(let H16 \def (eq_ind nat (minus i n) (\lambda (n: nat).(csubst0 n v x1 x2)) H14 (S (minus i (S n))) (minus_x_Sy i n H)) in (let H17 \def (csubst0_clear_S x1 x2 v (minus i (S n)) H16 e (clear_gen_flat f x1 e x3 H15)) in (or4_ind (clear x2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (H18: (clear x2 e)).(or4_intro0 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e (CHead x2 (Flat f) x4) H12 (clear_flat x2 e H18 f x4)))) (\lambda (H18: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5) x7))).(\lambda (H20: (clear x2 (CHead x6 (Bind x5) x8))).(\lambda (H21: (subst0 (minus i (S n)) v x7 x8)).(eq_ind_r C (CHead x6 (Bind x5) x7) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: 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B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x7) (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))) x5 x6 x7 x8 (refl_equal C (CHead x6 (Bind x5) x7)) (getl_intro n c2 (CHead x6 (Bind x5) x8) (CHead x2 (Flat f) x4) H12 (clear_flat x2 (CHead x6 (Bind x5) x8) H20 f x4)) H21)) e H19)))))))) H18)) (\lambda (H18: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5) x8))).(\lambda (H20: (clear x2 (CHead x7 (Bind x5) x8))).(\lambda (H21: (csubst0 (minus i (S n)) v x6 x7)).(eq_ind_r C (CHead x6 (Bind x5) x8) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v 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(CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) x5 x6 x7 x8 (refl_equal C (CHead x6 (Bind x5) x8)) (getl_intro n c2 (CHead x7 (Bind x5) x8) (CHead x2 (Flat f) x4) H12 (clear_flat x2 (CHead x7 (Bind x5) x8) H20 f x4)) H21)) e H19)))))))) H18)) (\lambda (H18: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: C).(\lambda (x8: T).(\lambda (x9: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5) x8))).(\lambda (H20: (clear x2 (CHead x7 (Bind x5) x9))).(\lambda (H21: (subst0 (minus i (S n)) v x8 x9)).(\lambda (H22: (csubst0 (minus i (S n)) v x6 x7)).(eq_ind_r C (CHead x6 (Bind x5) x8) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 (getl n c2 (CHead x6 (Bind x5) x8)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) x5 x6 x7 x8 x9 (refl_equal C (CHead x6 (Bind x5) x8)) (getl_intro n c2 (CHead x7 (Bind x5) x9) (CHead x2 (Flat f) x4) H12 (clear_flat x2 (CHead x7 (Bind x5) x9) H20 f x4)) H21 H22)) e H19)))))))))) H18)) H17)))))))]) H8 H9 H10 H11))))))))))) H6)) H5))))) H2)))))))))).
+ \lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H0: (csubst0 i v c1 c2)).(\lambda (e: C).(\lambda (H1: (getl n c1 e)).(let H2 \def (getl_gen_all c1 e n H1) in (ex2_ind C (\lambda (e0: C).(drop n O c1 e0)) (\lambda (e0: C).(clear e0 e)) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x: C).(\lambda (H3: (drop n O c1 x)).(\lambda (H4: (clear x e)).(let H5 \def (csubst0_drop_lt n i H c1 c2 v H0 x H3) in (or4_ind (drop n O c2 x) (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (H6: (drop n O c2 x)).(or4_intro0 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e x H6 H4))) (\lambda (H6: (ex3_4 K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w))))))).(ex3_4_ind K C T T (\lambda (k: K).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e0 k u)))))) (\lambda (k: K).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e0 k w)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H7: (eq C x (CHead x1 x0 x2))).(\lambda (H8: (drop n O c2 (CHead x1 x0 x3))).(\lambda (H9: (subst0 (minus i (s x0 n)) v x2 x3)).(let H10 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 (CHead x1 x0 x2) H7) in ((match x0 return (\lambda (_: ?).(\lambda (k: K).((drop n O c2 (CHead x1 k x3)) \to ((subst0 (minus i (s k n)) v x2 x3) \to ((clear (CHead x1 k x2) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))))))) with [(Bind b) \Rightarrow (\lambda (H11: (drop n O c2 (CHead x1 (Bind b) x3))).(\lambda (H12: (subst0 (minus i (s (Bind b) n)) v x2 x3)).(\lambda (H13: (clear (CHead x1 (Bind b) x2) e)).(eq_ind_r C (CHead x1 (Bind b) x2) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1 (getl n c2 (CHead x1 (Bind b) x2)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x2) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x2) (CHead e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))) b x1 x2 x3 (refl_equal C (CHead x1 (Bind b) x2)) (getl_intro n c2 (CHead x1 (Bind b) x3) (CHead x1 (Bind b) x3) H11 (clear_bind b x1 x3)) H12)) e (clear_gen_bind b x1 e x2 H13))))) | (Flat f) \Rightarrow (\lambda (H11: (drop n O c2 (CHead x1 (Flat f) x3))).(\lambda (_: (subst0 (minus i (s (Flat f) n)) v x2 x3)).(\lambda (H13: (clear (CHead x1 (Flat f) x2) e)).(or4_intro0 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e (CHead x1 (Flat f) x3) H11 (clear_flat x1 e (clear_gen_flat f x1 e x2 H13) f x3))))))]) H8 H9 H10))))))))) H6)) (\lambda (H6: (ex3_4 K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2))))))).(ex3_4_ind K C C T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C x (CHead e1 k u)))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(drop n O c2 (CHead e2 k u)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H7: (eq C x (CHead x1 x0 x3))).(\lambda (H8: (drop n O c2 (CHead x2 x0 x3))).(\lambda (H9: (csubst0 (minus i (s x0 n)) v x1 x2)).(let H10 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 (CHead x1 x0 x3) H7) in ((match x0 return (\lambda (_: ?).(\lambda (k: K).((drop n O c2 (CHead x2 k x3)) \to ((csubst0 (minus i (s k n)) v x1 x2) \to ((clear (CHead x1 k x3) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))))))) with [(Bind b) \Rightarrow (\lambda (H11: (drop n O c2 (CHead x2 (Bind b) x3))).(\lambda (H12: (csubst0 (minus i (s (Bind b) n)) v x1 x2)).(\lambda (H13: (clear (CHead x1 (Bind b) x3) e)).(eq_ind_r C (CHead x1 (Bind b) x3) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 (CHead x1 (Bind b) x3)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex3_4_intro B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) b x1 x2 x3 (refl_equal C (CHead x1 (Bind b) x3)) (getl_intro n c2 (CHead x2 (Bind b) x3) (CHead x2 (Bind b) x3) H11 (clear_bind b x2 x3)) H12)) e (clear_gen_bind b x1 e x3 H13))))) | (Flat f) \Rightarrow (\lambda (H11: (drop n O c2 (CHead x2 (Flat f) x3))).(\lambda (H12: (csubst0 (minus i (s (Flat f) n)) v x1 x2)).(\lambda (H13: (clear (CHead x1 (Flat f) x3) e)).(let H14 \def (eq_ind nat (minus i n) (\lambda (n: nat).(csubst0 n v x1 x2)) H12 (S (minus i (S n))) (minus_x_Sy i n H)) in (let H15 \def (csubst0_clear_S x1 x2 v (minus i (S n)) H14 e (clear_gen_flat f x1 e x3 H13)) in (or4_ind (clear x2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (H16: (clear x2 e)).(or4_intro0 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e (CHead x2 (Flat f) x3) H11 (clear_flat x2 e H16 f x3)))) (\lambda (H16: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x4: B).(\lambda (x5: C).(\lambda (x6: T).(\lambda (x7: T).(\lambda (H17: (eq C e (CHead x5 (Bind x4) x6))).(\lambda (H18: (clear x2 (CHead x5 (Bind x4) x7))).(\lambda (H19: (subst0 (minus i (S n)) v x6 x7)).(eq_ind_r C (CHead x5 (Bind x4) x6) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1 (getl n c2 (CHead x5 (Bind x4) x6)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x6) (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x5 (Bind x4) x6) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x6) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x6) (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))) x4 x5 x6 x7 (refl_equal C (CHead x5 (Bind x4) x6)) (getl_intro n c2 (CHead x5 (Bind x4) x7) (CHead x2 (Flat f) x3) H11 (clear_flat x2 (CHead x5 (Bind x4) x7) H18 f x3)) H19)) e H17)))))))) H16)) (\lambda (H16: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x4: B).(\lambda (x5: C).(\lambda (x6: C).(\lambda (x7: T).(\lambda (H17: (eq C e (CHead x5 (Bind x4) x7))).(\lambda (H18: (clear x2 (CHead x6 (Bind x4) x7))).(\lambda (H19: (csubst0 (minus i (S n)) v x5 x6)).(eq_ind_r C (CHead x5 (Bind x4) x7) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 (CHead x5 (Bind x4) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) x4 x5 x6 x7 (refl_equal C (CHead x5 (Bind x4) x7)) (getl_intro n c2 (CHead x6 (Bind x4) x7) (CHead x2 (Flat f) x3) H11 (clear_flat x2 (CHead x6 (Bind x4) x7) H18 f x3)) H19)) e H17)))))))) H16)) (\lambda (H16: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x4: B).(\lambda (x5: C).(\lambda (x6: C).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H17: (eq C e (CHead x5 (Bind x4) x7))).(\lambda (H18: (clear x2 (CHead x6 (Bind x4) x8))).(\lambda (H19: (subst0 (minus i (S n)) v x7 x8)).(\lambda (H20: (csubst0 (minus i (S n)) v x5 x6)).(eq_ind_r C (CHead x5 (Bind x4) x7) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 (getl n c2 (CHead x5 (Bind x4) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x5 (Bind x4) x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) x4 x5 x6 x7 x8 (refl_equal C (CHead x5 (Bind x4) x7)) (getl_intro n c2 (CHead x6 (Bind x4) x8) (CHead x2 (Flat f) x3) H11 (clear_flat x2 (CHead x6 (Bind x4) x8) H18 f x3)) H19 H20)) e H17)))))))))) H16)) H15))))))]) H8 H9 H10))))))))) H6)) (\lambda (H6: (ex4_5 K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))))).(ex4_5_ind K C C T T (\lambda (k: K).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C x (CHead e1 k u))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(drop n O c2 (CHead e2 k w))))))) (\lambda (k: K).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (s k n)) v u w)))))) (\lambda (k: K).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (s k n)) v e1 e2)))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H7: (eq C x (CHead x1 x0 x3))).(\lambda (H8: (drop n O c2 (CHead x2 x0 x4))).(\lambda (H9: (subst0 (minus i (s x0 n)) v x3 x4)).(\lambda (H10: (csubst0 (minus i (s x0 n)) v x1 x2)).(let H11 \def (eq_ind C x (\lambda (c: C).(clear c e)) H4 (CHead x1 x0 x3) H7) in ((match x0 return (\lambda (_: ?).(\lambda (k: K).((drop n O c2 (CHead x2 k x4)) \to ((subst0 (minus i (s k n)) v x3 x4) \to ((csubst0 (minus i (s k n)) v x1 x2) \to ((clear (CHead x1 k x3) e) \to (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))))))))) with [(Bind b) \Rightarrow (\lambda (H12: (drop n O c2 (CHead x2 (Bind b) x4))).(\lambda (H13: (subst0 (minus i (s (Bind b) n)) v x3 x4)).(\lambda (H14: (csubst0 (minus i (s (Bind b) n)) v x1 x2)).(\lambda (H15: (clear (CHead x1 (Bind b) x3) e)).(eq_ind_r C (CHead x1 (Bind b) x3) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 (getl n c2 (CHead x1 (Bind b) x3)) (ex3_4 B C T T (\lambda (b0: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e0 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b0) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u)))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b0) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b0: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x1 (Bind b) x3) (CHead e1 (Bind b0) u))))))) (\lambda (b0: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b0) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) b x1 x2 x3 x4 (refl_equal C (CHead x1 (Bind b) x3)) (getl_intro n c2 (CHead x2 (Bind b) x4) (CHead x2 (Bind b) x4) H12 (clear_bind b x2 x4)) H13 H14)) e (clear_gen_bind b x1 e x3 H15)))))) | (Flat f) \Rightarrow (\lambda (H12: (drop n O c2 (CHead x2 (Flat f) x4))).(\lambda (_: (subst0 (minus i (s (Flat f) n)) v x3 x4)).(\lambda (H14: (csubst0 (minus i (s (Flat f) n)) v x1 x2)).(\lambda (H15: (clear (CHead x1 (Flat f) x3) e)).(let H16 \def (eq_ind nat (minus i n) (\lambda (n: nat).(csubst0 n v x1 x2)) H14 (S (minus i (S n))) (minus_x_Sy i n H)) in (let H17 \def (csubst0_clear_S x1 x2 v (minus i (S n)) H16 e (clear_gen_flat f x1 e x3 H15)) in (or4_ind (clear x2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (H18: (clear x2 e)).(or4_intro0 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (getl_intro n c2 e (CHead x2 (Flat f) x4) H12 (clear_flat x2 e H18 f x4)))) (\lambda (H18: (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2))))))).(ex3_4_ind B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u1)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e0 (Bind b) u2)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5) x7))).(\lambda (H20: (clear x2 (CHead x6 (Bind x5) x8))).(\lambda (H21: (subst0 (minus i (S n)) v x7 x8)).(eq_ind_r C (CHead x6 (Bind x5) x7) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro1 (getl n c2 (CHead x6 (Bind x5) x7)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x7) (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x6 (Bind x5) x7) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x7) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex3_4_intro B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x7) (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w))))) x5 x6 x7 x8 (refl_equal C (CHead x6 (Bind x5) x7)) (getl_intro n c2 (CHead x6 (Bind x5) x8) (CHead x2 (Flat f) x4) H12 (clear_flat x2 (CHead x6 (Bind x5) x8) H20 f x4)) H21)) e H19)))))))) H18)) (\lambda (H18: (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))).(ex3_4_ind B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(clear x2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: C).(\lambda (x8: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5) x8))).(\lambda (H20: (clear x2 (CHead x7 (Bind x5) x8))).(\lambda (H21: (csubst0 (minus i (S n)) v x6 x7)).(eq_ind_r C (CHead x6 (Bind x5) x8) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro2 (getl n c2 (CHead x6 (Bind x5) x8)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex3_4_intro B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))) x5 x6 x7 x8 (refl_equal C (CHead x6 (Bind x5) x8)) (getl_intro n c2 (CHead x7 (Bind x5) x8) (CHead x2 (Flat f) x4) H12 (clear_flat x2 (CHead x7 (Bind x5) x8) H20 f x4)) H21)) e H19)))))))) H18)) (\lambda (H18: (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))).(ex4_5_ind B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u1))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (u2: T).(clear x2 (CHead e2 (Bind b) u2))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u1: T).(\lambda (u2: T).(subst0 (minus i (S n)) v u1 u2)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (or4 (getl n c2 e) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C e (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C e (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: C).(\lambda (x8: T).(\lambda (x9: T).(\lambda (H19: (eq C e (CHead x6 (Bind x5) x8))).(\lambda (H20: (clear x2 (CHead x7 (Bind x5) x9))).(\lambda (H21: (subst0 (minus i (S n)) v x8 x9)).(\lambda (H22: (csubst0 (minus i (S n)) v x6 x7)).(eq_ind_r C (CHead x6 (Bind x5) x8) (\lambda (c: C).(or4 (getl n c2 c) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C c (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C c (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))))) (or4_intro3 (getl n c2 (CHead x6 (Bind x5) x8)) (ex3_4 B C T T (\lambda (b: B).(\lambda (e0: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e0 (Bind b) u)))))) (\lambda (b: B).(\lambda (e0: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e0 (Bind b) w)))))) (\lambda (_: B).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (ex3_4 B C C T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u)))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (u: T).(getl n c2 (CHead e2 (Bind b) u)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) (ex4_5 B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2))))))) (ex4_5_intro B C C T T (\lambda (b: B).(\lambda (e1: C).(\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(eq C (CHead x6 (Bind x5) x8) (CHead e1 (Bind b) u))))))) (\lambda (b: B).(\lambda (_: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (w: T).(getl n c2 (CHead e2 (Bind b) w))))))) (\lambda (_: B).(\lambda (_: C).(\lambda (_: C).(\lambda (u: T).(\lambda (w: T).(subst0 (minus i (S n)) v u w)))))) (\lambda (_: B).(\lambda (e1: C).(\lambda (e2: C).(\lambda (_: T).(\lambda (_: T).(csubst0 (minus i (S n)) v e1 e2)))))) x5 x6 x7 x8 x9 (refl_equal C (CHead x6 (Bind x5) x8)) (getl_intro n c2 (CHead x7 (Bind x5) x9) (CHead x2 (Flat f) x4) H12 (clear_flat x2 (CHead x7 (Bind x5) x9) H20 f x4)) H21 H22)) e H19)))))))))) H18)) H17)))))))]) H8 H9 H10 H11))))))))))) H6)) H5))))) H2)))))))))).
 
 theorem csubst0_getl_ge_back:
  \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst0 i v c1 c2) \to (\forall (e: C).((getl n c2 e) \to (getl n c1 e)))))))))
@@ -1305,7 +1305,7 @@ theorem csubst1_flat:
 theorem csubst1_gen_head:
  \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).(\forall (v: T).(\forall (i: nat).((csubst1 (s k i) v (CHead c1 k u1) x) \to (ex3_2 T C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 i v c1 c2))))))))))
 \def
- \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda (H: (csubst1 (s k i) v (CHead c1 k u1) x)).(let H0 \def (match H return (\lambda (c: C).((eq C c x) \to (ex3_2 T C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 i v c1 c2)))))) with [csubst1_refl \Rightarrow (\lambda (H0: (eq C (CHead c1 k u1) x)).(eq_ind C (CHead c1 k u1) (\lambda (c: C).(ex3_2 T C (\lambda (u2: T).(\lambda (c2: C).(eq C c (CHead c2 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 i v c1 c2))))) (ex3_2_intro T C (\lambda (u2: T).(\lambda (c2: C).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 i v c1 c2))) u1 c1 (refl_equal C (CHead c1 k u1)) (subst1_refl i v u1) (csubst1_refl i v c1)) x H0)) | (csubst1_sing c2 H0) \Rightarrow (\lambda (H1: (eq C c2 x)).(eq_ind C x (\lambda (c: C).((csubst0 (s k i) v (CHead c1 k u1) c) \to (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C x (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3)))))) (\lambda (H2: (csubst0 (s k i) v (CHead c1 k u1) x)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C x (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (H3: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C x (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H: (eq nat (s k i) (s k x1))).(\lambda (H4: (eq C x (CHead c1 k x0))).(\lambda (H5: (subst0 x1 v u1 x0)).(eq_ind_r C (CHead c1 k x0) (\lambda (c: C).(ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))))) (let H6 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v u1 x0)) H5 i (s_inj k i x1 H)) in (ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead c1 k x0) (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) x0 c1 (refl_equal C (CHead c1 k x0)) (subst1_single i v u1 x0 H6) (csubst1_refl i v c1))) x H4)))))) H3)) (\lambda (H3: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C x (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H: (eq nat (s k i) (s k x1))).(\lambda (H4: (eq C x (CHead x0 k u1))).(\lambda (H5: (csubst0 x1 v c1 x0)).(eq_ind_r C (CHead x0 k u1) (\lambda (c: C).(ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))))) (let H6 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c1 x0)) H5 i (s_inj k i x1 H)) in (ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x0 k u1) (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) u1 x0 (refl_equal C (CHead x0 k u1)) (subst1_refl i v u1) (csubst1_sing i v c1 x0 H6))) x H4)))))) H3)) (\lambda (H3: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C x (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H: (eq nat (s k i) (s k x2))).(\lambda (H4: (eq C x (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v u1 x0)).(\lambda (H6: (csubst0 x2 v c1 x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c: C).(ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))))) (let H7 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c1 x1)) H6 i (s_inj k i x2 H)) in (let H8 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v u1 x0)) H5 i (s_inj k i x2 H)) in (ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x1 k x0) (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) x0 x1 (refl_equal C (CHead x1 k x0)) (subst1_single i v u1 x0 H8) (csubst1_sing i v c1 x1 H7)))) x H4)))))))) H3)) (csubst0_gen_head k c1 x u1 v (s k i) H2))) c2 (sym_eq C c2 x H1) H0))]) in (H0 (refl_equal C x))))))))).
+ \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda (v: T).(\lambda (i: nat).(\lambda (H: (csubst1 (s k i) v (CHead c1 k u1) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c: C).((eq C c x) \to (ex3_2 T C (\lambda (u2: T).(\lambda (c2: C).(eq C x (CHead c2 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 i v c1 c2))))))) with [csubst1_refl \Rightarrow (\lambda (H0: (eq C (CHead c1 k u1) x)).(eq_ind C (CHead c1 k u1) (\lambda (c: C).(ex3_2 T C (\lambda (u2: T).(\lambda (c2: C).(eq C c (CHead c2 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 i v c1 c2))))) (ex3_2_intro T C (\lambda (u2: T).(\lambda (c2: C).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c2: C).(csubst1 i v c1 c2))) u1 c1 (refl_equal C (CHead c1 k u1)) (subst1_refl i v u1) (csubst1_refl i v c1)) x H0)) | (csubst1_sing c2 H0) \Rightarrow (\lambda (H1: (eq C c2 x)).(eq_ind C x (\lambda (c: C).((csubst0 (s k i) v (CHead c1 k u1) c) \to (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C x (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3)))))) (\lambda (H2: (csubst0 (s k i) v (CHead c1 k u1) x)).(or3_ind (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2)))) (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C x (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (H3: (ex3_2 T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2))))).(ex3_2_ind T nat (\lambda (_: T).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (u2: T).(\lambda (_: nat).(eq C x (CHead c1 k u2)))) (\lambda (u2: T).(\lambda (j: nat).(subst0 j v u1 u2))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C x (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1: nat).(\lambda (H: (eq nat (s k i) (s k x1))).(\lambda (H4: (eq C x (CHead c1 k x0))).(\lambda (H5: (subst0 x1 v u1 x0)).(eq_ind_r C (CHead c1 k x0) (\lambda (c: C).(ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))))) (let H6 \def (eq_ind_r nat x1 (\lambda (n: nat).(subst0 n v u1 x0)) H5 i (s_inj k i x1 H)) in (ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead c1 k x0) (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) x0 c1 (refl_equal C (CHead c1 k x0)) (subst1_single i v u1 x0 H6) (csubst1_refl i v c1))) x H4)))))) H3)) (\lambda (H3: (ex3_2 C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u1)))) (\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2))))).(ex3_2_ind C nat (\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j)))) (\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u1)))) (\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C x (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: C).(\lambda (x1: nat).(\lambda (H: (eq nat (s k i) (s k x1))).(\lambda (H4: (eq C x (CHead x0 k u1))).(\lambda (H5: (csubst0 x1 v c1 x0)).(eq_ind_r C (CHead x0 k u1) (\lambda (c: C).(ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))))) (let H6 \def (eq_ind_r nat x1 (\lambda (n: nat).(csubst0 n v c1 x0)) H5 i (s_inj k i x1 H)) in (ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x0 k u1) (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) u1 x0 (refl_equal C (CHead x0 k u1)) (subst1_refl i v u1) (csubst1_sing i v c1 x0 H6))) x H4)))))) H3)) (\lambda (H3: (ex4_3 T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda (c2: C).(\lambda (_: nat).(eq C x (CHead c2 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c2: C).(\lambda (j: nat).(csubst0 j v c1 c2)))))).(ex4_3_ind T C nat (\lambda (_: T).(\lambda (_: C).(\lambda (j: nat).(eq nat (s k i) (s k j))))) (\lambda (u2: T).(\lambda (c3: C).(\lambda (_: nat).(eq C x (CHead c3 k u2))))) (\lambda (u2: T).(\lambda (_: C).(\lambda (j: nat).(subst0 j v u1 u2)))) (\lambda (_: T).(\lambda (c3: C).(\lambda (j: nat).(csubst0 j v c1 c3)))) (ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C x (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3)))) (\lambda (x0: T).(\lambda (x1: C).(\lambda (x2: nat).(\lambda (H: (eq nat (s k i) (s k x2))).(\lambda (H4: (eq C x (CHead x1 k x0))).(\lambda (H5: (subst0 x2 v u1 x0)).(\lambda (H6: (csubst0 x2 v c1 x1)).(eq_ind_r C (CHead x1 k x0) (\lambda (c: C).(ex3_2 T C (\lambda (u2: T).(\lambda (c3: C).(eq C c (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))))) (let H7 \def (eq_ind_r nat x2 (\lambda (n: nat).(csubst0 n v c1 x1)) H6 i (s_inj k i x2 H)) in (let H8 \def (eq_ind_r nat x2 (\lambda (n: nat).(subst0 n v u1 x0)) H5 i (s_inj k i x2 H)) in (ex3_2_intro T C (\lambda (u2: T).(\lambda (c3: C).(eq C (CHead x1 k x0) (CHead c3 k u2)))) (\lambda (u2: T).(\lambda (_: C).(subst1 i v u1 u2))) (\lambda (_: T).(\lambda (c3: C).(csubst1 i v c1 c3))) x0 x1 (refl_equal C (CHead x1 k x0)) (subst1_single i v u1 x0 H8) (csubst1_sing i v c1 x1 H7)))) x H4)))))))) H3)) (csubst0_gen_head k c1 x u1 v (s k i) H2))) c2 (sym_eq C c2 x H1) H0))]) in (H0 (refl_equal C x))))))))).
 
 theorem csubst1_getl_ge:
  \forall (i: nat).(\forall (n: nat).((le i n) \to (\forall (c1: C).(\forall (c2: C).(\forall (v: T).((csubst1 i v c1 c2) \to (\forall (e: C).((getl n c1 e) \to (getl n c2 e)))))))))
@@ -1325,7 +1325,7 @@ theorem csubst1_getl_ge_back:
 theorem getl_csubst1:
  \forall (d: nat).(\forall (c: C).(\forall (e: C).(\forall (u: T).((getl d c (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 d u c a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) d a0 a))))))))
 \def
- \lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (c: C).(\forall (e: C).(\forall (u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) n a0 a))))))))) (\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a)))))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H: (getl O (CSort n) (CHead e (Bind Abbr) u))).(getl_gen_sort n O (CHead e (Bind Abbr) u) H (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CSort n) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))).(\lambda (k: K).(match k return (\lambda (k0: K).(\forall (t: T).(\forall (e: C).(\forall (u: T).((getl O (CHead c0 k0 t) (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CHead c0 k0 t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))) with [(Bind b) \Rightarrow (\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u))).(let H1 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow e | (CHead c _ _) \Rightarrow c])) (CHead e (Bind Abbr) u) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e (Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u) H0))) in ((let H2 \def (f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead e (Bind Abbr) u) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e (Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u) H0))) in ((let H3 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead e (Bind Abbr) u) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e (Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u) H0))) in (\lambda (H4: (eq B Abbr b)).(\lambda (_: (eq C e c0)).(eq_ind_r T t (\lambda (t0: T).(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O t0 (CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))))) (eq_ind B Abbr (\lambda (b0: B).(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O t (CHead c0 (Bind b0) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))))) (ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O t (CHead c0 (Bind Abbr) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))) (CHead c0 (Bind Abbr) t) c0 (csubst1_refl O t (CHead c0 (Bind Abbr) t)) (drop_drop (Bind Abbr) O c0 c0 (drop_refl c0) t)) b H4) u H3)))) H2)) H1)))))) | (Flat f) \Rightarrow (\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl O (CHead c0 (Flat f) t) (CHead e (Bind Abbr) u))).(let H_x \def (subst1_ex u t O) in (let H1 \def H_x in (ex_ind T (\lambda (t2: T).(subst1 O u t (lift (S O) O t2))) (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a)))) (\lambda (x: T).(\lambda (H2: (subst1 O u t (lift (S O) O x))).(let H3 \def (H e u (getl_intro O c0 (CHead e (Bind Abbr) u) c0 (drop_refl c0) (clear_gen_flat f c0 (CHead e (Bind Abbr) u) t (getl_gen_O (CHead c0 (Flat f) t) (CHead e (Bind Abbr) u) H0)))) in (ex2_2_ind C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))) (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a)))) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H4: (csubst1 O u c0 x0)).(\lambda (H5: (drop (S O) O x0 x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))) (CHead x0 (Flat f) (lift (S O) O x)) x1 (csubst1_flat f O u t (lift (S O) O x) H2 c0 x0 H4) (drop_drop (Flat f) O x0 x1 H5 (lift (S O) O x))))))) H3)))) H1)))))))])))) c)) (\lambda (n: nat).(\lambda (H: ((\forall (c: C).(\forall (e: C).(\forall (u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) n a0 a)))))))))).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: C).(\forall (u: T).((getl (S n) c0 (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))))))) (\lambda (n0: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl (S n) (CSort n0) (CHead e (Bind Abbr) u))).(getl_gen_sort n0 (S n) (CHead e (Bind Abbr) u) H0 (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CSort n0) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))))))))) (\lambda (c0: C).(\lambda (H0: ((\forall (e: C).(\forall (u: T).((getl (S n) c0 (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))))))))).(\lambda (k: K).(match k return (\lambda (k0: K).(\forall (t: T).(\forall (e: C).(\forall (u: T).((getl (S n) (CHead c0 k0 t) (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 k0 t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))))))))) with [(Bind b) \Rightarrow (\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (H1: (getl (S n) (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u))).(let H_x \def (subst1_ex u t n) in (let H2 \def H_x in (ex_ind T (\lambda (t2: T).(subst1 n u t (lift (S O) n t2))) (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x: T).(\lambda (H3: (subst1 n u t (lift (S O) n x))).(let H4 \def (H c0 e u (getl_gen_S (Bind b) c0 (CHead e (Bind Abbr) u) t n H1)) in (ex2_2_ind C C (\lambda (a0: C).(\lambda (_: C).(csubst1 n u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) n a0 a))) (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H5: (csubst1 n u c0 x0)).(\lambda (H6: (drop (S O) n x0 x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (CHead x0 (Bind b) (lift (S O) n x)) (CHead x1 (Bind b) x) (csubst1_bind b n u t (lift (S O) n x) H3 c0 x0 H5) (drop_skip_bind (S O) n x0 x1 H6 b x)))))) H4)))) H2))))))) | (Flat f) \Rightarrow (\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (H1: (getl (S n) (CHead c0 (Flat f) t) (CHead e (Bind Abbr) u))).(let H_x \def (subst1_ex u t (S n)) in (let H2 \def H_x in (ex_ind T (\lambda (t2: T).(subst1 (S n) u t (lift (S O) (S n) t2))) (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x: T).(\lambda (H3: (subst1 (S n) u t (lift (S O) (S n) x))).(let H4 \def (H0 e u (getl_gen_S (Flat f) c0 (CHead e (Bind Abbr) u) t n H1)) in (ex2_2_ind C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H5: (csubst1 (S n) u c0 x0)).(\lambda (H6: (drop (S O) (S n) x0 x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (CHead x0 (Flat f) (lift (S O) (S n) x)) (CHead x1 (Flat f) x) (csubst1_flat f (S n) u t (lift (S O) (S n) x) H3 c0 x0 H5) (drop_skip_flat (S O) n x0 x1 H6 f x)))))) H4)))) H2)))))))])))) c)))) d).
+ \lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (c: C).(\forall (e: C).(\forall (u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) n a0 a))))))))) (\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a)))))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H: (getl O (CSort n) (CHead e (Bind Abbr) u))).(getl_gen_sort n O (CHead e (Bind Abbr) u) H (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CSort n) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (e: C).(\forall (u: T).((getl O c0 (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))))))))).(\lambda (k: K).(match k return (\lambda (_: ?).(\lambda (k0: K).(\forall (t: T).(\forall (e: C).(\forall (u: T).((getl O (CHead c0 k0 t) (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CHead c0 k0 t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a)))))))))) with [(Bind b) \Rightarrow (\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u))).(let H1 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow e | (CHead c _ _) \Rightarrow c])) (CHead e (Bind Abbr) u) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e (Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u) H0))) in ((let H2 \def (f_equal C B (\lambda (e0: C).(match e0 return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead e (Bind Abbr) u) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e (Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u) H0))) in ((let H3 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead e (Bind Abbr) u) (CHead c0 (Bind b) t) (clear_gen_bind b c0 (CHead e (Bind Abbr) u) t (getl_gen_O (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u) H0))) in (\lambda (H4: (eq B Abbr b)).(\lambda (_: (eq C e c0)).(eq_ind_r T t (\lambda (t0: T).(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O t0 (CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))))) (eq_ind B Abbr (\lambda (b0: B).(ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O t (CHead c0 (Bind b0) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))))) (ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O t (CHead c0 (Bind Abbr) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))) (CHead c0 (Bind Abbr) t) c0 (csubst1_refl O t (CHead c0 (Bind Abbr) t)) (drop_drop (Bind Abbr) O c0 c0 (drop_refl c0) t)) b H4) u H3)))) H2)) H1)))))) | (Flat f) \Rightarrow (\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl O (CHead c0 (Flat f) t) (CHead e (Bind Abbr) u))).(let H_x \def (subst1_ex u t O) in (let H1 \def H_x in (ex_ind T (\lambda (t2: T).(subst1 O u t (lift (S O) O t2))) (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a)))) (\lambda (x: T).(\lambda (H2: (subst1 O u t (lift (S O) O x))).(let H3 \def (H e u (getl_intro O c0 (CHead e (Bind Abbr) u) c0 (drop_refl c0) (clear_gen_flat f c0 (CHead e (Bind Abbr) u) t (getl_gen_O (CHead c0 (Flat f) t) (CHead e (Bind Abbr) u) H0)))) in (ex2_2_ind C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))) (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a)))) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H4: (csubst1 O u c0 x0)).(\lambda (H5: (drop (S O) O x0 x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 O u (CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) O a0 a))) (CHead x0 (Flat f) (lift (S O) O x)) x1 (csubst1_flat f O u t (lift (S O) O x) H2 c0 x0 H4) (drop_drop (Flat f) O x0 x1 H5 (lift (S O) O x))))))) H3)))) H1)))))))])))) c)) (\lambda (n: nat).(\lambda (H: ((\forall (c: C).(\forall (e: C).(\forall (u: T).((getl n c (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 n u c a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) n a0 a)))))))))).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e: C).(\forall (u: T).((getl (S n) c0 (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))))))) (\lambda (n0: nat).(\lambda (e: C).(\lambda (u: T).(\lambda (H0: (getl (S n) (CSort n0) (CHead e (Bind Abbr) u))).(getl_gen_sort n0 (S n) (CHead e (Bind Abbr) u) H0 (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CSort n0) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))))))))) (\lambda (c0: C).(\lambda (H0: ((\forall (e: C).(\forall (u: T).((getl (S n) c0 (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))))))))).(\lambda (k: K).(match k return (\lambda (_: ?).(\lambda (k0: K).(\forall (t: T).(\forall (e: C).(\forall (u: T).((getl (S n) (CHead c0 k0 t) (CHead e (Bind Abbr) u)) \to (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 k0 t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))))))))) with [(Bind b) \Rightarrow (\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (H1: (getl (S n) (CHead c0 (Bind b) t) (CHead e (Bind Abbr) u))).(let H_x \def (subst1_ex u t n) in (let H2 \def H_x in (ex_ind T (\lambda (t2: T).(subst1 n u t (lift (S O) n t2))) (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x: T).(\lambda (H3: (subst1 n u t (lift (S O) n x))).(let H4 \def (H c0 e u (getl_gen_S (Bind b) c0 (CHead e (Bind Abbr) u) t n H1)) in (ex2_2_ind C C (\lambda (a0: C).(\lambda (_: C).(csubst1 n u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) n a0 a))) (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H5: (csubst1 n u c0 x0)).(\lambda (H6: (drop (S O) n x0 x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Bind b) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (CHead x0 (Bind b) (lift (S O) n x)) (CHead x1 (Bind b) x) (csubst1_bind b n u t (lift (S O) n x) H3 c0 x0 H5) (drop_skip_bind (S O) n x0 x1 H6 b x)))))) H4)))) H2))))))) | (Flat f) \Rightarrow (\lambda (t: T).(\lambda (e: C).(\lambda (u: T).(\lambda (H1: (getl (S n) (CHead c0 (Flat f) t) (CHead e (Bind Abbr) u))).(let H_x \def (subst1_ex u t (S n)) in (let H2 \def H_x in (ex_ind T (\lambda (t2: T).(subst1 (S n) u t (lift (S O) (S n) t2))) (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x: T).(\lambda (H3: (subst1 (S n) u t (lift (S O) (S n) x))).(let H4 \def (H0 e u (getl_gen_S (Flat f) c0 (CHead e (Bind Abbr) u) t n H1)) in (ex2_2_ind C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u c0 a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (ex2_2 C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a)))) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H5: (csubst1 (S n) u c0 x0)).(\lambda (H6: (drop (S O) (S n) x0 x1)).(ex2_2_intro C C (\lambda (a0: C).(\lambda (_: C).(csubst1 (S n) u (CHead c0 (Flat f) t) a0))) (\lambda (a0: C).(\lambda (a: C).(drop (S O) (S n) a0 a))) (CHead x0 (Flat f) (lift (S O) (S n) x)) (CHead x1 (Flat f) x) (csubst1_flat f (S n) u t (lift (S O) (S n) x) H3 c0 x0 H5) (drop_skip_flat (S O) n x0 x1 H6 f x)))))) H4)))) H2)))))))])))) c)))) d).
 
 inductive fsubst0 (i:nat) (v:T) (c1:C) (t1:T): C \to (T \to Prop) \def
 | fsubst0_snd: \forall (t2: T).((subst0 i v t1 t2) \to (fsubst0 i v c1 t1 c1 t2))
@@ -1335,7 +1335,7 @@ inductive fsubst0 (i:nat) (v:T) (c1:C) (t1:T): C \to (T \to Prop) \def
 theorem fsubst0_gen_base:
  \forall (c1: C).(\forall (c2: C).(\forall (t1: T).(\forall (t2: T).(\forall (v: T).(\forall (i: nat).((fsubst0 i v c1 t1 c2 t2) \to (or3 (land (eq C c1 c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c1 c2)) (land (subst0 i v t1 t2) (csubst0 i v c1 c2)))))))))
 \def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (v: T).(\lambda (i: nat).(\lambda (H: (fsubst0 i v c1 t1 c2 t2)).(let H0 \def (match H return (\lambda (c: C).(\lambda (t: T).((eq C c c2) \to ((eq T t t2) \to (or3 (land (eq C c1 c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c1 c2)) (land (subst0 i v t1 t2) (csubst0 i v c1 c2))))))) with [(fsubst0_snd t0 H0) \Rightarrow (\lambda (H1: (eq C c1 c2)).(\lambda (H2: (eq T t0 t2)).(eq_ind C c2 (\lambda (c: C).((eq T t0 t2) \to ((subst0 i v t1 t0) \to (or3 (land (eq C c c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c c2)) (land (subst0 i v t1 t2) (csubst0 i v c c2)))))) (\lambda (H3: (eq T t0 t2)).(eq_ind T t2 (\lambda (t: T).((subst0 i v t1 t) \to (or3 (land (eq C c2 c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c2 c2)) (land (subst0 i v t1 t2) (csubst0 i v c2 c2))))) (\lambda (H4: (subst0 i v t1 t2)).(or3_intro0 (land (eq C c2 c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c2 c2)) (land (subst0 i v t1 t2) (csubst0 i v c2 c2)) (conj (eq C c2 c2) (subst0 i v t1 t2) (refl_equal C c2) H4))) t0 (sym_eq T t0 t2 H3))) c1 (sym_eq C c1 c2 H1) H2 H0))) | (fsubst0_fst c0 H0) \Rightarrow (\lambda (H1: (eq C c0 c2)).(\lambda (H2: (eq T t1 t2)).(eq_ind C c2 (\lambda (c: C).((eq T t1 t2) \to ((csubst0 i v c1 c) \to (or3 (land (eq C c1 c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c1 c2)) (land (subst0 i v t1 t2) (csubst0 i v c1 c2)))))) (\lambda (H3: (eq T t1 t2)).(eq_ind T t2 (\lambda (t: T).((csubst0 i v c1 c2) \to (or3 (land (eq C c1 c2) (subst0 i v t t2)) (land (eq T t t2) (csubst0 i v c1 c2)) (land (subst0 i v t t2) (csubst0 i v c1 c2))))) (\lambda (H4: (csubst0 i v c1 c2)).(or3_intro1 (land (eq C c1 c2) (subst0 i v t2 t2)) (land (eq T t2 t2) (csubst0 i v c1 c2)) (land (subst0 i v t2 t2) (csubst0 i v c1 c2)) (conj (eq T t2 t2) (csubst0 i v c1 c2) (refl_equal T t2) H4))) t1 (sym_eq T t1 t2 H3))) c0 (sym_eq C c0 c2 H1) H2 H0))) | (fsubst0_both t0 H0 c0 H1) \Rightarrow (\lambda (H2: (eq C c0 c2)).(\lambda (H3: (eq T t0 t2)).(eq_ind C c2 (\lambda (c: C).((eq T t0 t2) \to ((subst0 i v t1 t0) \to ((csubst0 i v c1 c) \to (or3 (land (eq C c1 c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c1 c2)) (land (subst0 i v t1 t2) (csubst0 i v c1 c2))))))) (\lambda (H4: (eq T t0 t2)).(eq_ind T t2 (\lambda (t: T).((subst0 i v t1 t) \to ((csubst0 i v c1 c2) \to (or3 (land (eq C c1 c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c1 c2)) (land (subst0 i v t1 t2) (csubst0 i v c1 c2)))))) (\lambda (H5: (subst0 i v t1 t2)).(\lambda (H6: (csubst0 i v c1 c2)).(or3_intro2 (land (eq C c1 c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c1 c2)) (land (subst0 i v t1 t2) (csubst0 i v c1 c2)) (conj (subst0 i v t1 t2) (csubst0 i v c1 c2) H5 H6)))) t0 (sym_eq T t0 t2 H4))) c0 (sym_eq C c0 c2 H2) H3 H0 H1)))]) in (H0 (refl_equal C c2) (refl_equal T t2))))))))).
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (v: T).(\lambda (i: nat).(\lambda (H: (fsubst0 i v c1 t1 c2 t2)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c: C).(\lambda (t: T).((eq C c c2) \to ((eq T t t2) \to (or3 (land (eq C c1 c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c1 c2)) (land (subst0 i v t1 t2) (csubst0 i v c1 c2)))))))) with [(fsubst0_snd t0 H0) \Rightarrow (\lambda (H1: (eq C c1 c2)).(\lambda (H2: (eq T t0 t2)).(eq_ind C c2 (\lambda (c: C).((eq T t0 t2) \to ((subst0 i v t1 t0) \to (or3 (land (eq C c c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c c2)) (land (subst0 i v t1 t2) (csubst0 i v c c2)))))) (\lambda (H3: (eq T t0 t2)).(eq_ind T t2 (\lambda (t: T).((subst0 i v t1 t) \to (or3 (land (eq C c2 c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c2 c2)) (land (subst0 i v t1 t2) (csubst0 i v c2 c2))))) (\lambda (H4: (subst0 i v t1 t2)).(or3_intro0 (land (eq C c2 c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c2 c2)) (land (subst0 i v t1 t2) (csubst0 i v c2 c2)) (conj (eq C c2 c2) (subst0 i v t1 t2) (refl_equal C c2) H4))) t0 (sym_eq T t0 t2 H3))) c1 (sym_eq C c1 c2 H1) H2 H0))) | (fsubst0_fst c0 H0) \Rightarrow (\lambda (H1: (eq C c0 c2)).(\lambda (H2: (eq T t1 t2)).(eq_ind C c2 (\lambda (c: C).((eq T t1 t2) \to ((csubst0 i v c1 c) \to (or3 (land (eq C c1 c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c1 c2)) (land (subst0 i v t1 t2) (csubst0 i v c1 c2)))))) (\lambda (H3: (eq T t1 t2)).(eq_ind T t2 (\lambda (t: T).((csubst0 i v c1 c2) \to (or3 (land (eq C c1 c2) (subst0 i v t t2)) (land (eq T t t2) (csubst0 i v c1 c2)) (land (subst0 i v t t2) (csubst0 i v c1 c2))))) (\lambda (H4: (csubst0 i v c1 c2)).(or3_intro1 (land (eq C c1 c2) (subst0 i v t2 t2)) (land (eq T t2 t2) (csubst0 i v c1 c2)) (land (subst0 i v t2 t2) (csubst0 i v c1 c2)) (conj (eq T t2 t2) (csubst0 i v c1 c2) (refl_equal T t2) H4))) t1 (sym_eq T t1 t2 H3))) c0 (sym_eq C c0 c2 H1) H2 H0))) | (fsubst0_both t0 H0 c0 H1) \Rightarrow (\lambda (H2: (eq C c0 c2)).(\lambda (H3: (eq T t0 t2)).(eq_ind C c2 (\lambda (c: C).((eq T t0 t2) \to ((subst0 i v t1 t0) \to ((csubst0 i v c1 c) \to (or3 (land (eq C c1 c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c1 c2)) (land (subst0 i v t1 t2) (csubst0 i v c1 c2))))))) (\lambda (H4: (eq T t0 t2)).(eq_ind T t2 (\lambda (t: T).((subst0 i v t1 t) \to ((csubst0 i v c1 c2) \to (or3 (land (eq C c1 c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c1 c2)) (land (subst0 i v t1 t2) (csubst0 i v c1 c2)))))) (\lambda (H5: (subst0 i v t1 t2)).(\lambda (H6: (csubst0 i v c1 c2)).(or3_intro2 (land (eq C c1 c2) (subst0 i v t1 t2)) (land (eq T t1 t2) (csubst0 i v c1 c2)) (land (subst0 i v t1 t2) (csubst0 i v c1 c2)) (conj (subst0 i v t1 t2) (csubst0 i v c1 c2) H5 H6)))) t0 (sym_eq T t0 t2 H4))) c0 (sym_eq C c0 c2 H2) H3 H0 H1)))]) in (H0 (refl_equal C c2) (refl_equal T t2))))))))).
 
 record G : Set \def {
  next: (nat \to nat);
@@ -1445,12 +1445,12 @@ inductive leq (g:G): A \to (A \to Prop) \def
 theorem leq_gen_sort:
  \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq g (ASort h1 n1) a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k))))))))))
 \def
- \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2: A).(\lambda (H: (leq g (ASort h1 n1) a2)).(let H0 \def (match H return (\lambda (a: A).(\lambda (a0: A).((eq A a (ASort h1 n1)) \to ((eq A a0 a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)))))))))) with [(leq_sort h0 h2 n0 n2 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h0 n0) (ASort h1 n1))).(\lambda (H2: (eq A (ASort h2 n2) a2)).((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n0])) (ASort h0 n0) (ASort h1 n1) H1) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0) (ASort h1 n1) H1) in (eq_ind nat h1 (\lambda (n: nat).((eq nat n0 n1) \to ((eq A (ASort h2 n2) a2) \to ((eq A (aplus g (ASort n n0) k) (aplus g (ASort h2 n2) k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A a2 (ASort h3 n3))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 n3) k0)))))))))) (\lambda (H5: (eq nat n0 n1)).(eq_ind nat n1 (\lambda (n: nat).((eq A (ASort h2 n2) a2) \to ((eq A (aplus g (ASort h1 n) k) (aplus g (ASort h2 n2) k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A a2 (ASort h3 n3))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 n3) k0))))))))) (\lambda (H6: (eq A (ASort h2 n2) a2)).(eq_ind A (ASort h2 n2) (\lambda (a: A).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A a (ASort h3 n3))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 n3) k0)))))))) (\lambda (H7: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2) (ASort h3 n3))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 n3) k0))))) n2 h2 k (refl_equal A (ASort h2 n2)) H7)) a2 H6)) n0 (sym_eq nat n0 n1 H5))) h0 (sym_eq nat h0 h1 H4))) H3)) H2 H0))) | (leq_head a1 a0 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort h1 n1))).(\lambda (H3: (eq A (AHead a0 a4) a2)).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort h1 n1) H2) in (False_ind ((eq A (AHead a0 a4) a2) \to ((leq g a1 a0) \to ((leq g a3 a4) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k))))))))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (ASort h1 n1)) (refl_equal A a2))))))).
+ \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2: A).(\lambda (H: (leq g (ASort h1 n1) a2)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (a: A).(\lambda (a0: A).((eq A a (ASort h1 n1)) \to ((eq A a0 a2) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k))))))))))) with [(leq_sort h0 h2 n0 n2 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h0 n0) (ASort h1 n1))).(\lambda (H2: (eq A (ASort h2 n2) a2)).((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n0])) (ASort h0 n0) (ASort h1 n1) H1) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0) (ASort h1 n1) H1) in (eq_ind nat h1 (\lambda (n: nat).((eq nat n0 n1) \to ((eq A (ASort h2 n2) a2) \to ((eq A (aplus g (ASort n n0) k) (aplus g (ASort h2 n2) k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A a2 (ASort h3 n3))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 n3) k0)))))))))) (\lambda (H5: (eq nat n0 n1)).(eq_ind nat n1 (\lambda (n: nat).((eq A (ASort h2 n2) a2) \to ((eq A (aplus g (ASort h1 n) k) (aplus g (ASort h2 n2) k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A a2 (ASort h3 n3))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 n3) k0))))))))) (\lambda (H6: (eq A (ASort h2 n2) a2)).(eq_ind A (ASort h2 n2) (\lambda (a: A).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A a (ASort h3 n3))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 n3) k0)))))))) (\lambda (H7: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2) (ASort h3 n3))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 n3) k0))))) n2 h2 k (refl_equal A (ASort h2 n2)) H7)) a2 H6)) n0 (sym_eq nat n0 n1 H5))) h0 (sym_eq nat h0 h1 H4))) H3)) H2 H0))) | (leq_head a1 a0 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort h1 n1))).(\lambda (H3: (eq A (AHead a0 a4) a2)).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort h1 n1) H2) in (False_ind ((eq A (AHead a0 a4) a2) \to ((leq g a1 a0) \to ((leq g a3 a4) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2 (ASort h2 n2))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k))))))))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (ASort h1 n1)) (refl_equal A a2))))))).
 
 theorem leq_gen_head:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g (AHead a1 a2) a) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))))))))
 \def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda (H: (leq g (AHead a1 a2) a)).(let H0 \def (match H return (\lambda (a0: A).(\lambda (a3: A).((eq A a0 (AHead a1 a2)) \to ((eq A a3 a) \to (ex3_2 A A (\lambda (a4: A).(\lambda (a5: A).(eq A a (AHead a4 a5)))) (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda (_: A).(\lambda (a5: A).(leq g a2 a5)))))))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead a1 a2))).(\lambda (H2: (eq A (ASort h2 n2) a)).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a1 a2) H1) in (False_ind ((eq A (ASort h2 n2) a) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4)))))) H3)) H2 H0))) | (leq_head a0 a3 H0 a4 a5 H1) \Rightarrow (\lambda (H2: (eq A (AHead a0 a4) (AHead a1 a2))).(\lambda (H3: (eq A (AHead a3 a5) a)).((let H4 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a0 a4) (AHead a1 a2) H2) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a4) (AHead a1 a2) H2) in (eq_ind A a1 (\lambda (a6: A).((eq A a4 a2) \to ((eq A (AHead a3 a5) a) \to ((leq g a6 a3) \to ((leq g a4 a5) \to (ex3_2 A A (\lambda (a7: A).(\lambda (a8: A).(eq A a (AHead a7 a8)))) (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_: A).(\lambda (a8: A).(leq g a2 a8))))))))) (\lambda (H6: (eq A a4 a2)).(eq_ind A a2 (\lambda (a6: A).((eq A (AHead a3 a5) a) \to ((leq g a1 a3) \to ((leq g a6 a5) \to (ex3_2 A A (\lambda (a7: A).(\lambda (a8: A).(eq A a (AHead a7 a8)))) (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_: A).(\lambda (a8: A).(leq g a2 a8)))))))) (\lambda (H7: (eq A (AHead a3 a5) a)).(eq_ind A (AHead a3 a5) (\lambda (a: A).((leq g a1 a3) \to ((leq g a2 a5) \to (ex3_2 A A (\lambda (a6: A).(\lambda (a7: A).(eq A a (AHead a6 a7)))) (\lambda (a6: A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 a7))))))) (\lambda (H8: (leq g a1 a3)).(\lambda (H9: (leq g a2 a5)).(ex3_2_intro A A (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a3 a5) (AHead a6 a7)))) (\lambda (a6: A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 a7))) a3 a5 (refl_equal A (AHead a3 a5)) H8 H9))) a H7)) a4 (sym_eq A a4 a2 H6))) a0 (sym_eq A a0 a1 H5))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (AHead a1 a2)) (refl_equal A a))))))).
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda (H: (leq g (AHead a1 a2) a)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (a0: A).(\lambda (a3: A).((eq A a0 (AHead a1 a2)) \to ((eq A a3 a) \to (ex3_2 A A (\lambda (a4: A).(\lambda (a5: A).(eq A a (AHead a4 a5)))) (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda (_: A).(\lambda (a5: A).(leq g a2 a5))))))))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead a1 a2))).(\lambda (H2: (eq A (ASort h2 n2) a)).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a1 a2) H1) in (False_ind ((eq A (ASort h2 n2) a) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (ex3_2 A A (\lambda (a3: A).(\lambda (a4: A).(eq A a (AHead a3 a4)))) (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4)))))) H3)) H2 H0))) | (leq_head a0 a3 H0 a4 a5 H1) \Rightarrow (\lambda (H2: (eq A (AHead a0 a4) (AHead a1 a2))).(\lambda (H3: (eq A (AHead a3 a5) a)).((let H4 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a0 a4) (AHead a1 a2) H2) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a4) (AHead a1 a2) H2) in (eq_ind A a1 (\lambda (a6: A).((eq A a4 a2) \to ((eq A (AHead a3 a5) a) \to ((leq g a6 a3) \to ((leq g a4 a5) \to (ex3_2 A A (\lambda (a7: A).(\lambda (a8: A).(eq A a (AHead a7 a8)))) (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_: A).(\lambda (a8: A).(leq g a2 a8))))))))) (\lambda (H6: (eq A a4 a2)).(eq_ind A a2 (\lambda (a6: A).((eq A (AHead a3 a5) a) \to ((leq g a1 a3) \to ((leq g a6 a5) \to (ex3_2 A A (\lambda (a7: A).(\lambda (a8: A).(eq A a (AHead a7 a8)))) (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_: A).(\lambda (a8: A).(leq g a2 a8)))))))) (\lambda (H7: (eq A (AHead a3 a5) a)).(eq_ind A (AHead a3 a5) (\lambda (a: A).((leq g a1 a3) \to ((leq g a2 a5) \to (ex3_2 A A (\lambda (a6: A).(\lambda (a7: A).(eq A a (AHead a6 a7)))) (\lambda (a6: A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 a7))))))) (\lambda (H8: (leq g a1 a3)).(\lambda (H9: (leq g a2 a5)).(ex3_2_intro A A (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a3 a5) (AHead a6 a7)))) (\lambda (a6: A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 a7))) a3 a5 (refl_equal A (AHead a3 a5)) H8 H9))) a H7)) a4 (sym_eq A a4 a2 H6))) a0 (sym_eq A a0 a1 H5))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (AHead a1 a2)) (refl_equal A a))))))).
 
 theorem asucc_gen_sort:
  \forall (g: G).(\forall (h: nat).(\forall (n: nat).(\forall (a: A).((eq A (ASort h n) (asucc g a)) \to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A a (ASort h0 n0)))))))))
@@ -1490,12 +1490,12 @@ theorem aplus_sort_S_S_simpl:
 theorem asucc_repl:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g (asucc g a1) (asucc g a2)))))
 \def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g (asucc g a) (asucc g a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k))).((match h1 return (\lambda (n: nat).((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (match n with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) with [O \Rightarrow (\lambda (H1: (eq A (aplus g (ASort O n1) k) (aplus g (ASort h2 n2) k))).((match h2 return (\lambda (n: nat).((eq A (aplus g (ASort O n1) k) (aplus g (ASort n n2) k)) \to (leq g (ASort O (next g n1)) (match n with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) with [O \Rightarrow (\lambda (H2: (eq A (aplus g (ASort O n1) k) (aplus g (ASort O n2) k))).(leq_sort g O O (next g n1) (next g n2) k (eq_ind A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n2)) k))) (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort O n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O n2) k)))) (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort O n1) k) H2) (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k)) (aplus g (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))) | (S n) \Rightarrow (\lambda (H2: (eq A (aplus g (ASort O n1) k) (aplus g (ASort (S n) n2) k))).(leq_sort g O n (next g n1) n2 k (eq_ind A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort n n2) k))) (eq_ind A (aplus g (ASort (S n) n2) (S k)) (\lambda (a: A).(eq A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort (S n) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort (S n) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S n) n2) k))) (aplus g (ASort O n1) k) H2) (aplus g (ASort n n2) k) (aplus_sort_S_S_simpl g n2 n k)) (aplus g (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k))))]) H1)) | (S n) \Rightarrow (\lambda (H1: (eq A (aplus g (ASort (S n) n1) k) (aplus g (ASort h2 n2) k))).((match h2 return (\lambda (n0: nat).((eq A (aplus g (ASort (S n) n1) k) (aplus g (ASort n0 n2) k)) \to (leq g (ASort n n1) (match n0 with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) with [O \Rightarrow (\lambda (H2: (eq A (aplus g (ASort (S n) n1) k) (aplus g (ASort O n2) k))).(leq_sort g n O n1 (next g n2) k (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq A (aplus g (ASort n n1) k) a)) (eq_ind A (aplus g (ASort (S n) n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort O n2) (S k)))) (eq_ind_r A (aplus g (ASort O n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O n2) k)))) (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort (S n) n1) k) H2) (aplus g (ASort n n1) k) (aplus_sort_S_S_simpl g n1 n k)) (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k)))) | (S n0) \Rightarrow (\lambda (H2: (eq A (aplus g (ASort (S n) n1) k) (aplus g (ASort (S n0) n2) k))).(leq_sort g n n0 n1 n2 k (eq_ind A (aplus g (ASort (S n) n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort n0 n2) k))) (eq_ind A (aplus g (ASort (S n0) n2) (S k)) (\lambda (a: A).(eq A (aplus g (ASort (S n) n1) (S k)) a)) (eq_ind_r A (aplus g (ASort (S n0) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort (S n0) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S n0) n2) k))) (aplus g (ASort (S n) n1) k) H2) (aplus g (ASort n0 n2) k) (aplus_sort_S_S_simpl g n2 n0 k)) (aplus g (ASort n n1) k) (aplus_sort_S_S_simpl g n1 n k))))]) H1))]) H0))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (H0: (leq g a3 a4)).(\lambda (_: (leq g (asucc g a3) (asucc g a4))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g (asucc g a5) (asucc g a6))).(leq_head g a3 a4 H0 (asucc g a5) (asucc g a6) H3))))))))) a1 a2 H)))).
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g (asucc g a) (asucc g a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k))).((match h1 return (\lambda (_: ?).(\lambda (n: nat).((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (match n with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))) with [O \Rightarrow (\lambda (H1: (eq A (aplus g (ASort O n1) k) (aplus g (ASort h2 n2) k))).((match h2 return (\lambda (_: ?).(\lambda (n: nat).((eq A (aplus g (ASort O n1) k) (aplus g (ASort n n2) k)) \to (leq g (ASort O (next g n1)) (match n with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))) with [O \Rightarrow (\lambda (H2: (eq A (aplus g (ASort O n1) k) (aplus g (ASort O n2) k))).(leq_sort g O O (next g n1) (next g n2) k (eq_ind A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n2)) k))) (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort O n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O n2) k)))) (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort O n1) k) H2) (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k)) (aplus g (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))) | (S n) \Rightarrow (\lambda (H2: (eq A (aplus g (ASort O n1) k) (aplus g (ASort (S n) n2) k))).(leq_sort g O n (next g n1) n2 k (eq_ind A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort n n2) k))) (eq_ind A (aplus g (ASort (S n) n2) (S k)) (\lambda (a: A).(eq A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort (S n) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort (S n) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S n) n2) k))) (aplus g (ASort O n1) k) H2) (aplus g (ASort n n2) k) (aplus_sort_S_S_simpl g n2 n k)) (aplus g (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k))))]) H1)) | (S n) \Rightarrow (\lambda (H1: (eq A (aplus g (ASort (S n) n1) k) (aplus g (ASort h2 n2) k))).((match h2 return (\lambda (_: ?).(\lambda (n0: nat).((eq A (aplus g (ASort (S n) n1) k) (aplus g (ASort n0 n2) k)) \to (leq g (ASort n n1) (match n0 with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))) with [O \Rightarrow (\lambda (H2: (eq A (aplus g (ASort (S n) n1) k) (aplus g (ASort O n2) k))).(leq_sort g n O n1 (next g n2) k (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq A (aplus g (ASort n n1) k) a)) (eq_ind A (aplus g (ASort (S n) n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort O n2) (S k)))) (eq_ind_r A (aplus g (ASort O n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O n2) k)))) (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort (S n) n1) k) H2) (aplus g (ASort n n1) k) (aplus_sort_S_S_simpl g n1 n k)) (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k)))) | (S n0) \Rightarrow (\lambda (H2: (eq A (aplus g (ASort (S n) n1) k) (aplus g (ASort (S n0) n2) k))).(leq_sort g n n0 n1 n2 k (eq_ind A (aplus g (ASort (S n) n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort n0 n2) k))) (eq_ind A (aplus g (ASort (S n0) n2) (S k)) (\lambda (a: A).(eq A (aplus g (ASort (S n) n1) (S k)) a)) (eq_ind_r A (aplus g (ASort (S n0) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort (S n0) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S n0) n2) k))) (aplus g (ASort (S n) n1) k) H2) (aplus g (ASort n0 n2) k) (aplus_sort_S_S_simpl g n2 n0 k)) (aplus g (ASort n n1) k) (aplus_sort_S_S_simpl g n1 n k))))]) H1))]) H0))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (H0: (leq g a3 a4)).(\lambda (_: (leq g (asucc g a3) (asucc g a4))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g (asucc g a5) (asucc g a6))).(leq_head g a3 a4 H0 (asucc g a5) (asucc g a6) H3))))))))) a1 a2 H)))).
 
 theorem asucc_inj:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc g a2)) \to (leq g a1 a2))))
 \def
- \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (a2: A).(A_ind (\lambda (a: A).((leq g (asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a))) (\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0)) (asucc g (ASort n1 n2)))).((match n return (\lambda (n3: nat).((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2)))) with [O \Rightarrow (\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc g (ASort n1 n2)))).((match n1 return (\lambda (n3: nat).((leq g (asucc g (ASort O n0)) (asucc g (ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 n2)))) with [O \Rightarrow (\lambda (H1: (leq g (asucc g (ASort O n0)) (asucc g (ASort O n2)))).(let H2 \def (match H1 return (\lambda (a: A).(\lambda (a0: A).((eq A a (ASort O (next g n0))) \to ((eq A a0 (ASort O (next g n2))) \to (leq g (ASort O n0) (ASort O n2)))))) with [(leq_sort h1 h2 n1 n3 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (ASort O (next g n0)))).(\lambda (H2: (eq A (ASort h2 n3) (ASort O (next g n2)))).((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) (ASort h1 n1) (ASort O (next g n0)) H1) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort O (next g n0)) H1) in (eq_ind nat O (\lambda (n: nat).((eq nat n1 (next g n0)) \to ((eq A (ASort h2 n3) (ASort O (next g n2))) \to ((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort O n0) (ASort O n2)))))) (\lambda (H5: (eq nat n1 (next g n0))).(eq_ind nat (next g n0) (\lambda (n: nat).((eq A (ASort h2 n3) (ASort O (next g n2))) \to ((eq A (aplus g (ASort O n) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort O n0) (ASort O n2))))) (\lambda (H6: (eq A (ASort h2 n3) (ASort O (next g n2)))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n3])) (ASort h2 n3) (ASort O (next g n2)) H6) in ((let H8 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n3) (ASort O (next g n2)) H6) in (eq_ind nat O (\lambda (n: nat).((eq nat n3 (next g n2)) \to ((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n n3) k)) \to (leq g (ASort O n0) (ASort O n2))))) (\lambda (H9: (eq nat n3 (next g n2))).(eq_ind nat (next g n2) (\lambda (n: nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort O n) k)) \to (leq g (ASort O n0) (ASort O n2)))) (\lambda (H10: (eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort O (next g n2)) k))).(let H \def (eq_ind_r A (aplus g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n2)) k))) H10 (aplus g (ASort O n0) (S k)) (aplus_sort_O_S_simpl g n0 k)) in (let H11 \def (eq_ind_r A (aplus g (ASort O (next g n2)) k) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S k)) a)) H (aplus g (ASort O n2) (S k)) (aplus_sort_O_S_simpl g n2 k)) in (leq_sort g O O n0 n2 (S k) H11)))) n3 (sym_eq nat n3 (next g n2) H9))) h2 (sym_eq nat h2 O H8))) H7))) n1 (sym_eq nat n1 (next g n0) H5))) h1 (sym_eq nat h1 O H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort O (next g n0)))).(\lambda (H3: (eq A (AHead a2 a4) (ASort O (next g n2)))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H2) in (False_ind ((eq A (AHead a2 a4) (ASort O (next g n2))) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort O n0) (ASort O n2))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal A (ASort O (next g n0))) (refl_equal A (ASort O (next g n2)))))) | (S n3) \Rightarrow (\lambda (H1: (leq g (asucc g (ASort O n0)) (asucc g (ASort (S n3) n2)))).(let H2 \def (match H1 return (\lambda (a: A).(\lambda (a0: A).((eq A a (ASort O (next g n0))) \to ((eq A a0 (ASort n3 n2)) \to (leq g (ASort O n0) (ASort (S n3) n2)))))) with [(leq_sort h1 h2 n1 n3 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (ASort O (next g n0)))).(\lambda (H2: (eq A (ASort h2 n3) (ASort n3 n2))).((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) (ASort h1 n1) (ASort O (next g n0)) H1) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort O (next g n0)) H1) in (eq_ind nat O (\lambda (n: nat).((eq nat n1 (next g n0)) \to ((eq A (ASort h2 n3) (ASort n3 n2)) \to ((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort O n0) (ASort (S n3) n2)))))) (\lambda (H5: (eq nat n1 (next g n0))).(eq_ind nat (next g n0) (\lambda (n: nat).((eq A (ASort h2 n3) (ASort n3 n2)) \to ((eq A (aplus g (ASort O n) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort O n0) (ASort (S n3) n2))))) (\lambda (H6: (eq A (ASort h2 n3) (ASort n3 n2))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n3])) (ASort h2 n3) (ASort n3 n2) H6) in ((let H8 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n3) (ASort n3 n2) H6) in (eq_ind nat n3 (\lambda (n: nat).((eq nat n3 n2) \to ((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n n3) k)) \to (leq g (ASort O n0) (ASort (S n3) n2))))) (\lambda (H9: (eq nat n3 n2)).(eq_ind nat n2 (\lambda (n: nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n3 n) k)) \to (leq g (ASort O n0) (ASort (S n3) n2)))) (\lambda (H10: (eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n3 n2) k))).(let H \def (eq_ind_r A (aplus g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a (aplus g (ASort n3 n2) k))) H10 (aplus g (ASort O n0) (S k)) (aplus_sort_O_S_simpl g n0 k)) in (let H11 \def (eq_ind_r A (aplus g (ASort n3 n2) k) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S k)) a)) H (aplus g (ASort (S n3) n2) (S k)) (aplus_sort_S_S_simpl g n2 n3 k)) in (leq_sort g O (S n3) n0 n2 (S k) H11)))) n3 (sym_eq nat n3 n2 H9))) h2 (sym_eq nat h2 n3 H8))) H7))) n1 (sym_eq nat n1 (next g n0) H5))) h1 (sym_eq nat h1 O H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort O (next g n0)))).(\lambda (H3: (eq A (AHead a2 a4) (ASort n3 n2))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H2) in (False_ind ((eq A (AHead a2 a4) (ASort n3 n2)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort O n0) (ASort (S n3) n2))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal A (ASort O (next g n0))) (refl_equal A (ASort n3 n2)))))]) H0)) | (S n3) \Rightarrow (\lambda (H0: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).((match n1 return (\lambda (n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 n2))) \to (leq g (ASort (S n3) n0) (ASort n4 n2)))) with [O \Rightarrow (\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort O n2)))).(let H2 \def (match H1 return (\lambda (a: A).(\lambda (a0: A).((eq A a (ASort n3 n0)) \to ((eq A a0 (ASort O (next g n2))) \to (leq g (ASort (S n3) n0) (ASort O n2)))))) with [(leq_sort h1 h2 n1 n3 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (ASort n3 n0))).(\lambda (H2: (eq A (ASort h2 n3) (ASort O (next g n2)))).((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) (ASort h1 n1) (ASort n3 n0) H1) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort n3 n0) H1) in (eq_ind nat n3 (\lambda (n: nat).((eq nat n1 n0) \to ((eq A (ASort h2 n3) (ASort O (next g n2))) \to ((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort (S n3) n0) (ASort O n2)))))) (\lambda (H5: (eq nat n1 n0)).(eq_ind nat n0 (\lambda (n: nat).((eq A (ASort h2 n3) (ASort O (next g n2))) \to ((eq A (aplus g (ASort n3 n) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort (S n3) n0) (ASort O n2))))) (\lambda (H6: (eq A (ASort h2 n3) (ASort O (next g n2)))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n3])) (ASort h2 n3) (ASort O (next g n2)) H6) in ((let H8 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n3) (ASort O (next g n2)) H6) in (eq_ind nat O (\lambda (n: nat).((eq nat n3 (next g n2)) \to ((eq A (aplus g (ASort n3 n0) k) (aplus g (ASort n n3) k)) \to (leq g (ASort (S n3) n0) (ASort O n2))))) (\lambda (H9: (eq nat n3 (next g n2))).(eq_ind nat (next g n2) (\lambda (n: nat).((eq A (aplus g (ASort n3 n0) k) (aplus g (ASort O n) k)) \to (leq g (ASort (S n3) n0) (ASort O n2)))) (\lambda (H10: (eq A (aplus g (ASort n3 n0) k) (aplus g (ASort O (next g n2)) k))).(let H \def (eq_ind_r A (aplus g (ASort n3 n0) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n2)) k))) H10 (aplus g (ASort (S n3) n0) (S k)) (aplus_sort_S_S_simpl g n0 n3 k)) in (let H11 \def (eq_ind_r A (aplus g (ASort O (next g n2)) k) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S k)) a)) H (aplus g (ASort O n2) (S k)) (aplus_sort_O_S_simpl g n2 k)) in (leq_sort g (S n3) O n0 n2 (S k) H11)))) n3 (sym_eq nat n3 (next g n2) H9))) h2 (sym_eq nat h2 O H8))) H7))) n1 (sym_eq nat n1 n0 H5))) h1 (sym_eq nat h1 n3 H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort n3 n0))).(\lambda (H3: (eq A (AHead a2 a4) (ASort O (next g n2)))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n3 n0) H2) in (False_ind ((eq A (AHead a2 a4) (ASort O (next g n2))) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort (S n3) n0) (ASort O n2))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal A (ASort n3 n0)) (refl_equal A (ASort O (next g n2)))))) | (S n4) \Rightarrow (\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort (S n4) n2)))).(let H2 \def (match H1 return (\lambda (a: A).(\lambda (a0: A).((eq A a (ASort n3 n0)) \to ((eq A a0 (ASort n4 n2)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2)))))) with [(leq_sort h1 h2 n3 n4 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n3) (ASort n3 n0))).(\lambda (H2: (eq A (ASort h2 n4) (ASort n4 n2))).((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n3])) (ASort h1 n3) (ASort n3 n0) H1) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n3) (ASort n3 n0) H1) in (eq_ind nat n3 (\lambda (n: nat).((eq nat n3 n0) \to ((eq A (ASort h2 n4) (ASort n4 n2)) \to ((eq A (aplus g (ASort n n3) k) (aplus g (ASort h2 n4) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2)))))) (\lambda (H5: (eq nat n3 n0)).(eq_ind nat n0 (\lambda (n: nat).((eq A (ASort h2 n4) (ASort n4 n2)) \to ((eq A (aplus g (ASort n3 n) k) (aplus g (ASort h2 n4) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))) (\lambda (H6: (eq A (ASort h2 n4) (ASort n4 n2))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n4])) (ASort h2 n4) (ASort n4 n2) H6) in ((let H8 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n4) (ASort n4 n2) H6) in (eq_ind nat n4 (\lambda (n: nat).((eq nat n4 n2) \to ((eq A (aplus g (ASort n3 n0) k) (aplus g (ASort n n4) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))) (\lambda (H9: (eq nat n4 n2)).(eq_ind nat n2 (\lambda (n: nat).((eq A (aplus g (ASort n3 n0) k) (aplus g (ASort n4 n) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2)))) (\lambda (H10: (eq A (aplus g (ASort n3 n0) k) (aplus g (ASort n4 n2) k))).(let H \def (eq_ind_r A (aplus g (ASort n3 n0) k) (\lambda (a: A).(eq A a (aplus g (ASort n4 n2) k))) H10 (aplus g (ASort (S n3) n0) (S k)) (aplus_sort_S_S_simpl g n0 n3 k)) in (let H11 \def (eq_ind_r A (aplus g (ASort n4 n2) k) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S k)) a)) H (aplus g (ASort (S n4) n2) (S k)) (aplus_sort_S_S_simpl g n2 n4 k)) in (leq_sort g (S n3) (S n4) n0 n2 (S k) H11)))) n4 (sym_eq nat n4 n2 H9))) h2 (sym_eq nat h2 n4 H8))) H7))) n3 (sym_eq nat n3 n0 H5))) h1 (sym_eq nat h1 n3 H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort n3 n0))).(\lambda (H3: (eq A (AHead a2 a4) (ASort n4 n2))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n3 n0) H2) in (False_ind ((eq A (AHead a2 a4) (ASort n4 n2)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal A (ASort n3 n0)) (refl_equal A (ASort n4 n2)))))]) H0))]) H)))) (\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda (H0: (((leq g (asucc g (ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0) a0)))).(\lambda (H1: (leq g (asucc g (ASort n n0)) (asucc g (AHead a a0)))).((match n return (\lambda (n1: nat).((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a a0)))))) with [O \Rightarrow (\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O n0) a)))).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq g (ASort O n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g (AHead a a0)))).(let H5 \def (match H4 return (\lambda (a1: A).(\lambda (a2: A).((eq A a1 (ASort O (next g n0))) \to ((eq A a2 (AHead a (asucc g a0))) \to (leq g (ASort O n0) (AHead a a0)))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (ASort O (next g n0)))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a (asucc g a0)))).((let H5 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) (ASort h1 n1) (ASort O (next g n0)) H3) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort O (next g n0)) H3) in (eq_ind nat O (\lambda (n: nat).((eq nat n1 (next g n0)) \to ((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0) (AHead a a0)))))) (\lambda (H7: (eq nat n1 (next g n0))).(eq_ind nat (next g n0) (\lambda (n: nat).((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort O n) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0) (AHead a a0))))) (\lambda (H8: (eq A (ASort h2 n2) (AHead a (asucc g a0)))).(let H9 \def (eq_ind A (ASort h2 n2) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a (asucc g a0)) H8) in (False_ind ((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0) (AHead a a0))) H9))) n1 (sym_eq nat n1 (next g n0) H7))) h1 (sym_eq nat h1 O H6))) H5)) H4 H2))) | (leq_head a1 a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4: (eq A (AHead a1 a3) (ASort O (next g n0)))).(\lambda (H5: (eq A (AHead a2 a4) (AHead a (asucc g a0)))).((let H6 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H4) in (False_ind ((eq A (AHead a2 a4) (AHead a (asucc g a0))) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort O n0) (AHead a a0))))) H6)) H5 H2 H3)))]) in (H5 (refl_equal A (ASort O (next g n0))) (refl_equal A (AHead a (asucc g a0)))))))) | (S n1) \Rightarrow (\lambda (_: (((leq g (asucc g (ASort (S n1) n0)) (asucc g a)) \to (leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g (asucc g (ASort (S n1) n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort (S n1) n0)) (asucc g (AHead a a0)))).(let H5 \def (match H4 return (\lambda (a1: A).(\lambda (a2: A).((eq A a1 (ASort n1 n0)) \to ((eq A a2 (AHead a (asucc g a0))) \to (leq g (ASort (S n1) n0) (AHead a a0)))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (ASort n1 n0))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a (asucc g a0)))).((let H5 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) (ASort h1 n1) (ASort n1 n0) H3) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort n1 n0) H3) in (eq_ind nat n1 (\lambda (n: nat).((eq nat n1 n0) \to ((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort (S n1) n0) (AHead a a0)))))) (\lambda (H7: (eq nat n1 n0)).(eq_ind nat n0 (\lambda (n: nat).((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n1 n) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort (S n1) n0) (AHead a a0))))) (\lambda (H8: (eq A (ASort h2 n2) (AHead a (asucc g a0)))).(let H9 \def (eq_ind A (ASort h2 n2) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a (asucc g a0)) H8) in (False_ind ((eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort (S n1) n0) (AHead a a0))) H9))) n1 (sym_eq nat n1 n0 H7))) h1 (sym_eq nat h1 n1 H6))) H5)) H4 H2))) | (leq_head a1 a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4: (eq A (AHead a1 a3) (ASort n1 n0))).(\lambda (H5: (eq A (AHead a2 a4) (AHead a (asucc g a0)))).((let H6 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1 n0) H4) in (False_ind ((eq A (AHead a2 a4) (AHead a (asucc g a0))) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort (S n1) n0) (AHead a a0))))) H6)) H5 H2 H3)))]) in (H5 (refl_equal A (ASort n1 n0)) (refl_equal A (AHead a (asucc g a0))))))))]) H H0 H1)))))) a2)))) (\lambda (a: A).(\lambda (_: ((\forall (a2: A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0) (asucc g a2)) \to (leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3: A).((leq g (asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a a0)) (asucc g (ASort n n0)))).((match n return (\lambda (n1: nat).((leq g (asucc g (AHead a a0)) (asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1 n0)))) with [O \Rightarrow (\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O n0)))).(let H3 \def (match H2 return (\lambda (a1: A).(\lambda (a2: A).((eq A a1 (AHead a (asucc g a0))) \to ((eq A a2 (ASort O (next g n0))) \to (leq g (AHead a a0) (ASort O n0)))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (AHead a (asucc g a0)))).(\lambda (H4: (eq A (ASort h2 n2) (ASort O (next g n0)))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a (asucc g a0)) H3) in (False_ind ((eq A (ASort h2 n2) (ASort O (next g n0))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (AHead a a0) (ASort O n0)))) H5)) H4 H2))) | (leq_head a1 a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4: (eq A (AHead a1 a3) (AHead a (asucc g a0)))).(\lambda (H5: (eq A (AHead a2 a4) (ASort O (next g n0)))).((let H6 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a1 a3) (AHead a (asucc g a0)) H4) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead a (asucc g a0)) H4) in (eq_ind A a (\lambda (a5: A).((eq A a3 (asucc g a0)) \to ((eq A (AHead a2 a4) (ASort O (next g n0))) \to ((leq g a5 a2) \to ((leq g a3 a4) \to (leq g (AHead a a0) (ASort O n0))))))) (\lambda (H8: (eq A a3 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a5: A).((eq A (AHead a2 a4) (ASort O (next g n0))) \to ((leq g a a2) \to ((leq g a5 a4) \to (leq g (AHead a a0) (ASort O n0)))))) (\lambda (H9: (eq A (AHead a2 a4) (ASort O (next g n0)))).(let H10 \def (eq_ind A (AHead a2 a4) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H9) in (False_ind ((leq g a a2) \to ((leq g (asucc g a0) a4) \to (leq g (AHead a a0) (ASort O n0)))) H10))) a3 (sym_eq A a3 (asucc g a0) H8))) a1 (sym_eq A a1 a H7))) H6)) H5 H2 H3)))]) in (H3 (refl_equal A (AHead a (asucc g a0))) (refl_equal A (ASort O (next g n0)))))) | (S n1) \Rightarrow (\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort (S n1) n0)))).(let H3 \def (match H2 return (\lambda (a1: A).(\lambda (a2: A).((eq A a1 (AHead a (asucc g a0))) \to ((eq A a2 (ASort n1 n0)) \to (leq g (AHead a a0) (ASort (S n1) n0)))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (AHead a (asucc g a0)))).(\lambda (H4: (eq A (ASort h2 n2) (ASort n1 n0))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a (asucc g a0)) H3) in (False_ind ((eq A (ASort h2 n2) (ASort n1 n0)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (AHead a a0) (ASort (S n1) n0)))) H5)) H4 H2))) | (leq_head a1 a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4: (eq A (AHead a1 a3) (AHead a (asucc g a0)))).(\lambda (H5: (eq A (AHead a2 a4) (ASort n1 n0))).((let H6 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a1 a3) (AHead a (asucc g a0)) H4) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead a (asucc g a0)) H4) in (eq_ind A a (\lambda (a5: A).((eq A a3 (asucc g a0)) \to ((eq A (AHead a2 a4) (ASort n1 n0)) \to ((leq g a5 a2) \to ((leq g a3 a4) \to (leq g (AHead a a0) (ASort (S n1) n0))))))) (\lambda (H8: (eq A a3 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a5: A).((eq A (AHead a2 a4) (ASort n1 n0)) \to ((leq g a a2) \to ((leq g a5 a4) \to (leq g (AHead a a0) (ASort (S n1) n0)))))) (\lambda (H9: (eq A (AHead a2 a4) (ASort n1 n0))).(let H10 \def (eq_ind A (AHead a2 a4) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1 n0) H9) in (False_ind ((leq g a a2) \to ((leq g (asucc g a0) a4) \to (leq g (AHead a a0) (ASort (S n1) n0)))) H10))) a3 (sym_eq A a3 (asucc g a0) H8))) a1 (sym_eq A a1 a H7))) H6)) H5 H2 H3)))]) in (H3 (refl_equal A (AHead a (asucc g a0))) (refl_equal A (ASort n1 n0)))))]) H1)))) (\lambda (a3: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))).(\lambda (a4: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a4)) \to (leq g (AHead a a0) a4)))).(\lambda (H3: (leq g (asucc g (AHead a a0)) (asucc g (AHead a3 a4)))).(let H4 \def (match H3 return (\lambda (a1: A).(\lambda (a2: A).((eq A a1 (AHead a (asucc g a0))) \to ((eq A a2 (AHead a3 (asucc g a4))) \to (leq g (AHead a a0) (AHead a3 a4)))))) with [(leq_sort h1 h2 n1 n2 k H4) \Rightarrow (\lambda (H5: (eq A (ASort h1 n1) (AHead a (asucc g a0)))).(\lambda (H6: (eq A (ASort h2 n2) (AHead a3 (asucc g a4)))).((let H7 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a (asucc g a0)) H5) in (False_ind ((eq A (ASort h2 n2) (AHead a3 (asucc g a4))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (AHead a a0) (AHead a3 a4)))) H7)) H6 H4))) | (leq_head a3 a4 H4 a5 a6 H5) \Rightarrow (\lambda (H6: (eq A (AHead a3 a5) (AHead a (asucc g a0)))).(\lambda (H7: (eq A (AHead a4 a6) (AHead a3 (asucc g a4)))).((let H8 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a) \Rightarrow a])) (AHead a3 a5) (AHead a (asucc g a0)) H6) in ((let H9 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead a _) \Rightarrow a])) (AHead a3 a5) (AHead a (asucc g a0)) H6) in (eq_ind A a (\lambda (a1: A).((eq A a5 (asucc g a0)) \to ((eq A (AHead a4 a6) (AHead a3 (asucc g a4))) \to ((leq g a1 a4) \to ((leq g a5 a6) \to (leq g (AHead a a0) (AHead a3 a4))))))) (\lambda (H10: (eq A a5 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a1: A).((eq A (AHead a4 a6) (AHead a3 (asucc g a4))) \to ((leq g a a4) \to ((leq g a1 a6) \to (leq g (AHead a a0) (AHead a3 a4)))))) (\lambda (H11: (eq A (AHead a4 a6) (AHead a3 (asucc g a4)))).(let H12 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a6 | (AHead _ a) \Rightarrow a])) (AHead a4 a6) (AHead a3 (asucc g a4)) H11) in ((let H13 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a4 | (AHead a _) \Rightarrow a])) (AHead a4 a6) (AHead a3 (asucc g a4)) H11) in (eq_ind A a3 (\lambda (a1: A).((eq A a6 (asucc g a4)) \to ((leq g a a1) \to ((leq g (asucc g a0) a6) \to (leq g (AHead a a0) (AHead a3 a4)))))) (\lambda (H14: (eq A a6 (asucc g a4))).(eq_ind A (asucc g a4) (\lambda (a1: A).((leq g a a3) \to ((leq g (asucc g a0) a1) \to (leq g (AHead a a0) (AHead a3 a4))))) (\lambda (H15: (leq g a a3)).(\lambda (H16: (leq g (asucc g a0) (asucc g a4))).(leq_head g a a3 H15 a0 a4 (H0 a4 H16)))) a6 (sym_eq A a6 (asucc g a4) H14))) a4 (sym_eq A a4 a3 H13))) H12))) a5 (sym_eq A a5 (asucc g a0) H10))) a3 (sym_eq A a3 a H9))) H8)) H7 H4 H5)))]) in (H4 (refl_equal A (AHead a (asucc g a0))) (refl_equal A (AHead a3 (asucc g a4)))))))))) a2)))))) a1)).
+ \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (a2: A).(A_ind (\lambda (a: A).((leq g (asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a))) (\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0)) (asucc g (ASort n1 n2)))).((match n return (\lambda (_: ?).(\lambda (n3: nat).((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))) with [O \Rightarrow (\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc g (ASort n1 n2)))).((match n1 return (\lambda (_: ?).(\lambda (n3: nat).((leq g (asucc g (ASort O n0)) (asucc g (ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 n2))))) with [O \Rightarrow (\lambda (H1: (leq g (asucc g (ASort O n0)) (asucc g (ASort O n2)))).(let H2 \def (match H1 return (\lambda (_: ?).(\lambda (a: A).(\lambda (a0: A).((eq A a (ASort O (next g n0))) \to ((eq A a0 (ASort O (next g n2))) \to (leq g (ASort O n0) (ASort O n2))))))) with [(leq_sort h1 h2 n1 n3 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (ASort O (next g n0)))).(\lambda (H2: (eq A (ASort h2 n3) (ASort O (next g n2)))).((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) (ASort h1 n1) (ASort O (next g n0)) H1) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort O (next g n0)) H1) in (eq_ind nat O (\lambda (n: nat).((eq nat n1 (next g n0)) \to ((eq A (ASort h2 n3) (ASort O (next g n2))) \to ((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort O n0) (ASort O n2)))))) (\lambda (H5: (eq nat n1 (next g n0))).(eq_ind nat (next g n0) (\lambda (n: nat).((eq A (ASort h2 n3) (ASort O (next g n2))) \to ((eq A (aplus g (ASort O n) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort O n0) (ASort O n2))))) (\lambda (H6: (eq A (ASort h2 n3) (ASort O (next g n2)))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n3])) (ASort h2 n3) (ASort O (next g n2)) H6) in ((let H8 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n3) (ASort O (next g n2)) H6) in (eq_ind nat O (\lambda (n: nat).((eq nat n3 (next g n2)) \to ((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n n3) k)) \to (leq g (ASort O n0) (ASort O n2))))) (\lambda (H9: (eq nat n3 (next g n2))).(eq_ind nat (next g n2) (\lambda (n: nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort O n) k)) \to (leq g (ASort O n0) (ASort O n2)))) (\lambda (H10: (eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort O (next g n2)) k))).(let H \def (eq_ind_r A (aplus g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n2)) k))) H10 (aplus g (ASort O n0) (S k)) (aplus_sort_O_S_simpl g n0 k)) in (let H11 \def (eq_ind_r A (aplus g (ASort O (next g n2)) k) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S k)) a)) H (aplus g (ASort O n2) (S k)) (aplus_sort_O_S_simpl g n2 k)) in (leq_sort g O O n0 n2 (S k) H11)))) n3 (sym_eq nat n3 (next g n2) H9))) h2 (sym_eq nat h2 O H8))) H7))) n1 (sym_eq nat n1 (next g n0) H5))) h1 (sym_eq nat h1 O H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort O (next g n0)))).(\lambda (H3: (eq A (AHead a2 a4) (ASort O (next g n2)))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H2) in (False_ind ((eq A (AHead a2 a4) (ASort O (next g n2))) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort O n0) (ASort O n2))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal A (ASort O (next g n0))) (refl_equal A (ASort O (next g n2)))))) | (S n3) \Rightarrow (\lambda (H1: (leq g (asucc g (ASort O n0)) (asucc g (ASort (S n3) n2)))).(let H2 \def (match H1 return (\lambda (_: ?).(\lambda (a: A).(\lambda (a0: A).((eq A a (ASort O (next g n0))) \to ((eq A a0 (ASort n3 n2)) \to (leq g (ASort O n0) (ASort (S n3) n2))))))) with [(leq_sort h1 h2 n1 n3 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (ASort O (next g n0)))).(\lambda (H2: (eq A (ASort h2 n3) (ASort n3 n2))).((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) (ASort h1 n1) (ASort O (next g n0)) H1) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort O (next g n0)) H1) in (eq_ind nat O (\lambda (n: nat).((eq nat n1 (next g n0)) \to ((eq A (ASort h2 n3) (ASort n3 n2)) \to ((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort O n0) (ASort (S n3) n2)))))) (\lambda (H5: (eq nat n1 (next g n0))).(eq_ind nat (next g n0) (\lambda (n: nat).((eq A (ASort h2 n3) (ASort n3 n2)) \to ((eq A (aplus g (ASort O n) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort O n0) (ASort (S n3) n2))))) (\lambda (H6: (eq A (ASort h2 n3) (ASort n3 n2))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n3])) (ASort h2 n3) (ASort n3 n2) H6) in ((let H8 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n3) (ASort n3 n2) H6) in (eq_ind nat n3 (\lambda (n: nat).((eq nat n3 n2) \to ((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n n3) k)) \to (leq g (ASort O n0) (ASort (S n3) n2))))) (\lambda (H9: (eq nat n3 n2)).(eq_ind nat n2 (\lambda (n: nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n3 n) k)) \to (leq g (ASort O n0) (ASort (S n3) n2)))) (\lambda (H10: (eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n3 n2) k))).(let H \def (eq_ind_r A (aplus g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a (aplus g (ASort n3 n2) k))) H10 (aplus g (ASort O n0) (S k)) (aplus_sort_O_S_simpl g n0 k)) in (let H11 \def (eq_ind_r A (aplus g (ASort n3 n2) k) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S k)) a)) H (aplus g (ASort (S n3) n2) (S k)) (aplus_sort_S_S_simpl g n2 n3 k)) in (leq_sort g O (S n3) n0 n2 (S k) H11)))) n3 (sym_eq nat n3 n2 H9))) h2 (sym_eq nat h2 n3 H8))) H7))) n1 (sym_eq nat n1 (next g n0) H5))) h1 (sym_eq nat h1 O H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort O (next g n0)))).(\lambda (H3: (eq A (AHead a2 a4) (ASort n3 n2))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H2) in (False_ind ((eq A (AHead a2 a4) (ASort n3 n2)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort O n0) (ASort (S n3) n2))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal A (ASort O (next g n0))) (refl_equal A (ASort n3 n2)))))]) H0)) | (S n3) \Rightarrow (\lambda (H0: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).((match n1 return (\lambda (_: ?).(\lambda (n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 n2))) \to (leq g (ASort (S n3) n0) (ASort n4 n2))))) with [O \Rightarrow (\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort O n2)))).(let H2 \def (match H1 return (\lambda (_: ?).(\lambda (a: A).(\lambda (a0: A).((eq A a (ASort n3 n0)) \to ((eq A a0 (ASort O (next g n2))) \to (leq g (ASort (S n3) n0) (ASort O n2))))))) with [(leq_sort h1 h2 n1 n3 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (ASort n3 n0))).(\lambda (H2: (eq A (ASort h2 n3) (ASort O (next g n2)))).((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) (ASort h1 n1) (ASort n3 n0) H1) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort n3 n0) H1) in (eq_ind nat n3 (\lambda (n: nat).((eq nat n1 n0) \to ((eq A (ASort h2 n3) (ASort O (next g n2))) \to ((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort (S n3) n0) (ASort O n2)))))) (\lambda (H5: (eq nat n1 n0)).(eq_ind nat n0 (\lambda (n: nat).((eq A (ASort h2 n3) (ASort O (next g n2))) \to ((eq A (aplus g (ASort n3 n) k) (aplus g (ASort h2 n3) k)) \to (leq g (ASort (S n3) n0) (ASort O n2))))) (\lambda (H6: (eq A (ASort h2 n3) (ASort O (next g n2)))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n3])) (ASort h2 n3) (ASort O (next g n2)) H6) in ((let H8 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n3) (ASort O (next g n2)) H6) in (eq_ind nat O (\lambda (n: nat).((eq nat n3 (next g n2)) \to ((eq A (aplus g (ASort n3 n0) k) (aplus g (ASort n n3) k)) \to (leq g (ASort (S n3) n0) (ASort O n2))))) (\lambda (H9: (eq nat n3 (next g n2))).(eq_ind nat (next g n2) (\lambda (n: nat).((eq A (aplus g (ASort n3 n0) k) (aplus g (ASort O n) k)) \to (leq g (ASort (S n3) n0) (ASort O n2)))) (\lambda (H10: (eq A (aplus g (ASort n3 n0) k) (aplus g (ASort O (next g n2)) k))).(let H \def (eq_ind_r A (aplus g (ASort n3 n0) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n2)) k))) H10 (aplus g (ASort (S n3) n0) (S k)) (aplus_sort_S_S_simpl g n0 n3 k)) in (let H11 \def (eq_ind_r A (aplus g (ASort O (next g n2)) k) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S k)) a)) H (aplus g (ASort O n2) (S k)) (aplus_sort_O_S_simpl g n2 k)) in (leq_sort g (S n3) O n0 n2 (S k) H11)))) n3 (sym_eq nat n3 (next g n2) H9))) h2 (sym_eq nat h2 O H8))) H7))) n1 (sym_eq nat n1 n0 H5))) h1 (sym_eq nat h1 n3 H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort n3 n0))).(\lambda (H3: (eq A (AHead a2 a4) (ASort O (next g n2)))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n3 n0) H2) in (False_ind ((eq A (AHead a2 a4) (ASort O (next g n2))) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort (S n3) n0) (ASort O n2))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal A (ASort n3 n0)) (refl_equal A (ASort O (next g n2)))))) | (S n4) \Rightarrow (\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort (S n4) n2)))).(let H2 \def (match H1 return (\lambda (_: ?).(\lambda (a: A).(\lambda (a0: A).((eq A a (ASort n3 n0)) \to ((eq A a0 (ASort n4 n2)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))))) with [(leq_sort h1 h2 n3 n4 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n3) (ASort n3 n0))).(\lambda (H2: (eq A (ASort h2 n4) (ASort n4 n2))).((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n3])) (ASort h1 n3) (ASort n3 n0) H1) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n3) (ASort n3 n0) H1) in (eq_ind nat n3 (\lambda (n: nat).((eq nat n3 n0) \to ((eq A (ASort h2 n4) (ASort n4 n2)) \to ((eq A (aplus g (ASort n n3) k) (aplus g (ASort h2 n4) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2)))))) (\lambda (H5: (eq nat n3 n0)).(eq_ind nat n0 (\lambda (n: nat).((eq A (ASort h2 n4) (ASort n4 n2)) \to ((eq A (aplus g (ASort n3 n) k) (aplus g (ASort h2 n4) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))) (\lambda (H6: (eq A (ASort h2 n4) (ASort n4 n2))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n4])) (ASort h2 n4) (ASort n4 n2) H6) in ((let H8 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n4) (ASort n4 n2) H6) in (eq_ind nat n4 (\lambda (n: nat).((eq nat n4 n2) \to ((eq A (aplus g (ASort n3 n0) k) (aplus g (ASort n n4) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))) (\lambda (H9: (eq nat n4 n2)).(eq_ind nat n2 (\lambda (n: nat).((eq A (aplus g (ASort n3 n0) k) (aplus g (ASort n4 n) k)) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2)))) (\lambda (H10: (eq A (aplus g (ASort n3 n0) k) (aplus g (ASort n4 n2) k))).(let H \def (eq_ind_r A (aplus g (ASort n3 n0) k) (\lambda (a: A).(eq A a (aplus g (ASort n4 n2) k))) H10 (aplus g (ASort (S n3) n0) (S k)) (aplus_sort_S_S_simpl g n0 n3 k)) in (let H11 \def (eq_ind_r A (aplus g (ASort n4 n2) k) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S k)) a)) H (aplus g (ASort (S n4) n2) (S k)) (aplus_sort_S_S_simpl g n2 n4 k)) in (leq_sort g (S n3) (S n4) n0 n2 (S k) H11)))) n4 (sym_eq nat n4 n2 H9))) h2 (sym_eq nat h2 n4 H8))) H7))) n3 (sym_eq nat n3 n0 H5))) h1 (sym_eq nat h1 n3 H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort n3 n0))).(\lambda (H3: (eq A (AHead a2 a4) (ASort n4 n2))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n3 n0) H2) in (False_ind ((eq A (AHead a2 a4) (ASort n4 n2)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort (S n3) n0) (ASort (S n4) n2))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal A (ASort n3 n0)) (refl_equal A (ASort n4 n2)))))]) H0))]) H)))) (\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda (H0: (((leq g (asucc g (ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0) a0)))).(\lambda (H1: (leq g (asucc g (ASort n n0)) (asucc g (AHead a a0)))).((match n return (\lambda (_: ?).(\lambda (n1: nat).((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a a0))))))) with [O \Rightarrow (\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O n0) a)))).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq g (ASort O n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g (AHead a a0)))).(let H5 \def (match H4 return (\lambda (_: ?).(\lambda (a1: A).(\lambda (a2: A).((eq A a1 (ASort O (next g n0))) \to ((eq A a2 (AHead a (asucc g a0))) \to (leq g (ASort O n0) (AHead a a0))))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (ASort O (next g n0)))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a (asucc g a0)))).((let H5 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) (ASort h1 n1) (ASort O (next g n0)) H3) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort O (next g n0)) H3) in (eq_ind nat O (\lambda (n: nat).((eq nat n1 (next g n0)) \to ((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0) (AHead a a0)))))) (\lambda (H7: (eq nat n1 (next g n0))).(eq_ind nat (next g n0) (\lambda (n: nat).((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort O n) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0) (AHead a a0))))) (\lambda (H8: (eq A (ASort h2 n2) (AHead a (asucc g a0)))).(let H9 \def (eq_ind A (ASort h2 n2) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a (asucc g a0)) H8) in (False_ind ((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort O n0) (AHead a a0))) H9))) n1 (sym_eq nat n1 (next g n0) H7))) h1 (sym_eq nat h1 O H6))) H5)) H4 H2))) | (leq_head a1 a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4: (eq A (AHead a1 a3) (ASort O (next g n0)))).(\lambda (H5: (eq A (AHead a2 a4) (AHead a (asucc g a0)))).((let H6 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H4) in (False_ind ((eq A (AHead a2 a4) (AHead a (asucc g a0))) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort O n0) (AHead a a0))))) H6)) H5 H2 H3)))]) in (H5 (refl_equal A (ASort O (next g n0))) (refl_equal A (AHead a (asucc g a0)))))))) | (S n1) \Rightarrow (\lambda (_: (((leq g (asucc g (ASort (S n1) n0)) (asucc g a)) \to (leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g (asucc g (ASort (S n1) n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort (S n1) n0)) (asucc g (AHead a a0)))).(let H5 \def (match H4 return (\lambda (_: ?).(\lambda (a1: A).(\lambda (a2: A).((eq A a1 (ASort n1 n0)) \to ((eq A a2 (AHead a (asucc g a0))) \to (leq g (ASort (S n1) n0) (AHead a a0))))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (ASort n1 n0))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a (asucc g a0)))).((let H5 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) (ASort h1 n1) (ASort n1 n0) H3) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort n1 n0) H3) in (eq_ind nat n1 (\lambda (n: nat).((eq nat n1 n0) \to ((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort (S n1) n0) (AHead a a0)))))) (\lambda (H7: (eq nat n1 n0)).(eq_ind nat n0 (\lambda (n: nat).((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort n1 n) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort (S n1) n0) (AHead a a0))))) (\lambda (H8: (eq A (ASort h2 n2) (AHead a (asucc g a0)))).(let H9 \def (eq_ind A (ASort h2 n2) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a (asucc g a0)) H8) in (False_ind ((eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) k)) \to (leq g (ASort (S n1) n0) (AHead a a0))) H9))) n1 (sym_eq nat n1 n0 H7))) h1 (sym_eq nat h1 n1 H6))) H5)) H4 H2))) | (leq_head a1 a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4: (eq A (AHead a1 a3) (ASort n1 n0))).(\lambda (H5: (eq A (AHead a2 a4) (AHead a (asucc g a0)))).((let H6 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1 n0) H4) in (False_ind ((eq A (AHead a2 a4) (AHead a (asucc g a0))) \to ((leq g a1 a2) \to ((leq g a3 a4) \to (leq g (ASort (S n1) n0) (AHead a a0))))) H6)) H5 H2 H3)))]) in (H5 (refl_equal A (ASort n1 n0)) (refl_equal A (AHead a (asucc g a0))))))))]) H H0 H1)))))) a2)))) (\lambda (a: A).(\lambda (_: ((\forall (a2: A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0) (asucc g a2)) \to (leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3: A).((leq g (asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a a0)) (asucc g (ASort n n0)))).((match n return (\lambda (_: ?).(\lambda (n1: nat).((leq g (asucc g (AHead a a0)) (asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1 n0))))) with [O \Rightarrow (\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O n0)))).(let H3 \def (match H2 return (\lambda (_: ?).(\lambda (a1: A).(\lambda (a2: A).((eq A a1 (AHead a (asucc g a0))) \to ((eq A a2 (ASort O (next g n0))) \to (leq g (AHead a a0) (ASort O n0))))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (AHead a (asucc g a0)))).(\lambda (H4: (eq A (ASort h2 n2) (ASort O (next g n0)))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a (asucc g a0)) H3) in (False_ind ((eq A (ASort h2 n2) (ASort O (next g n0))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (AHead a a0) (ASort O n0)))) H5)) H4 H2))) | (leq_head a1 a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4: (eq A (AHead a1 a3) (AHead a (asucc g a0)))).(\lambda (H5: (eq A (AHead a2 a4) (ASort O (next g n0)))).((let H6 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a1 a3) (AHead a (asucc g a0)) H4) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead a (asucc g a0)) H4) in (eq_ind A a (\lambda (a5: A).((eq A a3 (asucc g a0)) \to ((eq A (AHead a2 a4) (ASort O (next g n0))) \to ((leq g a5 a2) \to ((leq g a3 a4) \to (leq g (AHead a a0) (ASort O n0))))))) (\lambda (H8: (eq A a3 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a5: A).((eq A (AHead a2 a4) (ASort O (next g n0))) \to ((leq g a a2) \to ((leq g a5 a4) \to (leq g (AHead a a0) (ASort O n0)))))) (\lambda (H9: (eq A (AHead a2 a4) (ASort O (next g n0)))).(let H10 \def (eq_ind A (AHead a2 a4) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H9) in (False_ind ((leq g a a2) \to ((leq g (asucc g a0) a4) \to (leq g (AHead a a0) (ASort O n0)))) H10))) a3 (sym_eq A a3 (asucc g a0) H8))) a1 (sym_eq A a1 a H7))) H6)) H5 H2 H3)))]) in (H3 (refl_equal A (AHead a (asucc g a0))) (refl_equal A (ASort O (next g n0)))))) | (S n1) \Rightarrow (\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort (S n1) n0)))).(let H3 \def (match H2 return (\lambda (_: ?).(\lambda (a1: A).(\lambda (a2: A).((eq A a1 (AHead a (asucc g a0))) \to ((eq A a2 (ASort n1 n0)) \to (leq g (AHead a a0) (ASort (S n1) n0))))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (AHead a (asucc g a0)))).(\lambda (H4: (eq A (ASort h2 n2) (ASort n1 n0))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a (asucc g a0)) H3) in (False_ind ((eq A (ASort h2 n2) (ASort n1 n0)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (AHead a a0) (ASort (S n1) n0)))) H5)) H4 H2))) | (leq_head a1 a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4: (eq A (AHead a1 a3) (AHead a (asucc g a0)))).(\lambda (H5: (eq A (AHead a2 a4) (ASort n1 n0))).((let H6 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a1 a3) (AHead a (asucc g a0)) H4) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead a (asucc g a0)) H4) in (eq_ind A a (\lambda (a5: A).((eq A a3 (asucc g a0)) \to ((eq A (AHead a2 a4) (ASort n1 n0)) \to ((leq g a5 a2) \to ((leq g a3 a4) \to (leq g (AHead a a0) (ASort (S n1) n0))))))) (\lambda (H8: (eq A a3 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a5: A).((eq A (AHead a2 a4) (ASort n1 n0)) \to ((leq g a a2) \to ((leq g a5 a4) \to (leq g (AHead a a0) (ASort (S n1) n0)))))) (\lambda (H9: (eq A (AHead a2 a4) (ASort n1 n0))).(let H10 \def (eq_ind A (AHead a2 a4) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1 n0) H9) in (False_ind ((leq g a a2) \to ((leq g (asucc g a0) a4) \to (leq g (AHead a a0) (ASort (S n1) n0)))) H10))) a3 (sym_eq A a3 (asucc g a0) H8))) a1 (sym_eq A a1 a H7))) H6)) H5 H2 H3)))]) in (H3 (refl_equal A (AHead a (asucc g a0))) (refl_equal A (ASort n1 n0)))))]) H1)))) (\lambda (a3: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))).(\lambda (a4: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a4)) \to (leq g (AHead a a0) a4)))).(\lambda (H3: (leq g (asucc g (AHead a a0)) (asucc g (AHead a3 a4)))).(let H4 \def (match H3 return (\lambda (_: ?).(\lambda (a1: A).(\lambda (a2: A).((eq A a1 (AHead a (asucc g a0))) \to ((eq A a2 (AHead a3 (asucc g a4))) \to (leq g (AHead a a0) (AHead a3 a4))))))) with [(leq_sort h1 h2 n1 n2 k H4) \Rightarrow (\lambda (H5: (eq A (ASort h1 n1) (AHead a (asucc g a0)))).(\lambda (H6: (eq A (ASort h2 n2) (AHead a3 (asucc g a4)))).((let H7 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a (asucc g a0)) H5) in (False_ind ((eq A (ASort h2 n2) (AHead a3 (asucc g a4))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (AHead a a0) (AHead a3 a4)))) H7)) H6 H4))) | (leq_head a3 a4 H4 a5 a6 H5) \Rightarrow (\lambda (H6: (eq A (AHead a3 a5) (AHead a (asucc g a0)))).(\lambda (H7: (eq A (AHead a4 a6) (AHead a3 (asucc g a4)))).((let H8 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a) \Rightarrow a])) (AHead a3 a5) (AHead a (asucc g a0)) H6) in ((let H9 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead a _) \Rightarrow a])) (AHead a3 a5) (AHead a (asucc g a0)) H6) in (eq_ind A a (\lambda (a1: A).((eq A a5 (asucc g a0)) \to ((eq A (AHead a4 a6) (AHead a3 (asucc g a4))) \to ((leq g a1 a4) \to ((leq g a5 a6) \to (leq g (AHead a a0) (AHead a3 a4))))))) (\lambda (H10: (eq A a5 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a1: A).((eq A (AHead a4 a6) (AHead a3 (asucc g a4))) \to ((leq g a a4) \to ((leq g a1 a6) \to (leq g (AHead a a0) (AHead a3 a4)))))) (\lambda (H11: (eq A (AHead a4 a6) (AHead a3 (asucc g a4)))).(let H12 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a6 | (AHead _ a) \Rightarrow a])) (AHead a4 a6) (AHead a3 (asucc g a4)) H11) in ((let H13 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a4 | (AHead a _) \Rightarrow a])) (AHead a4 a6) (AHead a3 (asucc g a4)) H11) in (eq_ind A a3 (\lambda (a1: A).((eq A a6 (asucc g a4)) \to ((leq g a a1) \to ((leq g (asucc g a0) a6) \to (leq g (AHead a a0) (AHead a3 a4)))))) (\lambda (H14: (eq A a6 (asucc g a4))).(eq_ind A (asucc g a4) (\lambda (a1: A).((leq g a a3) \to ((leq g (asucc g a0) a1) \to (leq g (AHead a a0) (AHead a3 a4))))) (\lambda (H15: (leq g a a3)).(\lambda (H16: (leq g (asucc g a0) (asucc g a4))).(leq_head g a a3 H15 a0 a4 (H0 a4 H16)))) a6 (sym_eq A a6 (asucc g a4) H14))) a4 (sym_eq A a4 a3 H13))) H12))) a5 (sym_eq A a5 (asucc g a0) H10))) a3 (sym_eq A a3 a H9))) H8)) H7 H4 H5)))]) in (H4 (refl_equal A (AHead a (asucc g a0))) (refl_equal A (AHead a3 (asucc g a4)))))))))) a2)))))) a1)).
 
 theorem aplus_asort_O_simpl:
  \forall (g: G).(\forall (h: nat).(\forall (n: nat).(eq A (aplus g (ASort O n) h) (ASort O (next_plus g n h)))))
@@ -1520,7 +1520,7 @@ theorem aplus_ahead_simpl:
 theorem aplus_asucc_false:
  \forall (g: G).(\forall (a: A).(\forall (h: nat).((eq A (aplus g (asucc g a) h) a) \to (\forall (P: Prop).P))))
 \def
- \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: Prop).P)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (h: nat).(\lambda (H: (eq A (aplus g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) h) (ASort n n0))).(\lambda (P: Prop).((match n return (\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) h) (ASort n1 n0)) \to P)) with [O \Rightarrow (\lambda (H0: (eq A (aplus g (ASort O (next g n0)) h) (ASort O n0))).(let H1 \def (eq_ind A (aplus g (ASort O (next g n0)) h) (\lambda (a: A).(eq A a (ASort O n0))) H0 (ASort (minus O h) (next_plus g (next g n0) (minus h O))) (aplus_asort_simpl g h O (next g n0))) in (let H2 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec next_plus (g: G) (n: nat) (i: nat) on i: nat \def (match i with [O \Rightarrow n | (S i0) \Rightarrow (next g (next_plus g n i0))]) in next_plus) g (next g n0) (minus h O))])) (ASort (minus O h) (next_plus g (next g n0) (minus h O))) (ASort O n0) H1) in (let H3 \def (eq_ind_r nat (minus h O) (\lambda (n: nat).(eq nat (next_plus g (next g n0) n) n0)) H2 h (minus_n_O h)) in (le_lt_false (next_plus g (next g n0) h) n0 (eq_ind nat (next_plus g (next g n0) h) (\lambda (n1: nat).(le (next_plus g (next g n0) h) n1)) (le_n (next_plus g (next g n0) h)) n0 H3) (next_plus_lt g h n0) P))))) | (S n1) \Rightarrow (\lambda (H0: (eq A (aplus g (ASort n1 n0) h) (ASort (S n1) n0))).(let H1 \def (eq_ind A (aplus g (ASort n1 n0) h) (\lambda (a: A).(eq A a (ASort (S n1) n0))) H0 (ASort (minus n1 h) (next_plus g n0 (minus h n1))) (aplus_asort_simpl g h n1 n0)) in (let H2 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec minus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O \Rightarrow O | (S k) \Rightarrow (match m with [O \Rightarrow (S k) | (S l) \Rightarrow (minus k l)])])) in minus) n1 h)])) (ASort (minus n1 h) (next_plus g n0 (minus h n1))) (ASort (S n1) n0) H1) in ((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec next_plus (g: G) (n: nat) (i: nat) on i: nat \def (match i with [O \Rightarrow n | (S i0) \Rightarrow (next g (next_plus g n i0))]) in next_plus) g n0 (minus h n1))])) (ASort (minus n1 h) (next_plus g n0 (minus h n1))) (ASort (S n1) n0) H1) in (\lambda (H4: (eq nat (minus n1 h) (S n1))).(le_Sx_x n1 (eq_ind nat (minus n1 h) (\lambda (n2: nat).(le n2 n1)) (minus_le n1 h) (S n1) H4) P))) H2))))]) H)))))) (\lambda (a0: A).(\lambda (_: ((\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h: nat).((eq A (aplus g (asucc g a1) h) a1) \to (\forall (P: Prop).P))))).(\lambda (h: nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc g a1)) h) (AHead a0 a1))).(\lambda (P: Prop).(let H2 \def (eq_ind A (aplus g (AHead a0 (asucc g a1)) h) (\lambda (a: A).(eq A a (AHead a0 a1))) H1 (AHead a0 (aplus g (asucc g a1) h)) (aplus_ahead_simpl g h a0 (asucc g a1))) in (let H3 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow ((let rec aplus (g: G) (a: A) (n: nat) on n: A \def (match n with [O \Rightarrow a | (S n0) \Rightarrow (asucc g (aplus g a n0))]) in aplus) g (asucc g a1) h) | (AHead _ a) \Rightarrow a])) (AHead a0 (aplus g (asucc g a1) h)) (AHead a0 a1) H2) in (H0 h H3 P)))))))))) a)).
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: Prop).P)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (h: nat).(\lambda (H: (eq A (aplus g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) h) (ASort n n0))).(\lambda (P: Prop).((match n return (\lambda (_: ?).(\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) h) (ASort n1 n0)) \to P))) with [O \Rightarrow (\lambda (H0: (eq A (aplus g (ASort O (next g n0)) h) (ASort O n0))).(let H1 \def (eq_ind A (aplus g (ASort O (next g n0)) h) (\lambda (a: A).(eq A a (ASort O n0))) H0 (ASort (minus O h) (next_plus g (next g n0) (minus h O))) (aplus_asort_simpl g h O (next g n0))) in (let H2 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec next_plus (g: G) (n: nat) (i: nat) on i: nat \def (match i with [O \Rightarrow n | (S i0) \Rightarrow (next g (next_plus g n i0))]) in next_plus) g (next g n0) (minus h O))])) (ASort (minus O h) (next_plus g (next g n0) (minus h O))) (ASort O n0) H1) in (let H3 \def (eq_ind_r nat (minus h O) (\lambda (n: nat).(eq nat (next_plus g (next g n0) n) n0)) H2 h (minus_n_O h)) in (le_lt_false (next_plus g (next g n0) h) n0 (eq_ind nat (next_plus g (next g n0) h) (\lambda (n1: nat).(le (next_plus g (next g n0) h) n1)) (le_n (next_plus g (next g n0) h)) n0 H3) (next_plus_lt g h n0) P))))) | (S n1) \Rightarrow (\lambda (H0: (eq A (aplus g (ASort n1 n0) h) (ASort (S n1) n0))).(let H1 \def (eq_ind A (aplus g (ASort n1 n0) h) (\lambda (a: A).(eq A a (ASort (S n1) n0))) H0 (ASort (minus n1 h) (next_plus g n0 (minus h n1))) (aplus_asort_simpl g h n1 n0)) in (let H2 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec minus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O \Rightarrow O | (S k) \Rightarrow (match m with [O \Rightarrow (S k) | (S l) \Rightarrow (minus k l)])])) in minus) n1 h)])) (ASort (minus n1 h) (next_plus g n0 (minus h n1))) (ASort (S n1) n0) H1) in ((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec next_plus (g: G) (n: nat) (i: nat) on i: nat \def (match i with [O \Rightarrow n | (S i0) \Rightarrow (next g (next_plus g n i0))]) in next_plus) g n0 (minus h n1))])) (ASort (minus n1 h) (next_plus g n0 (minus h n1))) (ASort (S n1) n0) H1) in (\lambda (H4: (eq nat (minus n1 h) (S n1))).(le_Sx_x n1 (eq_ind nat (minus n1 h) (\lambda (n2: nat).(le n2 n1)) (minus_le n1 h) (S n1) H4) P))) H2))))]) H)))))) (\lambda (a0: A).(\lambda (_: ((\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h: nat).((eq A (aplus g (asucc g a1) h) a1) \to (\forall (P: Prop).P))))).(\lambda (h: nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc g a1)) h) (AHead a0 a1))).(\lambda (P: Prop).(let H2 \def (eq_ind A (aplus g (AHead a0 (asucc g a1)) h) (\lambda (a: A).(eq A a (AHead a0 a1))) H1 (AHead a0 (aplus g (asucc g a1) h)) (aplus_ahead_simpl g h a0 (asucc g a1))) in (let H3 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow ((let rec aplus (g: G) (a: A) (n: nat) on n: A \def (match n with [O \Rightarrow a | (S n0) \Rightarrow (asucc g (aplus g a n0))]) in aplus) g (asucc g a1) h) | (AHead _ a) \Rightarrow a])) (AHead a0 (aplus g (asucc g a1) h)) (AHead a0 a1) H2) in (H0 h H3 P)))))))))) a)).
 
 theorem aplus_inj:
  \forall (g: G).(\forall (h1: nat).(\forall (h2: nat).(\forall (a: A).((eq A (aplus g a h1) (aplus g a h2)) \to (eq nat h1 h2)))))
@@ -1530,7 +1530,7 @@ theorem aplus_inj:
 theorem ahead_inj_snd:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall (a4: A).((leq g (AHead a1 a2) (AHead a3 a4)) \to (leq g a2 a4))))))
 \def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda (a4: A).(\lambda (H: (leq g (AHead a1 a2) (AHead a3 a4))).(let H0 \def (match H return (\lambda (a: A).(\lambda (a0: A).((eq A a (AHead a1 a2)) \to ((eq A a0 (AHead a3 a4)) \to (leq g a2 a4))))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead a1 a2))).(\lambda (H2: (eq A (ASort h2 n2) (AHead a3 a4))).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a1 a2) H1) in (False_ind ((eq A (ASort h2 n2) (AHead a3 a4)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g a2 a4))) H3)) H2 H0))) | (leq_head a0 a5 H0 a6 a7 H1) \Rightarrow (\lambda (H2: (eq A (AHead a0 a6) (AHead a1 a2))).(\lambda (H3: (eq A (AHead a5 a7) (AHead a3 a4))).((let H4 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a6 | (AHead _ a) \Rightarrow a])) (AHead a0 a6) (AHead a1 a2) H2) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a6) (AHead a1 a2) H2) in (eq_ind A a1 (\lambda (a: A).((eq A a6 a2) \to ((eq A (AHead a5 a7) (AHead a3 a4)) \to ((leq g a a5) \to ((leq g a6 a7) \to (leq g a2 a4)))))) (\lambda (H6: (eq A a6 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a5 a7) (AHead a3 a4)) \to ((leq g a1 a5) \to ((leq g a a7) \to (leq g a2 a4))))) (\lambda (H7: (eq A (AHead a5 a7) (AHead a3 a4))).(let H8 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a7 | (AHead _ a) \Rightarrow a])) (AHead a5 a7) (AHead a3 a4) H7) in ((let H9 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a5 | (AHead a _) \Rightarrow a])) (AHead a5 a7) (AHead a3 a4) H7) in (eq_ind A a3 (\lambda (a: A).((eq A a7 a4) \to ((leq g a1 a) \to ((leq g a2 a7) \to (leq g a2 a4))))) (\lambda (H10: (eq A a7 a4)).(eq_ind A a4 (\lambda (a: A).((leq g a1 a3) \to ((leq g a2 a) \to (leq g a2 a4)))) (\lambda (_: (leq g a1 a3)).(\lambda (H12: (leq g a2 a4)).H12)) a7 (sym_eq A a7 a4 H10))) a5 (sym_eq A a5 a3 H9))) H8))) a6 (sym_eq A a6 a2 H6))) a0 (sym_eq A a0 a1 H5))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (AHead a1 a2)) (refl_equal A (AHead a3 a4))))))))).
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda (a4: A).(\lambda (H: (leq g (AHead a1 a2) (AHead a3 a4))).(let H0 \def (match H return (\lambda (_: ?).(\lambda (a: A).(\lambda (a0: A).((eq A a (AHead a1 a2)) \to ((eq A a0 (AHead a3 a4)) \to (leq g a2 a4)))))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead a1 a2))).(\lambda (H2: (eq A (ASort h2 n2) (AHead a3 a4))).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a1 a2) H1) in (False_ind ((eq A (ASort h2 n2) (AHead a3 a4)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g a2 a4))) H3)) H2 H0))) | (leq_head a0 a5 H0 a6 a7 H1) \Rightarrow (\lambda (H2: (eq A (AHead a0 a6) (AHead a1 a2))).(\lambda (H3: (eq A (AHead a5 a7) (AHead a3 a4))).((let H4 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a6 | (AHead _ a) \Rightarrow a])) (AHead a0 a6) (AHead a1 a2) H2) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a6) (AHead a1 a2) H2) in (eq_ind A a1 (\lambda (a: A).((eq A a6 a2) \to ((eq A (AHead a5 a7) (AHead a3 a4)) \to ((leq g a a5) \to ((leq g a6 a7) \to (leq g a2 a4)))))) (\lambda (H6: (eq A a6 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a5 a7) (AHead a3 a4)) \to ((leq g a1 a5) \to ((leq g a a7) \to (leq g a2 a4))))) (\lambda (H7: (eq A (AHead a5 a7) (AHead a3 a4))).(let H8 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a7 | (AHead _ a) \Rightarrow a])) (AHead a5 a7) (AHead a3 a4) H7) in ((let H9 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a5 | (AHead a _) \Rightarrow a])) (AHead a5 a7) (AHead a3 a4) H7) in (eq_ind A a3 (\lambda (a: A).((eq A a7 a4) \to ((leq g a1 a) \to ((leq g a2 a7) \to (leq g a2 a4))))) (\lambda (H10: (eq A a7 a4)).(eq_ind A a4 (\lambda (a: A).((leq g a1 a3) \to ((leq g a2 a) \to (leq g a2 a4)))) (\lambda (_: (leq g a1 a3)).(\lambda (H12: (leq g a2 a4)).H12)) a7 (sym_eq A a7 a4 H10))) a5 (sym_eq A a5 a3 H9))) H8))) a6 (sym_eq A a6 a2 H6))) a0 (sym_eq A a0 a1 H5))) H4)) H3 H0 H1)))]) in (H0 (refl_equal A (AHead a1 a2)) (refl_equal A (AHead a3 a4))))))))).
 
 theorem leq_refl:
  \forall (g: G).(\forall (a: A).(leq g a a))
@@ -1555,22 +1555,22 @@ theorem leq_sym:
 theorem leq_trans:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall (a3: A).((leq g a2 a3) \to (leq g a1 a3))))))
 \def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (a3: A).((leq g a0 a3) \to (leq g a a3))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (a3: A).(\lambda (H1: (leq g (ASort h2 n2) a3)).(let H2 \def (match H1 return (\lambda (a: A).(\lambda (a0: A).((eq A a (ASort h2 n2)) \to ((eq A a0 a3) \to (leq g (ASort h1 n1) a3))))) with [(leq_sort h0 h3 n0 n3 k0 H1) \Rightarrow (\lambda (H2: (eq A (ASort h0 n0) (ASort h2 n2))).(\lambda (H3: (eq A (ASort h3 n3) a3)).((let H4 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n0])) (ASort h0 n0) (ASort h2 n2) H2) in ((let H5 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0) (ASort h2 n2) H2) in (eq_ind nat h2 (\lambda (n: nat).((eq nat n0 n2) \to ((eq A (ASort h3 n3) a3) \to ((eq A (aplus g (ASort n n0) k0) (aplus g (ASort h3 n3) k0)) \to (leq g (ASort h1 n1) a3))))) (\lambda (H6: (eq nat n0 n2)).(eq_ind nat n2 (\lambda (n: nat).((eq A (ASort h3 n3) a3) \to ((eq A (aplus g (ASort h2 n) k0) (aplus g (ASort h3 n3) k0)) \to (leq g (ASort h1 n1) a3)))) (\lambda (H7: (eq A (ASort h3 n3) a3)).(eq_ind A (ASort h3 n3) (\lambda (a: A).((eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 n3) k0)) \to (leq g (ASort h1 n1) a))) (\lambda (H8: (eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 n3) k0))).(lt_le_e k k0 (leq g (ASort h1 n1) (ASort h3 n3)) (\lambda (H9: (lt k k0)).(let H_y \def (aplus_reg_r g (ASort h1 n1) (ASort h2 n2) k k H0 (minus k0 k)) in (let H10 \def (eq_ind_r nat (plus (minus k0 k) k) (\lambda (n: nat).(eq A (aplus g (ASort h1 n1) n) (aplus g (ASort h2 n2) n))) H_y k0 (le_plus_minus_sym k k0 (le_S_n k k0 (le_S (S k) k0 H9)))) in (leq_sort g h1 h3 n1 n3 k0 (trans_eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 n3) k0) H10 H8))))) (\lambda (H9: (le k0 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2) (ASort h3 n3) k0 k0 H8 (minus k k0)) in (let H10 \def (eq_ind_r nat (plus (minus k k0) k0) (\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g (ASort h3 n3) n))) H_y k (le_plus_minus_sym k0 k H9)) in (leq_sort g h1 h3 n1 n3 k (trans_eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k) (aplus g (ASort h3 n3) k) H0 H10))))))) a3 H7)) n0 (sym_eq nat n0 n2 H6))) h0 (sym_eq nat h0 h2 H5))) H4)) H3 H1))) | (leq_head a1 a2 H1 a0 a4 H2) \Rightarrow (\lambda (H3: (eq A (AHead a1 a0) (ASort h2 n2))).(\lambda (H4: (eq A (AHead a2 a4) a3)).((let H5 \def (eq_ind A (AHead a1 a0) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort h2 n2) H3) in (False_ind ((eq A (AHead a2 a4) a3) \to ((leq g a1 a2) \to ((leq g a0 a4) \to (leq g (ASort h1 n1) a3)))) H5)) H4 H1 H2)))]) in (H2 (refl_equal A (ASort h2 n2)) (refl_equal A a3))))))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_: (leq g a3 a4)).(\lambda (H1: ((\forall (a5: A).((leq g a4 a5) \to (leq g a3 a5))))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: ((\forall (a3: A).((leq g a6 a3) \to (leq g a5 a3))))).(\lambda (a0: A).(\lambda (H4: (leq g (AHead a4 a6) a0)).(let H5 \def (match H4 return (\lambda (a: A).(\lambda (a1: A).((eq A a (AHead a4 a6)) \to ((eq A a1 a0) \to (leq g (AHead a3 a5) a0))))) with [(leq_sort h1 h2 n1 n2 k H4) \Rightarrow (\lambda (H5: (eq A (ASort h1 n1) (AHead a4 a6))).(\lambda (H6: (eq A (ASort h2 n2) a0)).((let H7 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a4 a6) H5) in (False_ind ((eq A (ASort h2 n2) a0) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (AHead a3 a5) a0))) H7)) H6 H4))) | (leq_head a5 a6 H4 a7 a8 H5) \Rightarrow (\lambda (H6: (eq A (AHead a5 a7) (AHead a4 a6))).(\lambda (H7: (eq A (AHead a6 a8) a0)).((let H8 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a7 | (AHead _ a) \Rightarrow a])) (AHead a5 a7) (AHead a4 a6) H6) in ((let H9 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a5 | (AHead a _) \Rightarrow a])) (AHead a5 a7) (AHead a4 a6) H6) in (eq_ind A a4 (\lambda (a: A).((eq A a7 a6) \to ((eq A (AHead a6 a8) a0) \to ((leq g a a6) \to ((leq g a7 a8) \to (leq g (AHead a3 a5) a0)))))) (\lambda (H10: (eq A a7 a6)).(eq_ind A a6 (\lambda (a: A).((eq A (AHead a6 a8) a0) \to ((leq g a4 a6) \to ((leq g a a8) \to (leq g (AHead a3 a5) a0))))) (\lambda (H11: (eq A (AHead a6 a8) a0)).(eq_ind A (AHead a6 a8) (\lambda (a: A).((leq g a4 a6) \to ((leq g a6 a8) \to (leq g (AHead a3 a5) a)))) (\lambda (H12: (leq g a4 a6)).(\lambda (H13: (leq g a6 a8)).(leq_head g a3 a6 (H1 a6 H12) a5 a8 (H3 a8 H13)))) a0 H11)) a7 (sym_eq A a7 a6 H10))) a5 (sym_eq A a5 a4 H9))) H8)) H7 H4 H5)))]) in (H5 (refl_equal A (AHead a4 a6)) (refl_equal A a0))))))))))))) a1 a2 H)))).
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (a3: A).((leq g a0 a3) \to (leq g a a3))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (a3: A).(\lambda (H1: (leq g (ASort h2 n2) a3)).(let H2 \def (match H1 return (\lambda (_: ?).(\lambda (a: A).(\lambda (a0: A).((eq A a (ASort h2 n2)) \to ((eq A a0 a3) \to (leq g (ASort h1 n1) a3)))))) with [(leq_sort h0 h3 n0 n3 k0 H1) \Rightarrow (\lambda (H2: (eq A (ASort h0 n0) (ASort h2 n2))).(\lambda (H3: (eq A (ASort h3 n3) a3)).((let H4 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n0])) (ASort h0 n0) (ASort h2 n2) H2) in ((let H5 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0) (ASort h2 n2) H2) in (eq_ind nat h2 (\lambda (n: nat).((eq nat n0 n2) \to ((eq A (ASort h3 n3) a3) \to ((eq A (aplus g (ASort n n0) k0) (aplus g (ASort h3 n3) k0)) \to (leq g (ASort h1 n1) a3))))) (\lambda (H6: (eq nat n0 n2)).(eq_ind nat n2 (\lambda (n: nat).((eq A (ASort h3 n3) a3) \to ((eq A (aplus g (ASort h2 n) k0) (aplus g (ASort h3 n3) k0)) \to (leq g (ASort h1 n1) a3)))) (\lambda (H7: (eq A (ASort h3 n3) a3)).(eq_ind A (ASort h3 n3) (\lambda (a: A).((eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 n3) k0)) \to (leq g (ASort h1 n1) a))) (\lambda (H8: (eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 n3) k0))).(lt_le_e k k0 (leq g (ASort h1 n1) (ASort h3 n3)) (\lambda (H9: (lt k k0)).(let H_y \def (aplus_reg_r g (ASort h1 n1) (ASort h2 n2) k k H0 (minus k0 k)) in (let H10 \def (eq_ind_r nat (plus (minus k0 k) k) (\lambda (n: nat).(eq A (aplus g (ASort h1 n1) n) (aplus g (ASort h2 n2) n))) H_y k0 (le_plus_minus_sym k k0 (le_S_n k k0 (le_S (S k) k0 H9)))) in (leq_sort g h1 h3 n1 n3 k0 (trans_eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 n3) k0) H10 H8))))) (\lambda (H9: (le k0 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2) (ASort h3 n3) k0 k0 H8 (minus k k0)) in (let H10 \def (eq_ind_r nat (plus (minus k k0) k0) (\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g (ASort h3 n3) n))) H_y k (le_plus_minus_sym k0 k H9)) in (leq_sort g h1 h3 n1 n3 k (trans_eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k) (aplus g (ASort h3 n3) k) H0 H10))))))) a3 H7)) n0 (sym_eq nat n0 n2 H6))) h0 (sym_eq nat h0 h2 H5))) H4)) H3 H1))) | (leq_head a1 a2 H1 a0 a4 H2) \Rightarrow (\lambda (H3: (eq A (AHead a1 a0) (ASort h2 n2))).(\lambda (H4: (eq A (AHead a2 a4) a3)).((let H5 \def (eq_ind A (AHead a1 a0) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort h2 n2) H3) in (False_ind ((eq A (AHead a2 a4) a3) \to ((leq g a1 a2) \to ((leq g a0 a4) \to (leq g (ASort h1 n1) a3)))) H5)) H4 H1 H2)))]) in (H2 (refl_equal A (ASort h2 n2)) (refl_equal A a3))))))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_: (leq g a3 a4)).(\lambda (H1: ((\forall (a5: A).((leq g a4 a5) \to (leq g a3 a5))))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: ((\forall (a3: A).((leq g a6 a3) \to (leq g a5 a3))))).(\lambda (a0: A).(\lambda (H4: (leq g (AHead a4 a6) a0)).(let H5 \def (match H4 return (\lambda (_: ?).(\lambda (a: A).(\lambda (a1: A).((eq A a (AHead a4 a6)) \to ((eq A a1 a0) \to (leq g (AHead a3 a5) a0)))))) with [(leq_sort h1 h2 n1 n2 k H4) \Rightarrow (\lambda (H5: (eq A (ASort h1 n1) (AHead a4 a6))).(\lambda (H6: (eq A (ASort h2 n2) a0)).((let H7 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a4 a6) H5) in (False_ind ((eq A (ASort h2 n2) a0) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (AHead a3 a5) a0))) H7)) H6 H4))) | (leq_head a5 a6 H4 a7 a8 H5) \Rightarrow (\lambda (H6: (eq A (AHead a5 a7) (AHead a4 a6))).(\lambda (H7: (eq A (AHead a6 a8) a0)).((let H8 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a7 | (AHead _ a) \Rightarrow a])) (AHead a5 a7) (AHead a4 a6) H6) in ((let H9 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a5 | (AHead a _) \Rightarrow a])) (AHead a5 a7) (AHead a4 a6) H6) in (eq_ind A a4 (\lambda (a: A).((eq A a7 a6) \to ((eq A (AHead a6 a8) a0) \to ((leq g a a6) \to ((leq g a7 a8) \to (leq g (AHead a3 a5) a0)))))) (\lambda (H10: (eq A a7 a6)).(eq_ind A a6 (\lambda (a: A).((eq A (AHead a6 a8) a0) \to ((leq g a4 a6) \to ((leq g a a8) \to (leq g (AHead a3 a5) a0))))) (\lambda (H11: (eq A (AHead a6 a8) a0)).(eq_ind A (AHead a6 a8) (\lambda (a: A).((leq g a4 a6) \to ((leq g a6 a8) \to (leq g (AHead a3 a5) a)))) (\lambda (H12: (leq g a4 a6)).(\lambda (H13: (leq g a6 a8)).(leq_head g a3 a6 (H1 a6 H12) a5 a8 (H3 a8 H13)))) a0 H11)) a7 (sym_eq A a7 a6 H10))) a5 (sym_eq A a5 a4 H9))) H8)) H7 H4 H5)))]) in (H5 (refl_equal A (AHead a4 a6)) (refl_equal A a0))))))))))))) a1 a2 H)))).
 
 theorem leq_ahead_false:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1) \to (\forall (P: Prop).P))))
 \def
- \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead (ASort n n0) a2) (ASort n n0))).(\lambda (P: Prop).((match n return (\lambda (n1: nat).((leq g (AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) with [O \Rightarrow (\lambda (H0: (leq g (AHead (ASort O n0) a2) (ASort O n0))).(let H1 \def (match H0 return (\lambda (a: A).(\lambda (a0: A).((eq A a (AHead (ASort O n0) a2)) \to ((eq A a0 (ASort O n0)) \to P)))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead (ASort O n0) a2))).(\lambda (H2: (eq A (ASort h2 n2) (ASort O n0))).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead (ASort O n0) a2) H1) in (False_ind ((eq A (ASort h2 n2) (ASort O n0)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H3)) H2 H0))) | (leq_head a1 a0 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (AHead (ASort O n0) a2))).(\lambda (H3: (eq A (AHead a0 a4) (ASort O n0))).((let H4 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a1 a3) (AHead (ASort O n0) a2) H2) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead (ASort O n0) a2) H2) in (eq_ind A (ASort O n0) (\lambda (a: A).((eq A a3 a2) \to ((eq A (AHead a0 a4) (ASort O n0)) \to ((leq g a a0) \to ((leq g a3 a4) \to P))))) (\lambda (H6: (eq A a3 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a0 a4) (ASort O n0)) \to ((leq g (ASort O n0) a0) \to ((leq g a a4) \to P)))) (\lambda (H7: (eq A (AHead a0 a4) (ASort O n0))).(let H8 \def (eq_ind A (AHead a0 a4) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O n0) H7) in (False_ind ((leq g (ASort O n0) a0) \to ((leq g a2 a4) \to P)) H8))) a3 (sym_eq A a3 a2 H6))) a1 (sym_eq A a1 (ASort O n0) H5))) H4)) H3 H0 H1)))]) in (H1 (refl_equal A (AHead (ASort O n0) a2)) (refl_equal A (ASort O n0))))) | (S n1) \Rightarrow (\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort (S n1) n0))).(let H1 \def (match H0 return (\lambda (a: A).(\lambda (a0: A).((eq A a (AHead (ASort (S n1) n0) a2)) \to ((eq A a0 (ASort (S n1) n0)) \to P)))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead (ASort (S n1) n0) a2))).(\lambda (H2: (eq A (ASort h2 n2) (ASort (S n1) n0))).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead (ASort (S n1) n0) a2) H1) in (False_ind ((eq A (ASort h2 n2) (ASort (S n1) n0)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H3)) H2 H0))) | (leq_head a1 a0 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (AHead (ASort (S n1) n0) a2))).(\lambda (H3: (eq A (AHead a0 a4) (ASort (S n1) n0))).((let H4 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a1 a3) (AHead (ASort (S n1) n0) a2) H2) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead (ASort (S n1) n0) a2) H2) in (eq_ind A (ASort (S n1) n0) (\lambda (a: A).((eq A a3 a2) \to ((eq A (AHead a0 a4) (ASort (S n1) n0)) \to ((leq g a a0) \to ((leq g a3 a4) \to P))))) (\lambda (H6: (eq A a3 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a0 a4) (ASort (S n1) n0)) \to ((leq g (ASort (S n1) n0) a0) \to ((leq g a a4) \to P)))) (\lambda (H7: (eq A (AHead a0 a4) (ASort (S n1) n0))).(let H8 \def (eq_ind A (AHead a0 a4) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort (S n1) n0) H7) in (False_ind ((leq g (ASort (S n1) n0) a0) \to ((leq g a2 a4) \to P)) H8))) a3 (sym_eq A a3 a2 H6))) a1 (sym_eq A a1 (ASort (S n1) n0) H5))) H4)) H3 H0 H1)))]) in (H1 (refl_equal A (AHead (ASort (S n1) n0) a2)) (refl_equal A (ASort (S n1) n0)))))]) H)))))) (\lambda (a: A).(\lambda (H: ((\forall (a2: A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a0 a2) a0) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead (AHead a a0) a2) (AHead a a0))).(\lambda (P: Prop).(let H2 \def (match H1 return (\lambda (a1: A).(\lambda (a3: A).((eq A a1 (AHead (AHead a a0) a2)) \to ((eq A a3 (AHead a a0)) \to P)))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (AHead (AHead a a0) a2))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a a0))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead (AHead a a0) a2) H3) in (False_ind ((eq A (ASort h2 n2) (AHead a a0)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head a1 a3 H2 a4 a5 H3) \Rightarrow (\lambda (H4: (eq A (AHead a1 a4) (AHead (AHead a a0) a2))).(\lambda (H5: (eq A (AHead a3 a5) (AHead a a0))).((let H6 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a1 a4) (AHead (AHead a a0) a2) H4) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a4) (AHead (AHead a a0) a2) H4) in (eq_ind A (AHead a a0) (\lambda (a6: A).((eq A a4 a2) \to ((eq A (AHead a3 a5) (AHead a a0)) \to ((leq g a6 a3) \to ((leq g a4 a5) \to P))))) (\lambda (H8: (eq A a4 a2)).(eq_ind A a2 (\lambda (a2: A).((eq A (AHead a3 a5) (AHead a a0)) \to ((leq g (AHead a a0) a3) \to ((leq g a2 a5) \to P)))) (\lambda (H9: (eq A (AHead a3 a5) (AHead a a0))).(let H10 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a) \Rightarrow a])) (AHead a3 a5) (AHead a a0) H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead a _) \Rightarrow a])) (AHead a3 a5) (AHead a a0) H9) in (eq_ind A a (\lambda (a6: A).((eq A a5 a0) \to ((leq g (AHead a a0) a6) \to ((leq g a2 a5) \to P)))) (\lambda (H12: (eq A a5 a0)).(eq_ind A a0 (\lambda (a6: A).((leq g (AHead a a0) a) \to ((leq g a2 a6) \to P))) (\lambda (H13: (leq g (AHead a a0) a)).(\lambda (_: (leq g a2 a0)).(H a0 H13 P))) a5 (sym_eq A a5 a0 H12))) a3 (sym_eq A a3 a H11))) H10))) a4 (sym_eq A a4 a2 H8))) a1 (sym_eq A a1 (AHead a a0) H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead (AHead a a0) a2)) (refl_equal A (AHead a a0))))))))))) a1)).
+ \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead (ASort n n0) a2) (ASort n n0))).(\lambda (P: Prop).((match n return (\lambda (_: ?).(\lambda (n1: nat).((leq g (AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P))) with [O \Rightarrow (\lambda (H0: (leq g (AHead (ASort O n0) a2) (ASort O n0))).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (a: A).(\lambda (a0: A).((eq A a (AHead (ASort O n0) a2)) \to ((eq A a0 (ASort O n0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead (ASort O n0) a2))).(\lambda (H2: (eq A (ASort h2 n2) (ASort O n0))).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead (ASort O n0) a2) H1) in (False_ind ((eq A (ASort h2 n2) (ASort O n0)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H3)) H2 H0))) | (leq_head a1 a0 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (AHead (ASort O n0) a2))).(\lambda (H3: (eq A (AHead a0 a4) (ASort O n0))).((let H4 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a1 a3) (AHead (ASort O n0) a2) H2) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead (ASort O n0) a2) H2) in (eq_ind A (ASort O n0) (\lambda (a: A).((eq A a3 a2) \to ((eq A (AHead a0 a4) (ASort O n0)) \to ((leq g a a0) \to ((leq g a3 a4) \to P))))) (\lambda (H6: (eq A a3 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a0 a4) (ASort O n0)) \to ((leq g (ASort O n0) a0) \to ((leq g a a4) \to P)))) (\lambda (H7: (eq A (AHead a0 a4) (ASort O n0))).(let H8 \def (eq_ind A (AHead a0 a4) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O n0) H7) in (False_ind ((leq g (ASort O n0) a0) \to ((leq g a2 a4) \to P)) H8))) a3 (sym_eq A a3 a2 H6))) a1 (sym_eq A a1 (ASort O n0) H5))) H4)) H3 H0 H1)))]) in (H1 (refl_equal A (AHead (ASort O n0) a2)) (refl_equal A (ASort O n0))))) | (S n1) \Rightarrow (\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort (S n1) n0))).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (a: A).(\lambda (a0: A).((eq A a (AHead (ASort (S n1) n0) a2)) \to ((eq A a0 (ASort (S n1) n0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead (ASort (S n1) n0) a2))).(\lambda (H2: (eq A (ASort h2 n2) (ASort (S n1) n0))).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead (ASort (S n1) n0) a2) H1) in (False_ind ((eq A (ASort h2 n2) (ASort (S n1) n0)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H3)) H2 H0))) | (leq_head a1 a0 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (AHead (ASort (S n1) n0) a2))).(\lambda (H3: (eq A (AHead a0 a4) (ASort (S n1) n0))).((let H4 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a1 a3) (AHead (ASort (S n1) n0) a2) H2) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead (ASort (S n1) n0) a2) H2) in (eq_ind A (ASort (S n1) n0) (\lambda (a: A).((eq A a3 a2) \to ((eq A (AHead a0 a4) (ASort (S n1) n0)) \to ((leq g a a0) \to ((leq g a3 a4) \to P))))) (\lambda (H6: (eq A a3 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a0 a4) (ASort (S n1) n0)) \to ((leq g (ASort (S n1) n0) a0) \to ((leq g a a4) \to P)))) (\lambda (H7: (eq A (AHead a0 a4) (ASort (S n1) n0))).(let H8 \def (eq_ind A (AHead a0 a4) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort (S n1) n0) H7) in (False_ind ((leq g (ASort (S n1) n0) a0) \to ((leq g a2 a4) \to P)) H8))) a3 (sym_eq A a3 a2 H6))) a1 (sym_eq A a1 (ASort (S n1) n0) H5))) H4)) H3 H0 H1)))]) in (H1 (refl_equal A (AHead (ASort (S n1) n0) a2)) (refl_equal A (ASort (S n1) n0)))))]) H)))))) (\lambda (a: A).(\lambda (H: ((\forall (a2: A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a0 a2) a0) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead (AHead a a0) a2) (AHead a a0))).(\lambda (P: Prop).(let H2 \def (match H1 return (\lambda (_: ?).(\lambda (a1: A).(\lambda (a3: A).((eq A a1 (AHead (AHead a a0) a2)) \to ((eq A a3 (AHead a a0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (AHead (AHead a a0) a2))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a a0))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead (AHead a a0) a2) H3) in (False_ind ((eq A (ASort h2 n2) (AHead a a0)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head a1 a3 H2 a4 a5 H3) \Rightarrow (\lambda (H4: (eq A (AHead a1 a4) (AHead (AHead a a0) a2))).(\lambda (H5: (eq A (AHead a3 a5) (AHead a a0))).((let H6 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a1 a4) (AHead (AHead a a0) a2) H4) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a4) (AHead (AHead a a0) a2) H4) in (eq_ind A (AHead a a0) (\lambda (a6: A).((eq A a4 a2) \to ((eq A (AHead a3 a5) (AHead a a0)) \to ((leq g a6 a3) \to ((leq g a4 a5) \to P))))) (\lambda (H8: (eq A a4 a2)).(eq_ind A a2 (\lambda (a2: A).((eq A (AHead a3 a5) (AHead a a0)) \to ((leq g (AHead a a0) a3) \to ((leq g a2 a5) \to P)))) (\lambda (H9: (eq A (AHead a3 a5) (AHead a a0))).(let H10 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a) \Rightarrow a])) (AHead a3 a5) (AHead a a0) H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead a _) \Rightarrow a])) (AHead a3 a5) (AHead a a0) H9) in (eq_ind A a (\lambda (a6: A).((eq A a5 a0) \to ((leq g (AHead a a0) a6) \to ((leq g a2 a5) \to P)))) (\lambda (H12: (eq A a5 a0)).(eq_ind A a0 (\lambda (a6: A).((leq g (AHead a a0) a) \to ((leq g a2 a6) \to P))) (\lambda (H13: (leq g (AHead a a0) a)).(\lambda (_: (leq g a2 a0)).(H a0 H13 P))) a5 (sym_eq A a5 a0 H12))) a3 (sym_eq A a3 a H11))) H10))) a4 (sym_eq A a4 a2 H8))) a1 (sym_eq A a1 (AHead a a0) H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead (AHead a a0) a2)) (refl_equal A (AHead a a0))))))))))) a1)).
 
 theorem leq_ahead_asucc_false:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) (asucc g a1)) \to (\forall (P: Prop).P))))
 \def
- \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead (ASort n n0) a2) (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]))).(\lambda (P: Prop).((match n return (\lambda (n1: nat).((leq g (AHead (ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P)) with [O \Rightarrow (\lambda (H0: (leq g (AHead (ASort O n0) a2) (ASort O (next g n0)))).(let H1 \def (match H0 return (\lambda (a: A).(\lambda (a0: A).((eq A a (AHead (ASort O n0) a2)) \to ((eq A a0 (ASort O (next g n0))) \to P)))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead (ASort O n0) a2))).(\lambda (H2: (eq A (ASort h2 n2) (ASort O (next g n0)))).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead (ASort O n0) a2) H1) in (False_ind ((eq A (ASort h2 n2) (ASort O (next g n0))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H3)) H2 H0))) | (leq_head a1 a0 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (AHead (ASort O n0) a2))).(\lambda (H3: (eq A (AHead a0 a4) (ASort O (next g n0)))).((let H4 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a1 a3) (AHead (ASort O n0) a2) H2) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead (ASort O n0) a2) H2) in (eq_ind A (ASort O n0) (\lambda (a: A).((eq A a3 a2) \to ((eq A (AHead a0 a4) (ASort O (next g n0))) \to ((leq g a a0) \to ((leq g a3 a4) \to P))))) (\lambda (H6: (eq A a3 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a0 a4) (ASort O (next g n0))) \to ((leq g (ASort O n0) a0) \to ((leq g a a4) \to P)))) (\lambda (H7: (eq A (AHead a0 a4) (ASort O (next g n0)))).(let H8 \def (eq_ind A (AHead a0 a4) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H7) in (False_ind ((leq g (ASort O n0) a0) \to ((leq g a2 a4) \to P)) H8))) a3 (sym_eq A a3 a2 H6))) a1 (sym_eq A a1 (ASort O n0) H5))) H4)) H3 H0 H1)))]) in (H1 (refl_equal A (AHead (ASort O n0) a2)) (refl_equal A (ASort O (next g n0)))))) | (S n1) \Rightarrow (\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort n1 n0))).(let H1 \def (match H0 return (\lambda (a: A).(\lambda (a0: A).((eq A a (AHead (ASort (S n1) n0) a2)) \to ((eq A a0 (ASort n1 n0)) \to P)))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead (ASort (S n1) n0) a2))).(\lambda (H2: (eq A (ASort h2 n2) (ASort n1 n0))).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead (ASort (S n1) n0) a2) H1) in (False_ind ((eq A (ASort h2 n2) (ASort n1 n0)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H3)) H2 H0))) | (leq_head a1 a0 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (AHead (ASort (S n1) n0) a2))).(\lambda (H3: (eq A (AHead a0 a4) (ASort n1 n0))).((let H4 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a1 a3) (AHead (ASort (S n1) n0) a2) H2) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead (ASort (S n1) n0) a2) H2) in (eq_ind A (ASort (S n1) n0) (\lambda (a: A).((eq A a3 a2) \to ((eq A (AHead a0 a4) (ASort n1 n0)) \to ((leq g a a0) \to ((leq g a3 a4) \to P))))) (\lambda (H6: (eq A a3 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a0 a4) (ASort n1 n0)) \to ((leq g (ASort (S n1) n0) a0) \to ((leq g a a4) \to P)))) (\lambda (H7: (eq A (AHead a0 a4) (ASort n1 n0))).(let H8 \def (eq_ind A (AHead a0 a4) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1 n0) H7) in (False_ind ((leq g (ASort (S n1) n0) a0) \to ((leq g a2 a4) \to P)) H8))) a3 (sym_eq A a3 a2 H6))) a1 (sym_eq A a1 (ASort (S n1) n0) H5))) H4)) H3 H0 H1)))]) in (H1 (refl_equal A (AHead (ASort (S n1) n0) a2)) (refl_equal A (ASort n1 n0)))))]) H)))))) (\lambda (a: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a0 a2) (asucc g a0)) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead (AHead a a0) a2) (AHead a (asucc g a0)))).(\lambda (P: Prop).(let H2 \def (match H1 return (\lambda (a1: A).(\lambda (a3: A).((eq A a1 (AHead (AHead a a0) a2)) \to ((eq A a3 (AHead a (asucc g a0))) \to P)))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (AHead (AHead a a0) a2))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a (asucc g a0)))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead (AHead a a0) a2) H3) in (False_ind ((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head a1 a3 H2 a4 a5 H3) \Rightarrow (\lambda (H4: (eq A (AHead a1 a4) (AHead (AHead a a0) a2))).(\lambda (H5: (eq A (AHead a3 a5) (AHead a (asucc g a0)))).((let H6 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a1 a4) (AHead (AHead a a0) a2) H4) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a4) (AHead (AHead a a0) a2) H4) in (eq_ind A (AHead a a0) (\lambda (a6: A).((eq A a4 a2) \to ((eq A (AHead a3 a5) (AHead a (asucc g a0))) \to ((leq g a6 a3) \to ((leq g a4 a5) \to P))))) (\lambda (H8: (eq A a4 a2)).(eq_ind A a2 (\lambda (a2: A).((eq A (AHead a3 a5) (AHead a (asucc g a0))) \to ((leq g (AHead a a0) a3) \to ((leq g a2 a5) \to P)))) (\lambda (H9: (eq A (AHead a3 a5) (AHead a (asucc g a0)))).(let H10 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a) \Rightarrow a])) (AHead a3 a5) (AHead a (asucc g a0)) H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead a _) \Rightarrow a])) (AHead a3 a5) (AHead a (asucc g a0)) H9) in (eq_ind A a (\lambda (a6: A).((eq A a5 (asucc g a0)) \to ((leq g (AHead a a0) a6) \to ((leq g a2 a5) \to P)))) (\lambda (H12: (eq A a5 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a6: A).((leq g (AHead a a0) a) \to ((leq g a2 a6) \to P))) (\lambda (H13: (leq g (AHead a a0) a)).(\lambda (_: (leq g a2 (asucc g a0))).(leq_ahead_false g a a0 H13 P))) a5 (sym_eq A a5 (asucc g a0) H12))) a3 (sym_eq A a3 a H11))) H10))) a4 (sym_eq A a4 a2 H8))) a1 (sym_eq A a1 (AHead a a0) H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead (AHead a a0) a2)) (refl_equal A (AHead a (asucc g a0)))))))))))) a1)).
+ \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead (ASort n n0) a2) (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]))).(\lambda (P: Prop).((match n return (\lambda (_: ?).(\lambda (n1: nat).((leq g (AHead (ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P))) with [O \Rightarrow (\lambda (H0: (leq g (AHead (ASort O n0) a2) (ASort O (next g n0)))).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (a: A).(\lambda (a0: A).((eq A a (AHead (ASort O n0) a2)) \to ((eq A a0 (ASort O (next g n0))) \to P))))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead (ASort O n0) a2))).(\lambda (H2: (eq A (ASort h2 n2) (ASort O (next g n0)))).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead (ASort O n0) a2) H1) in (False_ind ((eq A (ASort h2 n2) (ASort O (next g n0))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H3)) H2 H0))) | (leq_head a1 a0 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (AHead (ASort O n0) a2))).(\lambda (H3: (eq A (AHead a0 a4) (ASort O (next g n0)))).((let H4 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a1 a3) (AHead (ASort O n0) a2) H2) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead (ASort O n0) a2) H2) in (eq_ind A (ASort O n0) (\lambda (a: A).((eq A a3 a2) \to ((eq A (AHead a0 a4) (ASort O (next g n0))) \to ((leq g a a0) \to ((leq g a3 a4) \to P))))) (\lambda (H6: (eq A a3 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a0 a4) (ASort O (next g n0))) \to ((leq g (ASort O n0) a0) \to ((leq g a a4) \to P)))) (\lambda (H7: (eq A (AHead a0 a4) (ASort O (next g n0)))).(let H8 \def (eq_ind A (AHead a0 a4) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H7) in (False_ind ((leq g (ASort O n0) a0) \to ((leq g a2 a4) \to P)) H8))) a3 (sym_eq A a3 a2 H6))) a1 (sym_eq A a1 (ASort O n0) H5))) H4)) H3 H0 H1)))]) in (H1 (refl_equal A (AHead (ASort O n0) a2)) (refl_equal A (ASort O (next g n0)))))) | (S n1) \Rightarrow (\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort n1 n0))).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (a: A).(\lambda (a0: A).((eq A a (AHead (ASort (S n1) n0) a2)) \to ((eq A a0 (ASort n1 n0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (AHead (ASort (S n1) n0) a2))).(\lambda (H2: (eq A (ASort h2 n2) (ASort n1 n0))).((let H3 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead (ASort (S n1) n0) a2) H1) in (False_ind ((eq A (ASort h2 n2) (ASort n1 n0)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H3)) H2 H0))) | (leq_head a1 a0 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (AHead (ASort (S n1) n0) a2))).(\lambda (H3: (eq A (AHead a0 a4) (ASort n1 n0))).((let H4 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a1 a3) (AHead (ASort (S n1) n0) a2) H2) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead (ASort (S n1) n0) a2) H2) in (eq_ind A (ASort (S n1) n0) (\lambda (a: A).((eq A a3 a2) \to ((eq A (AHead a0 a4) (ASort n1 n0)) \to ((leq g a a0) \to ((leq g a3 a4) \to P))))) (\lambda (H6: (eq A a3 a2)).(eq_ind A a2 (\lambda (a: A).((eq A (AHead a0 a4) (ASort n1 n0)) \to ((leq g (ASort (S n1) n0) a0) \to ((leq g a a4) \to P)))) (\lambda (H7: (eq A (AHead a0 a4) (ASort n1 n0))).(let H8 \def (eq_ind A (AHead a0 a4) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1 n0) H7) in (False_ind ((leq g (ASort (S n1) n0) a0) \to ((leq g a2 a4) \to P)) H8))) a3 (sym_eq A a3 a2 H6))) a1 (sym_eq A a1 (ASort (S n1) n0) H5))) H4)) H3 H0 H1)))]) in (H1 (refl_equal A (AHead (ASort (S n1) n0) a2)) (refl_equal A (ASort n1 n0)))))]) H)))))) (\lambda (a: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a0 a2) (asucc g a0)) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead (AHead a a0) a2) (AHead a (asucc g a0)))).(\lambda (P: Prop).(let H2 \def (match H1 return (\lambda (_: ?).(\lambda (a1: A).(\lambda (a3: A).((eq A a1 (AHead (AHead a a0) a2)) \to ((eq A a3 (AHead a (asucc g a0))) \to P))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (AHead (AHead a a0) a2))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a (asucc g a0)))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead (AHead a a0) a2) H3) in (False_ind ((eq A (ASort h2 n2) (AHead a (asucc g a0))) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head a1 a3 H2 a4 a5 H3) \Rightarrow (\lambda (H4: (eq A (AHead a1 a4) (AHead (AHead a a0) a2))).(\lambda (H5: (eq A (AHead a3 a5) (AHead a (asucc g a0)))).((let H6 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a1 a4) (AHead (AHead a a0) a2) H4) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a4) (AHead (AHead a a0) a2) H4) in (eq_ind A (AHead a a0) (\lambda (a6: A).((eq A a4 a2) \to ((eq A (AHead a3 a5) (AHead a (asucc g a0))) \to ((leq g a6 a3) \to ((leq g a4 a5) \to P))))) (\lambda (H8: (eq A a4 a2)).(eq_ind A a2 (\lambda (a2: A).((eq A (AHead a3 a5) (AHead a (asucc g a0))) \to ((leq g (AHead a a0) a3) \to ((leq g a2 a5) \to P)))) (\lambda (H9: (eq A (AHead a3 a5) (AHead a (asucc g a0)))).(let H10 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a) \Rightarrow a])) (AHead a3 a5) (AHead a (asucc g a0)) H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead a _) \Rightarrow a])) (AHead a3 a5) (AHead a (asucc g a0)) H9) in (eq_ind A a (\lambda (a6: A).((eq A a5 (asucc g a0)) \to ((leq g (AHead a a0) a6) \to ((leq g a2 a5) \to P)))) (\lambda (H12: (eq A a5 (asucc g a0))).(eq_ind A (asucc g a0) (\lambda (a6: A).((leq g (AHead a a0) a) \to ((leq g a2 a6) \to P))) (\lambda (H13: (leq g (AHead a a0) a)).(\lambda (_: (leq g a2 (asucc g a0))).(leq_ahead_false g a a0 H13 P))) a5 (sym_eq A a5 (asucc g a0) H12))) a3 (sym_eq A a3 a H11))) H10))) a4 (sym_eq A a4 a2 H8))) a1 (sym_eq A a1 (AHead a a0) H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead (AHead a a0) a2)) (refl_equal A (AHead a (asucc g a0)))))))))))) a1)).
 
 theorem leq_asucc_false:
  \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P: Prop).P)))
 \def
- \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).((leq g (asucc g a0) a0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (H: (leq g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n n0))).(\lambda (P: Prop).((match n return (\lambda (n1: nat).((leq g (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P)) with [O \Rightarrow (\lambda (H0: (leq g (ASort O (next g n0)) (ASort O n0))).(let H1 \def (match H0 return (\lambda (a: A).(\lambda (a0: A).((eq A a (ASort O (next g n0))) \to ((eq A a0 (ASort O n0)) \to P)))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (ASort O (next g n0)))).(\lambda (H2: (eq A (ASort h2 n2) (ASort O n0))).((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) (ASort h1 n1) (ASort O (next g n0)) H1) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort O (next g n0)) H1) in (eq_ind nat O (\lambda (n: nat).((eq nat n1 (next g n0)) \to ((eq A (ASort h2 n2) (ASort O n0)) \to ((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n2) k)) \to P)))) (\lambda (H5: (eq nat n1 (next g n0))).(eq_ind nat (next g n0) (\lambda (n: nat).((eq A (ASort h2 n2) (ASort O n0)) \to ((eq A (aplus g (ASort O n) k) (aplus g (ASort h2 n2) k)) \to P))) (\lambda (H6: (eq A (ASort h2 n2) (ASort O n0))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n2])) (ASort h2 n2) (ASort O n0) H6) in ((let H8 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n2) (ASort O n0) H6) in (eq_ind nat O (\lambda (n: nat).((eq nat n2 n0) \to ((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n n2) k)) \to P))) (\lambda (H9: (eq nat n2 n0)).(eq_ind nat n0 (\lambda (n: nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort O n) k)) \to P)) (\lambda (H10: (eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort O n0) k))).(let H \def (eq_ind_r A (aplus g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a (aplus g (ASort O n0) k))) H10 (aplus g (ASort O n0) (S k)) (aplus_sort_O_S_simpl g n0 k)) in (let H_y \def (aplus_inj g (S k) k (ASort O n0) H) in (le_Sx_x k (eq_ind_r nat k (\lambda (n: nat).(le n k)) (le_n k) (S k) H_y) P)))) n2 (sym_eq nat n2 n0 H9))) h2 (sym_eq nat h2 O H8))) H7))) n1 (sym_eq nat n1 (next g n0) H5))) h1 (sym_eq nat h1 O H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort O (next g n0)))).(\lambda (H3: (eq A (AHead a2 a4) (ASort O n0))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H2) in (False_ind ((eq A (AHead a2 a4) (ASort O n0)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to P))) H4)) H3 H0 H1)))]) in (H1 (refl_equal A (ASort O (next g n0))) (refl_equal A (ASort O n0))))) | (S n1) \Rightarrow (\lambda (H0: (leq g (ASort n1 n0) (ASort (S n1) n0))).(let H1 \def (match H0 return (\lambda (a: A).(\lambda (a0: A).((eq A a (ASort n1 n0)) \to ((eq A a0 (ASort (S n1) n0)) \to P)))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (ASort n1 n0))).(\lambda (H2: (eq A (ASort h2 n2) (ASort (S n1) n0))).((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) (ASort h1 n1) (ASort n1 n0) H1) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort n1 n0) H1) in (eq_ind nat n1 (\lambda (n: nat).((eq nat n1 n0) \to ((eq A (ASort h2 n2) (ASort (S n1) n0)) \to ((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n2) k)) \to P)))) (\lambda (H5: (eq nat n1 n0)).(eq_ind nat n0 (\lambda (n: nat).((eq A (ASort h2 n2) (ASort (S n1) n0)) \to ((eq A (aplus g (ASort n1 n) k) (aplus g (ASort h2 n2) k)) \to P))) (\lambda (H6: (eq A (ASort h2 n2) (ASort (S n1) n0))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n2])) (ASort h2 n2) (ASort (S n1) n0) H6) in ((let H8 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n2) (ASort (S n1) n0) H6) in (eq_ind nat (S n1) (\lambda (n: nat).((eq nat n2 n0) \to ((eq A (aplus g (ASort n1 n0) k) (aplus g (ASort n n2) k)) \to P))) (\lambda (H9: (eq nat n2 n0)).(eq_ind nat n0 (\lambda (n: nat).((eq A (aplus g (ASort n1 n0) k) (aplus g (ASort (S n1) n) k)) \to P)) (\lambda (H10: (eq A (aplus g (ASort n1 n0) k) (aplus g (ASort (S n1) n0) k))).(let H \def (eq_ind_r A (aplus g (ASort n1 n0) k) (\lambda (a: A).(eq A a (aplus g (ASort (S n1) n0) k))) H10 (aplus g (ASort (S n1) n0) (S k)) (aplus_sort_S_S_simpl g n0 n1 k)) in (let H_y \def (aplus_inj g (S k) k (ASort (S n1) n0) H) in (le_Sx_x k (eq_ind_r nat k (\lambda (n: nat).(le n k)) (le_n k) (S k) H_y) P)))) n2 (sym_eq nat n2 n0 H9))) h2 (sym_eq nat h2 (S n1) H8))) H7))) n1 (sym_eq nat n1 n0 H5))) h1 (sym_eq nat h1 n1 H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort n1 n0))).(\lambda (H3: (eq A (AHead a2 a4) (ASort (S n1) n0))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1 n0) H2) in (False_ind ((eq A (AHead a2 a4) (ASort (S n1) n0)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to P))) H4)) H3 H0 H1)))]) in (H1 (refl_equal A (ASort n1 n0)) (refl_equal A (ASort (S n1) n0)))))]) H))))) (\lambda (a0: A).(\lambda (_: (((leq g (asucc g a0) a0) \to (\forall (P: Prop).P)))).(\lambda (a1: A).(\lambda (H0: (((leq g (asucc g a1) a1) \to (\forall (P: Prop).P)))).(\lambda (H1: (leq g (AHead a0 (asucc g a1)) (AHead a0 a1))).(\lambda (P: Prop).(let H2 \def (match H1 return (\lambda (a: A).(\lambda (a2: A).((eq A a (AHead a0 (asucc g a1))) \to ((eq A a2 (AHead a0 a1)) \to P)))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (AHead a0 (asucc g a1)))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a0 a1))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a0 (asucc g a1)) H3) in (False_ind ((eq A (ASort h2 n2) (AHead a0 a1)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head a1 a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4: (eq A (AHead a1 a3) (AHead a0 (asucc g a1)))).(\lambda (H5: (eq A (AHead a2 a4) (AHead a0 a1))).((let H6 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a1 a3) (AHead a0 (asucc g a1)) H4) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead a0 (asucc g a1)) H4) in (eq_ind A a0 (\lambda (a: A).((eq A a3 (asucc g a1)) \to ((eq A (AHead a2 a4) (AHead a0 a1)) \to ((leq g a a2) \to ((leq g a3 a4) \to P))))) (\lambda (H8: (eq A a3 (asucc g a1))).(eq_ind A (asucc g a1) (\lambda (a: A).((eq A (AHead a2 a4) (AHead a0 a1)) \to ((leq g a0 a2) \to ((leq g a a4) \to P)))) (\lambda (H9: (eq A (AHead a2 a4) (AHead a0 a1))).(let H10 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a2 a4) (AHead a0 a1) H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a2 | (AHead a _) \Rightarrow a])) (AHead a2 a4) (AHead a0 a1) H9) in (eq_ind A a0 (\lambda (a: A).((eq A a4 a1) \to ((leq g a0 a) \to ((leq g (asucc g a1) a4) \to P)))) (\lambda (H12: (eq A a4 a1)).(eq_ind A a1 (\lambda (a: A).((leq g a0 a0) \to ((leq g (asucc g a1) a) \to P))) (\lambda (_: (leq g a0 a0)).(\lambda (H14: (leq g (asucc g a1) a1)).(H0 H14 P))) a4 (sym_eq A a4 a1 H12))) a2 (sym_eq A a2 a0 H11))) H10))) a3 (sym_eq A a3 (asucc g a1) H8))) a1 (sym_eq A a1 a0 H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead a0 (asucc g a1))) (refl_equal A (AHead a0 a1)))))))))) a)).
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).((leq g (asucc g a0) a0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (H: (leq g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n n0))).(\lambda (P: Prop).((match n return (\lambda (_: ?).(\lambda (n1: nat).((leq g (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P))) with [O \Rightarrow (\lambda (H0: (leq g (ASort O (next g n0)) (ASort O n0))).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (a: A).(\lambda (a0: A).((eq A a (ASort O (next g n0))) \to ((eq A a0 (ASort O n0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (ASort O (next g n0)))).(\lambda (H2: (eq A (ASort h2 n2) (ASort O n0))).((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) (ASort h1 n1) (ASort O (next g n0)) H1) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort O (next g n0)) H1) in (eq_ind nat O (\lambda (n: nat).((eq nat n1 (next g n0)) \to ((eq A (ASort h2 n2) (ASort O n0)) \to ((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n2) k)) \to P)))) (\lambda (H5: (eq nat n1 (next g n0))).(eq_ind nat (next g n0) (\lambda (n: nat).((eq A (ASort h2 n2) (ASort O n0)) \to ((eq A (aplus g (ASort O n) k) (aplus g (ASort h2 n2) k)) \to P))) (\lambda (H6: (eq A (ASort h2 n2) (ASort O n0))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n2])) (ASort h2 n2) (ASort O n0) H6) in ((let H8 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n2) (ASort O n0) H6) in (eq_ind nat O (\lambda (n: nat).((eq nat n2 n0) \to ((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort n n2) k)) \to P))) (\lambda (H9: (eq nat n2 n0)).(eq_ind nat n0 (\lambda (n: nat).((eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort O n) k)) \to P)) (\lambda (H10: (eq A (aplus g (ASort O (next g n0)) k) (aplus g (ASort O n0) k))).(let H \def (eq_ind_r A (aplus g (ASort O (next g n0)) k) (\lambda (a: A).(eq A a (aplus g (ASort O n0) k))) H10 (aplus g (ASort O n0) (S k)) (aplus_sort_O_S_simpl g n0 k)) in (let H_y \def (aplus_inj g (S k) k (ASort O n0) H) in (le_Sx_x k (eq_ind_r nat k (\lambda (n: nat).(le n k)) (le_n k) (S k) H_y) P)))) n2 (sym_eq nat n2 n0 H9))) h2 (sym_eq nat h2 O H8))) H7))) n1 (sym_eq nat n1 (next g n0) H5))) h1 (sym_eq nat h1 O H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort O (next g n0)))).(\lambda (H3: (eq A (AHead a2 a4) (ASort O n0))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) H2) in (False_ind ((eq A (AHead a2 a4) (ASort O n0)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to P))) H4)) H3 H0 H1)))]) in (H1 (refl_equal A (ASort O (next g n0))) (refl_equal A (ASort O n0))))) | (S n1) \Rightarrow (\lambda (H0: (leq g (ASort n1 n0) (ASort (S n1) n0))).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (a: A).(\lambda (a0: A).((eq A a (ASort n1 n0)) \to ((eq A a0 (ASort (S n1) n0)) \to P))))) with [(leq_sort h1 h2 n1 n2 k H0) \Rightarrow (\lambda (H1: (eq A (ASort h1 n1) (ASort n1 n0))).(\lambda (H2: (eq A (ASort h2 n2) (ASort (S n1) n0))).((let H3 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) (ASort h1 n1) (ASort n1 n0) H1) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h1])) (ASort h1 n1) (ASort n1 n0) H1) in (eq_ind nat n1 (\lambda (n: nat).((eq nat n1 n0) \to ((eq A (ASort h2 n2) (ASort (S n1) n0)) \to ((eq A (aplus g (ASort n n1) k) (aplus g (ASort h2 n2) k)) \to P)))) (\lambda (H5: (eq nat n1 n0)).(eq_ind nat n0 (\lambda (n: nat).((eq A (ASort h2 n2) (ASort (S n1) n0)) \to ((eq A (aplus g (ASort n1 n) k) (aplus g (ASort h2 n2) k)) \to P))) (\lambda (H6: (eq A (ASort h2 n2) (ASort (S n1) n0))).(let H7 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n2])) (ASort h2 n2) (ASort (S n1) n0) H6) in ((let H8 \def (f_equal A nat (\lambda (e: A).(match e return (\lambda (_: ?).nat) with [(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n2) (ASort (S n1) n0) H6) in (eq_ind nat (S n1) (\lambda (n: nat).((eq nat n2 n0) \to ((eq A (aplus g (ASort n1 n0) k) (aplus g (ASort n n2) k)) \to P))) (\lambda (H9: (eq nat n2 n0)).(eq_ind nat n0 (\lambda (n: nat).((eq A (aplus g (ASort n1 n0) k) (aplus g (ASort (S n1) n) k)) \to P)) (\lambda (H10: (eq A (aplus g (ASort n1 n0) k) (aplus g (ASort (S n1) n0) k))).(let H \def (eq_ind_r A (aplus g (ASort n1 n0) k) (\lambda (a: A).(eq A a (aplus g (ASort (S n1) n0) k))) H10 (aplus g (ASort (S n1) n0) (S k)) (aplus_sort_S_S_simpl g n0 n1 k)) in (let H_y \def (aplus_inj g (S k) k (ASort (S n1) n0) H) in (le_Sx_x k (eq_ind_r nat k (\lambda (n: nat).(le n k)) (le_n k) (S k) H_y) P)))) n2 (sym_eq nat n2 n0 H9))) h2 (sym_eq nat h2 (S n1) H8))) H7))) n1 (sym_eq nat n1 n0 H5))) h1 (sym_eq nat h1 n1 H4))) H3)) H2 H0))) | (leq_head a1 a2 H0 a3 a4 H1) \Rightarrow (\lambda (H2: (eq A (AHead a1 a3) (ASort n1 n0))).(\lambda (H3: (eq A (AHead a2 a4) (ASort (S n1) n0))).((let H4 \def (eq_ind A (AHead a1 a3) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort n1 n0) H2) in (False_ind ((eq A (AHead a2 a4) (ASort (S n1) n0)) \to ((leq g a1 a2) \to ((leq g a3 a4) \to P))) H4)) H3 H0 H1)))]) in (H1 (refl_equal A (ASort n1 n0)) (refl_equal A (ASort (S n1) n0)))))]) H))))) (\lambda (a0: A).(\lambda (_: (((leq g (asucc g a0) a0) \to (\forall (P: Prop).P)))).(\lambda (a1: A).(\lambda (H0: (((leq g (asucc g a1) a1) \to (\forall (P: Prop).P)))).(\lambda (H1: (leq g (AHead a0 (asucc g a1)) (AHead a0 a1))).(\lambda (P: Prop).(let H2 \def (match H1 return (\lambda (_: ?).(\lambda (a: A).(\lambda (a2: A).((eq A a (AHead a0 (asucc g a1))) \to ((eq A a2 (AHead a0 a1)) \to P))))) with [(leq_sort h1 h2 n1 n2 k H2) \Rightarrow (\lambda (H3: (eq A (ASort h1 n1) (AHead a0 (asucc g a1)))).(\lambda (H4: (eq A (ASort h2 n2) (AHead a0 a1))).((let H5 \def (eq_ind A (ASort h1 n1) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead a0 (asucc g a1)) H3) in (False_ind ((eq A (ASort h2 n2) (AHead a0 a1)) \to ((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to P)) H5)) H4 H2))) | (leq_head a1 a2 H2 a3 a4 H3) \Rightarrow (\lambda (H4: (eq A (AHead a1 a3) (AHead a0 (asucc g a1)))).(\lambda (H5: (eq A (AHead a2 a4) (AHead a0 a1))).((let H6 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a1 a3) (AHead a0 (asucc g a1)) H4) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a1 | (AHead a _) \Rightarrow a])) (AHead a1 a3) (AHead a0 (asucc g a1)) H4) in (eq_ind A a0 (\lambda (a: A).((eq A a3 (asucc g a1)) \to ((eq A (AHead a2 a4) (AHead a0 a1)) \to ((leq g a a2) \to ((leq g a3 a4) \to P))))) (\lambda (H8: (eq A a3 (asucc g a1))).(eq_ind A (asucc g a1) (\lambda (a: A).((eq A (AHead a2 a4) (AHead a0 a1)) \to ((leq g a0 a2) \to ((leq g a a4) \to P)))) (\lambda (H9: (eq A (AHead a2 a4) (AHead a0 a1))).(let H10 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a2 a4) (AHead a0 a1) H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a2 | (AHead a _) \Rightarrow a])) (AHead a2 a4) (AHead a0 a1) H9) in (eq_ind A a0 (\lambda (a: A).((eq A a4 a1) \to ((leq g a0 a) \to ((leq g (asucc g a1) a4) \to P)))) (\lambda (H12: (eq A a4 a1)).(eq_ind A a1 (\lambda (a: A).((leq g a0 a0) \to ((leq g (asucc g a1) a) \to P))) (\lambda (_: (leq g a0 a0)).(\lambda (H14: (leq g (asucc g a1) a1)).(H0 H14 P))) a4 (sym_eq A a4 a1 H12))) a2 (sym_eq A a2 a0 H11))) H10))) a3 (sym_eq A a3 (asucc g a1) H8))) a1 (sym_eq A a1 a0 H7))) H6)) H5 H2 H3)))]) in (H2 (refl_equal A (AHead a0 (asucc g a1))) (refl_equal A (AHead a0 a1)))))))))) a)).
 
 definition lweight:
  A \to nat
@@ -1624,7 +1624,7 @@ inductive aprem: nat \to (A \to (A \to Prop)) \def
 theorem aprem_repl:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall (i: nat).(\forall (b2: A).((aprem i a2 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem i a1 b1)))))))))
 \def
- \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (i: nat).(\forall (b2: A).((aprem i a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem i a b1)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (i: nat).(\lambda (b2: A).(\lambda (H1: (aprem i (ASort h2 n2) b2)).(let H2 \def (match H1 return (\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).((eq nat n i) \to ((eq A a (ASort h2 n2)) \to ((eq A a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem i (ASort h1 n1) b1))))))))) with [(aprem_zero a0 a3) \Rightarrow (\lambda (H1: (eq nat O i)).(\lambda (H2: (eq A (AHead a0 a3) (ASort h2 n2))).(\lambda (H3: (eq A a0 b2)).(eq_ind nat O (\lambda (n: nat).((eq A (AHead a0 a3) (ASort h2 n2)) \to ((eq A a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem n (ASort h1 n1) b1)))))) (\lambda (H4: (eq A (AHead a0 a3) (ASort h2 n2))).(let H5 \def (eq_ind A (AHead a0 a3) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort h2 n2) H4) in (False_ind ((eq A a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem O (ASort h1 n1) b1)))) H5))) i H1 H2 H3)))) | (aprem_succ a0 a i0 H1 a3) \Rightarrow (\lambda (H2: (eq nat (S i0) i)).(\lambda (H3: (eq A (AHead a3 a0) (ASort h2 n2))).(\lambda (H4: (eq A a b2)).(eq_ind nat (S i0) (\lambda (n: nat).((eq A (AHead a3 a0) (ASort h2 n2)) \to ((eq A a b2) \to ((aprem i0 a0 a) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem n (ASort h1 n1) b1))))))) (\lambda (H5: (eq A (AHead a3 a0) (ASort h2 n2))).(let H6 \def (eq_ind A (AHead a3 a0) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort h2 n2) H5) in (False_ind ((eq A a b2) \to ((aprem i0 a0 a) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (ASort h1 n1) b1))))) H6))) i H2 H3 H4 H1))))]) in (H2 (refl_equal nat i) (refl_equal A (ASort h2 n2)) (refl_equal A b2)))))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H0: (leq g a0 a3)).(\lambda (_: ((\forall (i: nat).(\forall (b2: A).((aprem i a3 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem i a0 b1)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq g a4 a5)).(\lambda (H3: ((\forall (i: nat).(\forall (b2: A).((aprem i a5 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem i a4 b1)))))))).(\lambda (i: nat).(\lambda (b2: A).(\lambda (H4: (aprem i (AHead a3 a5) b2)).((match i return (\lambda (n: nat).((aprem n (AHead a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem n (AHead a0 a4) b1))))) with [O \Rightarrow (\lambda (H5: (aprem O (AHead a3 a5) b2)).(let H6 \def (match H5 return (\lambda (n: nat).(\lambda (a: A).(\lambda (a1: A).((eq nat n O) \to ((eq A a (AHead a3 a5)) \to ((eq A a1 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1))))))))) with [(aprem_zero a6 a7) \Rightarrow (\lambda (_: (eq nat O O)).(\lambda (H5: (eq A (AHead a6 a7) (AHead a3 a5))).(\lambda (H6: (eq A a6 b2)).((let H7 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a7 | (AHead _ a) \Rightarrow a])) (AHead a6 a7) (AHead a3 a5) H5) in ((let H8 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a6 | (AHead a _) \Rightarrow a])) (AHead a6 a7) (AHead a3 a5) H5) in (eq_ind A a3 (\lambda (a: A).((eq A a7 a5) \to ((eq A a b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1)))))) (\lambda (H9: (eq A a7 a5)).(eq_ind A a5 (\lambda (_: A).((eq A a3 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1))))) (\lambda (H10: (eq A a3 b2)).(eq_ind A b2 (\lambda (_: A).(ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1)))) (eq_ind A a3 (\lambda (a: A).(ex2 A (\lambda (b1: A).(leq g b1 a)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1)))) (ex_intro2 A (\lambda (b1: A).(leq g b1 a3)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1)) a0 H0 (aprem_zero a0 a4)) b2 H10) a3 (sym_eq A a3 b2 H10))) a7 (sym_eq A a7 a5 H9))) a6 (sym_eq A a6 a3 H8))) H7)) H6)))) | (aprem_succ a6 a i H4 a7) \Rightarrow (\lambda (H5: (eq nat (S i) O)).(\lambda (H6: (eq A (AHead a7 a6) (AHead a3 a5))).(\lambda (H7: (eq A a b2)).((let H8 \def (eq_ind nat (S i) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5) in (False_ind ((eq A (AHead a7 a6) (AHead a3 a5)) \to ((eq A a b2) \to ((aprem i a6 a) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1)))))) H8)) H6 H7 H4))))]) in (H6 (refl_equal nat O) (refl_equal A (AHead a3 a5)) (refl_equal A b2)))) | (S n) \Rightarrow (\lambda (H5: (aprem (S n) (AHead a3 a5) b2)).(let H6 \def (match H5 return (\lambda (n0: nat).(\lambda (a: A).(\lambda (a1: A).((eq nat n0 (S n)) \to ((eq A a (AHead a3 a5)) \to ((eq A a1 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1))))))))) with [(aprem_zero a6 a7) \Rightarrow (\lambda (H4: (eq nat O (S n))).(\lambda (H5: (eq A (AHead a6 a7) (AHead a3 a5))).(\lambda (H6: (eq A a6 b2)).((let H7 \def (eq_ind nat O (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S n) H4) in (False_ind ((eq A (AHead a6 a7) (AHead a3 a5)) \to ((eq A a6 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1))))) H7)) H5 H6)))) | (aprem_succ a6 a i H4 a7) \Rightarrow (\lambda (H5: (eq nat (S i) (S n))).(\lambda (H6: (eq A (AHead a7 a6) (AHead a3 a5))).(\lambda (H7: (eq A a b2)).((let H8 \def (f_equal nat nat (\lambda (e: nat).(match e return (\lambda (_: ?).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S n) H5) in (eq_ind nat n (\lambda (n0: nat).((eq A (AHead a7 a6) (AHead a3 a5)) \to ((eq A a b2) \to ((aprem n0 a6 a) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1))))))) (\lambda (H9: (eq A (AHead a7 a6) (AHead a3 a5))).(let H10 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a6 | (AHead _ a) \Rightarrow a])) (AHead a7 a6) (AHead a3 a5) H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a7 | (AHead a _) \Rightarrow a])) (AHead a7 a6) (AHead a3 a5) H9) in (eq_ind A a3 (\lambda (_: A).((eq A a6 a5) \to ((eq A a b2) \to ((aprem n a6 a) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1))))))) (\lambda (H12: (eq A a6 a5)).(eq_ind A a5 (\lambda (a1: A).((eq A a b2) \to ((aprem n a1 a) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1)))))) (\lambda (H13: (eq A a b2)).(eq_ind A b2 (\lambda (a1: A).((aprem n a5 a1) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1))))) (\lambda (H14: (aprem n a5 b2)).(let H_x \def (H3 n b2 H14) in (let H3 \def H_x in (ex2_ind A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem n a4 b1)) (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1))) (\lambda (x: A).(\lambda (H15: (leq g x b2)).(\lambda (H16: (aprem n a4 x)).(ex_intro2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1)) x H15 (aprem_succ a4 x n H16 a0))))) H3)))) a (sym_eq A a b2 H13))) a6 (sym_eq A a6 a5 H12))) a7 (sym_eq A a7 a3 H11))) H10))) i (sym_eq nat i n H8))) H6 H7 H4))))]) in (H6 (refl_equal nat (S n)) (refl_equal A (AHead a3 a5)) (refl_equal A b2))))]) H4)))))))))))) a1 a2 H)))).
+ \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (i: nat).(\forall (b2: A).((aprem i a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem i a b1)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (i: nat).(\lambda (b2: A).(\lambda (H1: (aprem i (ASort h2 n2) b2)).(let H2 \def (match H1 return (\lambda (_: ?).(\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).((eq nat n i) \to ((eq A a (ASort h2 n2)) \to ((eq A a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem i (ASort h1 n1) b1)))))))))) with [(aprem_zero a0 a3) \Rightarrow (\lambda (H1: (eq nat O i)).(\lambda (H2: (eq A (AHead a0 a3) (ASort h2 n2))).(\lambda (H3: (eq A a0 b2)).(eq_ind nat O (\lambda (n: nat).((eq A (AHead a0 a3) (ASort h2 n2)) \to ((eq A a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem n (ASort h1 n1) b1)))))) (\lambda (H4: (eq A (AHead a0 a3) (ASort h2 n2))).(let H5 \def (eq_ind A (AHead a0 a3) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort h2 n2) H4) in (False_ind ((eq A a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem O (ASort h1 n1) b1)))) H5))) i H1 H2 H3)))) | (aprem_succ a0 a i0 H1 a3) \Rightarrow (\lambda (H2: (eq nat (S i0) i)).(\lambda (H3: (eq A (AHead a3 a0) (ASort h2 n2))).(\lambda (H4: (eq A a b2)).(eq_ind nat (S i0) (\lambda (n: nat).((eq A (AHead a3 a0) (ASort h2 n2)) \to ((eq A a b2) \to ((aprem i0 a0 a) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem n (ASort h1 n1) b1))))))) (\lambda (H5: (eq A (AHead a3 a0) (ASort h2 n2))).(let H6 \def (eq_ind A (AHead a3 a0) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort h2 n2) H5) in (False_ind ((eq A a b2) \to ((aprem i0 a0 a) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (ASort h1 n1) b1))))) H6))) i H2 H3 H4 H1))))]) in (H2 (refl_equal nat i) (refl_equal A (ASort h2 n2)) (refl_equal A b2)))))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H0: (leq g a0 a3)).(\lambda (_: ((\forall (i: nat).(\forall (b2: A).((aprem i a3 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem i a0 b1)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq g a4 a5)).(\lambda (H3: ((\forall (i: nat).(\forall (b2: A).((aprem i a5 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem i a4 b1)))))))).(\lambda (i: nat).(\lambda (b2: A).(\lambda (H4: (aprem i (AHead a3 a5) b2)).((match i return (\lambda (_: ?).(\lambda (n: nat).((aprem n (AHead a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem n (AHead a0 a4) b1)))))) with [O \Rightarrow (\lambda (H5: (aprem O (AHead a3 a5) b2)).(let H6 \def (match H5 return (\lambda (_: ?).(\lambda (n: nat).(\lambda (a: A).(\lambda (a1: A).((eq nat n O) \to ((eq A a (AHead a3 a5)) \to ((eq A a1 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1)))))))))) with [(aprem_zero a6 a7) \Rightarrow (\lambda (_: (eq nat O O)).(\lambda (H5: (eq A (AHead a6 a7) (AHead a3 a5))).(\lambda (H6: (eq A a6 b2)).((let H7 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a7 | (AHead _ a) \Rightarrow a])) (AHead a6 a7) (AHead a3 a5) H5) in ((let H8 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a6 | (AHead a _) \Rightarrow a])) (AHead a6 a7) (AHead a3 a5) H5) in (eq_ind A a3 (\lambda (a: A).((eq A a7 a5) \to ((eq A a b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1)))))) (\lambda (H9: (eq A a7 a5)).(eq_ind A a5 (\lambda (_: A).((eq A a3 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1))))) (\lambda (H10: (eq A a3 b2)).(eq_ind A b2 (\lambda (_: A).(ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1)))) (eq_ind A a3 (\lambda (a: A).(ex2 A (\lambda (b1: A).(leq g b1 a)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1)))) (ex_intro2 A (\lambda (b1: A).(leq g b1 a3)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1)) a0 H0 (aprem_zero a0 a4)) b2 H10) a3 (sym_eq A a3 b2 H10))) a7 (sym_eq A a7 a5 H9))) a6 (sym_eq A a6 a3 H8))) H7)) H6)))) | (aprem_succ a6 a i H4 a7) \Rightarrow (\lambda (H5: (eq nat (S i) O)).(\lambda (H6: (eq A (AHead a7 a6) (AHead a3 a5))).(\lambda (H7: (eq A a b2)).((let H8 \def (eq_ind nat (S i) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5) in (False_ind ((eq A (AHead a7 a6) (AHead a3 a5)) \to ((eq A a b2) \to ((aprem i a6 a) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1)))))) H8)) H6 H7 H4))))]) in (H6 (refl_equal nat O) (refl_equal A (AHead a3 a5)) (refl_equal A b2)))) | (S n) \Rightarrow (\lambda (H5: (aprem (S n) (AHead a3 a5) b2)).(let H6 \def (match H5 return (\lambda (_: ?).(\lambda (n0: nat).(\lambda (a: A).(\lambda (a1: A).((eq nat n0 (S n)) \to ((eq A a (AHead a3 a5)) \to ((eq A a1 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1)))))))))) with [(aprem_zero a6 a7) \Rightarrow (\lambda (H4: (eq nat O (S n))).(\lambda (H5: (eq A (AHead a6 a7) (AHead a3 a5))).(\lambda (H6: (eq A a6 b2)).((let H7 \def (eq_ind nat O (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S n) H4) in (False_ind ((eq A (AHead a6 a7) (AHead a3 a5)) \to ((eq A a6 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1))))) H7)) H5 H6)))) | (aprem_succ a6 a i H4 a7) \Rightarrow (\lambda (H5: (eq nat (S i) (S n))).(\lambda (H6: (eq A (AHead a7 a6) (AHead a3 a5))).(\lambda (H7: (eq A a b2)).((let H8 \def (f_equal nat nat (\lambda (e: nat).(match e return (\lambda (_: ?).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S n) H5) in (eq_ind nat n (\lambda (n0: nat).((eq A (AHead a7 a6) (AHead a3 a5)) \to ((eq A a b2) \to ((aprem n0 a6 a) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1))))))) (\lambda (H9: (eq A (AHead a7 a6) (AHead a3 a5))).(let H10 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a6 | (AHead _ a) \Rightarrow a])) (AHead a7 a6) (AHead a3 a5) H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a7 | (AHead a _) \Rightarrow a])) (AHead a7 a6) (AHead a3 a5) H9) in (eq_ind A a3 (\lambda (_: A).((eq A a6 a5) \to ((eq A a b2) \to ((aprem n a6 a) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1))))))) (\lambda (H12: (eq A a6 a5)).(eq_ind A a5 (\lambda (a1: A).((eq A a b2) \to ((aprem n a1 a) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1)))))) (\lambda (H13: (eq A a b2)).(eq_ind A b2 (\lambda (a1: A).((aprem n a5 a1) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1))))) (\lambda (H14: (aprem n a5 b2)).(let H_x \def (H3 n b2 H14) in (let H3 \def H_x in (ex2_ind A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem n a4 b1)) (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1))) (\lambda (x: A).(\lambda (H15: (leq g x b2)).(\lambda (H16: (aprem n a4 x)).(ex_intro2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S n) (AHead a0 a4) b1)) x H15 (aprem_succ a4 x n H16 a0))))) H3)))) a (sym_eq A a b2 H13))) a6 (sym_eq A a6 a5 H12))) a7 (sym_eq A a7 a3 H11))) H10))) i (sym_eq nat i n H8))) H6 H7 H4))))]) in (H6 (refl_equal nat (S n)) (refl_equal A (AHead a3 a5)) (refl_equal A b2))))]) H4)))))))))))) a1 a2 H)))).
 
 theorem aprem_asucc:
  \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (i: nat).((aprem i a1 a2) \to (aprem i (asucc g a1) a2)))))
@@ -1738,7 +1738,7 @@ theorem arity_lift:
 theorem arity_lift1:
  \forall (g: G).(\forall (a: A).(\forall (c2: C).(\forall (hds: PList).(\forall (c1: C).(\forall (t: T).((drop1 hds c1 c2) \to ((arity g c2 t a) \to (arity g c1 (lift1 hds t) a))))))))
 \def
- \lambda (g: G).(\lambda (a: A).(\lambda (c2: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c1: C).(\forall (t: T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1 (lift1 p t) a)))))) (\lambda (c1: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c1 c2)).(\lambda (H0: (arity g c2 t a)).(let H1 \def (match H return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p PNil) \to ((eq C c c1) \to ((eq C c0 c2) \to (arity g c1 t a))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c c1)).(\lambda (H3: (eq C c c2)).(eq_ind C c1 (\lambda (c0: C).((eq C c0 c2) \to (arity g c1 t a))) (\lambda (H4: (eq C c1 c2)).(eq_ind C c2 (\lambda (c0: C).(arity g c0 t a)) H0 c1 (sym_eq C c1 c2 H4))) c (sym_eq C c c1 H2) H3)))) | (drop1_cons c0 c3 h d H1 c4 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c0 c1)).(\lambda (H5: (eq C c4 c2)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e: PList).(match e return (\lambda (_: ?).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c0 c1) \to ((eq C c4 c2) \to ((drop h d c0 c3) \to ((drop1 hds c3 c4) \to (arity g c1 t a))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c1) (refl_equal C c2))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c1: C).(\forall (t: T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1 (lift1 p t) a))))))).(\lambda (c1: C).(\lambda (t: T).(\lambda (H0: (drop1 (PCons n n0 p) c1 c2)).(\lambda (H1: (arity g c2 t a)).(let H2 \def (match H0 return (\lambda (p0: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p0 (PCons n n0 p)) \to ((eq C c c1) \to ((eq C c0 c2) \to (arity g c1 (lift n n0 (lift1 p t)) a))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c c1)).(\lambda (H4: (eq C c c2)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e return (\lambda (_: ?).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c c1) \to ((eq C c c2) \to (arity g c1 (lift n n0 (lift1 p t)) a))) H5)) H3 H4)))) | (drop1_cons c0 c3 h d H2 c4 hds H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds) (PCons n n0 p))).(\lambda (H5: (eq C c0 c1)).(\lambda (H6: (eq C c4 c2)).((let H7 \def (f_equal PList PList (\lambda (e: PList).(match e return (\lambda (_: ?).PList) with [PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_: ?).nat) with [PNil \Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_: ?).nat) with [PNil \Rightarrow h | (PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c0 c1) \to ((eq C c4 c2) \to ((drop n1 d c0 c3) \to ((drop1 hds c3 c4) \to (arity g c1 (lift n n0 (lift1 p t)) a)))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq C c0 c1) \to ((eq C c4 c2) \to ((drop n n1 c0 c3) \to ((drop1 hds c3 c4) \to (arity g c1 (lift n n0 (lift1 p t)) a))))))) (\lambda (H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c0 c1) \to ((eq C c4 c2) \to ((drop n n0 c0 c3) \to ((drop1 p0 c3 c4) \to (arity g c1 (lift n n0 (lift1 p t)) a)))))) (\lambda (H12: (eq C c0 c1)).(eq_ind C c1 (\lambda (c: C).((eq C c4 c2) \to ((drop n n0 c c3) \to ((drop1 p c3 c4) \to (arity g c1 (lift n n0 (lift1 p t)) a))))) (\lambda (H13: (eq C c4 c2)).(eq_ind C c2 (\lambda (c: C).((drop n n0 c1 c3) \to ((drop1 p c3 c) \to (arity g c1 (lift n n0 (lift1 p t)) a)))) (\lambda (H14: (drop n n0 c1 c3)).(\lambda (H15: (drop1 p c3 c2)).(arity_lift g c3 (lift1 p t) a (H c3 t H15 H1) c1 n n0 H14))) c4 (sym_eq C c4 c2 H13))) c0 (sym_eq C c0 c1 H12))) hds (sym_eq PList hds p H11))) d (sym_eq nat d n0 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p)) (refl_equal C c1) (refl_equal C c2))))))))))) hds)))).
+ \lambda (g: G).(\lambda (a: A).(\lambda (c2: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c1: C).(\forall (t: T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1 (lift1 p t) a)))))) (\lambda (c1: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c1 c2)).(\lambda (H0: (arity g c2 t a)).(let H1 \def (match H return (\lambda (_: ?).(\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p PNil) \to ((eq C c c1) \to ((eq C c0 c2) \to (arity g c1 t a)))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c c1)).(\lambda (H3: (eq C c c2)).(eq_ind C c1 (\lambda (c0: C).((eq C c0 c2) \to (arity g c1 t a))) (\lambda (H4: (eq C c1 c2)).(eq_ind C c2 (\lambda (c0: C).(arity g c0 t a)) H0 c1 (sym_eq C c1 c2 H4))) c (sym_eq C c c1 H2) H3)))) | (drop1_cons c0 c3 h d H1 c4 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c0 c1)).(\lambda (H5: (eq C c4 c2)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e: PList).(match e return (\lambda (_: ?).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c0 c1) \to ((eq C c4 c2) \to ((drop h d c0 c3) \to ((drop1 hds c3 c4) \to (arity g c1 t a))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c1) (refl_equal C c2))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c1: C).(\forall (t: T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1 (lift1 p t) a))))))).(\lambda (c1: C).(\lambda (t: T).(\lambda (H0: (drop1 (PCons n n0 p) c1 c2)).(\lambda (H1: (arity g c2 t a)).(let H2 \def (match H0 return (\lambda (_: ?).(\lambda (p0: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p0 (PCons n n0 p)) \to ((eq C c c1) \to ((eq C c0 c2) \to (arity g c1 (lift n n0 (lift1 p t)) a)))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c c1)).(\lambda (H4: (eq C c c2)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e return (\lambda (_: ?).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c c1) \to ((eq C c c2) \to (arity g c1 (lift n n0 (lift1 p t)) a))) H5)) H3 H4)))) | (drop1_cons c0 c3 h d H2 c4 hds H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds) (PCons n n0 p))).(\lambda (H5: (eq C c0 c1)).(\lambda (H6: (eq C c4 c2)).((let H7 \def (f_equal PList PList (\lambda (e: PList).(match e return (\lambda (_: ?).PList) with [PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_: ?).nat) with [PNil \Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_: ?).nat) with [PNil \Rightarrow h | (PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c0 c1) \to ((eq C c4 c2) \to ((drop n1 d c0 c3) \to ((drop1 hds c3 c4) \to (arity g c1 (lift n n0 (lift1 p t)) a)))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq C c0 c1) \to ((eq C c4 c2) \to ((drop n n1 c0 c3) \to ((drop1 hds c3 c4) \to (arity g c1 (lift n n0 (lift1 p t)) a))))))) (\lambda (H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c0 c1) \to ((eq C c4 c2) \to ((drop n n0 c0 c3) \to ((drop1 p0 c3 c4) \to (arity g c1 (lift n n0 (lift1 p t)) a)))))) (\lambda (H12: (eq C c0 c1)).(eq_ind C c1 (\lambda (c: C).((eq C c4 c2) \to ((drop n n0 c c3) \to ((drop1 p c3 c4) \to (arity g c1 (lift n n0 (lift1 p t)) a))))) (\lambda (H13: (eq C c4 c2)).(eq_ind C c2 (\lambda (c: C).((drop n n0 c1 c3) \to ((drop1 p c3 c) \to (arity g c1 (lift n n0 (lift1 p t)) a)))) (\lambda (H14: (drop n n0 c1 c3)).(\lambda (H15: (drop1 p c3 c2)).(arity_lift g c3 (lift1 p t) a (H c3 t H15 H1) c1 n n0 H14))) c4 (sym_eq C c4 c2 H13))) c0 (sym_eq C c0 c1 H12))) hds (sym_eq PList hds p H11))) d (sym_eq nat d n0 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p)) (refl_equal C c1) (refl_equal C c2))))))))))) hds)))).
 
 theorem arity_mono:
  \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a1: A).((arity g c t a1) \to (\forall (a2: A).((arity g c t a2) \to (leq g a1 a2)))))))
@@ -1753,7 +1753,7 @@ theorem arity_cimp_conf:
 theorem arity_aprem:
  \forall (g: G).(\forall (c: C).(\forall (t: T).(\forall (a: A).((arity g c t a) \to (\forall (i: nat).(\forall (b: A).((aprem i a b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))))))))
 \def
- \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: (arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (a0: A).(\forall (i: nat).(\forall (b: A).((aprem i a0 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (i: nat).(\lambda (b: A).(\lambda (H0: (aprem i (ASort O n) b)).(let H1 \def (match H0 return (\lambda (n0: nat).(\lambda (a: A).(\lambda (a0: A).((eq nat n0 i) \to ((eq A a (ASort O n)) \to ((eq A a0 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))))))) with [(aprem_zero a1 a2) \Rightarrow (\lambda (H0: (eq nat O i)).(\lambda (H1: (eq A (AHead a1 a2) (ASort O n))).(\lambda (H2: (eq A a1 b)).(eq_ind nat O (\lambda (n0: nat).((eq A (AHead a1 a2) (ASort O n)) \to ((eq A a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus n0 j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))) (\lambda (H3: (eq A (AHead a1 a2) (ASort O n))).(let H4 \def (eq_ind A (AHead a1 a2) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O n) H3) in (False_ind ((eq A a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))) H4))) i H0 H1 H2)))) | (aprem_succ a2 a i0 H0 a1) \Rightarrow (\lambda (H1: (eq nat (S i0) i)).(\lambda (H2: (eq A (AHead a1 a2) (ASort O n))).(\lambda (H3: (eq A a b)).(eq_ind nat (S i0) (\lambda (n0: nat).((eq A (AHead a1 a2) (ASort O n)) \to ((eq A a b) \to ((aprem i0 a2 a) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus n0 j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))))) (\lambda (H4: (eq A (AHead a1 a2) (ASort O n))).(let H5 \def (eq_ind A (AHead a1 a2) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O n) H4) in (False_ind ((eq A a b) \to ((aprem i0 a2 a) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S i0) j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))) H5))) i H1 H2 H3 H0))))]) in (H1 (refl_equal nat i) (refl_equal A (ASort O n)) (refl_equal A b)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (i: nat).(\forall (b: A).((aprem i a0 b) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 d)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b: A).(\lambda (H3: (aprem i0 a0 b)).(let H_x \def (H2 i0 b H3) in (let H4 \def H_x in (ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H5: (drop (plus i0 x2) O x0 d)).(\lambda (H6: (arity g x0 x1 (asucc g b))).(let H_x0 \def (getl_drop_conf_rev (plus i0 x2) x0 d H5 Abbr c0 u i H0) in (let H7 \def H_x0 in (ex2_ind C (\lambda (c1: C).(drop (plus i0 x2) O c1 c0)) (\lambda (c1: C).(drop (S i) (plus i0 x2) c1 x0)) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x: C).(\lambda (H8: (drop (plus i0 x2) O x c0)).(\lambda (H9: (drop (S i) (plus i0 x2) x x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus i0 x2) x1) x2 H8 (arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2) H9))))) H7)))))))) H4)))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (H2: ((\forall (i: nat).(\forall (b: A).((aprem i (asucc g a0) b) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 d)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b: A).(\lambda (H3: (aprem i0 a0 b)).(let H4 \def (H2 i0 b (aprem_asucc g a0 b i0 H3)) in (ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H5: (drop (plus i0 x2) O x0 d)).(\lambda (H6: (arity g x0 x1 (asucc g b))).(let H_x \def (getl_drop_conf_rev (plus i0 x2) x0 d H5 Abst c0 u i H0) in (let H7 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (plus i0 x2) O c1 c0)) (\lambda (c1: C).(drop (S i) (plus i0 x2) c1 x0)) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x: C).(\lambda (H8: (drop (plus i0 x2) O x c0)).(\lambda (H9: (drop (S i) (plus i0 x2) x x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus i0 x2) x1) x2 H8 (arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2) H9))))) H7)))))))) H4))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: ((\forall (i: nat).(\forall (b: A).((aprem i a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (i: nat).(\forall (b0: A).((aprem i a2 b0) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead c0 (Bind b) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b0))))))))))).(\lambda (i: nat).(\lambda (b0: A).(\lambda (H5: (aprem i a2 b0)).(let H_x \def (H4 i b0 H5) in (let H6 \def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead c0 (Bind b) u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b0))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b0)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H7: (drop (plus i x2) O x0 (CHead c0 (Bind b) u))).(\lambda (H8: (arity g x0 x1 (asucc g b0))).(let H9 \def (eq_ind nat (S (plus i x2)) (\lambda (n: nat).(drop n O x0 c0)) (drop_S b x0 c0 u (plus i x2) H7) (plus i (S x2)) (plus_n_Sm i x2)) in (ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b0))))) x0 x1 (S x2) H9 H8))))))) H6))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c0 u (asucc g a1))).(\lambda (_: ((\forall (i: nat).(\forall (b: A).((aprem i (asucc g a1) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (i: nat).(\forall (b: A).((aprem i a2 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead c0 (Bind Abst) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (i: nat).(\lambda (b: A).(\lambda (H4: (aprem i (AHead a1 a2) b)).((match i return (\lambda (n: nat).((aprem n (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus n j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))))) with [O \Rightarrow (\lambda (H5: (aprem O (AHead a1 a2) b)).(let H6 \def (match H5 return (\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).((eq nat n O) \to ((eq A a (AHead a1 a2)) \to ((eq A a0 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))))))) with [(aprem_zero a0 a3) \Rightarrow (\lambda (_: (eq nat O O)).(\lambda (H5: (eq A (AHead a0 a3) (AHead a1 a2))).(\lambda (H6: (eq A a0 b)).((let H7 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H5) in ((let H8 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H5) in (eq_ind A a1 (\lambda (a: A).((eq A a3 a2) \to ((eq A a b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))) (\lambda (H9: (eq A a3 a2)).(eq_ind A a2 (\lambda (_: A).((eq A a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))) (\lambda (H10: (eq A a1 b)).(eq_ind A b (\lambda (_: A).(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))) (eq_ind A a1 (\lambda (a: A).(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g a))))))) (ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g a1))))) c0 u O (drop_refl c0) H0) b H10) a1 (sym_eq A a1 b H10))) a3 (sym_eq A a3 a2 H9))) a0 (sym_eq A a0 a1 H8))) H7)) H6)))) | (aprem_succ a0 a i H4 a3) \Rightarrow (\lambda (H5: (eq nat (S i) O)).(\lambda (H6: (eq A (AHead a3 a0) (AHead a1 a2))).(\lambda (H7: (eq A a b)).((let H8 \def (eq_ind nat (S i) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5) in (False_ind ((eq A (AHead a3 a0) (AHead a1 a2)) \to ((eq A a b) \to ((aprem i a0 a) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))) H8)) H6 H7 H4))))]) in (H6 (refl_equal nat O) (refl_equal A (AHead a1 a2)) (refl_equal A b)))) | (S n) \Rightarrow (\lambda (H5: (aprem (S n) (AHead a1 a2) b)).(let H6 \def (match H5 return (\lambda (n0: nat).(\lambda (a: A).(\lambda (a0: A).((eq nat n0 (S n)) \to ((eq A a (AHead a1 a2)) \to ((eq A a0 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))))))) with [(aprem_zero a0 a3) \Rightarrow (\lambda (H4: (eq nat O (S n))).(\lambda (H5: (eq A (AHead a0 a3) (AHead a1 a2))).(\lambda (H6: (eq A a0 b)).((let H7 \def (eq_ind nat O (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S n) H4) in (False_ind ((eq A (AHead a0 a3) (AHead a1 a2)) \to ((eq A a0 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))) H7)) H5 H6)))) | (aprem_succ a0 a i H4 a3) \Rightarrow (\lambda (H5: (eq nat (S i) (S n))).(\lambda (H6: (eq A (AHead a3 a0) (AHead a1 a2))).(\lambda (H7: (eq A a b)).((let H8 \def (f_equal nat nat (\lambda (e: nat).(match e return (\lambda (_: ?).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S n) H5) in (eq_ind nat n (\lambda (n0: nat).((eq A (AHead a3 a0) (AHead a1 a2)) \to ((eq A a b) \to ((aprem n0 a0 a) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))))) (\lambda (H9: (eq A (AHead a3 a0) (AHead a1 a2))).(let H10 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ a) \Rightarrow a])) (AHead a3 a0) (AHead a1 a2) H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead a _) \Rightarrow a])) (AHead a3 a0) (AHead a1 a2) H9) in (eq_ind A a1 (\lambda (_: A).((eq A a0 a2) \to ((eq A a b) \to ((aprem n a0 a) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))))) (\lambda (H12: (eq A a0 a2)).(eq_ind A a2 (\lambda (a1: A).((eq A a b) \to ((aprem n a1 a) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))) (\lambda (H13: (eq A a b)).(eq_ind A b (\lambda (a1: A).((aprem n a2 a1) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))) (\lambda (H14: (aprem n a2 b)).(let H_x \def (H3 n b H14) in (let H3 \def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus n j) O d (CHead c0 (Bind Abst) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H15: (drop (plus n x2) O x0 (CHead c0 (Bind Abst) u))).(\lambda (H16: (arity g x0 x1 (asucc g b))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))) x0 x1 x2 (drop_S Abst x0 c0 u (plus n x2) H15) H16)))))) H3)))) a (sym_eq A a b H13))) a0 (sym_eq A a0 a2 H12))) a3 (sym_eq A a3 a1 H11))) H10))) i (sym_eq nat i n H8))) H6 H7 H4))))]) in (H6 (refl_equal nat (S n)) (refl_equal A (AHead a1 a2)) (refl_equal A b))))]) H4))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: ((\forall (i: nat).(\forall (b: A).((aprem i a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda (H3: ((\forall (i: nat).(\forall (b: A).((aprem i (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (i: nat).(\lambda (b: A).(\lambda (H4: (aprem i a2 b)).(let H5 \def (H3 (S i) b (aprem_succ a2 b i H4 a1)) in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (S (plus i j)) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H6: (drop (S (plus i x2)) O x0 c0)).(\lambda (H7: (arity g x0 x1 (asucc g b))).(C_ind (\lambda (c1: C).((drop (S (plus i x2)) O c1 c0) \to ((arity g c1 x1 (asucc g b)) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))) (\lambda (n: nat).(\lambda (H8: (drop (S (plus i x2)) O (CSort n) c0)).(\lambda (_: (arity g (CSort n) x1 (asucc g b))).(and3_ind (eq C c0 (CSort n)) (eq nat (S (plus i x2)) O) (eq nat O O) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))) (\lambda (_: (eq C c0 (CSort n))).(\lambda (H11: (eq nat (S (plus i x2)) O)).(\lambda (_: (eq nat O O)).(let H13 \def (eq_ind nat (S (plus i x2)) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H11) in (False_ind (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))) H13))))) (drop_gen_sort n (S (plus i x2)) O c0 H8))))) (\lambda (d: C).(\lambda (IHd: (((drop (S (plus i x2)) O d c0) \to ((arity g d x1 (asucc g b)) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (H8: (drop (S (plus i x2)) O (CHead d k t1) c0)).(\lambda (H9: (arity g (CHead d k t1) x1 (asucc g b))).((match k return (\lambda (k0: K).((arity g (CHead d k0 t1) x1 (asucc g b)) \to ((drop (r k0 (plus i x2)) O d c0) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))))))) with [(Bind b0) \Rightarrow (\lambda (H10: (arity g (CHead d (Bind b0) t1) x1 (asucc g b))).(\lambda (H11: (drop (r (Bind b0) (plus i x2)) O d c0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) (CHead d (Bind b0) t1) x1 (S x2) (eq_ind nat (S (plus i x2)) (\lambda (n: nat).(drop n O (CHead d (Bind b0) t1) c0)) (drop_drop (Bind b0) (plus i x2) d c0 H11 t1) (plus i (S x2)) (plus_n_Sm i x2)) H10))) | (Flat f) \Rightarrow (\lambda (H10: (arity g (CHead d (Flat f) t1) x1 (asucc g b))).(\lambda (H11: (drop (r (Flat f) (plus i x2)) O d c0)).(let H12 \def (IHd H11 (arity_cimp_conf g (CHead d (Flat f) t1) x1 (asucc g b) H10 d (cimp_flat_sx f d t1))) in (ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: nat).(\lambda (H13: (drop (plus i x5) O x3 c0)).(\lambda (H14: (arity g x3 x4 (asucc g b))).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) x3 x4 x5 H13 H14)))))) H12))))]) H9 (drop_gen_drop k d c0 t1 (plus i x2) H8)))))))) x0 H6 H7)))))) H5)))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda (_: ((\forall (i: nat).(\forall (b: A).((aprem i (asucc g a0) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (t0: T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: ((\forall (i: nat).(\forall (b: A).((aprem i a0 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (i: nat).(\lambda (b: A).(\lambda (H4: (aprem i a0 b)).(let H_x \def (H3 i b H4) in (let H5 \def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H6: (drop (plus i x2) O x0 c0)).(\lambda (H7: (arity g x0 x1 (asucc g b))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) x0 x1 x2 H6 H7)))))) H5)))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 t0 a1)).(\lambda (H1: ((\forall (i: nat).(\forall (b: A).((aprem i a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (i: nat).(\lambda (b: A).(\lambda (H3: (aprem i a2 b)).(let H_x \def (aprem_repl g a1 a2 H2 i b H3) in (let H4 \def H_x in (ex2_ind A (\lambda (b1: A).(leq g b1 b)) (\lambda (b1: A).(aprem i a1 b1)) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))) (\lambda (x: A).(\lambda (H5: (leq g x b)).(\lambda (H6: (aprem i a1 x)).(let H_x0 \def (H1 i x H6) in (let H7 \def H_x0 in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g x))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H8: (drop (plus i x2) O x0 c0)).(\lambda (H9: (arity g x0 x1 (asucc g x))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))) x0 x1 x2 H8 (arity_repl g x0 x1 (asucc g x) H9 (asucc g b) (asucc_repl g x b H5)))))))) H7)))))) H4))))))))))))) c t a H))))).
+ \lambda (g: G).(\lambda (c: C).(\lambda (t: T).(\lambda (a: A).(\lambda (H: (arity g c t a)).(arity_ind g (\lambda (c0: C).(\lambda (_: T).(\lambda (a0: A).(\forall (i: nat).(\forall (b: A).((aprem i a0 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))))))) (\lambda (c0: C).(\lambda (n: nat).(\lambda (i: nat).(\lambda (b: A).(\lambda (H0: (aprem i (ASort O n) b)).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (n0: nat).(\lambda (a: A).(\lambda (a0: A).((eq nat n0 i) \to ((eq A a (ASort O n)) \to ((eq A a0 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))))) with [(aprem_zero a1 a2) \Rightarrow (\lambda (H0: (eq nat O i)).(\lambda (H1: (eq A (AHead a1 a2) (ASort O n))).(\lambda (H2: (eq A a1 b)).(eq_ind nat O (\lambda (n0: nat).((eq A (AHead a1 a2) (ASort O n)) \to ((eq A a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus n0 j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))) (\lambda (H3: (eq A (AHead a1 a2) (ASort O n))).(let H4 \def (eq_ind A (AHead a1 a2) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O n) H3) in (False_ind ((eq A a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))) H4))) i H0 H1 H2)))) | (aprem_succ a2 a i0 H0 a1) \Rightarrow (\lambda (H1: (eq nat (S i0) i)).(\lambda (H2: (eq A (AHead a1 a2) (ASort O n))).(\lambda (H3: (eq A a b)).(eq_ind nat (S i0) (\lambda (n0: nat).((eq A (AHead a1 a2) (ASort O n)) \to ((eq A a b) \to ((aprem i0 a2 a) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus n0 j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))))) (\lambda (H4: (eq A (AHead a1 a2) (ASort O n))).(let H5 \def (eq_ind A (AHead a1 a2) (\lambda (e: A).(match e return (\lambda (_: ?).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O n) H4) in (False_ind ((eq A a b) \to ((aprem i0 a2 a) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S i0) j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))) H5))) i H1 H2 H3 H0))))]) in (H1 (refl_equal nat i) (refl_equal A (ASort O n)) (refl_equal A b)))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abbr) u))).(\lambda (a0: A).(\lambda (_: (arity g d u a0)).(\lambda (H2: ((\forall (i: nat).(\forall (b: A).((aprem i a0 b) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 d)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b: A).(\lambda (H3: (aprem i0 a0 b)).(let H_x \def (H2 i0 b H3) in (let H4 \def H_x in (ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H5: (drop (plus i0 x2) O x0 d)).(\lambda (H6: (arity g x0 x1 (asucc g b))).(let H_x0 \def (getl_drop_conf_rev (plus i0 x2) x0 d H5 Abbr c0 u i H0) in (let H7 \def H_x0 in (ex2_ind C (\lambda (c1: C).(drop (plus i0 x2) O c1 c0)) (\lambda (c1: C).(drop (S i) (plus i0 x2) c1 x0)) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x: C).(\lambda (H8: (drop (plus i0 x2) O x c0)).(\lambda (H9: (drop (S i) (plus i0 x2) x x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus i0 x2) x1) x2 H8 (arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2) H9))))) H7)))))))) H4)))))))))))))) (\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind Abst) u))).(\lambda (a0: A).(\lambda (_: (arity g d u (asucc g a0))).(\lambda (H2: ((\forall (i: nat).(\forall (b: A).((aprem i (asucc g a0) b) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 d)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (i0: nat).(\lambda (b: A).(\lambda (H3: (aprem i0 a0 b)).(let H4 \def (H2 i0 b (aprem_asucc g a0 b i0 H3)) in (ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 d)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H5: (drop (plus i0 x2) O x0 d)).(\lambda (H6: (arity g x0 x1 (asucc g b))).(let H_x \def (getl_drop_conf_rev (plus i0 x2) x0 d H5 Abst c0 u i H0) in (let H7 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (plus i0 x2) O c1 c0)) (\lambda (c1: C).(drop (S i) (plus i0 x2) c1 x0)) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x: C).(\lambda (H8: (drop (plus i0 x2) O x c0)).(\lambda (H9: (drop (S i) (plus i0 x2) x x0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i0 j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) x (lift (S i) (plus i0 x2) x1) x2 H8 (arity_lift g x0 x1 (asucc g b) H6 x (S i) (plus i0 x2) H9))))) H7)))))))) H4))))))))))))) (\lambda (b: B).(\lambda (_: (not (eq B b Abst))).(\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: ((\forall (i: nat).(\forall (b: A).((aprem i a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind b) u) t0 a2)).(\lambda (H4: ((\forall (i: nat).(\forall (b0: A).((aprem i a2 b0) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead c0 (Bind b) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b0))))))))))).(\lambda (i: nat).(\lambda (b0: A).(\lambda (H5: (aprem i a2 b0)).(let H_x \def (H4 i b0 H5) in (let H6 \def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead c0 (Bind b) u))))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b0))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b0)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H7: (drop (plus i x2) O x0 (CHead c0 (Bind b) u))).(\lambda (H8: (arity g x0 x1 (asucc g b0))).(let H9 \def (eq_ind nat (S (plus i x2)) (\lambda (n: nat).(drop n O x0 c0)) (drop_S b x0 c0 u (plus i x2) H7) (plus i (S x2)) (plus_n_Sm i x2)) in (ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b0))))) x0 x1 (S x2) H9 H8))))))) H6))))))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (H0: (arity g c0 u (asucc g a1))).(\lambda (_: ((\forall (i: nat).(\forall (b: A).((aprem i (asucc g a1) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g (CHead c0 (Bind Abst) u) t0 a2)).(\lambda (H3: ((\forall (i: nat).(\forall (b: A).((aprem i a2 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d (CHead c0 (Bind Abst) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (i: nat).(\lambda (b: A).(\lambda (H4: (aprem i (AHead a1 a2) b)).((match i return (\lambda (_: ?).(\lambda (n: nat).((aprem n (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus n j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))) with [O \Rightarrow (\lambda (H5: (aprem O (AHead a1 a2) b)).(let H6 \def (match H5 return (\lambda (_: ?).(\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).((eq nat n O) \to ((eq A a (AHead a1 a2)) \to ((eq A a0 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))))) with [(aprem_zero a0 a3) \Rightarrow (\lambda (_: (eq nat O O)).(\lambda (H5: (eq A (AHead a0 a3) (AHead a1 a2))).(\lambda (H6: (eq A a0 b)).((let H7 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H5) in ((let H8 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H5) in (eq_ind A a1 (\lambda (a: A).((eq A a3 a2) \to ((eq A a b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))) (\lambda (H9: (eq A a3 a2)).(eq_ind A a2 (\lambda (_: A).((eq A a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))) (\lambda (H10: (eq A a1 b)).(eq_ind A b (\lambda (_: A).(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))) (eq_ind A a1 (\lambda (a: A).(ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g a))))))) (ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g a1))))) c0 u O (drop_refl c0) H0) b H10) a1 (sym_eq A a1 b H10))) a3 (sym_eq A a3 a2 H9))) a0 (sym_eq A a0 a1 H8))) H7)) H6)))) | (aprem_succ a0 a i H4 a3) \Rightarrow (\lambda (H5: (eq nat (S i) O)).(\lambda (H6: (eq A (AHead a3 a0) (AHead a1 a2))).(\lambda (H7: (eq A a b)).((let H8 \def (eq_ind nat (S i) (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H5) in (False_ind ((eq A (AHead a3 a0) (AHead a1 a2)) \to ((eq A a b) \to ((aprem i a0 a) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus O j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))) H8)) H6 H7 H4))))]) in (H6 (refl_equal nat O) (refl_equal A (AHead a1 a2)) (refl_equal A b)))) | (S n) \Rightarrow (\lambda (H5: (aprem (S n) (AHead a1 a2) b)).(let H6 \def (match H5 return (\lambda (_: ?).(\lambda (n0: nat).(\lambda (a: A).(\lambda (a0: A).((eq nat n0 (S n)) \to ((eq A a (AHead a1 a2)) \to ((eq A a0 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))))) with [(aprem_zero a0 a3) \Rightarrow (\lambda (H4: (eq nat O (S n))).(\lambda (H5: (eq A (AHead a0 a3) (AHead a1 a2))).(\lambda (H6: (eq A a0 b)).((let H7 \def (eq_ind nat O (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S n) H4) in (False_ind ((eq A (AHead a0 a3) (AHead a1 a2)) \to ((eq A a0 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))) H7)) H5 H6)))) | (aprem_succ a0 a i H4 a3) \Rightarrow (\lambda (H5: (eq nat (S i) (S n))).(\lambda (H6: (eq A (AHead a3 a0) (AHead a1 a2))).(\lambda (H7: (eq A a b)).((let H8 \def (f_equal nat nat (\lambda (e: nat).(match e return (\lambda (_: ?).nat) with [O \Rightarrow i | (S n) \Rightarrow n])) (S i) (S n) H5) in (eq_ind nat n (\lambda (n0: nat).((eq A (AHead a3 a0) (AHead a1 a2)) \to ((eq A a b) \to ((aprem n0 a0 a) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))))) (\lambda (H9: (eq A (AHead a3 a0) (AHead a1 a2))).(let H10 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ a) \Rightarrow a])) (AHead a3 a0) (AHead a1 a2) H9) in ((let H11 \def (f_equal A A (\lambda (e: A).(match e return (\lambda (_: ?).A) with [(ASort _ _) \Rightarrow a3 | (AHead a _) \Rightarrow a])) (AHead a3 a0) (AHead a1 a2) H9) in (eq_ind A a1 (\lambda (_: A).((eq A a0 a2) \to ((eq A a b) \to ((aprem n a0 a) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))))) (\lambda (H12: (eq A a0 a2)).(eq_ind A a2 (\lambda (a1: A).((eq A a b) \to ((aprem n a1 a) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))) (\lambda (H13: (eq A a b)).(eq_ind A b (\lambda (a1: A).((aprem n a2 a1) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))) (\lambda (H14: (aprem n a2 b)).(let H_x \def (H3 n b H14) in (let H3 \def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus n j) O d (CHead c0 (Bind Abst) u))))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H15: (drop (plus n x2) O x0 (CHead c0 (Bind Abst) u))).(\lambda (H16: (arity g x0 x1 (asucc g b))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus (S n) j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))) x0 x1 x2 (drop_S Abst x0 c0 u (plus n x2) H15) H16)))))) H3)))) a (sym_eq A a b H13))) a0 (sym_eq A a0 a2 H12))) a3 (sym_eq A a3 a1 H11))) H10))) i (sym_eq nat i n H8))) H6 H7 H4))))]) in (H6 (refl_equal nat (S n)) (refl_equal A (AHead a1 a2)) (refl_equal A b))))]) H4))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a1: A).(\lambda (_: (arity g c0 u a1)).(\lambda (_: ((\forall (i: nat).(\forall (b: A).((aprem i a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (t0: T).(\lambda (a2: A).(\lambda (_: (arity g c0 t0 (AHead a1 a2))).(\lambda (H3: ((\forall (i: nat).(\forall (b: A).((aprem i (AHead a1 a2) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (i: nat).(\lambda (b: A).(\lambda (H4: (aprem i a2 b)).(let H5 \def (H3 (S i) b (aprem_succ a2 b i H4 a1)) in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (S (plus i j)) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H6: (drop (S (plus i x2)) O x0 c0)).(\lambda (H7: (arity g x0 x1 (asucc g b))).(C_ind (\lambda (c1: C).((drop (S (plus i x2)) O c1 c0) \to ((arity g c1 x1 (asucc g b)) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))))))) (\lambda (n: nat).(\lambda (H8: (drop (S (plus i x2)) O (CSort n) c0)).(\lambda (_: (arity g (CSort n) x1 (asucc g b))).(and3_ind (eq C c0 (CSort n)) (eq nat (S (plus i x2)) O) (eq nat O O) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))) (\lambda (_: (eq C c0 (CSort n))).(\lambda (H11: (eq nat (S (plus i x2)) O)).(\lambda (_: (eq nat O O)).(let H13 \def (eq_ind nat (S (plus i x2)) (\lambda (ee: nat).(match ee return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H11) in (False_ind (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))) H13))))) (drop_gen_sort n (S (plus i x2)) O c0 H8))))) (\lambda (d: C).(\lambda (IHd: (((drop (S (plus i x2)) O d c0) \to ((arity g d x1 (asucc g b)) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))))))).(\lambda (k: K).(\lambda (t1: T).(\lambda (H8: (drop (S (plus i x2)) O (CHead d k t1) c0)).(\lambda (H9: (arity g (CHead d k t1) x1 (asucc g b))).((match k return (\lambda (_: ?).(\lambda (k0: K).((arity g (CHead d k0 t1) x1 (asucc g b)) \to ((drop (r k0 (plus i x2)) O d c0) \to (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))))))) with [(Bind b0) \Rightarrow (\lambda (H10: (arity g (CHead d (Bind b0) t1) x1 (asucc g b))).(\lambda (H11: (drop (r (Bind b0) (plus i x2)) O d c0)).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) (CHead d (Bind b0) t1) x1 (S x2) (eq_ind nat (S (plus i x2)) (\lambda (n: nat).(drop n O (CHead d (Bind b0) t1) c0)) (drop_drop (Bind b0) (plus i x2) d c0 H11 t1) (plus i (S x2)) (plus_n_Sm i x2)) H10))) | (Flat f) \Rightarrow (\lambda (H10: (arity g (CHead d (Flat f) t1) x1 (asucc g b))).(\lambda (H11: (drop (r (Flat f) (plus i x2)) O d c0)).(let H12 \def (IHd H11 (arity_cimp_conf g (CHead d (Flat f) t1) x1 (asucc g b) H10 d (cimp_flat_sx f d t1))) in (ex2_3_ind C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: nat).(\lambda (H13: (drop (plus i x5) O x3 c0)).(\lambda (H14: (arity g x3 x4 (asucc g b))).(ex2_3_intro C T nat (\lambda (d0: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d0 c0)))) (\lambda (d0: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d0 u0 (asucc g b))))) x3 x4 x5 H13 H14)))))) H12))))]) H9 (drop_gen_drop k d c0 t1 (plus i x2) H8)))))))) x0 H6 H7)))))) H5)))))))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (a0: A).(\lambda (_: (arity g c0 u (asucc g a0))).(\lambda (_: ((\forall (i: nat).(\forall (b: A).((aprem i (asucc g a0) b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (t0: T).(\lambda (_: (arity g c0 t0 a0)).(\lambda (H3: ((\forall (i: nat).(\forall (b: A).((aprem i a0 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (i: nat).(\lambda (b: A).(\lambda (H4: (aprem i a0 b)).(let H_x \def (H3 i b H4) in (let H5 \def H_x in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H6: (drop (plus i x2) O x0 c0)).(\lambda (H7: (arity g x0 x1 (asucc g b))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u0: T).(\lambda (_: nat).(arity g d u0 (asucc g b))))) x0 x1 x2 H6 H7)))))) H5)))))))))))))) (\lambda (c0: C).(\lambda (t0: T).(\lambda (a1: A).(\lambda (_: (arity g c0 t0 a1)).(\lambda (H1: ((\forall (i: nat).(\forall (b: A).((aprem i a1 b) \to (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))))))))).(\lambda (a2: A).(\lambda (H2: (leq g a1 a2)).(\lambda (i: nat).(\lambda (b: A).(\lambda (H3: (aprem i a2 b)).(let H_x \def (aprem_repl g a1 a2 H2 i b H3) in (let H4 \def H_x in (ex2_ind A (\lambda (b1: A).(leq g b1 b)) (\lambda (b1: A).(aprem i a1 b1)) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))) (\lambda (x: A).(\lambda (H5: (leq g x b)).(\lambda (H6: (aprem i a1 x)).(let H_x0 \def (H1 i x H6) in (let H7 \def H_x0 in (ex2_3_ind C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g x))))) (ex2_3 C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: nat).(\lambda (H8: (drop (plus i x2) O x0 c0)).(\lambda (H9: (arity g x0 x1 (asucc g x))).(ex2_3_intro C T nat (\lambda (d: C).(\lambda (_: T).(\lambda (j: nat).(drop (plus i j) O d c0)))) (\lambda (d: C).(\lambda (u: T).(\lambda (_: nat).(arity g d u (asucc g b))))) x0 x1 x2 H8 (arity_repl g x0 x1 (asucc g x) H9 (asucc g b) (asucc_repl g x b H5)))))))) H7)))))) H4))))))))))))) c t a H))))).
 
 theorem arity_appls_cast:
  \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t: T).(\forall (vs: TList).(\forall (a: A).((arity g c (THeads (Flat Appl) vs u) (asucc g a)) \to ((arity g c (THeads (Flat Appl) vs t) a) \to (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) a))))))))
@@ -1792,27 +1792,27 @@ inductive pr0: T \to (T \to Prop) \def
 theorem pr0_gen_sort:
  \forall (x: T).(\forall (n: nat).((pr0 (TSort n) x) \to (eq T x (TSort n))))
 \def
- \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TSort n) x)).(let H0 \def (match H return (\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n)) \to ((eq T t0 x) \to (eq T x (TSort n)))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (TSort n))).(\lambda (H1: (eq T t x)).(eq_ind T (TSort n) (\lambda (t0: T).((eq T t0 x) \to (eq T x (TSort n)))) (\lambda (H2: (eq T (TSort n) x)).(eq_ind T (TSort n) (\lambda (t0: T).(eq T t0 (TSort n))) (refl_equal T (TSort n)) x H2)) t (sym_eq T t (TSort n) H0) H1))) | (pr0_comp u1 u2 H0 t1 t2 H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u1 t1) (TSort n))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H2) in (False_ind ((eq T (THead k u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to (eq T x (TSort n))))) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t1 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) (TSort n))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t1 t2) \to (eq T x (TSort n))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u1 u2 H2 t1 t2 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (TSort n))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to (eq T x (TSort n))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u1 u2 H0 t1 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u1 t1) (TSort n))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to ((subst0 O u2 t2 w) \to (eq T x (TSort n)))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t1 t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t1)) (TSort n))).(\lambda (H3: (eq T t2 x)).((let H4 \def (eq_ind T (THead (Bind b) u (lift (S O) O t1)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H2) in (False_ind ((eq T t2 x) \to ((not (eq B b Abst)) \to ((pr0 t1 t2) \to (eq T x (TSort n))))) H4)) H3 H0 H1))) | (pr0_epsilon t1 t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t1) (TSort n))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H1) in (False_ind ((eq T t2 x) \to ((pr0 t1 t2) \to (eq T x (TSort n)))) H3)) H2 H0)))]) in (H0 (refl_equal T (TSort n)) (refl_equal T x))))).
+ \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TSort n) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (t: T).(\lambda (t0: T).((eq T t (TSort n)) \to ((eq T t0 x) \to (eq T x (TSort n))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (TSort n))).(\lambda (H1: (eq T t x)).(eq_ind T (TSort n) (\lambda (t0: T).((eq T t0 x) \to (eq T x (TSort n)))) (\lambda (H2: (eq T (TSort n) x)).(eq_ind T (TSort n) (\lambda (t0: T).(eq T t0 (TSort n))) (refl_equal T (TSort n)) x H2)) t (sym_eq T t (TSort n) H0) H1))) | (pr0_comp u1 u2 H0 t1 t2 H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u1 t1) (TSort n))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H2) in (False_ind ((eq T (THead k u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to (eq T x (TSort n))))) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t1 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) (TSort n))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t1 t2) \to (eq T x (TSort n))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u1 u2 H2 t1 t2 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (TSort n))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to (eq T x (TSort n))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u1 u2 H0 t1 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u1 t1) (TSort n))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to ((subst0 O u2 t2 w) \to (eq T x (TSort n)))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t1 t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t1)) (TSort n))).(\lambda (H3: (eq T t2 x)).((let H4 \def (eq_ind T (THead (Bind b) u (lift (S O) O t1)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H2) in (False_ind ((eq T t2 x) \to ((not (eq B b Abst)) \to ((pr0 t1 t2) \to (eq T x (TSort n))))) H4)) H3 H0 H1))) | (pr0_epsilon t1 t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t1) (TSort n))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort n) H1) in (False_ind ((eq T t2 x) \to ((pr0 t1 t2) \to (eq T x (TSort n)))) H3)) H2 H0)))]) in (H0 (refl_equal T (TSort n)) (refl_equal T x))))).
 
 theorem pr0_gen_lref:
  \forall (x: T).(\forall (n: nat).((pr0 (TLRef n) x) \to (eq T x (TLRef n))))
 \def
- \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TLRef n) x)).(let H0 \def (match H return (\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to ((eq T t0 x) \to (eq T x (TLRef n)))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (TLRef n))).(\lambda (H1: (eq T t x)).(eq_ind T (TLRef n) (\lambda (t0: T).((eq T t0 x) \to (eq T x (TLRef n)))) (\lambda (H2: (eq T (TLRef n) x)).(eq_ind T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) x H2)) t (sym_eq T t (TLRef n) H0) H1))) | (pr0_comp u1 u2 H0 t1 t2 H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u1 t1) (TLRef n))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in (False_ind ((eq T (THead k u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to (eq T x (TLRef n))))) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t1 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) (TLRef n))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t1 t2) \to (eq T x (TLRef n))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u1 u2 H2 t1 t2 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (TLRef n))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to (eq T x (TLRef n))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u1 u2 H0 t1 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u1 t1) (TLRef n))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to ((subst0 O u2 t2 w) \to (eq T x (TLRef n)))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t1 t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t1)) (TLRef n))).(\lambda (H3: (eq T t2 x)).((let H4 \def (eq_ind T (THead (Bind b) u (lift (S O) O t1)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in (False_ind ((eq T t2 x) \to ((not (eq B b Abst)) \to ((pr0 t1 t2) \to (eq T x (TLRef n))))) H4)) H3 H0 H1))) | (pr0_epsilon t1 t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t1) (TLRef n))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind ((eq T t2 x) \to ((pr0 t1 t2) \to (eq T x (TLRef n)))) H3)) H2 H0)))]) in (H0 (refl_equal T (TLRef n)) (refl_equal T x))))).
+ \lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr0 (TLRef n) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (t: T).(\lambda (t0: T).((eq T t (TLRef n)) \to ((eq T t0 x) \to (eq T x (TLRef n))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (TLRef n))).(\lambda (H1: (eq T t x)).(eq_ind T (TLRef n) (\lambda (t0: T).((eq T t0 x) \to (eq T x (TLRef n)))) (\lambda (H2: (eq T (TLRef n) x)).(eq_ind T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) x H2)) t (sym_eq T t (TLRef n) H0) H1))) | (pr0_comp u1 u2 H0 t1 t2 H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u1 t1) (TLRef n))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def (eq_ind T (THead k u1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in (False_ind ((eq T (THead k u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to (eq T x (TLRef n))))) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t1 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) (TLRef n))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t1)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t1 t2) \to (eq T x (TLRef n))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u1 u2 H2 t1 t2 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (TLRef n))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to (eq T x (TLRef n))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u1 u2 H0 t1 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u1 t1) (TLRef n))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u1 u2) \to ((pr0 t1 t2) \to ((subst0 O u2 t2 w) \to (eq T x (TLRef n)))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t1 t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t1)) (TLRef n))).(\lambda (H3: (eq T t2 x)).((let H4 \def (eq_ind T (THead (Bind b) u (lift (S O) O t1)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in (False_ind ((eq T t2 x) \to ((not (eq B b Abst)) \to ((pr0 t1 t2) \to (eq T x (TLRef n))))) H4)) H3 H0 H1))) | (pr0_epsilon t1 t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t1) (TLRef n))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind ((eq T t2 x) \to ((pr0 t1 t2) \to (eq T x (TLRef n)))) H3)) H2 H0)))]) in (H0 (refl_equal T (TLRef n)) (refl_equal T x))))).
 
 theorem pr0_gen_abst:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))))))
 \def
- \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead (Bind Abst) u1 t1) x)).(let H0 \def (match H return (\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Bind Abst) u1 t1)) \to ((eq T t0 x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Bind Abst) u1 t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Bind Abst) u1 t1) (\lambda (t0: T).((eq T t0 x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))))) (\lambda (H2: (eq T (THead (Bind Abst) u1 t1) x)).(eq_ind T (THead (Bind Abst) u1 t1) (\lambda (t0: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Bind Abst) u1 t1)) (pr0_refl u1) (pr0_refl t1)) x H2)) t (sym_eq T t (THead (Bind Abst) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abst) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abst) u1 t1) H2) in ((let H6 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u0 t0) (THead (Bind Abst) u1 t1) H2) in (eq_ind K (Bind Abst) (\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k0 u2 t2) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))))))) (\lambda (H7: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Bind Abst) u2 t2) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))))))) (\lambda (H8: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Bind Abst) u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))))) (\lambda (H9: (eq T (THead (Bind Abst) u2 t2) x)).(eq_ind T (THead (Bind Abst) u2 t2) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))))) (\lambda (H10: (pr0 u1 u2)).(\lambda (H11: (pr0 t1 t2)).(ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abst) u2 t2) (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Bind Abst) u2 t2)) H10 H11))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) k (sym_eq K k (Bind Abst) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1 t1) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Bind Abst) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1 t1) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abst) u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind Abst) u1 t1) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1) H2) in ((let H6 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1) H2) in (eq_ind B Abst (\lambda (b0: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not (eq B b0 Abst)) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))))))) (\lambda (H7: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not (eq B Abst Abst)) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))))))) (\lambda (H8: (eq T (lift (S O) O t0) t1)).(eq_ind T (lift (S O) O t0) (\lambda (t: T).((eq T t2 x) \to ((not (eq B Abst Abst)) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))))))) (\lambda (H9: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((not (eq B Abst Abst)) \to ((pr0 t0 t) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift (S O) O t0) t3))))))) (\lambda (H10: (not (eq B Abst Abst))).(\lambda (_: (pr0 t0 x)).(False_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift (S O) O t0) t3)))) (H10 (refl_equal B Abst))))) t2 (sym_eq T t2 x H9))) t1 H8)) u (sym_eq T u u1 H7))) b (sym_eq B b Abst H6))) H5)) H4)) H3 H0 H1))) | (pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Bind Abst) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1 t1) H1) in (False_ind ((eq T t2 x) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))) H3)) H2 H0)))]) in (H0 (refl_equal T (THead (Bind Abst) u1 t1)) (refl_equal T x)))))).
+ \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead (Bind Abst) u1 t1) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Bind Abst) u1 t1)) \to ((eq T t0 x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Bind Abst) u1 t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Bind Abst) u1 t1) (\lambda (t0: T).((eq T t0 x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))))) (\lambda (H2: (eq T (THead (Bind Abst) u1 t1) x)).(eq_ind T (THead (Bind Abst) u1 t1) (\lambda (t0: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abst) u1 t1) (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Bind Abst) u1 t1)) (pr0_refl u1) (pr0_refl t1)) x H2)) t (sym_eq T t (THead (Bind Abst) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abst) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abst) u1 t1) H2) in ((let H6 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u0 t0) (THead (Bind Abst) u1 t1) H2) in (eq_ind K (Bind Abst) (\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k0 u2 t2) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))))))) (\lambda (H7: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Bind Abst) u2 t2) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))))))) (\lambda (H8: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Bind Abst) u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))))) (\lambda (H9: (eq T (THead (Bind Abst) u2 t2) x)).(eq_ind T (THead (Bind Abst) u2 t2) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))))) (\lambda (H10: (pr0 u1 u2)).(\lambda (H11: (pr0 t1 t2)).(ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abst) u2 t2) (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Bind Abst) u2 t2)) H10 H11))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) k (sym_eq K k (Bind Abst) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1 t1) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Bind Abst) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1 t1) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abst) u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind Abst) u1 t1) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1) H2) in ((let H6 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1) H2) in (eq_ind B Abst (\lambda (b0: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not (eq B b0 Abst)) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))))))) (\lambda (H7: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not (eq B Abst Abst)) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))))))))) (\lambda (H8: (eq T (lift (S O) O t0) t1)).(eq_ind T (lift (S O) O t0) (\lambda (t: T).((eq T t2 x) \to ((not (eq B Abst Abst)) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))))))) (\lambda (H9: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((not (eq B Abst Abst)) \to ((pr0 t0 t) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift (S O) O t0) t3))))))) (\lambda (H10: (not (eq B Abst Abst))).(\lambda (_: (pr0 t0 x)).(False_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift (S O) O t0) t3)))) (H10 (refl_equal B Abst))))) t2 (sym_eq T t2 x H9))) t1 H8)) u (sym_eq T u u1 H7))) b (sym_eq B b Abst H6))) H5)) H4)) H3 H0 H1))) | (pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Bind Abst) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1 t1) H1) in (False_ind ((eq T t2 x) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))))) H3)) H2 H0)))]) in (H0 (refl_equal T (THead (Bind Abst) u1 t1)) (refl_equal T x)))))).
 
 theorem pr0_gen_appl:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))))))))
 \def
- \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead (Flat Appl) u1 t1) x)).(let H0 \def (match H return (\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl) u1 t1)) \to ((eq T t0 x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Flat Appl) u1 t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Flat Appl) u1 t1) (\lambda (t0: T).((eq T t0 x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))))))) (\lambda (H2: (eq T (THead (Flat Appl) u1 t1) x)).(eq_ind T (THead (Flat Appl) u1 t1) (\lambda (t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Flat Appl) u1 t1)) (pr0_refl u1) (pr0_refl t1))) x H2)) t (sym_eq T t (THead (Flat Appl) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Flat Appl) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Flat Appl) u1 t1) H2) in ((let H6 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u0 t0) (THead (Flat Appl) u1 t1) H2) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k0 u2 t2) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))))))))) (\lambda (H7: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Flat Appl) u2 t2) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))))) (\lambda (H8: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Flat Appl) u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))))))) (\lambda (H9: (eq T (THead (Flat Appl) u2 t2) x)).(eq_ind T (THead (Flat Appl) u2 t2) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))) (\lambda (H10: (pr0 u1 u2)).(\lambda (H11: (pr0 t1 t2)).(or3_intro0 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Flat Appl) u2 t2)) H10 H11)))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) k (sym_eq K k (Flat Appl) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow (THead (Bind Abst) u t0) | (TLRef _) \Rightarrow (THead (Bind Abst) u t0) | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) u1 t1) H2) in (eq_ind T u1 (\lambda (t: T).((eq T (THead (Bind Abst) u t0) t1) \to ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 t v2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T x (THead (Bind b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))))) (\lambda (H6: (eq T (THead (Bind Abst) u t0) t1)).(eq_ind T (THead (Bind Abst) u t0) (\lambda (t: T).((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 u1 v2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T x (THead (Bind b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))))))) (\lambda (H7: (eq T (THead (Bind Abbr) v2 t2) x)).(eq_ind T (THead (Bind Abbr) v2 t2) (\lambda (t: T).((pr0 u1 v2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (THead (Bind Abst) u t0) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t (THead (Bind b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))) (\lambda (H8: (pr0 u1 v2)).(\lambda (H9: (pr0 t0 t2)).(or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (THead (Bind Abst) u t0) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) u t0 v2 t2 (refl_equal T (THead (Bind Abst) u t0)) (refl_equal T (THead (Bind Abbr) v2 t2)) H8 H9)))) x H7)) t1 H6)) v1 (sym_eq T v1 u1 H5))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow (THead (Bind b) u0 t0) | (TLRef _) \Rightarrow (THead (Bind b) u0 t0) | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1) H4) in (eq_ind T u1 (\lambda (t: T).((eq T (THead (Bind b) u0 t0) t1) \to ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 t v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T x (THead (Bind b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))))))) (\lambda (H8: (eq T (THead (Bind b) u0 t0) t1)).(eq_ind T (THead (Bind b) u0 t0) (\lambda (t: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 u1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T x (THead (Bind b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))))))))) (\lambda (H9: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 u1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 (THead (Bind b) u0 t0) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t (THead (Bind b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))))) (\lambda (H10: (not (eq B b Abst))).(\lambda (H11: (pr0 u1 v2)).(\lambda (H12: (pr0 u0 u2)).(\lambda (H13: (pr0 t0 t2)).(or3_intro2 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 (THead (Bind b) u0 t0) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (ex6_6_intro B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))) b u0 t0 v2 u2 t2 H10 (refl_equal T (THead (Bind b) u0 t0)) (refl_equal T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2))) H11 H12 H13)))))) x H9)) t1 H8)) v1 (sym_eq T v1 u1 H7))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Flat Appl) u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1 t1) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1 t1) H2) in (False_ind ((eq T t2 x) \to ((not (eq B b Abst)) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b0) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))) H4)) H3 H0 H1))) | (pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Flat Appl) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u1 t1) H1) in (False_ind ((eq T t2 x) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H3)) H2 H0)))]) in (H0 (refl_equal T (THead (Flat Appl) u1 t1)) (refl_equal T x)))))).
+ \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead (Flat Appl) u1 t1) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl) u1 t1)) \to ((eq T t0 x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Flat Appl) u1 t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Flat Appl) u1 t1) (\lambda (t0: T).((eq T t0 x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))))))) (\lambda (H2: (eq T (THead (Flat Appl) u1 t1) x)).(eq_ind T (THead (Flat Appl) u1 t1) (\lambda (t0: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T t0 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u1 t1) (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Flat Appl) u1 t1)) (pr0_refl u1) (pr0_refl t1))) x H2)) t (sym_eq T t (THead (Flat Appl) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Flat Appl) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Flat Appl) u1 t1) H2) in ((let H6 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u0 t0) (THead (Flat Appl) u1 t1) H2) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k0 u2 t2) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))))))))) (\lambda (H7: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Flat Appl) u2 t2) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))))) (\lambda (H8: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Flat Appl) u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))))))) (\lambda (H9: (eq T (THead (Flat Appl) u2 t2) x)).(eq_ind T (THead (Flat Appl) u2 t2) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))) (\lambda (H10: (pr0 u1 u2)).(\lambda (H11: (pr0 t1 t2)).(or3_intro0 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Flat Appl) u2 t2) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Flat Appl) u2 t2)) H10 H11)))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) k (sym_eq K k (Flat Appl) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow (THead (Bind Abst) u t0) | (TLRef _) \Rightarrow (THead (Bind Abst) u t0) | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) u1 t1) H2) in (eq_ind T u1 (\lambda (t: T).((eq T (THead (Bind Abst) u t0) t1) \to ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 t v2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T x (THead (Bind b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))))) (\lambda (H6: (eq T (THead (Bind Abst) u t0) t1)).(eq_ind T (THead (Bind Abst) u t0) (\lambda (t: T).((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 u1 v2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T x (THead (Bind b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))))))) (\lambda (H7: (eq T (THead (Bind Abbr) v2 t2) x)).(eq_ind T (THead (Bind Abbr) v2 t2) (\lambda (t: T).((pr0 u1 v2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (THead (Bind Abst) u t0) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t (THead (Bind b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))) (\lambda (H8: (pr0 u1 v2)).(\lambda (H9: (pr0 t0 t2)).(or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (THead (Bind Abst) u t0) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind b) v3 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) u t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) v2 t2) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) u t0 v2 t2 (refl_equal T (THead (Bind Abst) u t0)) (refl_equal T (THead (Bind Abbr) v2 t2)) H8 H9)))) x H7)) t1 H6)) v1 (sym_eq T v1 u1 H5))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow (THead (Bind b) u0 t0) | (TLRef _) \Rightarrow (THead (Bind b) u0 t0) | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1) H4) in ((let H7 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Appl) u1 t1) H4) in (eq_ind T u1 (\lambda (t: T).((eq T (THead (Bind b) u0 t0) t1) \to ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 t v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T x (THead (Bind b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))))))) (\lambda (H8: (eq T (THead (Bind b) u0 t0) t1)).(eq_ind T (THead (Bind b) u0 t0) (\lambda (t: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 u1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T x (THead (Bind b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))))))))) (\lambda (H9: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 u1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 (THead (Bind b) u0 t0) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T t (THead (Bind b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))))) (\lambda (H10: (not (eq B b Abst))).(\lambda (H11: (pr0 u1 v2)).(\lambda (H12: (pr0 u0 u2)).(\lambda (H13: (pr0 t0 t2)).(or3_intro2 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 (THead (Bind b) u0 t0) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (ex6_6_intro B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind b) u0 t0) (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v3: T).(\lambda (t3: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) (THead (Bind b0) v3 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))) b u0 t0 v2 u2 t2 H10 (refl_equal T (THead (Bind b) u0 t0)) (refl_equal T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2))) H11 H12 H13)))))) x H9)) t1 H8)) v1 (sym_eq T v1 u1 H7))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Flat Appl) u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1 t1) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u3))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1 t1) H2) in (False_ind ((eq T t2 x) \to ((not (eq B b Abst)) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b0 Abst)))))))) (\lambda (b0: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b0) y1 z1)))))))) (\lambda (b0: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b0) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))))))) H4)) H3 H0 H1))) | (pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Flat Appl) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) u1 t1) H1) in (False_ind ((eq T t2 x) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))))))) H3)) H2 H0)))]) in (H0 (refl_equal T (THead (Flat Appl) u1 t1)) (refl_equal T x)))))).
 
 theorem pr0_gen_cast:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x)))))
 \def
- \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead (Flat Cast) u1 t1) x)).(let H0 \def (match H return (\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Cast) u1 t1)) \to ((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x)))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Flat Cast) u1 t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Flat Cast) u1 t1) (\lambda (t0: T).((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x)))) (\lambda (H2: (eq T (THead (Flat Cast) u1 t1) x)).(eq_ind T (THead (Flat Cast) u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t0))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) u1 t1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead (Flat Cast) u1 t1)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) u1 t1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Flat Cast) u1 t1)) (pr0_refl u1) (pr0_refl t1))) x H2)) t (sym_eq T t (THead (Flat Cast) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in ((let H6 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in (eq_ind K (Flat Cast) (\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k0 u2 t2) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x)))))))) (\lambda (H7: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Flat Cast) u2 t2) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x))))))) (\lambda (H8: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Flat Cast) u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x)))))) (\lambda (H9: (eq T (THead (Flat Cast) u2 t2) x)).(eq_ind T (THead (Flat Cast) u2 t2) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t))))) (\lambda (H10: (pr0 u1 u2)).(\lambda (H11: (pr0 t1 t2)).(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Flat Cast) u2 t2)) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Flat Cast) u2 t2)) H10 H11)))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) k (sym_eq K k (Flat Cast) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Cast) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Flat Cast) u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u1 t1) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x)))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u1 t1) H2) in (False_ind ((eq T t2 x) \to ((not (eq B b Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x))))) H4)) H3 H0 H1))) | (pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Flat Cast) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) u t0) (THead (Flat Cast) u1 t1) H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u t0) (THead (Flat Cast) u1 t1) H1) in (eq_ind T u1 (\lambda (_: T).((eq T t0 t1) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x)))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x))))) (\lambda (H6: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 t1 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x)))) (\lambda (H7: (pr0 t1 x)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x) H7)) t2 (sym_eq T t2 x H6))) t0 (sym_eq T t0 t1 H5))) u (sym_eq T u u1 H4))) H3)) H2 H0)))]) in (H0 (refl_equal T (THead (Flat Cast) u1 t1)) (refl_equal T x)))))).
+ \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead (Flat Cast) u1 t1) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Cast) u1 t1)) \to ((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Flat Cast) u1 t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Flat Cast) u1 t1) (\lambda (t0: T).((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 x)))) (\lambda (H2: (eq T (THead (Flat Cast) u1 t1) x)).(eq_ind T (THead (Flat Cast) u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t0))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) u1 t1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead (Flat Cast) u1 t1)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Flat Cast) u1 t1) (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Flat Cast) u1 t1)) (pr0_refl u1) (pr0_refl t1))) x H2)) t (sym_eq T t (THead (Flat Cast) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in ((let H6 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in (eq_ind K (Flat Cast) (\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k0 u2 t2) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x)))))))) (\lambda (H7: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Flat Cast) u2 t2) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x))))))) (\lambda (H8: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Flat Cast) u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x)))))) (\lambda (H9: (eq T (THead (Flat Cast) u2 t2) x)).(eq_ind T (THead (Flat Cast) u2 t2) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t))))) (\lambda (H10: (pr0 u1 u2)).(\lambda (H11: (pr0 t1 t2)).(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (THead (Flat Cast) u2 t2)) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Flat Cast) u2 t2) (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Flat Cast) u2 t2)) H10 H11)))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) k (sym_eq K k (Flat Cast) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Flat Cast) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Flat Cast) u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u1 t1) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x)))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u1 t1) H2) in (False_ind ((eq T t2 x) \to ((not (eq B b Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x))))) H4)) H3 H0 H1))) | (pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Flat Cast) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) u t0) (THead (Flat Cast) u1 t1) H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u t0) (THead (Flat Cast) u1 t1) H1) in (eq_ind T u1 (\lambda (_: T).((eq T t0 t1) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x)))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x))))) (\lambda (H6: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 t1 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x)))) (\lambda (H7: (pr0 t1 x)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x) H7)) t2 (sym_eq T t2 x H6))) t0 (sym_eq T t0 t1 H5))) u (sym_eq T u u1 H4))) H3)) H2 H0)))]) in (H0 (refl_equal T (THead (Flat Cast) u1 t1)) (refl_equal T x)))))).
 
 theorem pr0_lift:
  \forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (h: nat).(\forall (d: nat).(pr0 (lift h d t1) (lift h d t2))))))
@@ -1822,12 +1822,12 @@ theorem pr0_lift:
 theorem pr0_gen_abbr:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O x))))))
 \def
- \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead (Bind Abbr) u1 t1) x)).(let H0 \def (match H return (\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to ((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O x))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Bind Abbr) u1 t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (t0: T).((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O x))))) (\lambda (H2: (eq T (THead (Bind Abbr) u1 t1) x)).(eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u1 t1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2)))))) u1 t1 (refl_equal T (THead (Bind Abbr) u1 t1)) (pr0_refl u1) (or_introl (pr0 t1 t1) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u1 y t1))) (pr0_refl t1)))) x H2)) t (sym_eq T t (THead (Bind Abbr) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) H2) in ((let H6 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) H2) in (eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k0 u2 t2) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Bind Abbr) u2 t2) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Bind Abbr) u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x))))))) (\lambda (H9: (eq T (THead (Bind Abbr) u2 t2) x)).(eq_ind T (THead (Bind Abbr) u2 t2) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O t)))))) (\lambda (H10: (pr0 u1 u2)).(\lambda (H11: (pr0 t1 t2)).(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 t2) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u2 t2))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 t2) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3)))))) u2 t2 (refl_equal T (THead (Bind Abbr) u2 t2)) H10 (or_introl (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))) H11))))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) k (sym_eq K k (Bind Abbr) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t1 (lift (S O) O x)))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Bind Abbr) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x)))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H3) in ((let H6 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H3) in (eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T (THead (Bind Abbr) u2 w) x)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O t))))))) (\lambda (H9: (pr0 u1 u2)).(\lambda (H10: (pr0 t1 t2)).(\lambda (H11: (subst0 O u2 t2 w)).(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u2 w))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3)))))) u2 w (refl_equal T (THead (Bind Abbr) u2 w)) H9 (or_intror (pr0 t1 w) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y w))) (ex_intro2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y w)) t2 H10 H11))))))) x H8)) t0 (sym_eq T t0 t1 H7))) u0 (sym_eq T u0 u1 H6))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H2) in ((let H6 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H2) in (eq_ind B Abbr (\lambda (b0: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not (eq B b0 Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not (eq B Abbr Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T (lift (S O) O t0) t1)).(eq_ind T (lift (S O) O t0) (\lambda (t: T).((eq T t2 x) \to ((not (eq B Abbr Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t t3) (ex2 T (\lambda (y: T).(pr0 t y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t (lift (S O) O x))))))) (\lambda (H9: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((not (eq B Abbr Abst)) \to ((pr0 t0 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y: T).(pr0 (lift (S O) O t0) y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 (lift (S O) O t0) (lift (S O) O x)))))) (\lambda (_: (not (eq B Abbr Abst))).(\lambda (H11: (pr0 t0 x)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y: T).(pr0 (lift (S O) O t0) y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 (lift (S O) O t0) (lift (S O) O x)) (pr0_lift t0 x H11 (S O) O)))) t2 (sym_eq T t2 x H9))) t1 H8)) u (sym_eq T u u1 H7))) b (sym_eq B b Abbr H6))) H5)) H4)) H3 H0 H1))) | (pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Bind Abbr) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H1) in (False_ind ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t1 (lift (S O) O x))))) H3)) H2 H0)))]) in (H0 (refl_equal T (THead (Bind Abbr) u1 t1)) (refl_equal T x)))))).
+ \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead (Bind Abbr) u1 t1) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Bind Abbr) u1 t1)) \to ((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O x)))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Bind Abbr) u1 t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (t0: T).((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O x))))) (\lambda (H2: (eq T (THead (Bind Abbr) u1 t1) x)).(eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u1 t1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2)))))) u1 t1 (refl_equal T (THead (Bind Abbr) u1 t1)) (pr0_refl u1) (or_introl (pr0 t1 t1) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u1 y t1))) (pr0_refl t1)))) x H2)) t (sym_eq T t (THead (Bind Abbr) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) H2) in ((let H6 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u0 t0) (THead (Bind Abbr) u1 t1) H2) in (eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k0 u2 t2) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Bind Abbr) u2 t2) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Bind Abbr) u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x))))))) (\lambda (H9: (eq T (THead (Bind Abbr) u2 t2) x)).(eq_ind T (THead (Bind Abbr) u2 t2) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O t)))))) (\lambda (H10: (pr0 u1 u2)).(\lambda (H11: (pr0 t1 t2)).(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 t2) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u2 t2))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 t2) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3)))))) u2 t2 (refl_equal T (THead (Bind Abbr) u2 t2)) H10 (or_introl (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2))) H11))))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) k (sym_eq K k (Bind Abbr) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t1 (lift (S O) O x)))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Bind Abbr) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x)))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H3) in ((let H6 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead (Bind Abbr) u0 t0) (THead (Bind Abbr) u1 t1) H3) in (eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T (THead (Bind Abbr) u2 w) x)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O t))))))) (\lambda (H9: (pr0 u1 u2)).(\lambda (H10: (pr0 t1 t2)).(\lambda (H11: (subst0 O u2 t2 w)).(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3))))))) (pr0 t1 (lift (S O) O (THead (Bind Abbr) u2 w))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (u3: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u3 y t3)))))) u2 w (refl_equal T (THead (Bind Abbr) u2 w)) H9 (or_intror (pr0 t1 w) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y w))) (ex_intro2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y w)) t2 H10 H11))))))) x H8)) t0 (sym_eq T t0 t1 H7))) u0 (sym_eq T u0 u1 H6))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H2) in ((let H6 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t1) H2) in (eq_ind B Abbr (\lambda (b0: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not (eq B b0 Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not (eq B Abbr Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T (lift (S O) O t0) t1)).(eq_ind T (lift (S O) O t0) (\lambda (t: T).((eq T t2 x) \to ((not (eq B Abbr Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t t3) (ex2 T (\lambda (y: T).(pr0 t y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t (lift (S O) O x))))))) (\lambda (H9: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((not (eq B Abbr Abst)) \to ((pr0 t0 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y: T).(pr0 (lift (S O) O t0) y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 (lift (S O) O t0) (lift (S O) O x)))))) (\lambda (_: (not (eq B Abbr Abst))).(\lambda (H11: (pr0 t0 x)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 (lift (S O) O t0) t3) (ex2 T (\lambda (y: T).(pr0 (lift (S O) O t0) y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 (lift (S O) O t0) (lift (S O) O x)) (pr0_lift t0 x H11 (S O) O)))) t2 (sym_eq T t2 x H9))) t1 H8)) u (sym_eq T u u1 H7))) b (sym_eq B b Abbr H6))) H5)) H4)) H3 H0 H1))) | (pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Bind Abbr) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t1) H1) in (False_ind ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t1 (lift (S O) O x))))) H3)) H2 H0)))]) in (H0 (refl_equal T (THead (Bind Abbr) u1 t1)) (refl_equal T x)))))).
 
 theorem pr0_gen_void:
  \forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr0 (THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O x))))))
 \def
- \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead (Bind Void) u1 t1) x)).(let H0 \def (match H return (\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to ((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O x))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Bind Void) u1 t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Bind Void) u1 t1) (\lambda (t0: T).((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O x))))) (\lambda (H2: (eq T (THead (Bind Void) u1 t1) x)).(eq_ind T (THead (Bind Void) u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Void) u1 t1) (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O (THead (Bind Void) u1 t1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Void) u1 t1) (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Bind Void) u1 t1)) (pr0_refl u1) (pr0_refl t1))) x H2)) t (sym_eq T t (THead (Bind Void) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead (Bind Void) u1 t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind Void) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind Void) u1 t1) H2) in ((let H6 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u0 t0) (THead (Bind Void) u1 t1) H2) in (eq_ind K (Bind Void) (\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k0 u2 t2) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Bind Void) u2 t2) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Bind Void) u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x))))))) (\lambda (H9: (eq T (THead (Bind Void) u2 t2) x)).(eq_ind T (THead (Bind Void) u2 t2) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t)))))) (\lambda (H10: (pr0 u1 u2)).(\lambda (H11: (pr0 t1 t2)).(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Void) u2 t2) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead (Bind Void) u2 t2))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Void) u2 t2) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Bind Void) u2 t2)) H10 H11)))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) k (sym_eq K k (Bind Void) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Void) u1 t1))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Void) u1 t1) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x)))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Bind Void) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Void) u1 t1) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x)))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Void) u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind Void) u1 t1) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x))))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 t1) H2) in ((let H6 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 t1) H2) in (eq_ind B Void (\lambda (b0: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not (eq B b0 Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not (eq B Void Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T (lift (S O) O t0) t1)).(eq_ind T (lift (S O) O t0) (\lambda (t: T).((eq T t2 x) \to ((not (eq B Void Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))) (pr0 t (lift (S O) O x))))))) (\lambda (H9: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((not (eq B Void Abst)) \to ((pr0 t0 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift (S O) O t0) t3)))) (pr0 (lift (S O) O t0) (lift (S O) O x)))))) (\lambda (_: (not (eq B Void Abst))).(\lambda (H11: (pr0 t0 x)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift (S O) O t0) t3)))) (pr0 (lift (S O) O t0) (lift (S O) O x)) (pr0_lift t0 x H11 (S O) O)))) t2 (sym_eq T t2 x H9))) t1 H8)) u (sym_eq T u u1 H7))) b (sym_eq B b Void H6))) H5)) H4)) H3 H0 H1))) | (pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Bind Void) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Void) u1 t1) H1) in (False_ind ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x))))) H3)) H2 H0)))]) in (H0 (refl_equal T (THead (Bind Void) u1 t1)) (refl_equal T x)))))).
+ \lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr0 (THead (Bind Void) u1 t1) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Bind Void) u1 t1)) \to ((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O x)))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Bind Void) u1 t1))).(\lambda (H1: (eq T t x)).(eq_ind T (THead (Bind Void) u1 t1) (\lambda (t0: T).((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O x))))) (\lambda (H2: (eq T (THead (Bind Void) u1 t1) x)).(eq_ind T (THead (Bind Void) u1 t1) (\lambda (t0: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t0 (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O t0)))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Void) u1 t1) (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (lift (S O) O (THead (Bind Void) u1 t1))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Void) u1 t1) (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u1 t1 (refl_equal T (THead (Bind Void) u1 t1)) (pr0_refl u1) (pr0_refl t1))) x H2)) t (sym_eq T t (THead (Bind Void) u1 t1) H0) H1))) | (pr0_comp u0 u2 H0 t0 t2 H1 k) \Rightarrow (\lambda (H2: (eq T (THead k u0 t0) (THead (Bind Void) u1 t1))).(\lambda (H3: (eq T (THead k u2 t2) x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Bind Void) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Bind Void) u1 t1) H2) in ((let H6 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u0 t0) (THead (Bind Void) u1 t1) H2) in (eq_ind K (Bind Void) (\lambda (k0: K).((eq T u0 u1) \to ((eq T t0 t1) \to ((eq T (THead k0 u2 t2) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t0 t1) \to ((eq T (THead (Bind Void) u2 t2) x) \to ((pr0 t u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T (THead (Bind Void) u2 t2) x) \to ((pr0 u1 u2) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x))))))) (\lambda (H9: (eq T (THead (Bind Void) u2 t2) x)).(eq_ind T (THead (Bind Void) u2 t2) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t1 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t)))))) (\lambda (H10: (pr0 u1 u2)).(\lambda (H11: (pr0 t1 t2)).(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Void) u2 t2) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O (THead (Bind Void) u2 t2))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t3: T).(eq T (THead (Bind Void) u2 t2) (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) u2 t2 (refl_equal T (THead (Bind Void) u2 t2)) H10 H11)))) x H9)) t0 (sym_eq T t0 t1 H8))) u0 (sym_eq T u0 u1 H7))) k (sym_eq K k (Bind Void) H6))) H5)) H4)) H3 H0 H1))) | (pr0_beta u v1 v2 H0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Void) u1 t1))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t2) x)).((let H4 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Void) u1 t1) H2) in (False_ind ((eq T (THead (Bind Abbr) v2 t2) x) \to ((pr0 v1 v2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x)))))) H4)) H3 H0 H1))) | (pr0_upsilon b H0 v1 v2 H1 u0 u2 H2 t0 t2 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (THead (Bind Void) u1 t1))).(\lambda (H5: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x)).((let H6 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Void) u1 t1) H4) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)) x) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x)))))))) H6)) H5 H0 H1 H2 H3))) | (pr0_delta u0 u2 H0 t0 t2 H1 w H2) \Rightarrow (\lambda (H3: (eq T (THead (Bind Abbr) u0 t0) (THead (Bind Void) u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).((let H5 \def (eq_ind T (THead (Bind Abbr) u0 t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind Void) u1 t1) H3) in (False_ind ((eq T (THead (Bind Abbr) u2 w) x) \to ((pr0 u0 u2) \to ((pr0 t0 t2) \to ((subst0 O u2 t2 w) \to (or (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x))))))) H5)) H4 H0 H1 H2))) | (pr0_zeta b H0 t0 t2 H1 u) \Rightarrow (\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 t1))).(\lambda (H3: (eq T t2 x)).((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 t1) H2) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 t1) H2) in ((let H6 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 t1) H2) in (eq_ind B Void (\lambda (b0: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not (eq B b0 Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x))))))))) (\lambda (H7: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O t0) t1) \to ((eq T t2 x) \to ((not (eq B Void Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x)))))))) (\lambda (H8: (eq T (lift (S O) O t0) t1)).(eq_ind T (lift (S O) O t0) (\lambda (t: T).((eq T t2 x) \to ((not (eq B Void Abst)) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t t3)))) (pr0 t (lift (S O) O x))))))) (\lambda (H9: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((not (eq B Void Abst)) \to ((pr0 t0 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift (S O) O t0) t3)))) (pr0 (lift (S O) O t0) (lift (S O) O x)))))) (\lambda (_: (not (eq B Void Abst))).(\lambda (H11: (pr0 t0 x)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 (lift (S O) O t0) t3)))) (pr0 (lift (S O) O t0) (lift (S O) O x)) (pr0_lift t0 x H11 (S O) O)))) t2 (sym_eq T t2 x H9))) t1 H8)) u (sym_eq T u u1 H7))) b (sym_eq B b Void H6))) H5)) H4)) H3 H0 H1))) | (pr0_epsilon t0 t2 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Bind Void) u1 t1))).(\lambda (H2: (eq T t2 x)).((let H3 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Void) u1 t1) H1) in (False_ind ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x))))) H3)) H2 H0)))]) in (H0 (refl_equal T (THead (Bind Void) u1 t1)) (refl_equal T x)))))).
 
 theorem pr0_gen_lift:
  \forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr0 (lift h d t1) x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr0 t1 t2)))))))
@@ -1892,7 +1892,7 @@ theorem pr0_confluence__pr0_delta_epsilon:
 theorem pr0_confluence:
  \forall (t0: T).(\forall (t1: T).((pr0 t0 t1) \to (\forall (t2: T).((pr0 t0 t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))
 \def
- \lambda (t0: T).(tlt_wf_ind (\lambda (t: T).(\forall (t1: T).((pr0 t t1) \to (\forall (t2: T).((pr0 t t2) \to (ex2 T (\lambda (t3: T).(pr0 t1 t3)) (\lambda (t3: T).(pr0 t2 t3)))))))) (\lambda (t: T).(\lambda (H: ((\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))))))))))).(\lambda (t1: T).(\lambda (H0: (pr0 t t1)).(\lambda (t2: T).(\lambda (H1: (pr0 t t2)).(let H2 \def (match H0 return (\lambda (t0: T).(\lambda (t3: T).((eq T t0 t) \to ((eq T t3 t1) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))))))) with [(pr0_refl t0) \Rightarrow (\lambda (H2: (eq T t0 t)).(\lambda (H3: (eq T t0 t1)).(eq_ind T t (\lambda (t: T).((eq T t t1) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1))))) (\lambda (H4: (eq T t t1)).(eq_ind T t1 (\lambda (_: T).(ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1)))) (let H5 \def (match H1 return (\lambda (t0: T).(\lambda (t3: T).((eq T t0 t) \to ((eq T t3 t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))))))) with [(pr0_refl t3) \Rightarrow (\lambda (H5: (eq T t3 t)).(\lambda (H6: (eq T t3 t2)).(eq_ind T t (\lambda (t: T).((eq T t t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind T t (\lambda (t: T).(eq T t3 t)) H5 t2 H7) in (let H1 \def (eq_ind T t (\lambda (t: T).(eq T t t1)) H4 t2 H7) in (let H2 \def (eq_ind T t (\lambda (t: T).(eq T t0 t)) H2 t2 H7) in (let H3 \def (eq_ind T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H t2 H7) in (let H4 \def (eq_ind T t2 (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H3 t1 H1) in (eq_ind_r T t1 (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))) (let H8 \def (eq_ind T t2 (\lambda (t: T).(eq T t0 t)) H2 t1 H1) in (ex_intro2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t1 t)) t1 (pr0_refl t1) (pr0_refl t1))) t2 H1)))))) t (sym_eq T t t2 H7))) t3 (sym_eq T t3 t H5) H6))) | (pr0_comp u1 u2 H4 t3 t4 H5 k) \Rightarrow (\lambda (H6: (eq T (THead k u1 t3) t)).(\lambda (H7: (eq T (THead k u2 t4) t2)).(eq_ind T (THead k u1 t3) (\lambda (_: T).((eq T (THead k u2 t4) t2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H8: (eq T (THead k u2 t4) t2)).(eq_ind T (THead k u2 t4) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (H9: (pr0 u1 u2)).(\lambda (H10: (pr0 t3 t4)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t1)) H4 (THead k u1 t3) H6) in (eq_ind T (THead k u1 t3) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t t0)) (\lambda (t0: T).(pr0 (THead k u2 t4) t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t0 t)) H2 (THead k u1 t3) H6) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead k u1 t3) H6) in (ex_intro2 T (\lambda (t: T).(pr0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead k u2 t4) t)) (THead k u2 t4) (pr0_comp u1 u2 H9 t3 t4 H10 k) (pr0_refl (THead k u2 t4))))) t1 H0)))) t2 H8)) t H6 H7 H4 H5))) | (pr0_beta u v1 v2 H4 t3 t4 H5) \Rightarrow (\lambda (H6: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t)).(\lambda (H7: (eq T (THead (Bind Abbr) v2 t4) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t4) t2) \to ((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H8: (eq T (THead (Bind Abbr) v2 t4) t2)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda (t: T).((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (H9: (pr0 v1 v2)).(\lambda (H10: (pr0 t3 t4)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t1)) H4 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H6) in (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t4) t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t0 t)) H2 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H6) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H6) in (ex_intro2 T (\lambda (t: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t4) t)) (THead (Bind Abbr) v2 t4) (pr0_beta u v1 v2 H9 t3 t4 H10) (pr0_refl (THead (Bind Abbr) v2 t4))))) t1 H0)))) t2 H8)) t H6 H7 H4 H5))) | (pr0_upsilon b H4 v1 v2 H5 u1 u2 H6 t3 t4 H7) \Rightarrow (\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t)).(\lambda (H9: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))))) (\lambda (H10: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))))))) (\lambda (H11: (not (eq B b Abst))).(\lambda (H12: (pr0 v1 v2)).(\lambda (H13: (pr0 u1 u2)).(\lambda (H14: (pr0 t3 t4)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t1)) H4 (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) H8) in (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t0 t)) H2 (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) H8) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) H8) in (pr0_confluence__pr0_cong_upsilon_refl b H11 u1 u2 H13 t3 t4 H14 v1 v2 v2 H12 (pr0_refl v2)))) t1 H0)))))) t2 H10)) t H8 H9 H4 H5 H6 H7))) | (pr0_delta u1 u2 H4 t3 t4 H5 w H6) \Rightarrow (\lambda (H7: (eq T (THead (Bind Abbr) u1 t3) t)).(\lambda (H8: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))) (\lambda (H9: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0))))))) (\lambda (H10: (pr0 u1 u2)).(\lambda (H11: (pr0 t3 t4)).(\lambda (H12: (subst0 O u2 t4 w)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t1)) H4 (THead (Bind Abbr) u1 t3) H7) in (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t0 t)) H2 (THead (Bind Abbr) u1 t3) H7) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind Abbr) u1 t3) H7) in (ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u1 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (THead (Bind Abbr) u2 w) (pr0_delta u1 u2 H10 t3 t4 H11 w H12) (pr0_refl (THead (Bind Abbr) u2 w))))) t1 H0))))) t2 H9)) t H7 H8 H4 H5 H6))) | (pr0_zeta b H4 t3 t4 H5 u) \Rightarrow (\lambda (H6: (eq T (THead (Bind b) u (lift (S O) O t3)) t)).(\lambda (H7: (eq T t4 t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t3)) (\lambda (_: T).((eq T t4 t2) \to ((not (eq B b Abst)) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H8: (eq T t4 t2)).(eq_ind T t2 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t3 t) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H9: (not (eq B b Abst))).(\lambda (H10: (pr0 t3 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t1)) H4 (THead (Bind b) u (lift (S O) O t3)) H6) in (eq_ind T (THead (Bind b) u (lift (S O) O t3)) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t0 t)) H2 (THead (Bind b) u (lift (S O) O t3)) H6) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind b) u (lift (S O) O t3)) H6) in (ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind b) u (lift (S O) O t3)) t)) (\lambda (t: T).(pr0 t2 t)) t2 (pr0_zeta b H9 t3 t2 H10 u) (pr0_refl t2)))) t1 H0)))) t4 (sym_eq T t4 t2 H8))) t H6 H7 H4 H5))) | (pr0_epsilon t3 t4 H4 u) \Rightarrow (\lambda (H5: (eq T (THead (Flat Cast) u t3) t)).(\lambda (H6: (eq T t4 t2)).(eq_ind T (THead (Flat Cast) u t3) (\lambda (_: T).((eq T t4 t2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H7: (eq T t4 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t3 t) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H8: (pr0 t3 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t1)) H4 (THead (Flat Cast) u t3) H5) in (eq_ind T (THead (Flat Cast) u t3) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t0 t)) H2 (THead (Flat Cast) u t3) H5) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Cast) u t3) H5) in (ex_intro2 T (\lambda (t: T).(pr0 (THead (Flat Cast) u t3) t)) (\lambda (t: T).(pr0 t2 t)) t2 (pr0_epsilon t3 t2 H8 u) (pr0_refl t2)))) t1 H0))) t4 (sym_eq T t4 t2 H7))) t H5 H6 H4)))]) in (H5 (refl_equal T t) (refl_equal T t2))) t (sym_eq T t t1 H4))) t0 (sym_eq T t0 t H2) H3))) | (pr0_comp u1 u2 H2 t0 t3 H3 k) \Rightarrow (\lambda (H4: (eq T (THead k u1 t0) t)).(\lambda (H5: (eq T (THead k u2 t3) t1)).(eq_ind T (THead k u1 t0) (\lambda (_: T).((eq T (THead k u2 t3) t1) \to ((pr0 u1 u2) \to ((pr0 t0 t3) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1))))))) (\lambda (H6: (eq T (THead k u2 t3) t1)).(eq_ind T (THead k u2 t3) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t0 t3) \to (ex2 T (\lambda (t1: T).(pr0 t t1)) (\lambda (t1: T).(pr0 t2 t1)))))) (\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (pr0 t0 t3)).(let H9 \def (match H1 return (\lambda (t0: T).(\lambda (t1: T).((eq T t0 t) \to ((eq T t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 t2 t))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H6: (eq T t4 t)).(\lambda (H9: (eq T t4 t2)).(eq_ind T t (\lambda (t: T).((eq T t t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H10: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t2)) H10 (THead k u1 t0) H4) in (eq_ind T (THead k u1 t0) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t4 t)) H6 (THead k u1 t0) H4) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead k u1 t0) H4) in (ex_intro2 T (\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 (THead k u1 t0) t)) (THead k u2 t3) (pr0_refl (THead k u2 t3)) (pr0_comp u1 u2 H7 t0 t3 H8 k)))) t2 H0)) t (sym_eq T t t2 H10))) t4 (sym_eq T t4 t H6) H9))) | (pr0_comp u0 u3 H6 t4 t5 H7 k0) \Rightarrow (\lambda (H9: (eq T (THead k0 u0 t4) t)).(\lambda (H10: (eq T (THead k0 u3 t5) t2)).(eq_ind T (THead k0 u0 t4) (\lambda (_: T).((eq T (THead k0 u3 t5) t2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H11: (eq T (THead k0 u3 t5) t2)).(eq_ind T (THead k0 u3 t5) (\lambda (t: T).((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (H12: (pr0 u0 u3)).(\lambda (H13: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead k u1 t0) t)) H4 (THead k0 u0 t4) H9) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead k0 u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 (THead k0 u3 t5) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead k u1 t0) (THead k0 u0 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u1 t0) (THead k0 u0 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead k u1 t0) (THead k0 u0 t4) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u1 t0) (THead k0 u0 t4) H0) in (eq_ind K k0 (\lambda (k: K).((eq T u1 u0) \to ((eq T t0 t4) \to (ex2 T (\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 (THead k0 u3 t5) t)))))) (\lambda (H10: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead k0 u2 t3) t0)) (\lambda (t0: T).(pr0 (THead k0 u3 t5) t0))))) (\lambda (H11: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead k0 u2 t3) t0)) (\lambda (t0: T).(pr0 (THead k0 u3 t5) t0)))) (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead k0 u0 t4) H9) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8 t4 H11) in (let H6 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 u0 H10) in (ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead k0 u2 t3) t)) (\lambda (t: T).(pr0 (THead k0 u3 t5) t))) (\lambda (x: T).(\lambda (H7: (pr0 u2 x)).(\lambda (H8: (pr0 u3 x)).(ex2_ind T (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead k0 u2 t3) t)) (\lambda (t: T).(pr0 (THead k0 u3 t5) t))) (\lambda (x0: T).(\lambda (H9: (pr0 t3 x0)).(\lambda (H12: (pr0 t5 x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead k0 u2 t3) t)) (\lambda (t: T).(pr0 (THead k0 u3 t5) t)) (THead k0 x x0) (pr0_comp u2 x H7 t3 x0 H9 k0) (pr0_comp u3 x H8 t5 x0 H12 k0))))) (H4 t4 (tlt_head_dx k0 u0 t4) t3 H5 t5 H13))))) (H4 u0 (tlt_head_sx k0 u0 t4) u2 H6 u3 H12))))) t0 (sym_eq T t0 t4 H11))) u1 (sym_eq T u1 u0 H10))) k (sym_eq K k k0 H3))) H2)) H1)))]) in (H1 (refl_equal T (THead k0 u0 t4))))))) t2 H11)) t H9 H10 H6 H7))) | (pr0_beta u v1 v2 H6 t4 t5 H7) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) t)).(\lambda (H10: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H11: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) (\lambda (t: T).((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (H12: (pr0 v1 v2)).(\lambda (H13: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead k u1 t0) t)) H4 (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H9) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Flat Appl) v1 (THead (Bind Abst) u t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead k u1 t0) (THead (Flat Appl) v1 (THead (Bind Abst) u t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u1 t0) (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead k u1 t0) (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u1 t0) (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H0) in (eq_ind K (Flat Appl) (\lambda (k: K).((eq T u1 v1) \to ((eq T t0 (THead (Bind Abst) u t4)) \to (ex2 T (\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t)))))) (\lambda (H10: (eq T u1 v1)).(eq_ind T v1 (\lambda (_: T).((eq T t0 (THead (Bind Abst) u t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0))))) (\lambda (H11: (eq T t0 (THead (Bind Abst) u t4))).(eq_ind T (THead (Bind Abst) u t4) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)))) (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H9) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8 (THead (Bind Abst) u t4) H11) in (let H6 \def (match H5 return (\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Bind Abst) u t4)) \to ((eq T t0 t3) \to (ex2 T (\lambda (t1: T).(pr0 (THead (Flat Appl) u2 t3) t1)) (\lambda (t1: T).(pr0 (THead (Bind Abbr) v2 t5) t1))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Bind Abst) u t4))).(\lambda (H5: (eq T t t3)).(eq_ind T (THead (Bind Abst) u t4) (\lambda (t0: T).((eq T t0 t3) \to (ex2 T (\lambda (t1: T).(pr0 (THead (Flat Appl) u2 t3) t1)) (\lambda (t1: T).(pr0 (THead (Bind Abbr) v2 t5) t1))))) (\lambda (H6: (eq T (THead (Bind Abst) u t4) t3)).(eq_ind T (THead (Bind Abst) u t4) (\lambda (t0: T).(ex2 T (\lambda (t1: T).(pr0 (THead (Flat Appl) u2 t0) t1)) (\lambda (t1: T).(pr0 (THead (Bind Abbr) v2 t5) t1)))) (let H1 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 v1 H10) in (ex2_ind T (\lambda (t0: T).(pr0 u2 t0)) (\lambda (t0: T).(pr0 v2 t0)) (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u t4)) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0))) (\lambda (x: T).(\lambda (H2: (pr0 u2 x)).(\lambda (H3: (pr0 v2 x)).(ex_intro2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u t4)) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)) (THead (Bind Abbr) x t5) (pr0_beta u u2 x H2 t4 t5 H13) (pr0_comp v2 x H3 t5 t5 (pr0_refl t5) (Bind Abbr)))))) (H4 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t4)) u2 H1 v2 H12))) t3 H6)) t (sym_eq T t (THead (Bind Abst) u t4) H0) H5))) | (pr0_comp u0 u3 H0 t1 t2 H4 k0) \Rightarrow (\lambda (H5: (eq T (THead k0 u0 t1) (THead (Bind Abst) u t4))).(\lambda (H8: (eq T (THead k0 u3 t2) t3)).((let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t])) (THead k0 u0 t1) (THead (Bind Abst) u t4) H5) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k0 u0 t1) (THead (Bind Abst) u t4) H5) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k _ _) \Rightarrow k])) (THead k0 u0 t1) (THead (Bind Abst) u t4) H5) in (eq_ind K (Bind Abst) (\lambda (k: K).((eq T u0 u) \to ((eq T t1 t4) \to ((eq T (THead k u3 t2) t3) \to ((pr0 u0 u3) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))))))))) (\lambda (H6: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq T t1 t4) \to ((eq T (THead (Bind Abst) u3 t2) t3) \to ((pr0 t u3) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)))))))) (\lambda (H9: (eq T t1 t4)).(eq_ind T t4 (\lambda (t: T).((eq T (THead (Bind Abst) u3 t2) t3) \to ((pr0 u u3) \to ((pr0 t t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0))))))) (\lambda (H11: (eq T (THead (Bind Abst) u3 t2) t3)).(eq_ind T (THead (Bind Abst) u3 t2) (\lambda (t: T).((pr0 u u3) \to ((pr0 t4 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)))))) (\lambda (_: (pr0 u u3)).(\lambda (H15: (pr0 t4 t2)).(let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 v1 H10) in (ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 v2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t2)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))) (\lambda (x: T).(\lambda (H10: (pr0 u2 x)).(\lambda (H12: (pr0 v2 x)).(ex2_ind T (\lambda (t: T).(pr0 t2 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t2)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))) (\lambda (x0: T).(\lambda (H13: (pr0 t2 x0)).(\lambda (H16: (pr0 t5 x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t2)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t)) (THead (Bind Abbr) x x0) (pr0_beta u3 u2 x H10 t2 x0 H13) (pr0_comp v2 x H12 t5 x0 H16 (Bind Abbr)))))) (H4 t4 (tlt_trans (THead (Bind Abst) u t4) t4 (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) (tlt_head_dx (Bind Abst) u t4) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abst) u t4))) t2 H15 t5 H13))))) (H4 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t4)) u2 H7 v2 H12))))) t3 H11)) t1 (sym_eq T t1 t4 H9))) u0 (sym_eq T u0 u H6))) k0 (sym_eq K k0 (Bind Abst) H3))) H2)) H1)) H8 H0 H4))) | (pr0_beta u0 v0 v3 H0 t1 t2 H4) \Rightarrow (\lambda (H5: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t1)) (THead (Bind Abst) u t4))).(\lambda (H8: (eq T (THead (Bind Abbr) v3 t2) t3)).((let H1 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t1)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t4) H5) in (False_ind ((eq T (THead (Bind Abbr) v3 t2) t3) \to ((pr0 v0 v3) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t)))))) H1)) H8 H0 H4))) | (pr0_upsilon b H0 v0 v3 H4 u0 u3 H5 t1 t2 H8) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v0 (THead (Bind b) u0 t1)) (THead (Bind Abst) u t4))).(\lambda (H12: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t2)) t3)).((let H1 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u0 t1)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t4) H11) in (False_ind ((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t2)) t3) \to ((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t)))))))) H1)) H12 H0 H4 H5 H8))) | (pr0_delta u0 u3 H0 t1 t2 H4 w H5) \Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u0 t1) (THead (Bind Abst) u t4))).(\lambda (H11: (eq T (THead (Bind Abbr) u3 w) t3)).((let H1 \def (eq_ind T (THead (Bind Abbr) u0 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind Abst) u t4) H8) in (False_ind ((eq T (THead (Bind Abbr) u3 w) t3) \to ((pr0 u0 u3) \to ((pr0 t1 t2) \to ((subst0 O u3 t2 w) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))))))) H1)) H11 H0 H4 H5))) | (pr0_zeta b H0 t1 t2 H4 u0) \Rightarrow (\lambda (H5: (eq T (THead (Bind b) u0 (lift (S O) O t1)) (THead (Bind Abst) u t4))).(\lambda (H8: (eq T t2 t3)).((let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t1) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t1) | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u0 (lift (S O) O t1)) (THead (Bind Abst) u t4) H5) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead (Bind b) u0 (lift (S O) O t1)) (THead (Bind Abst) u t4) H5) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O t1)) (THead (Bind Abst) u t4) H5) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S O) O t1) t4) \to ((eq T t2 t3) \to ((not (eq B b0 Abst)) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))))))))) (\lambda (H6: (eq T u0 u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t1) t4) \to ((eq T t2 t3) \to ((not (eq B Abst Abst)) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)))))))) (\lambda (H9: (eq T (lift (S O) O t1) t4)).(eq_ind T (lift (S O) O t1) (\lambda (_: T).((eq T t2 t3) \to ((not (eq B Abst Abst)) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0))))))) (\lambda (H7: (eq T t2 t3)).(eq_ind T t3 (\lambda (t: T).((not (eq B Abst Abst)) \to ((pr0 t1 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)))))) (\lambda (H11: (not (eq B Abst Abst))).(\lambda (_: (pr0 t1 t3)).(let H10 \def (match (H11 (refl_equal B Abst)) return (\lambda (_: ?).(ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t)))) with []) in H10))) t2 (sym_eq T t2 t3 H7))) t4 H9)) u0 (sym_eq T u0 u H6))) b (sym_eq B b Abst H3))) H2)) H1)) H8 H0 H4))) | (pr0_epsilon t1 t2 H0 u0) \Rightarrow (\lambda (H4: (eq T (THead (Flat Cast) u0 t1) (THead (Bind Abst) u t4))).(\lambda (H5: (eq T t2 t3)).((let H1 \def (eq_ind T (THead (Flat Cast) u0 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t4) H4) in (False_ind ((eq T t2 t3) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))))) H1)) H5 H0)))]) in (H6 (refl_equal T (THead (Bind Abst) u t4)) (refl_equal T t3))))) t0 (sym_eq T t0 (THead (Bind Abst) u t4) H11))) u1 (sym_eq T u1 v1 H10))) k (sym_eq K k (Flat Appl) H3))) H2)) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)))))))) t2 H11)) t H9 H10 H6 H7))) | (pr0_upsilon b H6 v1 v2 H7 u0 u3 H8 t4 t5 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) t)).(\lambda (H11: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) (\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))))) (\lambda (H12: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t t0)))))))) (\lambda (H13: (not (eq B b Abst))).(\lambda (H14: (pr0 v1 v2)).(\lambda (H15: (pr0 u0 u3)).(\lambda (H16: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead k u1 t0) t)) H4 (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) H10) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Flat Appl) v1 (THead (Bind b) u0 t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead k u1 t0) (THead (Flat Appl) v1 (THead (Bind b) u0 t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u1 t0) (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead k u1 t0) (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u1 t0) (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) H0) in (eq_ind K (Flat Appl) (\lambda (k: K).((eq T u1 v1) \to ((eq T t0 (THead (Bind b) u0 t4)) \to (ex2 T (\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)))))) (\lambda (H11: (eq T u1 v1)).(eq_ind T v1 (\lambda (_: T).((eq T t0 (THead (Bind b) u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0))))) (\lambda (H12: (eq T t0 (THead (Bind b) u0 t4))).(eq_ind T (THead (Bind b) u0 t4) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0)))) (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) H10) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8 (THead (Bind b) u0 t4) H12) in (let H6 \def (match H5 return (\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Bind b) u0 t4)) \to ((eq T t0 t3) \to (ex2 T (\lambda (t1: T).(pr0 (THead (Flat Appl) u2 t3) t1)) (\lambda (t1: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t1))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Bind b) u0 t4))).(\lambda (H5: (eq T t t3)).(eq_ind T (THead (Bind b) u0 t4) (\lambda (t0: T).((eq T t0 t3) \to (ex2 T (\lambda (t1: T).(pr0 (THead (Flat Appl) u2 t3) t1)) (\lambda (t1: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t1))))) (\lambda (H6: (eq T (THead (Bind b) u0 t4) t3)).(eq_ind T (THead (Bind b) u0 t4) (\lambda (t0: T).(ex2 T (\lambda (t1: T).(pr0 (THead (Flat Appl) u2 t0) t1)) (\lambda (t1: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t1)))) (let H1 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 v1 H11) in (ex2_ind T (\lambda (t0: T).(pr0 u2 t0)) (\lambda (t0: T).(pr0 v2 t0)) (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u0 t4)) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0))) (\lambda (x: T).(\lambda (H2: (pr0 u2 x)).(\lambda (H3: (pr0 v2 x)).(pr0_confluence__pr0_cong_upsilon_refl b H13 u0 u3 H15 t4 t5 H16 u2 v2 x H2 H3)))) (H4 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind b) u0 t4)) u2 H1 v2 H14))) t3 H6)) t (sym_eq T t (THead (Bind b) u0 t4) H0) H5))) | (pr0_comp u4 u5 H0 t1 t2 H4 k0) \Rightarrow (\lambda (H5: (eq T (THead k0 u4 t1) (THead (Bind b) u0 t4))).(\lambda (H10: (eq T (THead k0 u5 t2) t3)).((let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t])) (THead k0 u4 t1) (THead (Bind b) u0 t4) H5) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t _) \Rightarrow t])) (THead k0 u4 t1) (THead (Bind b) u0 t4) H5) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k _ _) \Rightarrow k])) (THead k0 u4 t1) (THead (Bind b) u0 t4) H5) in (eq_ind K (Bind b) (\lambda (k: K).((eq T u4 u0) \to ((eq T t1 t4) \to ((eq T (THead k u5 t2) t3) \to ((pr0 u4 u5) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))))))))) (\lambda (H6: (eq T u4 u0)).(eq_ind T u0 (\lambda (t: T).((eq T t1 t4) \to ((eq T (THead (Bind b) u5 t2) t3) \to ((pr0 t u5) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0)))))))) (\lambda (H12: (eq T t1 t4)).(eq_ind T t4 (\lambda (t: T).((eq T (THead (Bind b) u5 t2) t3) \to ((pr0 u0 u5) \to ((pr0 t t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0))))))) (\lambda (H8: (eq T (THead (Bind b) u5 t2) t3)).(eq_ind T (THead (Bind b) u5 t2) (\lambda (t: T).((pr0 u0 u5) \to ((pr0 t4 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0)))))) (\lambda (H17: (pr0 u0 u5)).(\lambda (H18: (pr0 t4 t2)).(let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 v1 H11) in (ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 v2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x: T).(\lambda (H9: (pr0 u2 x)).(\lambda (H11: (pr0 v2 x)).(ex2_ind T (\lambda (t: T).(pr0 t2 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x0: T).(\lambda (H14: (pr0 t2 x0)).(\lambda (H16: (pr0 t5 x0)).(ex2_ind T (\lambda (t: T).(pr0 u5 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x1: T).(\lambda (H15: (pr0 u5 x1)).(\lambda (H19: (pr0 u3 x1)).(pr0_confluence__pr0_cong_upsilon_cong b H13 u2 v2 x H9 H11 t2 t5 x0 H14 H16 u5 u3 x1 H15 H19)))) (H4 u0 (tlt_trans (THead (Bind b) u0 t4) u0 (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) (tlt_head_sx (Bind b) u0 t4) (tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 t4))) u5 H17 u3 H15))))) (H4 t4 (tlt_trans (THead (Bind b) u0 t4) t4 (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) (tlt_head_dx (Bind b) u0 t4) (tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 t4))) t2 H18 t5 H16))))) (H4 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind b) u0 t4)) u2 H7 v2 H14))))) t3 H8)) t1 (sym_eq T t1 t4 H12))) u4 (sym_eq T u4 u0 H6))) k0 (sym_eq K k0 (Bind b) H3))) H2)) H1)) H10 H0 H4))) | (pr0_beta u v0 v3 H0 t1 t2 H4) \Rightarrow (\lambda (H5: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t1)) (THead (Bind b) u0 t4))).(\lambda (H10: (eq T (THead (Bind Abbr) v3 t2) t3)).((let H1 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t1)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 t4) H5) in (False_ind ((eq T (THead (Bind Abbr) v3 t2) t3) \to ((pr0 v0 v3) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)))))) H1)) H10 H0 H4))) | (pr0_upsilon b0 H0 v0 v3 H4 u4 u5 H5 t1 t2 H10) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 t1)) (THead (Bind b) u0 t4))).(\lambda (H14: (eq T (THead (Bind b0) u5 (THead (Flat Appl) (lift (S O) O v3) t2)) t3)).((let H1 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b0) u4 t1)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 t4) H13) in (False_ind ((eq T (THead (Bind b0) u5 (THead (Flat Appl) (lift (S O) O v3) t2)) t3) \to ((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u4 u5) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)))))))) H1)) H14 H0 H4 H5 H10))) | (pr0_delta u4 u5 H0 t1 t2 H4 w H5) \Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr) u4 t1) (THead (Bind b) u0 t4))).(\lambda (H17: (eq T (THead (Bind Abbr) u5 w) t3)).((let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abbr) u4 t1) (THead (Bind b) u0 t4) H10) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t _) \Rightarrow t])) (THead (Bind Abbr) u4 t1) (THead (Bind b) u0 t4) H10) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t1) (THead (Bind b) u0 t4) H10) in (eq_ind B Abbr (\lambda (b: B).((eq T u4 u0) \to ((eq T t1 t4) \to ((eq T (THead (Bind Abbr) u5 w) t3) \to ((pr0 u4 u5) \to ((pr0 t1 t2) \to ((subst0 O u5 t2 w) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)))))))))) (\lambda (H6: (eq T u4 u0)).(eq_ind T u0 (\lambda (t: T).((eq T t1 t4) \to ((eq T (THead (Bind Abbr) u5 w) t3) \to ((pr0 t u5) \to ((pr0 t1 t2) \to ((subst0 O u5 t2 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0))))))))) (\lambda (H8: (eq T t1 t4)).(eq_ind T t4 (\lambda (t: T).((eq T (THead (Bind Abbr) u5 w) t3) \to ((pr0 u0 u5) \to ((pr0 t t2) \to ((subst0 O u5 t2 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0)))))))) (\lambda (H18: (eq T (THead (Bind Abbr) u5 w) t3)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t: T).((pr0 u0 u5) \to ((pr0 t4 t2) \to ((subst0 O u5 t2 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0))))))) (\lambda (H19: (pr0 u0 u5)).(\lambda (H20: (pr0 t4 t2)).(\lambda (H21: (subst0 O u5 t2 w)).(let H9 \def (eq_ind_r B b (\lambda (b: B).(\forall (v: T).((tlt v (THead (Flat Appl) v1 (THead (Bind b) u0 t4))) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))))) H4 Abbr H3) in (let H12 \def (eq_ind_r B b (\lambda (b: B).(eq T t0 (THead (Bind b) u0 t4))) H12 Abbr H3) in (let H13 \def (eq_ind_r B b (\lambda (b: B).(not (eq B b Abst))) H13 Abbr H3) in (let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 v1 H11) in (ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 v2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x: T).(\lambda (H11: (pr0 u2 x)).(\lambda (H14: (pr0 v2 x)).(ex2_ind T (\lambda (t: T).(pr0 t2 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x0: T).(\lambda (H16: (pr0 t2 x0)).(\lambda (H22: (pr0 t5 x0)).(ex2_ind T (\lambda (t: T).(pr0 u5 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x1: T).(\lambda (H15: (pr0 u5 x1)).(\lambda (H23: (pr0 u3 x1)).(pr0_confluence__pr0_cong_upsilon_delta H13 u5 t2 w H21 u2 v2 x H11 H14 t5 x0 H16 H22 u3 x1 H15 H23)))) (H9 u0 (tlt_trans (THead (Bind Abbr) u0 t4) u0 (THead (Flat Appl) v1 (THead (Bind Abbr) u0 t4)) (tlt_head_sx (Bind Abbr) u0 t4) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t4))) u5 H19 u3 H15))))) (H9 t4 (tlt_trans (THead (Bind Abbr) u0 t4) t4 (THead (Flat Appl) v1 (THead (Bind Abbr) u0 t4)) (tlt_head_dx (Bind Abbr) u0 t4) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t4))) t2 H20 t5 H16))))) (H9 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind Abbr) u0 t4)) u2 H7 v2 H14))))))))) t3 H18)) t1 (sym_eq T t1 t4 H8))) u4 (sym_eq T u4 u0 H6))) b H3)) H2)) H1)) H17 H0 H4 H5))) | (pr0_zeta b0 H0 t1 t2 H4 u) \Rightarrow (\lambda (H5: (eq T (THead (Bind b0) u (lift (S O) O t1)) (THead (Bind b) u0 t4))).(\lambda (H10: (eq T t2 t3)).((let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t1) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t1) | (THead _ _ t) \Rightarrow t])) (THead (Bind b0) u (lift (S O) O t1)) (THead (Bind b) u0 t4) H5) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b0) u (lift (S O) O t1)) (THead (Bind b) u0 t4) H5) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u (lift (S O) O t1)) (THead (Bind b) u0 t4) H5) in (eq_ind B b (\lambda (b1: B).((eq T u u0) \to ((eq T (lift (S O) O t1) t4) \to ((eq T t2 t3) \to ((not (eq B b1 Abst)) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))))))))) (\lambda (H6: (eq T u u0)).(eq_ind T u0 (\lambda (_: T).((eq T (lift (S O) O t1) t4) \to ((eq T t2 t3) \to ((not (eq B b Abst)) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0)))))))) (\lambda (H13: (eq T (lift (S O) O t1) t4)).(eq_ind T (lift (S O) O t1) (\lambda (_: T).((eq T t2 t3) \to ((not (eq B b Abst)) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0))))))) (\lambda (H8: (eq T t2 t3)).(eq_ind T t3 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t1 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0)))))) (\lambda (H17: (not (eq B b Abst))).(\lambda (H18: (pr0 t1 t3)).(let H9 \def (eq_ind_r T t4 (\lambda (t: T).(\forall (v: T).((tlt v (THead (Flat Appl) v1 (THead (Bind b) u0 t))) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H4 (lift (S O) O t1) H13) in (let H12 \def (eq_ind_r T t4 (\lambda (t: T).(eq T t0 (THead (Bind b) u0 t))) H12 (lift (S O) O t1) H13) in (let H16 \def (eq_ind_r T t4 (\lambda (t: T).(pr0 t t5)) H16 (lift (S O) O t1) H13) in (ex2_ind T (\lambda (t3: T).(eq T t5 (lift (S O) O t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x: T).(\lambda (H19: (eq T t5 (lift (S O) O x))).(\lambda (H20: (pr0 t1 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t)) t0)))) (let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 v1 H11) in (ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 v2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O x))) t))) (\lambda (x0: T).(\lambda (H11: (pr0 u2 x0)).(\lambda (H14: (pr0 v2 x0)).(ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O x))) t))) (\lambda (x1: T).(\lambda (H21: (pr0 x x1)).(\lambda (H22: (pr0 t3 x1)).(pr0_confluence__pr0_cong_upsilon_zeta b H17 u0 u3 H15 u2 v2 x0 H11 H14 x t3 x1 H21 H22)))) (H9 t1 (tlt_trans (THead (Bind b) u0 (lift (S O) O t1)) t1 (THead (Flat Appl) v1 (THead (Bind b) u0 (lift (S O) O t1))) (lift_tlt_dx (Bind b) u0 t1 (S O) O) (tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 (lift (S O) O t1)))) x H20 t3 H18))))) (H9 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind b) u0 (lift (S O) O t1))) u2 H7 v2 H14))) t5 H19)))) (pr0_gen_lift t1 t5 (S O) O H16))))))) t2 (sym_eq T t2 t3 H8))) t4 H13)) u (sym_eq T u u0 H6))) b0 (sym_eq B b0 b H3))) H2)) H1)) H10 H0 H4))) | (pr0_epsilon t1 t2 H0 u) \Rightarrow (\lambda (H4: (eq T (THead (Flat Cast) u t1) (THead (Bind b) u0 t4))).(\lambda (H5: (eq T t2 t3)).((let H1 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 t4) H4) in (False_ind ((eq T t2 t3) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))))) H1)) H5 H0)))]) in (H6 (refl_equal T (THead (Bind b) u0 t4)) (refl_equal T t3))))) t0 (sym_eq T t0 (THead (Bind b) u0 t4) H12))) u1 (sym_eq T u1 v1 H11))) k (sym_eq K k (Flat Appl) H3))) H2)) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind b) u0 t4)))))))))) t2 H12)) t H10 H11 H6 H7 H8 H9))) | (pr0_delta u0 u3 H6 t4 t5 H7 w H8) \Rightarrow (\lambda (H9: (eq T (THead (Bind Abbr) u0 t4) t)).(\lambda (H10: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind Abbr) u0 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u3 w) t2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to ((subst0 O u3 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))))) (\lambda (H11: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t: T).((pr0 u0 u3) \to ((pr0 t4 t5) \to ((subst0 O u3 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t t0))))))) (\lambda (H12: (pr0 u0 u3)).(\lambda (H13: (pr0 t4 t5)).(\lambda (H14: (subst0 O u3 t5 w)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead k u1 t0) t)) H4 (THead (Bind Abbr) u0 t4) H9) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Bind Abbr) u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 w) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead k u1 t0) (THead (Bind Abbr) u0 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u1 t0) (THead (Bind Abbr) u0 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead k u1 t0) (THead (Bind Abbr) u0 t4) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u1 t0) (THead (Bind Abbr) u0 t4) H0) in (eq_ind K (Bind Abbr) (\lambda (k: K).((eq T u1 u0) \to ((eq T t0 t4) \to (ex2 T (\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t)))))) (\lambda (H10: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 w) t0))))) (\lambda (H11: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 w) t0)))) (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind Abbr) u0 t4) H9) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8 t4 H11) in (let H6 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 u0 H10) in (ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t))) (\lambda (x: T).(\lambda (H7: (pr0 u2 x)).(\lambda (H8: (pr0 u3 x)).(ex2_ind T (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t))) (\lambda (x0: T).(\lambda (H9: (pr0 t3 x0)).(\lambda (H12: (pr0 t5 x0)).(pr0_confluence__pr0_cong_delta u3 t5 w H14 u2 x H7 H8 t3 x0 H9 H12)))) (H4 t4 (tlt_head_dx (Bind Abbr) u0 t4) t3 H5 t5 H13))))) (H4 u0 (tlt_head_sx (Bind Abbr) u0 t4) u2 H6 u3 H12))))) t0 (sym_eq T t0 t4 H11))) u1 (sym_eq T u1 u0 H10))) k (sym_eq K k (Bind Abbr) H3))) H2)) H1)))]) in (H1 (refl_equal T (THead (Bind Abbr) u0 t4)))))))) t2 H11)) t H9 H10 H6 H7 H8))) | (pr0_zeta b H6 t4 t5 H7 u) \Rightarrow (\lambda (H9: (eq T (THead (Bind b) u (lift (S O) O t4)) t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (_: T).((eq T t5 t2) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H11: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H12: (not (eq B b Abst))).(\lambda (H13: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead k u1 t0) t)) H4 (THead (Bind b) u (lift (S O) O t4)) H9) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Bind b) u (lift (S O) O t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead k u1 t0) (THead (Bind b) u (lift (S O) O t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u1 t0) (THead (Bind b) u (lift (S O) O t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead k u1 t0) (THead (Bind b) u (lift (S O) O t4)) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u1 t0) (THead (Bind b) u (lift (S O) O t4)) H0) in (eq_ind K (Bind b) (\lambda (k: K).((eq T u1 u) \to ((eq T t0 (lift (S O) O t4)) \to (ex2 T (\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 t2 t)))))) (\lambda (H10: (eq T u1 u)).(eq_ind T u (\lambda (_: T).((eq T t0 (lift (S O) O t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H11: (eq T t0 (lift (S O) O t4))).(eq_ind T (lift (S O) O t4) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind b) u (lift (S O) O t4)) H9) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8 (lift (S O) O t4) H11) in (ex2_ind T (\lambda (t2: T).(eq T t3 (lift (S O) O t2))) (\lambda (t2: T).(pr0 t4 t2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 t3) t)) (\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H6: (eq T t3 (lift (S O) O x))).(\lambda (H8: (pr0 t4 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 t) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 u H10) in (ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t)) (\lambda (t: T).(pr0 t2 t))) (\lambda (x0: T).(\lambda (H9: (pr0 x x0)).(\lambda (H13: (pr0 t2 x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t)) (\lambda (t: T).(pr0 t2 t)) x0 (pr0_zeta b H12 x x0 H9 u2) H13)))) (H4 t4 (lift_tlt_dx (Bind b) u t4 (S O) O) x H8 t2 H13))) t3 H6)))) (pr0_gen_lift t4 t3 (S O) O H5)))) t0 (sym_eq T t0 (lift (S O) O t4) H11))) u1 (sym_eq T u1 u H10))) k (sym_eq K k (Bind b) H3))) H2)) H1)))]) in (H1 (refl_equal T (THead (Bind b) u (lift (S O) O t4)))))))) t5 (sym_eq T t5 t2 H11))) t H9 H10 H6 H7))) | (pr0_epsilon t4 t5 H6 u) \Rightarrow (\lambda (H9: (eq T (THead (Flat Cast) u t4) t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u t4) (\lambda (_: T).((eq T t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H11: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H12: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead k u1 t0) t)) H4 (THead (Flat Cast) u t4) H9) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Flat Cast) u t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead k u1 t0) (THead (Flat Cast) u t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u1 t0) (THead (Flat Cast) u t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead k u1 t0) (THead (Flat Cast) u t4) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u1 t0) (THead (Flat Cast) u t4) H0) in (eq_ind K (Flat Cast) (\lambda (k: K).((eq T u1 u) \to ((eq T t0 t4) \to (ex2 T (\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 t2 t)))))) (\lambda (H10: (eq T u1 u)).(eq_ind T u (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Cast) u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H11: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Cast) u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Cast) u t4) H9) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8 t4 H11) in (let H6 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 u H10) in (ex2_ind T (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Cast) u2 t3) t)) (\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H7: (pr0 t3 x)).(\lambda (H8: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Flat Cast) u2 t3) t)) (\lambda (t: T).(pr0 t2 t)) x (pr0_epsilon t3 x H7 u2) H8)))) (H4 t4 (tlt_head_dx (Flat Cast) u t4) t3 H5 t2 H12))))) t0 (sym_eq T t0 t4 H11))) u1 (sym_eq T u1 u H10))) k (sym_eq K k (Flat Cast) H3))) H2)) H1)))]) in (H1 (refl_equal T (THead (Flat Cast) u t4)))))) t5 (sym_eq T t5 t2 H11))) t H9 H10 H6)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t1 H6)) t H4 H5 H2 H3))) | (pr0_beta u v1 v2 H2 t0 t3 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t)).(\lambda (H5: (eq T (THead (Bind Abbr) v2 t3) t1)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t3) t1) \to ((pr0 v1 v2) \to ((pr0 t0 t3) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1))))))) (\lambda (H6: (eq T (THead (Bind Abbr) v2 t3) t1)).(eq_ind T (THead (Bind Abbr) v2 t3) (\lambda (t: T).((pr0 v1 v2) \to ((pr0 t0 t3) \to (ex2 T (\lambda (t1: T).(pr0 t t1)) (\lambda (t1: T).(pr0 t2 t1)))))) (\lambda (H7: (pr0 v1 v2)).(\lambda (H8: (pr0 t0 t3)).(let H9 \def (match H1 return (\lambda (t0: T).(\lambda (t1: T).((eq T t0 t) \to ((eq T t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 t2 t))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H6: (eq T t4 t)).(\lambda (H9: (eq T t4 t2)).(eq_ind T t (\lambda (t: T).((eq T t t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H10: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t2)) H10 (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) H4) in (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t4 t)) H6 (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) H4) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) H4) in (ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t)) (THead (Bind Abbr) v2 t3) (pr0_refl (THead (Bind Abbr) v2 t3)) (pr0_beta u v1 v2 H7 t0 t3 H8)))) t2 H0)) t (sym_eq T t t2 H10))) t4 (sym_eq T t4 t H6) H9))) | (pr0_comp u1 u2 H6 t4 t5 H7 k) \Rightarrow (\lambda (H9: (eq T (THead k u1 t4) t)).(\lambda (H10: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u1 t4) (\lambda (_: T).((eq T (THead k u2 t5) t2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H11: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u2 t5) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (H12: (pr0 u1 u2)).(\lambda (H13: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t)) H4 (THead k u1 t4) H9) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead k u1 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead k u2 t5) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead k u1 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow (THead (Bind Abst) u t0) | (TLRef _) \Rightarrow (THead (Bind Abst) u t0) | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead k u1 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead k u1 t4) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) | (THead k _ _) \Rightarrow k])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead k u1 t4) H0) in (eq_ind K (Flat Appl) (\lambda (k: K).((eq T v1 u1) \to ((eq T (THead (Bind Abst) u t0) t4) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead k u2 t5) t)))))) (\lambda (H10: (eq T v1 u1)).(eq_ind T u1 (\lambda (_: T).((eq T (THead (Bind Abst) u t0) t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0))))) (\lambda (H11: (eq T (THead (Bind Abst) u t0) t4)).(eq_ind T (THead (Bind Abst) u t0) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0)))) (let H4 \def (eq_ind_r K k (\lambda (k: K).(eq T (THead k u1 t4) t)) H9 (Flat Appl) H3) in (let H5 \def (eq_ind_r T t4 (\lambda (t: T).(pr0 t t5)) H13 (THead (Bind Abst) u t0) H11) in (let H6 \def (match H5 return (\lambda (t: T).(\lambda (t1: T).((eq T t (THead (Bind Abst) u t0)) \to ((eq T t1 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0))))))) with [(pr0_refl t2) \Rightarrow (\lambda (H0: (eq T t2 (THead (Bind Abst) u t0))).(\lambda (H1: (eq T t2 t5)).(eq_ind T (THead (Bind Abst) u t0) (\lambda (t: T).((eq T t t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0))))) (\lambda (H2: (eq T (THead (Bind Abst) u t0) t5)).(eq_ind T (THead (Bind Abst) u t0) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t) t0)))) (let H3 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead (Flat Appl) u1 t0) t)) H4 (THead (Bind Abst) u t0) H11) in (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t0)) H3) in (let H5 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H7 u1 H10) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 u2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u t0)) t))) (\lambda (x: T).(\lambda (H6: (pr0 v2 x)).(\lambda (H7: (pr0 u2 x)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u t0)) t)) (THead (Bind Abbr) x t3) (pr0_comp v2 x H6 t3 t3 (pr0_refl t3) (Bind Abbr)) (pr0_beta u u2 x H7 t0 t3 H8))))) (H4 u1 (tlt_head_sx (Flat Appl) u1 (THead (Bind Abst) u t0)) v2 H5 u2 H12))))) t5 H2)) t2 (sym_eq T t2 (THead (Bind Abst) u t0) H0) H1))) | (pr0_comp u0 u3 H0 t2 t6 H1 k) \Rightarrow (\lambda (H5: (eq T (THead k u0 t2) (THead (Bind Abst) u t0))).(\lambda (H13: (eq T (THead k u3 t6) t5)).((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ _ t) \Rightarrow t])) (THead k u0 t2) (THead (Bind Abst) u t0) H5) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t2) (THead (Bind Abst) u t0) H5) in ((let H6 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u0 t2) (THead (Bind Abst) u t0) H5) in (eq_ind K (Bind Abst) (\lambda (k0: K).((eq T u0 u) \to ((eq T t2 t0) \to ((eq T (THead k0 u3 t6) t5) \to ((pr0 u0 u3) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t5) t))))))))) (\lambda (H9: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq T t2 t0) \to ((eq T (THead (Bind Abst) u3 t6) t5) \to ((pr0 t u3) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0)))))))) (\lambda (H14: (eq T t2 t0)).(eq_ind T t0 (\lambda (t: T).((eq T (THead (Bind Abst) u3 t6) t5) \to ((pr0 u u3) \to ((pr0 t t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0))))))) (\lambda (H15: (eq T (THead (Bind Abst) u3 t6) t5)).(eq_ind T (THead (Bind Abst) u3 t6) (\lambda (t: T).((pr0 u u3) \to ((pr0 t0 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t) t0)))))) (\lambda (_: (pr0 u u3)).(\lambda (H17: (pr0 t0 t6)).(let H4 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead (Flat Appl) u1 t0) t)) H4 (THead (Bind Abst) u t0) H11) in (let H11 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t0)) H4) in (let H7 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H7 u1 H10) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 u2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t6)) t))) (\lambda (x: T).(\lambda (H10: (pr0 v2 x)).(\lambda (H12: (pr0 u2 x)).(ex2_ind T (\lambda (t: T).(pr0 t6 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t6)) t))) (\lambda (x0: T).(\lambda (H8: (pr0 t6 x0)).(\lambda (H18: (pr0 t3 x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t6)) t)) (THead (Bind Abbr) x x0) (pr0_comp v2 x H10 t3 x0 H18 (Bind Abbr)) (pr0_beta u3 u2 x H12 t6 x0 H8))))) (H11 t0 (tlt_trans (THead (Bind Abst) u t0) t0 (THead (Flat Appl) u1 (THead (Bind Abst) u t0)) (tlt_head_dx (Bind Abst) u t0) (tlt_head_dx (Flat Appl) u1 (THead (Bind Abst) u t0))) t6 H17 t3 H8))))) (H11 u1 (tlt_head_sx (Flat Appl) u1 (THead (Bind Abst) u t0)) v2 H7 u2 H12))))))) t5 H15)) t2 (sym_eq T t2 t0 H14))) u0 (sym_eq T u0 u H9))) k (sym_eq K k (Bind Abst) H6))) H3)) H2)) H13 H0 H1))) | (pr0_beta u0 v0 v3 H0 t2 t6 H1) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t2)) (THead (Bind Abst) u t0))).(\lambda (H11: (eq T (THead (Bind Abbr) v3 t6) t5)).((let H2 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t2)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t0) H4) in (False_ind ((eq T (THead (Bind Abbr) v3 t6) t5) \to ((pr0 v0 v3) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t5) t)))))) H2)) H11 H0 H1))) | (pr0_upsilon b H0 v0 v3 H1 u0 u3 H4 t2 t6 H11) \Rightarrow (\lambda (H12: (eq T (THead (Flat Appl) v0 (THead (Bind b) u0 t2)) (THead (Bind Abst) u t0))).(\lambda (H13: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t5)).((let H2 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u0 t2)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t0) H12) in (False_ind ((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t5) \to ((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t5) t)))))))) H2)) H13 H0 H1 H4 H11))) | (pr0_delta u0 u3 H0 t2 t6 H1 w H4) \Rightarrow (\lambda (H11: (eq T (THead (Bind Abbr) u0 t2) (THead (Bind Abst) u t0))).(\lambda (H12: (eq T (THead (Bind Abbr) u3 w) t5)).((let H2 \def (eq_ind T (THead (Bind Abbr) u0 t2) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind Abst) u t0) H11) in (False_ind ((eq T (THead (Bind Abbr) u3 w) t5) \to ((pr0 u0 u3) \to ((pr0 t2 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t5) t))))))) H2)) H12 H0 H1 H4))) | (pr0_zeta b H0 t2 t6 H1 u0) \Rightarrow (\lambda (H4: (eq T (THead (Bind b) u0 (lift (S O) O t2)) (THead (Bind Abst) u t0))).(\lambda (H11: (eq T t6 t5)).((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t2) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t2) | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u0 (lift (S O) O t2)) (THead (Bind Abst) u t0) H4) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead (Bind b) u0 (lift (S O) O t2)) (THead (Bind Abst) u t0) H4) in ((let H5 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O t2)) (THead (Bind Abst) u t0) H4) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S O) O t2) t0) \to ((eq T t6 t5) \to ((not (eq B b0 Abst)) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t5) t))))))))) (\lambda (H6: (eq T u0 u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t2) t0) \to ((eq T t6 t5) \to ((not (eq B Abst Abst)) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0)))))))) (\lambda (H12: (eq T (lift (S O) O t2) t0)).(eq_ind T (lift (S O) O t2) (\lambda (_: T).((eq T t6 t5) \to ((not (eq B Abst Abst)) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0))))))) (\lambda (H7: (eq T t6 t5)).(eq_ind T t5 (\lambda (t: T).((not (eq B Abst Abst)) \to ((pr0 t2 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0)))))) (\lambda (H13: (not (eq B Abst Abst))).(\lambda (_: (pr0 t2 t5)).(let H8 \def (match (H13 (refl_equal B Abst)) return (\lambda (_: ?).(ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t5) t)))) with []) in H8))) t6 (sym_eq T t6 t5 H7))) t0 H12)) u0 (sym_eq T u0 u H6))) b (sym_eq B b Abst H5))) H3)) H2)) H11 H0 H1))) | (pr0_epsilon t2 t6 H0 u0) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u0 t2) (THead (Bind Abst) u t0))).(\lambda (H4: (eq T t6 t5)).((let H2 \def (eq_ind T (THead (Flat Cast) u0 t2) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t0) H1) in (False_ind ((eq T t6 t5) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t5) t))))) H2)) H4 H0)))]) in (H6 (refl_equal T (THead (Bind Abst) u t0)) (refl_equal T t5))))) t4 H11)) v1 (sym_eq T v1 u1 H10))) k H3)) H2)) H1)))]) in (H1 (refl_equal T (THead k u1 t4))))))) t2 H11)) t H9 H10 H6 H7))) | (pr0_beta u0 v0 v3 H6 t4 t5 H7) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)) t)).(\lambda (H10: (eq T (THead (Bind Abbr) v3 t5) t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)) (\lambda (_: T).((eq T (THead (Bind Abbr) v3 t5) t2) \to ((pr0 v0 v3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H11: (eq T (THead (Bind Abbr) v3 t5) t2)).(eq_ind T (THead (Bind Abbr) v3 t5) (\lambda (t: T).((pr0 v0 v3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (H12: (pr0 v0 v3)).(\lambda (H13: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t)) H4 (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)) H9) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v3 t5) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t0) \Rightarrow t0])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t0 _) \Rightarrow t0])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)) H0) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)) H0) in (eq_ind T v0 (\lambda (_: T).((eq T u u0) \to ((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v3 t5) t0)))))) (\lambda (H10: (eq T u u0)).(eq_ind T u0 (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v3 t5) t0))))) (\lambda (H11: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v3 t5) t0)))) (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)) H9) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8 t4 H11) in (let H6 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H7 v0 H3) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 v3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v3 t5) t))) (\lambda (x: T).(\lambda (H7: (pr0 v2 x)).(\lambda (H8: (pr0 v3 x)).(ex2_ind T (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v3 t5) t))) (\lambda (x0: T).(\lambda (H9: (pr0 t3 x0)).(\lambda (H12: (pr0 t5 x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v3 t5) t)) (THead (Bind Abbr) x x0) (pr0_comp v2 x H7 t3 x0 H9 (Bind Abbr)) (pr0_comp v3 x H8 t5 x0 H12 (Bind Abbr)))))) (H4 t4 (tlt_trans (THead (Bind Abst) u0 t4) t4 (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)) (tlt_head_dx (Bind Abst) u0 t4) (tlt_head_dx (Flat Appl) v0 (THead (Bind Abst) u0 t4))) t3 H5 t5 H13))))) (H4 v0 (tlt_head_sx (Flat Appl) v0 (THead (Bind Abst) u0 t4)) v2 H6 v3 H12))))) t0 (sym_eq T t0 t4 H11))) u (sym_eq T u u0 H10))) v1 (sym_eq T v1 v0 H3))) H2)) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)))))))) t2 H11)) t H9 H10 H6 H7))) | (pr0_upsilon b H6 v0 v3 H7 u1 u2 H8 t4 t5 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v0 (THead (Bind b) u1 t4)) t)).(\lambda (H11: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u1 t4)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t2) \to ((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))))) (\lambda (H12: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t t0)))))))) (\lambda (H13: (not (eq B b Abst))).(\lambda (_: (pr0 v0 v3)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t)) H4 (THead (Flat Appl) v0 (THead (Bind b) u1 t4)) H10) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Flat Appl) v0 (THead (Bind b) u1 t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind b) u1 t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t0) \Rightarrow t0])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind b) u1 t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t0 _) \Rightarrow t0])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind b) u1 t4)) H0) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).B) with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind b) u1 t4)) H0) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind b) u1 t4)) H0) in (eq_ind T v0 (\lambda (_: T).((eq B Abst b) \to ((eq T u u1) \to ((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t0))))))) (\lambda (H11: (eq B Abst b)).(eq_ind B Abst (\lambda (b: B).((eq T u u1) \to ((eq T t0 t4) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t)))))) (\lambda (H12: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t0))))) (\lambda (H14: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t0)))) (let H5 \def (eq_ind_r B b (\lambda (b: B).(not (eq B b Abst))) H13 Abst H11) in (let H6 \def (match (H5 (refl_equal B Abst)) return (\lambda (_: ?).(ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t)))) with []) in H6)) t0 (sym_eq T t0 t4 H14))) u (sym_eq T u u1 H12))) b H11)) v1 (sym_eq T v1 v0 H4))) H3)) H2)) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v0 (THead (Bind b) u1 t4)))))))))) t2 H12)) t H10 H11 H6 H7 H8 H9))) | (pr0_delta u1 u2 H6 t4 t5 H7 w H8) \Rightarrow (\lambda (H9: (eq T (THead (Bind Abbr) u1 t4) t)).(\lambda (H10: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u1 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))))) (\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t t0))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t4 t5)).(\lambda (_: (subst0 O u2 t5 w)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t)) H4 (THead (Bind Abbr) u1 t4) H9) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Abbr) u1 t4))).(let H1 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t4) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t))) H1)))]) in (H1 (refl_equal T (THead (Bind Abbr) u1 t4)))))))) t2 H11)) t H9 H10 H6 H7 H8))) | (pr0_zeta b H6 t4 t5 H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Bind b) u0 (lift (S O) O t4)) t)).(\lambda (H9: (eq T t5 t2)).(eq_ind T (THead (Bind b) u0 (lift (S O) O t4)) (\lambda (_: T).((eq T t5 t2) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H10: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t)) H4 (THead (Bind b) u0 (lift (S O) O t4)) H8) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Bind b) u0 (lift (S O) O t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind b) u0 (lift (S O) O t4)))).(let H1 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 (lift (S O) O t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 t2 t))) H1)))]) in (H1 (refl_equal T (THead (Bind b) u0 (lift (S O) O t4)))))))) t5 (sym_eq T t5 t2 H10))) t H8 H9 H6 H7))) | (pr0_epsilon t4 t5 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 t4) t)).(\lambda (H8: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u0 t4) (\lambda (_: T).((eq T t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H9: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (_: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t)) H4 (THead (Flat Cast) u0 t4) H7) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Flat Cast) u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Cast) u0 t4))).(let H1 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u0 t4) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 t2 t))) H1)))]) in (H1 (refl_equal T (THead (Flat Cast) u0 t4)))))) t5 (sym_eq T t5 t2 H9))) t H7 H8 H6)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t1 H6)) t H4 H5 H2 H3))) | (pr0_upsilon b H2 v1 v2 H3 u1 u2 H4 t0 t3 H5) \Rightarrow (\lambda (H6: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t)).(\lambda (H7: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t1)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t1) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t0 t3) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1))))))))) (\lambda (H8: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t1)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t0 t3) \to (ex2 T (\lambda (t1: T).(pr0 t t1)) (\lambda (t1: T).(pr0 t2 t1)))))))) (\lambda (H9: (not (eq B b Abst))).(\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 u1 u2)).(\lambda (H12: (pr0 t0 t3)).(let H13 \def (match H1 return (\lambda (t0: T).(\lambda (t1: T).((eq T t0 t) \to ((eq T t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 t2 t))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H8: (eq T t4 t)).(\lambda (H13: (eq T t4 t2)).(eq_ind T t (\lambda (t: T).((eq T t t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H14: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t2)) H14 (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) H6) in (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t4 t)) H8 (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) H6) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) H6) in (ex2_sym T (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 t0))) (pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3))) (pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11 t0 t3 H12 v1 v2 v2 H10 (pr0_refl v2))))) t2 H0)) t (sym_eq T t t2 H14))) t4 (sym_eq T t4 t H8) H13))) | (pr0_comp u0 u3 H8 t4 t5 H9 k) \Rightarrow (\lambda (H13: (eq T (THead k u0 t4) t)).(\lambda (H14: (eq T (THead k u3 t5) t2)).(eq_ind T (THead k u0 t4) (\lambda (_: T).((eq T (THead k u3 t5) t2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H15: (eq T (THead k u3 t5) t2)).(eq_ind T (THead k u3 t5) (\lambda (t: T).((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (H16: (pr0 u0 u3)).(\lambda (H17: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t)) H6 (THead k u0 t4) H13) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead k u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead k u3 t5) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead k u0 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow (THead (Bind b) u1 t0) | (TLRef _) \Rightarrow (THead (Bind b) u1 t0) | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead k u0 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead k u0 t4) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) | (THead k _ _) \Rightarrow k])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead k u0 t4) H0) in (eq_ind K (Flat Appl) (\lambda (k: K).((eq T v1 u0) \to ((eq T (THead (Bind b) u1 t0) t4) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead k u3 t5) t)))))) (\lambda (H14: (eq T v1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T (THead (Bind b) u1 t0) t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0))))) (\lambda (H15: (eq T (THead (Bind b) u1 t0) t4)).(eq_ind T (THead (Bind b) u1 t0) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0)))) (let H4 \def (eq_ind_r K k (\lambda (k: K).(eq T (THead k u0 t4) t)) H13 (Flat Appl) H3) in (let H5 \def (eq_ind_r T t4 (\lambda (t: T).(pr0 t t5)) H17 (THead (Bind b) u1 t0) H15) in (let H6 \def (match H5 return (\lambda (t: T).(\lambda (t1: T).((eq T t (THead (Bind b) u1 t0)) \to ((eq T t1 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0))))))) with [(pr0_refl t2) \Rightarrow (\lambda (H0: (eq T t2 (THead (Bind b) u1 t0))).(\lambda (H1: (eq T t2 t5)).(eq_ind T (THead (Bind b) u1 t0) (\lambda (t: T).((eq T t t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0))))) (\lambda (H2: (eq T (THead (Bind b) u1 t0) t5)).(eq_ind T (THead (Bind b) u1 t0) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t) t0)))) (let H3 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead (Flat Appl) u0 t0) t)) H4 (THead (Bind b) u1 t0) H15) in (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) u0 (THead (Bind b) u1 t0)) H3) in (let H5 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H10 u0 H14) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u1 t0)) t))) (\lambda (x: T).(\lambda (H6: (pr0 v2 x)).(\lambda (H7: (pr0 u3 x)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead (Bind b) u1 t0))) (pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3))) (pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11 t0 t3 H12 u3 v2 x H7 H6))))) (H4 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 t0)) v2 H5 u3 H16))))) t5 H2)) t2 (sym_eq T t2 (THead (Bind b) u1 t0) H0) H1))) | (pr0_comp u4 u5 H0 t2 t6 H1 k) \Rightarrow (\lambda (H6: (eq T (THead k u4 t2) (THead (Bind b) u1 t0))).(\lambda (H13: (eq T (THead k u5 t6) t5)).((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ _ t) \Rightarrow t])) (THead k u4 t2) (THead (Bind b) u1 t0) H6) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t _) \Rightarrow t])) (THead k u4 t2) (THead (Bind b) u1 t0) H6) in ((let H5 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u4 t2) (THead (Bind b) u1 t0) H6) in (eq_ind K (Bind b) (\lambda (k0: K).((eq T u4 u1) \to ((eq T t2 t0) \to ((eq T (THead k0 u5 t6) t5) \to ((pr0 u4 u5) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 t5) t))))))))) (\lambda (H7: (eq T u4 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t2 t0) \to ((eq T (THead (Bind b) u5 t6) t5) \to ((pr0 t u5) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0)))))))) (\lambda (H17: (eq T t2 t0)).(eq_ind T t0 (\lambda (t: T).((eq T (THead (Bind b) u5 t6) t5) \to ((pr0 u1 u5) \to ((pr0 t t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0))))))) (\lambda (H8: (eq T (THead (Bind b) u5 t6) t5)).(eq_ind T (THead (Bind b) u5 t6) (\lambda (t: T).((pr0 u1 u5) \to ((pr0 t0 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t) t0)))))) (\lambda (H18: (pr0 u1 u5)).(\lambda (H19: (pr0 t0 t6)).(let H15 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead (Flat Appl) u0 t0) t)) H4 (THead (Bind b) u1 t0) H15) in (let H20 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) u0 (THead (Bind b) u1 t0)) H15) in (let H4 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H10 u0 H14) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u5 t6)) t))) (\lambda (x: T).(\lambda (H10: (pr0 v2 x)).(\lambda (H14: (pr0 u3 x)).(ex2_ind T (\lambda (t: T).(pr0 t6 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u5 t6)) t))) (\lambda (x0: T).(\lambda (H12: (pr0 t6 x0)).(\lambda (H16: (pr0 t3 x0)).(ex2_ind T (\lambda (t: T).(pr0 u5 t)) (\lambda (t: T).(pr0 u2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u5 t6)) t))) (\lambda (x1: T).(\lambda (H11: (pr0 u5 x1)).(\lambda (H21: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead (Bind b) u5 t6))) (pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3))) (pr0_confluence__pr0_cong_upsilon_cong b H9 u3 v2 x H14 H10 t6 t3 x0 H12 H16 u5 u2 x1 H11 H21))))) (H20 u1 (tlt_trans (THead (Bind b) u1 t0) u1 (THead (Flat Appl) u0 (THead (Bind b) u1 t0)) (tlt_head_sx (Bind b) u1 t0) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 t0))) u5 H18 u2 H11))))) (H20 t0 (tlt_trans (THead (Bind b) u1 t0) t0 (THead (Flat Appl) u0 (THead (Bind b) u1 t0)) (tlt_head_dx (Bind b) u1 t0) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 t0))) t6 H19 t3 H12))))) (H20 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 t0)) v2 H4 u3 H16))))))) t5 H8)) t2 (sym_eq T t2 t0 H17))) u4 (sym_eq T u4 u1 H7))) k (sym_eq K k (Bind b) H5))) H3)) H2)) H13 H0 H1))) | (pr0_beta u v0 v3 H0 t2 t6 H1) \Rightarrow (\lambda (H6: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t2)) (THead (Bind b) u1 t0))).(\lambda (H13: (eq T (THead (Bind Abbr) v3 t6) t5)).((let H2 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t2)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u1 t0) H6) in (False_ind ((eq T (THead (Bind Abbr) v3 t6) t5) \to ((pr0 v0 v3) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 t5) t)))))) H2)) H13 H0 H1))) | (pr0_upsilon b0 H0 v0 v3 H1 u4 u5 H6 t2 t6 H13) \Rightarrow (\lambda (H14: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 t2)) (THead (Bind b) u1 t0))).(\lambda (H15: (eq T (THead (Bind b0) u5 (THead (Flat Appl) (lift (S O) O v3) t6)) t5)).((let H2 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b0) u4 t2)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u1 t0) H14) in (False_ind ((eq T (THead (Bind b0) u5 (THead (Flat Appl) (lift (S O) O v3) t6)) t5) \to ((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u4 u5) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 t5) t)))))))) H2)) H15 H0 H1 H6 H13))) | (pr0_delta u4 u5 H0 t2 t6 H1 w H6) \Rightarrow (\lambda (H13: (eq T (THead (Bind Abbr) u4 t2) (THead (Bind b) u1 t0))).(\lambda (H17: (eq T (THead (Bind Abbr) u5 w) t5)).((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abbr) u4 t2) (THead (Bind b) u1 t0) H13) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t _) \Rightarrow t])) (THead (Bind Abbr) u4 t2) (THead (Bind b) u1 t0) H13) in ((let H5 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t2) (THead (Bind b) u1 t0) H13) in (eq_ind B Abbr (\lambda (b: B).((eq T u4 u1) \to ((eq T t2 t0) \to ((eq T (THead (Bind Abbr) u5 w) t5) \to ((pr0 u4 u5) \to ((pr0 t2 t6) \to ((subst0 O u5 t6 w) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 t5) t)))))))))) (\lambda (H7: (eq T u4 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t2 t0) \to ((eq T (THead (Bind Abbr) u5 w) t5) \to ((pr0 t u5) \to ((pr0 t2 t6) \to ((subst0 O u5 t6 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0))))))))) (\lambda (H18: (eq T t2 t0)).(eq_ind T t0 (\lambda (t: T).((eq T (THead (Bind Abbr) u5 w) t5) \to ((pr0 u1 u5) \to ((pr0 t t6) \to ((subst0 O u5 t6 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0)))))))) (\lambda (H19: (eq T (THead (Bind Abbr) u5 w) t5)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t: T).((pr0 u1 u5) \to ((pr0 t0 t6) \to ((subst0 O u5 t6 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t) t0))))))) (\lambda (H20: (pr0 u1 u5)).(\lambda (H21: (pr0 t0 t6)).(\lambda (H22: (subst0 O u5 t6 w)).(let H15 \def (eq_ind_r B b (\lambda (b: B).(eq T (THead (Bind b) u1 t0) t4)) H15 Abbr H5) in (let H9 \def (eq_ind_r B b (\lambda (b: B).(not (eq B b Abst))) H9 Abbr H5) in (let H23 \def (eq_ind_r B b (\lambda (b: B).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t1)) H8 Abbr H5) in (let H4 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead (Flat Appl) u0 t0) t)) H4 (THead (Bind Abbr) u1 t0) H15) in (let H8 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) u0 (THead (Bind Abbr) u1 t0)) H4) in (let H10 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H10 u0 H14) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t))) (\lambda (x: T).(\lambda (H14: (pr0 v2 x)).(\lambda (H16: (pr0 u3 x)).(ex2_ind T (\lambda (t: T).(pr0 t6 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t))) (\lambda (x0: T).(\lambda (H12: (pr0 t6 x0)).(\lambda (H24: (pr0 t3 x0)).(ex2_ind T (\lambda (t: T).(pr0 u5 t)) (\lambda (t: T).(pr0 u2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t))) (\lambda (x1: T).(\lambda (H11: (pr0 u5 x1)).(\lambda (H25: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w))) (pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t3))) (pr0_confluence__pr0_cong_upsilon_delta H9 u5 t6 w H22 u3 v2 x H16 H14 t3 x0 H12 H24 u2 x1 H11 H25))))) (H8 u1 (tlt_trans (THead (Bind Abbr) u1 t0) u1 (THead (Flat Appl) u0 (THead (Bind Abbr) u1 t0)) (tlt_head_sx (Bind Abbr) u1 t0) (tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t0))) u5 H20 u2 H11))))) (H8 t0 (tlt_trans (THead (Bind Abbr) u1 t0) t0 (THead (Flat Appl) u0 (THead (Bind Abbr) u1 t0)) (tlt_head_dx (Bind Abbr) u1 t0) (tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t0))) t6 H21 t3 H12))))) (H8 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind Abbr) u1 t0)) v2 H10 u3 H16))))))))))) t5 H19)) t2 (sym_eq T t2 t0 H18))) u4 (sym_eq T u4 u1 H7))) b H5)) H3)) H2)) H17 H0 H1 H6))) | (pr0_zeta b0 H0 t2 t6 H1 u) \Rightarrow (\lambda (H6: (eq T (THead (Bind b0) u (lift (S O) O t2)) (THead (Bind b) u1 t0))).(\lambda (H13: (eq T t6 t5)).((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t2) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t2) | (THead _ _ t) \Rightarrow t])) (THead (Bind b0) u (lift (S O) O t2)) (THead (Bind b) u1 t0) H6) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b0) u (lift (S O) O t2)) (THead (Bind b) u1 t0) H6) in ((let H5 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u (lift (S O) O t2)) (THead (Bind b) u1 t0) H6) in (eq_ind B b (\lambda (b1: B).((eq T u u1) \to ((eq T (lift (S O) O t2) t0) \to ((eq T t6 t5) \to ((not (eq B b1 Abst)) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 t5) t))))))))) (\lambda (H7: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O t2) t0) \to ((eq T t6 t5) \to ((not (eq B b Abst)) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0)))))))) (\lambda (H17: (eq T (lift (S O) O t2) t0)).(eq_ind T (lift (S O) O t2) (\lambda (_: T).((eq T t6 t5) \to ((not (eq B b Abst)) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0))))))) (\lambda (H8: (eq T t6 t5)).(eq_ind T t5 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t2 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0)))))) (\lambda (H18: (not (eq B b Abst))).(\lambda (H19: (pr0 t2 t5)).(let H9 \def (eq_ind_r T t0 (\lambda (t: T).(eq T (THead (Bind b) u1 t) t4)) H15 (lift (S O) O t2) H17) in (let H15 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead (Flat Appl) u0 t0) t)) H4 (THead (Bind b) u1 (lift (S O) O t2)) H9) in (let H20 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t2))) H15) in (let H12 \def (eq_ind_r T t0 (\lambda (t: T).(pr0 t t3)) H12 (lift (S O) O t2) H17) in (ex2_ind T (\lambda (t4: T).(eq T t3 (lift (S O) O t4))) (\lambda (t3: T).(pr0 t2 t3)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 t5) t))) (\lambda (x: T).(\lambda (H21: (eq T t3 (lift (S O) O x))).(\lambda (H22: (pr0 t2 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0)))) (let H4 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H10 u0 H14) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O x))) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 t5) t))) (\lambda (x0: T).(\lambda (H10: (pr0 v2 x0)).(\lambda (H14: (pr0 u3 x0)).(ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O x))) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 t5) t))) (\lambda (x1: T).(\lambda (H16: (pr0 x x1)).(\lambda (H23: (pr0 t5 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 t5)) (pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O x)))) (pr0_confluence__pr0_cong_upsilon_zeta b H18 u1 u2 H11 u3 v2 x0 H14 H10 x t5 x1 H16 H23))))) (H20 t2 (tlt_trans (THead (Bind b) u1 (lift (S O) O t2)) t2 (THead (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t2))) (lift_tlt_dx (Bind b) u1 t2 (S O) O) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t2)))) x H22 t5 H19))))) (H20 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t2))) v2 H4 u3 H16))) t3 H21)))) (pr0_gen_lift t2 t3 (S O) O H12)))))))) t6 (sym_eq T t6 t5 H8))) t0 H17)) u (sym_eq T u u1 H7))) b0 (sym_eq B b0 b H5))) H3)) H2)) H13 H0 H1))) | (pr0_epsilon t2 t6 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t2) (THead (Bind b) u1 t0))).(\lambda (H6: (eq T t6 t5)).((let H2 \def (eq_ind T (THead (Flat Cast) u t2) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u1 t0) H1) in (False_ind ((eq T t6 t5) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 t5) t))))) H2)) H6 H0)))]) in (H6 (refl_equal T (THead (Bind b) u1 t0)) (refl_equal T t5))))) t4 H15)) v1 (sym_eq T v1 u0 H14))) k H3)) H2)) H1)))]) in (H1 (refl_equal T (THead k u0 t4))))))) t2 H15)) t H13 H14 H8 H9))) | (pr0_beta u v0 v3 H8 t4 t5 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t4)) t)).(\lambda (H13: (eq T (THead (Bind Abbr) v3 t5) t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t4)) (\lambda (_: T).((eq T (THead (Bind Abbr) v3 t5) t2) \to ((pr0 v0 v3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H14: (eq T (THead (Bind Abbr) v3 t5) t2)).(eq_ind T (THead (Bind Abbr) v3 t5) (\lambda (t: T).((pr0 v0 v3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (_: (pr0 v0 v3)).(\lambda (_: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t)) H6 (THead (Flat Appl) v0 (THead (Bind Abst) u t4)) H10) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Flat Appl) v0 (THead (Bind Abst) u t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v3 t5) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind Abst) u t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t0) \Rightarrow t0])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind Abst) u t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind Abst) u t4)) H0) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind Abst) u t4)) H0) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind Abst) u t4)) H0) in (eq_ind T v0 (\lambda (_: T).((eq B b Abst) \to ((eq T u1 u) \to ((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v3 t5) t0))))))) (\lambda (H13: (eq B b Abst)).(eq_ind B Abst (\lambda (b: B).((eq T u1 u) \to ((eq T t0 t4) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v3 t5) t)))))) (\lambda (H14: (eq T u1 u)).(eq_ind T u (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v3 t5) t0))))) (\lambda (H15: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v3 t5) t0)))) (let H5 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v0 (THead (Bind Abst) u t4)) H10) in (let H6 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H12 t4 H15) in (let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H11 u H14) in (let H8 \def (eq_ind B b (\lambda (b: B).(not (eq B b Abst))) H9 Abst H13) in (let H9 \def (match (H8 (refl_equal B Abst)) return (\lambda (_: ?).(ex2 T (\lambda (t: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v3 t5) t)))) with []) in H9))))) t0 (sym_eq T t0 t4 H15))) u1 (sym_eq T u1 u H14))) b (sym_eq B b Abst H13))) v1 (sym_eq T v1 v0 H4))) H3)) H2)) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v0 (THead (Bind Abst) u t4)))))))) t2 H14)) t H10 H13 H8 H9))) | (pr0_upsilon b0 H8 v0 v3 H9 u0 u3 H10 t4 t5 H11) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) t)).(\lambda (H14: (eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) (\lambda (_: T).((eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0))))))))) (\lambda (H15: (eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t2)).(eq_ind T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) (\lambda (t: T).((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t t0)))))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (H17: (pr0 v0 v3)).(\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t)) H6 (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) H13) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Flat Appl) v0 (THead (Bind b0) u0 t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t0) \Rightarrow t0])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) H0) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) H0) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) H0) in (eq_ind T v0 (\lambda (_: T).((eq B b b0) \to ((eq T u1 u0) \to ((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t0))))))) (\lambda (H14: (eq B b b0)).(eq_ind B b0 (\lambda (b: B).((eq T u1 u0) \to ((eq T t0 t4) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t)))))) (\lambda (H15: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t0))))) (\lambda (H16: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t0)))) (let H5 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) H13) in (let H6 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H12 t4 H16) in (let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H11 u0 H15) in (let H8 \def (eq_ind B b (\lambda (b: B).(not (eq B b Abst))) H9 b0 H14) in (let H9 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H10 v0 H4) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 v3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t))) (\lambda (x: T).(\lambda (H10: (pr0 v2 x)).(\lambda (H11: (pr0 v3 x)).(ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t))) (\lambda (x0: T).(\lambda (H12: (pr0 u2 x0)).(\lambda (H13: (pr0 u3 x0)).(ex2_ind T (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t))) (\lambda (x1: T).(\lambda (H17: (pr0 t3 x1)).(\lambda (H18: (pr0 t5 x1)).(pr0_confluence__pr0_upsilon_upsilon b0 H8 v2 v3 x H10 H11 u2 u3 x0 H12 H13 t3 t5 x1 H17 H18)))) (H5 t4 (tlt_trans (THead (Bind b0) u0 t4) t4 (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) (tlt_head_dx (Bind b0) u0 t4) (tlt_head_dx (Flat Appl) v0 (THead (Bind b0) u0 t4))) t3 H6 t5 H19))))) (H5 u0 (tlt_trans (THead (Bind b0) u0 t4) u0 (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) (tlt_head_sx (Bind b0) u0 t4) (tlt_head_dx (Flat Appl) v0 (THead (Bind b0) u0 t4))) u2 H7 u3 H18))))) (H5 v0 (tlt_head_sx (Flat Appl) v0 (THead (Bind b0) u0 t4)) v2 H9 v3 H17))))))) t0 (sym_eq T t0 t4 H16))) u1 (sym_eq T u1 u0 H15))) b (sym_eq B b b0 H14))) v1 (sym_eq T v1 v0 H4))) H3)) H2)) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)))))))))) t2 H15)) t H13 H14 H8 H9 H10 H11))) | (pr0_delta u0 u3 H8 t4 t5 H9 w H10) \Rightarrow (\lambda (H11: (eq T (THead (Bind Abbr) u0 t4) t)).(\lambda (H12: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind Abbr) u0 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u3 w) t2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to ((subst0 O u3 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0)))))))) (\lambda (H13: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t: T).((pr0 u0 u3) \to ((pr0 t4 t5) \to ((subst0 O u3 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t t0))))))) (\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t4 t5)).(\lambda (_: (subst0 O u3 t5 w)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t)) H6 (THead (Bind Abbr) u0 t4) H11) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Bind Abbr) u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 w) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Bind Abbr) u0 t4))).(let H1 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u0 t4) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t))) H1)))]) in (H1 (refl_equal T (THead (Bind Abbr) u0 t4)))))))) t2 H13)) t H11 H12 H8 H9 H10))) | (pr0_zeta b0 H8 t4 t5 H9 u) \Rightarrow (\lambda (H10: (eq T (THead (Bind b0) u (lift (S O) O t4)) t)).(\lambda (H11: (eq T t5 t2)).(eq_ind T (THead (Bind b0) u (lift (S O) O t4)) (\lambda (_: T).((eq T t5 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H12: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((not (eq B b0 Abst)) \to ((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (_: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t)) H6 (THead (Bind b0) u (lift (S O) O t4)) H10) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Bind b0) u (lift (S O) O t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Bind b0) u (lift (S O) O t4)))).(let H1 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b0) u (lift (S O) O t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 t2 t))) H1)))]) in (H1 (refl_equal T (THead (Bind b0) u (lift (S O) O t4)))))))) t5 (sym_eq T t5 t2 H12))) t H10 H11 H8 H9))) | (pr0_epsilon t4 t5 H8 u) \Rightarrow (\lambda (H9: (eq T (THead (Flat Cast) u t4) t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u t4) (\lambda (_: T).((eq T t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H11: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (_: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t)) H6 (THead (Flat Cast) u t4) H9) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Flat Cast) u t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Cast) u t4))).(let H1 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u t4) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 t2 t))) H1)))]) in (H1 (refl_equal T (THead (Flat Cast) u t4)))))) t5 (sym_eq T t5 t2 H11))) t H9 H10 H8)))]) in (H13 (refl_equal T t) (refl_equal T t2))))))) t1 H8)) t H6 H7 H2 H3 H4 H5))) | (pr0_delta u1 u2 H2 t0 t3 H3 w H4) \Rightarrow (\lambda (H5: (eq T (THead (Bind Abbr) u1 t0) t)).(\lambda (H6: (eq T (THead (Bind Abbr) u2 w) t1)).(eq_ind T (THead (Bind Abbr) u1 t0) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t1) \to ((pr0 u1 u2) \to ((pr0 t0 t3) \to ((subst0 O u2 t3 w) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1)))))))) (\lambda (H7: (eq T (THead (Bind Abbr) u2 w) t1)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t0 t3) \to ((subst0 O u2 t3 w) \to (ex2 T (\lambda (t1: T).(pr0 t t1)) (\lambda (t1: T).(pr0 t2 t1))))))) (\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (pr0 t0 t3)).(\lambda (H10: (subst0 O u2 t3 w)).(let H11 \def (match H1 return (\lambda (t0: T).(\lambda (t1: T).((eq T t0 t) \to ((eq T t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H7: (eq T t4 t)).(\lambda (H11: (eq T t4 t2)).(eq_ind T t (\lambda (t: T).((eq T t t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H12: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t2)) H12 (THead (Bind Abbr) u1 t0) H5) in (eq_ind T (THead (Bind Abbr) u1 t0) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t4 t)) H7 (THead (Bind Abbr) u1 t0) H5) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind Abbr) u1 t0) H5) in (ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u1 t0) t)) (THead (Bind Abbr) u2 w) (pr0_refl (THead (Bind Abbr) u2 w)) (pr0_delta u1 u2 H8 t0 t3 H9 w H10)))) t2 H0)) t (sym_eq T t t2 H12))) t4 (sym_eq T t4 t H7) H11))) | (pr0_comp u0 u3 H7 t4 t5 H8 k) \Rightarrow (\lambda (H11: (eq T (THead k u0 t4) t)).(\lambda (H12: (eq T (THead k u3 t5) t2)).(eq_ind T (THead k u0 t4) (\lambda (_: T).((eq T (THead k u3 t5) t2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H13: (eq T (THead k u3 t5) t2)).(eq_ind T (THead k u3 t5) (\lambda (t: T).((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (H14: (pr0 u0 u3)).(\lambda (H15: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind Abbr) u1 t0) t)) H5 (THead k u0 t4) H11) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead k u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead k u3 t5) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind Abbr) u1 t0) (THead k u0 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abbr) u1 t0) (THead k u0 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead (Bind Abbr) u1 t0) (THead k u0 t4) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow (Bind Abbr) | (TLRef _) \Rightarrow (Bind Abbr) | (THead k _ _) \Rightarrow k])) (THead (Bind Abbr) u1 t0) (THead k u0 t4) H0) in (eq_ind K (Bind Abbr) (\lambda (k: K).((eq T u1 u0) \to ((eq T t0 t4) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead k u3 t5) t)))))) (\lambda (H12: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 t5) t0))))) (\lambda (H13: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 t5) t0)))) (let H4 \def (eq_ind_r K k (\lambda (k: K).(eq T (THead k u0 t4) t)) H11 (Bind Abbr) H3) in (let H5 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind Abbr) u0 t4) H4) in (let H6 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H9 t4 H13) in (let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H8 u0 H12) in (ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 t5) t))) (\lambda (x: T).(\lambda (H8: (pr0 u2 x)).(\lambda (H9: (pr0 u3 x)).(ex2_ind T (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 t5) t))) (\lambda (x0: T).(\lambda (H11: (pr0 t3 x0)).(\lambda (H14: (pr0 t5 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u3 t5)) (pr0 (THead (Bind Abbr) u2 w)) (pr0_confluence__pr0_cong_delta u2 t3 w H10 u3 x H9 H8 t5 x0 H14 H11))))) (H5 t4 (tlt_head_dx (Bind Abbr) u0 t4) t3 H6 t5 H15))))) (H5 u0 (tlt_head_sx (Bind Abbr) u0 t4) u2 H7 u3 H14)))))) t0 (sym_eq T t0 t4 H13))) u1 (sym_eq T u1 u0 H12))) k H3)) H2)) H1)))]) in (H1 (refl_equal T (THead k u0 t4))))))) t2 H13)) t H11 H12 H7 H8))) | (pr0_beta u v1 v2 H7 t4 t5 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) t)).(\lambda (H10: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H11: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) (\lambda (t: T).((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind Abbr) u1 t0) t)) H5 (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H9) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Flat Appl) v1 (THead (Bind Abst) u t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind Abbr) u1 t0) (THead (Flat Appl) v1 (THead (Bind Abst) u t4)))).(let H1 \def (eq_ind T (THead (Bind Abbr) u1 t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)))))))) t2 H11)) t H9 H10 H7 H8))) | (pr0_upsilon b H7 v1 v2 H8 u0 u3 H9 t4 t5 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) t)).(\lambda (H12: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) (\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0))))))))) (\lambda (H13: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t t0)))))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind Abbr) u1 t0) t)) H5 (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) H11) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Flat Appl) v1 (THead (Bind b) u0 t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind Abbr) u1 t0) (THead (Flat Appl) v1 (THead (Bind b) u0 t4)))).(let H1 \def (eq_ind T (THead (Bind Abbr) u1 t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind b) u0 t4)))))))))) t2 H13)) t H11 H12 H7 H8 H9 H10))) | (pr0_delta u0 u3 H7 t4 t5 H8 w0 H9) \Rightarrow (\lambda (H11: (eq T (THead (Bind Abbr) u0 t4) t)).(\lambda (H12: (eq T (THead (Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u0 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u3 w0) t2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to ((subst0 O u3 t5 w0) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0)))))))) (\lambda (H13: (eq T (THead (Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u3 w0) (\lambda (t: T).((pr0 u0 u3) \to ((pr0 t4 t5) \to ((subst0 O u3 t5 w0) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t t0))))))) (\lambda (H14: (pr0 u0 u3)).(\lambda (H15: (pr0 t4 t5)).(\lambda (H16: (subst0 O u3 t5 w0)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind Abbr) u1 t0) t)) H5 (THead (Bind Abbr) u0 t4) H11) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Bind Abbr) u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 w0) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind Abbr) u1 t0) (THead (Bind Abbr) u0 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abbr) u1 t0) (THead (Bind Abbr) u0 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead (Bind Abbr) u1 t0) (THead (Bind Abbr) u0 t4) H0) in (eq_ind T u0 (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 w0) t0))))) (\lambda (H12: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 w0) t0)))) (let H3 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind Abbr) u0 t4) H11) in (let H4 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H9 t4 H12) in (let H5 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H8 u0 H2) in (ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x: T).(\lambda (H6: (pr0 u2 x)).(\lambda (H7: (pr0 u3 x)).(ex2_ind T (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x0: T).(\lambda (H8: (pr0 t3 x0)).(\lambda (H9: (pr0 t5 x0)).(pr0_confluence__pr0_delta_delta u2 t3 w H10 u3 t5 w0 H16 x H6 H7 x0 H8 H9)))) (H3 t4 (tlt_head_dx (Bind Abbr) u0 t4) t3 H4 t5 H15))))) (H3 u0 (tlt_head_sx (Bind Abbr) u0 t4) u2 H5 u3 H14))))) t0 (sym_eq T t0 t4 H12))) u1 (sym_eq T u1 u0 H2))) H1)))]) in (H1 (refl_equal T (THead (Bind Abbr) u0 t4)))))))) t2 H13)) t H11 H12 H7 H8 H9))) | (pr0_zeta b H7 t4 t5 H8 u) \Rightarrow (\lambda (H11: (eq T (THead (Bind b) u (lift (S O) O t4)) t)).(\lambda (H12: (eq T t5 t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (_: T).((eq T t5 t2) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H13: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H14: (not (eq B b Abst))).(\lambda (H15: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind Abbr) u1 t0) t)) H5 (THead (Bind b) u (lift (S O) O t4)) H11) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Bind b) u (lift (S O) O t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind Abbr) u1 t0) (THead (Bind b) u (lift (S O) O t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abbr) u1 t0) (THead (Bind b) u (lift (S O) O t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead (Bind Abbr) u1 t0) (THead (Bind b) u (lift (S O) O t4)) H0) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u1 t0) (THead (Bind b) u (lift (S O) O t4)) H0) in (eq_ind B Abbr (\lambda (_: B).((eq T u1 u) \to ((eq T t0 (lift (S O) O t4)) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t)))))) (\lambda (H12: (eq T u1 u)).(eq_ind T u (\lambda (_: T).((eq T t0 (lift (S O) O t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H13: (eq T t0 (lift (S O) O t4))).(eq_ind T (lift (S O) O t4) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H4 \def (eq_ind_r B b (\lambda (b: B).(not (eq B b Abst))) H14 Abbr H3) in (let H5 \def (eq_ind_r B b (\lambda (b: B).(eq T (THead (Bind b) u (lift (S O) O t4)) t)) H11 Abbr H3) in (let H6 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind Abbr) u (lift (S O) O t4)) H5) in (let H7 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H9 (lift (S O) O t4) H13) in (ex2_ind T (\lambda (t2: T).(eq T t3 (lift (S O) O t2))) (\lambda (t2: T).(pr0 t4 t2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H9: (eq T t3 (lift (S O) O x))).(\lambda (H11: (pr0 t4 x)).(let H10 \def (eq_ind T t3 (\lambda (t: T).(subst0 O u2 t w)) H10 (lift (S O) O x) H9) in (let H8 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H8 u H12) in (ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t2 x0)).(pr0_confluence__pr0_delta_epsilon u2 (lift (S O) O x) w H10 x (pr0_refl (lift (S O) O x)) t2)))) (H6 t4 (lift_tlt_dx (Bind Abbr) u t4 (S O) O) x H11 t2 H15))))))) (pr0_gen_lift t4 t3 (S O) O H7)))))) t0 (sym_eq T t0 (lift (S O) O t4) H13))) u1 (sym_eq T u1 u H12))) b H3)) H2)) H1)))]) in (H1 (refl_equal T (THead (Bind b) u (lift (S O) O t4)))))))) t5 (sym_eq T t5 t2 H13))) t H11 H12 H7 H8))) | (pr0_epsilon t4 t5 H7 u) \Rightarrow (\lambda (H8: (eq T (THead (Flat Cast) u t4) t)).(\lambda (H9: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u t4) (\lambda (_: T).((eq T t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H10: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (_: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind Abbr) u1 t0) t)) H5 (THead (Flat Cast) u t4) H8) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Flat Cast) u t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind Abbr) u1 t0) (THead (Flat Cast) u t4))).(let H1 \def (eq_ind T (THead (Bind Abbr) u1 t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t4) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t))) H1)))]) in (H1 (refl_equal T (THead (Flat Cast) u t4)))))) t5 (sym_eq T t5 t2 H10))) t H8 H9 H7)))]) in (H11 (refl_equal T t) (refl_equal T t2)))))) t1 H7)) t H5 H6 H2 H3 H4))) | (pr0_zeta b H2 t0 t3 H3 u) \Rightarrow (\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t0)) t)).(\lambda (H5: (eq T t3 t1)).(eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (_: T).((eq T t3 t1) \to ((not (eq B b Abst)) \to ((pr0 t0 t3) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1))))))) (\lambda (H6: (eq T t3 t1)).(eq_ind T t1 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t0 t) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1)))))) (\lambda (H7: (not (eq B b Abst))).(\lambda (H8: (pr0 t0 t1)).(let H9 \def (match H1 return (\lambda (t0: T).(\lambda (t3: T).((eq T t0 t) \to ((eq T t3 t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H6: (eq T t4 t)).(\lambda (H9: (eq T t4 t2)).(eq_ind T t (\lambda (t: T).((eq T t t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H10: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t2)) H10 (THead (Bind b) u (lift (S O) O t0)) H4) in (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t4 t)) H6 (THead (Bind b) u (lift (S O) O t0)) H4) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind b) u (lift (S O) O t0)) H4) in (ex_intro2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind b) u (lift (S O) O t0)) t)) t1 (pr0_refl t1) (pr0_zeta b H7 t0 t1 H8 u)))) t2 H0)) t (sym_eq T t t2 H10))) t4 (sym_eq T t4 t H6) H9))) | (pr0_comp u1 u2 H6 t4 t5 H7 k) \Rightarrow (\lambda (H9: (eq T (THead k u1 t4) t)).(\lambda (H10: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u1 t4) (\lambda (_: T).((eq T (THead k u2 t5) t2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H11: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u2 t5) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H13: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind b) u (lift (S O) O t0)) t)) H4 (THead k u1 t4) H9) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead k u1 t4)) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead k u2 t5) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead k u1 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead k u1 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead k u1 t4) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef _) \Rightarrow (Bind b) | (THead k _ _) \Rightarrow k])) (THead (Bind b) u (lift (S O) O t0)) (THead k u1 t4) H0) in (eq_ind K (Bind b) (\lambda (k: K).((eq T u u1) \to ((eq T (lift (S O) O t0) t4) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead k u2 t5) t)))))) (\lambda (H10: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O t0) t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u2 t5) t0))))) (\lambda (H11: (eq T (lift (S O) O t0) t4)).(eq_ind T (lift (S O) O t0) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u2 t5) t0)))) (let H4 \def (eq_ind_r K k (\lambda (k: K).(eq T (THead k u1 t4) t)) H9 (Bind b) H3) in (let H5 \def (eq_ind_r T t4 (\lambda (t: T).(pr0 t t5)) H13 (lift (S O) O t0) H11) in (ex2_ind T (\lambda (t2: T).(eq T t5 (lift (S O) O t2))) (\lambda (t2: T).(pr0 t0 t2)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 t5) t))) (\lambda (x: T).(\lambda (H6: (eq T t5 (lift (S O) O x))).(\lambda (H9: (pr0 t0 x)).(let H12 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead (Bind b) u1 t0) t)) H4 (lift (S O) O t0) H11) in (let H13 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind b) u1 (lift (S O) O t0)) H12) in (eq_ind_r T (lift (S O) O x) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u2 t) t0)))) (ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t))) (\lambda (x0: T).(\lambda (H8: (pr0 x x0)).(\lambda (H14: (pr0 t1 x0)).(ex_intro2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t)) x0 H14 (pr0_zeta b H7 x x0 H8 u2))))) (H13 t0 (lift_tlt_dx (Bind b) u1 t0 (S O) O) x H9 t1 H8)) t5 H6)))))) (pr0_gen_lift t0 t5 (S O) O H5)))) t4 H11)) u (sym_eq T u u1 H10))) k H3)) H2)) H1)))]) in (H1 (refl_equal T (THead k u1 t4))))))) t2 H11)) t H9 H10 H6 H7))) | (pr0_beta u0 v1 v2 H6 t4 t5 H7) \Rightarrow (\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) t)).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H10: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) (\lambda (t: T).((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind b) u (lift (S O) O t0)) t)) H4 (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) H8) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4))) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)))).(let H1 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)))))))) t2 H10)) t H8 H9 H6 H7))) | (pr0_upsilon b0 H6 v1 v2 H7 u1 u2 H8 t4 t5 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1 (THead (Bind b0) u1 t4)) t)).(\lambda (H11: (eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b0) u1 t4)) (\lambda (_: T).((eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))))) (\lambda (H12: (eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) (\lambda (t: T).((not (eq B b0 Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind b) u (lift (S O) O t0)) t)) H4 (THead (Flat Appl) v1 (THead (Bind b0) u1 t4)) H10) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Flat Appl) v1 (THead (Bind b0) u1 t4))) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat Appl) v1 (THead (Bind b0) u1 t4)))).(let H1 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 (THead (Bind b0) u1 t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t))) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind b0) u1 t4)))))))))) t2 H12)) t H10 H11 H6 H7 H8 H9))) | (pr0_delta u1 u2 H6 t4 t5 H7 w H8) \Rightarrow (\lambda (H9: (eq T (THead (Bind Abbr) u1 t4) t)).(\lambda (H10: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u1 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))) (\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H13: (pr0 t4 t5)).(\lambda (H14: (subst0 O u2 t5 w)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind b) u (lift (S O) O t0)) t)) H4 (THead (Bind Abbr) u1 t4) H9) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t4)) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t4) H0) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t4) H0) in (eq_ind B Abbr (\lambda (_: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t4) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)))))) (\lambda (H10: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O t0) t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0))))) (\lambda (H11: (eq T (lift (S O) O t0) t4)).(eq_ind T (lift (S O) O t0) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)))) (let H4 \def (eq_ind_r T t4 (\lambda (t: T).(pr0 t t5)) H13 (lift (S O) O t0) H11) in (ex2_ind T (\lambda (t2: T).(eq T t5 (lift (S O) O t2))) (\lambda (t2: T).(pr0 t0 t2)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t))) (\lambda (x: T).(\lambda (H5: (eq T t5 (lift (S O) O x))).(\lambda (H6: (pr0 t0 x)).(let H9 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead (Bind Abbr) u1 t0) t)) H9 (lift (S O) O t0) H11) in (let H12 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind Abbr) u1 (lift (S O) O t0)) H9) in (let H13 \def (eq_ind T t5 (\lambda (t: T).(subst0 O u2 t w)) H14 (lift (S O) O x) H5) in (let H7 \def (eq_ind B b (\lambda (b: B).(not (eq B b Abst))) H7 Abbr H3) in (ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t1 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u2 w)) (pr0 t1) (pr0_confluence__pr0_delta_epsilon u2 (lift (S O) O x) w H13 x (pr0_refl (lift (S O) O x)) t1))))) (H12 t0 (lift_tlt_dx (Bind Abbr) u1 t0 (S O) O) x H6 t1 H8))))))))) (pr0_gen_lift t0 t5 (S O) O H4))) t4 H11)) u (sym_eq T u u1 H10))) b (sym_eq B b Abbr H3))) H2)) H1)))]) in (H1 (refl_equal T (THead (Bind Abbr) u1 t4)))))))) t2 H11)) t H9 H10 H6 H7 H8))) | (pr0_zeta b0 H6 t4 t5 H7 u0) \Rightarrow (\lambda (H9: (eq T (THead (Bind b0) u0 (lift (S O) O t4)) t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T (THead (Bind b0) u0 (lift (S O) O t4)) (\lambda (_: T).((eq T t5 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H11: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((not (eq B b0 Abst)) \to ((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (H13: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind b) u (lift (S O) O t0)) t)) H4 (THead (Bind b0) u0 (lift (S O) O t4)) H9) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Bind b0) u0 (lift (S O) O t4))) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind b0) u0 (lift (S O) O t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind b0) u0 (lift (S O) O t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind b0) u0 (lift (S O) O t4)) H0) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind b0) u0 (lift (S O) O t4)) H0) in (eq_ind B b0 (\lambda (_: B).((eq T u u0) \to ((eq T (lift (S O) O t0) (lift (S O) O t4)) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))) (\lambda (H10: (eq T u u0)).(eq_ind T u0 (\lambda (_: T).((eq T (lift (S O) O t0) (lift (S O) O t4)) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H11: (eq T (lift (S O) O t0) (lift (S O) O t4))).(eq_ind T (lift (S O) O t0) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind b0) u0 (lift (S O) O t4)) H9) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t1)) H8 t4 (lift_inj t0 t4 (S O) O H11)) in (let H6 \def (eq_ind B b (\lambda (b: B).(not (eq B b Abst))) H7 b0 H3) in (ex2_ind T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H7: (pr0 t1 x)).(\lambda (H8: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)) x H7 H8)))) (H4 t4 (lift_tlt_dx (Bind b0) u0 t4 (S O) O) t1 H5 t2 H13))))) (lift (S O) O t4) H11)) u (sym_eq T u u0 H10))) b (sym_eq B b b0 H3))) H2)) H1)))]) in (H1 (refl_equal T (THead (Bind b0) u0 (lift (S O) O t4)))))))) t5 (sym_eq T t5 t2 H11))) t H9 H10 H6 H7))) | (pr0_epsilon t4 t5 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 t4) t)).(\lambda (H8: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u0 t4) (\lambda (_: T).((eq T t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H9: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (_: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind b) u (lift (S O) O t0)) t)) H4 (THead (Flat Cast) u0 t4) H7) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Flat Cast) u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat Cast) u0 t4))).(let H1 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u0 t4) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))) H1)))]) in (H1 (refl_equal T (THead (Flat Cast) u0 t4)))))) t5 (sym_eq T t5 t2 H9))) t H7 H8 H6)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t3 (sym_eq T t3 t1 H6))) t H4 H5 H2 H3))) | (pr0_epsilon t0 t3 H2 u) \Rightarrow (\lambda (H3: (eq T (THead (Flat Cast) u t0) t)).(\lambda (H4: (eq T t3 t1)).(eq_ind T (THead (Flat Cast) u t0) (\lambda (_: T).((eq T t3 t1) \to ((pr0 t0 t3) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1)))))) (\lambda (H5: (eq T t3 t1)).(eq_ind T t1 (\lambda (t: T).((pr0 t0 t) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1))))) (\lambda (H6: (pr0 t0 t1)).(let H7 \def (match H1 return (\lambda (t0: T).(\lambda (t3: T).((eq T t0 t) \to ((eq T t3 t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H5: (eq T t4 t)).(\lambda (H7: (eq T t4 t2)).(eq_ind T t (\lambda (t: T).((eq T t t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t2)) H8 (THead (Flat Cast) u t0) H3) in (eq_ind T (THead (Flat Cast) u t0) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t4 t)) H5 (THead (Flat Cast) u t0) H3) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Cast) u t0) H3) in (ex_intro2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Flat Cast) u t0) t)) t1 (pr0_refl t1) (pr0_epsilon t0 t1 H6 u)))) t2 H0)) t (sym_eq T t t2 H8))) t4 (sym_eq T t4 t H5) H7))) | (pr0_comp u1 u2 H5 t4 t5 H6 k) \Rightarrow (\lambda (H7: (eq T (THead k u1 t4) t)).(\lambda (H8: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u1 t4) (\lambda (_: T).((eq T (THead k u2 t5) t2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H9: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u2 t5) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H11: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Cast) u t0) t)) H3 (THead k u1 t4) H7) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead k u1 t4)) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead k u2 t5) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Cast) u t0) (THead k u1 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) u t0) (THead k u1 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u t0) (THead k u1 t4) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow (Flat Cast) | (TLRef _) \Rightarrow (Flat Cast) | (THead k _ _) \Rightarrow k])) (THead (Flat Cast) u t0) (THead k u1 t4) H0) in (eq_ind K (Flat Cast) (\lambda (k: K).((eq T u u1) \to ((eq T t0 t4) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead k u2 t5) t)))))) (\lambda (H8: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Flat Cast) u2 t5) t0))))) (\lambda (H9: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Flat Cast) u2 t5) t0)))) (let H4 \def (eq_ind_r K k (\lambda (k: K).(eq T (THead k u1 t4) t)) H7 (Flat Cast) H3) in (let H5 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Cast) u1 t4) H4) in (let H6 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t1)) H6 t4 H9) in (ex2_ind T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Flat Cast) u2 t5) t))) (\lambda (x: T).(\lambda (H7: (pr0 t1 x)).(\lambda (H10: (pr0 t5 x)).(ex_intro2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Flat Cast) u2 t5) t)) x H7 (pr0_epsilon t5 x H10 u2))))) (H5 t4 (tlt_head_dx (Flat Cast) u1 t4) t1 H6 t5 H11))))) t0 (sym_eq T t0 t4 H9))) u (sym_eq T u u1 H8))) k H3)) H2)) H1)))]) in (H1 (refl_equal T (THead k u1 t4))))))) t2 H9)) t H7 H8 H5 H6))) | (pr0_beta u0 v1 v2 H5 t4 t5 H6) \Rightarrow (\lambda (H7: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) t)).(\lambda (H8: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H9: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) (\lambda (t: T).((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Cast) u t0) t)) H3 (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) H7) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4))) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Cast) u t0) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)))).(let H1 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)))))))) t2 H9)) t H7 H8 H5 H6))) | (pr0_upsilon b H5 v1 v2 H6 u1 u2 H7 t4 t5 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) t)).(\lambda (H10: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))))) (\lambda (H11: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Cast) u t0) t)) H3 (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H9) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Flat Appl) v1 (THead (Bind b) u1 t4))) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Cast) u t0) (THead (Flat Appl) v1 (THead (Bind b) u1 t4)))).(let H1 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t))) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind b) u1 t4)))))))))) t2 H11)) t H9 H10 H5 H6 H7 H8))) | (pr0_delta u1 u2 H5 t4 t5 H6 w H7) \Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u1 t4) t)).(\lambda (H9: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u1 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))) (\lambda (H10: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t4 t5)).(\lambda (_: (subst0 O u2 t5 w)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Cast) u t0) t)) H3 (THead (Bind Abbr) u1 t4) H8) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t4)) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Cast) u t0) (THead (Bind Abbr) u1 t4))).(let H1 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t4) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t))) H1)))]) in (H1 (refl_equal T (THead (Bind Abbr) u1 t4)))))))) t2 H10)) t H8 H9 H5 H6 H7))) | (pr0_zeta b H5 t4 t5 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead (Bind b) u0 (lift (S O) O t4)) t)).(\lambda (H8: (eq T t5 t2)).(eq_ind T (THead (Bind b) u0 (lift (S O) O t4)) (\lambda (_: T).((eq T t5 t2) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H9: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Cast) u t0) t)) H3 (THead (Bind b) u0 (lift (S O) O t4)) H7) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Bind b) u0 (lift (S O) O t4))) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Cast) u t0) (THead (Bind b) u0 (lift (S O) O t4)))).(let H1 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 (lift (S O) O t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))) H1)))]) in (H1 (refl_equal T (THead (Bind b) u0 (lift (S O) O t4)))))))) t5 (sym_eq T t5 t2 H9))) t H7 H8 H5 H6))) | (pr0_epsilon t4 t5 H5 u0) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 t4) t)).(\lambda (H8: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u0 t4) (\lambda (_: T).((eq T t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H9: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H10: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Cast) u t0) t)) H3 (THead (Flat Cast) u0 t4) H7) in (let H1 \def (match H0 return (\lambda (t: T).((eq T t (THead (Flat Cast) u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Cast) u t0) (THead (Flat Cast) u0 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) u t0) (THead (Flat Cast) u0 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u t0) (THead (Flat Cast) u0 t4) H0) in (eq_ind T u0 (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H8: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H3 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Cast) u0 t4) H7) in (let H4 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t1)) H6 t4 H8) in (ex2_ind T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H5: (pr0 t1 x)).(\lambda (H6: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)) x H5 H6)))) (H3 t4 (tlt_head_dx (Flat Cast) u0 t4) t1 H4 t2 H10)))) t0 (sym_eq T t0 t4 H8))) u (sym_eq T u u0 H2))) H1)))]) in (H1 (refl_equal T (THead (Flat Cast) u0 t4)))))) t5 (sym_eq T t5 t2 H9))) t H7 H8 H5)))]) in (H7 (refl_equal T t) (refl_equal T t2)))) t3 (sym_eq T t3 t1 H5))) t H3 H4 H2)))]) in (H2 (refl_equal T t) (refl_equal T t1))))))))) t0).
+ \lambda (t0: T).(tlt_wf_ind (\lambda (t: T).(\forall (t1: T).((pr0 t t1) \to (\forall (t2: T).((pr0 t t2) \to (ex2 T (\lambda (t3: T).(pr0 t1 t3)) (\lambda (t3: T).(pr0 t2 t3)))))))) (\lambda (t: T).(\lambda (H: ((\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))))))))))).(\lambda (t1: T).(\lambda (H0: (pr0 t t1)).(\lambda (t2: T).(\lambda (H1: (pr0 t t2)).(let H2 \def (match H0 return (\lambda (_: ?).(\lambda (t0: T).(\lambda (t3: T).((eq T t0 t) \to ((eq T t3 t1) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))) with [(pr0_refl t0) \Rightarrow (\lambda (H2: (eq T t0 t)).(\lambda (H3: (eq T t0 t1)).(eq_ind T t (\lambda (t: T).((eq T t t1) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1))))) (\lambda (H4: (eq T t t1)).(eq_ind T t1 (\lambda (_: T).(ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1)))) (let H5 \def (match H1 return (\lambda (_: ?).(\lambda (t0: T).(\lambda (t3: T).((eq T t0 t) \to ((eq T t3 t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))) with [(pr0_refl t3) \Rightarrow (\lambda (H5: (eq T t3 t)).(\lambda (H6: (eq T t3 t2)).(eq_ind T t (\lambda (t: T).((eq T t t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind T t (\lambda (t: T).(eq T t3 t)) H5 t2 H7) in (let H1 \def (eq_ind T t (\lambda (t: T).(eq T t t1)) H4 t2 H7) in (let H2 \def (eq_ind T t (\lambda (t: T).(eq T t0 t)) H2 t2 H7) in (let H3 \def (eq_ind T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H t2 H7) in (let H4 \def (eq_ind T t2 (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H3 t1 H1) in (eq_ind_r T t1 (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))) (let H8 \def (eq_ind T t2 (\lambda (t: T).(eq T t0 t)) H2 t1 H1) in (ex_intro2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t1 t)) t1 (pr0_refl t1) (pr0_refl t1))) t2 H1)))))) t (sym_eq T t t2 H7))) t3 (sym_eq T t3 t H5) H6))) | (pr0_comp u1 u2 H4 t3 t4 H5 k) \Rightarrow (\lambda (H6: (eq T (THead k u1 t3) t)).(\lambda (H7: (eq T (THead k u2 t4) t2)).(eq_ind T (THead k u1 t3) (\lambda (_: T).((eq T (THead k u2 t4) t2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H8: (eq T (THead k u2 t4) t2)).(eq_ind T (THead k u2 t4) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (H9: (pr0 u1 u2)).(\lambda (H10: (pr0 t3 t4)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t1)) H4 (THead k u1 t3) H6) in (eq_ind T (THead k u1 t3) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t t0)) (\lambda (t0: T).(pr0 (THead k u2 t4) t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t0 t)) H2 (THead k u1 t3) H6) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead k u1 t3) H6) in (ex_intro2 T (\lambda (t: T).(pr0 (THead k u1 t3) t)) (\lambda (t: T).(pr0 (THead k u2 t4) t)) (THead k u2 t4) (pr0_comp u1 u2 H9 t3 t4 H10 k) (pr0_refl (THead k u2 t4))))) t1 H0)))) t2 H8)) t H6 H7 H4 H5))) | (pr0_beta u v1 v2 H4 t3 t4 H5) \Rightarrow (\lambda (H6: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t)).(\lambda (H7: (eq T (THead (Bind Abbr) v2 t4) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t4) t2) \to ((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H8: (eq T (THead (Bind Abbr) v2 t4) t2)).(eq_ind T (THead (Bind Abbr) v2 t4) (\lambda (t: T).((pr0 v1 v2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (H9: (pr0 v1 v2)).(\lambda (H10: (pr0 t3 t4)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t1)) H4 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H6) in (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t4) t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t0 t)) H2 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H6) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) H6) in (ex_intro2 T (\lambda (t: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u t3)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t4) t)) (THead (Bind Abbr) v2 t4) (pr0_beta u v1 v2 H9 t3 t4 H10) (pr0_refl (THead (Bind Abbr) v2 t4))))) t1 H0)))) t2 H8)) t H6 H7 H4 H5))) | (pr0_upsilon b H4 v1 v2 H5 u1 u2 H6 t3 t4 H7) \Rightarrow (\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) t)).(\lambda (H9: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))))) (\lambda (H10: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))))))) (\lambda (H11: (not (eq B b Abst))).(\lambda (H12: (pr0 v1 v2)).(\lambda (H13: (pr0 u1 u2)).(\lambda (H14: (pr0 t3 t4)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t1)) H4 (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) H8) in (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t4)) t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t0 t)) H2 (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) H8) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t3)) H8) in (pr0_confluence__pr0_cong_upsilon_refl b H11 u1 u2 H13 t3 t4 H14 v1 v2 v2 H12 (pr0_refl v2)))) t1 H0)))))) t2 H10)) t H8 H9 H4 H5 H6 H7))) | (pr0_delta u1 u2 H4 t3 t4 H5 w H6) \Rightarrow (\lambda (H7: (eq T (THead (Bind Abbr) u1 t3) t)).(\lambda (H8: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))) (\lambda (H9: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t3 t4) \to ((subst0 O u2 t4 w) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0))))))) (\lambda (H10: (pr0 u1 u2)).(\lambda (H11: (pr0 t3 t4)).(\lambda (H12: (subst0 O u2 t4 w)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t1)) H4 (THead (Bind Abbr) u1 t3) H7) in (eq_ind T (THead (Bind Abbr) u1 t3) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t0 t)) H2 (THead (Bind Abbr) u1 t3) H7) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind Abbr) u1 t3) H7) in (ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u1 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (THead (Bind Abbr) u2 w) (pr0_delta u1 u2 H10 t3 t4 H11 w H12) (pr0_refl (THead (Bind Abbr) u2 w))))) t1 H0))))) t2 H9)) t H7 H8 H4 H5 H6))) | (pr0_zeta b H4 t3 t4 H5 u) \Rightarrow (\lambda (H6: (eq T (THead (Bind b) u (lift (S O) O t3)) t)).(\lambda (H7: (eq T t4 t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t3)) (\lambda (_: T).((eq T t4 t2) \to ((not (eq B b Abst)) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H8: (eq T t4 t2)).(eq_ind T t2 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t3 t) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H9: (not (eq B b Abst))).(\lambda (H10: (pr0 t3 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t1)) H4 (THead (Bind b) u (lift (S O) O t3)) H6) in (eq_ind T (THead (Bind b) u (lift (S O) O t3)) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t0 t)) H2 (THead (Bind b) u (lift (S O) O t3)) H6) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind b) u (lift (S O) O t3)) H6) in (ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind b) u (lift (S O) O t3)) t)) (\lambda (t: T).(pr0 t2 t)) t2 (pr0_zeta b H9 t3 t2 H10 u) (pr0_refl t2)))) t1 H0)))) t4 (sym_eq T t4 t2 H8))) t H6 H7 H4 H5))) | (pr0_epsilon t3 t4 H4 u) \Rightarrow (\lambda (H5: (eq T (THead (Flat Cast) u t3) t)).(\lambda (H6: (eq T t4 t2)).(eq_ind T (THead (Flat Cast) u t3) (\lambda (_: T).((eq T t4 t2) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H7: (eq T t4 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t3 t) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H8: (pr0 t3 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t1)) H4 (THead (Flat Cast) u t3) H5) in (eq_ind T (THead (Flat Cast) u t3) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t0 t)) H2 (THead (Flat Cast) u t3) H5) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Cast) u t3) H5) in (ex_intro2 T (\lambda (t: T).(pr0 (THead (Flat Cast) u t3) t)) (\lambda (t: T).(pr0 t2 t)) t2 (pr0_epsilon t3 t2 H8 u) (pr0_refl t2)))) t1 H0))) t4 (sym_eq T t4 t2 H7))) t H5 H6 H4)))]) in (H5 (refl_equal T t) (refl_equal T t2))) t (sym_eq T t t1 H4))) t0 (sym_eq T t0 t H2) H3))) | (pr0_comp u1 u2 H2 t0 t3 H3 k) \Rightarrow (\lambda (H4: (eq T (THead k u1 t0) t)).(\lambda (H5: (eq T (THead k u2 t3) t1)).(eq_ind T (THead k u1 t0) (\lambda (_: T).((eq T (THead k u2 t3) t1) \to ((pr0 u1 u2) \to ((pr0 t0 t3) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1))))))) (\lambda (H6: (eq T (THead k u2 t3) t1)).(eq_ind T (THead k u2 t3) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t0 t3) \to (ex2 T (\lambda (t1: T).(pr0 t t1)) (\lambda (t1: T).(pr0 t2 t1)))))) (\lambda (H7: (pr0 u1 u2)).(\lambda (H8: (pr0 t0 t3)).(let H9 \def (match H1 return (\lambda (_: ?).(\lambda (t0: T).(\lambda (t1: T).((eq T t0 t) \to ((eq T t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 t2 t)))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H6: (eq T t4 t)).(\lambda (H9: (eq T t4 t2)).(eq_ind T t (\lambda (t: T).((eq T t t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H10: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t2)) H10 (THead k u1 t0) H4) in (eq_ind T (THead k u1 t0) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t4 t)) H6 (THead k u1 t0) H4) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead k u1 t0) H4) in (ex_intro2 T (\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 (THead k u1 t0) t)) (THead k u2 t3) (pr0_refl (THead k u2 t3)) (pr0_comp u1 u2 H7 t0 t3 H8 k)))) t2 H0)) t (sym_eq T t t2 H10))) t4 (sym_eq T t4 t H6) H9))) | (pr0_comp u0 u3 H6 t4 t5 H7 k0) \Rightarrow (\lambda (H9: (eq T (THead k0 u0 t4) t)).(\lambda (H10: (eq T (THead k0 u3 t5) t2)).(eq_ind T (THead k0 u0 t4) (\lambda (_: T).((eq T (THead k0 u3 t5) t2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H11: (eq T (THead k0 u3 t5) t2)).(eq_ind T (THead k0 u3 t5) (\lambda (t: T).((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (H12: (pr0 u0 u3)).(\lambda (H13: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead k u1 t0) t)) H4 (THead k0 u0 t4) H9) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead k0 u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 (THead k0 u3 t5) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead k u1 t0) (THead k0 u0 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u1 t0) (THead k0 u0 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead k u1 t0) (THead k0 u0 t4) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u1 t0) (THead k0 u0 t4) H0) in (eq_ind K k0 (\lambda (k: K).((eq T u1 u0) \to ((eq T t0 t4) \to (ex2 T (\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 (THead k0 u3 t5) t)))))) (\lambda (H10: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead k0 u2 t3) t0)) (\lambda (t0: T).(pr0 (THead k0 u3 t5) t0))))) (\lambda (H11: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead k0 u2 t3) t0)) (\lambda (t0: T).(pr0 (THead k0 u3 t5) t0)))) (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead k0 u0 t4) H9) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8 t4 H11) in (let H6 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 u0 H10) in (ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead k0 u2 t3) t)) (\lambda (t: T).(pr0 (THead k0 u3 t5) t))) (\lambda (x: T).(\lambda (H7: (pr0 u2 x)).(\lambda (H8: (pr0 u3 x)).(ex2_ind T (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead k0 u2 t3) t)) (\lambda (t: T).(pr0 (THead k0 u3 t5) t))) (\lambda (x0: T).(\lambda (H9: (pr0 t3 x0)).(\lambda (H12: (pr0 t5 x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead k0 u2 t3) t)) (\lambda (t: T).(pr0 (THead k0 u3 t5) t)) (THead k0 x x0) (pr0_comp u2 x H7 t3 x0 H9 k0) (pr0_comp u3 x H8 t5 x0 H12 k0))))) (H4 t4 (tlt_head_dx k0 u0 t4) t3 H5 t5 H13))))) (H4 u0 (tlt_head_sx k0 u0 t4) u2 H6 u3 H12))))) t0 (sym_eq T t0 t4 H11))) u1 (sym_eq T u1 u0 H10))) k (sym_eq K k k0 H3))) H2)) H1)))]) in (H1 (refl_equal T (THead k0 u0 t4))))))) t2 H11)) t H9 H10 H6 H7))) | (pr0_beta u v1 v2 H6 t4 t5 H7) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) t)).(\lambda (H10: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H11: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) (\lambda (t: T).((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (H12: (pr0 v1 v2)).(\lambda (H13: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead k u1 t0) t)) H4 (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H9) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Flat Appl) v1 (THead (Bind Abst) u t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead k u1 t0) (THead (Flat Appl) v1 (THead (Bind Abst) u t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u1 t0) (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead k u1 t0) (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u1 t0) (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H0) in (eq_ind K (Flat Appl) (\lambda (k: K).((eq T u1 v1) \to ((eq T t0 (THead (Bind Abst) u t4)) \to (ex2 T (\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t)))))) (\lambda (H10: (eq T u1 v1)).(eq_ind T v1 (\lambda (_: T).((eq T t0 (THead (Bind Abst) u t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0))))) (\lambda (H11: (eq T t0 (THead (Bind Abst) u t4))).(eq_ind T (THead (Bind Abst) u t4) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)))) (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H9) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8 (THead (Bind Abst) u t4) H11) in (let H6 \def (match H5 return (\lambda (_: ?).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Bind Abst) u t4)) \to ((eq T t0 t3) \to (ex2 T (\lambda (t1: T).(pr0 (THead (Flat Appl) u2 t3) t1)) (\lambda (t1: T).(pr0 (THead (Bind Abbr) v2 t5) t1)))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Bind Abst) u t4))).(\lambda (H5: (eq T t t3)).(eq_ind T (THead (Bind Abst) u t4) (\lambda (t0: T).((eq T t0 t3) \to (ex2 T (\lambda (t1: T).(pr0 (THead (Flat Appl) u2 t3) t1)) (\lambda (t1: T).(pr0 (THead (Bind Abbr) v2 t5) t1))))) (\lambda (H6: (eq T (THead (Bind Abst) u t4) t3)).(eq_ind T (THead (Bind Abst) u t4) (\lambda (t0: T).(ex2 T (\lambda (t1: T).(pr0 (THead (Flat Appl) u2 t0) t1)) (\lambda (t1: T).(pr0 (THead (Bind Abbr) v2 t5) t1)))) (let H1 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 v1 H10) in (ex2_ind T (\lambda (t0: T).(pr0 u2 t0)) (\lambda (t0: T).(pr0 v2 t0)) (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u t4)) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0))) (\lambda (x: T).(\lambda (H2: (pr0 u2 x)).(\lambda (H3: (pr0 v2 x)).(ex_intro2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u t4)) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)) (THead (Bind Abbr) x t5) (pr0_beta u u2 x H2 t4 t5 H13) (pr0_comp v2 x H3 t5 t5 (pr0_refl t5) (Bind Abbr)))))) (H4 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t4)) u2 H1 v2 H12))) t3 H6)) t (sym_eq T t (THead (Bind Abst) u t4) H0) H5))) | (pr0_comp u0 u3 H0 t1 t2 H4 k0) \Rightarrow (\lambda (H5: (eq T (THead k0 u0 t1) (THead (Bind Abst) u t4))).(\lambda (H8: (eq T (THead k0 u3 t2) t3)).((let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t])) (THead k0 u0 t1) (THead (Bind Abst) u t4) H5) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k0 u0 t1) (THead (Bind Abst) u t4) H5) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k _ _) \Rightarrow k])) (THead k0 u0 t1) (THead (Bind Abst) u t4) H5) in (eq_ind K (Bind Abst) (\lambda (k: K).((eq T u0 u) \to ((eq T t1 t4) \to ((eq T (THead k u3 t2) t3) \to ((pr0 u0 u3) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))))))))) (\lambda (H6: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq T t1 t4) \to ((eq T (THead (Bind Abst) u3 t2) t3) \to ((pr0 t u3) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)))))))) (\lambda (H9: (eq T t1 t4)).(eq_ind T t4 (\lambda (t: T).((eq T (THead (Bind Abst) u3 t2) t3) \to ((pr0 u u3) \to ((pr0 t t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0))))))) (\lambda (H11: (eq T (THead (Bind Abst) u3 t2) t3)).(eq_ind T (THead (Bind Abst) u3 t2) (\lambda (t: T).((pr0 u u3) \to ((pr0 t4 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)))))) (\lambda (_: (pr0 u u3)).(\lambda (H15: (pr0 t4 t2)).(let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 v1 H10) in (ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 v2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t2)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))) (\lambda (x: T).(\lambda (H10: (pr0 u2 x)).(\lambda (H12: (pr0 v2 x)).(ex2_ind T (\lambda (t: T).(pr0 t2 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t2)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))) (\lambda (x0: T).(\lambda (H13: (pr0 t2 x0)).(\lambda (H16: (pr0 t5 x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t2)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t)) (THead (Bind Abbr) x x0) (pr0_beta u3 u2 x H10 t2 x0 H13) (pr0_comp v2 x H12 t5 x0 H16 (Bind Abbr)))))) (H4 t4 (tlt_trans (THead (Bind Abst) u t4) t4 (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) (tlt_head_dx (Bind Abst) u t4) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abst) u t4))) t2 H15 t5 H13))))) (H4 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind Abst) u t4)) u2 H7 v2 H12))))) t3 H11)) t1 (sym_eq T t1 t4 H9))) u0 (sym_eq T u0 u H6))) k0 (sym_eq K k0 (Bind Abst) H3))) H2)) H1)) H8 H0 H4))) | (pr0_beta u0 v0 v3 H0 t1 t2 H4) \Rightarrow (\lambda (H5: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t1)) (THead (Bind Abst) u t4))).(\lambda (H8: (eq T (THead (Bind Abbr) v3 t2) t3)).((let H1 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t1)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t4) H5) in (False_ind ((eq T (THead (Bind Abbr) v3 t2) t3) \to ((pr0 v0 v3) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t)))))) H1)) H8 H0 H4))) | (pr0_upsilon b H0 v0 v3 H4 u0 u3 H5 t1 t2 H8) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v0 (THead (Bind b) u0 t1)) (THead (Bind Abst) u t4))).(\lambda (H12: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t2)) t3)).((let H1 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u0 t1)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t4) H11) in (False_ind ((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t2)) t3) \to ((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t)))))))) H1)) H12 H0 H4 H5 H8))) | (pr0_delta u0 u3 H0 t1 t2 H4 w H5) \Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u0 t1) (THead (Bind Abst) u t4))).(\lambda (H11: (eq T (THead (Bind Abbr) u3 w) t3)).((let H1 \def (eq_ind T (THead (Bind Abbr) u0 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind Abst) u t4) H8) in (False_ind ((eq T (THead (Bind Abbr) u3 w) t3) \to ((pr0 u0 u3) \to ((pr0 t1 t2) \to ((subst0 O u3 t2 w) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))))))) H1)) H11 H0 H4 H5))) | (pr0_zeta b H0 t1 t2 H4 u0) \Rightarrow (\lambda (H5: (eq T (THead (Bind b) u0 (lift (S O) O t1)) (THead (Bind Abst) u t4))).(\lambda (H8: (eq T t2 t3)).((let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t1) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t1) | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u0 (lift (S O) O t1)) (THead (Bind Abst) u t4) H5) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead (Bind b) u0 (lift (S O) O t1)) (THead (Bind Abst) u t4) H5) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O t1)) (THead (Bind Abst) u t4) H5) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S O) O t1) t4) \to ((eq T t2 t3) \to ((not (eq B b0 Abst)) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))))))))) (\lambda (H6: (eq T u0 u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t1) t4) \to ((eq T t2 t3) \to ((not (eq B Abst Abst)) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)))))))) (\lambda (H9: (eq T (lift (S O) O t1) t4)).(eq_ind T (lift (S O) O t1) (\lambda (_: T).((eq T t2 t3) \to ((not (eq B Abst Abst)) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0))))))) (\lambda (H7: (eq T t2 t3)).(eq_ind T t3 (\lambda (t: T).((not (eq B Abst Abst)) \to ((pr0 t1 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)))))) (\lambda (H11: (not (eq B Abst Abst))).(\lambda (_: (pr0 t1 t3)).(let H10 \def (match (H11 (refl_equal B Abst)) return (\lambda (_: ?).(ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t)))) with []) in H10))) t2 (sym_eq T t2 t3 H7))) t4 H9)) u0 (sym_eq T u0 u H6))) b (sym_eq B b Abst H3))) H2)) H1)) H8 H0 H4))) | (pr0_epsilon t1 t2 H0 u0) \Rightarrow (\lambda (H4: (eq T (THead (Flat Cast) u0 t1) (THead (Bind Abst) u t4))).(\lambda (H5: (eq T t2 t3)).((let H1 \def (eq_ind T (THead (Flat Cast) u0 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t4) H4) in (False_ind ((eq T t2 t3) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))))) H1)) H5 H0)))]) in (H6 (refl_equal T (THead (Bind Abst) u t4)) (refl_equal T t3))))) t0 (sym_eq T t0 (THead (Bind Abst) u t4) H11))) u1 (sym_eq T u1 v1 H10))) k (sym_eq K k (Flat Appl) H3))) H2)) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)))))))) t2 H11)) t H9 H10 H6 H7))) | (pr0_upsilon b H6 v1 v2 H7 u0 u3 H8 t4 t5 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) t)).(\lambda (H11: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) (\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))))) (\lambda (H12: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t t0)))))))) (\lambda (H13: (not (eq B b Abst))).(\lambda (H14: (pr0 v1 v2)).(\lambda (H15: (pr0 u0 u3)).(\lambda (H16: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead k u1 t0) t)) H4 (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) H10) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Flat Appl) v1 (THead (Bind b) u0 t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead k u1 t0) (THead (Flat Appl) v1 (THead (Bind b) u0 t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u1 t0) (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead k u1 t0) (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u1 t0) (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) H0) in (eq_ind K (Flat Appl) (\lambda (k: K).((eq T u1 v1) \to ((eq T t0 (THead (Bind b) u0 t4)) \to (ex2 T (\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)))))) (\lambda (H11: (eq T u1 v1)).(eq_ind T v1 (\lambda (_: T).((eq T t0 (THead (Bind b) u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0))))) (\lambda (H12: (eq T t0 (THead (Bind b) u0 t4))).(eq_ind T (THead (Bind b) u0 t4) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0)))) (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) H10) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8 (THead (Bind b) u0 t4) H12) in (let H6 \def (match H5 return (\lambda (_: ?).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Bind b) u0 t4)) \to ((eq T t0 t3) \to (ex2 T (\lambda (t1: T).(pr0 (THead (Flat Appl) u2 t3) t1)) (\lambda (t1: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t1)))))))) with [(pr0_refl t) \Rightarrow (\lambda (H0: (eq T t (THead (Bind b) u0 t4))).(\lambda (H5: (eq T t t3)).(eq_ind T (THead (Bind b) u0 t4) (\lambda (t0: T).((eq T t0 t3) \to (ex2 T (\lambda (t1: T).(pr0 (THead (Flat Appl) u2 t3) t1)) (\lambda (t1: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t1))))) (\lambda (H6: (eq T (THead (Bind b) u0 t4) t3)).(eq_ind T (THead (Bind b) u0 t4) (\lambda (t0: T).(ex2 T (\lambda (t1: T).(pr0 (THead (Flat Appl) u2 t0) t1)) (\lambda (t1: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t1)))) (let H1 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 v1 H11) in (ex2_ind T (\lambda (t0: T).(pr0 u2 t0)) (\lambda (t0: T).(pr0 v2 t0)) (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u0 t4)) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0))) (\lambda (x: T).(\lambda (H2: (pr0 u2 x)).(\lambda (H3: (pr0 v2 x)).(pr0_confluence__pr0_cong_upsilon_refl b H13 u0 u3 H15 t4 t5 H16 u2 v2 x H2 H3)))) (H4 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind b) u0 t4)) u2 H1 v2 H14))) t3 H6)) t (sym_eq T t (THead (Bind b) u0 t4) H0) H5))) | (pr0_comp u4 u5 H0 t1 t2 H4 k0) \Rightarrow (\lambda (H5: (eq T (THead k0 u4 t1) (THead (Bind b) u0 t4))).(\lambda (H10: (eq T (THead k0 u5 t2) t3)).((let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t])) (THead k0 u4 t1) (THead (Bind b) u0 t4) H5) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t _) \Rightarrow t])) (THead k0 u4 t1) (THead (Bind b) u0 t4) H5) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k0 | (TLRef _) \Rightarrow k0 | (THead k _ _) \Rightarrow k])) (THead k0 u4 t1) (THead (Bind b) u0 t4) H5) in (eq_ind K (Bind b) (\lambda (k: K).((eq T u4 u0) \to ((eq T t1 t4) \to ((eq T (THead k u5 t2) t3) \to ((pr0 u4 u5) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))))))))) (\lambda (H6: (eq T u4 u0)).(eq_ind T u0 (\lambda (t: T).((eq T t1 t4) \to ((eq T (THead (Bind b) u5 t2) t3) \to ((pr0 t u5) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0)))))))) (\lambda (H12: (eq T t1 t4)).(eq_ind T t4 (\lambda (t: T).((eq T (THead (Bind b) u5 t2) t3) \to ((pr0 u0 u5) \to ((pr0 t t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0))))))) (\lambda (H8: (eq T (THead (Bind b) u5 t2) t3)).(eq_ind T (THead (Bind b) u5 t2) (\lambda (t: T).((pr0 u0 u5) \to ((pr0 t4 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0)))))) (\lambda (H17: (pr0 u0 u5)).(\lambda (H18: (pr0 t4 t2)).(let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 v1 H11) in (ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 v2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x: T).(\lambda (H9: (pr0 u2 x)).(\lambda (H11: (pr0 v2 x)).(ex2_ind T (\lambda (t: T).(pr0 t2 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x0: T).(\lambda (H14: (pr0 t2 x0)).(\lambda (H16: (pr0 t5 x0)).(ex2_ind T (\lambda (t: T).(pr0 u5 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind b) u5 t2)) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x1: T).(\lambda (H15: (pr0 u5 x1)).(\lambda (H19: (pr0 u3 x1)).(pr0_confluence__pr0_cong_upsilon_cong b H13 u2 v2 x H9 H11 t2 t5 x0 H14 H16 u5 u3 x1 H15 H19)))) (H4 u0 (tlt_trans (THead (Bind b) u0 t4) u0 (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) (tlt_head_sx (Bind b) u0 t4) (tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 t4))) u5 H17 u3 H15))))) (H4 t4 (tlt_trans (THead (Bind b) u0 t4) t4 (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) (tlt_head_dx (Bind b) u0 t4) (tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 t4))) t2 H18 t5 H16))))) (H4 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind b) u0 t4)) u2 H7 v2 H14))))) t3 H8)) t1 (sym_eq T t1 t4 H12))) u4 (sym_eq T u4 u0 H6))) k0 (sym_eq K k0 (Bind b) H3))) H2)) H1)) H10 H0 H4))) | (pr0_beta u v0 v3 H0 t1 t2 H4) \Rightarrow (\lambda (H5: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t1)) (THead (Bind b) u0 t4))).(\lambda (H10: (eq T (THead (Bind Abbr) v3 t2) t3)).((let H1 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t1)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 t4) H5) in (False_ind ((eq T (THead (Bind Abbr) v3 t2) t3) \to ((pr0 v0 v3) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)))))) H1)) H10 H0 H4))) | (pr0_upsilon b0 H0 v0 v3 H4 u4 u5 H5 t1 t2 H10) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 t1)) (THead (Bind b) u0 t4))).(\lambda (H14: (eq T (THead (Bind b0) u5 (THead (Flat Appl) (lift (S O) O v3) t2)) t3)).((let H1 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b0) u4 t1)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 t4) H13) in (False_ind ((eq T (THead (Bind b0) u5 (THead (Flat Appl) (lift (S O) O v3) t2)) t3) \to ((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u4 u5) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)))))))) H1)) H14 H0 H4 H5 H10))) | (pr0_delta u4 u5 H0 t1 t2 H4 w H5) \Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr) u4 t1) (THead (Bind b) u0 t4))).(\lambda (H17: (eq T (THead (Bind Abbr) u5 w) t3)).((let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abbr) u4 t1) (THead (Bind b) u0 t4) H10) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t _) \Rightarrow t])) (THead (Bind Abbr) u4 t1) (THead (Bind b) u0 t4) H10) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t1) (THead (Bind b) u0 t4) H10) in (eq_ind B Abbr (\lambda (b: B).((eq T u4 u0) \to ((eq T t1 t4) \to ((eq T (THead (Bind Abbr) u5 w) t3) \to ((pr0 u4 u5) \to ((pr0 t1 t2) \to ((subst0 O u5 t2 w) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t)))))))))) (\lambda (H6: (eq T u4 u0)).(eq_ind T u0 (\lambda (t: T).((eq T t1 t4) \to ((eq T (THead (Bind Abbr) u5 w) t3) \to ((pr0 t u5) \to ((pr0 t1 t2) \to ((subst0 O u5 t2 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0))))))))) (\lambda (H8: (eq T t1 t4)).(eq_ind T t4 (\lambda (t: T).((eq T (THead (Bind Abbr) u5 w) t3) \to ((pr0 u0 u5) \to ((pr0 t t2) \to ((subst0 O u5 t2 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0)))))))) (\lambda (H18: (eq T (THead (Bind Abbr) u5 w) t3)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t: T).((pr0 u0 u5) \to ((pr0 t4 t2) \to ((subst0 O u5 t2 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0))))))) (\lambda (H19: (pr0 u0 u5)).(\lambda (H20: (pr0 t4 t2)).(\lambda (H21: (subst0 O u5 t2 w)).(let H9 \def (eq_ind_r B b (\lambda (b: B).(\forall (v: T).((tlt v (THead (Flat Appl) v1 (THead (Bind b) u0 t4))) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))))) H4 Abbr H3) in (let H12 \def (eq_ind_r B b (\lambda (b: B).(eq T t0 (THead (Bind b) u0 t4))) H12 Abbr H3) in (let H13 \def (eq_ind_r B b (\lambda (b: B).(not (eq B b Abst))) H13 Abbr H3) in (let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 v1 H11) in (ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 v2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x: T).(\lambda (H11: (pr0 u2 x)).(\lambda (H14: (pr0 v2 x)).(ex2_ind T (\lambda (t: T).(pr0 t2 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x0: T).(\lambda (H16: (pr0 t2 x0)).(\lambda (H22: (pr0 t5 x0)).(ex2_ind T (\lambda (t: T).(pr0 u5 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abbr) u5 w)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x1: T).(\lambda (H15: (pr0 u5 x1)).(\lambda (H23: (pr0 u3 x1)).(pr0_confluence__pr0_cong_upsilon_delta H13 u5 t2 w H21 u2 v2 x H11 H14 t5 x0 H16 H22 u3 x1 H15 H23)))) (H9 u0 (tlt_trans (THead (Bind Abbr) u0 t4) u0 (THead (Flat Appl) v1 (THead (Bind Abbr) u0 t4)) (tlt_head_sx (Bind Abbr) u0 t4) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t4))) u5 H19 u3 H15))))) (H9 t4 (tlt_trans (THead (Bind Abbr) u0 t4) t4 (THead (Flat Appl) v1 (THead (Bind Abbr) u0 t4)) (tlt_head_dx (Bind Abbr) u0 t4) (tlt_head_dx (Flat Appl) v1 (THead (Bind Abbr) u0 t4))) t2 H20 t5 H16))))) (H9 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind Abbr) u0 t4)) u2 H7 v2 H14))))))))) t3 H18)) t1 (sym_eq T t1 t4 H8))) u4 (sym_eq T u4 u0 H6))) b H3)) H2)) H1)) H17 H0 H4 H5))) | (pr0_zeta b0 H0 t1 t2 H4 u) \Rightarrow (\lambda (H5: (eq T (THead (Bind b0) u (lift (S O) O t1)) (THead (Bind b) u0 t4))).(\lambda (H10: (eq T t2 t3)).((let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t1) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t1) | (THead _ _ t) \Rightarrow t])) (THead (Bind b0) u (lift (S O) O t1)) (THead (Bind b) u0 t4) H5) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b0) u (lift (S O) O t1)) (THead (Bind b) u0 t4) H5) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u (lift (S O) O t1)) (THead (Bind b) u0 t4) H5) in (eq_ind B b (\lambda (b1: B).((eq T u u0) \to ((eq T (lift (S O) O t1) t4) \to ((eq T t2 t3) \to ((not (eq B b1 Abst)) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))))))))) (\lambda (H6: (eq T u u0)).(eq_ind T u0 (\lambda (_: T).((eq T (lift (S O) O t1) t4) \to ((eq T t2 t3) \to ((not (eq B b Abst)) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0)))))))) (\lambda (H13: (eq T (lift (S O) O t1) t4)).(eq_ind T (lift (S O) O t1) (\lambda (_: T).((eq T t2 t3) \to ((not (eq B b Abst)) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0))))))) (\lambda (H8: (eq T t2 t3)).(eq_ind T t3 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t1 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0)))))) (\lambda (H17: (not (eq B b Abst))).(\lambda (H18: (pr0 t1 t3)).(let H9 \def (eq_ind_r T t4 (\lambda (t: T).(\forall (v: T).((tlt v (THead (Flat Appl) v1 (THead (Bind b) u0 t))) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H4 (lift (S O) O t1) H13) in (let H12 \def (eq_ind_r T t4 (\lambda (t: T).(eq T t0 (THead (Bind b) u0 t))) H12 (lift (S O) O t1) H13) in (let H16 \def (eq_ind_r T t4 (\lambda (t: T).(pr0 t t5)) H16 (lift (S O) O t1) H13) in (ex2_ind T (\lambda (t3: T).(eq T t5 (lift (S O) O t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))) (\lambda (x: T).(\lambda (H19: (eq T t5 (lift (S O) O x))).(\lambda (H20: (pr0 t1 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t)) t0)))) (let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 v1 H11) in (ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 v2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O x))) t))) (\lambda (x0: T).(\lambda (H11: (pr0 u2 x0)).(\lambda (H14: (pr0 v2 x0)).(ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O x))) t))) (\lambda (x1: T).(\lambda (H21: (pr0 x x1)).(\lambda (H22: (pr0 t3 x1)).(pr0_confluence__pr0_cong_upsilon_zeta b H17 u0 u3 H15 u2 v2 x0 H11 H14 x t3 x1 H21 H22)))) (H9 t1 (tlt_trans (THead (Bind b) u0 (lift (S O) O t1)) t1 (THead (Flat Appl) v1 (THead (Bind b) u0 (lift (S O) O t1))) (lift_tlt_dx (Bind b) u0 t1 (S O) O) (tlt_head_dx (Flat Appl) v1 (THead (Bind b) u0 (lift (S O) O t1)))) x H20 t3 H18))))) (H9 v1 (tlt_head_sx (Flat Appl) v1 (THead (Bind b) u0 (lift (S O) O t1))) u2 H7 v2 H14))) t5 H19)))) (pr0_gen_lift t1 t5 (S O) O H16))))))) t2 (sym_eq T t2 t3 H8))) t4 H13)) u (sym_eq T u u0 H6))) b0 (sym_eq B b0 b H3))) H2)) H1)) H10 H0 H4))) | (pr0_epsilon t1 t2 H0 u) \Rightarrow (\lambda (H4: (eq T (THead (Flat Cast) u t1) (THead (Bind b) u0 t4))).(\lambda (H5: (eq T t2 t3)).((let H1 \def (eq_ind T (THead (Flat Cast) u t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 t4) H4) in (False_ind ((eq T t2 t3) \to ((pr0 t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))))) H1)) H5 H0)))]) in (H6 (refl_equal T (THead (Bind b) u0 t4)) (refl_equal T t3))))) t0 (sym_eq T t0 (THead (Bind b) u0 t4) H12))) u1 (sym_eq T u1 v1 H11))) k (sym_eq K k (Flat Appl) H3))) H2)) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind b) u0 t4)))))))))) t2 H12)) t H10 H11 H6 H7 H8 H9))) | (pr0_delta u0 u3 H6 t4 t5 H7 w H8) \Rightarrow (\lambda (H9: (eq T (THead (Bind Abbr) u0 t4) t)).(\lambda (H10: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind Abbr) u0 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u3 w) t2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to ((subst0 O u3 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))))) (\lambda (H11: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t: T).((pr0 u0 u3) \to ((pr0 t4 t5) \to ((subst0 O u3 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t t0))))))) (\lambda (H12: (pr0 u0 u3)).(\lambda (H13: (pr0 t4 t5)).(\lambda (H14: (subst0 O u3 t5 w)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead k u1 t0) t)) H4 (THead (Bind Abbr) u0 t4) H9) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Bind Abbr) u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 w) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead k u1 t0) (THead (Bind Abbr) u0 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u1 t0) (THead (Bind Abbr) u0 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead k u1 t0) (THead (Bind Abbr) u0 t4) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u1 t0) (THead (Bind Abbr) u0 t4) H0) in (eq_ind K (Bind Abbr) (\lambda (k: K).((eq T u1 u0) \to ((eq T t0 t4) \to (ex2 T (\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t)))))) (\lambda (H10: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 w) t0))))) (\lambda (H11: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 w) t0)))) (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind Abbr) u0 t4) H9) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8 t4 H11) in (let H6 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 u0 H10) in (ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t))) (\lambda (x: T).(\lambda (H7: (pr0 u2 x)).(\lambda (H8: (pr0 u3 x)).(ex2_ind T (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t))) (\lambda (x0: T).(\lambda (H9: (pr0 t3 x0)).(\lambda (H12: (pr0 t5 x0)).(pr0_confluence__pr0_cong_delta u3 t5 w H14 u2 x H7 H8 t3 x0 H9 H12)))) (H4 t4 (tlt_head_dx (Bind Abbr) u0 t4) t3 H5 t5 H13))))) (H4 u0 (tlt_head_sx (Bind Abbr) u0 t4) u2 H6 u3 H12))))) t0 (sym_eq T t0 t4 H11))) u1 (sym_eq T u1 u0 H10))) k (sym_eq K k (Bind Abbr) H3))) H2)) H1)))]) in (H1 (refl_equal T (THead (Bind Abbr) u0 t4)))))))) t2 H11)) t H9 H10 H6 H7 H8))) | (pr0_zeta b H6 t4 t5 H7 u) \Rightarrow (\lambda (H9: (eq T (THead (Bind b) u (lift (S O) O t4)) t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (_: T).((eq T t5 t2) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H11: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H12: (not (eq B b Abst))).(\lambda (H13: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead k u1 t0) t)) H4 (THead (Bind b) u (lift (S O) O t4)) H9) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Bind b) u (lift (S O) O t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead k u1 t0) (THead (Bind b) u (lift (S O) O t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u1 t0) (THead (Bind b) u (lift (S O) O t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead k u1 t0) (THead (Bind b) u (lift (S O) O t4)) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u1 t0) (THead (Bind b) u (lift (S O) O t4)) H0) in (eq_ind K (Bind b) (\lambda (k: K).((eq T u1 u) \to ((eq T t0 (lift (S O) O t4)) \to (ex2 T (\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 t2 t)))))) (\lambda (H10: (eq T u1 u)).(eq_ind T u (\lambda (_: T).((eq T t0 (lift (S O) O t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H11: (eq T t0 (lift (S O) O t4))).(eq_ind T (lift (S O) O t4) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind b) u (lift (S O) O t4)) H9) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8 (lift (S O) O t4) H11) in (ex2_ind T (\lambda (t2: T).(eq T t3 (lift (S O) O t2))) (\lambda (t2: T).(pr0 t4 t2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 t3) t)) (\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H6: (eq T t3 (lift (S O) O x))).(\lambda (H8: (pr0 t4 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 t) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 u H10) in (ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t)) (\lambda (t: T).(pr0 t2 t))) (\lambda (x0: T).(\lambda (H9: (pr0 x x0)).(\lambda (H13: (pr0 t2 x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t)) (\lambda (t: T).(pr0 t2 t)) x0 (pr0_zeta b H12 x x0 H9 u2) H13)))) (H4 t4 (lift_tlt_dx (Bind b) u t4 (S O) O) x H8 t2 H13))) t3 H6)))) (pr0_gen_lift t4 t3 (S O) O H5)))) t0 (sym_eq T t0 (lift (S O) O t4) H11))) u1 (sym_eq T u1 u H10))) k (sym_eq K k (Bind b) H3))) H2)) H1)))]) in (H1 (refl_equal T (THead (Bind b) u (lift (S O) O t4)))))))) t5 (sym_eq T t5 t2 H11))) t H9 H10 H6 H7))) | (pr0_epsilon t4 t5 H6 u) \Rightarrow (\lambda (H9: (eq T (THead (Flat Cast) u t4) t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u t4) (\lambda (_: T).((eq T t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H11: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H12: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead k u1 t0) t)) H4 (THead (Flat Cast) u t4) H9) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Flat Cast) u t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead k u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead k u1 t0) (THead (Flat Cast) u t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u1 t0) (THead (Flat Cast) u t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead k u1 t0) (THead (Flat Cast) u t4) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u1 t0) (THead (Flat Cast) u t4) H0) in (eq_ind K (Flat Cast) (\lambda (k: K).((eq T u1 u) \to ((eq T t0 t4) \to (ex2 T (\lambda (t: T).(pr0 (THead k u2 t3) t)) (\lambda (t: T).(pr0 t2 t)))))) (\lambda (H10: (eq T u1 u)).(eq_ind T u (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Flat Cast) u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H11: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Flat Cast) u2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Cast) u t4) H9) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8 t4 H11) in (let H6 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H7 u H10) in (ex2_ind T (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Flat Cast) u2 t3) t)) (\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H7: (pr0 t3 x)).(\lambda (H8: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Flat Cast) u2 t3) t)) (\lambda (t: T).(pr0 t2 t)) x (pr0_epsilon t3 x H7 u2) H8)))) (H4 t4 (tlt_head_dx (Flat Cast) u t4) t3 H5 t2 H12))))) t0 (sym_eq T t0 t4 H11))) u1 (sym_eq T u1 u H10))) k (sym_eq K k (Flat Cast) H3))) H2)) H1)))]) in (H1 (refl_equal T (THead (Flat Cast) u t4)))))) t5 (sym_eq T t5 t2 H11))) t H9 H10 H6)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t1 H6)) t H4 H5 H2 H3))) | (pr0_beta u v1 v2 H2 t0 t3 H3) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t)).(\lambda (H5: (eq T (THead (Bind Abbr) v2 t3) t1)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t3) t1) \to ((pr0 v1 v2) \to ((pr0 t0 t3) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1))))))) (\lambda (H6: (eq T (THead (Bind Abbr) v2 t3) t1)).(eq_ind T (THead (Bind Abbr) v2 t3) (\lambda (t: T).((pr0 v1 v2) \to ((pr0 t0 t3) \to (ex2 T (\lambda (t1: T).(pr0 t t1)) (\lambda (t1: T).(pr0 t2 t1)))))) (\lambda (H7: (pr0 v1 v2)).(\lambda (H8: (pr0 t0 t3)).(let H9 \def (match H1 return (\lambda (_: ?).(\lambda (t0: T).(\lambda (t1: T).((eq T t0 t) \to ((eq T t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 t2 t)))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H6: (eq T t4 t)).(\lambda (H9: (eq T t4 t2)).(eq_ind T t (\lambda (t: T).((eq T t t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H10: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t2)) H10 (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) H4) in (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t4 t)) H6 (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) H4) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) H4) in (ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t)) (THead (Bind Abbr) v2 t3) (pr0_refl (THead (Bind Abbr) v2 t3)) (pr0_beta u v1 v2 H7 t0 t3 H8)))) t2 H0)) t (sym_eq T t t2 H10))) t4 (sym_eq T t4 t H6) H9))) | (pr0_comp u1 u2 H6 t4 t5 H7 k) \Rightarrow (\lambda (H9: (eq T (THead k u1 t4) t)).(\lambda (H10: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u1 t4) (\lambda (_: T).((eq T (THead k u2 t5) t2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H11: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u2 t5) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (H12: (pr0 u1 u2)).(\lambda (H13: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t)) H4 (THead k u1 t4) H9) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead k u1 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead k u2 t5) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead k u1 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow (THead (Bind Abst) u t0) | (TLRef _) \Rightarrow (THead (Bind Abst) u t0) | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead k u1 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead k u1 t4) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) | (THead k _ _) \Rightarrow k])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead k u1 t4) H0) in (eq_ind K (Flat Appl) (\lambda (k: K).((eq T v1 u1) \to ((eq T (THead (Bind Abst) u t0) t4) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead k u2 t5) t)))))) (\lambda (H10: (eq T v1 u1)).(eq_ind T u1 (\lambda (_: T).((eq T (THead (Bind Abst) u t0) t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0))))) (\lambda (H11: (eq T (THead (Bind Abst) u t0) t4)).(eq_ind T (THead (Bind Abst) u t0) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0)))) (let H4 \def (eq_ind_r K k (\lambda (k: K).(eq T (THead k u1 t4) t)) H9 (Flat Appl) H3) in (let H5 \def (eq_ind_r T t4 (\lambda (t: T).(pr0 t t5)) H13 (THead (Bind Abst) u t0) H11) in (let H6 \def (match H5 return (\lambda (_: ?).(\lambda (t: T).(\lambda (t1: T).((eq T t (THead (Bind Abst) u t0)) \to ((eq T t1 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0)))))))) with [(pr0_refl t2) \Rightarrow (\lambda (H0: (eq T t2 (THead (Bind Abst) u t0))).(\lambda (H1: (eq T t2 t5)).(eq_ind T (THead (Bind Abst) u t0) (\lambda (t: T).((eq T t t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0))))) (\lambda (H2: (eq T (THead (Bind Abst) u t0) t5)).(eq_ind T (THead (Bind Abst) u t0) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t) t0)))) (let H3 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead (Flat Appl) u1 t0) t)) H4 (THead (Bind Abst) u t0) H11) in (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t0)) H3) in (let H5 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H7 u1 H10) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 u2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u t0)) t))) (\lambda (x: T).(\lambda (H6: (pr0 v2 x)).(\lambda (H7: (pr0 u2 x)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u t0)) t)) (THead (Bind Abbr) x t3) (pr0_comp v2 x H6 t3 t3 (pr0_refl t3) (Bind Abbr)) (pr0_beta u u2 x H7 t0 t3 H8))))) (H4 u1 (tlt_head_sx (Flat Appl) u1 (THead (Bind Abst) u t0)) v2 H5 u2 H12))))) t5 H2)) t2 (sym_eq T t2 (THead (Bind Abst) u t0) H0) H1))) | (pr0_comp u0 u3 H0 t2 t6 H1 k) \Rightarrow (\lambda (H5: (eq T (THead k u0 t2) (THead (Bind Abst) u t0))).(\lambda (H13: (eq T (THead k u3 t6) t5)).((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ _ t) \Rightarrow t])) (THead k u0 t2) (THead (Bind Abst) u t0) H5) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t2) (THead (Bind Abst) u t0) H5) in ((let H6 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u0 t2) (THead (Bind Abst) u t0) H5) in (eq_ind K (Bind Abst) (\lambda (k0: K).((eq T u0 u) \to ((eq T t2 t0) \to ((eq T (THead k0 u3 t6) t5) \to ((pr0 u0 u3) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t5) t))))))))) (\lambda (H9: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq T t2 t0) \to ((eq T (THead (Bind Abst) u3 t6) t5) \to ((pr0 t u3) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0)))))))) (\lambda (H14: (eq T t2 t0)).(eq_ind T t0 (\lambda (t: T).((eq T (THead (Bind Abst) u3 t6) t5) \to ((pr0 u u3) \to ((pr0 t t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0))))))) (\lambda (H15: (eq T (THead (Bind Abst) u3 t6) t5)).(eq_ind T (THead (Bind Abst) u3 t6) (\lambda (t: T).((pr0 u u3) \to ((pr0 t0 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t) t0)))))) (\lambda (_: (pr0 u u3)).(\lambda (H17: (pr0 t0 t6)).(let H4 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead (Flat Appl) u1 t0) t)) H4 (THead (Bind Abst) u t0) H11) in (let H11 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) u1 (THead (Bind Abst) u t0)) H4) in (let H7 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H7 u1 H10) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 u2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t6)) t))) (\lambda (x: T).(\lambda (H10: (pr0 v2 x)).(\lambda (H12: (pr0 u2 x)).(ex2_ind T (\lambda (t: T).(pr0 t6 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t6)) t))) (\lambda (x0: T).(\lambda (H8: (pr0 t6 x0)).(\lambda (H18: (pr0 t3 x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 (THead (Bind Abst) u3 t6)) t)) (THead (Bind Abbr) x x0) (pr0_comp v2 x H10 t3 x0 H18 (Bind Abbr)) (pr0_beta u3 u2 x H12 t6 x0 H8))))) (H11 t0 (tlt_trans (THead (Bind Abst) u t0) t0 (THead (Flat Appl) u1 (THead (Bind Abst) u t0)) (tlt_head_dx (Bind Abst) u t0) (tlt_head_dx (Flat Appl) u1 (THead (Bind Abst) u t0))) t6 H17 t3 H8))))) (H11 u1 (tlt_head_sx (Flat Appl) u1 (THead (Bind Abst) u t0)) v2 H7 u2 H12))))))) t5 H15)) t2 (sym_eq T t2 t0 H14))) u0 (sym_eq T u0 u H9))) k (sym_eq K k (Bind Abst) H6))) H3)) H2)) H13 H0 H1))) | (pr0_beta u0 v0 v3 H0 t2 t6 H1) \Rightarrow (\lambda (H4: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t2)) (THead (Bind Abst) u t0))).(\lambda (H11: (eq T (THead (Bind Abbr) v3 t6) t5)).((let H2 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t2)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t0) H4) in (False_ind ((eq T (THead (Bind Abbr) v3 t6) t5) \to ((pr0 v0 v3) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t5) t)))))) H2)) H11 H0 H1))) | (pr0_upsilon b H0 v0 v3 H1 u0 u3 H4 t2 t6 H11) \Rightarrow (\lambda (H12: (eq T (THead (Flat Appl) v0 (THead (Bind b) u0 t2)) (THead (Bind Abst) u t0))).(\lambda (H13: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t5)).((let H2 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u0 t2)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t0) H12) in (False_ind ((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v3) t6)) t5) \to ((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t5) t)))))))) H2)) H13 H0 H1 H4 H11))) | (pr0_delta u0 u3 H0 t2 t6 H1 w H4) \Rightarrow (\lambda (H11: (eq T (THead (Bind Abbr) u0 t2) (THead (Bind Abst) u t0))).(\lambda (H12: (eq T (THead (Bind Abbr) u3 w) t5)).((let H2 \def (eq_ind T (THead (Bind Abbr) u0 t2) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (THead (Bind Abst) u t0) H11) in (False_ind ((eq T (THead (Bind Abbr) u3 w) t5) \to ((pr0 u0 u3) \to ((pr0 t2 t6) \to ((subst0 O u3 t6 w) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t5) t))))))) H2)) H12 H0 H1 H4))) | (pr0_zeta b H0 t2 t6 H1 u0) \Rightarrow (\lambda (H4: (eq T (THead (Bind b) u0 (lift (S O) O t2)) (THead (Bind Abst) u t0))).(\lambda (H11: (eq T t6 t5)).((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t2) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t2) | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u0 (lift (S O) O t2)) (THead (Bind Abst) u t0) H4) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead (Bind b) u0 (lift (S O) O t2)) (THead (Bind Abst) u t0) H4) in ((let H5 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u0 (lift (S O) O t2)) (THead (Bind Abst) u t0) H4) in (eq_ind B Abst (\lambda (b0: B).((eq T u0 u) \to ((eq T (lift (S O) O t2) t0) \to ((eq T t6 t5) \to ((not (eq B b0 Abst)) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t5) t))))))))) (\lambda (H6: (eq T u0 u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t2) t0) \to ((eq T t6 t5) \to ((not (eq B Abst Abst)) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0)))))))) (\lambda (H12: (eq T (lift (S O) O t2) t0)).(eq_ind T (lift (S O) O t2) (\lambda (_: T).((eq T t6 t5) \to ((not (eq B Abst Abst)) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0))))))) (\lambda (H7: (eq T t6 t5)).(eq_ind T t5 (\lambda (t: T).((not (eq B Abst Abst)) \to ((pr0 t2 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u2 t5) t0)))))) (\lambda (H13: (not (eq B Abst Abst))).(\lambda (_: (pr0 t2 t5)).(let H8 \def (match (H13 (refl_equal B Abst)) return (\lambda (_: ?).(ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t5) t)))) with []) in H8))) t6 (sym_eq T t6 t5 H7))) t0 H12)) u0 (sym_eq T u0 u H6))) b (sym_eq B b Abst H5))) H3)) H2)) H11 H0 H1))) | (pr0_epsilon t2 t6 H0 u0) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u0 t2) (THead (Bind Abst) u t0))).(\lambda (H4: (eq T t6 t5)).((let H2 \def (eq_ind T (THead (Flat Cast) u0 t2) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u t0) H1) in (False_ind ((eq T t6 t5) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u2 t5) t))))) H2)) H4 H0)))]) in (H6 (refl_equal T (THead (Bind Abst) u t0)) (refl_equal T t5))))) t4 H11)) v1 (sym_eq T v1 u1 H10))) k H3)) H2)) H1)))]) in (H1 (refl_equal T (THead k u1 t4))))))) t2 H11)) t H9 H10 H6 H7))) | (pr0_beta u0 v0 v3 H6 t4 t5 H7) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)) t)).(\lambda (H10: (eq T (THead (Bind Abbr) v3 t5) t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)) (\lambda (_: T).((eq T (THead (Bind Abbr) v3 t5) t2) \to ((pr0 v0 v3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H11: (eq T (THead (Bind Abbr) v3 t5) t2)).(eq_ind T (THead (Bind Abbr) v3 t5) (\lambda (t: T).((pr0 v0 v3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (H12: (pr0 v0 v3)).(\lambda (H13: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t)) H4 (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)) H9) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v3 t5) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t0) \Rightarrow t0])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t0 _) \Rightarrow t0])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)) H0) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)) H0) in (eq_ind T v0 (\lambda (_: T).((eq T u u0) \to ((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v3 t5) t0)))))) (\lambda (H10: (eq T u u0)).(eq_ind T u0 (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v3 t5) t0))))) (\lambda (H11: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v3 t5) t0)))) (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)) H9) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H8 t4 H11) in (let H6 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H7 v0 H3) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 v3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v3 t5) t))) (\lambda (x: T).(\lambda (H7: (pr0 v2 x)).(\lambda (H8: (pr0 v3 x)).(ex2_ind T (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v3 t5) t))) (\lambda (x0: T).(\lambda (H9: (pr0 t3 x0)).(\lambda (H12: (pr0 t5 x0)).(ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v3 t5) t)) (THead (Bind Abbr) x x0) (pr0_comp v2 x H7 t3 x0 H9 (Bind Abbr)) (pr0_comp v3 x H8 t5 x0 H12 (Bind Abbr)))))) (H4 t4 (tlt_trans (THead (Bind Abst) u0 t4) t4 (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)) (tlt_head_dx (Bind Abst) u0 t4) (tlt_head_dx (Flat Appl) v0 (THead (Bind Abst) u0 t4))) t3 H5 t5 H13))))) (H4 v0 (tlt_head_sx (Flat Appl) v0 (THead (Bind Abst) u0 t4)) v2 H6 v3 H12))))) t0 (sym_eq T t0 t4 H11))) u (sym_eq T u u0 H10))) v1 (sym_eq T v1 v0 H3))) H2)) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v0 (THead (Bind Abst) u0 t4)))))))) t2 H11)) t H9 H10 H6 H7))) | (pr0_upsilon b H6 v0 v3 H7 u1 u2 H8 t4 t5 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v0 (THead (Bind b) u1 t4)) t)).(\lambda (H11: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind b) u1 t4)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t2) \to ((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))))) (\lambda (H12: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 v0 v3) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t t0)))))))) (\lambda (H13: (not (eq B b Abst))).(\lambda (_: (pr0 v0 v3)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t)) H4 (THead (Flat Appl) v0 (THead (Bind b) u1 t4)) H10) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Flat Appl) v0 (THead (Bind b) u1 t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind b) u1 t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t0) \Rightarrow t0])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind b) u1 t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t0 _) \Rightarrow t0])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind b) u1 t4)) H0) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).B) with [(TSort _) \Rightarrow Abst | (TLRef _) \Rightarrow Abst | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind b) u1 t4)) H0) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Appl) v0 (THead (Bind b) u1 t4)) H0) in (eq_ind T v0 (\lambda (_: T).((eq B Abst b) \to ((eq T u u1) \to ((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t0))))))) (\lambda (H11: (eq B Abst b)).(eq_ind B Abst (\lambda (b: B).((eq T u u1) \to ((eq T t0 t4) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t)))))) (\lambda (H12: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t0))))) (\lambda (H14: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t0)))) (let H5 \def (eq_ind_r B b (\lambda (b: B).(not (eq B b Abst))) H13 Abst H11) in (let H6 \def (match (H5 (refl_equal B Abst)) return (\lambda (_: ?).(ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v3) t5)) t)))) with []) in H6)) t0 (sym_eq T t0 t4 H14))) u (sym_eq T u u1 H12))) b H11)) v1 (sym_eq T v1 v0 H4))) H3)) H2)) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v0 (THead (Bind b) u1 t4)))))))))) t2 H12)) t H10 H11 H6 H7 H8 H9))) | (pr0_delta u1 u2 H6 t4 t5 H7 w H8) \Rightarrow (\lambda (H9: (eq T (THead (Bind Abbr) u1 t4) t)).(\lambda (H10: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u1 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))))) (\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t t0))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t4 t5)).(\lambda (_: (subst0 O u2 t5 w)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t)) H4 (THead (Bind Abbr) u1 t4) H9) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind Abbr) u1 t4))).(let H1 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t4) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t))) H1)))]) in (H1 (refl_equal T (THead (Bind Abbr) u1 t4)))))))) t2 H11)) t H9 H10 H6 H7 H8))) | (pr0_zeta b H6 t4 t5 H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Bind b) u0 (lift (S O) O t4)) t)).(\lambda (H9: (eq T t5 t2)).(eq_ind T (THead (Bind b) u0 (lift (S O) O t4)) (\lambda (_: T).((eq T t5 t2) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H10: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t)) H4 (THead (Bind b) u0 (lift (S O) O t4)) H8) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Bind b) u0 (lift (S O) O t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Bind b) u0 (lift (S O) O t4)))).(let H1 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 (lift (S O) O t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 t2 t))) H1)))]) in (H1 (refl_equal T (THead (Bind b) u0 (lift (S O) O t4)))))))) t5 (sym_eq T t5 t2 H10))) t H8 H9 H6 H7))) | (pr0_epsilon t4 t5 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 t4) t)).(\lambda (H8: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u0 t4) (\lambda (_: T).((eq T t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H9: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (_: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t)) H4 (THead (Flat Cast) u0 t4) H7) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Flat Cast) u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t3) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (THead (Flat Cast) u0 t4))).(let H1 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u0 t4) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t3) t)) (\lambda (t: T).(pr0 t2 t))) H1)))]) in (H1 (refl_equal T (THead (Flat Cast) u0 t4)))))) t5 (sym_eq T t5 t2 H9))) t H7 H8 H6)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t1 H6)) t H4 H5 H2 H3))) | (pr0_upsilon b H2 v1 v2 H3 u1 u2 H4 t0 t3 H5) \Rightarrow (\lambda (H6: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t)).(\lambda (H7: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t1)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t1) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t0 t3) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1))))))))) (\lambda (H8: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t1)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t0 t3) \to (ex2 T (\lambda (t1: T).(pr0 t t1)) (\lambda (t1: T).(pr0 t2 t1)))))))) (\lambda (H9: (not (eq B b Abst))).(\lambda (H10: (pr0 v1 v2)).(\lambda (H11: (pr0 u1 u2)).(\lambda (H12: (pr0 t0 t3)).(let H13 \def (match H1 return (\lambda (_: ?).(\lambda (t0: T).(\lambda (t1: T).((eq T t0 t) \to ((eq T t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 t2 t)))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H8: (eq T t4 t)).(\lambda (H13: (eq T t4 t2)).(eq_ind T t (\lambda (t: T).((eq T t t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H14: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t2)) H14 (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) H6) in (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t4 t)) H8 (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) H6) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) H6) in (ex2_sym T (pr0 (THead (Flat Appl) v1 (THead (Bind b) u1 t0))) (pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3))) (pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11 t0 t3 H12 v1 v2 v2 H10 (pr0_refl v2))))) t2 H0)) t (sym_eq T t t2 H14))) t4 (sym_eq T t4 t H8) H13))) | (pr0_comp u0 u3 H8 t4 t5 H9 k) \Rightarrow (\lambda (H13: (eq T (THead k u0 t4) t)).(\lambda (H14: (eq T (THead k u3 t5) t2)).(eq_ind T (THead k u0 t4) (\lambda (_: T).((eq T (THead k u3 t5) t2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H15: (eq T (THead k u3 t5) t2)).(eq_ind T (THead k u3 t5) (\lambda (t: T).((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (H16: (pr0 u0 u3)).(\lambda (H17: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t)) H6 (THead k u0 t4) H13) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead k u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead k u3 t5) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead k u0 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow (THead (Bind b) u1 t0) | (TLRef _) \Rightarrow (THead (Bind b) u1 t0) | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead k u0 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead k u0 t4) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow (Flat Appl) | (TLRef _) \Rightarrow (Flat Appl) | (THead k _ _) \Rightarrow k])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead k u0 t4) H0) in (eq_ind K (Flat Appl) (\lambda (k: K).((eq T v1 u0) \to ((eq T (THead (Bind b) u1 t0) t4) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead k u3 t5) t)))))) (\lambda (H14: (eq T v1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T (THead (Bind b) u1 t0) t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0))))) (\lambda (H15: (eq T (THead (Bind b) u1 t0) t4)).(eq_ind T (THead (Bind b) u1 t0) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0)))) (let H4 \def (eq_ind_r K k (\lambda (k: K).(eq T (THead k u0 t4) t)) H13 (Flat Appl) H3) in (let H5 \def (eq_ind_r T t4 (\lambda (t: T).(pr0 t t5)) H17 (THead (Bind b) u1 t0) H15) in (let H6 \def (match H5 return (\lambda (_: ?).(\lambda (t: T).(\lambda (t1: T).((eq T t (THead (Bind b) u1 t0)) \to ((eq T t1 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0)))))))) with [(pr0_refl t2) \Rightarrow (\lambda (H0: (eq T t2 (THead (Bind b) u1 t0))).(\lambda (H1: (eq T t2 t5)).(eq_ind T (THead (Bind b) u1 t0) (\lambda (t: T).((eq T t t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0))))) (\lambda (H2: (eq T (THead (Bind b) u1 t0) t5)).(eq_ind T (THead (Bind b) u1 t0) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t) t0)))) (let H3 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead (Flat Appl) u0 t0) t)) H4 (THead (Bind b) u1 t0) H15) in (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) u0 (THead (Bind b) u1 t0)) H3) in (let H5 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H10 u0 H14) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u1 t0)) t))) (\lambda (x: T).(\lambda (H6: (pr0 v2 x)).(\lambda (H7: (pr0 u3 x)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead (Bind b) u1 t0))) (pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3))) (pr0_confluence__pr0_cong_upsilon_refl b H9 u1 u2 H11 t0 t3 H12 u3 v2 x H7 H6))))) (H4 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 t0)) v2 H5 u3 H16))))) t5 H2)) t2 (sym_eq T t2 (THead (Bind b) u1 t0) H0) H1))) | (pr0_comp u4 u5 H0 t2 t6 H1 k) \Rightarrow (\lambda (H6: (eq T (THead k u4 t2) (THead (Bind b) u1 t0))).(\lambda (H13: (eq T (THead k u5 t6) t5)).((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ _ t) \Rightarrow t])) (THead k u4 t2) (THead (Bind b) u1 t0) H6) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t _) \Rightarrow t])) (THead k u4 t2) (THead (Bind b) u1 t0) H6) in ((let H5 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u4 t2) (THead (Bind b) u1 t0) H6) in (eq_ind K (Bind b) (\lambda (k0: K).((eq T u4 u1) \to ((eq T t2 t0) \to ((eq T (THead k0 u5 t6) t5) \to ((pr0 u4 u5) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 t5) t))))))))) (\lambda (H7: (eq T u4 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t2 t0) \to ((eq T (THead (Bind b) u5 t6) t5) \to ((pr0 t u5) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0)))))))) (\lambda (H17: (eq T t2 t0)).(eq_ind T t0 (\lambda (t: T).((eq T (THead (Bind b) u5 t6) t5) \to ((pr0 u1 u5) \to ((pr0 t t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0))))))) (\lambda (H8: (eq T (THead (Bind b) u5 t6) t5)).(eq_ind T (THead (Bind b) u5 t6) (\lambda (t: T).((pr0 u1 u5) \to ((pr0 t0 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t) t0)))))) (\lambda (H18: (pr0 u1 u5)).(\lambda (H19: (pr0 t0 t6)).(let H15 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead (Flat Appl) u0 t0) t)) H4 (THead (Bind b) u1 t0) H15) in (let H20 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) u0 (THead (Bind b) u1 t0)) H15) in (let H4 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H10 u0 H14) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u5 t6)) t))) (\lambda (x: T).(\lambda (H10: (pr0 v2 x)).(\lambda (H14: (pr0 u3 x)).(ex2_ind T (\lambda (t: T).(pr0 t6 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u5 t6)) t))) (\lambda (x0: T).(\lambda (H12: (pr0 t6 x0)).(\lambda (H16: (pr0 t3 x0)).(ex2_ind T (\lambda (t: T).(pr0 u5 t)) (\lambda (t: T).(pr0 u2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 (THead (Bind b) u5 t6)) t))) (\lambda (x1: T).(\lambda (H11: (pr0 u5 x1)).(\lambda (H21: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead (Bind b) u5 t6))) (pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3))) (pr0_confluence__pr0_cong_upsilon_cong b H9 u3 v2 x H14 H10 t6 t3 x0 H12 H16 u5 u2 x1 H11 H21))))) (H20 u1 (tlt_trans (THead (Bind b) u1 t0) u1 (THead (Flat Appl) u0 (THead (Bind b) u1 t0)) (tlt_head_sx (Bind b) u1 t0) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 t0))) u5 H18 u2 H11))))) (H20 t0 (tlt_trans (THead (Bind b) u1 t0) t0 (THead (Flat Appl) u0 (THead (Bind b) u1 t0)) (tlt_head_dx (Bind b) u1 t0) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 t0))) t6 H19 t3 H12))))) (H20 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 t0)) v2 H4 u3 H16))))))) t5 H8)) t2 (sym_eq T t2 t0 H17))) u4 (sym_eq T u4 u1 H7))) k (sym_eq K k (Bind b) H5))) H3)) H2)) H13 H0 H1))) | (pr0_beta u v0 v3 H0 t2 t6 H1) \Rightarrow (\lambda (H6: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t2)) (THead (Bind b) u1 t0))).(\lambda (H13: (eq T (THead (Bind Abbr) v3 t6) t5)).((let H2 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t2)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u1 t0) H6) in (False_ind ((eq T (THead (Bind Abbr) v3 t6) t5) \to ((pr0 v0 v3) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 t5) t)))))) H2)) H13 H0 H1))) | (pr0_upsilon b0 H0 v0 v3 H1 u4 u5 H6 t2 t6 H13) \Rightarrow (\lambda (H14: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u4 t2)) (THead (Bind b) u1 t0))).(\lambda (H15: (eq T (THead (Bind b0) u5 (THead (Flat Appl) (lift (S O) O v3) t6)) t5)).((let H2 \def (eq_ind T (THead (Flat Appl) v0 (THead (Bind b0) u4 t2)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u1 t0) H14) in (False_ind ((eq T (THead (Bind b0) u5 (THead (Flat Appl) (lift (S O) O v3) t6)) t5) \to ((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u4 u5) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 t5) t)))))))) H2)) H15 H0 H1 H6 H13))) | (pr0_delta u4 u5 H0 t2 t6 H1 w H6) \Rightarrow (\lambda (H13: (eq T (THead (Bind Abbr) u4 t2) (THead (Bind b) u1 t0))).(\lambda (H17: (eq T (THead (Bind Abbr) u5 w) t5)).((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t2 | (TLRef _) \Rightarrow t2 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abbr) u4 t2) (THead (Bind b) u1 t0) H13) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u4 | (TLRef _) \Rightarrow u4 | (THead _ t _) \Rightarrow t])) (THead (Bind Abbr) u4 t2) (THead (Bind b) u1 t0) H13) in ((let H5 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u4 t2) (THead (Bind b) u1 t0) H13) in (eq_ind B Abbr (\lambda (b: B).((eq T u4 u1) \to ((eq T t2 t0) \to ((eq T (THead (Bind Abbr) u5 w) t5) \to ((pr0 u4 u5) \to ((pr0 t2 t6) \to ((subst0 O u5 t6 w) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 t5) t)))))))))) (\lambda (H7: (eq T u4 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t2 t0) \to ((eq T (THead (Bind Abbr) u5 w) t5) \to ((pr0 t u5) \to ((pr0 t2 t6) \to ((subst0 O u5 t6 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0))))))))) (\lambda (H18: (eq T t2 t0)).(eq_ind T t0 (\lambda (t: T).((eq T (THead (Bind Abbr) u5 w) t5) \to ((pr0 u1 u5) \to ((pr0 t t6) \to ((subst0 O u5 t6 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0)))))))) (\lambda (H19: (eq T (THead (Bind Abbr) u5 w) t5)).(eq_ind T (THead (Bind Abbr) u5 w) (\lambda (t: T).((pr0 u1 u5) \to ((pr0 t0 t6) \to ((subst0 O u5 t6 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t) t0))))))) (\lambda (H20: (pr0 u1 u5)).(\lambda (H21: (pr0 t0 t6)).(\lambda (H22: (subst0 O u5 t6 w)).(let H15 \def (eq_ind_r B b (\lambda (b: B).(eq T (THead (Bind b) u1 t0) t4)) H15 Abbr H5) in (let H9 \def (eq_ind_r B b (\lambda (b: B).(not (eq B b Abst))) H9 Abbr H5) in (let H23 \def (eq_ind_r B b (\lambda (b: B).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t1)) H8 Abbr H5) in (let H4 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead (Flat Appl) u0 t0) t)) H4 (THead (Bind Abbr) u1 t0) H15) in (let H8 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) u0 (THead (Bind Abbr) u1 t0)) H4) in (let H10 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H10 u0 H14) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t))) (\lambda (x: T).(\lambda (H14: (pr0 v2 x)).(\lambda (H16: (pr0 u3 x)).(ex2_ind T (\lambda (t: T).(pr0 t6 t)) (\lambda (t: T).(pr0 t3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t))) (\lambda (x0: T).(\lambda (H12: (pr0 t6 x0)).(\lambda (H24: (pr0 t3 x0)).(ex2_ind T (\lambda (t: T).(pr0 u5 t)) (\lambda (t: T).(pr0 u2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w)) t))) (\lambda (x1: T).(\lambda (H11: (pr0 u5 x1)).(\lambda (H25: (pr0 u2 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 (THead (Bind Abbr) u5 w))) (pr0 (THead (Bind Abbr) u2 (THead (Flat Appl) (lift (S O) O v2) t3))) (pr0_confluence__pr0_cong_upsilon_delta H9 u5 t6 w H22 u3 v2 x H16 H14 t3 x0 H12 H24 u2 x1 H11 H25))))) (H8 u1 (tlt_trans (THead (Bind Abbr) u1 t0) u1 (THead (Flat Appl) u0 (THead (Bind Abbr) u1 t0)) (tlt_head_sx (Bind Abbr) u1 t0) (tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t0))) u5 H20 u2 H11))))) (H8 t0 (tlt_trans (THead (Bind Abbr) u1 t0) t0 (THead (Flat Appl) u0 (THead (Bind Abbr) u1 t0)) (tlt_head_dx (Bind Abbr) u1 t0) (tlt_head_dx (Flat Appl) u0 (THead (Bind Abbr) u1 t0))) t6 H21 t3 H12))))) (H8 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind Abbr) u1 t0)) v2 H10 u3 H16))))))))))) t5 H19)) t2 (sym_eq T t2 t0 H18))) u4 (sym_eq T u4 u1 H7))) b H5)) H3)) H2)) H17 H0 H1 H6))) | (pr0_zeta b0 H0 t2 t6 H1 u) \Rightarrow (\lambda (H6: (eq T (THead (Bind b0) u (lift (S O) O t2)) (THead (Bind b) u1 t0))).(\lambda (H13: (eq T t6 t5)).((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t2) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t2) | (THead _ _ t) \Rightarrow t])) (THead (Bind b0) u (lift (S O) O t2)) (THead (Bind b) u1 t0) H6) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b0) u (lift (S O) O t2)) (THead (Bind b) u1 t0) H6) in ((let H5 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u (lift (S O) O t2)) (THead (Bind b) u1 t0) H6) in (eq_ind B b (\lambda (b1: B).((eq T u u1) \to ((eq T (lift (S O) O t2) t0) \to ((eq T t6 t5) \to ((not (eq B b1 Abst)) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 t5) t))))))))) (\lambda (H7: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O t2) t0) \to ((eq T t6 t5) \to ((not (eq B b Abst)) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0)))))))) (\lambda (H17: (eq T (lift (S O) O t2) t0)).(eq_ind T (lift (S O) O t2) (\lambda (_: T).((eq T t6 t5) \to ((not (eq B b Abst)) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0))))))) (\lambda (H8: (eq T t6 t5)).(eq_ind T t5 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t2 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0)))))) (\lambda (H18: (not (eq B b Abst))).(\lambda (H19: (pr0 t2 t5)).(let H9 \def (eq_ind_r T t0 (\lambda (t: T).(eq T (THead (Bind b) u1 t) t4)) H15 (lift (S O) O t2) H17) in (let H15 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead (Flat Appl) u0 t0) t)) H4 (THead (Bind b) u1 (lift (S O) O t2)) H9) in (let H20 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t2))) H15) in (let H12 \def (eq_ind_r T t0 (\lambda (t: T).(pr0 t t3)) H12 (lift (S O) O t2) H17) in (ex2_ind T (\lambda (t4: T).(eq T t3 (lift (S O) O t4))) (\lambda (t3: T).(pr0 t2 t3)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 t5) t))) (\lambda (x: T).(\lambda (H21: (eq T t3 (lift (S O) O x))).(\lambda (H22: (pr0 t2 x)).(eq_ind_r T (lift (S O) O x) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t)) t0)) (\lambda (t0: T).(pr0 (THead (Flat Appl) u3 t5) t0)))) (let H4 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H10 u0 H14) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O x))) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 t5) t))) (\lambda (x0: T).(\lambda (H10: (pr0 v2 x0)).(\lambda (H14: (pr0 u3 x0)).(ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O x))) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 t5) t))) (\lambda (x1: T).(\lambda (H16: (pr0 x x1)).(\lambda (H23: (pr0 t5 x1)).(ex2_sym T (pr0 (THead (Flat Appl) u3 t5)) (pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O x)))) (pr0_confluence__pr0_cong_upsilon_zeta b H18 u1 u2 H11 u3 v2 x0 H14 H10 x t5 x1 H16 H23))))) (H20 t2 (tlt_trans (THead (Bind b) u1 (lift (S O) O t2)) t2 (THead (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t2))) (lift_tlt_dx (Bind b) u1 t2 (S O) O) (tlt_head_dx (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t2)))) x H22 t5 H19))))) (H20 u0 (tlt_head_sx (Flat Appl) u0 (THead (Bind b) u1 (lift (S O) O t2))) v2 H4 u3 H16))) t3 H21)))) (pr0_gen_lift t2 t3 (S O) O H12)))))))) t6 (sym_eq T t6 t5 H8))) t0 H17)) u (sym_eq T u u1 H7))) b0 (sym_eq B b0 b H5))) H3)) H2)) H13 H0 H1))) | (pr0_epsilon t2 t6 H0 u) \Rightarrow (\lambda (H1: (eq T (THead (Flat Cast) u t2) (THead (Bind b) u1 t0))).(\lambda (H6: (eq T t6 t5)).((let H2 \def (eq_ind T (THead (Flat Cast) u t2) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u1 t0) H1) in (False_ind ((eq T t6 t5) \to ((pr0 t2 t6) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Flat Appl) u3 t5) t))))) H2)) H6 H0)))]) in (H6 (refl_equal T (THead (Bind b) u1 t0)) (refl_equal T t5))))) t4 H15)) v1 (sym_eq T v1 u0 H14))) k H3)) H2)) H1)))]) in (H1 (refl_equal T (THead k u0 t4))))))) t2 H15)) t H13 H14 H8 H9))) | (pr0_beta u v0 v3 H8 t4 t5 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v0 (THead (Bind Abst) u t4)) t)).(\lambda (H13: (eq T (THead (Bind Abbr) v3 t5) t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind Abst) u t4)) (\lambda (_: T).((eq T (THead (Bind Abbr) v3 t5) t2) \to ((pr0 v0 v3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H14: (eq T (THead (Bind Abbr) v3 t5) t2)).(eq_ind T (THead (Bind Abbr) v3 t5) (\lambda (t: T).((pr0 v0 v3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (_: (pr0 v0 v3)).(\lambda (_: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t)) H6 (THead (Flat Appl) v0 (THead (Bind Abst) u t4)) H10) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Flat Appl) v0 (THead (Bind Abst) u t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v3 t5) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind Abst) u t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t0) \Rightarrow t0])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind Abst) u t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind Abst) u t4)) H0) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind Abst) u t4)) H0) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind Abst) u t4)) H0) in (eq_ind T v0 (\lambda (_: T).((eq B b Abst) \to ((eq T u1 u) \to ((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v3 t5) t0))))))) (\lambda (H13: (eq B b Abst)).(eq_ind B Abst (\lambda (b: B).((eq T u1 u) \to ((eq T t0 t4) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v3 t5) t)))))) (\lambda (H14: (eq T u1 u)).(eq_ind T u (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v3 t5) t0))))) (\lambda (H15: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v3 t5) t0)))) (let H5 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v0 (THead (Bind Abst) u t4)) H10) in (let H6 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H12 t4 H15) in (let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H11 u H14) in (let H8 \def (eq_ind B b (\lambda (b: B).(not (eq B b Abst))) H9 Abst H13) in (let H9 \def (match (H8 (refl_equal B Abst)) return (\lambda (_: ?).(ex2 T (\lambda (t: T).(pr0 (THead (Bind Abst) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v3 t5) t)))) with []) in H9))))) t0 (sym_eq T t0 t4 H15))) u1 (sym_eq T u1 u H14))) b (sym_eq B b Abst H13))) v1 (sym_eq T v1 v0 H4))) H3)) H2)) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v0 (THead (Bind Abst) u t4)))))))) t2 H14)) t H10 H13 H8 H9))) | (pr0_upsilon b0 H8 v0 v3 H9 u0 u3 H10 t4 t5 H11) \Rightarrow (\lambda (H13: (eq T (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) t)).(\lambda (H14: (eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t2)).(eq_ind T (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) (\lambda (_: T).((eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0))))))))) (\lambda (H15: (eq T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t2)).(eq_ind T (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) (\lambda (t: T).((not (eq B b0 Abst)) \to ((pr0 v0 v3) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t t0)))))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (H17: (pr0 v0 v3)).(\lambda (H18: (pr0 u0 u3)).(\lambda (H19: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t)) H6 (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) H13) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Flat Appl) v0 (THead (Bind b0) u0 t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t0) \Rightarrow t0])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t0 _) \Rightarrow t0])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) H0) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) H0) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) H0) in (eq_ind T v0 (\lambda (_: T).((eq B b b0) \to ((eq T u1 u0) \to ((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t0))))))) (\lambda (H14: (eq B b b0)).(eq_ind B b0 (\lambda (b: B).((eq T u1 u0) \to ((eq T t0 t4) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t)))))) (\lambda (H15: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t0))))) (\lambda (H16: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t0)))) (let H5 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) H13) in (let H6 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H12 t4 H16) in (let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H11 u0 H15) in (let H8 \def (eq_ind B b (\lambda (b: B).(not (eq B b Abst))) H9 b0 H14) in (let H9 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H10 v0 H4) in (ex2_ind T (\lambda (t: T).(pr0 v2 t)) (\lambda (t: T).(pr0 v3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t))) (\lambda (x: T).(\lambda (H10: (pr0 v2 x)).(\lambda (H11: (pr0 v3 x)).(ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t))) (\lambda (x0: T).(\lambda (H12: (pr0 u2 x0)).(\lambda (H13: (pr0 u3 x0)).(ex2_ind T (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Bind b0) u3 (THead (Flat Appl) (lift (S O) O v3) t5)) t))) (\lambda (x1: T).(\lambda (H17: (pr0 t3 x1)).(\lambda (H18: (pr0 t5 x1)).(pr0_confluence__pr0_upsilon_upsilon b0 H8 v2 v3 x H10 H11 u2 u3 x0 H12 H13 t3 t5 x1 H17 H18)))) (H5 t4 (tlt_trans (THead (Bind b0) u0 t4) t4 (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) (tlt_head_dx (Bind b0) u0 t4) (tlt_head_dx (Flat Appl) v0 (THead (Bind b0) u0 t4))) t3 H6 t5 H19))))) (H5 u0 (tlt_trans (THead (Bind b0) u0 t4) u0 (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)) (tlt_head_sx (Bind b0) u0 t4) (tlt_head_dx (Flat Appl) v0 (THead (Bind b0) u0 t4))) u2 H7 u3 H18))))) (H5 v0 (tlt_head_sx (Flat Appl) v0 (THead (Bind b0) u0 t4)) v2 H9 v3 H17))))))) t0 (sym_eq T t0 t4 H16))) u1 (sym_eq T u1 u0 H15))) b (sym_eq B b b0 H14))) v1 (sym_eq T v1 v0 H4))) H3)) H2)) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v0 (THead (Bind b0) u0 t4)))))))))) t2 H15)) t H13 H14 H8 H9 H10 H11))) | (pr0_delta u0 u3 H8 t4 t5 H9 w H10) \Rightarrow (\lambda (H11: (eq T (THead (Bind Abbr) u0 t4) t)).(\lambda (H12: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind Abbr) u0 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u3 w) t2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to ((subst0 O u3 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0)))))))) (\lambda (H13: (eq T (THead (Bind Abbr) u3 w) t2)).(eq_ind T (THead (Bind Abbr) u3 w) (\lambda (t: T).((pr0 u0 u3) \to ((pr0 t4 t5) \to ((subst0 O u3 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t t0))))))) (\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t4 t5)).(\lambda (_: (subst0 O u3 t5 w)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t)) H6 (THead (Bind Abbr) u0 t4) H11) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Bind Abbr) u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 w) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Bind Abbr) u0 t4))).(let H1 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u0 t4) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w) t))) H1)))]) in (H1 (refl_equal T (THead (Bind Abbr) u0 t4)))))))) t2 H13)) t H11 H12 H8 H9 H10))) | (pr0_zeta b0 H8 t4 t5 H9 u) \Rightarrow (\lambda (H10: (eq T (THead (Bind b0) u (lift (S O) O t4)) t)).(\lambda (H11: (eq T t5 t2)).(eq_ind T (THead (Bind b0) u (lift (S O) O t4)) (\lambda (_: T).((eq T t5 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H12: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((not (eq B b0 Abst)) \to ((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (_: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t)) H6 (THead (Bind b0) u (lift (S O) O t4)) H10) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Bind b0) u (lift (S O) O t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Bind b0) u (lift (S O) O t4)))).(let H1 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b0) u (lift (S O) O t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 t2 t))) H1)))]) in (H1 (refl_equal T (THead (Bind b0) u (lift (S O) O t4)))))))) t5 (sym_eq T t5 t2 H12))) t H10 H11 H8 H9))) | (pr0_epsilon t4 t5 H8 u) \Rightarrow (\lambda (H9: (eq T (THead (Flat Cast) u t4) t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u t4) (\lambda (_: T).((eq T t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H11: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (_: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t)) H6 (THead (Flat Cast) u t4) H9) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Flat Cast) u t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (THead (Flat Cast) u t4))).(let H1 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u t4) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t)) (\lambda (t: T).(pr0 t2 t))) H1)))]) in (H1 (refl_equal T (THead (Flat Cast) u t4)))))) t5 (sym_eq T t5 t2 H11))) t H9 H10 H8)))]) in (H13 (refl_equal T t) (refl_equal T t2))))))) t1 H8)) t H6 H7 H2 H3 H4 H5))) | (pr0_delta u1 u2 H2 t0 t3 H3 w H4) \Rightarrow (\lambda (H5: (eq T (THead (Bind Abbr) u1 t0) t)).(\lambda (H6: (eq T (THead (Bind Abbr) u2 w) t1)).(eq_ind T (THead (Bind Abbr) u1 t0) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t1) \to ((pr0 u1 u2) \to ((pr0 t0 t3) \to ((subst0 O u2 t3 w) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1)))))))) (\lambda (H7: (eq T (THead (Bind Abbr) u2 w) t1)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t0 t3) \to ((subst0 O u2 t3 w) \to (ex2 T (\lambda (t1: T).(pr0 t t1)) (\lambda (t1: T).(pr0 t2 t1))))))) (\lambda (H8: (pr0 u1 u2)).(\lambda (H9: (pr0 t0 t3)).(\lambda (H10: (subst0 O u2 t3 w)).(let H11 \def (match H1 return (\lambda (_: ?).(\lambda (t0: T).(\lambda (t1: T).((eq T t0 t) \to ((eq T t1 t2) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t)))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H7: (eq T t4 t)).(\lambda (H11: (eq T t4 t2)).(eq_ind T t (\lambda (t: T).((eq T t t2) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H12: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t2)) H12 (THead (Bind Abbr) u1 t0) H5) in (eq_ind T (THead (Bind Abbr) u1 t0) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t4 t)) H7 (THead (Bind Abbr) u1 t0) H5) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind Abbr) u1 t0) H5) in (ex_intro2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u1 t0) t)) (THead (Bind Abbr) u2 w) (pr0_refl (THead (Bind Abbr) u2 w)) (pr0_delta u1 u2 H8 t0 t3 H9 w H10)))) t2 H0)) t (sym_eq T t t2 H12))) t4 (sym_eq T t4 t H7) H11))) | (pr0_comp u0 u3 H7 t4 t5 H8 k) \Rightarrow (\lambda (H11: (eq T (THead k u0 t4) t)).(\lambda (H12: (eq T (THead k u3 t5) t2)).(eq_ind T (THead k u0 t4) (\lambda (_: T).((eq T (THead k u3 t5) t2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H13: (eq T (THead k u3 t5) t2)).(eq_ind T (THead k u3 t5) (\lambda (t: T).((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (H14: (pr0 u0 u3)).(\lambda (H15: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind Abbr) u1 t0) t)) H5 (THead k u0 t4) H11) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead k u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead k u3 t5) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind Abbr) u1 t0) (THead k u0 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abbr) u1 t0) (THead k u0 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead (Bind Abbr) u1 t0) (THead k u0 t4) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow (Bind Abbr) | (TLRef _) \Rightarrow (Bind Abbr) | (THead k _ _) \Rightarrow k])) (THead (Bind Abbr) u1 t0) (THead k u0 t4) H0) in (eq_ind K (Bind Abbr) (\lambda (k: K).((eq T u1 u0) \to ((eq T t0 t4) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead k u3 t5) t)))))) (\lambda (H12: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 t5) t0))))) (\lambda (H13: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 t5) t0)))) (let H4 \def (eq_ind_r K k (\lambda (k: K).(eq T (THead k u0 t4) t)) H11 (Bind Abbr) H3) in (let H5 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind Abbr) u0 t4) H4) in (let H6 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H9 t4 H13) in (let H7 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H8 u0 H12) in (ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 t5) t))) (\lambda (x: T).(\lambda (H8: (pr0 u2 x)).(\lambda (H9: (pr0 u3 x)).(ex2_ind T (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 t5) t))) (\lambda (x0: T).(\lambda (H11: (pr0 t3 x0)).(\lambda (H14: (pr0 t5 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u3 t5)) (pr0 (THead (Bind Abbr) u2 w)) (pr0_confluence__pr0_cong_delta u2 t3 w H10 u3 x H9 H8 t5 x0 H14 H11))))) (H5 t4 (tlt_head_dx (Bind Abbr) u0 t4) t3 H6 t5 H15))))) (H5 u0 (tlt_head_sx (Bind Abbr) u0 t4) u2 H7 u3 H14)))))) t0 (sym_eq T t0 t4 H13))) u1 (sym_eq T u1 u0 H12))) k H3)) H2)) H1)))]) in (H1 (refl_equal T (THead k u0 t4))))))) t2 H13)) t H11 H12 H7 H8))) | (pr0_beta u v1 v2 H7 t4 t5 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) t)).(\lambda (H10: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H11: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) (\lambda (t: T).((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind Abbr) u1 t0) t)) H5 (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H9) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Flat Appl) v1 (THead (Bind Abst) u t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind Abbr) u1 t0) (THead (Flat Appl) v1 (THead (Bind Abst) u t4)))).(let H1 \def (eq_ind T (THead (Bind Abbr) u1 t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)))))))) t2 H11)) t H9 H10 H7 H8))) | (pr0_upsilon b H7 v1 v2 H8 u0 u3 H9 t4 t5 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) t)).(\lambda (H12: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) (\lambda (_: T).((eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0))))))))) (\lambda (H13: (eq T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t t0)))))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u0 u3)).(\lambda (_: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind Abbr) u1 t0) t)) H5 (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) H11) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Flat Appl) v1 (THead (Bind b) u0 t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind Abbr) u1 t0) (THead (Flat Appl) v1 (THead (Bind b) u0 t4)))).(let H1 \def (eq_ind T (THead (Bind Abbr) u1 t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 (THead (Bind b) u0 t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind b) u3 (THead (Flat Appl) (lift (S O) O v2) t5)) t))) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind b) u0 t4)))))))))) t2 H13)) t H11 H12 H7 H8 H9 H10))) | (pr0_delta u0 u3 H7 t4 t5 H8 w0 H9) \Rightarrow (\lambda (H11: (eq T (THead (Bind Abbr) u0 t4) t)).(\lambda (H12: (eq T (THead (Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u0 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u3 w0) t2) \to ((pr0 u0 u3) \to ((pr0 t4 t5) \to ((subst0 O u3 t5 w0) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0)))))))) (\lambda (H13: (eq T (THead (Bind Abbr) u3 w0) t2)).(eq_ind T (THead (Bind Abbr) u3 w0) (\lambda (t: T).((pr0 u0 u3) \to ((pr0 t4 t5) \to ((subst0 O u3 t5 w0) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t t0))))))) (\lambda (H14: (pr0 u0 u3)).(\lambda (H15: (pr0 t4 t5)).(\lambda (H16: (subst0 O u3 t5 w0)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind Abbr) u1 t0) t)) H5 (THead (Bind Abbr) u0 t4) H11) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Bind Abbr) u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 w0) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind Abbr) u1 t0) (THead (Bind Abbr) u0 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abbr) u1 t0) (THead (Bind Abbr) u0 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead (Bind Abbr) u1 t0) (THead (Bind Abbr) u0 t4) H0) in (eq_ind T u0 (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 w0) t0))))) (\lambda (H12: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u3 w0) t0)))) (let H3 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind Abbr) u0 t4) H11) in (let H4 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H9 t4 H12) in (let H5 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H8 u0 H2) in (ex2_ind T (\lambda (t: T).(pr0 u2 t)) (\lambda (t: T).(pr0 u3 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x: T).(\lambda (H6: (pr0 u2 x)).(\lambda (H7: (pr0 u3 x)).(ex2_ind T (\lambda (t: T).(pr0 t3 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u3 w0) t))) (\lambda (x0: T).(\lambda (H8: (pr0 t3 x0)).(\lambda (H9: (pr0 t5 x0)).(pr0_confluence__pr0_delta_delta u2 t3 w H10 u3 t5 w0 H16 x H6 H7 x0 H8 H9)))) (H3 t4 (tlt_head_dx (Bind Abbr) u0 t4) t3 H4 t5 H15))))) (H3 u0 (tlt_head_sx (Bind Abbr) u0 t4) u2 H5 u3 H14))))) t0 (sym_eq T t0 t4 H12))) u1 (sym_eq T u1 u0 H2))) H1)))]) in (H1 (refl_equal T (THead (Bind Abbr) u0 t4)))))))) t2 H13)) t H11 H12 H7 H8 H9))) | (pr0_zeta b H7 t4 t5 H8 u) \Rightarrow (\lambda (H11: (eq T (THead (Bind b) u (lift (S O) O t4)) t)).(\lambda (H12: (eq T t5 t2)).(eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (_: T).((eq T t5 t2) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H13: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H14: (not (eq B b Abst))).(\lambda (H15: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind Abbr) u1 t0) t)) H5 (THead (Bind b) u (lift (S O) O t4)) H11) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Bind b) u (lift (S O) O t4))) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind Abbr) u1 t0) (THead (Bind b) u (lift (S O) O t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abbr) u1 t0) (THead (Bind b) u (lift (S O) O t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead (Bind Abbr) u1 t0) (THead (Bind b) u (lift (S O) O t4)) H0) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u1 t0) (THead (Bind b) u (lift (S O) O t4)) H0) in (eq_ind B Abbr (\lambda (_: B).((eq T u1 u) \to ((eq T t0 (lift (S O) O t4)) \to (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t)))))) (\lambda (H12: (eq T u1 u)).(eq_ind T u (\lambda (_: T).((eq T t0 (lift (S O) O t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H13: (eq T t0 (lift (S O) O t4))).(eq_ind T (lift (S O) O t4) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H4 \def (eq_ind_r B b (\lambda (b: B).(not (eq B b Abst))) H14 Abbr H3) in (let H5 \def (eq_ind_r B b (\lambda (b: B).(eq T (THead (Bind b) u (lift (S O) O t4)) t)) H11 Abbr H3) in (let H6 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind Abbr) u (lift (S O) O t4)) H5) in (let H7 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H9 (lift (S O) O t4) H13) in (ex2_ind T (\lambda (t2: T).(eq T t3 (lift (S O) O t2))) (\lambda (t2: T).(pr0 t4 t2)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H9: (eq T t3 (lift (S O) O x))).(\lambda (H11: (pr0 t4 x)).(let H10 \def (eq_ind T t3 (\lambda (t: T).(subst0 O u2 t w)) H10 (lift (S O) O x) H9) in (let H8 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H8 u H12) in (ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t2 x0)).(pr0_confluence__pr0_delta_epsilon u2 (lift (S O) O x) w H10 x (pr0_refl (lift (S O) O x)) t2)))) (H6 t4 (lift_tlt_dx (Bind Abbr) u t4 (S O) O) x H11 t2 H15))))))) (pr0_gen_lift t4 t3 (S O) O H7)))))) t0 (sym_eq T t0 (lift (S O) O t4) H13))) u1 (sym_eq T u1 u H12))) b H3)) H2)) H1)))]) in (H1 (refl_equal T (THead (Bind b) u (lift (S O) O t4)))))))) t5 (sym_eq T t5 t2 H13))) t H11 H12 H7 H8))) | (pr0_epsilon t4 t5 H7 u) \Rightarrow (\lambda (H8: (eq T (THead (Flat Cast) u t4) t)).(\lambda (H9: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u t4) (\lambda (_: T).((eq T t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H10: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (_: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind Abbr) u1 t0) t)) H5 (THead (Flat Cast) u t4) H8) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Flat Cast) u t4)) \to (ex2 T (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind Abbr) u1 t0) (THead (Flat Cast) u t4))).(let H1 \def (eq_ind T (THead (Bind Abbr) u1 t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u t4) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)) (\lambda (t: T).(pr0 t2 t))) H1)))]) in (H1 (refl_equal T (THead (Flat Cast) u t4)))))) t5 (sym_eq T t5 t2 H10))) t H8 H9 H7)))]) in (H11 (refl_equal T t) (refl_equal T t2)))))) t1 H7)) t H5 H6 H2 H3 H4))) | (pr0_zeta b H2 t0 t3 H3 u) \Rightarrow (\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t0)) t)).(\lambda (H5: (eq T t3 t1)).(eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (_: T).((eq T t3 t1) \to ((not (eq B b Abst)) \to ((pr0 t0 t3) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1))))))) (\lambda (H6: (eq T t3 t1)).(eq_ind T t1 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t0 t) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1)))))) (\lambda (H7: (not (eq B b Abst))).(\lambda (H8: (pr0 t0 t1)).(let H9 \def (match H1 return (\lambda (_: ?).(\lambda (t0: T).(\lambda (t3: T).((eq T t0 t) \to ((eq T t3 t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H6: (eq T t4 t)).(\lambda (H9: (eq T t4 t2)).(eq_ind T t (\lambda (t: T).((eq T t t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H10: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t2)) H10 (THead (Bind b) u (lift (S O) O t0)) H4) in (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t4 t)) H6 (THead (Bind b) u (lift (S O) O t0)) H4) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind b) u (lift (S O) O t0)) H4) in (ex_intro2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind b) u (lift (S O) O t0)) t)) t1 (pr0_refl t1) (pr0_zeta b H7 t0 t1 H8 u)))) t2 H0)) t (sym_eq T t t2 H10))) t4 (sym_eq T t4 t H6) H9))) | (pr0_comp u1 u2 H6 t4 t5 H7 k) \Rightarrow (\lambda (H9: (eq T (THead k u1 t4) t)).(\lambda (H10: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u1 t4) (\lambda (_: T).((eq T (THead k u2 t5) t2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H11: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u2 t5) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H13: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind b) u (lift (S O) O t0)) t)) H4 (THead k u1 t4) H9) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead k u1 t4)) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead k u2 t5) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead k u1 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead k u1 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead k u1 t4) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow (Bind b) | (TLRef _) \Rightarrow (Bind b) | (THead k _ _) \Rightarrow k])) (THead (Bind b) u (lift (S O) O t0)) (THead k u1 t4) H0) in (eq_ind K (Bind b) (\lambda (k: K).((eq T u u1) \to ((eq T (lift (S O) O t0) t4) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead k u2 t5) t)))))) (\lambda (H10: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O t0) t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u2 t5) t0))))) (\lambda (H11: (eq T (lift (S O) O t0) t4)).(eq_ind T (lift (S O) O t0) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u2 t5) t0)))) (let H4 \def (eq_ind_r K k (\lambda (k: K).(eq T (THead k u1 t4) t)) H9 (Bind b) H3) in (let H5 \def (eq_ind_r T t4 (\lambda (t: T).(pr0 t t5)) H13 (lift (S O) O t0) H11) in (ex2_ind T (\lambda (t2: T).(eq T t5 (lift (S O) O t2))) (\lambda (t2: T).(pr0 t0 t2)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 t5) t))) (\lambda (x: T).(\lambda (H6: (eq T t5 (lift (S O) O x))).(\lambda (H9: (pr0 t0 x)).(let H12 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead (Bind b) u1 t0) t)) H4 (lift (S O) O t0) H11) in (let H13 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind b) u1 (lift (S O) O t0)) H12) in (eq_ind_r T (lift (S O) O x) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u2 t) t0)))) (ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t))) (\lambda (x0: T).(\lambda (H8: (pr0 x x0)).(\lambda (H14: (pr0 t1 x0)).(ex_intro2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (lift (S O) O x)) t)) x0 H14 (pr0_zeta b H7 x x0 H8 u2))))) (H13 t0 (lift_tlt_dx (Bind b) u1 t0 (S O) O) x H9 t1 H8)) t5 H6)))))) (pr0_gen_lift t0 t5 (S O) O H5)))) t4 H11)) u (sym_eq T u u1 H10))) k H3)) H2)) H1)))]) in (H1 (refl_equal T (THead k u1 t4))))))) t2 H11)) t H9 H10 H6 H7))) | (pr0_beta u0 v1 v2 H6 t4 t5 H7) \Rightarrow (\lambda (H8: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) t)).(\lambda (H9: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H10: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) (\lambda (t: T).((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind b) u (lift (S O) O t0)) t)) H4 (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) H8) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4))) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)))).(let H1 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)))))))) t2 H10)) t H8 H9 H6 H7))) | (pr0_upsilon b0 H6 v1 v2 H7 u1 u2 H8 t4 t5 H9) \Rightarrow (\lambda (H10: (eq T (THead (Flat Appl) v1 (THead (Bind b0) u1 t4)) t)).(\lambda (H11: (eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b0) u1 t4)) (\lambda (_: T).((eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))))) (\lambda (H12: (eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) (\lambda (t: T).((not (eq B b0 Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind b) u (lift (S O) O t0)) t)) H4 (THead (Flat Appl) v1 (THead (Bind b0) u1 t4)) H10) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Flat Appl) v1 (THead (Bind b0) u1 t4))) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat Appl) v1 (THead (Bind b0) u1 t4)))).(let H1 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) v1 (THead (Bind b0) u1 t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t))) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind b0) u1 t4)))))))))) t2 H12)) t H10 H11 H6 H7 H8 H9))) | (pr0_delta u1 u2 H6 t4 t5 H7 w H8) \Rightarrow (\lambda (H9: (eq T (THead (Bind Abbr) u1 t4) t)).(\lambda (H10: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u1 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))) (\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H13: (pr0 t4 t5)).(\lambda (H14: (subst0 O u2 t5 w)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind b) u (lift (S O) O t0)) t)) H4 (THead (Bind Abbr) u1 t4) H9) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t4)) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t4) H0) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abbr) u1 t4) H0) in (eq_ind B Abbr (\lambda (_: B).((eq T u u1) \to ((eq T (lift (S O) O t0) t4) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t)))))) (\lambda (H10: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T (lift (S O) O t0) t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0))))) (\lambda (H11: (eq T (lift (S O) O t0) t4)).(eq_ind T (lift (S O) O t0) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)))) (let H4 \def (eq_ind_r T t4 (\lambda (t: T).(pr0 t t5)) H13 (lift (S O) O t0) H11) in (ex2_ind T (\lambda (t2: T).(eq T t5 (lift (S O) O t2))) (\lambda (t2: T).(pr0 t0 t2)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t))) (\lambda (x: T).(\lambda (H5: (eq T t5 (lift (S O) O x))).(\lambda (H6: (pr0 t0 x)).(let H9 \def (eq_ind_r T t4 (\lambda (t0: T).(eq T (THead (Bind Abbr) u1 t0) t)) H9 (lift (S O) O t0) H11) in (let H12 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind Abbr) u1 (lift (S O) O t0)) H9) in (let H13 \def (eq_ind T t5 (\lambda (t: T).(subst0 O u2 t w)) H14 (lift (S O) O x) H5) in (let H7 \def (eq_ind B b (\lambda (b: B).(not (eq B b Abst))) H7 Abbr H3) in (ex2_ind T (\lambda (t: T).(pr0 x t)) (\lambda (t: T).(pr0 t1 t)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t))) (\lambda (x0: T).(\lambda (_: (pr0 x x0)).(\lambda (_: (pr0 t1 x0)).(ex2_sym T (pr0 (THead (Bind Abbr) u2 w)) (pr0 t1) (pr0_confluence__pr0_delta_epsilon u2 (lift (S O) O x) w H13 x (pr0_refl (lift (S O) O x)) t1))))) (H12 t0 (lift_tlt_dx (Bind Abbr) u1 t0 (S O) O) x H6 t1 H8))))))))) (pr0_gen_lift t0 t5 (S O) O H4))) t4 H11)) u (sym_eq T u u1 H10))) b (sym_eq B b Abbr H3))) H2)) H1)))]) in (H1 (refl_equal T (THead (Bind Abbr) u1 t4)))))))) t2 H11)) t H9 H10 H6 H7 H8))) | (pr0_zeta b0 H6 t4 t5 H7 u0) \Rightarrow (\lambda (H9: (eq T (THead (Bind b0) u0 (lift (S O) O t4)) t)).(\lambda (H10: (eq T t5 t2)).(eq_ind T (THead (Bind b0) u0 (lift (S O) O t4)) (\lambda (_: T).((eq T t5 t2) \to ((not (eq B b0 Abst)) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H11: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((not (eq B b0 Abst)) \to ((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (_: (not (eq B b0 Abst))).(\lambda (H13: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind b) u (lift (S O) O t0)) t)) H4 (THead (Bind b0) u0 (lift (S O) O t4)) H9) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Bind b0) u0 (lift (S O) O t4))) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind b0) u0 (lift (S O) O t4)))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t0) | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind b0) u0 (lift (S O) O t4)) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind b0) u0 (lift (S O) O t4)) H0) in ((let H3 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b | (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind b0) u0 (lift (S O) O t4)) H0) in (eq_ind B b0 (\lambda (_: B).((eq T u u0) \to ((eq T (lift (S O) O t0) (lift (S O) O t4)) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))) (\lambda (H10: (eq T u u0)).(eq_ind T u0 (\lambda (_: T).((eq T (lift (S O) O t0) (lift (S O) O t4)) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H11: (eq T (lift (S O) O t0) (lift (S O) O t4))).(eq_ind T (lift (S O) O t0) (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H4 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Bind b0) u0 (lift (S O) O t4)) H9) in (let H5 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t1)) H8 t4 (lift_inj t0 t4 (S O) O H11)) in (let H6 \def (eq_ind B b (\lambda (b: B).(not (eq B b Abst))) H7 b0 H3) in (ex2_ind T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H7: (pr0 t1 x)).(\lambda (H8: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)) x H7 H8)))) (H4 t4 (lift_tlt_dx (Bind b0) u0 t4 (S O) O) t1 H5 t2 H13))))) (lift (S O) O t4) H11)) u (sym_eq T u u0 H10))) b (sym_eq B b b0 H3))) H2)) H1)))]) in (H1 (refl_equal T (THead (Bind b0) u0 (lift (S O) O t4)))))))) t5 (sym_eq T t5 t2 H11))) t H9 H10 H6 H7))) | (pr0_epsilon t4 t5 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 t4) t)).(\lambda (H8: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u0 t4) (\lambda (_: T).((eq T t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H9: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (_: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Bind b) u (lift (S O) O t0)) t)) H4 (THead (Flat Cast) u0 t4) H7) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Flat Cast) u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Flat Cast) u0 t4))).(let H1 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u0 t4) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))) H1)))]) in (H1 (refl_equal T (THead (Flat Cast) u0 t4)))))) t5 (sym_eq T t5 t2 H9))) t H7 H8 H6)))]) in (H9 (refl_equal T t) (refl_equal T t2))))) t3 (sym_eq T t3 t1 H6))) t H4 H5 H2 H3))) | (pr0_epsilon t0 t3 H2 u) \Rightarrow (\lambda (H3: (eq T (THead (Flat Cast) u t0) t)).(\lambda (H4: (eq T t3 t1)).(eq_ind T (THead (Flat Cast) u t0) (\lambda (_: T).((eq T t3 t1) \to ((pr0 t0 t3) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1)))))) (\lambda (H5: (eq T t3 t1)).(eq_ind T t1 (\lambda (t: T).((pr0 t0 t) \to (ex2 T (\lambda (t2: T).(pr0 t1 t2)) (\lambda (t1: T).(pr0 t2 t1))))) (\lambda (H6: (pr0 t0 t1)).(let H7 \def (match H1 return (\lambda (_: ?).(\lambda (t0: T).(\lambda (t3: T).((eq T t0 t) \to ((eq T t3 t2) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H5: (eq T t4 t)).(\lambda (H7: (eq T t4 t2)).(eq_ind T t (\lambda (t: T).((eq T t t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T t t2)) H8 (THead (Flat Cast) u t0) H3) in (eq_ind T (THead (Flat Cast) u t0) (\lambda (t: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))) (let H1 \def (eq_ind_r T t (\lambda (t: T).(eq T t4 t)) H5 (THead (Flat Cast) u t0) H3) in (let H2 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Cast) u t0) H3) in (ex_intro2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Flat Cast) u t0) t)) t1 (pr0_refl t1) (pr0_epsilon t0 t1 H6 u)))) t2 H0)) t (sym_eq T t t2 H8))) t4 (sym_eq T t4 t H5) H7))) | (pr0_comp u1 u2 H5 t4 t5 H6 k) \Rightarrow (\lambda (H7: (eq T (THead k u1 t4) t)).(\lambda (H8: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u1 t4) (\lambda (_: T).((eq T (THead k u2 t5) t2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H9: (eq T (THead k u2 t5) t2)).(eq_ind T (THead k u2 t5) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (_: (pr0 u1 u2)).(\lambda (H11: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Cast) u t0) t)) H3 (THead k u1 t4) H7) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead k u1 t4)) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead k u2 t5) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Cast) u t0) (THead k u1 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) u t0) (THead k u1 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u t0) (THead k u1 t4) H0) in ((let H3 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow (Flat Cast) | (TLRef _) \Rightarrow (Flat Cast) | (THead k _ _) \Rightarrow k])) (THead (Flat Cast) u t0) (THead k u1 t4) H0) in (eq_ind K (Flat Cast) (\lambda (k: K).((eq T u u1) \to ((eq T t0 t4) \to (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead k u2 t5) t)))))) (\lambda (H8: (eq T u u1)).(eq_ind T u1 (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Flat Cast) u2 t5) t0))))) (\lambda (H9: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Flat Cast) u2 t5) t0)))) (let H4 \def (eq_ind_r K k (\lambda (k: K).(eq T (THead k u1 t4) t)) H7 (Flat Cast) H3) in (let H5 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Cast) u1 t4) H4) in (let H6 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t1)) H6 t4 H9) in (ex2_ind T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t5 t)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Flat Cast) u2 t5) t))) (\lambda (x: T).(\lambda (H7: (pr0 t1 x)).(\lambda (H10: (pr0 t5 x)).(ex_intro2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Flat Cast) u2 t5) t)) x H7 (pr0_epsilon t5 x H10 u2))))) (H5 t4 (tlt_head_dx (Flat Cast) u1 t4) t1 H6 t5 H11))))) t0 (sym_eq T t0 t4 H9))) u (sym_eq T u u1 H8))) k H3)) H2)) H1)))]) in (H1 (refl_equal T (THead k u1 t4))))))) t2 H9)) t H7 H8 H5 H6))) | (pr0_beta u0 v1 v2 H5 t4 t5 H6) \Rightarrow (\lambda (H7: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) t)).(\lambda (H8: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t5) t2) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H9: (eq T (THead (Bind Abbr) v2 t5) t2)).(eq_ind T (THead (Bind Abbr) v2 t5) (\lambda (t: T).((pr0 v1 v2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))))) (\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Cast) u t0) t)) H3 (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) H7) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4))) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) v2 t5) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Cast) u t0) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)))).(let H1 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) v2 t5) t))) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)))))))) t2 H9)) t H7 H8 H5 H6))) | (pr0_upsilon b H5 v1 v2 H6 u1 u2 H7 t4 t5 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) t)).(\lambda (H10: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))))) (\lambda (H11: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0)))))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t4 t5)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Cast) u t0) t)) H3 (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H9) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Flat Appl) v1 (THead (Bind b) u1 t4))) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Cast) u t0) (THead (Flat Appl) v1 (THead (Bind b) u1 t4)))).(let H1 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t))) H1)))]) in (H1 (refl_equal T (THead (Flat Appl) v1 (THead (Bind b) u1 t4)))))))))) t2 H11)) t H9 H10 H5 H6 H7 H8))) | (pr0_delta u1 u2 H5 t4 t5 H6 w H7) \Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u1 t4) t)).(\lambda (H9: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u1 t4) (\lambda (_: T).((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))) (\lambda (H10: (eq T (THead (Bind Abbr) u2 w) t2)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t t0))))))) (\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 t4 t5)).(\lambda (_: (subst0 O u2 t5 w)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Cast) u t0) t)) H3 (THead (Bind Abbr) u1 t4) H8) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Bind Abbr) u1 t4)) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 (THead (Bind Abbr) u2 w) t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Cast) u t0) (THead (Bind Abbr) u1 t4))).(let H1 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u1 t4) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 (THead (Bind Abbr) u2 w) t))) H1)))]) in (H1 (refl_equal T (THead (Bind Abbr) u1 t4)))))))) t2 H10)) t H8 H9 H5 H6 H7))) | (pr0_zeta b H5 t4 t5 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead (Bind b) u0 (lift (S O) O t4)) t)).(\lambda (H8: (eq T t5 t2)).(eq_ind T (THead (Bind b) u0 (lift (S O) O t4)) (\lambda (_: T).((eq T t5 t2) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))))) (\lambda (H9: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Cast) u t0) t)) H3 (THead (Bind b) u0 (lift (S O) O t4)) H7) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Bind b) u0 (lift (S O) O t4))) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Cast) u t0) (THead (Bind b) u0 (lift (S O) O t4)))).(let H1 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 (lift (S O) O t4)) H0) in (False_ind (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))) H1)))]) in (H1 (refl_equal T (THead (Bind b) u0 (lift (S O) O t4)))))))) t5 (sym_eq T t5 t2 H9))) t H7 H8 H5 H6))) | (pr0_epsilon t4 t5 H5 u0) \Rightarrow (\lambda (H7: (eq T (THead (Flat Cast) u0 t4) t)).(\lambda (H8: (eq T t5 t2)).(eq_ind T (THead (Flat Cast) u0 t4) (\lambda (_: T).((eq T t5 t2) \to ((pr0 t4 t5) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))) (\lambda (H9: (eq T t5 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t4 t) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H10: (pr0 t4 t2)).(let H0 \def (eq_ind_r T t (\lambda (t: T).(eq T (THead (Flat Cast) u t0) t)) H3 (THead (Flat Cast) u0 t4) H7) in (let H1 \def (match H0 return (\lambda (_: ?).(\lambda (t: T).((eq T t (THead (Flat Cast) u0 t4)) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))) with [refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Flat Cast) u t0) (THead (Flat Cast) u0 t4))).(let H1 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) u t0) (THead (Flat Cast) u0 t4) H0) in ((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u t0) (THead (Flat Cast) u0 t4) H0) in (eq_ind T u0 (\lambda (_: T).((eq T t0 t4) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0))))) (\lambda (H8: (eq T t0 t4)).(eq_ind T t4 (\lambda (_: T).(ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))) (let H3 \def (eq_ind_r T t (\lambda (t: T).(\forall (v: T).((tlt v t) \to (\forall (t1: T).((pr0 v t1) \to (\forall (t2: T).((pr0 v t2) \to (ex2 T (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(pr0 t2 t0)))))))))) H (THead (Flat Cast) u0 t4) H7) in (let H4 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t1)) H6 t4 H8) in (ex2_ind T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)) (ex2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t))) (\lambda (x: T).(\lambda (H5: (pr0 t1 x)).(\lambda (H6: (pr0 t2 x)).(ex_intro2 T (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(pr0 t2 t)) x H5 H6)))) (H3 t4 (tlt_head_dx (Flat Cast) u0 t4) t1 H4 t2 H10)))) t0 (sym_eq T t0 t4 H8))) u (sym_eq T u u0 H2))) H1)))]) in (H1 (refl_equal T (THead (Flat Cast) u0 t4)))))) t5 (sym_eq T t5 t2 H9))) t H7 H8 H5)))]) in (H7 (refl_equal T t) (refl_equal T t2)))) t3 (sym_eq T t3 t1 H5))) t H3 H4 H2)))]) in (H2 (refl_equal T t) (refl_equal T t1))))))))) t0).
 
 theorem pr0_delta1:
  \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (w: T).((subst1 O u2 t2 w) \to (pr0 (THead (Bind Abbr) u1 t1) (THead (Bind Abbr) u2 w)))))))))
@@ -1917,7 +1917,7 @@ theorem pr0_subst1:
 theorem nf0_dec:
  \forall (t1: T).(or (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2))))
 \def
- \lambda (t1: T).(T_ind (\lambda (t: T).(or (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2))))) (\lambda (n: nat).(or_introl (\forall (t2: T).((pr0 (TSort n) t2) \to (eq T (TSort n) t2))) (ex2 T (\lambda (t2: T).((eq T (TSort n) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (TSort n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TSort n) t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T (TSort n)) t2 (pr0_gen_sort t2 n H)))))) (\lambda (n: nat).(or_introl (\forall (t2: T).((pr0 (TLRef n) t2) \to (eq T (TLRef n) t2))) (ex2 T (\lambda (t2: T).((eq T (TLRef n) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (TLRef n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TLRef n) t2)).(eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t)) (refl_equal T (TLRef n)) t2 (pr0_gen_lref t2 n H)))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H: (or (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2))))).(\lambda (t0: T).(\lambda (H0: (or (\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))))).(match k return (\lambda (k0: K).(or (\forall (t2: T).((pr0 (THead k0 t t0) t2) \to (eq T (THead k0 t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead k0 t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead k0 t t0) t2))))) with [(Bind b) \Rightarrow (match b return (\lambda (b0: B).(or (\forall (t2: T).((pr0 (THead (Bind b0) t t0) t2) \to (eq T (THead (Bind b0) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind b0) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind b0) t t0) t2))))) with [Abbr \Rightarrow (or_intror (\forall (t2: T).((pr0 (THead (Bind Abbr) t t0) t2) \to (eq T (THead (Bind Abbr) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t t0) t2))) (let H_x \def (dnf_dec t t0 O) in (let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 O t t0 (lift (S O) O v)) (eq T t0 (lift (S O) O v)))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t t0) t2))) (\lambda (x: T).(\lambda (H2: (or (subst0 O t t0 (lift (S O) O x)) (eq T t0 (lift (S O) O x)))).(or_ind (subst0 O t t0 (lift (S O) O x)) (eq T t0 (lift (S O) O x)) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t t0) t2))) (\lambda (H3: (subst0 O t t0 (lift (S O) O x))).(ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t t0) t2)) (THead (Bind Abbr) t (lift (S O) O x)) (\lambda (H4: (eq T (THead (Bind Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O x)))).(\lambda (P: Prop).(let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O x)) H4) in (let H6 \def (eq_ind T t0 (\lambda (t0: T).(subst0 O t t0 (lift (S O) O x))) H3 (lift (S O) O x) H5) in (subst0_refl t (lift (S O) O x) O H6 P))))) (pr0_delta t t (pr0_refl t) t0 t0 (pr0_refl t0) (lift (S O) O x) H3))) (\lambda (H3: (eq T t0 (lift (S O) O x))).(eq_ind_r T (lift (S O) O x) (\lambda (t2: T).(ex2 T (\lambda (t3: T).((eq T (THead (Bind Abbr) t t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Abbr) t t2) t3)))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t (lift (S O) O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t (lift (S O) O x)) t2)) x (\lambda (H4: (eq T (THead (Bind Abbr) t (lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y (Bind Abbr) x t (S O) O H4 P))) (pr0_zeta Abbr not_abbr_abst x x (pr0_refl x) t)) t0 H3)) H2))) H1)))) | Abst \Rightarrow (let H1 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda (H2: ((\forall (t2: T).((pr0 t t2) \to (eq T t t2))))).(let H3 \def H0 in (or_ind (\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda (H4: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (\lambda (t2: T).(\lambda (H5: (pr0 (THead (Bind Abst) t t0) t2)).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))) (eq T (THead (Bind Abst) t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H7: (pr0 t x0)).(\lambda (H8: (pr0 t0 x1)).(let H_y \def (H4 x1 H8) in (let H_y0 \def (H2 x0 H7) in (let H9 \def (eq_ind_r T x1 (\lambda (t: T).(pr0 t0 t)) H8 t0 H_y) in (let H10 \def (eq_ind_r T x1 (\lambda (t: T).(eq T t2 (THead (Bind Abst) x0 t))) H6 t0 H_y) in (let H11 \def (eq_ind_r T x0 (\lambda (t0: T).(pr0 t t0)) H7 t H_y0) in (let H12 \def (eq_ind_r T x0 (\lambda (t: T).(eq T t2 (THead (Bind Abst) t t0))) H10 t H_y0) in (eq_ind_r T (THead (Bind Abst) t t0) (\lambda (t3: T).(eq T (THead (Bind Abst) t t0) t3)) (refl_equal T (THead (Bind Abst) t t0)) t2 H12)))))))))))) (pr0_gen_abst t t0 t2 H5)))))) (\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) (or (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda (x: T).(\lambda (H5: (((eq T t0 x) \to (\forall (P: Prop).P)))).(\lambda (H6: (pr0 t0 x)).(or_intror (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind Abst) t x) (\lambda (H7: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t x))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abst) t t0) (THead (Bind Abst) t x) H7) in (let H9 \def (eq_ind_r T x (\lambda (t: T).(pr0 t0 t)) H6 t0 H8) in (let H10 \def (eq_ind_r T x (\lambda (t: T).((eq T t0 t) \to (\forall (P: Prop).P))) H5 t0 H8) in (H10 (refl_equal T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H6 (Bind Abst))))))) H4)) H3))) (\lambda (H2: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda (x: T).(\lambda (H3: (((eq T t x) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind Abst) x t0) (\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) x t0))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t _) \Rightarrow t])) (THead (Bind Abst) t t0) (THead (Bind Abst) x t0) H5) in (let H7 \def (eq_ind_r T x (\lambda (t0: T).(pr0 t t0)) H4 t H6) in (let H8 \def (eq_ind_r T x (\lambda (t0: T).((eq T t t0) \to (\forall (P: Prop).P))) H3 t H6) in (H8 (refl_equal T t) P)))))) (pr0_comp t x H4 t0 t0 (pr0_refl t0) (Bind Abst))))))) H2)) H1)) | Void \Rightarrow (let H_x \def (dnf_dec t t0 O) in (let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 O t t0 (lift (S O) O v)) (eq T t0 (lift (S O) O v)))) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda (x: T).(\lambda (H2: (or (subst0 O t t0 (lift (S O) O x)) (eq T t0 (lift (S O) O x)))).(or_ind (subst0 O t t0 (lift (S O) O x)) (eq T t0 (lift (S O) O x)) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda (H3: (subst0 O t t0 (lift (S O) O x))).(let H4 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda (H5: ((\forall (t2: T).((pr0 t t2) \to (eq T t t2))))).(let H6 \def H0 in (or_ind (\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda (H7: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (\lambda (t2: T).(\lambda (H8: (pr0 (THead (Bind Void) t t0) t2)).(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3)))) (pr0 t0 (lift (S O) O t2)) (eq T (THead (Bind Void) t t0) t2) (\lambda (H9: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t0 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))) (eq T (THead (Bind Void) t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H11: (pr0 t x0)).(\lambda (H12: (pr0 t0 x1)).(let H_y \def (H7 x1 H12) in (let H_y0 \def (H5 x0 H11) in (let H13 \def (eq_ind_r T x1 (\lambda (t: T).(pr0 t0 t)) H12 t0 H_y) in (let H14 \def (eq_ind_r T x1 (\lambda (t: T).(eq T t2 (THead (Bind Void) x0 t))) H10 t0 H_y) in (let H15 \def (eq_ind_r T x0 (\lambda (t0: T).(pr0 t t0)) H11 t H_y0) in (let H16 \def (eq_ind_r T x0 (\lambda (t: T).(eq T t2 (THead (Bind Void) t t0))) H14 t H_y0) in (eq_ind_r T (THead (Bind Void) t t0) (\lambda (t3: T).(eq T (THead (Bind Void) t t0) t3)) (refl_equal T (THead (Bind Void) t t0)) t2 H16)))))))))))) H9)) (\lambda (H9: (pr0 t0 (lift (S O) O t2))).(let H_y \def (H7 (lift (S O) O t2) H9) in (let H10 \def (eq_ind T t0 (\lambda (t0: T).(subst0 O t t0 (lift (S O) O x))) H3 (lift (S O) O t2) H_y) in (eq_ind_r T (lift (S O) O t2) (\lambda (t3: T).(eq T (THead (Bind Void) t t3) t2)) (subst0_gen_lift_false t2 t (lift (S O) O x) (S O) O O (le_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n)) (le_n (plus (S O) O)) (plus O (S O)) (plus_comm O (S O))) H10 (eq T (THead (Bind Void) t (lift (S O) O t2)) t2)) t0 H_y)))) (pr0_gen_void t t0 t2 H8)))))) (\lambda (H7: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda (x0: T).(\lambda (H8: (((eq T t0 x0) \to (\forall (P: Prop).P)))).(\lambda (H9: (pr0 t0 x0)).(or_intror (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind Void) t x0) (\lambda (H10: (eq T (THead (Bind Void) t t0) (THead (Bind Void) t x0))).(\lambda (P: Prop).(let H11 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Void) t t0) (THead (Bind Void) t x0) H10) in (let H12 \def (eq_ind_r T x0 (\lambda (t: T).(pr0 t0 t)) H9 t0 H11) in (let H13 \def (eq_ind_r T x0 (\lambda (t: T).((eq T t0 t) \to (\forall (P: Prop).P))) H8 t0 H11) in (H13 (refl_equal T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x0 H9 (Bind Void))))))) H7)) H6))) (\lambda (H5: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda (x0: T).(\lambda (H6: (((eq T t x0) \to (\forall (P: Prop).P)))).(\lambda (H7: (pr0 t x0)).(or_intror (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind Void) x0 t0) (\lambda (H8: (eq T (THead (Bind Void) t t0) (THead (Bind Void) x0 t0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t _) \Rightarrow t])) (THead (Bind Void) t t0) (THead (Bind Void) x0 t0) H8) in (let H10 \def (eq_ind_r T x0 (\lambda (t0: T).(pr0 t t0)) H7 t H9) in (let H11 \def (eq_ind_r T x0 (\lambda (t0: T).((eq T t t0) \to (\forall (P: Prop).P))) H6 t H9) in (H11 (refl_equal T t) P)))))) (pr0_comp t x0 H7 t0 t0 (pr0_refl t0) (Bind Void))))))) H5)) H4))) (\lambda (H3: (eq T t0 (lift (S O) O x))).(let H4 \def (eq_ind T t0 (\lambda (t: T).(or (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2))))) H0 (lift (S O) O x) H3) in (eq_ind_r T (lift (S O) O x) (\lambda (t2: T).(or (\forall (t3: T).((pr0 (THead (Bind Void) t t2) t3) \to (eq T (THead (Bind Void) t t2) t3))) (ex2 T (\lambda (t3: T).((eq T (THead (Bind Void) t t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Void) t t2) t3))))) (or_intror (\forall (t2: T).((pr0 (THead (Bind Void) t (lift (S O) O x)) t2) \to (eq T (THead (Bind Void) t (lift (S O) O x)) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t (lift (S O) O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t (lift (S O) O x)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Void) t (lift (S O) O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t (lift (S O) O x)) t2)) x (\lambda (H5: (eq T (THead (Bind Void) t (lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y (Bind Void) x t (S O) O H5 P))) (pr0_zeta Void not_void_abst x x (pr0_refl x) t))) t0 H3))) H2))) H1)))]) | (Flat f) \Rightarrow (match f return (\lambda (f0: F).(or (\forall (t2: T).((pr0 (THead (Flat f0) t t0) t2) \to (eq T (THead (Flat f0) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat f0) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat f0) t t0) t2))))) with [Appl \Rightarrow (let H_x \def (binder_dec t0) in (let H1 \def H_x in (or_ind (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (H2: (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u))))))).(ex_3_ind B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H3: (eq T t0 (THead (Bind x0) x1 x2))).(let H4 \def (eq_ind T t0 (\lambda (t: T).(or (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2))))) H0 (THead (Bind x0) x1 x2) H3) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t2: T).(or (\forall (t3: T).((pr0 (THead (Flat Appl) t t2) t3) \to (eq T (THead (Flat Appl) t t2) t3))) (ex2 T (\lambda (t3: T).((eq T (THead (Flat Appl) t t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Flat Appl) t t2) t3))))) ((match x0 return (\lambda (b: B).((or (\forall (t2: T).((pr0 (THead (Bind b) x1 x2) t2) \to (eq T (THead (Bind b) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind b) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind b) x1 x2) t2)))) \to (or (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2)))))) with [Abbr \Rightarrow (\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Abbr) x1 x2) t2) \to (eq T (THead (Bind Abbr) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) x1 x2) t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2)) (THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) (THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)))).(\lambda (P: Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) (\lambda (ee: T).(match ee return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])])) I (THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)) H6) in (False_ind P H7)))) (pr0_upsilon Abbr not_abbr_abst t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2))))) | Abst \Rightarrow (\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Abst) x1 x2) t2) \to (eq T (THead (Bind Abst) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) x1 x2) t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2)) (THead (Bind Abbr) t x2) (\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) (THead (Bind Abbr) t x2))).(\lambda (P: Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) (\lambda (ee: T).(match ee return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) t x2) H6) in (False_ind P H7)))) (pr0_beta x1 t t (pr0_refl t) x2 x2 (pr0_refl x2))))) | Void \Rightarrow (\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Void) x1 x2) t2) \to (eq T (THead (Bind Void) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) x1 x2) t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2)) (THead (Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) (THead (Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)))).(\lambda (P: Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) (\lambda (ee: T).(match ee return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])])) I (THead (Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)) H6) in (False_ind P H7)))) (pr0_upsilon Void not_void_abst t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2)))))]) H4) t0 H3)))))) H2)) (\lambda (H2: ((\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))).(let H3 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (H4: ((\forall (t2: T).((pr0 t t2) \to (eq T t t2))))).(let H5 \def H0 in (or_ind (\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (H6: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) (\lambda (t2: T).(\lambda (H7: (pr0 (THead (Flat Appl) t t0) t2)).(or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t0 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H10: (pr0 t x0)).(\lambda (H11: (pr0 t0 x1)).(let H_y \def (H6 x1 H11) in (let H_y0 \def (H4 x0 H10) in (let H12 \def (eq_ind_r T x1 (\lambda (t: T).(pr0 t0 t)) H11 t0 H_y) in (let H13 \def (eq_ind_r T x1 (\lambda (t: T).(eq T t2 (THead (Flat Appl) x0 t))) H9 t0 H_y) in (let H14 \def (eq_ind_r T x0 (\lambda (t0: T).(pr0 t t0)) H10 t H_y0) in (let H15 \def (eq_ind_r T x0 (\lambda (t: T).(eq T t2 (THead (Flat Appl) t t0))) H13 t H_y0) in (eq_ind_r T (THead (Flat Appl) t t0) (\lambda (t3: T).(eq T (THead (Flat Appl) t t0) t3)) (refl_equal T (THead (Flat Appl) t t0)) t2 H15)))))))))))) H8)) (\lambda (H8: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H9: (eq T t0 (THead (Bind Abst) x0 x1))).(\lambda (H10: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr0 t x2)).(\lambda (_: (pr0 x1 x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t3: T).(eq T (THead (Flat Appl) t t0) t3)) (let H13 \def (eq_ind T t0 (\lambda (t: T).(\forall (t2: T).((pr0 t t2) \to (eq T t t2)))) H6 (THead (Bind Abst) x0 x1) H9) in (let H14 \def (eq_ind T t0 (\lambda (t: T).(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t (THead (Bind b) w u)) \to (\forall (P: Prop).P)))))) H2 (THead (Bind Abst) x0 x1) H9) in (eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(eq T (THead (Flat Appl) t t3) (THead (Bind Abbr) x2 x3))) (H14 Abst x0 x1 (H13 (THead (Bind Abst) x0 x1) (pr0_refl (THead (Bind Abst) x0 x1))) (eq T (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) x2 x3))) t0 H9))) t2 H10))))))))) H8)) (\lambda (H8: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H10: (eq T t0 (THead (Bind x0) x1 x2))).(\lambda (H11: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda (_: (pr0 t x3)).(\lambda (_: (pr0 x1 x4)).(\lambda (_: (pr0 x2 x5)).(eq_ind_r T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t3: T).(eq T (THead (Flat Appl) t t0) t3)) (let H15 \def (eq_ind T t0 (\lambda (t: T).(\forall (t2: T).((pr0 t t2) \to (eq T t t2)))) H6 (THead (Bind x0) x1 x2) H10) in (let H16 \def (eq_ind T t0 (\lambda (t: T).(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t (THead (Bind b) w u)) \to (\forall (P: Prop).P)))))) H2 (THead (Bind x0) x1 x2) H10) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(eq T (THead (Flat Appl) t t3) (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))) (H16 x0 x1 x2 (H15 (THead (Bind x0) x1 x2) (pr0_refl (THead (Bind x0) x1 x2))) (eq T (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))) t0 H10))) t2 H11))))))))))))) H8)) (pr0_gen_appl t t0 t2 H7)))))) (\lambda (H6: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (x: T).(\lambda (H7: (((eq T t0 x) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr0 t0 x)).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) (THead (Flat Appl) t x) (\lambda (H9: (eq T (THead (Flat Appl) t t0) (THead (Flat Appl) t x))).(\lambda (P: Prop).(let H10 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) t t0) (THead (Flat Appl) t x) H9) in (let H11 \def (eq_ind_r T x (\lambda (t: T).(pr0 t0 t)) H8 t0 H10) in (let H12 \def (eq_ind_r T x (\lambda (t: T).((eq T t0 t) \to (\forall (P: Prop).P))) H7 t0 H10) in (H12 (refl_equal T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H8 (Flat Appl))))))) H6)) H5))) (\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (x: T).(\lambda (H5: (((eq T t x) \to (\forall (P: Prop).P)))).(\lambda (H6: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) (THead (Flat Appl) x t0) (\lambda (H7: (eq T (THead (Flat Appl) t t0) (THead (Flat Appl) x t0))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) t t0) (THead (Flat Appl) x t0) H7) in (let H9 \def (eq_ind_r T x (\lambda (t0: T).(pr0 t t0)) H6 t H8) in (let H10 \def (eq_ind_r T x (\lambda (t0: T).((eq T t t0) \to (\forall (P: Prop).P))) H5 t H8) in (H10 (refl_equal T t) P)))))) (pr0_comp t x H6 t0 t0 (pr0_refl t0) (Flat Appl))))))) H4)) H3))) H1))) | Cast \Rightarrow (or_intror (\forall (t2: T).((pr0 (THead (Flat Cast) t t0) t2) \to (eq T (THead (Flat Cast) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Cast) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Cast) t t0) t2)) t0 (\lambda (H1: (eq T (THead (Flat Cast) t t0) t0)).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) t t0 H1 P))) (pr0_epsilon t0 t0 (pr0_refl t0) t)))])])))))) t1).
+ \lambda (t1: T).(T_ind (\lambda (t: T).(or (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2))))) (\lambda (n: nat).(or_introl (\forall (t2: T).((pr0 (TSort n) t2) \to (eq T (TSort n) t2))) (ex2 T (\lambda (t2: T).((eq T (TSort n) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (TSort n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TSort n) t2)).(eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T (TSort n)) t2 (pr0_gen_sort t2 n H)))))) (\lambda (n: nat).(or_introl (\forall (t2: T).((pr0 (TLRef n) t2) \to (eq T (TLRef n) t2))) (ex2 T (\lambda (t2: T).((eq T (TLRef n) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (TLRef n) t2))) (\lambda (t2: T).(\lambda (H: (pr0 (TLRef n) t2)).(eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t)) (refl_equal T (TLRef n)) t2 (pr0_gen_lref t2 n H)))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H: (or (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2))))).(\lambda (t0: T).(\lambda (H0: (or (\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))))).(match k return (\lambda (_: ?).(\lambda (k0: K).(or (\forall (t2: T).((pr0 (THead k0 t t0) t2) \to (eq T (THead k0 t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead k0 t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead k0 t t0) t2)))))) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).(\lambda (b0: B).(or (\forall (t2: T).((pr0 (THead (Bind b0) t t0) t2) \to (eq T (THead (Bind b0) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind b0) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind b0) t t0) t2)))))) with [Abbr \Rightarrow (or_intror (\forall (t2: T).((pr0 (THead (Bind Abbr) t t0) t2) \to (eq T (THead (Bind Abbr) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t t0) t2))) (let H_x \def (dnf_dec t t0 O) in (let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 O t t0 (lift (S O) O v)) (eq T t0 (lift (S O) O v)))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t t0) t2))) (\lambda (x: T).(\lambda (H2: (or (subst0 O t t0 (lift (S O) O x)) (eq T t0 (lift (S O) O x)))).(or_ind (subst0 O t t0 (lift (S O) O x)) (eq T t0 (lift (S O) O x)) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t t0) t2))) (\lambda (H3: (subst0 O t t0 (lift (S O) O x))).(ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t t0) t2)) (THead (Bind Abbr) t (lift (S O) O x)) (\lambda (H4: (eq T (THead (Bind Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O x)))).(\lambda (P: Prop).(let H5 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abbr) t t0) (THead (Bind Abbr) t (lift (S O) O x)) H4) in (let H6 \def (eq_ind T t0 (\lambda (t0: T).(subst0 O t t0 (lift (S O) O x))) H3 (lift (S O) O x) H5) in (subst0_refl t (lift (S O) O x) O H6 P))))) (pr0_delta t t (pr0_refl t) t0 t0 (pr0_refl t0) (lift (S O) O x) H3))) (\lambda (H3: (eq T t0 (lift (S O) O x))).(eq_ind_r T (lift (S O) O x) (\lambda (t2: T).(ex2 T (\lambda (t3: T).((eq T (THead (Bind Abbr) t t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Abbr) t t2) t3)))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) t (lift (S O) O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) t (lift (S O) O x)) t2)) x (\lambda (H4: (eq T (THead (Bind Abbr) t (lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y (Bind Abbr) x t (S O) O H4 P))) (pr0_zeta Abbr not_abbr_abst x x (pr0_refl x) t)) t0 H3)) H2))) H1)))) | Abst \Rightarrow (let H1 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda (H2: ((\forall (t2: T).((pr0 t t2) \to (eq T t t2))))).(let H3 \def H0 in (or_ind (\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda (H4: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (\lambda (t2: T).(\lambda (H5: (pr0 (THead (Bind Abst) t t0) t2)).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))) (eq T (THead (Bind Abst) t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H6: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H7: (pr0 t x0)).(\lambda (H8: (pr0 t0 x1)).(let H_y \def (H4 x1 H8) in (let H_y0 \def (H2 x0 H7) in (let H9 \def (eq_ind_r T x1 (\lambda (t: T).(pr0 t0 t)) H8 t0 H_y) in (let H10 \def (eq_ind_r T x1 (\lambda (t: T).(eq T t2 (THead (Bind Abst) x0 t))) H6 t0 H_y) in (let H11 \def (eq_ind_r T x0 (\lambda (t0: T).(pr0 t t0)) H7 t H_y0) in (let H12 \def (eq_ind_r T x0 (\lambda (t: T).(eq T t2 (THead (Bind Abst) t t0))) H10 t H_y0) in (eq_ind_r T (THead (Bind Abst) t t0) (\lambda (t3: T).(eq T (THead (Bind Abst) t t0) t3)) (refl_equal T (THead (Bind Abst) t t0)) t2 H12)))))))))))) (pr0_gen_abst t t0 t2 H5)))))) (\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) (or (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda (x: T).(\lambda (H5: (((eq T t0 x) \to (\forall (P: Prop).P)))).(\lambda (H6: (pr0 t0 x)).(or_intror (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind Abst) t x) (\lambda (H7: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) t x))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abst) t t0) (THead (Bind Abst) t x) H7) in (let H9 \def (eq_ind_r T x (\lambda (t: T).(pr0 t0 t)) H6 t0 H8) in (let H10 \def (eq_ind_r T x (\lambda (t: T).((eq T t0 t) \to (\forall (P: Prop).P))) H5 t0 H8) in (H10 (refl_equal T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H6 (Bind Abst))))))) H4)) H3))) (\lambda (H2: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)))) (\lambda (x: T).(\lambda (H3: (((eq T t x) \to (\forall (P: Prop).P)))).(\lambda (H4: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead (Bind Abst) t t0) t2) \to (eq T (THead (Bind Abst) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Abst) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) t t0) t2)) (THead (Bind Abst) x t0) (\lambda (H5: (eq T (THead (Bind Abst) t t0) (THead (Bind Abst) x t0))).(\lambda (P: Prop).(let H6 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t _) \Rightarrow t])) (THead (Bind Abst) t t0) (THead (Bind Abst) x t0) H5) in (let H7 \def (eq_ind_r T x (\lambda (t0: T).(pr0 t t0)) H4 t H6) in (let H8 \def (eq_ind_r T x (\lambda (t0: T).((eq T t t0) \to (\forall (P: Prop).P))) H3 t H6) in (H8 (refl_equal T t) P)))))) (pr0_comp t x H4 t0 t0 (pr0_refl t0) (Bind Abst))))))) H2)) H1)) | Void \Rightarrow (let H_x \def (dnf_dec t t0 O) in (let H1 \def H_x in (ex_ind T (\lambda (v: T).(or (subst0 O t t0 (lift (S O) O v)) (eq T t0 (lift (S O) O v)))) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda (x: T).(\lambda (H2: (or (subst0 O t t0 (lift (S O) O x)) (eq T t0 (lift (S O) O x)))).(or_ind (subst0 O t t0 (lift (S O) O x)) (eq T t0 (lift (S O) O x)) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda (H3: (subst0 O t t0 (lift (S O) O x))).(let H4 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda (H5: ((\forall (t2: T).((pr0 t t2) \to (eq T t t2))))).(let H6 \def H0 in (or_ind (\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda (H7: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (\lambda (t2: T).(\lambda (H8: (pr0 (THead (Bind Void) t t0) t2)).(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3)))) (pr0 t0 (lift (S O) O t2)) (eq T (THead (Bind Void) t t0) t2) (\lambda (H9: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t0 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))) (eq T (THead (Bind Void) t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H10: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H11: (pr0 t x0)).(\lambda (H12: (pr0 t0 x1)).(let H_y \def (H7 x1 H12) in (let H_y0 \def (H5 x0 H11) in (let H13 \def (eq_ind_r T x1 (\lambda (t: T).(pr0 t0 t)) H12 t0 H_y) in (let H14 \def (eq_ind_r T x1 (\lambda (t: T).(eq T t2 (THead (Bind Void) x0 t))) H10 t0 H_y) in (let H15 \def (eq_ind_r T x0 (\lambda (t0: T).(pr0 t t0)) H11 t H_y0) in (let H16 \def (eq_ind_r T x0 (\lambda (t: T).(eq T t2 (THead (Bind Void) t t0))) H14 t H_y0) in (eq_ind_r T (THead (Bind Void) t t0) (\lambda (t3: T).(eq T (THead (Bind Void) t t0) t3)) (refl_equal T (THead (Bind Void) t t0)) t2 H16)))))))))))) H9)) (\lambda (H9: (pr0 t0 (lift (S O) O t2))).(let H_y \def (H7 (lift (S O) O t2) H9) in (let H10 \def (eq_ind T t0 (\lambda (t0: T).(subst0 O t t0 (lift (S O) O x))) H3 (lift (S O) O t2) H_y) in (eq_ind_r T (lift (S O) O t2) (\lambda (t3: T).(eq T (THead (Bind Void) t t3) t2)) (subst0_gen_lift_false t2 t (lift (S O) O x) (S O) O O (le_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n)) (le_n (plus (S O) O)) (plus O (S O)) (plus_comm O (S O))) H10 (eq T (THead (Bind Void) t (lift (S O) O t2)) t2)) t0 H_y)))) (pr0_gen_void t t0 t2 H8)))))) (\lambda (H7: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda (x0: T).(\lambda (H8: (((eq T t0 x0) \to (\forall (P: Prop).P)))).(\lambda (H9: (pr0 t0 x0)).(or_intror (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind Void) t x0) (\lambda (H10: (eq T (THead (Bind Void) t t0) (THead (Bind Void) t x0))).(\lambda (P: Prop).(let H11 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Bind Void) t t0) (THead (Bind Void) t x0) H10) in (let H12 \def (eq_ind_r T x0 (\lambda (t: T).(pr0 t0 t)) H9 t0 H11) in (let H13 \def (eq_ind_r T x0 (\lambda (t: T).((eq T t0 t) \to (\forall (P: Prop).P))) H8 t0 H11) in (H13 (refl_equal T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x0 H9 (Bind Void))))))) H7)) H6))) (\lambda (H5: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)))) (\lambda (x0: T).(\lambda (H6: (((eq T t x0) \to (\forall (P: Prop).P)))).(\lambda (H7: (pr0 t x0)).(or_intror (\forall (t2: T).((pr0 (THead (Bind Void) t t0) t2) \to (eq T (THead (Bind Void) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Void) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t t0) t2)) (THead (Bind Void) x0 t0) (\lambda (H8: (eq T (THead (Bind Void) t t0) (THead (Bind Void) x0 t0))).(\lambda (P: Prop).(let H9 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t _) \Rightarrow t])) (THead (Bind Void) t t0) (THead (Bind Void) x0 t0) H8) in (let H10 \def (eq_ind_r T x0 (\lambda (t0: T).(pr0 t t0)) H7 t H9) in (let H11 \def (eq_ind_r T x0 (\lambda (t0: T).((eq T t t0) \to (\forall (P: Prop).P))) H6 t H9) in (H11 (refl_equal T t) P)))))) (pr0_comp t x0 H7 t0 t0 (pr0_refl t0) (Bind Void))))))) H5)) H4))) (\lambda (H3: (eq T t0 (lift (S O) O x))).(let H4 \def (eq_ind T t0 (\lambda (t: T).(or (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2))))) H0 (lift (S O) O x) H3) in (eq_ind_r T (lift (S O) O x) (\lambda (t2: T).(or (\forall (t3: T).((pr0 (THead (Bind Void) t t2) t3) \to (eq T (THead (Bind Void) t t2) t3))) (ex2 T (\lambda (t3: T).((eq T (THead (Bind Void) t t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Bind Void) t t2) t3))))) (or_intror (\forall (t2: T).((pr0 (THead (Bind Void) t (lift (S O) O x)) t2) \to (eq T (THead (Bind Void) t (lift (S O) O x)) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) t (lift (S O) O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t (lift (S O) O x)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Bind Void) t (lift (S O) O x)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) t (lift (S O) O x)) t2)) x (\lambda (H5: (eq T (THead (Bind Void) t (lift (S O) O x)) x)).(\lambda (P: Prop).(thead_x_lift_y_y (Bind Void) x t (S O) O H5 P))) (pr0_zeta Void not_void_abst x x (pr0_refl x) t))) t0 H3))) H2))) H1)))]) | (Flat f) \Rightarrow (match f return (\lambda (_: ?).(\lambda (f0: F).(or (\forall (t2: T).((pr0 (THead (Flat f0) t t0) t2) \to (eq T (THead (Flat f0) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat f0) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat f0) t t0) t2)))))) with [Appl \Rightarrow (let H_x \def (binder_dec t0) in (let H1 \def H_x in (or_ind (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u)))))) (\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (H2: (ex_3 B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u))))))).(ex_3_ind B T T (\lambda (b: B).(\lambda (w: T).(\lambda (u: T).(eq T t0 (THead (Bind b) w u))))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (H3: (eq T t0 (THead (Bind x0) x1 x2))).(let H4 \def (eq_ind T t0 (\lambda (t: T).(or (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2))))) H0 (THead (Bind x0) x1 x2) H3) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t2: T).(or (\forall (t3: T).((pr0 (THead (Flat Appl) t t2) t3) \to (eq T (THead (Flat Appl) t t2) t3))) (ex2 T (\lambda (t3: T).((eq T (THead (Flat Appl) t t2) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(pr0 (THead (Flat Appl) t t2) t3))))) ((match x0 return (\lambda (_: ?).(\lambda (b: B).((or (\forall (t2: T).((pr0 (THead (Bind b) x1 x2) t2) \to (eq T (THead (Bind b) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind b) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind b) x1 x2) t2)))) \to (or (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind b) x1 x2)) t2))))))) with [Abbr \Rightarrow (\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Abbr) x1 x2) t2) \to (eq T (THead (Bind Abbr) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abbr) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abbr) x1 x2) t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) t2)) (THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) (THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)))).(\lambda (P: Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Abbr) x1 x2)) (\lambda (ee: T).(match ee return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])])) I (THead (Bind Abbr) x1 (THead (Flat Appl) (lift (S O) O t) x2)) H6) in (False_ind P H7)))) (pr0_upsilon Abbr not_abbr_abst t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2))))) | Abst \Rightarrow (\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Abst) x1 x2) t2) \to (eq T (THead (Bind Abst) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Abst) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Abst) x1 x2) t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) t2)) (THead (Bind Abbr) t x2) (\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) (THead (Bind Abbr) t x2))).(\lambda (P: Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Abst) x1 x2)) (\lambda (ee: T).(match ee return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) t x2) H6) in (False_ind P H7)))) (pr0_beta x1 t t (pr0_refl t) x2 x2 (pr0_refl x2))))) | Void \Rightarrow (\lambda (_: (or (\forall (t2: T).((pr0 (THead (Bind Void) x1 x2) t2) \to (eq T (THead (Bind Void) x1 x2) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Bind Void) x1 x2) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Bind Void) x1 x2) t2))))).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2) \to (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t (THead (Bind Void) x1 x2)) t2)) (THead (Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)) (\lambda (H6: (eq T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) (THead (Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)))).(\lambda (P: Prop).(let H7 \def (eq_ind T (THead (Flat Appl) t (THead (Bind Void) x1 x2)) (\lambda (ee: T).(match ee return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ t) \Rightarrow (match t return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])])) I (THead (Bind Void) x1 (THead (Flat Appl) (lift (S O) O t) x2)) H6) in (False_ind P H7)))) (pr0_upsilon Void not_void_abst t t (pr0_refl t) x1 x1 (pr0_refl x1) x2 x2 (pr0_refl x2)))))]) H4) t0 H3)))))) H2)) (\lambda (H2: ((\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t0 (THead (Bind b) w u)) \to (\forall (P: Prop).P))))))).(let H3 \def H in (or_ind (\forall (t2: T).((pr0 t t2) \to (eq T t t2))) (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (H4: ((\forall (t2: T).((pr0 t t2) \to (eq T t t2))))).(let H5 \def H0 in (or_ind (\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2))) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (H6: ((\forall (t2: T).((pr0 t0 t2) \to (eq T t0 t2))))).(or_introl (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) (\lambda (t2: T).(\lambda (H7: (pr0 (THead (Flat Appl) t t0) t2)).(or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (H8: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t0 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 t u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t0 t3))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H9: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H10: (pr0 t x0)).(\lambda (H11: (pr0 t0 x1)).(let H_y \def (H6 x1 H11) in (let H_y0 \def (H4 x0 H10) in (let H12 \def (eq_ind_r T x1 (\lambda (t: T).(pr0 t0 t)) H11 t0 H_y) in (let H13 \def (eq_ind_r T x1 (\lambda (t: T).(eq T t2 (THead (Flat Appl) x0 t))) H9 t0 H_y) in (let H14 \def (eq_ind_r T x0 (\lambda (t0: T).(pr0 t t0)) H10 t H_y0) in (let H15 \def (eq_ind_r T x0 (\lambda (t: T).(eq T t2 (THead (Flat Appl) t t0))) H13 t H_y0) in (eq_ind_r T (THead (Flat Appl) t t0) (\lambda (t3: T).(eq T (THead (Flat Appl) t t0) t3)) (refl_equal T (THead (Flat Appl) t t0)) t2 H15)))))))))))) H8)) (\lambda (H8: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 t u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H9: (eq T t0 (THead (Bind Abst) x0 x1))).(\lambda (H10: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (_: (pr0 t x2)).(\lambda (_: (pr0 x1 x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t3: T).(eq T (THead (Flat Appl) t t0) t3)) (let H13 \def (eq_ind T t0 (\lambda (t: T).(\forall (t2: T).((pr0 t t2) \to (eq T t t2)))) H6 (THead (Bind Abst) x0 x1) H9) in (let H14 \def (eq_ind T t0 (\lambda (t: T).(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t (THead (Bind b) w u)) \to (\forall (P: Prop).P)))))) H2 (THead (Bind Abst) x0 x1) H9) in (eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t3: T).(eq T (THead (Flat Appl) t t3) (THead (Bind Abbr) x2 x3))) (H14 Abst x0 x1 (H13 (THead (Bind Abst) x0 x1) (pr0_refl (THead (Bind Abst) x0 x1))) (eq T (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) (THead (Bind Abbr) x2 x3))) t0 H9))) t2 H10))))))))) H8)) (\lambda (H8: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 t u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))) (eq T (THead (Flat Appl) t t0) t2) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (not (eq B x0 Abst))).(\lambda (H10: (eq T t0 (THead (Bind x0) x1 x2))).(\lambda (H11: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda (_: (pr0 t x3)).(\lambda (_: (pr0 x1 x4)).(\lambda (_: (pr0 x2 x5)).(eq_ind_r T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t3: T).(eq T (THead (Flat Appl) t t0) t3)) (let H15 \def (eq_ind T t0 (\lambda (t: T).(\forall (t2: T).((pr0 t t2) \to (eq T t t2)))) H6 (THead (Bind x0) x1 x2) H10) in (let H16 \def (eq_ind T t0 (\lambda (t: T).(\forall (b: B).(\forall (w: T).(\forall (u: T).((eq T t (THead (Bind b) w u)) \to (\forall (P: Prop).P)))))) H2 (THead (Bind x0) x1 x2) H10) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t3: T).(eq T (THead (Flat Appl) t t3) (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))) (H16 x0 x1 x2 (H15 (THead (Bind x0) x1 x2) (pr0_refl (THead (Bind x0) x1 x2))) (eq T (THead (Flat Appl) t (THead (Bind x0) x1 x2)) (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))) t0 H10))) t2 H11))))))))))))) H8)) (pr0_gen_appl t t0 t2 H7)))))) (\lambda (H6: (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t0 t2)) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (x: T).(\lambda (H7: (((eq T t0 x) \to (\forall (P: Prop).P)))).(\lambda (H8: (pr0 t0 x)).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) (THead (Flat Appl) t x) (\lambda (H9: (eq T (THead (Flat Appl) t t0) (THead (Flat Appl) t x))).(\lambda (P: Prop).(let H10 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) t t0) (THead (Flat Appl) t x) H9) in (let H11 \def (eq_ind_r T x (\lambda (t: T).(pr0 t0 t)) H8 t0 H10) in (let H12 \def (eq_ind_r T x (\lambda (t: T).((eq T t0 t) \to (\forall (P: Prop).P))) H7 t0 H10) in (H12 (refl_equal T t0) P)))))) (pr0_comp t t (pr0_refl t) t0 x H8 (Flat Appl))))))) H6)) H5))) (\lambda (H4: (ex2 T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)))).(ex2_ind T (\lambda (t2: T).((eq T t t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t t2)) (or (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)))) (\lambda (x: T).(\lambda (H5: (((eq T t x) \to (\forall (P: Prop).P)))).(\lambda (H6: (pr0 t x)).(or_intror (\forall (t2: T).((pr0 (THead (Flat Appl) t t0) t2) \to (eq T (THead (Flat Appl) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Appl) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Appl) t t0) t2)) (THead (Flat Appl) x t0) (\lambda (H7: (eq T (THead (Flat Appl) t t0) (THead (Flat Appl) x t0))).(\lambda (P: Prop).(let H8 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) t t0) (THead (Flat Appl) x t0) H7) in (let H9 \def (eq_ind_r T x (\lambda (t0: T).(pr0 t t0)) H6 t H8) in (let H10 \def (eq_ind_r T x (\lambda (t0: T).((eq T t t0) \to (\forall (P: Prop).P))) H5 t H8) in (H10 (refl_equal T t) P)))))) (pr0_comp t x H6 t0 t0 (pr0_refl t0) (Flat Appl))))))) H4)) H3))) H1))) | Cast \Rightarrow (or_intror (\forall (t2: T).((pr0 (THead (Flat Cast) t t0) t2) \to (eq T (THead (Flat Cast) t t0) t2))) (ex2 T (\lambda (t2: T).((eq T (THead (Flat Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Cast) t t0) t2))) (ex_intro2 T (\lambda (t2: T).((eq T (THead (Flat Cast) t t0) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 (THead (Flat Cast) t t0) t2)) t0 (\lambda (H1: (eq T (THead (Flat Cast) t t0) t0)).(\lambda (P: Prop).(thead_x_y_y (Flat Cast) t t0 H1 P))) (pr0_epsilon t0 t0 (pr0_refl t0) t)))])])))))) t1).
 
 inductive pr1: T \to (T \to Prop) \def
 | pr1_r: \forall (t: T).(pr1 t t)
@@ -1960,32 +1960,32 @@ inductive wcpr0: C \to (C \to Prop) \def
 theorem wcpr0_gen_sort:
  \forall (x: C).(\forall (n: nat).((wcpr0 (CSort n) x) \to (eq C x (CSort n))))
 \def
- \lambda (x: C).(\lambda (n: nat).(\lambda (H: (wcpr0 (CSort n) x)).(let H0 \def (match H return (\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n)) \to ((eq C c0 x) \to (eq C x (CSort n)))))) with [(wcpr0_refl c) \Rightarrow (\lambda (H0: (eq C c (CSort n))).(\lambda (H1: (eq C c x)).(eq_ind C (CSort n) (\lambda (c0: C).((eq C c0 x) \to (eq C x (CSort n)))) (\lambda (H2: (eq C (CSort n) x)).(eq_ind C (CSort n) (\lambda (c0: C).(eq C c0 (CSort n))) (refl_equal C (CSort n)) x H2)) c (sym_eq C c (CSort n) H0) H1))) | (wcpr0_comp c1 c2 H0 u1 u2 H1 k) \Rightarrow (\lambda (H2: (eq C (CHead c1 k u1) (CSort n))).(\lambda (H3: (eq C (CHead c2 k u2) x)).((let H4 \def (eq_ind C (CHead c1 k u1) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H2) in (False_ind ((eq C (CHead c2 k u2) x) \to ((wcpr0 c1 c2) \to ((pr0 u1 u2) \to (eq C x (CSort n))))) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CSort n)) (refl_equal C x))))).
+ \lambda (x: C).(\lambda (n: nat).(\lambda (H: (wcpr0 (CSort n) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n)) \to ((eq C c0 x) \to (eq C x (CSort n))))))) with [(wcpr0_refl c) \Rightarrow (\lambda (H0: (eq C c (CSort n))).(\lambda (H1: (eq C c x)).(eq_ind C (CSort n) (\lambda (c0: C).((eq C c0 x) \to (eq C x (CSort n)))) (\lambda (H2: (eq C (CSort n) x)).(eq_ind C (CSort n) (\lambda (c0: C).(eq C c0 (CSort n))) (refl_equal C (CSort n)) x H2)) c (sym_eq C c (CSort n) H0) H1))) | (wcpr0_comp c1 c2 H0 u1 u2 H1 k) \Rightarrow (\lambda (H2: (eq C (CHead c1 k u1) (CSort n))).(\lambda (H3: (eq C (CHead c2 k u2) x)).((let H4 \def (eq_ind C (CHead c1 k u1) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n) H2) in (False_ind ((eq C (CHead c2 k u2) x) \to ((wcpr0 c1 c2) \to ((pr0 u1 u2) \to (eq C x (CSort n))))) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CSort n)) (refl_equal C x))))).
 
 theorem wcpr0_gen_head:
  \forall (k: K).(\forall (c1: C).(\forall (x: C).(\forall (u1: T).((wcpr0 (CHead c1 k u1) x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))
 \def
- \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda (H: (wcpr0 (CHead c1 k u1) x)).(let H0 \def (match H return (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead c1 k u1)) \to ((eq C c0 x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))) with [(wcpr0_refl c) \Rightarrow (\lambda (H0: (eq C c (CHead c1 k u1))).(\lambda (H1: (eq C c x)).(eq_ind C (CHead c1 k u1) (\lambda (c0: C).((eq C c0 x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))) (\lambda (H2: (eq C (CHead c1 k u1) x)).(eq_ind C (CHead c1 k u1) (\lambda (c0: C).(or (eq C c0 (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (or_introl (eq C (CHead c1 k u1) (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))) (refl_equal C (CHead c1 k u1))) x H2)) c (sym_eq C c (CHead c1 k u1) H0) H1))) | (wcpr0_comp c0 c2 H0 u0 u2 H1 k0) \Rightarrow (\lambda (H2: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(\lambda (H3: (eq C (CHead c2 k0 u2) x)).((let H4 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c1 k u1) H2) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c0 k0 u0) (CHead c1 k u1) H2) in ((let H6 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k u1) H2) in (eq_ind C c1 (\lambda (c: C).((eq K k0 k) \to ((eq T u0 u1) \to ((eq C (CHead c2 k0 u2) x) \to ((wcpr0 c c2) \to ((pr0 u0 u2) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C x (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))))))))))) (\lambda (H7: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to ((eq C (CHead c2 k1 u2) x) \to ((wcpr0 c1 c2) \to ((pr0 u0 u2) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C x (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))))))) (\lambda (H8: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 k u2) x) \to ((wcpr0 c1 c2) \to ((pr0 t u2) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C x (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))))))))) (\lambda (H9: (eq C (CHead c2 k u2) x)).(eq_ind C (CHead c2 k u2) (\lambda (c: C).((wcpr0 c1 c2) \to ((pr0 u1 u2) \to (or (eq C c (CHead c1 k u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C c (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))))) (\lambda (H10: (wcpr0 c1 c2)).(\lambda (H11: (pr0 u1 u2)).(or_intror (eq C (CHead c2 k u2) (CHead c1 k u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c2 u2 (refl_equal C (CHead c2 k u2)) H10 H11)))) x H9)) u0 (sym_eq T u0 u1 H8))) k0 (sym_eq K k0 k H7))) c0 (sym_eq C c0 c1 H6))) H5)) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CHead c1 k u1)) (refl_equal C x))))))).
+ \lambda (k: K).(\lambda (c1: C).(\lambda (x: C).(\lambda (u1: T).(\lambda (H: (wcpr0 (CHead c1 k u1) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead c1 k u1)) \to ((eq C c0 x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))) with [(wcpr0_refl c) \Rightarrow (\lambda (H0: (eq C c (CHead c1 k u1))).(\lambda (H1: (eq C c x)).(eq_ind C (CHead c1 k u1) (\lambda (c0: C).((eq C c0 x) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq C x (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))) (\lambda (H2: (eq C (CHead c1 k u1) x)).(eq_ind C (CHead c1 k u1) (\lambda (c0: C).(or (eq C c0 (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq C c0 (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))) (or_introl (eq C (CHead c1 k u1) (CHead c1 k u1)) (ex3_2 C T (\lambda (c2: C).(\lambda (u2: T).(eq C (CHead c1 k u1) (CHead c2 k u2)))) (\lambda (c2: C).(\lambda (_: T).(wcpr0 c1 c2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))) (refl_equal C (CHead c1 k u1))) x H2)) c (sym_eq C c (CHead c1 k u1) H0) H1))) | (wcpr0_comp c0 c2 H0 u0 u2 H1 k0) \Rightarrow (\lambda (H2: (eq C (CHead c0 k0 u0) (CHead c1 k u1))).(\lambda (H3: (eq C (CHead c2 k0 u2) x)).((let H4 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 u0) (CHead c1 k u1) H2) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c0 k0 u0) (CHead c1 k u1) H2) in ((let H6 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k0 u0) (CHead c1 k u1) H2) in (eq_ind C c1 (\lambda (c: C).((eq K k0 k) \to ((eq T u0 u1) \to ((eq C (CHead c2 k0 u2) x) \to ((wcpr0 c c2) \to ((pr0 u0 u2) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C x (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))))))))))) (\lambda (H7: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T u0 u1) \to ((eq C (CHead c2 k1 u2) x) \to ((wcpr0 c1 c2) \to ((pr0 u0 u2) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C x (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))))))) (\lambda (H8: (eq T u0 u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 k u2) x) \to ((wcpr0 c1 c2) \to ((pr0 t u2) \to (or (eq C x (CHead c1 k u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C x (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))))))))) (\lambda (H9: (eq C (CHead c2 k u2) x)).(eq_ind C (CHead c2 k u2) (\lambda (c: C).((wcpr0 c1 c2) \to ((pr0 u1 u2) \to (or (eq C c (CHead c1 k u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C c (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))))))) (\lambda (H10: (wcpr0 c1 c2)).(\lambda (H11: (pr0 u1 u2)).(or_intror (eq C (CHead c2 k u2) (CHead c1 k u1)) (ex3_2 C T (\lambda (c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3)))) (ex3_2_intro C T (\lambda (c3: C).(\lambda (u3: T).(eq C (CHead c2 k u2) (CHead c3 k u3)))) (\lambda (c3: C).(\lambda (_: T).(wcpr0 c1 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c2 u2 (refl_equal C (CHead c2 k u2)) H10 H11)))) x H9)) u0 (sym_eq T u0 u1 H8))) k0 (sym_eq K k0 k H7))) c0 (sym_eq C c0 c1 H6))) H5)) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CHead c1 k u1)) (refl_equal C x))))))).
 
 theorem wcpr0_drop:
  \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))))
 \def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind (\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c0 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda (u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c3 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c0 k u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 k u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c0 k u1) (CHead e1 k0 u0))).(let H4 \def (match (drop_gen_refl (CHead c0 k u1) (CHead e1 k0 u0) H3) return (\lambda (c: C).((eq C c (CHead e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u0 u2)))))) with [refl_equal \Rightarrow (\lambda (H3: (eq C (CHead c0 k u1) (CHead e1 k0 u0))).(let H4 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k u1) (CHead e1 k0 u0) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c0 k u1) (CHead e1 k0 u0) H3) in ((let H6 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k u1) (CHead e1 k0 u0) H3) in (eq_ind C e1 (\lambda (_: C).((eq K k k0) \to ((eq T u1 u0) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u0 u2))))))) (\lambda (H7: (eq K k k0)).(eq_ind K k0 (\lambda (k: K).((eq T u1 u0) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u0 u2)))))) (\lambda (H8: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k0 u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u0 u2))))) (let H9 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H2 u0 H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(wcpr0 c c3)) H0 e1 H6) in (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k0 u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u0 u2))) c3 u2 (drop_refl (CHead c3 k0 u2)) H10 H9))) u1 (sym_eq T u1 u0 H8))) k (sym_eq K k k0 H7))) c0 (sym_eq C c0 e1 H6))) H5)) H4)))]) in (H4 (refl_equal C (CHead e1 k0 u0)))))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((drop n O (CHead c0 k0 u1) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 k0 u2) (CHead e2 k u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u3 u2))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c0 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O (CHead c3 k0 u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((drop n O (CHead c0 (Bind b) u1) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Bind b) u2) (CHead e2 k u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u3 u2)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c0 (Bind b) u1) (CHead e1 k0 u0))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (drop n O c3 (CHead x0 k0 x1))).(\lambda (H6: (wcpr0 e1 x0)).(\lambda (H7: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop (Bind b) n c3 (CHead x0 k0 x1) H5 u2) H6 H7)))))) (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c0 (CHead e1 k0 u0) u1 n H4)))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((drop n O (CHead c0 (Flat f) u1) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Flat f) u2) (CHead e2 k u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u3 u2)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c0 (Flat f) u1) (CHead e1 k0 u0))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (drop (S n) O c3 (CHead x0 k0 x1))).(\lambda (H6: (wcpr0 e1 x0)).(\lambda (H7: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop (Flat f) n c3 (CHead x0 k0 x1) H5 u2) H6 H7)))))) (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c0 (CHead e1 k0 u0) u1 n H4)))))))))) k) h)))))))))) c1 c2 H))).
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind (\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c0 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda (u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c3 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c0 k u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 k u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c0 k u1) (CHead e1 k0 u0))).(let H4 \def (match (drop_gen_refl (CHead c0 k u1) (CHead e1 k0 u0) H3) return (\lambda (_: ?).(\lambda (c: C).((eq C c (CHead e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u0 u2))))))) with [refl_equal \Rightarrow (\lambda (H3: (eq C (CHead c0 k u1) (CHead e1 k0 u0))).(let H4 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c0 k u1) (CHead e1 k0 u0) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c0 k u1) (CHead e1 k0 u0) H3) in ((let H6 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k u1) (CHead e1 k0 u0) H3) in (eq_ind C e1 (\lambda (_: C).((eq K k k0) \to ((eq T u1 u0) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u0 u2))))))) (\lambda (H7: (eq K k k0)).(eq_ind K k0 (\lambda (k: K).((eq T u1 u0) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u0 u2)))))) (\lambda (H8: (eq T u1 u0)).(eq_ind T u0 (\lambda (_: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k0 u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u0 u2))))) (let H9 \def (eq_ind T u1 (\lambda (t: T).(pr0 t u2)) H2 u0 H8) in (let H10 \def (eq_ind C c0 (\lambda (c: C).(wcpr0 c c3)) H0 e1 H6) in (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop O O (CHead c3 k0 u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u0 u2))) c3 u2 (drop_refl (CHead c3 k0 u2)) H10 H9))) u1 (sym_eq T u1 u0 H8))) k (sym_eq K k k0 H7))) c0 (sym_eq C c0 e1 H6))) H5)) H4)))]) in (H4 (refl_equal C (CHead e1 k0 u0)))))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((drop n O (CHead c0 k0 u1) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 k0 u2) (CHead e2 k u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u3 u2))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c0 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O (CHead c3 k0 u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((drop n O (CHead c0 (Bind b) u1) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Bind b) u2) (CHead e2 k u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u3 u2)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c0 (Bind b) u1) (CHead e1 k0 u0))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (drop n O c3 (CHead x0 k0 x1))).(\lambda (H6: (wcpr0 e1 x0)).(\lambda (H7: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop (Bind b) n c3 (CHead x0 k0 x1) H5 u2) H6 H7)))))) (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c0 (CHead e1 k0 u0) u1 n H4)))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((drop n O (CHead c0 (Flat f) u1) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c3 (Flat f) u2) (CHead e2 k u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u3 u2)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c0 (Flat f) u1) (CHead e1 k0 u0))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (drop (S n) O c3 (CHead x0 k0 x1))).(\lambda (H6: (wcpr0 e1 x0)).(\lambda (H7: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (drop_drop (Flat f) n c3 (CHead x0 k0 x1) H5 u2) H6 H7)))))) (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c0 (CHead e1 k0 u0) u1 n H4)))))))))) k) h)))))))))) c1 c2 H))).
 
 theorem wcpr0_drop_back:
  \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c1 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c2 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1)))))))))))
 \def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind (\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda (u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c0 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 k u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c0 k u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c3 k u2) (CHead e1 k0 u0))).(let H4 \def (match (drop_gen_refl (CHead c3 k u2) (CHead e1 k0 u0) H3) return (\lambda (c: C).((eq C c (CHead e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop O O (CHead c0 k u1) (CHead e2 k0 u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u0)))))) with [refl_equal \Rightarrow (\lambda (H3: (eq C (CHead c3 k u2) (CHead e1 k0 u0))).(let H4 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c3 k u2) (CHead e1 k0 u0) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c3 k u2) (CHead e1 k0 u0) H3) in ((let H6 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 k u2) (CHead e1 k0 u0) H3) in (eq_ind C e1 (\lambda (_: C).((eq K k k0) \to ((eq T u2 u0) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop O O (CHead c0 k u1) (CHead e2 k0 u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u0))))))) (\lambda (H7: (eq K k k0)).(eq_ind K k0 (\lambda (k: K).((eq T u2 u0) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop O O (CHead c0 k u1) (CHead e2 k0 u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u0)))))) (\lambda (H8: (eq T u2 u0)).(eq_ind T u0 (\lambda (_: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop O O (CHead c0 k0 u1) (CHead e2 k0 u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u0))))) (let H9 \def (eq_ind T u2 (\lambda (t: T).(pr0 u1 t)) H2 u0 H8) in (let H10 \def (eq_ind C c3 (\lambda (c: C).(wcpr0 c0 c)) H0 e1 H6) in (ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop O O (CHead c0 k0 u1) (CHead e2 k0 u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u0))) c0 u1 (drop_refl (CHead c0 k0 u1)) H10 H9))) u2 (sym_eq T u2 u0 H8))) k (sym_eq K k k0 H7))) c3 (sym_eq C c3 e1 H6))) H5)) H4)))]) in (H4 (refl_equal C (CHead e1 k0 u0)))))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((drop n O (CHead c3 k0 u2) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop n O (CHead c0 k0 u1) (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u3))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c3 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O (CHead c0 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((drop n O (CHead c3 (Bind b) u2) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop n O (CHead c0 (Bind b) u1) (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Bind b) u2) (CHead e1 k0 u0))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n O c0 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c0 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (drop n O c0 (CHead x0 k0 x1))).(\lambda (H6: (wcpr0 x0 e1)).(\lambda (H7: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c0 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop (Bind b) n c0 (CHead x0 k0 x1) H5 u1) H6 H7)))))) (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c3 (CHead e1 k0 u0) u2 n H4)))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((drop n O (CHead c3 (Flat f) u2) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop n O (CHead c0 (Flat f) u1) (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Flat f) u2) (CHead e1 k0 u0))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O c0 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (drop (S n) O c0 (CHead x0 k0 x1))).(\lambda (H6: (wcpr0 x0 e1)).(\lambda (H7: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop (Flat f) n c0 (CHead x0 k0 x1) H5 u1) H6 H7)))))) (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c3 (CHead e1 k0 u0) u2 n H4)))))))))) k) h)))))))))) c2 c1 H))).
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind (\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda (u1: T).(\lambda (k: K).(\lambda (H0: (drop h O c (CHead e1 k u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((drop h O c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop h O c0 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((drop n O (CHead c3 k u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop n O (CHead c0 k u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (drop O O (CHead c3 k u2) (CHead e1 k0 u0))).(let H4 \def (match (drop_gen_refl (CHead c3 k u2) (CHead e1 k0 u0) H3) return (\lambda (_: ?).(\lambda (c: C).((eq C c (CHead e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop O O (CHead c0 k u1) (CHead e2 k0 u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u0))))))) with [refl_equal \Rightarrow (\lambda (H3: (eq C (CHead c3 k u2) (CHead e1 k0 u0))).(let H4 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u2 | (CHead _ _ t) \Rightarrow t])) (CHead c3 k u2) (CHead e1 k0 u0) H3) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c3 k u2) (CHead e1 k0 u0) H3) in ((let H6 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c3 | (CHead c _ _) \Rightarrow c])) (CHead c3 k u2) (CHead e1 k0 u0) H3) in (eq_ind C e1 (\lambda (_: C).((eq K k k0) \to ((eq T u2 u0) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop O O (CHead c0 k u1) (CHead e2 k0 u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u0))))))) (\lambda (H7: (eq K k k0)).(eq_ind K k0 (\lambda (k: K).((eq T u2 u0) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop O O (CHead c0 k u1) (CHead e2 k0 u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u0)))))) (\lambda (H8: (eq T u2 u0)).(eq_ind T u0 (\lambda (_: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop O O (CHead c0 k0 u1) (CHead e2 k0 u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u0))))) (let H9 \def (eq_ind T u2 (\lambda (t: T).(pr0 u1 t)) H2 u0 H8) in (let H10 \def (eq_ind C c3 (\lambda (c: C).(wcpr0 c0 c)) H0 e1 H6) in (ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(drop O O (CHead c0 k0 u1) (CHead e2 k0 u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u0))) c0 u1 (drop_refl (CHead c0 k0 u1)) H10 H9))) u2 (sym_eq T u2 u0 H8))) k (sym_eq K k k0 H7))) c3 (sym_eq C c3 e1 H6))) H5)) H4)))]) in (H4 (refl_equal C (CHead e1 k0 u0)))))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((drop n O (CHead c3 k0 u2) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop n O (CHead c0 k0 u1) (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u3))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((drop (S n) O (CHead c3 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(drop (S n) O (CHead c0 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((drop n O (CHead c3 (Bind b) u2) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop n O (CHead c0 (Bind b) u1) (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Bind b) u2) (CHead e1 k0 u0))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop n O c0 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c0 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (drop n O c0 (CHead x0 k0 x1))).(\lambda (H6: (wcpr0 x0 e1)).(\lambda (H7: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c0 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop (Bind b) n c0 (CHead x0 k0 x1) H5 u1) H6 H7)))))) (H1 n e1 u0 k0 (drop_gen_drop (Bind b) c3 (CHead e1 k0 u0) u2 n H4)))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((drop n O (CHead c3 (Flat f) u2) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(drop n O (CHead c0 (Flat f) u1) (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: (drop (S n) O (CHead c3 (Flat f) u2) (CHead e1 k0 u0))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O c0 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (drop (S n) O c0 (CHead x0 k0 x1))).(\lambda (H6: (wcpr0 x0 e1)).(\lambda (H7: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(drop (S n) O (CHead c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (drop_drop (Flat f) n c0 (CHead x0 k0 x1) H5 u1) H6 H7)))))) (H1 (S n) e1 u0 k0 (drop_gen_drop (Flat f) c3 (CHead e1 k0 u0) u2 n H4)))))))))) k) h)))))))))) c2 c1 H))).
 
 theorem wcpr0_getl:
  \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2)))))))))))
 \def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind (\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c0 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda (u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c3 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c0 k u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c0 k u1) (CHead e1 k0 u0))).((match k return (\lambda (k1: K).((clear (CHead c0 k1 u1) (CHead e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 k1 u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))))) with [(Bind b) \Rightarrow (\lambda (H4: (clear (CHead c0 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c0 (Bind b) u1) (clear_gen_bind b c0 (CHead e1 k0 u0) u1 H4)) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead e1 k0 u0) (CHead c0 (Bind b) u1) (clear_gen_bind b c0 (CHead e1 k0 u0) u1 H4)) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0 u0) (CHead c0 (Bind b) u1) (clear_gen_bind b c0 (CHead e1 k0 u0) u1 H4)) in (\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c0)).(eq_ind_r K (Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Bind b) u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind_r T u1 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 t u3))))) (eq_ind_r C c0 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 c0 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c3 u2 (getl_refl b c3 u2) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5))) | (Flat f) \Rightarrow (\lambda (H4: (clear (CHead c0 (Flat f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 O e1 u0 k0 (getl_intro O c0 (CHead e1 k0 u0) c0 (drop_refl c0) (clear_gen_flat f c0 (CHead e1 k0 u0) u1 H4))) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (getl_flat c3 (CHead x0 k0 x1) O H6 f u2) H7 H8)))))) H5)))]) (getl_gen_O (CHead c0 k u1) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((getl n (CHead c0 k0 u1) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k0 u2) (CHead e2 k u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u3 u2))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl (S n) (CHead c0 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl (S n) (CHead c3 k0 u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((getl n (CHead c0 (Bind b) u1) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 (Bind b) u2) (CHead e2 k u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u3 u2)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: (getl (S n) (CHead c0 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (getl_gen_S (Bind b) c0 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl n c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl n c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Bind b) n c3 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((getl n (CHead c0 (Flat f) u1) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 (Flat f) u2) (CHead e2 k u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u3 u2)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: (getl (S n) (CHead c0 (Flat f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (getl_gen_S (Flat f) c0 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Flat f) n c3 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) k) h)))))))))) c1 c2 H))).
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(wcpr0_ind (\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c0 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda (u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c3 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u1 u2))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c0 k u1) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k u2) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))) (\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c0 k u1) (CHead e1 k0 u0))).((match k return (\lambda (_: ?).(\lambda (k1: K).((clear (CHead c0 k1 u1) (CHead e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 k1 u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))))) with [(Bind b) \Rightarrow (\lambda (H4: (clear (CHead c0 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c0 (Bind b) u1) (clear_gen_bind b c0 (CHead e1 k0 u0) u1 H4)) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead e1 k0 u0) (CHead c0 (Bind b) u1) (clear_gen_bind b c0 (CHead e1 k0 u0) u1 H4)) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0 u0) (CHead c0 (Bind b) u1) (clear_gen_bind b c0 (CHead e1 k0 u0) u1 H4)) in (\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c0)).(eq_ind_r K (Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Bind b) u2) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))))) (eq_ind_r T u1 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 t u3))))) (eq_ind_r C c0 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 c e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Bind b) u2) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 c0 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u1 u3))) c3 u2 (getl_refl b c3 u2) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5))) | (Flat f) \Rightarrow (\lambda (H4: (clear (CHead c0 (Flat f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 O e1 u0 k0 (getl_intro O c0 (CHead e1 k0 u0) c0 (drop_refl c0) (clear_gen_flat f c0 (CHead e1 k0 u0) u1 H4))) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl O c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl O c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (getl_flat c3 (CHead x0 k0 x1) O H6 f u2) H7 H8)))))) H5)))]) (getl_gen_O (CHead c0 k u1) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((getl n (CHead c0 k0 u1) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 k0 u2) (CHead e2 k u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u3 u2))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl (S n) (CHead c0 k0 u1) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl (S n) (CHead c3 k0 u2) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u4: T).(pr0 u3 u4))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((getl n (CHead c0 (Bind b) u1) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 (Bind b) u2) (CHead e2 k u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u3 u2)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: (getl (S n) (CHead c0 (Bind b) u1) (CHead e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (getl_gen_S (Bind b) c0 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl n c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl n c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 (Bind b) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Bind b) n c3 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((getl n (CHead c0 (Flat f) u1) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c3 (Flat f) u2) (CHead e2 k u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u3 u2)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: (getl (S n) (CHead c0 (Flat f) u1) (CHead e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (getl_gen_S (Flat f) c0 (CHead e1 k0 u0) u1 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) c3 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c3 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 e1 x0)).(\lambda (H8: (pr0 u0 x1)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c3 (Flat f) u2) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e1 e2))) (\lambda (_: C).(\lambda (u3: T).(pr0 u0 u3))) x0 x1 (getl_head (Flat f) n c3 (CHead x0 k0 x1) H6 u2) H7 H8)))))) H5))))))))) k) h)))))))))) c1 c2 H))).
 
 theorem wcpr0_getl_back:
  \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c1 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c2 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1)))))))))))
 \def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind (\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda (u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c0 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 k u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c0 k u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c3 k u2) (CHead e1 k0 u0))).((match k return (\lambda (k1: K).((clear (CHead c3 k1 u2) (CHead e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 k1 u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))))) with [(Bind b) \Rightarrow (\lambda (H4: (clear (CHead c3 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c3 (Bind b) u2) (clear_gen_bind b c3 (CHead e1 k0 u0) u2 H4)) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead e1 k0 u0) (CHead c3 (Bind b) u2) (clear_gen_bind b c3 (CHead e1 k0 u0) u2 H4)) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0 u0) (CHead c3 (Bind b) u2) (clear_gen_bind b c3 (CHead e1 k0 u0) u2 H4)) in (\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c3)).(eq_ind_r K (Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 (Bind b) u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind_r T u2 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 t))))) (eq_ind_r C c3 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))) c0 u1 (getl_refl b c0 u1) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5))) | (Flat f) \Rightarrow (\lambda (H4: (clear (CHead c3 (Flat f) u2) (CHead e1 k0 u0))).(let H5 \def (H1 O e1 u0 k0 (getl_intro O c3 (CHead e1 k0 u0) c3 (drop_refl c3) (clear_gen_flat f c3 (CHead e1 k0 u0) u2 H4))) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl O c0 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl O c0 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (getl_flat c0 (CHead x0 k0 x1) O H6 f u1) H7 H8)))))) H5)))]) (getl_gen_O (CHead c3 k u2) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((getl n (CHead c3 k0 u2) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl n (CHead c0 k0 u1) (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u3))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl (S n) (CHead c3 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl (S n) (CHead c0 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((getl n (CHead c3 (Bind b) u2) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl n (CHead c0 (Bind b) u1) (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: (getl (S n) (CHead c3 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (getl_gen_S (Bind b) c3 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl n c0 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c0 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl n c0 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c0 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Bind b) n c0 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((getl n (CHead c3 (Flat f) u2) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl n (CHead c0 (Flat f) u1) (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: (getl (S n) (CHead c3 (Flat f) u2) (CHead e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (getl_gen_S (Flat f) c3 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) c0 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c0 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Flat f) n c0 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) k) h)))))))))) c2 c1 H))).
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind (\lambda (c: C).(\lambda (c0: C).(\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c0 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))))))))))) (\lambda (c: C).(\lambda (h: nat).(\lambda (e1: C).(\lambda (u1: T).(\lambda (k: K).(\lambda (H0: (getl h c (CHead e1 k u1))).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u2: T).(getl h c (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))) e1 u1 H0 (wcpr0_refl e1) (pr0_refl u1)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (H1: ((\forall (h: nat).(\forall (e1: C).(\forall (u1: T).(\forall (k: K).((getl h c3 (CHead e1 k u1)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl h c0 (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u1))))))))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (e1: C).(\forall (u3: T).(\forall (k0: K).((getl n (CHead c3 k u2) (CHead e1 k0 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl n (CHead c0 k u1) (CHead e2 k0 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))) (\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H3: (getl O (CHead c3 k u2) (CHead e1 k0 u0))).((match k return (\lambda (_: ?).(\lambda (k1: K).((clear (CHead c3 k1 u2) (CHead e1 k0 u0)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 k1 u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))))) with [(Bind b) \Rightarrow (\lambda (H4: (clear (CHead c3 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow e1 | (CHead c _ _) \Rightarrow c])) (CHead e1 k0 u0) (CHead c3 (Bind b) u2) (clear_gen_bind b c3 (CHead e1 k0 u0) u2 H4)) in ((let H6 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead e1 k0 u0) (CHead c3 (Bind b) u2) (clear_gen_bind b c3 (CHead e1 k0 u0) u2 H4)) in ((let H7 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead e1 k0 u0) (CHead c3 (Bind b) u2) (clear_gen_bind b c3 (CHead e1 k0 u0) u2 H4)) in (\lambda (H8: (eq K k0 (Bind b))).(\lambda (H9: (eq C e1 c3)).(eq_ind_r K (Bind b) (\lambda (k1: K).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 (Bind b) u1) (CHead e2 k1 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))))) (eq_ind_r T u2 (\lambda (t: T).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 t))))) (eq_ind_r C c3 (\lambda (c: C).(ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 (Bind b) u1) (CHead e2 (Bind b) u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 c3))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u2))) c0 u1 (getl_refl b c0 u1) H0 H2) e1 H9) u0 H7) k0 H8)))) H6)) H5))) | (Flat f) \Rightarrow (\lambda (H4: (clear (CHead c3 (Flat f) u2) (CHead e1 k0 u0))).(let H5 \def (H1 O e1 u0 k0 (getl_intro O c3 (CHead e1 k0 u0) c3 (drop_refl c3) (clear_gen_flat f c3 (CHead e1 k0 u0) u2 H4))) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl O c0 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl O c0 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl O (CHead c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (getl_flat c0 (CHead x0 k0 x1) O H6 f u1) H7 H8)))))) H5)))]) (getl_gen_O (CHead c3 k u2) (CHead e1 k0 u0) H3)))))) (K_ind (\lambda (k0: K).(\forall (n: nat).(((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((getl n (CHead c3 k0 u2) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl n (CHead c0 k0 u1) (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u3))))))))) \to (\forall (e1: C).(\forall (u3: T).(\forall (k1: K).((getl (S n) (CHead c3 k0 u2) (CHead e1 k1 u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u4: T).(getl (S n) (CHead c0 k0 u1) (CHead e2 k1 u4)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u4: T).(pr0 u4 u3))))))))))) (\lambda (b: B).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((getl n (CHead c3 (Bind b) u2) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl n (CHead c0 (Bind b) u1) (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: (getl (S n) (CHead c3 (Bind b) u2) (CHead e1 k0 u0))).(let H5 \def (H1 n e1 u0 k0 (getl_gen_S (Bind b) c3 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl n c0 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c0 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl n c0 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c0 (Bind b) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Bind b) n c0 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) (\lambda (f: F).(\lambda (n: nat).(\lambda (_: ((\forall (e1: C).(\forall (u3: T).(\forall (k: K).((getl n (CHead c3 (Flat f) u2) (CHead e1 k u3)) \to (ex3_2 C T (\lambda (e2: C).(\lambda (u2: T).(getl n (CHead c0 (Flat f) u1) (CHead e2 k u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u3)))))))))).(\lambda (e1: C).(\lambda (u0: T).(\lambda (k0: K).(\lambda (H4: (getl (S n) (CHead c3 (Flat f) u2) (CHead e1 k0 u0))).(let H5 \def (H1 (S n) e1 u0 k0 (getl_gen_S (Flat f) c3 (CHead e1 k0 u0) u2 n H4)) in (ex3_2_ind C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) c0 (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) (ex3_2 C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (getl (S n) c0 (CHead x0 k0 x1))).(\lambda (H7: (wcpr0 x0 e1)).(\lambda (H8: (pr0 x1 u0)).(ex3_2_intro C T (\lambda (e2: C).(\lambda (u3: T).(getl (S n) (CHead c0 (Flat f) u1) (CHead e2 k0 u3)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 e1))) (\lambda (_: C).(\lambda (u3: T).(pr0 u3 u0))) x0 x1 (getl_head (Flat f) n c0 (CHead x0 k0 x1) H6 u1) H7 H8)))))) H5))))))))) k) h)))))))))) c2 c1 H))).
 
 inductive pr2: C \to (T \to (T \to Prop)) \def
 | pr2_free: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (pr2 c t1 t2))))
@@ -1994,27 +1994,27 @@ inductive pr2: C \to (T \to (T \to Prop)) \def
 theorem pr2_gen_sort:
  \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TSort n) x) \to (eq T x (TSort n)))))
 \def
- \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TSort n) x)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to ((eq T t (TSort n)) \to ((eq T t0 x) \to (eq T x (TSort n)))))))) with [(pr2_free c0 t1 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1 (TSort n))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t1 (TSort n)) \to ((eq T t2 x) \to ((pr0 t1 t2) \to (eq T x (TSort n)))))) (\lambda (H4: (eq T t1 (TSort n))).(eq_ind T (TSort n) (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (eq T x (TSort n))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (TSort n) t) \to (eq T x (TSort n)))) (\lambda (H6: (pr0 (TSort n) x)).(let H7 \def (eq_ind T x (\lambda (t: T).(pr2 c (TSort n) t)) H (TSort n) (pr0_gen_sort x n H6)) in (eq_ind_r T (TSort n) (\lambda (t: T).(eq T t (TSort n))) (refl_equal T (TSort n)) x (pr0_gen_sort x n H6)))) t2 (sym_eq T t2 x H5))) t1 (sym_eq T t1 (TSort n) H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t1 (TSort n))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c: C).((eq T t1 (TSort n)) \to ((eq T t x) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t1 t2) \to ((subst0 i u t2 t) \to (eq T x (TSort n)))))))) (\lambda (H6: (eq T t1 (TSort n))).(eq_ind T (TSort n) (\lambda (t0: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (eq T x (TSort n))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t0: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (TSort n) t2) \to ((subst0 i u t2 t0) \to (eq T x (TSort n)))))) (\lambda (_: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 (TSort n) t2)).(\lambda (H10: (subst0 i u t2 x)).(let H11 \def (eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H10 (TSort n) (pr0_gen_sort t2 n H9)) in (subst0_gen_sort u x i n H11 (eq T x (TSort n))))))) t (sym_eq T t x H7))) t1 (sym_eq T t1 (TSort n) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T (TSort n)) (refl_equal T x)))))).
+ \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TSort n) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to ((eq T t (TSort n)) \to ((eq T t0 x) \to (eq T x (TSort n))))))))) with [(pr2_free c0 t1 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1 (TSort n))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t1 (TSort n)) \to ((eq T t2 x) \to ((pr0 t1 t2) \to (eq T x (TSort n)))))) (\lambda (H4: (eq T t1 (TSort n))).(eq_ind T (TSort n) (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (eq T x (TSort n))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (TSort n) t) \to (eq T x (TSort n)))) (\lambda (H6: (pr0 (TSort n) x)).(let H7 \def (eq_ind T x (\lambda (t: T).(pr2 c (TSort n) t)) H (TSort n) (pr0_gen_sort x n H6)) in (eq_ind_r T (TSort n) (\lambda (t: T).(eq T t (TSort n))) (refl_equal T (TSort n)) x (pr0_gen_sort x n H6)))) t2 (sym_eq T t2 x H5))) t1 (sym_eq T t1 (TSort n) H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t1 (TSort n))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c: C).((eq T t1 (TSort n)) \to ((eq T t x) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t1 t2) \to ((subst0 i u t2 t) \to (eq T x (TSort n)))))))) (\lambda (H6: (eq T t1 (TSort n))).(eq_ind T (TSort n) (\lambda (t0: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (eq T x (TSort n))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t0: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (TSort n) t2) \to ((subst0 i u t2 t0) \to (eq T x (TSort n)))))) (\lambda (_: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 (TSort n) t2)).(\lambda (H10: (subst0 i u t2 x)).(let H11 \def (eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H10 (TSort n) (pr0_gen_sort t2 n H9)) in (subst0_gen_sort u x i n H11 (eq T x (TSort n))))))) t (sym_eq T t x H7))) t1 (sym_eq T t1 (TSort n) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T (TSort n)) (refl_equal T x)))))).
 
 theorem pr2_gen_lref:
  \forall (c: C).(\forall (x: T).(\forall (n: nat).((pr2 c (TLRef n) x) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S n) O u)))))))))
 \def
- \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TLRef n) x)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to ((eq T t (TLRef n)) \to ((eq T t0 x) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S n) O u)))))))))))) with [(pr2_free c0 t1 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1 (TLRef n))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t1 (TLRef n)) \to ((eq T t2 x) \to ((pr0 t1 t2) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S n) O u)))))))))) (\lambda (H4: (eq T t1 (TLRef n))).(eq_ind T (TLRef n) (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S n) O u))))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (TLRef n) t) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S n) O u)))))))) (\lambda (H6: (pr0 (TLRef n) x)).(let H7 \def (eq_ind T x (\lambda (t: T).(pr2 c (TLRef n) t)) H (TLRef n) (pr0_gen_lref x n H6)) in (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t (lift (S n) O u))))))) (or_introl (eq T (TLRef n) (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T (TLRef n) (lift (S n) O u))))) (refl_equal T (TLRef n))) x (pr0_gen_lref x n H6)))) t2 (sym_eq T t2 x H5))) t1 (sym_eq T t1 (TLRef n) H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t1 (TLRef n))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t1 (TLRef n)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1 t2) \to ((subst0 i u t2 t) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T x (lift (S n) O u0)))))))))))) (\lambda (H6: (eq T t1 (TLRef n))).(eq_ind T (TLRef n) (\lambda (t0: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T x (lift (S n) O u0))))))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t0: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (TLRef n) t2) \to ((subst0 i u t2 t0) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T x (lift (S n) O u0)))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 (TLRef n) t2)).(\lambda (H10: (subst0 i u t2 x)).(let H11 \def (eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H10 (TLRef n) (pr0_gen_lref t2 n H9)) in (and_ind (eq nat n i) (eq T x (lift (S n) O u)) (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T x (lift (S n) O u0)))))) (\lambda (H12: (eq nat n i)).(\lambda (H13: (eq T x (lift (S n) O u))).(let H14 \def (eq_ind_r nat i (\lambda (n: nat).(getl n c (CHead d (Bind Abbr) u))) H8 n H12) in (let H15 \def (eq_ind T x (\lambda (t: T).(pr2 c (TLRef n) t)) H (lift (S n) O u) H13) in (eq_ind_r T (lift (S n) O u) (\lambda (t0: T).(or (eq T t0 (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t0 (lift (S n) O u0))))))) (or_intror (eq T (lift (S n) O u) (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S n) O u0))))) (ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S n) O u0)))) d u H14 (refl_equal T (lift (S n) O u)))) x H13))))) (subst0_gen_lref u x i n H11)))))) t (sym_eq T t x H7))) t1 (sym_eq T t1 (TLRef n) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T (TLRef n)) (refl_equal T x)))))).
+ \lambda (c: C).(\lambda (x: T).(\lambda (n: nat).(\lambda (H: (pr2 c (TLRef n) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to ((eq T t (TLRef n)) \to ((eq T t0 x) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S n) O u))))))))))))) with [(pr2_free c0 t1 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1 (TLRef n))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t1 (TLRef n)) \to ((eq T t2 x) \to ((pr0 t1 t2) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S n) O u)))))))))) (\lambda (H4: (eq T t1 (TLRef n))).(eq_ind T (TLRef n) (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S n) O u))))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (TLRef n) t) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T x (lift (S n) O u)))))))) (\lambda (H6: (pr0 (TLRef n) x)).(let H7 \def (eq_ind T x (\lambda (t: T).(pr2 c (TLRef n) t)) H (TLRef n) (pr0_gen_lref x n H6)) in (eq_ind_r T (TLRef n) (\lambda (t: T).(or (eq T t (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T t (lift (S n) O u))))))) (or_introl (eq T (TLRef n) (TLRef n)) (ex2_2 C T (\lambda (d: C).(\lambda (u: T).(getl n c (CHead d (Bind Abbr) u)))) (\lambda (_: C).(\lambda (u: T).(eq T (TLRef n) (lift (S n) O u))))) (refl_equal T (TLRef n))) x (pr0_gen_lref x n H6)))) t2 (sym_eq T t2 x H5))) t1 (sym_eq T t1 (TLRef n) H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t1 (TLRef n))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t1 (TLRef n)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1 t2) \to ((subst0 i u t2 t) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T x (lift (S n) O u0)))))))))))) (\lambda (H6: (eq T t1 (TLRef n))).(eq_ind T (TLRef n) (\lambda (t0: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T x (lift (S n) O u0))))))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t0: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (TLRef n) t2) \to ((subst0 i u t2 t0) \to (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T x (lift (S n) O u0)))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 (TLRef n) t2)).(\lambda (H10: (subst0 i u t2 x)).(let H11 \def (eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H10 (TLRef n) (pr0_gen_lref t2 n H9)) in (and_ind (eq nat n i) (eq T x (lift (S n) O u)) (or (eq T x (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T x (lift (S n) O u0)))))) (\lambda (H12: (eq nat n i)).(\lambda (H13: (eq T x (lift (S n) O u))).(let H14 \def (eq_ind_r nat i (\lambda (n: nat).(getl n c (CHead d (Bind Abbr) u))) H8 n H12) in (let H15 \def (eq_ind T x (\lambda (t: T).(pr2 c (TLRef n) t)) H (lift (S n) O u) H13) in (eq_ind_r T (lift (S n) O u) (\lambda (t0: T).(or (eq T t0 (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T t0 (lift (S n) O u0))))))) (or_intror (eq T (lift (S n) O u) (TLRef n)) (ex2_2 C T (\lambda (d0: C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S n) O u0))))) (ex2_2_intro C T (\lambda (d0: C).(\lambda (u0: T).(getl n c (CHead d0 (Bind Abbr) u0)))) (\lambda (_: C).(\lambda (u0: T).(eq T (lift (S n) O u) (lift (S n) O u0)))) d u H14 (refl_equal T (lift (S n) O u)))) x H13))))) (subst0_gen_lref u x i n H11)))))) t (sym_eq T t x H7))) t1 (sym_eq T t1 (TLRef n) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T (TLRef n)) (refl_equal T x)))))).
 
 theorem pr2_gen_abst:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c (THead (Bind Abst) u1 t1) x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))))))))))
 \def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr2 c (THead (Bind Abst) u1 t1) x)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to ((eq T t (THead (Bind Abst) u1 t1)) \to ((eq T t0 x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t0 (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Abst) u1 t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))))))) (\lambda (H4: (eq T t0 (THead (Bind Abst) u1 t1))).(eq_ind T (THead (Bind Abst) u1 t1) (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead (Bind Abst) u1 t1) t) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))))) (\lambda (H6: (pr0 (THead (Bind Abst) u1 t1) x)).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T x (\lambda (t: T).(pr2 c (THead (Bind Abst) u1 t1) t)) H (THead (Bind Abst) x0 x1) H7) in (eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind Abst) x0 x1)) (pr2_free c u1 x0 H8) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c (Bind b) u) t1 x1 H9)))) x H7))))))) (pr0_gen_abst u1 t1 x H6))) t2 (sym_eq T t2 x H5))) t0 (sym_eq T t0 (THead (Bind Abst) u1 t1) H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Bind Abst) u1 t1))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t0 (THead (Bind Abst) u1 t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))))))))) (\lambda (H6: (eq T t0 (THead (Bind Abst) u1 t1))).(eq_ind T (THead (Bind Abst) u1 t1) (\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4))))))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (THead (Bind Abst) u1 t1) t2) \to ((subst0 i u t2 t3) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4)))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 (THead (Bind Abst) u1 t1) t2)).(\lambda (H10: (subst0 i u t2 x)).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H12: (pr0 u1 x0)).(\lambda (H13: (pr0 t1 x1)).(let H14 \def (eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H10 (THead (Bind Abst) x0 x1) H11) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (H15: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2: T).(\lambda (H16: (eq T x (THead (Bind Abst) x2 x1))).(\lambda (H17: (subst0 i u x0 x2)).(let H18 \def (eq_ind T x (\lambda (t: T).(pr2 c (THead (Bind Abst) u1 t1) t)) H (THead (Bind Abst) x2 x1) H16) in (eq_ind_r T (THead (Bind Abst) x2 x1) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x2 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x2 x1 (refl_equal T (THead (Bind Abst) x2 x1)) (pr2_delta c d u i H8 u1 x0 H12 x2 H17) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) t1 x1 H13)))) x H16))))) H15)) (\lambda (H15: (ex2 T (\lambda (t2: T).(eq T x (THead (Bind Abst) x0 t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) u x1 t2)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2: T).(\lambda (H16: (eq T x (THead (Bind Abst) x0 x2))).(\lambda (H17: (subst0 (s (Bind Abst) i) u x1 x2)).(let H18 \def (eq_ind T x (\lambda (t: T).(pr2 c (THead (Bind Abst) u1 t1) t)) H (THead (Bind Abst) x0 x2) H16) in (eq_ind_r T (THead (Bind Abst) x0 x2) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x0 x2) (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x0 x2 (refl_equal T (THead (Bind Abst) x0 x2)) (pr2_free c u1 x0 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 x1 H13 x2 H17)))) x H16))))) H15)) (\lambda (H15: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) u x1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H16: (eq T x (THead (Bind Abst) x2 x3))).(\lambda (H17: (subst0 i u x0 x2)).(\lambda (H18: (subst0 (s (Bind Abst) i) u x1 x3)).(let H19 \def (eq_ind T x (\lambda (t: T).(pr2 c (THead (Bind Abst) u1 t1) t)) H (THead (Bind Abst) x2 x3) H16) in (eq_ind_r T (THead (Bind Abst) x2 x3) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x2 x3) (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x2 x3 (refl_equal T (THead (Bind Abst) x2 x3)) (pr2_delta c d u i H8 u1 x0 H12 x2 H17) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 x1 H13 x3 H18)))) x H16))))))) H15)) (subst0_gen_head (Bind Abst) u x0 x1 x i H14)))))))) (pr0_gen_abst u1 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Bind Abst) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T (THead (Bind Abst) u1 t1)) (refl_equal T x))))))).
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr2 c (THead (Bind Abst) u1 t1) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to ((eq T t (THead (Bind Abst) u1 t1)) \to ((eq T t0 x) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t0 (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Abst) u1 t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))))))) (\lambda (H4: (eq T t0 (THead (Bind Abst) u1 t1))).(eq_ind T (THead (Bind Abst) u1 t1) (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead (Bind Abst) u1 t1) t) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))))) (\lambda (H6: (pr0 (THead (Bind Abst) u1 t1) x)).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H7: (eq T x (THead (Bind Abst) x0 x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1 x1)).(let H10 \def (eq_ind T x (\lambda (t: T).(pr2 c (THead (Bind Abst) u1 t1) t)) H (THead (Bind Abst) x0 x1) H7) in (eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind Abst) x0 x1)) (pr2_free c u1 x0 H8) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c (Bind b) u) t1 x1 H9)))) x H7))))))) (pr0_gen_abst u1 t1 x H6))) t2 (sym_eq T t2 x H5))) t0 (sym_eq T t0 (THead (Bind Abst) u1 t1) H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Bind Abst) u1 t1))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t0 (THead (Bind Abst) u1 t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))))))))) (\lambda (H6: (eq T t0 (THead (Bind Abst) u1 t1))).(eq_ind T (THead (Bind Abst) u1 t1) (\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4))))))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (THead (Bind Abst) u1 t1) t2) \to ((subst0 i u t2 t3) \to (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4)))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 (THead (Bind Abst) u1 t1) t2)).(\lambda (H10: (subst0 i u t2 x)).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H11: (eq T t2 (THead (Bind Abst) x0 x1))).(\lambda (H12: (pr0 u1 x0)).(\lambda (H13: (pr0 t1 x1)).(let H14 \def (eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H10 (THead (Bind Abst) x0 x1) H11) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (H15: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Bind Abst) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2: T).(\lambda (H16: (eq T x (THead (Bind Abst) x2 x1))).(\lambda (H17: (subst0 i u x0 x2)).(let H18 \def (eq_ind T x (\lambda (t: T).(pr2 c (THead (Bind Abst) u1 t1) t)) H (THead (Bind Abst) x2 x1) H16) in (eq_ind_r T (THead (Bind Abst) x2 x1) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x2 x1) (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x2 x1 (refl_equal T (THead (Bind Abst) x2 x1)) (pr2_delta c d u i H8 u1 x0 H12 x2 H17) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) t1 x1 H13)))) x H16))))) H15)) (\lambda (H15: (ex2 T (\lambda (t2: T).(eq T x (THead (Bind Abst) x0 t2))) (\lambda (t2: T).(subst0 (s (Bind Abst) i) u x1 t2)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind Abst) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3)) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2: T).(\lambda (H16: (eq T x (THead (Bind Abst) x0 x2))).(\lambda (H17: (subst0 (s (Bind Abst) i) u x1 x2)).(let H18 \def (eq_ind T x (\lambda (t: T).(pr2 c (THead (Bind Abst) u1 t1) t)) H (THead (Bind Abst) x0 x2) H16) in (eq_ind_r T (THead (Bind Abst) x0 x2) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x0 x2) (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x0 x2 (refl_equal T (THead (Bind Abst) x0 x2)) (pr2_free c u1 x0 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 x1 H13 x2 H17)))) x H16))))) H15)) (\lambda (H15: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abst) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Bind Abst) i) u x1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abst) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H16: (eq T x (THead (Bind Abst) x2 x3))).(\lambda (H17: (subst0 i u x0 x2)).(\lambda (H18: (subst0 (s (Bind Abst) i) u x1 x3)).(let H19 \def (eq_ind T x (\lambda (t: T).(pr2 c (THead (Bind Abst) u1 t1) t)) H (THead (Bind Abst) x2 x3) H16) in (eq_ind_r T (THead (Bind Abst) x2 x3) (\lambda (t3: T).(ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abst) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abst) x2 x3) (THead (Bind Abst) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x2 x3 (refl_equal T (THead (Bind Abst) x2 x3)) (pr2_delta c d u i H8 u1 x0 H12 x2 H17) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 x1 H13 x3 H18)))) x H16))))))) H15)) (subst0_gen_head (Bind Abst) u x0 x1 x i H14)))))))) (pr0_gen_abst u1 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Bind Abst) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T (THead (Bind Abst) u1 t1)) (refl_equal T x))))))).
 
 theorem pr2_gen_cast:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c (THead (Flat Cast) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (pr2 c t1 x))))))
 \def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr2 c (THead (Flat Cast) u1 t1) x)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to ((eq T t (THead (Flat Cast) u1 t1)) \to ((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (pr2 c t1 x)))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t0 (THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Flat Cast) u1 t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)))))) (\lambda (H4: (eq T t0 (THead (Flat Cast) u1 t1))).(eq_ind T (THead (Flat Cast) u1 t1) (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead (Flat Cast) u1 t1) t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)))) (\lambda (H6: (pr0 (THead (Flat Cast) u1 t1) x)).(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq T x (THead (Flat Cast) x0 x1))).(\lambda (H9: (pr0 u1 x0)).(\lambda (H10: (pr0 t1 x1)).(let H11 \def (eq_ind T x (\lambda (t: T).(pr2 c (THead (Flat Cast) u1 t1) t)) H (THead (Flat Cast) x0 x1) H8) in (eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 t))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 (THead (Flat Cast) x0 x1)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x1 (refl_equal T (THead (Flat Cast) x0 x1)) (pr2_free c u1 x0 H9) (pr2_free c t1 x1 H10))) x H8))))))) H7)) (\lambda (H7: (pr0 t1 x)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x) (pr2_free c t1 x H7))) (pr0_gen_cast u1 t1 x H6))) t2 (sym_eq T t2 x H5))) t0 (sym_eq T t0 (THead (Flat Cast) u1 t1) H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u1 t1))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t0 (THead (Flat Cast) u1 t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)))))))) (\lambda (H6: (eq T t0 (THead (Flat Cast) u1 t1))).(eq_ind T (THead (Flat Cast) u1 t1) (\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 t4)))) (pr2 c t1 x))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (THead (Flat Cast) u1 t1) t2) \to ((subst0 i u t2 t3) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 t4)))) (pr2 c t1 x)))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 (THead (Flat Cast) u1 t1) t2)).(\lambda (H10: (subst0 i u t2 x)).(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Flat Cast) x0 x1))).(\lambda (H13: (pr0 u1 x0)).(\lambda (H14: (pr0 t1 x1)).(let H15 \def (eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H10 (THead (Flat Cast) x0 x1) H12) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead (Flat Cast) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (H16: (ex2 T (\lambda (u2: T).(eq T x (THead (Flat Cast) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Flat Cast) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x2: T).(\lambda (H17: (eq T x (THead (Flat Cast) x2 x1))).(\lambda (H18: (subst0 i u x0 x2)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3))) x2 x1 H17 (pr2_delta c d u i H8 u1 x0 H13 x2 H18) (pr2_free c t1 x1 H14)))))) H16)) (\lambda (H16: (ex2 T (\lambda (t2: T).(eq T x (THead (Flat Cast) x0 t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) u x1 t2)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x2: T).(\lambda (H17: (eq T x (THead (Flat Cast) x0 x2))).(\lambda (H18: (subst0 (s (Flat Cast) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x2 H17 (pr2_free c u1 x0 H13) (pr2_delta c d u i H8 t1 x1 H14 x2 H18)))))) H16)) (\lambda (H16: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Cast) i) u x1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H17: (eq T x (THead (Flat Cast) x2 x3))).(\lambda (H18: (subst0 i u x0 x2)).(\lambda (H19: (subst0 (s (Flat Cast) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3))) x2 x3 H17 (pr2_delta c d u i H8 u1 x0 H13 x2 H18) (pr2_delta c d u i H8 t1 x1 H14 x3 H19)))))))) H16)) (subst0_gen_head (Flat Cast) u x0 x1 x i H15)))))))) H11)) (\lambda (H11: (pr0 t1 t2)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x) (pr2_delta c d u i H8 t1 t2 H11 x H10))) (pr0_gen_cast u1 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Flat Cast) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T (THead (Flat Cast) u1 t1)) (refl_equal T x))))))).
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr2 c (THead (Flat Cast) u1 t1) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to ((eq T t (THead (Flat Cast) u1 t1)) \to ((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (pr2 c t1 x))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t0 (THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Flat Cast) u1 t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)))))) (\lambda (H4: (eq T t0 (THead (Flat Cast) u1 t1))).(eq_ind T (THead (Flat Cast) u1 t1) (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead (Flat Cast) u1 t1) t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)))) (\lambda (H6: (pr0 (THead (Flat Cast) u1 t1) x)).(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 x) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H8: (eq T x (THead (Flat Cast) x0 x1))).(\lambda (H9: (pr0 u1 x0)).(\lambda (H10: (pr0 t1 x1)).(let H11 \def (eq_ind T x (\lambda (t: T).(pr2 c (THead (Flat Cast) u1 t1) t)) H (THead (Flat Cast) x0 x1) H8) in (eq_ind_r T (THead (Flat Cast) x0 x1) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 t))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 (THead (Flat Cast) x0 x1)) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Cast) x0 x1) (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x1 (refl_equal T (THead (Flat Cast) x0 x1)) (pr2_free c u1 x0 H9) (pr2_free c t1 x1 H10))) x H8))))))) H7)) (\lambda (H7: (pr0 t1 x)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x) (pr2_free c t1 x H7))) (pr0_gen_cast u1 t1 x H6))) t2 (sym_eq T t2 x H5))) t0 (sym_eq T t0 (THead (Flat Cast) u1 t1) H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Flat Cast) u1 t1))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t0 (THead (Flat Cast) u1 t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)))))))) (\lambda (H6: (eq T t0 (THead (Flat Cast) u1 t1))).(eq_ind T (THead (Flat Cast) u1 t1) (\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 t4)))) (pr2 c t1 x))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (THead (Flat Cast) u1 t1) t2) \to ((subst0 i u t2 t3) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Cast) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 t4)))) (pr2 c t1 x)))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 (THead (Flat Cast) u1 t1) t2)).(\lambda (H10: (subst0 i u t2 x)).(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 t2) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H12: (eq T t2 (THead (Flat Cast) x0 x1))).(\lambda (H13: (pr0 u1 x0)).(\lambda (H14: (pr0 t1 x1)).(let H15 \def (eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H10 (THead (Flat Cast) x0 x1) H12) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead (Flat Cast) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (H16: (ex2 T (\lambda (u2: T).(eq T x (THead (Flat Cast) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Flat Cast) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x2: T).(\lambda (H17: (eq T x (THead (Flat Cast) x2 x1))).(\lambda (H18: (subst0 i u x0 x2)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3))) x2 x1 H17 (pr2_delta c d u i H8 u1 x0 H13 x2 H18) (pr2_free c t1 x1 H14)))))) H16)) (\lambda (H16: (ex2 T (\lambda (t2: T).(eq T x (THead (Flat Cast) x0 t2))) (\lambda (t2: T).(subst0 (s (Flat Cast) i) u x1 t2)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Flat Cast) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x2: T).(\lambda (H17: (eq T x (THead (Flat Cast) x0 x2))).(\lambda (H18: (subst0 (s (Flat Cast) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x2 H17 (pr2_free c u1 x0 H13) (pr2_delta c d u i H8 t1 x1 H14 x2 H18)))))) H16)) (\lambda (H16: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Cast) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Cast) i) u x1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Cast) i) u x1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x)) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H17: (eq T x (THead (Flat Cast) x2 x3))).(\lambda (H18: (subst0 i u x0 x2)).(\lambda (H19: (subst0 (s (Flat Cast) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3))) x2 x3 H17 (pr2_delta c d u i H8 u1 x0 H13 x2 H18) (pr2_delta c d u i H8 t1 x1 H14 x3 H19)))))))) H16)) (subst0_gen_head (Flat Cast) u x0 x1 x i H15)))))))) H11)) (\lambda (H11: (pr0 t1 t2)).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Cast) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (pr2 c t1 x) (pr2_delta c d u i H8 t1 t2 H11 x H10))) (pr0_gen_cast u1 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Flat Cast) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T (THead (Flat Cast) u1 t1)) (refl_equal T x))))))).
 
 theorem pr2_gen_csort:
  \forall (t1: T).(\forall (t2: T).(\forall (n: nat).((pr2 (CSort n) t1 t2) \to (pr0 t1 t2))))
 \def
- \lambda (t1: T).(\lambda (t2: T).(\lambda (n: nat).(\lambda (H: (pr2 (CSort n) t1 t2)).(let H0 \def (match H return (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CSort n)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr0 t1 t2))))))) with [(pr2_free c t0 t3 H0) \Rightarrow (\lambda (H1: (eq C c (CSort n))).(\lambda (H2: (eq T t0 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CSort n) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr0 t1 t2))))) (\lambda (H4: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr0 t1 t2)))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr0 t1 t2))) (\lambda (H6: (pr0 t1 t2)).H6) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c (sym_eq C c (CSort n) H1) H2 H3 H0)))) | (pr2_delta c d u i H0 t0 t3 H1 t H2) \Rightarrow (\lambda (H3: (eq C c (CSort n))).(\lambda (H4: (eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CSort n) (\lambda (c0: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c0 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr0 t1 t2))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CSort n) (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr0 t1 t2)))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CSort n) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr0 t1 t2))))) (\lambda (H8: (getl i (CSort n) (CHead d (Bind Abbr) u))).(\lambda (_: (pr0 t1 t3)).(\lambda (_: (subst0 i u t3 t2)).(getl_gen_sort n i (CHead d (Bind Abbr) u) H8 (pr0 t1 t2))))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c (sym_eq C c (CSort n) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C (CSort n)) (refl_equal T t1) (refl_equal T t2)))))).
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (n: nat).(\lambda (H: (pr2 (CSort n) t1 t2)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c (CSort n)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr0 t1 t2)))))))) with [(pr2_free c t0 t3 H0) \Rightarrow (\lambda (H1: (eq C c (CSort n))).(\lambda (H2: (eq T t0 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CSort n) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr0 t1 t2))))) (\lambda (H4: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr0 t1 t2)))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr0 t1 t2))) (\lambda (H6: (pr0 t1 t2)).H6) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c (sym_eq C c (CSort n) H1) H2 H3 H0)))) | (pr2_delta c d u i H0 t0 t3 H1 t H2) \Rightarrow (\lambda (H3: (eq C c (CSort n))).(\lambda (H4: (eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CSort n) (\lambda (c0: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c0 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr0 t1 t2))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CSort n) (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr0 t1 t2)))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CSort n) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr0 t1 t2))))) (\lambda (H8: (getl i (CSort n) (CHead d (Bind Abbr) u))).(\lambda (_: (pr0 t1 t3)).(\lambda (_: (subst0 i u t3 t2)).(getl_gen_sort n i (CHead d (Bind Abbr) u) H8 (pr0 t1 t2))))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c (sym_eq C c (CSort n) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C (CSort n)) (refl_equal T t1) (refl_equal T t2)))))).
 
 theorem pr2_gen_ctail:
  \forall (k: K).(\forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).((pr2 (CTail k u c) t1 t2) \to (or (pr2 c t1 t2) (ex3 T (\lambda (_: T).(eq K k (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t: T).(subst0 (clen c) u t t2)))))))))
@@ -2034,12 +2034,12 @@ theorem pr2_head_1:
 theorem pr2_head_2:
  \forall (c: C).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u) t1 t2) \to (pr2 c (THead k u t1) (THead k u t2)))))))
 \def
- \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(K_ind (\lambda (k0: K).((pr2 (CHead c k0 u) t1 t2) \to (pr2 c (THead k0 u t1) (THead k0 u t2)))) (\lambda (b: B).(\lambda (H: (pr2 (CHead c (Bind b) u) t1 t2)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Bind b) u)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u t2)))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow (\lambda (H1: (eq C c0 (CHead c (Bind b) u))).(\lambda (H2: (eq T t0 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Bind b) u) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u t2)))))) (\lambda (H4: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u t2))))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead (Bind b) u t1) (THead (Bind b) u t2) (pr0_comp u u (pr0_refl u) t1 t2 H6 (Bind b)))) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 (CHead c (Bind b) u) H1) H2 H3 H0)))) | (pr2_delta c0 d u0 i H0 t0 t3 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 (CHead c (Bind b) u))).(\lambda (H4: (eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead c (Bind b) u) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u0)) \to ((pr0 t0 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t4 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t1 t3) \to ((subst0 i u0 t3 t4) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u t2)))))) (\lambda (H8: (getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u0 t3 t2)).((match i return (\lambda (n: nat).((getl n (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((subst0 n u0 t3 t2) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u t2))))) with [O \Rightarrow (\lambda (H11: (getl O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0))).(\lambda (H12: (subst0 O u0 t3 t2)).(let H \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H11))) in ((let H13 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H11))) in ((let H14 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H11))) in (\lambda (H15: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H17 \def (eq_ind T u0 (\lambda (t: T).(subst0 O t t3 t2)) H12 u H14) in (eq_ind B Abbr (\lambda (b: B).(pr2 c (THead (Bind b) u t1) (THead (Bind b) u t2))) (pr2_free c (THead (Bind Abbr) u t1) (THead (Bind Abbr) u t2) (pr0_delta u u (pr0_refl u) t1 t3 H9 t2 H17)) b H15))))) H13)) H)))) | (S n) \Rightarrow (\lambda (H11: (getl (S n) (CHead c (Bind b) u) (CHead d (Bind Abbr) u0))).(\lambda (H12: (subst0 (S n) u0 t3 t2)).(pr2_delta c d u0 (r (Bind b) n) (getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n H11) (THead (Bind b) u t1) (THead (Bind b) u t3) (pr0_comp u u (pr0_refl u) t1 t3 H9 (Bind b)) (THead (Bind b) u t2) (subst0_snd (Bind b) u0 t2 t3 (r (Bind b) n) H12 u))))]) H8 H10)))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0 (sym_eq C c0 (CHead c (Bind b) u) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C (CHead c (Bind b) u)) (refl_equal T t1) (refl_equal T t2))))) (\lambda (f: F).(\lambda (H: (pr2 (CHead c (Flat f) u) t1 t2)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Flat f) u)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2)))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow (\lambda (H1: (eq C c0 (CHead c (Flat f) u))).(\lambda (H2: (eq T t0 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Flat f) u) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2)))))) (\lambda (H4: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead (Flat f) u t1) (THead (Flat f) u t2) (pr0_comp u u (pr0_refl u) t1 t2 H6 (Flat f)))) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 (CHead c (Flat f) u) H1) H2 H3 H0)))) | (pr2_delta c0 d u0 i H0 t0 t3 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 (CHead c (Flat f) u))).(\lambda (H4: (eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead c (Flat f) u) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u0)) \to ((pr0 t0 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c (Flat f) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t4 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CHead c (Flat f) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t1 t3) \to ((subst0 i u0 t3 t4) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2)))))) (\lambda (H8: (getl i (CHead c (Flat f) u) (CHead d (Bind Abbr) u0))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u0 t3 t2)).((match i return (\lambda (n: nat).((getl n (CHead c (Flat f) u) (CHead d (Bind Abbr) u0)) \to ((subst0 n u0 t3 t2) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))) with [O \Rightarrow (\lambda (H11: (getl O (CHead c (Flat f) u) (CHead d (Bind Abbr) u0))).(\lambda (H12: (subst0 O u0 t3 t2)).(pr2_delta c d u0 O (getl_intro O c (CHead d (Bind Abbr) u0) c (drop_refl c) (clear_gen_flat f c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Flat f) u) (CHead d (Bind Abbr) u0) H11))) (THead (Flat f) u t1) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t1 t3 H9 (Flat f)) (THead (Flat f) u t2) (subst0_snd (Flat f) u0 t2 t3 O H12 u)))) | (S n) \Rightarrow (\lambda (H11: (getl (S n) (CHead c (Flat f) u) (CHead d (Bind Abbr) u0))).(\lambda (H12: (subst0 (S n) u0 t3 t2)).(pr2_delta c d u0 (r (Flat f) n) (getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H11) (THead (Flat f) u t1) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t1 t3 H9 (Flat f)) (THead (Flat f) u t2) (subst0_snd (Flat f) u0 t2 t3 (r (Flat f) n) H12 u))))]) H8 H10)))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0 (sym_eq C c0 (CHead c (Flat f) u) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C (CHead c (Flat f) u)) (refl_equal T t1) (refl_equal T t2))))) k))))).
+ \lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(K_ind (\lambda (k0: K).((pr2 (CHead c k0 u) t1 t2) \to (pr2 c (THead k0 u t1) (THead k0 u t2)))) (\lambda (b: B).(\lambda (H: (pr2 (CHead c (Bind b) u) t1 t2)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Bind b) u)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u t2))))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow (\lambda (H1: (eq C c0 (CHead c (Bind b) u))).(\lambda (H2: (eq T t0 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Bind b) u) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u t2)))))) (\lambda (H4: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u t2))))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead (Bind b) u t1) (THead (Bind b) u t2) (pr0_comp u u (pr0_refl u) t1 t2 H6 (Bind b)))) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 (CHead c (Bind b) u) H1) H2 H3 H0)))) | (pr2_delta c0 d u0 i H0 t0 t3 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 (CHead c (Bind b) u))).(\lambda (H4: (eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead c (Bind b) u) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u0)) \to ((pr0 t0 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t4 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t1 t3) \to ((subst0 i u0 t3 t4) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u t2)))))) (\lambda (H8: (getl i (CHead c (Bind b) u) (CHead d (Bind Abbr) u0))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u0 t3 t2)).((match i return (\lambda (_: ?).(\lambda (n: nat).((getl n (CHead c (Bind b) u) (CHead d (Bind Abbr) u0)) \to ((subst0 n u0 t3 t2) \to (pr2 c (THead (Bind b) u t1) (THead (Bind b) u t2)))))) with [O \Rightarrow (\lambda (H11: (getl O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0))).(\lambda (H12: (subst0 O u0 t3 t2)).(let H \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H11))) in ((let H13 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H11))) in ((let H14 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u0) (CHead c (Bind b) u) (clear_gen_bind b c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Bind b) u) (CHead d (Bind Abbr) u0) H11))) in (\lambda (H15: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H17 \def (eq_ind T u0 (\lambda (t: T).(subst0 O t t3 t2)) H12 u H14) in (eq_ind B Abbr (\lambda (b: B).(pr2 c (THead (Bind b) u t1) (THead (Bind b) u t2))) (pr2_free c (THead (Bind Abbr) u t1) (THead (Bind Abbr) u t2) (pr0_delta u u (pr0_refl u) t1 t3 H9 t2 H17)) b H15))))) H13)) H)))) | (S n) \Rightarrow (\lambda (H11: (getl (S n) (CHead c (Bind b) u) (CHead d (Bind Abbr) u0))).(\lambda (H12: (subst0 (S n) u0 t3 t2)).(pr2_delta c d u0 (r (Bind b) n) (getl_gen_S (Bind b) c (CHead d (Bind Abbr) u0) u n H11) (THead (Bind b) u t1) (THead (Bind b) u t3) (pr0_comp u u (pr0_refl u) t1 t3 H9 (Bind b)) (THead (Bind b) u t2) (subst0_snd (Bind b) u0 t2 t3 (r (Bind b) n) H12 u))))]) H8 H10)))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0 (sym_eq C c0 (CHead c (Bind b) u) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C (CHead c (Bind b) u)) (refl_equal T t1) (refl_equal T t2))))) (\lambda (f: F).(\lambda (H: (pr2 (CHead c (Flat f) u) t1 t2)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Flat f) u)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow (\lambda (H1: (eq C c0 (CHead c (Flat f) u))).(\lambda (H2: (eq T t0 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Flat f) u) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2)))))) (\lambda (H4: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead (Flat f) u t1) (THead (Flat f) u t2) (pr0_comp u u (pr0_refl u) t1 t2 H6 (Flat f)))) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 (CHead c (Flat f) u) H1) H2 H3 H0)))) | (pr2_delta c0 d u0 i H0 t0 t3 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 (CHead c (Flat f) u))).(\lambda (H4: (eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead c (Flat f) u) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u0)) \to ((pr0 t0 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c (Flat f) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t4 t3) \to ((subst0 i u0 t3 t) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CHead c (Flat f) u) (CHead d (Bind Abbr) u0)) \to ((pr0 t1 t3) \to ((subst0 i u0 t3 t4) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2)))))) (\lambda (H8: (getl i (CHead c (Flat f) u) (CHead d (Bind Abbr) u0))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u0 t3 t2)).((match i return (\lambda (_: ?).(\lambda (n: nat).((getl n (CHead c (Flat f) u) (CHead d (Bind Abbr) u0)) \to ((subst0 n u0 t3 t2) \to (pr2 c (THead (Flat f) u t1) (THead (Flat f) u t2)))))) with [O \Rightarrow (\lambda (H11: (getl O (CHead c (Flat f) u) (CHead d (Bind Abbr) u0))).(\lambda (H12: (subst0 O u0 t3 t2)).(pr2_delta c d u0 O (getl_intro O c (CHead d (Bind Abbr) u0) c (drop_refl c) (clear_gen_flat f c (CHead d (Bind Abbr) u0) u (getl_gen_O (CHead c (Flat f) u) (CHead d (Bind Abbr) u0) H11))) (THead (Flat f) u t1) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t1 t3 H9 (Flat f)) (THead (Flat f) u t2) (subst0_snd (Flat f) u0 t2 t3 O H12 u)))) | (S n) \Rightarrow (\lambda (H11: (getl (S n) (CHead c (Flat f) u) (CHead d (Bind Abbr) u0))).(\lambda (H12: (subst0 (S n) u0 t3 t2)).(pr2_delta c d u0 (r (Flat f) n) (getl_gen_S (Flat f) c (CHead d (Bind Abbr) u0) u n H11) (THead (Flat f) u t1) (THead (Flat f) u t3) (pr0_comp u u (pr0_refl u) t1 t3 H9 (Flat f)) (THead (Flat f) u t2) (subst0_snd (Flat f) u0 t2 t3 (r (Flat f) n) H12 u))))]) H8 H10)))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0 (sym_eq C c0 (CHead c (Flat f) u) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C (CHead c (Flat f) u)) (refl_equal T t1) (refl_equal T t2))))) k))))).
 
 theorem clear_pr2_trans:
  \forall (c2: C).(\forall (t1: T).(\forall (t2: T).((pr2 c2 t1 t2) \to (\forall (c1: C).((clear c1 c2) \to (pr2 c1 t1 t2))))))
 \def
- \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c2 t1 t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def (match H return (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c c2) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c1 t1 t2))))))) with [(pr2_free c t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c c2)).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C c2 (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c1 t1 t2))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c1 t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c1 t1 t2))) (\lambda (H7: (pr0 t1 t2)).(pr2_free c1 t1 t2 H7)) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 H5))) c (sym_eq C c c2 H2) H3 H4 H1)))) | (pr2_delta c d u i H1 t0 t3 H2 t H3) \Rightarrow (\lambda (H4: (eq C c c2)).(\lambda (H5: (eq T t0 t1)).(\lambda (H6: (eq T t t2)).(eq_ind C c2 (\lambda (c0: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c0 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr2 c1 t1 t2))))))) (\lambda (H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i c2 (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c1 t1 t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i c2 (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c1 t1 t2))))) (\lambda (H9: (getl i c2 (CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda (H11: (subst0 i u t3 t2)).(pr2_delta c1 d u i (clear_getl_trans i c2 (CHead d (Bind Abbr) u) H9 c1 H0) t1 t3 H10 t2 H11)))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c (sym_eq C c c2 H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C c2) (refl_equal T t1) (refl_equal T t2)))))))).
+ \lambda (c2: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c2 t1 t2)).(\lambda (c1: C).(\lambda (H0: (clear c1 c2)).(let H1 \def (match H return (\lambda (_: ?).(\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c c2) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c1 t1 t2)))))))) with [(pr2_free c t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c c2)).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C c2 (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c1 t1 t2))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c1 t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c1 t1 t2))) (\lambda (H7: (pr0 t1 t2)).(pr2_free c1 t1 t2 H7)) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 H5))) c (sym_eq C c c2 H2) H3 H4 H1)))) | (pr2_delta c d u i H1 t0 t3 H2 t H3) \Rightarrow (\lambda (H4: (eq C c c2)).(\lambda (H5: (eq T t0 t1)).(\lambda (H6: (eq T t t2)).(eq_ind C c2 (\lambda (c0: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c0 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr2 c1 t1 t2))))))) (\lambda (H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i c2 (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c1 t1 t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i c2 (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c1 t1 t2))))) (\lambda (H9: (getl i c2 (CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda (H11: (subst0 i u t3 t2)).(pr2_delta c1 d u i (clear_getl_trans i c2 (CHead d (Bind Abbr) u) H9 c1 H0) t1 t3 H10 t2 H11)))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c (sym_eq C c c2 H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C c2) (refl_equal T t1) (refl_equal T t2)))))))).
 
 theorem pr2_cflat:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (f: F).(\forall (v: T).(pr2 (CHead c (Flat f) v) t1 t2))))))
@@ -2054,37 +2054,37 @@ theorem pr2_ctail:
 theorem pr2_gen_cbind:
  \forall (b: B).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall (t2: T).((pr2 (CHead c (Bind b) v) t1 t2) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2)))))))
 \def
- \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Bind b) v)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2)))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow (\lambda (H1: (eq C c0 (CHead c (Bind b) v))).(\lambda (H2: (eq T t0 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Bind b) v) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2)))))) (\lambda (H4: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2))))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead (Bind b) v t1) (THead (Bind b) v t2) (pr0_comp v v (pr0_refl v) t1 t2 H6 (Bind b)))) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 (CHead c (Bind b) v) H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t3 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 (CHead c (Bind b) v))).(\lambda (H4: (eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead c (Bind b) v) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c (Bind b) v) (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CHead c (Bind b) v) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2)))))) (\lambda (H8: (getl i (CHead c (Bind b) v) (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u t3 t2)).(let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u) v i H8) in (let H \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d (Bind Abbr) u) (CHead c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2)) (\lambda (H11: (land (eq nat i O) (eq C (CHead d (Bind Abbr) u) (CHead c (Bind b) v)))).(and_ind (eq nat i O) (eq C (CHead d (Bind Abbr) u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2)) (\lambda (H12: (eq nat i O)).(\lambda (H13: (eq C (CHead d (Bind Abbr) u) (CHead c (Bind b) v))).(let H14 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) H13) in ((let H15 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) H13) in ((let H16 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) H13) in (\lambda (H17: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H19 \def (eq_ind nat i (\lambda (n: nat).(subst0 n u t3 t2)) H10 O H12) in (let H20 \def (eq_ind T u (\lambda (t: T).(subst0 O t t3 t2)) H19 v H16) in (eq_ind B Abbr (\lambda (b: B).(pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2))) (pr2_free c (THead (Bind Abbr) v t1) (THead (Bind Abbr) v t2) (pr0_delta v v (pr0_refl v) t1 t3 H9 t2 H20)) b H17)))))) H15)) H14)))) H11)) (\lambda (H11: (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u))))).(ex2_ind nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u))) (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2)) (\lambda (x: nat).(\lambda (H12: (eq nat i (S x))).(\lambda (H13: (getl x c (CHead d (Bind Abbr) u))).(let H14 \def (f_equal nat nat (\lambda (e: nat).e) i (S x) H12) in (let H15 \def (eq_ind nat i (\lambda (n: nat).(subst0 n u t3 t2)) H10 (S x) H14) in (pr2_head_2 c v t1 t2 (Bind b) (pr2_delta (CHead c (Bind b) v) d u (S x) (getl_clear_bind b (CHead c (Bind b) v) c v (clear_bind b c v) (CHead d (Bind Abbr) u) x H13) t1 t3 H9 t2 H15))))))) H11)) H)))))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0 (sym_eq C c0 (CHead c (Bind b) v) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C (CHead c (Bind b) v)) (refl_equal T t1) (refl_equal T t2)))))))).
+ \lambda (b: B).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 (CHead c (Bind b) v) t1 t2)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Bind b) v)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2))))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow (\lambda (H1: (eq C c0 (CHead c (Bind b) v))).(\lambda (H2: (eq T t0 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Bind b) v) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2)))))) (\lambda (H4: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2))))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2)))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c (THead (Bind b) v t1) (THead (Bind b) v t2) (pr0_comp v v (pr0_refl v) t1 t2 H6 (Bind b)))) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 (CHead c (Bind b) v) H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t3 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 (CHead c (Bind b) v))).(\lambda (H4: (eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead c (Bind b) v) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2)))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c (Bind b) v) (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2))))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CHead c (Bind b) v) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2)))))) (\lambda (H8: (getl i (CHead c (Bind b) v) (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u t3 t2)).(let H_x \def (getl_gen_bind b c (CHead d (Bind Abbr) u) v i H8) in (let H \def H_x in (or_ind (land (eq nat i O) (eq C (CHead d (Bind Abbr) u) (CHead c (Bind b) v))) (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u)))) (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2)) (\lambda (H11: (land (eq nat i O) (eq C (CHead d (Bind Abbr) u) (CHead c (Bind b) v)))).(and_ind (eq nat i O) (eq C (CHead d (Bind Abbr) u) (CHead c (Bind b) v)) (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2)) (\lambda (H12: (eq nat i O)).(\lambda (H13: (eq C (CHead d (Bind Abbr) u) (CHead c (Bind b) v))).(let H14 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) H13) in ((let H15 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) H13) in ((let H16 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead c (Bind b) v) H13) in (\lambda (H17: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H19 \def (eq_ind nat i (\lambda (n: nat).(subst0 n u t3 t2)) H10 O H12) in (let H20 \def (eq_ind T u (\lambda (t: T).(subst0 O t t3 t2)) H19 v H16) in (eq_ind B Abbr (\lambda (b: B).(pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2))) (pr2_free c (THead (Bind Abbr) v t1) (THead (Bind Abbr) v t2) (pr0_delta v v (pr0_refl v) t1 t3 H9 t2 H20)) b H17)))))) H15)) H14)))) H11)) (\lambda (H11: (ex2 nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u))))).(ex2_ind nat (\lambda (j: nat).(eq nat i (S j))) (\lambda (j: nat).(getl j c (CHead d (Bind Abbr) u))) (pr2 c (THead (Bind b) v t1) (THead (Bind b) v t2)) (\lambda (x: nat).(\lambda (H12: (eq nat i (S x))).(\lambda (H13: (getl x c (CHead d (Bind Abbr) u))).(let H14 \def (f_equal nat nat (\lambda (e: nat).e) i (S x) H12) in (let H15 \def (eq_ind nat i (\lambda (n: nat).(subst0 n u t3 t2)) H10 (S x) H14) in (pr2_head_2 c v t1 t2 (Bind b) (pr2_delta (CHead c (Bind b) v) d u (S x) (getl_clear_bind b (CHead c (Bind b) v) c v (clear_bind b c v) (CHead d (Bind Abbr) u) x H13) t1 t3 H9 t2 H15))))))) H11)) H)))))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0 (sym_eq C c0 (CHead c (Bind b) v) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C (CHead c (Bind b) v)) (refl_equal T t1) (refl_equal T t2)))))))).
 
 theorem pr2_gen_cflat:
  \forall (f: F).(\forall (c: C).(\forall (v: T).(\forall (t1: T).(\forall (t2: T).((pr2 (CHead c (Flat f) v) t1 t2) \to (pr2 c t1 t2))))))
 \def
- \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Flat f) v)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c t1 t2))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow (\lambda (H1: (eq C c0 (CHead c (Flat f) v))).(\lambda (H2: (eq T t0 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Flat f) v) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c t1 t2))))) (\lambda (H4: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c t1 t2)))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c t1 t2))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c t1 t2 H6)) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 (CHead c (Flat f) v) H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t3 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 (CHead c (Flat f) v))).(\lambda (H4: (eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead c (Flat f) v) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr2 c t1 t2))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c (Flat f) v) (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c t1 t2)))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CHead c (Flat f) v) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c t1 t2))))) (\lambda (H8: (getl i (CHead c (Flat f) v) (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u t3 t2)).(let H_y \def (getl_gen_flat f c (CHead d (Bind Abbr) u) v i H8) in (pr2_delta c d u i H_y t1 t3 H9 t2 H10))))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0 (sym_eq C c0 (CHead c (Flat f) v) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C (CHead c (Flat f) v)) (refl_equal T t1) (refl_equal T t2)))))))).
+ \lambda (f: F).(\lambda (c: C).(\lambda (v: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 (CHead c (Flat f) v) t1 t2)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c (Flat f) v)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c t1 t2)))))))) with [(pr2_free c0 t0 t3 H0) \Rightarrow (\lambda (H1: (eq C c0 (CHead c (Flat f) v))).(\lambda (H2: (eq T t0 t1)).(\lambda (H3: (eq T t3 t2)).(eq_ind C (CHead c (Flat f) v) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c t1 t2))))) (\lambda (H4: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c t1 t2)))) (\lambda (H5: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c t1 t2))) (\lambda (H6: (pr0 t1 t2)).(pr2_free c t1 t2 H6)) t3 (sym_eq T t3 t2 H5))) t0 (sym_eq T t0 t1 H4))) c0 (sym_eq C c0 (CHead c (Flat f) v) H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t3 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 (CHead c (Flat f) v))).(\lambda (H4: (eq T t0 t1)).(\lambda (H5: (eq T t t2)).(eq_ind C (CHead c (Flat f) v) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr2 c t1 t2))))))) (\lambda (H6: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c (Flat f) v) (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c t1 t2)))))) (\lambda (H7: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CHead c (Flat f) v) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c t1 t2))))) (\lambda (H8: (getl i (CHead c (Flat f) v) (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 t1 t3)).(\lambda (H10: (subst0 i u t3 t2)).(let H_y \def (getl_gen_flat f c (CHead d (Bind Abbr) u) v i H8) in (pr2_delta c d u i H_y t1 t3 H9 t2 H10))))) t (sym_eq T t t2 H7))) t0 (sym_eq T t0 t1 H6))) c0 (sym_eq C c0 (CHead c (Flat f) v) H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C (CHead c (Flat f) v)) (refl_equal T t1) (refl_equal T t2)))))))).
 
 theorem pr2_lift:
  \forall (c: C).(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to (\forall (t1: T).(\forall (t2: T).((pr2 e t1 t2) \to (pr2 c (lift h d t1) (lift h d t2)))))))))
 \def
- \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 e t1 t2)).(let H1 \def (match H0 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 e) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c (lift h d t1) (lift h d t2)))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0 e)).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C e (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (lift h d t1) (lift h d t2)))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c (lift h d t1) (lift h d t2))))) (\lambda (H6: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (lift h d t1) (lift h d t2)))) (\lambda (H7: (pr0 t1 t2)).(pr2_free c (lift h d t1) (lift h d t2) (pr0_lift t1 t2 H7 h d))) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 e H2) H3 H4 H1)))) | (pr2_delta c0 d0 u i H1 t0 t3 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 e)).(\lambda (H5: (eq T t0 t1)).(\lambda (H6: (eq T t t2)).(eq_ind C e (\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d0 (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr2 c (lift h d t1) (lift h d t2)))))))) (\lambda (H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i e (CHead d0 (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c (lift h d t1) (lift h d t2))))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i e (CHead d0 (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c (lift h d t1) (lift h d t2)))))) (\lambda (H9: (getl i e (CHead d0 (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda (H11: (subst0 i u t3 t2)).(lt_le_e i d (pr2 c (lift h d t1) (lift h d t2)) (\lambda (H0: (lt i d)).(let H \def (drop_getl_trans_le i d (le_S_n i d (le_S (S i) d H0)) c e h H (CHead d0 (Bind Abbr) u) H9) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i O c e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d0 (Bind Abbr) u)))) (pr2 c (lift h d t1) (lift h d t2)) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H12: (drop i O c x0)).(\lambda (H13: (drop h (minus d i) x0 x1)).(\lambda (H14: (clear x1 (CHead d0 (Bind Abbr) u))).(let H15 \def (eq_ind nat (minus d i) (\lambda (n: nat).(drop h n x0 x1)) H13 (S (minus d (S i))) (minus_x_Sy d i H0)) in (let H16 \def (drop_clear_S x1 x0 h (minus d (S i)) H15 Abbr d0 u H14) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h (minus d (S i)) u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c (lift h d t1) (lift h d t2)) (\lambda (x: C).(\lambda (H17: (clear x0 (CHead x (Bind Abbr) (lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S i)) x d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead x (Bind Abbr) (lift h (minus d (S i)) u)) x0 H12 H17) (lift h d t1) (lift h d t3) (pr0_lift t1 t3 H10 h d) (lift h d t2) (subst0_lift_lt t3 t2 u i H11 d H0 h))))) H16)))))))) H))) (\lambda (H0: (le d i)).(pr2_delta c d0 u (plus i h) (drop_getl_trans_ge i c e d h H (CHead d0 (Bind Abbr) u) H9 H0) (lift h d t1) (lift h d t3) (pr0_lift t1 t3 H10 h d) (lift h d t2) (subst0_lift_ge t3 t2 u i h H11 d H0))))))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0 e H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C e) (refl_equal T t1) (refl_equal T t2)))))))))).
+ \lambda (c: C).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (drop h d c e)).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 e t1 t2)).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 e) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr2 c (lift h d t1) (lift h d t2))))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0 e)).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C e (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr2 c (lift h d t1) (lift h d t2)))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr2 c (lift h d t1) (lift h d t2))))) (\lambda (H6: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr2 c (lift h d t1) (lift h d t2)))) (\lambda (H7: (pr0 t1 t2)).(pr2_free c (lift h d t1) (lift h d t2) (pr0_lift t1 t2 H7 h d))) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 e H2) H3 H4 H1)))) | (pr2_delta c0 d0 u i H1 t0 t3 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 e)).(\lambda (H5: (eq T t0 t1)).(\lambda (H6: (eq T t t2)).(eq_ind C e (\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d0 (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr2 c (lift h d t1) (lift h d t2)))))))) (\lambda (H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i e (CHead d0 (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr2 c (lift h d t1) (lift h d t2))))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i e (CHead d0 (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr2 c (lift h d t1) (lift h d t2)))))) (\lambda (H9: (getl i e (CHead d0 (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda (H11: (subst0 i u t3 t2)).(lt_le_e i d (pr2 c (lift h d t1) (lift h d t2)) (\lambda (H0: (lt i d)).(let H \def (drop_getl_trans_le i d (le_S_n i d (le_S (S i) d H0)) c e h H (CHead d0 (Bind Abbr) u) H9) in (ex3_2_ind C C (\lambda (e0: C).(\lambda (_: C).(drop i O c e0))) (\lambda (e0: C).(\lambda (e1: C).(drop h (minus d i) e0 e1))) (\lambda (_: C).(\lambda (e1: C).(clear e1 (CHead d0 (Bind Abbr) u)))) (pr2 c (lift h d t1) (lift h d t2)) (\lambda (x0: C).(\lambda (x1: C).(\lambda (H12: (drop i O c x0)).(\lambda (H13: (drop h (minus d i) x0 x1)).(\lambda (H14: (clear x1 (CHead d0 (Bind Abbr) u))).(let H15 \def (eq_ind nat (minus d i) (\lambda (n: nat).(drop h n x0 x1)) H13 (S (minus d (S i))) (minus_x_Sy d i H0)) in (let H16 \def (drop_clear_S x1 x0 h (minus d (S i)) H15 Abbr d0 u H14) in (ex2_ind C (\lambda (c1: C).(clear x0 (CHead c1 (Bind Abbr) (lift h (minus d (S i)) u)))) (\lambda (c1: C).(drop h (minus d (S i)) c1 d0)) (pr2 c (lift h d t1) (lift h d t2)) (\lambda (x: C).(\lambda (H17: (clear x0 (CHead x (Bind Abbr) (lift h (minus d (S i)) u)))).(\lambda (_: (drop h (minus d (S i)) x d0)).(pr2_delta c x (lift h (minus d (S i)) u) i (getl_intro i c (CHead x (Bind Abbr) (lift h (minus d (S i)) u)) x0 H12 H17) (lift h d t1) (lift h d t3) (pr0_lift t1 t3 H10 h d) (lift h d t2) (subst0_lift_lt t3 t2 u i H11 d H0 h))))) H16)))))))) H))) (\lambda (H0: (le d i)).(pr2_delta c d0 u (plus i h) (drop_getl_trans_ge i c e d h H (CHead d0 (Bind Abbr) u) H9 H0) (lift h d t1) (lift h d t3) (pr0_lift t1 t3 H10 h d) (lift h d t2) (subst0_lift_ge t3 t2 u i h H11 d H0))))))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0 e H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C e) (refl_equal T t1) (refl_equal T t2)))))))))).
 
 theorem pr2_gen_appl:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c (THead (Flat Appl) u1 t1) x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))))
 \def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr2 c (THead (Flat Appl) u1 t1) x)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to ((eq T t (THead (Flat Appl) u1 t1)) \to ((eq T t0 x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t0 (THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Flat Appl) u1 t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))))) (\lambda (H4: (eq T t0 (THead (Flat Appl) u1 t1))).(eq_ind T (THead (Flat Appl) u1 t1) (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead (Flat Appl) u1 t1) t) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))) (\lambda (H6: (pr0 (THead (Flat Appl) u1 t1) x)).(or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H: (eq T x (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1 x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x0 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x1 (refl_equal T (THead (Flat Appl) x0 x1)) (pr2_free c u1 x0 H8) (pr2_free c t1 x1 H9))) x H)))))) H7)) (\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H: (eq T t1 (THead (Bind Abst) x0 x1))).(\lambda (H8: (eq T x (THead (Bind Abbr) x2 x3))).(\lambda (H9: (pr0 u1 x2)).(\lambda (H10: (pr0 x1 x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))) x0 x1 x2 x3 (refl_equal T (THead (Bind Abst) x0 x1)) (refl_equal T (THead (Bind Abbr) x2 x3)) (pr2_free c u1 x2 H9) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c (Bind b) u) x1 x3 H10))))) t1 H) x H8))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H: (not (eq B x0 Abst))).(\lambda (H8: (eq T t1 (THead (Bind x0) x1 x2))).(\lambda (H9: (eq T x (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda (H10: (pr0 u1 x3)).(\lambda (H11: (pr0 x1 x4)).(\lambda (H12: (pr0 x2 x5)).(eq_ind_r T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x3 x4 H (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))) (pr2_free c u1 x3 H10) (pr2_free c x1 x4 H11) (pr2_free (CHead c (Bind x0) x4) x2 x5 H12))) t1 H8) x H9))))))))))))) H7)) (pr0_gen_appl u1 t1 x H6))) t2 (sym_eq T t2 x H5))) t0 (sym_eq T t0 (THead (Flat Appl) u1 t1) H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u1 t1))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t0 (THead (Flat Appl) u1 t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))))))) (\lambda (H6: (eq T t0 (THead (Flat Appl) u1 t1))).(eq_ind T (THead (Flat Appl) u1 t1) (\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (THead (Flat Appl) u1 t1) t2) \to ((subst0 i u t2 t3) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 (THead (Flat Appl) u1 t1) t2)).(\lambda (H10: (subst0 i u t2 x)).(or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H12: (pr0 u1 x0)).(\lambda (H13: (pr0 t1 x1)).(let H14 \def (eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H10 (THead (Flat Appl) x0 x1) H) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Flat Appl) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H15: (ex2 T (\lambda (u2: T).(eq T x (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda (H16: (eq T x (THead (Flat Appl) x2 x1))).(\lambda (H17: (subst0 i u x0 x2)).(eq_ind_r T (THead (Flat Appl) x2 x1) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x2 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3))) x2 x1 (refl_equal T (THead (Flat Appl) x2 x1)) (pr2_delta c d u i H8 u1 x0 H12 x2 H17) (pr2_free c t1 x1 H13))) x H16)))) H15)) (\lambda (H15: (ex2 T (\lambda (t2: T).(eq T x (THead (Flat Appl) x0 t2))) (\lambda (t2: T).(subst0 (s (Flat Appl) i) u x1 t2)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Flat Appl) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda (H16: (eq T x (THead (Flat Appl) x0 x2))).(\lambda (H17: (subst0 (s (Flat Appl) i) u x1 x2)).(eq_ind_r T (THead (Flat Appl) x0 x2) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x2 (refl_equal T (THead (Flat Appl) x0 x2)) (pr2_free c u1 x0 H12) (pr2_delta c d u i H8 t1 x1 H13 x2 H17))) x H16)))) H15)) (\lambda (H15: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) u x1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H16: (eq T x (THead (Flat Appl) x2 x3))).(\lambda (H17: (subst0 i u x0 x2)).(\lambda (H18: (subst0 (s (Flat Appl) i) u x1 x3)).(eq_ind_r T (THead (Flat Appl) x2 x3) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3))) x2 x3 (refl_equal T (THead (Flat Appl) x2 x3)) (pr2_delta c d u i H8 u1 x0 H12 x2 H17) (pr2_delta c d u i H8 t1 x1 H13 x3 H18))) x H16)))))) H15)) (subst0_gen_head (Flat Appl) u x0 x1 x i H14)))))))) H11)) (\lambda (H11: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H: (eq T t1 (THead (Bind Abst) x0 x1))).(\lambda (H12: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (H13: (pr0 u1 x2)).(\lambda (H14: (pr0 x1 x3)).(let H15 \def (eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H10 (THead (Bind Abbr) x2 x3) H12) in (eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Bind Abbr) x2 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H16: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (H17: (eq T x (THead (Bind Abbr) x4 x3))).(\lambda (H18: (subst0 i u x2 x4)).(eq_ind_r T (THead (Bind Abbr) x4 x3) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t1 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))) x0 x1 x4 x3 (refl_equal T (THead (Bind Abst) x0 x1)) (refl_equal T (THead (Bind Abbr) x4 x3)) (pr2_delta c d u i H8 u1 x2 H13 x4 H18) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) x1 x3 H14))))) x H17)))) H16)) (\lambda (H16: (ex2 T (\lambda (t2: T).(eq T x (THead (Bind Abbr) x2 t2))) (\lambda (t2: T).(subst0 (s (Bind Abbr) i) u x3 t2)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind Abbr) x2 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (H17: (eq T x (THead (Bind Abbr) x2 x4))).(\lambda (H18: (subst0 (s (Bind Abbr) i) u x3 x4)).(eq_ind_r T (THead (Bind Abbr) x2 x4) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t1 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))) x0 x1 x2 x4 (refl_equal T (THead (Bind Abst) x0 x1)) (refl_equal T (THead (Bind Abbr) x2 x4)) (pr2_free c u1 x2 H13) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_clear_bind b (CHead c (Bind b) u0) c u0 (clear_bind b c u0) (CHead d (Bind Abbr) u) i H8) x1 x3 H14 x4 H18))))) x H17)))) H16)) (\lambda (H16: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Bind Abbr) i) u x3 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H17: (eq T x (THead (Bind Abbr) x4 x5))).(\lambda (H18: (subst0 i u x2 x4)).(\lambda (H19: (subst0 (s (Bind Abbr) i) u x3 x5)).(eq_ind_r T (THead (Bind Abbr) x4 x5) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t1 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))) x0 x1 x4 x5 (refl_equal T (THead (Bind Abst) x0 x1)) (refl_equal T (THead (Bind Abbr) x4 x5)) (pr2_delta c d u i H8 u1 x2 H13 x4 H18) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_clear_bind b (CHead c (Bind b) u0) c u0 (clear_bind b c u0) (CHead d (Bind Abbr) u) i H8) x1 x3 H14 x5 H19))))) x H17)))))) H16)) (subst0_gen_head (Bind Abbr) u x2 x3 x i H15)) t1 H)))))))))) H11)) (\lambda (H11: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 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(lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H: (not (eq B x0 Abst))).(\lambda (H12: (eq T t1 (THead (Bind x0) x1 x2))).(\lambda (H13: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda (H14: (pr0 u1 x3)).(\lambda (H15: (pr0 x1 x4)).(\lambda (H16: (pr0 x2 x5)).(let H17 \def (eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H10 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) H13) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead (Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2: T).(subst0 i u x4 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Bind x0) x4 t3))) (\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H18: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind x0) u2 (THead (Flat Appl) 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(eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda (H19: (eq T x (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda (H20: (subst0 i u x4 x6)).(eq_ind_r T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t1 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 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T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 t1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 t1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda 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T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x x3 x4 H (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal 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T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (x8: T).(\lambda (H22: (eq T x6 (THead (Flat Appl) x7 x8))).(\lambda (H23: (subst0 (s (Bind x0) i) u (lift (S O) O x3) x7)).(\lambda (H24: (subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 x8)).(eq_ind_r T (THead (Flat Appl) x7 x8) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 t1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 t1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 t1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x7 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 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(z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t1 x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x) x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x) x8)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 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(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x) x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x8 x x4 H (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x) x8))) (pr2_delta c d u i H8 u1 x3 H14 x H27) (pr2_free c x1 x4 H15) (pr2_delta (CHead c (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead c (Bind x0) x4) c x4 (clear_bind x0 c x4) (CHead d (Bind Abbr) u) i H8) x2 x5 H16 x8 H24))) x7 H25))))) (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O H23 (le_S_n (S O) (S i) (lt_le_S (S O) (S (S i)) (lt_n_S O (S i) (le_lt_n_Sm O i (le_O_n i))))))) x6 H22)))))) H21)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5 x6 (s (Bind x0) i) H20)) x H19)))) H18)) (\lambda (H18: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind x0) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (H19: (eq T x (THead (Bind x0) x6 x7))).(\lambda (H20: (subst0 i u x4 x6)).(\lambda (H21: (subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) x7)).(eq_ind_r T (THead (Bind x0) x6 x7) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t1 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T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (ex2 T (\lambda (t3: T).(eq T x7 (THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H22: (ex2 T (\lambda (u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2)))).(ex2_ind T (\lambda (u2: T).(eq T x7 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind 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T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x: T).(\lambda (H23: (eq T x7 (THead (Flat Appl) x x5))).(\lambda (H24: (subst0 (s (Bind x0) i) u (lift (S O) O x3) x)).(eq_ind_r T (THead (Flat Appl) x x5) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 t1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 t1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x x5)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x8: T).(\lambda (H25: (eq T x (lift (S O) O x8))).(\lambda (H26: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x8)).(let H27 \def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u x3 x8)) H26 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x8) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t1 x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t1 x5)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t1 x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T 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B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x8) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 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(\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 t1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t1) (THead 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y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda 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T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda 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(_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 t1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x8 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x: T).(\lambda (H26: (eq T x8 (lift (S O) O x))).(\lambda (H27: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x)).(let H28 \def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u x3 x)) H27 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t1 x9)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t1 x9)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t1 x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x) x9)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x) x9)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x) x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x) x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x9 x x6 H (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x) x9))) (pr2_delta c d u i H8 u1 x3 H14 x H28) (pr2_delta c d u i H8 x1 x4 H15 x6 H20) (pr2_delta (CHead c (Bind x0) x6) d u (S i) (getl_clear_bind x0 (CHead c (Bind x0) x6) c x6 (clear_bind x0 c x6) (CHead d (Bind Abbr) u) i H8) x2 x5 H16 x9 H25))) x8 H26))))) (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H24 (le_S_n (S O) (S i) (lt_le_S (S O) (S (S i)) (lt_n_S O (S i) (le_lt_n_Sm O i (le_O_n i))))))) x7 H23)))))) H22)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5 x7 (s (Bind x0) i) H21)) x H19)))))) H18)) (subst0_gen_head (Bind x0) u x4 (THead (Flat Appl) (lift (S O) O x3) x5) x i H17)) t1 H12)))))))))))))) H11)) (pr0_gen_appl u1 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Flat Appl) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T (THead (Flat Appl) u1 t1)) (refl_equal T x))))))).
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr2 c (THead (Flat Appl) u1 t1) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to ((eq T t (THead (Flat Appl) u1 t1)) \to ((eq T t0 x) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c t1 t2)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t2)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t0 (THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Flat Appl) u1 t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))))) (\lambda (H4: (eq T t0 (THead (Flat Appl) u1 t1))).(eq_ind T (THead (Flat Appl) u1 t1) (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead (Flat Appl) u1 t1) t) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))) (\lambda (H6: (pr0 (THead (Flat Appl) u1 t1) x)).(or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H: (eq T x (THead (Flat Appl) x0 x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1 x1)).(eq_ind_r T (THead (Flat Appl) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x0 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x1 (refl_equal T (THead (Flat Appl) x0 x1)) (pr2_free c u1 x0 H8) (pr2_free c t1 x1 H9))) x H)))))) H7)) (\lambda (H7: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H: (eq T t1 (THead (Bind Abst) x0 x1))).(\lambda (H8: (eq T x (THead (Bind Abbr) x2 x3))).(\lambda (H9: (pr0 u1 x2)).(\lambda (H10: (pr0 x1 x3)).(eq_ind_r T (THead (Bind Abbr) x2 x3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: 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Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3))))))) x0 x1 x2 x3 (refl_equal T (THead (Bind Abst) x0 x1)) (refl_equal T (THead (Bind Abbr) x2 x3)) (pr2_free c u1 x2 H9) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c (Bind b) u) x1 x3 H10))))) t1 H) x H8))))))))) H7)) (\lambda (H7: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t2: T).(eq T x (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T x (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H: (not (eq B x0 Abst))).(\lambda (H8: (eq T t1 (THead (Bind x0) x1 x2))).(\lambda (H9: (eq T x (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda (H10: (pr0 u1 x3)).(\lambda (H11: (pr0 x1 x4)).(\lambda (H12: (pr0 x2 x5)).(eq_ind_r T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x3 x4 H (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5))) (pr2_free c u1 x3 H10) (pr2_free c x1 x4 H11) (pr2_free (CHead c (Bind x0) x4) x2 x5 H12))) t1 H8) x H9))))))))))))) H7)) (pr0_gen_appl u1 t1 x H6))) t2 (sym_eq T t2 x H5))) t0 (sym_eq T t0 (THead (Flat Appl) u1 t1) H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Flat Appl) u1 t1))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t0 (THead (Flat Appl) u1 t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))))))) (\lambda (H6: (eq T t0 (THead (Flat Appl) u1 t1))).(eq_ind T (THead (Flat Appl) u1 t1) (\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (THead (Flat Appl) u1 t1) t2) \to ((subst0 i u t2 t3) \to (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 (THead (Flat Appl) u1 t1) t2)).(\lambda (H10: (subst0 i u t2 x)).(or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3)))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H: (eq T t2 (THead (Flat Appl) x0 x1))).(\lambda (H12: (pr0 u1 x0)).(\lambda (H13: (pr0 t1 x1)).(let H14 \def (eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H10 (THead (Flat Appl) x0 x1) H) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Flat Appl) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H15: (ex2 T (\lambda (u2: T).(eq T x (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Flat Appl) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda (H16: (eq T x (THead (Flat Appl) x2 x1))).(\lambda (H17: (subst0 i u x0 x2)).(eq_ind_r T (THead (Flat Appl) x2 x1) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x2 x1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3))) x2 x1 (refl_equal T (THead (Flat Appl) x2 x1)) (pr2_delta c d u i H8 u1 x0 H12 x2 H17) (pr2_free c t1 x1 H13))) x H16)))) H15)) (\lambda (H15: (ex2 T (\lambda (t2: T).(eq T x (THead (Flat Appl) x0 t2))) (\lambda (t2: T).(subst0 (s (Flat Appl) i) u x1 t2)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Flat Appl) x0 t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda (H16: (eq T x (THead (Flat Appl) x0 x2))).(\lambda (H17: (subst0 (s (Flat Appl) i) u x1 x2)).(eq_ind_r T (THead (Flat Appl) x0 x2) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x0 x2) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x0 x2) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3))) x0 x2 (refl_equal T (THead (Flat Appl) x0 x2)) (pr2_free c u1 x0 H12) (pr2_delta c d u i H8 t1 x1 H13 x2 H17))) x H16)))) H15)) (\lambda (H15: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) i) u x1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) i) u x1 t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H16: (eq T x (THead (Flat Appl) x2 x3))).(\lambda (H17: (subst0 i u x0 x2)).(\lambda (H18: (subst0 (s (Flat Appl) i) u x1 x3)).(eq_ind_r T (THead (Flat Appl) x2 x3) (\lambda (t3: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c t1 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro0 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Flat Appl) x2 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Flat Appl) x2 x3) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3))) x2 x3 (refl_equal T (THead (Flat Appl) x2 x3)) (pr2_delta c d u i H8 u1 x0 H12 x2 H17) (pr2_delta c d u i H8 t1 x1 H13 x3 H18))) x H16)))))) H15)) (subst0_gen_head (Flat Appl) u x0 x1 x i H14)))))))) H11)) (\lambda (H11: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (H: (eq T t1 (THead (Bind Abst) x0 x1))).(\lambda (H12: (eq T t2 (THead (Bind Abbr) x2 x3))).(\lambda (H13: (pr0 u1 x2)).(\lambda (H14: (pr0 x1 x3)).(let H15 \def (eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H10 (THead (Bind Abbr) x2 x3) H12) in (eq_ind_r T (THead (Bind Abst) x0 x1) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Bind Abbr) x2 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H16: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Bind Abbr) u2 x3))) (\lambda (u2: T).(subst0 i u x2 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (H17: (eq T x (THead (Bind Abbr) x4 x3))).(\lambda (H18: (subst0 i u x2 x4)).(eq_ind_r T (THead (Bind Abbr) x4 x3) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t1 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 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T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x3) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))) x0 x1 x4 x3 (refl_equal T (THead (Bind Abst) x0 x1)) (refl_equal T (THead (Bind Abbr) x4 x3)) (pr2_delta c d u i H8 u1 x2 H13 x4 H18) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) x1 x3 H14))))) x H17)))) H16)) (\lambda (H16: (ex2 T (\lambda (t2: T).(eq T x (THead (Bind Abbr) x2 t2))) (\lambda (t2: T).(subst0 (s (Bind Abbr) i) u x3 t2)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind Abbr) x2 t3))) (\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (H17: (eq T x (THead (Bind Abbr) x2 x4))).(\lambda (H18: (subst0 (s (Bind Abbr) i) u x3 x4)).(eq_ind_r T (THead (Bind Abbr) x2 x4) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t1 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x2 x4) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))) x0 x1 x2 x4 (refl_equal T (THead (Bind Abst) x0 x1)) (refl_equal T (THead (Bind Abbr) x2 x4)) (pr2_free c u1 x2 H13) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_clear_bind b (CHead c (Bind b) u0) c u0 (clear_bind b c u0) (CHead d (Bind Abbr) u) i H8) x1 x3 H14 x4 H18))))) x H17)))) H16)) (\lambda (H16: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Bind Abbr) i) u x3 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x2 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x3 t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H17: (eq T x (THead (Bind Abbr) x4 x5))).(\lambda (H18: (subst0 i u x2 x4)).(\lambda (H19: (subst0 (s (Bind Abbr) i) u x3 x5)).(eq_ind_r T (THead (Bind Abbr) x4 x5) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t1 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro1 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind Abst) x0 x1) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex4_4_intro T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind Abst) x0 x1) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x4 x5) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3))))))) x0 x1 x4 x5 (refl_equal T (THead (Bind Abst) x0 x1)) (refl_equal T (THead (Bind Abbr) x4 x5)) (pr2_delta c d u i H8 u1 x2 H13 x4 H18) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_clear_bind b (CHead c (Bind b) u0) c u0 (clear_bind b c u0) (CHead d (Bind Abbr) u) i H8) x1 x3 H14 x5 H19))))) x H17)))))) H16)) (subst0_gen_head (Bind Abbr) u x2 x3 x i H15)) t1 H)))))))))) H11)) (\lambda (H11: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v2: T).(\lambda (t3: T).(eq T t2 (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u2) t3))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(\lambda (_: T).(pr0 u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (v2: T).(\lambda (_: T).(pr0 y1 v2))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t3: T).(pr0 z1 t3))))))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x0: B).(\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H: (not (eq B x0 Abst))).(\lambda (H12: (eq T t1 (THead (Bind x0) x1 x2))).(\lambda (H13: (eq T t2 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda (H14: (pr0 u1 x3)).(\lambda (H15: (pr0 x1 x4)).(\lambda (H16: (pr0 x2 x5)).(let H17 \def (eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H10 (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x5)) H13) in (eq_ind_r T (THead (Bind x0) x1 x2) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c t1 t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T t1 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead (Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2: T).(subst0 i u x4 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Bind x0) x4 t3))) (\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (H18: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2: T).(subst0 i u x4 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Bind x0) u2 (THead (Flat Appl) (lift (S O) O x3) x5)))) (\lambda (u2: T).(subst0 i u x4 u2)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda (H19: (eq T x (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)))).(\lambda (H20: (subst0 i u x4 x6)).(eq_ind_r T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t1 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x5 x3 x6 H (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x5))) (pr2_free c u1 x3 H14) (pr2_delta c d u i H8 x1 x4 H15 x6 H20) (pr2_free (CHead c (Bind x0) x6) x2 x5 H16))) x H19)))) H18)) (\lambda (H18: (ex2 T (\lambda (t2: T).(eq T x (THead (Bind x0) x4 t2))) (\lambda (t2: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) t2)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind x0) x4 t3))) (\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x6: T).(\lambda (H19: (eq T x (THead (Bind x0) x4 x6))).(\lambda (H20: (subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) x6)).(eq_ind_r T (THead (Bind x0) x4 x6) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T t1 (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t1 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_ind (ex2 T (\lambda (u2: T).(eq T x6 (THead (Flat Appl) u2 x5))) (\lambda (u2: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (ex2 T (\lambda (t3: T).(eq T x6 (THead (Flat Appl) (lift (S O) O x3) t3))) (\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3)))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) 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T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 t1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x x5)) 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T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x x3 x4 H (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x3) x))) (pr2_free c u1 x3 H14) (pr2_free c x1 x4 H15) (pr2_delta (CHead c (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead c (Bind x0) x4) c x4 (clear_bind x0 c x4) (CHead d (Bind Abbr) u) i H8) x2 x5 H16 x H23))) x6 H22)))) H21)) (\lambda (H21: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x6 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x6 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 x6) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x7: T).(\lambda (x8: T).(\lambda (H22: (eq T x6 (THead (Flat Appl) x7 x8))).(\lambda (H23: (subst0 (s (Bind x0) i) u (lift (S O) O x3) x7)).(\lambda (H24: (subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 x8)).(eq_ind_r T (THead (Flat Appl) x7 x8) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 t1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 t1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 t1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x7 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) x7 x8)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: 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T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x: T).(\lambda (H25: (eq T x7 (lift (S O) O x))).(\lambda (H26: (subst0 (minus (s (Bind x0) i) (S O)) u x3 x)).(let H27 \def (eq_ind nat (minus (s (Bind x0) i) (S O)) (\lambda (n: nat).(subst0 n u x3 x)) H26 i (s_arith1 x0 i)) in (eq_ind_r T (lift (S O) O x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t1 x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t1 x8)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) t1 x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x) x8)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x) x8)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda 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(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x) x8)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x8 x x4 H (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x4 (THead (Flat Appl) (lift (S O) O x) x8))) (pr2_delta c d u i H8 u1 x3 H14 x H27) (pr2_free c x1 x4 H15) (pr2_delta (CHead c (Bind x0) x4) d u (S i) (getl_clear_bind x0 (CHead c (Bind x0) x4) c x4 (clear_bind x0 c x4) (CHead d (Bind Abbr) u) i H8) x2 x5 H16 x8 H24))) x7 H25))))) (subst0_gen_lift_ge u x3 x7 (s (Bind x0) i) (S O) O H23 (le_S_n (S O) (S i) (lt_le_S (S O) (S (S i)) (lt_n_S O (S i) (le_lt_n_Sm O i (le_O_n i))))))) x6 H22)))))) H21)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5 x6 (s (Bind x0) i) H20)) x H19)))) H18)) (\lambda (H18: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind x0) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind x0) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x4 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind x0) i) u (THead (Flat Appl) (lift (S O) O x3) x5) t3))) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: 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(s (Bind x0) i)) u x5 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 x7) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))) (\lambda (x: T).(\lambda (H23: (eq T x7 (THead (Flat Appl) (lift (S O) O x3) x))).(\lambda (H24: (subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 x)).(eq_ind_r T (THead (Flat Appl) (lift (S O) O x3) x) (\lambda (t1: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 t1) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 t1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t1) (THead 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y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x x3 x6 H (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x3) x))) (pr2_free c u1 x3 H14) (pr2_delta c d u i H8 x1 x4 H15 x6 H20) (pr2_delta (CHead c (Bind x0) x6) d u (S i) (getl_clear_bind x0 (CHead c (Bind x0) x6) c x6 (clear_bind x0 c x6) (CHead d (Bind Abbr) u) i H8) x2 x5 H16 x H24))) x7 H23)))) H22)) (\lambda (H22: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x7 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x7 (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 (s (Bind x0) i) u (lift (S O) O x3) u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Flat Appl) (s (Bind x0) i)) u x5 t3))) (or3 (ex3_2 T T (\lambda (u2: 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(_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 t1) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 t1) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (ex2_ind T (\lambda (t3: T).(eq T x8 (lift (S O) O t3))) (\lambda (t3: T).(subst0 (minus (s (Bind x0) i) (S O)) u x3 t3)) (or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) x8 x9)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda 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Appl) t1 x9)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) t1 x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x) x9)) (THead (Flat Appl) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr2 c (THead (Bind x0) x1 x2) t3)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x) x9)) (THead (Bind Abbr) u2 t3)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) z1 t3)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x) x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (ex6_6_intro B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Bind x0) x1 x2) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x) x9)) (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) x0 x1 x2 x9 x x6 H (refl_equal T (THead (Bind x0) x1 x2)) (refl_equal T (THead (Bind x0) x6 (THead (Flat Appl) (lift (S O) O x) x9))) (pr2_delta c d u i H8 u1 x3 H14 x H28) (pr2_delta c d u i H8 x1 x4 H15 x6 H20) (pr2_delta (CHead c (Bind x0) x6) d u (S i) (getl_clear_bind x0 (CHead c (Bind x0) x6) c x6 (clear_bind x0 c x6) (CHead d (Bind Abbr) u) i H8) x2 x5 H16 x9 H25))) x8 H26))))) (subst0_gen_lift_ge u x3 x8 (s (Bind x0) i) (S O) O H24 (le_S_n (S O) (S i) (lt_le_S (S O) (S (S i)) (lt_n_S O (S i) (le_lt_n_Sm O i (le_O_n i))))))) x7 H23)))))) H22)) (subst0_gen_head (Flat Appl) u (lift (S O) O x3) x5 x7 (s (Bind x0) i) H21)) x H19)))))) H18)) (subst0_gen_head (Bind x0) u x4 (THead (Flat Appl) (lift (S O) O x3) x5) x i H17)) t1 H12)))))))))))))) H11)) (pr0_gen_appl u1 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Flat Appl) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T (THead (Flat Appl) u1 t1)) (refl_equal T x))))))).
 
 theorem pr2_gen_abbr:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c (THead (Bind Abbr) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
 \def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr2 c (THead (Bind Abbr) u1 t1) x)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to ((eq T t (THead (Bind Abbr) u1 t1)) \to ((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t0 (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Abbr) u1 t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) (\lambda (H4: (eq T t0 (THead (Bind Abbr) u1 t1))).(eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead (Bind Abbr) u1 t1) t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))) (\lambda (H6: (pr0 (THead (Bind Abbr) u1 t1) x)).(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2)))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t3)))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H: (eq T x (THead (Bind Abbr) x0 x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H_x: (or (pr0 t1 x1) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O x0 y x1))))).(or_ind (pr0 t1 x1) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O x0 y x1))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))) (\lambda (H9: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O t)))))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O (THead (Bind Abbr) x0 x1))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 x1)) (pr2_free c u1 x0 H8) (or3_intro0 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 x1))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 x1))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z x1)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c (Bind b) u) t1 x1 H9)))))) x H)) (\lambda (H_x0: (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O x0 y x1)))).(ex2_ind T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O x0 y x1)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))) (\lambda (x2: T).(\lambda (H9: (pr0 t1 x2)).(\lambda (H10: (subst0 O x0 x2 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O t)))))) (ex2_ind T (\lambda (t: T).(subst0 O u1 x2 t)) (\lambda (t: T).(pr0 t x1)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Abbr) x0 x1) (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O (THead (Bind Abbr) x0 x1)))))) (\lambda (x: T).(\lambda (_: (subst0 O u1 x2 x)).(\lambda (_: (pr0 x x1)).(or_introl (ex3_2 T 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(ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))) x0 x1 (refl_equal T (THead (Bind Abbr) x0 x1)) (pr2_free c u1 x0 H8) (or3_intro1 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 x1))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 x1))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z x1)))) (ex_intro2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 x1)) x0 H8 (pr2_delta (CHead c (Bind Abbr) x0) c x0 O (getl_refl Abbr c x0) t1 x2 H9 x1 H10)))))))) (pr0_subst0_back x0 x2 x1 O H10 u1 H8)) x H)))) H_x0)) H_x)))))) H7)) (\lambda (H: (pr0 t1 (lift (S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x 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(b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H17: (eq T x (THead (Bind Abbr) x3 x4))).(\lambda (H18: (subst0 i u x0 x3)).(\lambda (H19: (subst0 (s (Bind Abbr) i) u x1 x4)).(ex2_ind T (\lambda (t1: T).(subst0 O u1 x2 t1)) (\lambda (t1: T).(pr0 t1 x1)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) (\lambda (x5: T).(\lambda (H20: (subst0 O u1 x2 x5)).(\lambda (H21: (pr0 x5 x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))) x3 x4 H17 (pr2_delta c d u i H8 u1 x0 H12 x3 H18) (or3_intro2 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 x4))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c (Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t1 x2 H13 x5 H20) H21 (pr2_delta (CHead c (Bind Abbr) u1) d u (S i) (getl_head (Bind Abbr) i c (CHead d (Bind Abbr) u) H8 u1) x1 x1 (pr0_refl x1) x4 H19)))))))) (pr0_subst0_back x0 x2 x1 O H14 u1 H12))))))) H16)) (subst0_gen_head (Bind Abbr) u x0 x1 x i H15)))))) H_x0)) H_x)))))) H11)) (\lambda (H: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 (lift (S O) O t2) H (lift (S O) O x) (subst0_lift_ge_S t2 x u i H10 O (le_O_n i))))))) (pr0_gen_abbr u1 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Bind Abbr) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T (THead (Bind Abbr) u1 t1)) (refl_equal T x))))))).
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr2 c (THead (Bind Abbr) u1 t1) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to ((eq T t (THead (Bind Abbr) u1 t1)) \to ((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t2))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t2)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t0 (THead (Bind Abbr) u1 t1))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Abbr) u1 t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) (\lambda (H4: (eq T t0 (THead (Bind Abbr) u1 t1))).(eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead (Bind Abbr) u1 t1) t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))) (\lambda (H6: (pr0 (THead (Bind Abbr) u1 t1) x)).(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t3))))))) (pr0 t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t2: T).(or (pr0 t1 t2) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t2)))))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (u2: T).(\lambda (t3: T).(or (pr0 t1 t3) (ex2 T (\lambda (y: T).(pr0 t1 y)) (\lambda (y: T).(subst0 O u2 y t3)))))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda 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(_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))) (\lambda (H9: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Abbr) x0 x1) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))) (ex2 T (\lambda (u: T).(pr0 u1 u)) (\lambda (u: T).(pr2 (CHead c (Bind Abbr) u) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O t)))))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T 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T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))) x0 x3 H17 (pr2_free c u1 x0 H12) (or3_intro2 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 x3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 x3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z x3)))) (ex3_2_intro T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z x3))) x4 x1 (pr2_delta (CHead c (Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t1 x2 H13 x4 H19) H20 (pr2_delta (CHead c (Bind Abbr) u1) d u (S i) (getl_head (Bind Abbr) i c (CHead d (Bind Abbr) u) H8 u1) x1 x1 (pr0_refl x1) x3 H18)))))))) (pr0_subst0_back x0 x2 x1 O H14 u1 H12))))) H16)) (\lambda (H16: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Abbr) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Bind Abbr) i) u x1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Abbr) i) u x1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H17: (eq T x (THead (Bind Abbr) x3 x4))).(\lambda (H18: (subst0 i u x0 x3)).(\lambda (H19: (subst0 (s (Bind Abbr) i) u x1 x4)).(ex2_ind T (\lambda (t1: T).(subst0 O u1 x2 t1)) (\lambda (t1: T).(pr0 t1 x1)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) (\lambda (x5: T).(\lambda (H20: (subst0 O u1 x2 x5)).(\lambda (H21: (pr0 x5 x1)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3))))))) x3 x4 H17 (pr2_delta c d u i H8 u1 x0 H12 x3 H18) (or3_intro2 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 x4))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 x4))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z x4)))) (ex3_2_intro T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z x4))) x5 x1 (pr2_delta (CHead c (Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t1 x2 H13 x5 H20) H21 (pr2_delta (CHead c (Bind Abbr) u1) d u (S i) (getl_head (Bind Abbr) i c (CHead d (Bind Abbr) u) H8 u1) x1 x1 (pr0_refl x1) x4 H19)))))))) (pr0_subst0_back x0 x2 x1 O H14 u1 H12))))))) H16)) (subst0_gen_head (Bind Abbr) u x0 x1 x i H15)))))) H_x0)) H_x)))))) H11)) (\lambda (H: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Abbr) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(or3 (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))) (ex2 T (\lambda (u0: T).(pr0 u1 u0)) (\lambda (u0: T).(pr2 (CHead c (Bind Abbr) u0) t1 t3))) (ex3_2 T T (\lambda (y: T).(\lambda (_: T).(pr2 (CHead c (Bind Abbr) u1) t1 y))) (\lambda (y: T).(\lambda (z: T).(pr0 y z))) (\lambda (_: T).(\lambda (z: T).(pr2 (CHead c (Bind Abbr) u1) z t3)))))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 (lift (S O) O t2) H (lift (S O) O x) (subst0_lift_ge_S t2 x u i H10 O (le_O_n i))))))) (pr0_gen_abbr u1 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Bind Abbr) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T (THead (Bind Abbr) u1 t1)) (refl_equal T x))))))).
 
 theorem pr2_gen_void:
  \forall (c: C).(\forall (u1: T).(\forall (t1: T).(\forall (x: T).((pr2 c (THead (Bind Void) u1 t1) x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))
 \def
- \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr2 c (THead (Bind Void) u1 t1) x)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to ((eq T t (THead (Bind Void) u1 t1)) \to ((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1 t1))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Void) u1 t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) (\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(eq_ind T (THead (Bind Void) u1 t1) (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead (Bind Void) u1 t1) t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))) (\lambda (H6: (pr0 (THead (Bind Void) u1 t1) x)).(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H: (eq T x (THead (Bind Void) x0 x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Void) x0 x1) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O t)))))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O (THead (Bind Void) x0 x1))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) (pr2_free c u1 x0 H8) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c (Bind b) u) t1 x1 H9))))) x H)))))) H7)) (\lambda (H: (pr0 t1 (lift (S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c (Bind b) u) t1 (lift (S O) O x) H))))) (pr0_gen_void u1 t1 x H6))) t2 (sym_eq T t2 x H5))) t0 (sym_eq T t0 (THead (Bind Void) u1 t1) H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t0 (THead (Bind Void) u1 t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))))))))) (\lambda (H6: (eq T t0 (THead (Bind Void) u1 t1))).(eq_ind T (THead (Bind Void) u1 t1) (\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind Void) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (THead (Bind Void) u1 t1) t2) \to ((subst0 i u t2 t3) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind Void) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 (THead (Bind Void) u1 t1) t2)).(\lambda (H10: (subst0 i u t2 x)).(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H12: (pr0 u1 x0)).(\lambda (H13: (pr0 t1 x1)).(let H14 \def (eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H10 (THead (Bind Void) x0 x1) H) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Bind Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) (\lambda (H15: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) (\lambda (x2: T).(\lambda (H16: (eq T x (THead (Bind Void) x2 x1))).(\lambda (H17: (subst0 i u x0 x2)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x2 x1 H16 (pr2_delta c d u i H8 u1 x0 H12 x2 H17) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) t1 x1 H13)))))))) H15)) (\lambda (H15: (ex2 T (\lambda (t2: T).(eq T x (THead (Bind Void) x0 t2))) (\lambda (t2: T).(subst0 (s (Bind Void) i) u x1 t2)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) (\lambda (x2: T).(\lambda (H16: (eq T x (THead (Bind Void) x0 x2))).(\lambda (H17: (subst0 (s (Bind Void) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x0 x2 H16 (pr2_free c u1 x0 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 x1 H13 x2 H17)))))))) H15)) (\lambda (H15: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Bind Void) i) u x1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H16: (eq T x (THead (Bind Void) x2 x3))).(\lambda (H17: (subst0 i u x0 x2)).(\lambda (H18: (subst0 (s (Bind Void) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x2 x3 H16 (pr2_delta c d u i H8 u1 x0 H12 x2 H17) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 x1 H13 x3 H18)))))))))) H15)) (subst0_gen_head (Bind Void) u x0 x1 x i H14)))))))) H11)) (\lambda (H: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 (lift (S O) O t2) H (lift (S O) O x) (subst0_lift_ge_S t2 x u i H10 O (le_O_n i))))))) (pr0_gen_void u1 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Bind Void) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T (THead (Bind Void) u1 t1)) (refl_equal T x))))))).
+ \lambda (c: C).(\lambda (u1: T).(\lambda (t1: T).(\lambda (x: T).(\lambda (H: (pr2 c (THead (Bind Void) u1 t1) x)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to ((eq T t (THead (Bind Void) u1 t1)) \to ((eq T t0 x) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t2)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))))))) with [(pr2_free c0 t0 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t0 (THead (Bind Void) u1 t1))).(\lambda (H3: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (THead (Bind Void) u1 t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))))) (\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(eq_ind T (THead (Bind Void) u1 t1) (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))))))) (\lambda (H5: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (THead (Bind Void) u1 t1) t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))))) (\lambda (H6: (pr0 (THead (Bind Void) u1 t1) x)).(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O x)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))) (\lambda (H7: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H: (eq T x (THead (Bind Void) x0 x1))).(\lambda (H8: (pr0 u1 x0)).(\lambda (H9: (pr0 t1 x1)).(eq_ind_r T (THead (Bind Void) x0 x1) (\lambda (t: T).(or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O t)))))) (or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O (THead (Bind Void) x0 x1))))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T (THead (Bind Void) x0 x1) (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3))))) x0 x1 (refl_equal T (THead (Bind Void) x0 x1)) (pr2_free c u1 x0 H8) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c (Bind b) u) t1 x1 H9))))) x H)))))) H7)) (\lambda (H: (pr0 t1 (lift (S O) O x))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 t3)))))) (\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) t1 (lift (S O) O x)))) (\lambda (b: B).(\lambda (u: T).(pr2_free (CHead c (Bind b) u) t1 (lift (S O) O x) H))))) (pr0_gen_void u1 t1 x H6))) t2 (sym_eq T t2 x H5))) t0 (sym_eq T t0 (THead (Bind Void) u1 t1) H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t0 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t0 (THead (Bind Void) u1 t1))).(\lambda (H5: (eq T t x)).(eq_ind C c (\lambda (c1: C).((eq T t0 (THead (Bind Void) u1 t1)) \to ((eq T t x) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))))))))) (\lambda (H6: (eq T t0 (THead (Bind Void) u1 t1))).(eq_ind T (THead (Bind Void) u1 t1) (\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind Void) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))))))))) (\lambda (H7: (eq T t x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 (THead (Bind Void) u1 t1) t2) \to ((subst0 i u t2 t3) \to (or (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x (THead (Bind Void) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t4)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 (THead (Bind Void) u1 t1) t2)).(\lambda (H10: (subst0 i u t2 x)).(or_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3)))) (pr0 t1 (lift (S O) O t2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) (\lambda (H11: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T t2 (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t3: T).(pr0 t1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H: (eq T t2 (THead (Bind Void) x0 x1))).(\lambda (H12: (pr0 u1 x0)).(\lambda (H13: (pr0 t1 x1)).(let H14 \def (eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H10 (THead (Bind Void) x0 x1) H) in (or3_ind (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2))) (ex2 T (\lambda (t3: T).(eq T x (THead (Bind Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) (\lambda (H15: (ex2 T (\lambda (u2: T).(eq T x (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)))).(ex2_ind T (\lambda (u2: T).(eq T x (THead (Bind Void) u2 x1))) (\lambda (u2: T).(subst0 i u x0 u2)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) (\lambda (x2: T).(\lambda (H16: (eq T x (THead (Bind Void) x2 x1))).(\lambda (H17: (subst0 i u x0 x2)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x2 x1 H16 (pr2_delta c d u i H8 u1 x0 H12 x2 H17) (\lambda (b: B).(\lambda (u0: T).(pr2_free (CHead c (Bind b) u0) t1 x1 H13)))))))) H15)) (\lambda (H15: (ex2 T (\lambda (t2: T).(eq T x (THead (Bind Void) x0 t2))) (\lambda (t2: T).(subst0 (s (Bind Void) i) u x1 t2)))).(ex2_ind T (\lambda (t3: T).(eq T x (THead (Bind Void) x0 t3))) (\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3)) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) (\lambda (x2: T).(\lambda (H16: (eq T x (THead (Bind Void) x0 x2))).(\lambda (H17: (subst0 (s (Bind Void) i) u x1 x2)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x0 x2 H16 (pr2_free c u1 x0 H12) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 x1 H13 x2 H17)))))))) H15)) (\lambda (H15: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x (THead (Bind Void) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s (Bind Void) i) u x1 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i u x0 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s (Bind Void) i) u x1 t3))) (or (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x))))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H16: (eq T x (THead (Bind Void) x2 x3))).(\lambda (H17: (subst0 i u x0 x2)).(\lambda (H18: (subst0 (s (Bind Void) i) u x1 x3)).(or_introl (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) (ex3_2_intro T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3))))) x2 x3 H16 (pr2_delta c d u i H8 u1 x0 H12 x2 H17) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 x1 H13 x3 H18)))))))))) H15)) (subst0_gen_head (Bind Void) u x0 x1 x i H14)))))))) H11)) (\lambda (H: (pr0 t1 (lift (S O) O t2))).(or_intror (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq T x (THead (Bind Void) u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c u1 u2))) (\lambda (_: T).(\lambda (t3: T).(\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 t3)))))) (\forall (b: B).(\forall (u0: T).(pr2 (CHead c (Bind b) u0) t1 (lift (S O) O x)))) (\lambda (b: B).(\lambda (u0: T).(pr2_delta (CHead c (Bind b) u0) d u (S i) (getl_head (Bind b) i c (CHead d (Bind Abbr) u) H8 u0) t1 (lift (S O) O t2) H (lift (S O) O x) (subst0_lift_ge_S t2 x u i H10 O (le_O_n i))))))) (pr0_gen_void u1 t1 t2 H9))))) t (sym_eq T t x H7))) t0 (sym_eq T t0 (THead (Bind Void) u1 t1) H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T (THead (Bind Void) u1 t1)) (refl_equal T x))))))).
 
 theorem pr2_gen_lift:
  \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr2 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e t1 t2))))))))))
 \def
- \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (pr2 c (lift h d t1) x)).(\lambda (e: C).(\lambda (H0: (drop h d c e)).(let H1 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to ((eq T t (lift h d t1)) \to ((eq T t0 x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e t1 t2))))))))) with [(pr2_free c0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq T t0 (lift h d t1))).(\lambda (H4: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (lift h d t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))))))) (\lambda (H5: (eq T t0 (lift h d t1))).(eq_ind T (lift h d t1) (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))))) (\lambda (H6: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (lift h d t1) t) \to (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))))) (\lambda (H7: (pr0 (lift h d t1) x)).(ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x0: T).(\lambda (H: (eq T x (lift h d x0))).(\lambda (H8: (pr0 t1 x0)).(eq_ind_r T (lift h d x0) (\lambda (t: T).(ex2 T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))) (ex_intro2 T (\lambda (t3: T).(eq T (lift h d x0) (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x0 (refl_equal T (lift h d x0)) (pr2_free e t1 x0 H8)) x H)))) (pr0_gen_lift t1 x h d H7))) t2 (sym_eq T t2 x H6))) t0 (sym_eq T t0 (lift h d t1) H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d0 u i H1 t0 t2 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t0 (lift h d t1))).(\lambda (H6: (eq T t x)).(eq_ind C c (\lambda (c: C).((eq T t0 (lift h d t1)) \to ((eq T t x) \to ((getl i c (CHead d0 (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))))))))) (\lambda (H7: (eq T t0 (lift h d t1))).(eq_ind T (lift h d t1) (\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d0 (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (ex2 T (\lambda (t4: T).(eq T x (lift h d t4))) (\lambda (t4: T).(pr2 e t1 t4)))))))) (\lambda (H8: (eq T t x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d0 (Bind Abbr) u)) \to ((pr0 (lift h d t1) t2) \to ((subst0 i u t2 t3) \to (ex2 T (\lambda (t4: T).(eq T x (lift h d t4))) (\lambda (t4: T).(pr2 e t1 t4))))))) (\lambda (H9: (getl i c (CHead d0 (Bind Abbr) u))).(\lambda (H10: (pr0 (lift h d t1) t2)).(\lambda (H11: (subst0 i u t2 x)).(ex2_ind T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x0: T).(\lambda (H: (eq T t2 (lift h d x0))).(\lambda (H12: (pr0 t1 x0)).(let H13 \def (eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H11 (lift h d x0) H) in (lt_le_e i d (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (H14: (lt i d)).(let H15 \def (eq_ind nat d (\lambda (n: nat).(drop h n c e)) H0 (S (plus i (minus d (S i)))) (lt_plus_minus i d H14)) in (let H16 \def (eq_ind nat d (\lambda (n: nat).(subst0 i u (lift h n x0) x)) H13 (S (plus i (minus d (S i)))) (lt_plus_minus i d H14)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus d (S i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus d (S i)) d0 e0))) (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H0: (eq T u (lift h (minus d (S i)) x1))).(\lambda (H17: (getl i e (CHead x2 (Bind Abbr) x1))).(\lambda (_: (drop h (minus d (S i)) d0 x2)).(let H19 \def (eq_ind T u (\lambda (t: T).(subst0 i t (lift h (S (plus i (minus d (S i)))) x0) x)) H16 (lift h (minus d (S i)) x1) H0) in (ex2_ind T (\lambda (t3: T).(eq T x (lift h (S (plus i (minus d (S i)))) t3))) (\lambda (t3: T).(subst0 i x1 x0 t3)) (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x3: T).(\lambda (H20: (eq T x (lift h (S (plus i (minus d (S i)))) x3))).(\lambda (H21: (subst0 i x1 x0 x3)).(let H22 \def (eq_ind_r nat (S (plus i (minus d (S i)))) (\lambda (n: nat).(eq T x (lift h n x3))) H20 d (lt_plus_minus i d H14)) in (ex_intro2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x3 H22 (pr2_delta e x2 x1 i H17 t1 x0 H12 x3 H21)))))) (subst0_gen_lift_lt x1 x0 x i h (minus d (S i)) H19)))))))) (getl_drop_conf_lt Abbr c d0 u i H9 e h (minus d (S i)) H15))))) (\lambda (H14: (le d i)).(lt_le_e i (plus d h) (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (H15: (lt i (plus d h))).(subst0_gen_lift_false x0 u x h d i H14 H15 H13 (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))))) (\lambda (H15: (le (plus d h) i)).(ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u x0 t3)) (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x1: T).(\lambda (H16: (eq T x (lift h d x1))).(\lambda (H17: (subst0 (minus i h) u x0 x1)).(ex_intro2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x1 H16 (pr2_delta e d0 u (minus i h) (getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c H9 e h d H0 H15) t1 x0 H12 x1 H17))))) (subst0_gen_lift_ge u x0 x i h d H13 H15)))))))))) (pr0_gen_lift t1 t2 h d H10))))) t (sym_eq T t x H8))) t0 (sym_eq T t0 (lift h d t1) H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C c) (refl_equal T (lift h d t1)) (refl_equal T x)))))))))).
+ \lambda (c: C).(\lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (pr2 c (lift h d t1) x)).(\lambda (e: C).(\lambda (H0: (drop h d c e)).(let H1 \def (match H return (\lambda (_: ?).(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to ((eq T t (lift h d t1)) \to ((eq T t0 x) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr2 e t1 t2)))))))))) with [(pr2_free c0 t0 t2 H1) \Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq T t0 (lift h d t1))).(\lambda (H4: (eq T t2 x)).(eq_ind C c (\lambda (_: C).((eq T t0 (lift h d t1)) \to ((eq T t2 x) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))))))) (\lambda (H5: (eq T t0 (lift h d t1))).(eq_ind T (lift h d t1) (\lambda (t: T).((eq T t2 x) \to ((pr0 t t2) \to (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))))) (\lambda (H6: (eq T t2 x)).(eq_ind T x (\lambda (t: T).((pr0 (lift h d t1) t) \to (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))))) (\lambda (H7: (pr0 (lift h d t1) x)).(ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x0: T).(\lambda (H: (eq T x (lift h d x0))).(\lambda (H8: (pr0 t1 x0)).(eq_ind_r T (lift h d x0) (\lambda (t: T).(ex2 T (\lambda (t3: T).(eq T t (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)))) (ex_intro2 T (\lambda (t3: T).(eq T (lift h d x0) (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x0 (refl_equal T (lift h d x0)) (pr2_free e t1 x0 H8)) x H)))) (pr0_gen_lift t1 x h d H7))) t2 (sym_eq T t2 x H6))) t0 (sym_eq T t0 (lift h d t1) H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d0 u i H1 t0 t2 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t0 (lift h d t1))).(\lambda (H6: (eq T t x)).(eq_ind C c (\lambda (c: C).((eq T t0 (lift h d t1)) \to ((eq T t x) \to ((getl i c (CHead d0 (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))))))))) (\lambda (H7: (eq T t0 (lift h d t1))).(eq_ind T (lift h d t1) (\lambda (t3: T).((eq T t x) \to ((getl i c (CHead d0 (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t) \to (ex2 T (\lambda (t4: T).(eq T x (lift h d t4))) (\lambda (t4: T).(pr2 e t1 t4)))))))) (\lambda (H8: (eq T t x)).(eq_ind T x (\lambda (t3: T).((getl i c (CHead d0 (Bind Abbr) u)) \to ((pr0 (lift h d t1) t2) \to ((subst0 i u t2 t3) \to (ex2 T (\lambda (t4: T).(eq T x (lift h d t4))) (\lambda (t4: T).(pr2 e t1 t4))))))) (\lambda (H9: (getl i c (CHead d0 (Bind Abbr) u))).(\lambda (H10: (pr0 (lift h d t1) t2)).(\lambda (H11: (subst0 i u t2 x)).(ex2_ind T (\lambda (t3: T).(eq T t2 (lift h d t3))) (\lambda (t3: T).(pr0 t1 t3)) (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x0: T).(\lambda (H: (eq T t2 (lift h d x0))).(\lambda (H12: (pr0 t1 x0)).(let H13 \def (eq_ind T t2 (\lambda (t: T).(subst0 i u t x)) H11 (lift h d x0) H) in (lt_le_e i d (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (H14: (lt i d)).(let H15 \def (eq_ind nat d (\lambda (n: nat).(drop h n c e)) H0 (S (plus i (minus d (S i)))) (lt_plus_minus i d H14)) in (let H16 \def (eq_ind nat d (\lambda (n: nat).(subst0 i u (lift h n x0) x)) H13 (S (plus i (minus d (S i)))) (lt_plus_minus i d H14)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (minus d (S i)) v)))) (\lambda (v: T).(\lambda (e0: C).(getl i e (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (minus d (S i)) d0 e0))) (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x1: T).(\lambda (x2: C).(\lambda (H0: (eq T u (lift h (minus d (S i)) x1))).(\lambda (H17: (getl i e (CHead x2 (Bind Abbr) x1))).(\lambda (_: (drop h (minus d (S i)) d0 x2)).(let H19 \def (eq_ind T u (\lambda (t: T).(subst0 i t (lift h (S (plus i (minus d (S i)))) x0) x)) H16 (lift h (minus d (S i)) x1) H0) in (ex2_ind T (\lambda (t3: T).(eq T x (lift h (S (plus i (minus d (S i)))) t3))) (\lambda (t3: T).(subst0 i x1 x0 t3)) (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x3: T).(\lambda (H20: (eq T x (lift h (S (plus i (minus d (S i)))) x3))).(\lambda (H21: (subst0 i x1 x0 x3)).(let H22 \def (eq_ind_r nat (S (plus i (minus d (S i)))) (\lambda (n: nat).(eq T x (lift h n x3))) H20 d (lt_plus_minus i d H14)) in (ex_intro2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x3 H22 (pr2_delta e x2 x1 i H17 t1 x0 H12 x3 H21)))))) (subst0_gen_lift_lt x1 x0 x i h (minus d (S i)) H19)))))))) (getl_drop_conf_lt Abbr c d0 u i H9 e h (minus d (S i)) H15))))) (\lambda (H14: (le d i)).(lt_le_e i (plus d h) (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (H15: (lt i (plus d h))).(subst0_gen_lift_false x0 u x h d i H14 H15 H13 (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))))) (\lambda (H15: (le (plus d h) i)).(ex2_ind T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(subst0 (minus i h) u x0 t3)) (ex2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3))) (\lambda (x1: T).(\lambda (H16: (eq T x (lift h d x1))).(\lambda (H17: (subst0 (minus i h) u x0 x1)).(ex_intro2 T (\lambda (t3: T).(eq T x (lift h d t3))) (\lambda (t3: T).(pr2 e t1 t3)) x1 H16 (pr2_delta e d0 u (minus i h) (getl_drop_conf_ge i (CHead d0 (Bind Abbr) u) c H9 e h d H0 H15) t1 x0 H12 x1 H17))))) (subst0_gen_lift_ge u x0 x i h d H13 H15)))))))))) (pr0_gen_lift t1 t2 h d H10))))) t (sym_eq T t x H8))) t0 (sym_eq T t0 (lift h d t1) H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C c) (refl_equal T (lift h d t1)) (refl_equal T x)))))))))).
 
 theorem pr2_confluence__pr2_free_free:
  \forall (c: C).(\forall (t0: T).(\forall (t1: T).(\forall (t2: T).((pr0 t0 t1) \to ((pr0 t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))))))))
@@ -2104,7 +2104,7 @@ theorem pr2_confluence__pr2_delta_delta:
 theorem pr2_confluence:
  \forall (c: C).(\forall (t0: T).(\forall (t1: T).((pr2 c t0 t1) \to (\forall (t2: T).((pr2 c t0 t2) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))))))))
 \def
- \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0 t1)).(\lambda (t2: T).(\lambda (H0: (pr2 c t0 t2)).(let H1 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda (t3: T).((eq C c0 c) \to ((eq T t t0) \to ((eq T t3 t1) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))))))) with [(pr2_free c0 t3 t4 H1) \Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq T t3 t0)).(\lambda (H4: (eq T t4 t1)).(eq_ind C c (\lambda (_: C).((eq T t3 t0) \to ((eq T t4 t1) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))))))) (\lambda (H5: (eq T t3 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t4 t1) \to ((pr0 t t4) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0)))))) (\lambda (H6: (eq T t4 t1)).(eq_ind T t1 (\lambda (t: T).((pr0 t0 t) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))) (\lambda (H7: (pr0 t0 t1)).(let H8 \def (match H0 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t3: T).((eq C c0 c) \to ((eq T t t0) \to ((eq T t3 t2) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))))))) with [(pr2_free c1 t5 t6 H5) \Rightarrow (\lambda (H6: (eq C c1 c)).(\lambda (H8: (eq T t5 t0)).(\lambda (H9: (eq T t6 t2)).(eq_ind C c (\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))))))) (\lambda (H10: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t6 t2) \to ((pr0 t t6) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0)))))) (\lambda (H11: (eq T t6 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t0 t) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))) (\lambda (H12: (pr0 t0 t2)).(pr2_confluence__pr2_free_free c t0 t1 t2 H7 H12)) t6 (sym_eq T t6 t2 H11))) t5 (sym_eq T t5 t0 H10))) c1 (sym_eq C c1 c H6) H8 H9 H5)))) | (pr2_delta c1 d u i H5 t5 t6 H6 t H7) \Rightarrow (\lambda (H8: (eq C c1 c)).(\lambda (H9: (eq T t5 t0)).(\lambda (H10: (eq T t t2)).(eq_ind C c (\lambda (c0: C).((eq T t5 t0) \to ((eq T t t2) \to ((getl i c0 (CHead d (Bind Abbr) u)) \to ((pr0 t5 t6) \to ((subst0 i u t6 t) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))))))) (\lambda (H11: (eq T t5 t0)).(eq_ind T t0 (\lambda (t0: T).((eq T t t2) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t6) \to ((subst0 i u t6 t) \to (ex2 T (\lambda (t2: T).(pr2 c t1 t2)) (\lambda (t1: T).(pr2 c t2 t1)))))))) (\lambda (H12: (eq T t t2)).(eq_ind T t2 (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t6) \to ((subst0 i u t6 t3) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))))) (\lambda (H13: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H14: (pr0 t0 t6)).(\lambda (H15: (subst0 i u t6 t2)).(pr2_confluence__pr2_free_delta c d t0 t1 t2 t6 u i H7 H13 H14 H15)))) t (sym_eq T t t2 H12))) t5 (sym_eq T t5 t0 H11))) c1 (sym_eq C c1 c H8) H9 H10 H5 H6 H7))))]) in (H8 (refl_equal C c) (refl_equal T t0) (refl_equal T t2)))) t4 (sym_eq T t4 t1 H6))) t3 (sym_eq T t3 t0 H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t3 t4 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t3 t0)).(\lambda (H6: (eq T t t1)).(eq_ind C c (\lambda (c1: C).((eq T t3 t0) \to ((eq T t t1) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t3 t4) \to ((subst0 i u t4 t) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))))))) (\lambda (H7: (eq T t3 t0)).(eq_ind T t0 (\lambda (t0: T).((eq T t t1) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) \to ((subst0 i u t4 t) \to (ex2 T (\lambda (t2: T).(pr2 c t1 t2)) (\lambda (t1: T).(pr2 c t2 t1)))))))) (\lambda (H8: (eq T t t1)).(eq_ind T t1 (\lambda (t5: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) \to ((subst0 i u t4 t5) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))))) (\lambda (H9: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t0 t4)).(\lambda (H11: (subst0 i u t4 t1)).(let H12 \def (match H0 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t3: T).((eq C c0 c) \to ((eq T t t0) \to ((eq T t3 t2) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))))))) with [(pr2_free c1 t5 t6 H7) \Rightarrow (\lambda (H8: (eq C c1 c)).(\lambda (H12: (eq T t5 t0)).(\lambda (H13: (eq T t6 t2)).(eq_ind C c (\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))))))) (\lambda (H14: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t6 t2) \to ((pr0 t t6) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0)))))) (\lambda (H15: (eq T t6 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t0 t) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))) (\lambda (H16: (pr0 t0 t2)).(ex2_sym T (pr2 c t2) (pr2 c t1) (pr2_confluence__pr2_free_delta c d t0 t2 t1 t4 u i H16 H9 H10 H11))) t6 (sym_eq T t6 t2 H15))) t5 (sym_eq T t5 t0 H14))) c1 (sym_eq C c1 c H8) H12 H13 H7)))) | (pr2_delta c1 d0 u0 i0 H7 t5 t6 H8 t7 H9) \Rightarrow (\lambda (H12: (eq C c1 c)).(\lambda (H13: (eq T t5 t0)).(\lambda (H14: (eq T t7 t2)).(eq_ind C c (\lambda (c0: C).((eq T t5 t0) \to ((eq T t7 t2) \to ((getl i0 c0 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t5 t6) \to ((subst0 i0 u0 t6 t7) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))))))))) (\lambda (H15: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t7 t2) \to ((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t t6) \to ((subst0 i0 u0 t6 t7) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0)))))))) (\lambda (H16: (eq T t7 t2)).(eq_ind T t2 (\lambda (t: T).((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t6) \to ((subst0 i0 u0 t6 t) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))))) (\lambda (H17: (getl i0 c (CHead d0 (Bind Abbr) u0))).(\lambda (H18: (pr0 t0 t6)).(\lambda (H19: (subst0 i0 u0 t6 t2)).(pr2_confluence__pr2_delta_delta c d d0 t0 t1 t2 t4 t6 u u0 i i0 H9 H10 H11 H17 H18 H19)))) t7 (sym_eq T t7 t2 H16))) t5 (sym_eq T t5 t0 H15))) c1 (sym_eq C c1 c H12) H13 H14 H7 H8 H9))))]) in (H12 (refl_equal C c) (refl_equal T t0) (refl_equal T t2)))))) t (sym_eq T t t1 H8))) t3 (sym_eq T t3 t0 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C c) (refl_equal T t0) (refl_equal T t1)))))))).
+ \lambda (c: C).(\lambda (t0: T).(\lambda (t1: T).(\lambda (H: (pr2 c t0 t1)).(\lambda (t2: T).(\lambda (H0: (pr2 c t0 t2)).(let H1 \def (match H return (\lambda (_: ?).(\lambda (c0: C).(\lambda (t: T).(\lambda (t3: T).((eq C c0 c) \to ((eq T t t0) \to ((eq T t3 t1) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0)))))))))) with [(pr2_free c0 t3 t4 H1) \Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq T t3 t0)).(\lambda (H4: (eq T t4 t1)).(eq_ind C c (\lambda (_: C).((eq T t3 t0) \to ((eq T t4 t1) \to ((pr0 t3 t4) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))))))) (\lambda (H5: (eq T t3 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t4 t1) \to ((pr0 t t4) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0)))))) (\lambda (H6: (eq T t4 t1)).(eq_ind T t1 (\lambda (t: T).((pr0 t0 t) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))) (\lambda (H7: (pr0 t0 t1)).(let H8 \def (match H0 return (\lambda (_: ?).(\lambda (c0: C).(\lambda (t: T).(\lambda (t3: T).((eq C c0 c) \to ((eq T t t0) \to ((eq T t3 t2) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0)))))))))) with [(pr2_free c1 t5 t6 H5) \Rightarrow (\lambda (H6: (eq C c1 c)).(\lambda (H8: (eq T t5 t0)).(\lambda (H9: (eq T t6 t2)).(eq_ind C c (\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))))))) (\lambda (H10: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t6 t2) \to ((pr0 t t6) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0)))))) (\lambda (H11: (eq T t6 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t0 t) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))) (\lambda (H12: (pr0 t0 t2)).(pr2_confluence__pr2_free_free c t0 t1 t2 H7 H12)) t6 (sym_eq T t6 t2 H11))) t5 (sym_eq T t5 t0 H10))) c1 (sym_eq C c1 c H6) H8 H9 H5)))) | (pr2_delta c1 d u i H5 t5 t6 H6 t H7) \Rightarrow (\lambda (H8: (eq C c1 c)).(\lambda (H9: (eq T t5 t0)).(\lambda (H10: (eq T t t2)).(eq_ind C c (\lambda (c0: C).((eq T t5 t0) \to ((eq T t t2) \to ((getl i c0 (CHead d (Bind Abbr) u)) \to ((pr0 t5 t6) \to ((subst0 i u t6 t) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))))))) (\lambda (H11: (eq T t5 t0)).(eq_ind T t0 (\lambda (t0: T).((eq T t t2) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t6) \to ((subst0 i u t6 t) \to (ex2 T (\lambda (t2: T).(pr2 c t1 t2)) (\lambda (t1: T).(pr2 c t2 t1)))))))) (\lambda (H12: (eq T t t2)).(eq_ind T t2 (\lambda (t3: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t6) \to ((subst0 i u t6 t3) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))))) (\lambda (H13: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H14: (pr0 t0 t6)).(\lambda (H15: (subst0 i u t6 t2)).(pr2_confluence__pr2_free_delta c d t0 t1 t2 t6 u i H7 H13 H14 H15)))) t (sym_eq T t t2 H12))) t5 (sym_eq T t5 t0 H11))) c1 (sym_eq C c1 c H8) H9 H10 H5 H6 H7))))]) in (H8 (refl_equal C c) (refl_equal T t0) (refl_equal T t2)))) t4 (sym_eq T t4 t1 H6))) t3 (sym_eq T t3 t0 H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t3 t4 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t3 t0)).(\lambda (H6: (eq T t t1)).(eq_ind C c (\lambda (c1: C).((eq T t3 t0) \to ((eq T t t1) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t3 t4) \to ((subst0 i u t4 t) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))))))) (\lambda (H7: (eq T t3 t0)).(eq_ind T t0 (\lambda (t0: T).((eq T t t1) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) \to ((subst0 i u t4 t) \to (ex2 T (\lambda (t2: T).(pr2 c t1 t2)) (\lambda (t1: T).(pr2 c t2 t1)))))))) (\lambda (H8: (eq T t t1)).(eq_ind T t1 (\lambda (t5: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t4) \to ((subst0 i u t4 t5) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))))) (\lambda (H9: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t0 t4)).(\lambda (H11: (subst0 i u t4 t1)).(let H12 \def (match H0 return (\lambda (_: ?).(\lambda (c0: C).(\lambda (t: T).(\lambda (t3: T).((eq C c0 c) \to ((eq T t t0) \to ((eq T t3 t2) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0)))))))))) with [(pr2_free c1 t5 t6 H7) \Rightarrow (\lambda (H8: (eq C c1 c)).(\lambda (H12: (eq T t5 t0)).(\lambda (H13: (eq T t6 t2)).(eq_ind C c (\lambda (_: C).((eq T t5 t0) \to ((eq T t6 t2) \to ((pr0 t5 t6) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))))))) (\lambda (H14: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t6 t2) \to ((pr0 t t6) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0)))))) (\lambda (H15: (eq T t6 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t0 t) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))) (\lambda (H16: (pr0 t0 t2)).(ex2_sym T (pr2 c t2) (pr2 c t1) (pr2_confluence__pr2_free_delta c d t0 t2 t1 t4 u i H16 H9 H10 H11))) t6 (sym_eq T t6 t2 H15))) t5 (sym_eq T t5 t0 H14))) c1 (sym_eq C c1 c H8) H12 H13 H7)))) | (pr2_delta c1 d0 u0 i0 H7 t5 t6 H8 t7 H9) \Rightarrow (\lambda (H12: (eq C c1 c)).(\lambda (H13: (eq T t5 t0)).(\lambda (H14: (eq T t7 t2)).(eq_ind C c (\lambda (c0: C).((eq T t5 t0) \to ((eq T t7 t2) \to ((getl i0 c0 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t5 t6) \to ((subst0 i0 u0 t6 t7) \to (ex2 T (\lambda (t: T).(pr2 c t1 t)) (\lambda (t: T).(pr2 c t2 t))))))))) (\lambda (H15: (eq T t5 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t7 t2) \to ((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t t6) \to ((subst0 i0 u0 t6 t7) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0)))))))) (\lambda (H16: (eq T t7 t2)).(eq_ind T t2 (\lambda (t: T).((getl i0 c (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t6) \to ((subst0 i0 u0 t6 t) \to (ex2 T (\lambda (t0: T).(pr2 c t1 t0)) (\lambda (t0: T).(pr2 c t2 t0))))))) (\lambda (H17: (getl i0 c (CHead d0 (Bind Abbr) u0))).(\lambda (H18: (pr0 t0 t6)).(\lambda (H19: (subst0 i0 u0 t6 t2)).(pr2_confluence__pr2_delta_delta c d d0 t0 t1 t2 t4 t6 u u0 i i0 H9 H10 H11 H17 H18 H19)))) t7 (sym_eq T t7 t2 H16))) t5 (sym_eq T t5 t0 H15))) c1 (sym_eq C c1 c H12) H13 H14 H7 H8 H9))))]) in (H12 (refl_equal C c) (refl_equal T t0) (refl_equal T t2)))))) t (sym_eq T t t1 H8))) t3 (sym_eq T t3 t0 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C c) (refl_equal T t0) (refl_equal T t1)))))))).
 
 theorem pr2_delta1:
  \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c (CHead d (Bind Abbr) u)) \to (\forall (t1: T).(\forall (t2: T).((pr0 t1 t2) \to (\forall (t: T).((subst1 i u t2 t) \to (pr2 c t1 t))))))))))
@@ -2114,7 +2114,7 @@ theorem pr2_delta1:
 theorem pr2_subst1:
  \forall (c: C).(\forall (e: C).(\forall (v: T).(\forall (i: nat).((getl i c (CHead e (Bind Abbr) v)) \to (\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))))))
 \def
- \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 c t1 t2)).(let H1 \def (match H0 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to ((eq T t t1) \to ((eq T t0 t2) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C c (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))))) (\lambda (H6: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))) (\lambda (H7: (pr0 t1 t2)).(\lambda (w1: T).(\lambda (H0: (subst1 i v t1 w1)).(ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (x: T).(\lambda (H8: (pr0 w1 x)).(\lambda (H9: (subst1 i v t2 x)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)) x (pr2_free c w1 x H8) H9)))) (pr0_subst1 t1 t2 H7 v w1 i H0 v (pr0_refl v)))))) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d u i0 H1 t0 t3 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t0 t1)).(\lambda (H6: (eq T t t2)).(eq_ind C c (\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i0 c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i0 u t3 t) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))))) (\lambda (H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i0 c (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i0 u t3 t) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i0 c (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i0 u t3 t4) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))) (\lambda (H9: (getl i0 c (CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda (H11: (subst0 i0 u t3 t2)).(\lambda (w1: T).(\lambda (H0: (subst1 i v t1 w1)).(ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v t3 w2)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (x: T).(\lambda (H12: (pr0 w1 x)).(\lambda (H13: (subst1 i v t3 x)).(neq_eq_e i i0 (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (H14: (not (eq nat i i0))).(ex2_ind T (\lambda (t1: T).(subst1 i v t2 t1)) (\lambda (t1: T).(subst1 i0 u x t1)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (x0: T).(\lambda (H15: (subst1 i v t2 x0)).(\lambda (H16: (subst1 i0 u x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)) x0 (pr2_delta1 c d u i0 H9 w1 x H12 x0 H16) H15)))) (subst1_confluence_neq t3 t2 u i0 (subst1_single i0 u t3 t2 H11) x v i H13 (sym_not_eq nat i i0 H14)))) (\lambda (H14: (eq nat i i0)).(let H15 \def (eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u t3 t2)) H11 i H14) in (let H16 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d (Bind Abbr) u))) H9 i H14) in (let H17 \def (eq_ind C (CHead e (Bind Abbr) v) (\lambda (c0: C).(getl i c c0)) H (CHead d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H16)) in (let H18 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow e | (CHead c _ _) \Rightarrow c])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H16)) in ((let H19 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H16)) in (\lambda (H20: (eq C e d)).(let H21 \def (eq_ind_r T u (\lambda (t: T).(getl i c (CHead d (Bind Abbr) t))) H17 v H19) in (let H22 \def (eq_ind_r T u (\lambda (t: T).(subst0 i t t3 t2)) H15 v H19) in (let H23 \def (eq_ind_r C d (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) v))) H21 e H20) in (ex2_ind T (\lambda (t1: T).(subst1 i v t2 t1)) (\lambda (t1: T).(subst1 i v x t1)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (x0: T).(\lambda (H24: (subst1 i v t2 x0)).(\lambda (H25: (subst1 i v x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)) x0 (pr2_delta1 c e v i H23 w1 x H12 x0 H25) H24)))) (subst1_confluence_eq t3 t2 v i (subst1_single i v t3 t2 H22) x H13))))))) H18)))))))))) (pr0_subst1 t1 t3 H10 v w1 i H0 v (pr0_refl v)))))))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C c) (refl_equal T t1) (refl_equal T t2)))))))))).
+ \lambda (c: C).(\lambda (e: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H: (getl i c (CHead e (Bind Abbr) v))).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr2 c t1 t2)).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to ((eq T t t1) \to ((eq T t0 t2) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C c (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))))) (\lambda (H6: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))) (\lambda (H7: (pr0 t1 t2)).(\lambda (w1: T).(\lambda (H0: (subst1 i v t1 w1)).(ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (x: T).(\lambda (H8: (pr0 w1 x)).(\lambda (H9: (subst1 i v t2 x)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)) x (pr2_free c w1 x H8) H9)))) (pr0_subst1 t1 t2 H7 v w1 i H0 v (pr0_refl v)))))) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 c H2) H3 H4 H1)))) | (pr2_delta c0 d u i0 H1 t0 t3 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 c)).(\lambda (H5: (eq T t0 t1)).(\lambda (H6: (eq T t t2)).(eq_ind C c (\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i0 c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i0 u t3 t) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))))) (\lambda (H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i0 c (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i0 u t3 t) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)))))))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i0 c (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i0 u t3 t4) \to (\forall (w1: T).((subst1 i v t1 w1) \to (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))))))))) (\lambda (H9: (getl i0 c (CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda (H11: (subst0 i0 u t3 t2)).(\lambda (w1: T).(\lambda (H0: (subst1 i v t1 w1)).(ex2_ind T (\lambda (w2: T).(pr0 w1 w2)) (\lambda (w2: T).(subst1 i v t3 w2)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (x: T).(\lambda (H12: (pr0 w1 x)).(\lambda (H13: (subst1 i v t3 x)).(neq_eq_e i i0 (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (H14: (not (eq nat i i0))).(ex2_ind T (\lambda (t1: T).(subst1 i v t2 t1)) (\lambda (t1: T).(subst1 i0 u x t1)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (x0: T).(\lambda (H15: (subst1 i v t2 x0)).(\lambda (H16: (subst1 i0 u x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)) x0 (pr2_delta1 c d u i0 H9 w1 x H12 x0 H16) H15)))) (subst1_confluence_neq t3 t2 u i0 (subst1_single i0 u t3 t2 H11) x v i H13 (sym_not_eq nat i i0 H14)))) (\lambda (H14: (eq nat i i0)).(let H15 \def (eq_ind_r nat i0 (\lambda (n: nat).(subst0 n u t3 t2)) H11 i H14) in (let H16 \def (eq_ind_r nat i0 (\lambda (n: nat).(getl n c (CHead d (Bind Abbr) u))) H9 i H14) in (let H17 \def (eq_ind C (CHead e (Bind Abbr) v) (\lambda (c0: C).(getl i c c0)) H (CHead d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H16)) in (let H18 \def (f_equal C C (\lambda (e0: C).(match e0 return (\lambda (_: ?).C) with [(CSort _) \Rightarrow e | (CHead c _ _) \Rightarrow c])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H16)) in ((let H19 \def (f_equal C T (\lambda (e0: C).(match e0 return (\lambda (_: ?).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead e (Bind Abbr) v) (CHead d (Bind Abbr) u) (getl_mono c (CHead e (Bind Abbr) v) i H (CHead d (Bind Abbr) u) H16)) in (\lambda (H20: (eq C e d)).(let H21 \def (eq_ind_r T u (\lambda (t: T).(getl i c (CHead d (Bind Abbr) t))) H17 v H19) in (let H22 \def (eq_ind_r T u (\lambda (t: T).(subst0 i t t3 t2)) H15 v H19) in (let H23 \def (eq_ind_r C d (\lambda (c0: C).(getl i c (CHead c0 (Bind Abbr) v))) H21 e H20) in (ex2_ind T (\lambda (t1: T).(subst1 i v t2 t1)) (\lambda (t1: T).(subst1 i v x t1)) (ex2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2))) (\lambda (x0: T).(\lambda (H24: (subst1 i v t2 x0)).(\lambda (H25: (subst1 i v x x0)).(ex_intro2 T (\lambda (w2: T).(pr2 c w1 w2)) (\lambda (w2: T).(subst1 i v t2 w2)) x0 (pr2_delta1 c e v i H23 w1 x H12 x0 H25) H24)))) (subst1_confluence_eq t3 t2 v i (subst1_single i v t3 t2 H22) x H13))))))) H18)))))))))) (pr0_subst1 t1 t3 H10 v w1 i H0 v (pr0_refl v)))))))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C c) (refl_equal T t1) (refl_equal T t2)))))))))).
 
 theorem pr2_gen_cabbr:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 x2))))))))))))))))
@@ -2193,12 +2193,12 @@ theorem pr3_cflat:
 theorem pr3_pr0_pr2_t:
  \forall (u1: T).(\forall (u2: T).((pr0 u1 u2) \to (\forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 (CHead c k u1) t1 t2))))))))
 \def
- \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2 (CHead c k u2) t1 t2)).(let H1 \def (match H0 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c k u2)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr3 (CHead c k u1) t1 t2))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0 (CHead c k u2))).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C (CHead c k u2) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr3 (CHead c k u1) t1 t2))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr3 (CHead c k u1) t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr3 (CHead c k u1) t1 t2))) (\lambda (H7: (pr0 t1 t2)).(pr3_pr2 (CHead c k u1) t1 t2 (pr2_free (CHead c k u1) t1 t2 H7))) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 (CHead c k u2) H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t0 t3 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T t0 t1)).(\lambda (H6: (eq T t t2)).(eq_ind C (CHead c k u2) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr3 (CHead c k u1) t1 t2))))))) (\lambda (H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr3 (CHead c k u1) t1 t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr3 (CHead c k u1) t1 t2))))) (\lambda (H9: (getl i (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda (H11: (subst0 i u t3 t2)).(nat_ind (\lambda (n: nat).((getl n (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 n u t3 t2) \to (pr3 (CHead c k u1) t1 t2)))) (\lambda (H12: (getl O (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H13: (subst0 O u t3 t2)).(K_ind (\lambda (k: K).((getl O (CHead c k u2) (CHead d (Bind Abbr) u)) \to (pr3 (CHead c k u1) t1 t2))) (\lambda (b: B).(\lambda (H14: (getl O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(let H0 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H14))) in ((let H15 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H14))) in ((let H16 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H14))) in (\lambda (H17: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H19 \def (eq_ind T u (\lambda (t: T).(subst0 O t t3 t2)) H13 u2 H16) in (eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c (Bind b0) u1) t1 t2)) (ex2_ind T (\lambda (t1: T).(subst0 O u1 t3 t1)) (\lambda (t1: T).(pr0 t1 t2)) (pr3 (CHead c (Bind Abbr) u1) t1 t2) (\lambda (x: T).(\lambda (H20: (subst0 O u1 t3 x)).(\lambda (H21: (pr0 x t2)).(pr3_sing (CHead c (Bind Abbr) u1) x t1 (pr2_delta (CHead c (Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t1 t3 H10 x H20) t2 (pr3_pr2 (CHead c (Bind Abbr) u1) x t2 (pr2_free (CHead c (Bind Abbr) u1) x t2 H21)))))) (pr0_subst0_back u2 t3 t2 O H19 u1 H)) b H17))))) H15)) H0)))) (\lambda (f: F).(\lambda (H14: (getl O (CHead c (Flat f) u2) (CHead d (Bind Abbr) u))).(pr3_pr2 (CHead c (Flat f) u1) t1 t2 (pr2_cflat c t1 t2 (pr2_delta c d u O (getl_intro O c (CHead d (Bind Abbr) u) c (drop_refl c) (clear_gen_flat f c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Flat f) u2) (CHead d (Bind Abbr) u) H14))) t1 t3 H10 t2 H13) f u1)))) k H12))) (\lambda (i0: nat).(\lambda (IHi: (((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pr3 (CHead c k u1) t1 t2))))).(\lambda (H12: (getl (S i0) (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H13: (subst0 (S i0) u t3 t2)).(K_ind (\lambda (k: K).((getl (S i0) (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pr3 (CHead c k u1) t1 t2)))) \to (pr3 (CHead c k u1) t1 t2)))) (\lambda (b: B).(\lambda (H14: (getl (S i0) (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(\lambda (_: (((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pr3 (CHead c (Bind b) u1) t1 t2))))).(pr3_pr2 (CHead c (Bind b) u1) t1 t2 (pr2_delta (CHead c (Bind b) u1) d u (S i0) (getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c (CHead d (Bind Abbr) u) u2 i0 H14) u1) t1 t3 H10 t2 H13))))) (\lambda (f: F).(\lambda (H14: (getl (S i0) (CHead c (Flat f) u2) (CHead d (Bind Abbr) u))).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pr3 (CHead c (Flat f) u1) t1 t2))))).(pr3_pr2 (CHead c (Flat f) u1) t1 t2 (pr2_cflat c t1 t2 (pr2_delta c d u (r (Flat f) i0) (getl_gen_S (Flat f) c (CHead d (Bind Abbr) u) u2 i0 H14) t1 t3 H10 t2 H13) f u1))))) k H12 IHi))))) i H9 H11)))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0 (CHead c k u2) H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C (CHead c k u2)) (refl_equal T t1) (refl_equal T t2)))))))))).
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u1 u2)).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2 (CHead c k u2) t1 t2)).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c k u2)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pr3 (CHead c k u1) t1 t2)))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0 (CHead c k u2))).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C (CHead c k u2) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pr3 (CHead c k u1) t1 t2))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pr3 (CHead c k u1) t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pr3 (CHead c k u1) t1 t2))) (\lambda (H7: (pr0 t1 t2)).(pr3_pr2 (CHead c k u1) t1 t2 (pr2_free (CHead c k u1) t1 t2 H7))) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 (CHead c k u2) H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t0 t3 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T t0 t1)).(\lambda (H6: (eq T t t2)).(eq_ind C (CHead c k u2) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pr3 (CHead c k u1) t1 t2))))))) (\lambda (H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pr3 (CHead c k u1) t1 t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pr3 (CHead c k u1) t1 t2))))) (\lambda (H9: (getl i (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda (H11: (subst0 i u t3 t2)).(nat_ind (\lambda (n: nat).((getl n (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 n u t3 t2) \to (pr3 (CHead c k u1) t1 t2)))) (\lambda (H12: (getl O (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H13: (subst0 O u t3 t2)).(K_ind (\lambda (k: K).((getl O (CHead c k u2) (CHead d (Bind Abbr) u)) \to (pr3 (CHead c k u1) t1 t2))) (\lambda (b: B).(\lambda (H14: (getl O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(let H0 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H14))) in ((let H15 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H14))) in ((let H16 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Bind b) u2) (CHead d (Bind Abbr) u) H14))) in (\lambda (H17: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H19 \def (eq_ind T u (\lambda (t: T).(subst0 O t t3 t2)) H13 u2 H16) in (eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c (Bind b0) u1) t1 t2)) (ex2_ind T (\lambda (t1: T).(subst0 O u1 t3 t1)) (\lambda (t1: T).(pr0 t1 t2)) (pr3 (CHead c (Bind Abbr) u1) t1 t2) (\lambda (x: T).(\lambda (H20: (subst0 O u1 t3 x)).(\lambda (H21: (pr0 x t2)).(pr3_sing (CHead c (Bind Abbr) u1) x t1 (pr2_delta (CHead c (Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t1 t3 H10 x H20) t2 (pr3_pr2 (CHead c (Bind Abbr) u1) x t2 (pr2_free (CHead c (Bind Abbr) u1) x t2 H21)))))) (pr0_subst0_back u2 t3 t2 O H19 u1 H)) b H17))))) H15)) H0)))) (\lambda (f: F).(\lambda (H14: (getl O (CHead c (Flat f) u2) (CHead d (Bind Abbr) u))).(pr3_pr2 (CHead c (Flat f) u1) t1 t2 (pr2_cflat c t1 t2 (pr2_delta c d u O (getl_intro O c (CHead d (Bind Abbr) u) c (drop_refl c) (clear_gen_flat f c (CHead d (Bind Abbr) u) u2 (getl_gen_O (CHead c (Flat f) u2) (CHead d (Bind Abbr) u) H14))) t1 t3 H10 t2 H13) f u1)))) k H12))) (\lambda (i0: nat).(\lambda (IHi: (((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pr3 (CHead c k u1) t1 t2))))).(\lambda (H12: (getl (S i0) (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H13: (subst0 (S i0) u t3 t2)).(K_ind (\lambda (k: K).((getl (S i0) (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pr3 (CHead c k u1) t1 t2)))) \to (pr3 (CHead c k u1) t1 t2)))) (\lambda (b: B).(\lambda (H14: (getl (S i0) (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(\lambda (_: (((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pr3 (CHead c (Bind b) u1) t1 t2))))).(pr3_pr2 (CHead c (Bind b) u1) t1 t2 (pr2_delta (CHead c (Bind b) u1) d u (S i0) (getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) (getl_gen_S (Bind b) c (CHead d (Bind Abbr) u) u2 i0 H14) u1) t1 t3 H10 t2 H13))))) (\lambda (f: F).(\lambda (H14: (getl (S i0) (CHead c (Flat f) u2) (CHead d (Bind Abbr) u))).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pr3 (CHead c (Flat f) u1) t1 t2))))).(pr3_pr2 (CHead c (Flat f) u1) t1 t2 (pr2_cflat c t1 t2 (pr2_delta c d u (r (Flat f) i0) (getl_gen_S (Flat f) c (CHead d (Bind Abbr) u) u2 i0 H14) t1 t3 H10 t2 H13) f u1))))) k H12 IHi))))) i H9 H11)))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0 (CHead c k u2) H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C (CHead c k u2)) (refl_equal T t1) (refl_equal T t2)))))))))).
 
 theorem pr3_pr2_pr2_t:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u1 u2) \to (\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 (CHead c k u1) t1 t2))))))))
 \def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1 u2)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to ((eq T t u1) \to ((eq T t0 u2) \to (\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 (CHead c k u1) t1 t2))))))))))) with [(pr2_free c0 t1 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1 u1)).(\lambda (H3: (eq T t2 u2)).(eq_ind C c (\lambda (_: C).((eq T t1 u1) \to ((eq T t2 u2) \to ((pr0 t1 t2) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4))))))))) (\lambda (H4: (eq T t1 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t2 u2) \to ((pr0 t t2) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4)))))))) (\lambda (H5: (eq T t2 u2)).(eq_ind T u2 (\lambda (t: T).((pr0 u1 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4))))))) (\lambda (H6: (pr0 u1 u2)).(\lambda (t3: T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H: (pr2 (CHead c k u2) t3 t4)).(pr3_pr0_pr2_t u1 u2 H6 c t3 t4 k H)))))) t2 (sym_eq T t2 u2 H5))) t1 (sym_eq T t1 u1 H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t1 u1)).(\lambda (H5: (eq T t u2)).(eq_ind C c (\lambda (c1: C).((eq T t1 u1) \to ((eq T t u2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1 t2) \to ((subst0 i u t2 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4))))))))))) (\lambda (H6: (eq T t1 u1)).(eq_ind T u1 (\lambda (t0: T).((eq T t u2) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4)))))))))) (\lambda (H7: (eq T t u2)).(eq_ind T u2 (\lambda (t0: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 u1 t2) \to ((subst0 i u t2 t0) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 u1 t2)).(\lambda (H10: (subst0 i u t2 u2)).(\lambda (t3: T).(\lambda (t0: T).(\lambda (k: K).(\lambda (H: (pr2 (CHead c k u2) t3 t0)).(let H11 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda (t1: T).((eq C c0 (CHead c k u2)) \to ((eq T t t3) \to ((eq T t1 t0) \to (pr3 (CHead c k u1) t3 t0))))))) with [(pr2_free c0 t3 t4 H3) \Rightarrow (\lambda (H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T t3 t3)).(\lambda (H6: (eq T t4 t0)).(eq_ind C (CHead c k u2) (\lambda (_: C).((eq T t3 t3) \to ((eq T t4 t0) \to ((pr0 t3 t4) \to (pr3 (CHead c k u1) t3 t0))))) (\lambda (H7: (eq T t3 t3)).(eq_ind T t3 (\lambda (t: T).((eq T t4 t0) \to ((pr0 t t4) \to (pr3 (CHead c k u1) t3 t0)))) (\lambda (H8: (eq T t4 t0)).(eq_ind T t0 (\lambda (t: T).((pr0 t3 t) \to (pr3 (CHead c k u1) t3 t0))) (\lambda (H9: (pr0 t3 t0)).(pr3_pr2 (CHead c k u1) t3 t0 (pr2_free (CHead c k u1) t3 t0 H9))) t4 (sym_eq T t4 t0 H8))) t3 (sym_eq T t3 t3 H7))) c0 (sym_eq C c0 (CHead c k u2) H4) H5 H6 H3)))) | (pr2_delta c0 d0 u0 i0 H3 t3 t4 H4 t H5) \Rightarrow (\lambda (H6: (eq C c0 (CHead c k u2))).(\lambda (H7: (eq T t3 t3)).(\lambda (H11: (eq T t t0)).(eq_ind C (CHead c k u2) (\lambda (c1: C).((eq T t3 t3) \to ((eq T t t0) \to ((getl i0 c1 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t3 t4) \to ((subst0 i0 u0 t4 t) \to (pr3 (CHead c k u1) t3 t0))))))) (\lambda (H12: (eq T t3 t3)).(eq_ind T t3 (\lambda (t1: T).((eq T t t0) \to ((getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to ((pr0 t1 t4) \to ((subst0 i0 u0 t4 t) \to (pr3 (CHead c k u1) t3 t0)))))) (\lambda (H13: (eq T t t0)).(eq_ind T t0 (\lambda (t1: T).((getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to ((pr0 t3 t4) \to ((subst0 i0 u0 t4 t1) \to (pr3 (CHead c k u1) t3 t0))))) (\lambda (H14: (getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0))).(\lambda (H15: (pr0 t3 t4)).(\lambda (H16: (subst0 i0 u0 t4 t0)).((match i0 return (\lambda (n: nat).((getl n (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t4 t0) \to (pr3 (CHead c k u1) t3 t0)))) with [O \Rightarrow (\lambda (H17: (getl O (CHead c k u2) (CHead d0 (Bind Abbr) u0))).(\lambda (H18: (subst0 O u0 t4 t0)).((match k return (\lambda (k: K).((clear (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to (pr3 (CHead c k u1) t3 t0))) with [(Bind b) \Rightarrow (\lambda (H19: (clear (CHead c (Bind b) u2) (CHead d0 (Bind Abbr) u0))).(let H \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d0 | (CHead c _ _) \Rightarrow c])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H19)) in ((let H0 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H19)) in ((let H1 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H19)) in (\lambda (H20: (eq B Abbr b)).(\lambda (_: (eq C d0 c)).(let H22 \def (eq_ind T u0 (\lambda (t: T).(subst0 O t t4 t0)) H18 u2 H1) in (eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c (Bind b0) u1) t3 t0)) (ex2_ind T (\lambda (t0: T).(subst0 O t2 t4 t0)) (\lambda (t1: T).(subst0 (S (plus i O)) u t1 t0)) (pr3 (CHead c (Bind Abbr) u1) t3 t0) (\lambda (x: T).(\lambda (H2: (subst0 O t2 t4 x)).(\lambda (H10: (subst0 (S (plus i O)) u x t0)).(let H23 \def (f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H24 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n u x t0)) H10 (S i) H23) in (ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(pr0 t0 x)) (pr3 (CHead c (Bind Abbr) u1) t3 t0) (\lambda (x0: T).(\lambda (H9: (subst0 O u1 t4 x0)).(\lambda (H25: (pr0 x0 x)).(pr3_sing (CHead c (Bind Abbr) u1) x0 t3 (pr2_delta (CHead c (Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t3 t4 H15 x0 H9) t0 (pr3_pr2 (CHead c (Bind Abbr) u1) x0 t0 (pr2_delta (CHead c (Bind Abbr) u1) d u (S i) (getl_clear_bind Abbr (CHead c (Bind Abbr) u1) c u1 (clear_bind Abbr c u1) (CHead d (Bind Abbr) u) i H8) x0 x H25 t0 H24)))))) (pr0_subst0_back t2 t4 x O H2 u1 H9))))))) (subst0_subst0 t4 t0 u2 O H22 t2 u i H10)) b H20))))) H0)) H))) | (Flat f) \Rightarrow (\lambda (H8: (clear (CHead c (Flat f) u2) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c (Flat f) u1) t3 t0 (pr2_cflat c t3 t0 (pr2_delta c d0 u0 O (getl_intro O c (CHead d0 (Bind Abbr) u0) c (drop_refl c) (clear_gen_flat f c (CHead d0 (Bind Abbr) u0) u2 H8)) t3 t4 H15 t0 H18) f u1)))]) (getl_gen_O (CHead c k u2) (CHead d0 (Bind Abbr) u0) H17)))) | (S n) \Rightarrow (\lambda (H8: (getl (S n) (CHead c k u2) (CHead d0 (Bind Abbr) u0))).(\lambda (H9: (subst0 (S n) u0 t4 t0)).((match k return (\lambda (k: K).((getl (S n) (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to (pr3 (CHead c k u1) t3 t0))) with [(Bind b) \Rightarrow (\lambda (H10: (getl (S n) (CHead c (Bind b) u2) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c (Bind b) u1) t3 t0 (pr2_delta (CHead c (Bind b) u1) d0 u0 (S n) (getl_head (Bind b) n c (CHead d0 (Bind Abbr) u0) (getl_gen_S (Bind b) c (CHead d0 (Bind Abbr) u0) u2 n H10) u1) t3 t4 H15 t0 H9))) | (Flat f) \Rightarrow (\lambda (H10: (getl (S n) (CHead c (Flat f) u2) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c (Flat f) u1) t3 t0 (pr2_cflat c t3 t0 (pr2_delta c d0 u0 (r (Flat f) n) (getl_gen_S (Flat f) c (CHead d0 (Bind Abbr) u0) u2 n H10) t3 t4 H15 t0 H9) f u1)))]) H8)))]) H14 H16)))) t (sym_eq T t t0 H13))) t3 (sym_eq T t3 t3 H12))) c0 (sym_eq C c0 (CHead c k u2) H6) H7 H11 H3 H4 H5))))]) in (H11 (refl_equal C (CHead c k u2)) (refl_equal T t3) (refl_equal T t0)))))))))) t (sym_eq T t u2 H7))) t1 (sym_eq T t1 u1 H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T u1) (refl_equal T u2)))))).
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u1 u2)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to ((eq T t u1) \to ((eq T t0 u2) \to (\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pr3 (CHead c k u1) t1 t2)))))))))))) with [(pr2_free c0 t1 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1 u1)).(\lambda (H3: (eq T t2 u2)).(eq_ind C c (\lambda (_: C).((eq T t1 u1) \to ((eq T t2 u2) \to ((pr0 t1 t2) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4))))))))) (\lambda (H4: (eq T t1 u1)).(eq_ind T u1 (\lambda (t: T).((eq T t2 u2) \to ((pr0 t t2) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4)))))))) (\lambda (H5: (eq T t2 u2)).(eq_ind T u2 (\lambda (t: T).((pr0 u1 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4))))))) (\lambda (H6: (pr0 u1 u2)).(\lambda (t3: T).(\lambda (t4: T).(\lambda (k: K).(\lambda (H: (pr2 (CHead c k u2) t3 t4)).(pr3_pr0_pr2_t u1 u2 H6 c t3 t4 k H)))))) t2 (sym_eq T t2 u2 H5))) t1 (sym_eq T t1 u1 H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t1 u1)).(\lambda (H5: (eq T t u2)).(eq_ind C c (\lambda (c1: C).((eq T t1 u1) \to ((eq T t u2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1 t2) \to ((subst0 i u t2 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4))))))))))) (\lambda (H6: (eq T t1 u1)).(eq_ind T u1 (\lambda (t0: T).((eq T t u2) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4)))))))))) (\lambda (H7: (eq T t u2)).(eq_ind T u2 (\lambda (t0: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 u1 t2) \to ((subst0 i u t2 t0) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pr3 (CHead c k u1) t3 t4))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 u1 t2)).(\lambda (H10: (subst0 i u t2 u2)).(\lambda (t3: T).(\lambda (t0: T).(\lambda (k: K).(\lambda (H: (pr2 (CHead c k u2) t3 t0)).(let H11 \def (match H return (\lambda (_: ?).(\lambda (c0: C).(\lambda (t: T).(\lambda (t1: T).((eq C c0 (CHead c k u2)) \to ((eq T t t3) \to ((eq T t1 t0) \to (pr3 (CHead c k u1) t3 t0)))))))) with [(pr2_free c0 t3 t4 H3) \Rightarrow (\lambda (H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T t3 t3)).(\lambda (H6: (eq T t4 t0)).(eq_ind C (CHead c k u2) (\lambda (_: C).((eq T t3 t3) \to ((eq T t4 t0) \to ((pr0 t3 t4) \to (pr3 (CHead c k u1) t3 t0))))) (\lambda (H7: (eq T t3 t3)).(eq_ind T t3 (\lambda (t: T).((eq T t4 t0) \to ((pr0 t t4) \to (pr3 (CHead c k u1) t3 t0)))) (\lambda (H8: (eq T t4 t0)).(eq_ind T t0 (\lambda (t: T).((pr0 t3 t) \to (pr3 (CHead c k u1) t3 t0))) (\lambda (H9: (pr0 t3 t0)).(pr3_pr2 (CHead c k u1) t3 t0 (pr2_free (CHead c k u1) t3 t0 H9))) t4 (sym_eq T t4 t0 H8))) t3 (sym_eq T t3 t3 H7))) c0 (sym_eq C c0 (CHead c k u2) H4) H5 H6 H3)))) | (pr2_delta c0 d0 u0 i0 H3 t3 t4 H4 t H5) \Rightarrow (\lambda (H6: (eq C c0 (CHead c k u2))).(\lambda (H7: (eq T t3 t3)).(\lambda (H11: (eq T t t0)).(eq_ind C (CHead c k u2) (\lambda (c1: C).((eq T t3 t3) \to ((eq T t t0) \to ((getl i0 c1 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t3 t4) \to ((subst0 i0 u0 t4 t) \to (pr3 (CHead c k u1) t3 t0))))))) (\lambda (H12: (eq T t3 t3)).(eq_ind T t3 (\lambda (t1: T).((eq T t t0) \to ((getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to ((pr0 t1 t4) \to ((subst0 i0 u0 t4 t) \to (pr3 (CHead c k u1) t3 t0)))))) (\lambda (H13: (eq T t t0)).(eq_ind T t0 (\lambda (t1: T).((getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to ((pr0 t3 t4) \to ((subst0 i0 u0 t4 t1) \to (pr3 (CHead c k u1) t3 t0))))) (\lambda (H14: (getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0))).(\lambda (H15: (pr0 t3 t4)).(\lambda (H16: (subst0 i0 u0 t4 t0)).((match i0 return (\lambda (_: ?).(\lambda (n: nat).((getl n (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t4 t0) \to (pr3 (CHead c k u1) t3 t0))))) with [O \Rightarrow (\lambda (H17: (getl O (CHead c k u2) (CHead d0 (Bind Abbr) u0))).(\lambda (H18: (subst0 O u0 t4 t0)).((match k return (\lambda (_: ?).(\lambda (k: K).((clear (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to (pr3 (CHead c k u1) t3 t0)))) with [(Bind b) \Rightarrow (\lambda (H19: (clear (CHead c (Bind b) u2) (CHead d0 (Bind Abbr) u0))).(let H \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d0 | (CHead c _ _) \Rightarrow c])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H19)) in ((let H0 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H19)) in ((let H1 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H19)) in (\lambda (H20: (eq B Abbr b)).(\lambda (_: (eq C d0 c)).(let H22 \def (eq_ind T u0 (\lambda (t: T).(subst0 O t t4 t0)) H18 u2 H1) in (eq_ind B Abbr (\lambda (b0: B).(pr3 (CHead c (Bind b0) u1) t3 t0)) (ex2_ind T (\lambda (t0: T).(subst0 O t2 t4 t0)) (\lambda (t1: T).(subst0 (S (plus i O)) u t1 t0)) (pr3 (CHead c (Bind Abbr) u1) t3 t0) (\lambda (x: T).(\lambda (H2: (subst0 O t2 t4 x)).(\lambda (H10: (subst0 (S (plus i O)) u x t0)).(let H23 \def (f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H24 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n u x t0)) H10 (S i) H23) in (ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(pr0 t0 x)) (pr3 (CHead c (Bind Abbr) u1) t3 t0) (\lambda (x0: T).(\lambda (H9: (subst0 O u1 t4 x0)).(\lambda (H25: (pr0 x0 x)).(pr3_sing (CHead c (Bind Abbr) u1) x0 t3 (pr2_delta (CHead c (Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t3 t4 H15 x0 H9) t0 (pr3_pr2 (CHead c (Bind Abbr) u1) x0 t0 (pr2_delta (CHead c (Bind Abbr) u1) d u (S i) (getl_clear_bind Abbr (CHead c (Bind Abbr) u1) c u1 (clear_bind Abbr c u1) (CHead d (Bind Abbr) u) i H8) x0 x H25 t0 H24)))))) (pr0_subst0_back t2 t4 x O H2 u1 H9))))))) (subst0_subst0 t4 t0 u2 O H22 t2 u i H10)) b H20))))) H0)) H))) | (Flat f) \Rightarrow (\lambda (H8: (clear (CHead c (Flat f) u2) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c (Flat f) u1) t3 t0 (pr2_cflat c t3 t0 (pr2_delta c d0 u0 O (getl_intro O c (CHead d0 (Bind Abbr) u0) c (drop_refl c) (clear_gen_flat f c (CHead d0 (Bind Abbr) u0) u2 H8)) t3 t4 H15 t0 H18) f u1)))]) (getl_gen_O (CHead c k u2) (CHead d0 (Bind Abbr) u0) H17)))) | (S n) \Rightarrow (\lambda (H8: (getl (S n) (CHead c k u2) (CHead d0 (Bind Abbr) u0))).(\lambda (H9: (subst0 (S n) u0 t4 t0)).((match k return (\lambda (_: ?).(\lambda (k: K).((getl (S n) (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to (pr3 (CHead c k u1) t3 t0)))) with [(Bind b) \Rightarrow (\lambda (H10: (getl (S n) (CHead c (Bind b) u2) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c (Bind b) u1) t3 t0 (pr2_delta (CHead c (Bind b) u1) d0 u0 (S n) (getl_head (Bind b) n c (CHead d0 (Bind Abbr) u0) (getl_gen_S (Bind b) c (CHead d0 (Bind Abbr) u0) u2 n H10) u1) t3 t4 H15 t0 H9))) | (Flat f) \Rightarrow (\lambda (H10: (getl (S n) (CHead c (Flat f) u2) (CHead d0 (Bind Abbr) u0))).(pr3_pr2 (CHead c (Flat f) u1) t3 t0 (pr2_cflat c t3 t0 (pr2_delta c d0 u0 (r (Flat f) n) (getl_gen_S (Flat f) c (CHead d0 (Bind Abbr) u0) u2 n H10) t3 t4 H15 t0 H9) f u1)))]) H8)))]) H14 H16)))) t (sym_eq T t t0 H13))) t3 (sym_eq T t3 t3 H12))) c0 (sym_eq C c0 (CHead c k u2) H6) H7 H11 H3 H4 H5))))]) in (H11 (refl_equal C (CHead c k u2)) (refl_equal T t3) (refl_equal T t0)))))))))) t (sym_eq T t u2 H7))) t1 (sym_eq T t1 u1 H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T u1) (refl_equal T u2)))))).
 
 theorem pr3_pr2_pr3_t:
  \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u1 u2) \to (pr3 (CHead c k u1) t1 t2))))))))
@@ -2218,7 +2218,7 @@ theorem pr3_lift:
 theorem pr3_wcpr0_t:
  \forall (c1: C).(\forall (c2: C).((wcpr0 c2 c1) \to (\forall (t1: T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pr3 c2 t1 t2))))))
 \def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind (\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c0 t1 t2) \to (pr3 c t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).H0)))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1: T).(\forall (t2: T).((pr3 c3 t1 t2) \to (pr3 c0 t1 t2)))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H3: (pr3 (CHead c3 k u2) t1 t2)).(pr3_ind (CHead c3 k u1) (\lambda (t: T).(\lambda (t0: T).(pr3 (CHead c0 k u1) t t0))) (\lambda (t: T).(pr3_refl (CHead c0 k u1) t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H4: (pr2 (CHead c3 k u1) t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead c3 k u1) t0 t4)).(\lambda (H6: (pr3 (CHead c0 k u1) t0 t4)).(pr3_t t0 t3 (CHead c0 k u1) (let H7 \def (match H4 return (\lambda (c: C).(\lambda (t: T).(\lambda (t1: T).((eq C c (CHead c3 k u1)) \to ((eq T t t3) \to ((eq T t1 t0) \to (pr3 (CHead c0 k u1) t3 t0))))))) with [(pr2_free c t1 t2 H2) \Rightarrow (\lambda (H3: (eq C c (CHead c3 k u1))).(\lambda (H4: (eq T t1 t3)).(\lambda (H5: (eq T t2 t0)).(eq_ind C (CHead c3 k u1) (\lambda (_: C).((eq T t1 t3) \to ((eq T t2 t0) \to ((pr0 t1 t2) \to (pr3 (CHead c0 k u1) t3 t0))))) (\lambda (H6: (eq T t1 t3)).(eq_ind T t3 (\lambda (t: T).((eq T t2 t0) \to ((pr0 t t2) \to (pr3 (CHead c0 k u1) t3 t0)))) (\lambda (H7: (eq T t2 t0)).(eq_ind T t0 (\lambda (t: T).((pr0 t3 t) \to (pr3 (CHead c0 k u1) t3 t0))) (\lambda (H8: (pr0 t3 t0)).(pr3_pr2 (CHead c0 k u1) t3 t0 (pr2_free (CHead c0 k u1) t3 t0 H8))) t2 (sym_eq T t2 t0 H7))) t1 (sym_eq T t1 t3 H6))) c (sym_eq C c (CHead c3 k u1) H3) H4 H5 H2)))) | (pr2_delta c d u i H2 t1 t2 H3 t H4) \Rightarrow (\lambda (H5: (eq C c (CHead c3 k u1))).(\lambda (H6: (eq T t1 t3)).(\lambda (H7: (eq T t t0)).(eq_ind C (CHead c3 k u1) (\lambda (c1: C).((eq T t1 t3) \to ((eq T t t0) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1 t2) \to ((subst0 i u t2 t) \to (pr3 (CHead c0 k u1) t3 t0))))))) (\lambda (H8: (eq T t1 t3)).(eq_ind T t3 (\lambda (t4: T).((eq T t t0) \to ((getl i (CHead c3 k u1) (CHead d (Bind Abbr) u)) \to ((pr0 t4 t2) \to ((subst0 i u t2 t) \to (pr3 (CHead c0 k u1) t3 t0)))))) (\lambda (H9: (eq T t t0)).(eq_ind T t0 (\lambda (t4: T).((getl i (CHead c3 k u1) (CHead d (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t4) \to (pr3 (CHead c0 k u1) t3 t0))))) (\lambda (H10: (getl i (CHead c3 k u1) (CHead d (Bind Abbr) u))).(\lambda (H11: (pr0 t3 t2)).(\lambda (H12: (subst0 i u t2 t0)).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u2: T).(getl i (CHead c0 k u1) (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 d))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u))) (pr3 (CHead c0 k u1) t3 t0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H1: (getl i (CHead c0 k u1) (CHead x0 (Bind Abbr) x1))).(\lambda (_: (wcpr0 x0 d)).(\lambda (H14: (pr0 x1 u)).(ex2_ind T (\lambda (t0: T).(subst0 i x1 t2 t0)) (\lambda (t3: T).(pr0 t3 t0)) (pr3 (CHead c0 k u1) t3 t0) (\lambda (x: T).(\lambda (H15: (subst0 i x1 t2 x)).(\lambda (H16: (pr0 x t0)).(pr3_sing (CHead c0 k u1) x t3 (pr2_delta (CHead c0 k u1) x0 x1 i H1 t3 t2 H11 x H15) t0 (pr3_pr2 (CHead c0 k u1) x t0 (pr2_free (CHead c0 k u1) x t0 H16)))))) (pr0_subst0_back u t2 t0 i H12 x1 H14))))))) (wcpr0_getl_back (CHead c3 k u1) (CHead c0 k u1) (wcpr0_comp c0 c3 H0 u1 u1 (pr0_refl u1) k) i d u (Bind Abbr) H10))))) t (sym_eq T t t0 H9))) t1 (sym_eq T t1 t3 H8))) c (sym_eq C c (CHead c3 k u1) H5) H6 H7 H2 H3 H4))))]) in (H7 (refl_equal C (CHead c3 k u1)) (refl_equal T t3) (refl_equal T t0))) t4 H6))))))) t1 t2 (pr3_pr2_pr3_t c3 u2 t1 t2 k H3 u1 (pr2_free c3 u1 u2 H2)))))))))))))) c2 c1 H))).
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c2 c1)).(wcpr0_ind (\lambda (c: C).(\lambda (c0: C).(\forall (t1: T).(\forall (t2: T).((pr3 c0 t1 t2) \to (pr3 c t1 t2)))))) (\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 c t1 t2)).H0)))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H0: (wcpr0 c0 c3)).(\lambda (_: ((\forall (t1: T).(\forall (t2: T).((pr3 c3 t1 t2) \to (pr3 c0 t1 t2)))))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H2: (pr0 u1 u2)).(\lambda (k: K).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H3: (pr3 (CHead c3 k u2) t1 t2)).(pr3_ind (CHead c3 k u1) (\lambda (t: T).(\lambda (t0: T).(pr3 (CHead c0 k u1) t t0))) (\lambda (t: T).(pr3_refl (CHead c0 k u1) t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H4: (pr2 (CHead c3 k u1) t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead c3 k u1) t0 t4)).(\lambda (H6: (pr3 (CHead c0 k u1) t0 t4)).(pr3_t t0 t3 (CHead c0 k u1) (let H7 \def (match H4 return (\lambda (_: ?).(\lambda (c: C).(\lambda (t: T).(\lambda (t1: T).((eq C c (CHead c3 k u1)) \to ((eq T t t3) \to ((eq T t1 t0) \to (pr3 (CHead c0 k u1) t3 t0)))))))) with [(pr2_free c t1 t2 H2) \Rightarrow (\lambda (H3: (eq C c (CHead c3 k u1))).(\lambda (H4: (eq T t1 t3)).(\lambda (H5: (eq T t2 t0)).(eq_ind C (CHead c3 k u1) (\lambda (_: C).((eq T t1 t3) \to ((eq T t2 t0) \to ((pr0 t1 t2) \to (pr3 (CHead c0 k u1) t3 t0))))) (\lambda (H6: (eq T t1 t3)).(eq_ind T t3 (\lambda (t: T).((eq T t2 t0) \to ((pr0 t t2) \to (pr3 (CHead c0 k u1) t3 t0)))) (\lambda (H7: (eq T t2 t0)).(eq_ind T t0 (\lambda (t: T).((pr0 t3 t) \to (pr3 (CHead c0 k u1) t3 t0))) (\lambda (H8: (pr0 t3 t0)).(pr3_pr2 (CHead c0 k u1) t3 t0 (pr2_free (CHead c0 k u1) t3 t0 H8))) t2 (sym_eq T t2 t0 H7))) t1 (sym_eq T t1 t3 H6))) c (sym_eq C c (CHead c3 k u1) H3) H4 H5 H2)))) | (pr2_delta c d u i H2 t1 t2 H3 t H4) \Rightarrow (\lambda (H5: (eq C c (CHead c3 k u1))).(\lambda (H6: (eq T t1 t3)).(\lambda (H7: (eq T t t0)).(eq_ind C (CHead c3 k u1) (\lambda (c1: C).((eq T t1 t3) \to ((eq T t t0) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1 t2) \to ((subst0 i u t2 t) \to (pr3 (CHead c0 k u1) t3 t0))))))) (\lambda (H8: (eq T t1 t3)).(eq_ind T t3 (\lambda (t4: T).((eq T t t0) \to ((getl i (CHead c3 k u1) (CHead d (Bind Abbr) u)) \to ((pr0 t4 t2) \to ((subst0 i u t2 t) \to (pr3 (CHead c0 k u1) t3 t0)))))) (\lambda (H9: (eq T t t0)).(eq_ind T t0 (\lambda (t4: T).((getl i (CHead c3 k u1) (CHead d (Bind Abbr) u)) \to ((pr0 t3 t2) \to ((subst0 i u t2 t4) \to (pr3 (CHead c0 k u1) t3 t0))))) (\lambda (H10: (getl i (CHead c3 k u1) (CHead d (Bind Abbr) u))).(\lambda (H11: (pr0 t3 t2)).(\lambda (H12: (subst0 i u t2 t0)).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u2: T).(getl i (CHead c0 k u1) (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 e2 d))) (\lambda (_: C).(\lambda (u2: T).(pr0 u2 u))) (pr3 (CHead c0 k u1) t3 t0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H1: (getl i (CHead c0 k u1) (CHead x0 (Bind Abbr) x1))).(\lambda (_: (wcpr0 x0 d)).(\lambda (H14: (pr0 x1 u)).(ex2_ind T (\lambda (t0: T).(subst0 i x1 t2 t0)) (\lambda (t3: T).(pr0 t3 t0)) (pr3 (CHead c0 k u1) t3 t0) (\lambda (x: T).(\lambda (H15: (subst0 i x1 t2 x)).(\lambda (H16: (pr0 x t0)).(pr3_sing (CHead c0 k u1) x t3 (pr2_delta (CHead c0 k u1) x0 x1 i H1 t3 t2 H11 x H15) t0 (pr3_pr2 (CHead c0 k u1) x t0 (pr2_free (CHead c0 k u1) x t0 H16)))))) (pr0_subst0_back u t2 t0 i H12 x1 H14))))))) (wcpr0_getl_back (CHead c3 k u1) (CHead c0 k u1) (wcpr0_comp c0 c3 H0 u1 u1 (pr0_refl u1) k) i d u (Bind Abbr) H10))))) t (sym_eq T t t0 H9))) t1 (sym_eq T t1 t3 H8))) c (sym_eq C c (CHead c3 k u1) H5) H6 H7 H2 H3 H4))))]) in (H7 (refl_equal C (CHead c3 k u1)) (refl_equal T t3) (refl_equal T t0))) t4 H6))))))) t1 t2 (pr3_pr2_pr3_t c3 u2 t1 t2 k H3 u1 (pr2_free c3 u1 u2 H2)))))))))))))) c2 c1 H))).
 
 theorem pr3_gen_lift:
  \forall (c: C).(\forall (t1: T).(\forall (x: T).(\forall (h: nat).(\forall (d: nat).((pr3 c (lift h d t1) x) \to (\forall (e: C).((drop h d c e) \to (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr3 e t1 t2))))))))))
@@ -2278,27 +2278,27 @@ inductive csuba (g:G): C \to (C \to Prop) \def
 theorem csuba_gen_abbr:
  \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g (CHead d1 (Bind Abbr) u) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))
 \def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (csuba g (CHead d1 (Bind Abbr) u) c)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to ((eq C c1 c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Bind Abbr) u))).(\lambda (H1: (eq C (CSort n) c)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abbr) u) H0) in (False_ind ((eq C (CSort n) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) H2)) H1))) | (csuba_head c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) (CHead d1 (Bind Abbr) u))).(\lambda (H2: (eq C (CHead c2 k u0) c)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H1) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind Abbr)) \to ((eq T u0 u) \to ((eq C (CHead c2 k u0) c) \to ((csuba g c0 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) (\lambda (H6: (eq K k (Bind Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead c2 k0 u0) c) \to ((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))) (\lambda (H7: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c2 (Bind Abbr) t) c) \to ((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (H8: (eq C (CHead c2 (Bind Abbr) u) c)).(eq_ind C (CHead c2 (Bind Abbr) u) (\lambda (c: C).((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (H9: (csuba g d1 c2)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C (CHead c2 (Bind Abbr) u)) H9)) c H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Bind Abbr) H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1 (Bind Abbr) u))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) c)).((let H5 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) u) H3) in (False_ind ((eq C (CHead c2 (Bind Abbr) u0) c) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))) H5)) H4 H0 H1 H2)))]) in (H0 (refl_equal C (CHead d1 (Bind Abbr) u)) (refl_equal C c))))))).
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (csuba g (CHead d1 (Bind Abbr) u) c)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c0: C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind Abbr) u)) \to ((eq C c1 c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Bind Abbr) u))).(\lambda (H1: (eq C (CSort n) c)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abbr) u) H0) in (False_ind ((eq C (CSort n) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) H2)) H1))) | (csuba_head c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) (CHead d1 (Bind Abbr) u))).(\lambda (H2: (eq C (CHead c2 k u0) c)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H1) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind Abbr)) \to ((eq T u0 u) \to ((eq C (CHead c2 k u0) c) \to ((csuba g c0 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) (\lambda (H6: (eq K k (Bind Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead c2 k0 u0) c) \to ((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))) (\lambda (H7: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c2 (Bind Abbr) t) c) \to ((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (H8: (eq C (CHead c2 (Bind Abbr) u) c)).(eq_ind C (CHead c2 (Bind Abbr) u) (\lambda (c: C).((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (H9: (csuba g d1 c2)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C (CHead c2 (Bind Abbr) u)) H9)) c H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Bind Abbr) H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1 (Bind Abbr) u))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) c)).((let H5 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) u) H3) in (False_ind ((eq C (CHead c2 (Bind Abbr) u0) c) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))) H5)) H4 H0 H1 H2)))]) in (H0 (refl_equal C (CHead d1 (Bind Abbr) u)) (refl_equal C c))))))).
 
 theorem csuba_gen_void:
  \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u: T).((csuba g (CHead d1 (Bind Void) u) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))))))
 \def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (csuba g (CHead d1 (Bind Void) u) c)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind Void) u)) \to ((eq C c1 c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Bind Void) u))).(\lambda (H1: (eq C (CSort n) c)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Void) u) H0) in (False_ind ((eq C (CSort n) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))) H2)) H1))) | (csuba_head c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) (CHead d1 (Bind Void) u))).(\lambda (H2: (eq C (CHead c2 k u0) c)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u0) (CHead d1 (Bind Void) u) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c1 k u0) (CHead d1 (Bind Void) u) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u0) (CHead d1 (Bind Void) u) H1) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind Void)) \to ((eq T u0 u) \to ((eq C (CHead c2 k u0) c) \to ((csuba g c0 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) (\lambda (H6: (eq K k (Bind Void))).(eq_ind K (Bind Void) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead c2 k0 u0) c) \to ((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2))))))) (\lambda (H7: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c2 (Bind Void) t) c) \to ((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (H8: (eq C (CHead c2 (Bind Void) u) c)).(eq_ind C (CHead c2 (Bind Void) u) (\lambda (c: C).((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (H9: (csuba g d1 c2)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Void) u) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C (CHead c2 (Bind Void) u)) H9)) c H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Bind Void) H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1 (Bind Void) u))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) c)).((let H5 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Void) u) H3) in (False_ind ((eq C (CHead c2 (Bind Abbr) u0) c) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2))))))) H5)) H4 H0 H1 H2)))]) in (H0 (refl_equal C (CHead d1 (Bind Void) u)) (refl_equal C c))))))).
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (csuba g (CHead d1 (Bind Void) u) c)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c0: C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind Void) u)) \to ((eq C c1 c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Bind Void) u))).(\lambda (H1: (eq C (CSort n) c)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Void) u) H0) in (False_ind ((eq C (CSort n) c) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))) H2)) H1))) | (csuba_head c1 c2 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) (CHead d1 (Bind Void) u))).(\lambda (H2: (eq C (CHead c2 k u0) c)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u0) (CHead d1 (Bind Void) u) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c1 k u0) (CHead d1 (Bind Void) u) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u0) (CHead d1 (Bind Void) u) H1) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind Void)) \to ((eq T u0 u) \to ((eq C (CHead c2 k u0) c) \to ((csuba g c0 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) (\lambda (H6: (eq K k (Bind Void))).(eq_ind K (Bind Void) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead c2 k0 u0) c) \to ((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2))))))) (\lambda (H7: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c2 (Bind Void) t) c) \to ((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (H8: (eq C (CHead c2 (Bind Void) u) c)).(eq_ind C (CHead c2 (Bind Void) u) (\lambda (c: C).((csuba g d1 c2) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (H9: (csuba g d1 c2)).(ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Void) u) (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C (CHead c2 (Bind Void) u)) H9)) c H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Bind Void) H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1 (Bind Void) u))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u0) c)).((let H5 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Void) u) H3) in (False_ind ((eq C (CHead c2 (Bind Abbr) u0) c) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u0 a) \to (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Void) u))) (\lambda (d2: C).(csuba g d1 d2))))))) H5)) H4 H0 H1 H2)))]) in (H0 (refl_equal C (CHead d1 (Bind Void) u)) (refl_equal C c))))))).
 
 theorem csuba_gen_abst:
  \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).((csuba g (CHead d1 (Bind Abst) u1) c) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))
 \def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda (H: (csuba g (CHead d1 (Bind Abst) u1) c)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind Abst) u1)) \to ((eq C c1 c) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Bind Abst) u1))).(\lambda (H1: (eq C (CSort n) c)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abst) u1) H0) in (False_ind ((eq C (CSort n) c) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) H2)) H1))) | (csuba_head c1 c2 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) (CHead d1 (Bind Abst) u1))).(\lambda (H2: (eq C (CHead c2 k u) c)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H1) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind Abst)) \to ((eq T u u1) \to ((eq C (CHead c2 k u) c) \to ((csuba g c0 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) (\lambda (H6: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) (\lambda (k0: K).((eq T u u1) \to ((eq C (CHead c2 k0 u) c) \to ((csuba g d1 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))) (\lambda (H7: (eq T u u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 (Bind Abst) t) c) \to ((csuba g d1 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) (\lambda (H8: (eq C (CHead c2 (Bind Abst) u1) c)).(eq_ind C (CHead c2 (Bind Abst) u1) (\lambda (c: C).((csuba g d1 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))) (\lambda (H9: (csuba g d1 c2)).(or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abst) u1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C (CHead c2 (Bind Abst) u1)) H9))) c H8)) u (sym_eq T u u1 H7))) k (sym_eq K k (Bind Abst) H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u) c)).((let H5 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow t | (CHead _ _ t) \Rightarrow t])) (CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1) H3) in ((let H6 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1) H3) in (eq_ind C d1 (\lambda (c0: C).((eq T t u1) \to ((eq C (CHead c2 (Bind Abbr) u) c) \to ((csuba g c0 c2) \to ((arity g c0 t (asucc g a)) \to ((arity g c2 u a) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0)))))))))))) (\lambda (H7: (eq T t u1)).(eq_ind T u1 (\lambda (t0: T).((eq C (CHead c2 (Bind Abbr) u) c) \to ((csuba g d1 c2) \to ((arity g d1 t0 (asucc g a)) \to ((arity g c2 u a) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0))))))))))) (\lambda (H8: (eq C (CHead c2 (Bind Abbr) u) c)).(eq_ind C (CHead c2 (Bind Abbr) u) (\lambda (c: C).((csuba g d1 c2) \to ((arity g d1 u1 (asucc g a)) \to ((arity g c2 u a) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0)))))))))) (\lambda (H9: (csuba g d1 c2)).(\lambda (H10: (arity g d1 u1 (asucc g a))).(\lambda (H11: (arity g c2 u a)).(or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0)))) c2 u a (refl_equal C (CHead c2 (Bind Abbr) u)) H9 H10 H11))))) c H8)) t (sym_eq T t u1 H7))) c1 (sym_eq C c1 d1 H6))) H5)) H4 H0 H1 H2)))]) in (H0 (refl_equal C (CHead d1 (Bind Abst) u1)) (refl_equal C c))))))).
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda (H: (csuba g (CHead d1 (Bind Abst) u1) c)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c0: C).(\lambda (c1: C).((eq C c0 (CHead d1 (Bind Abst) u1)) \to ((eq C c1 c) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Bind Abst) u1))).(\lambda (H1: (eq C (CSort n) c)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abst) u1) H0) in (False_ind ((eq C (CSort n) c) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) H2)) H1))) | (csuba_head c1 c2 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) (CHead d1 (Bind Abst) u1))).(\lambda (H2: (eq C (CHead c2 k u) c)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead d1 (Bind Abst) u1) H1) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Bind Abst)) \to ((eq T u u1) \to ((eq C (CHead c2 k u) c) \to ((csuba g c0 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) (\lambda (H6: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) (\lambda (k0: K).((eq T u u1) \to ((eq C (CHead c2 k0 u) c) \to ((csuba g d1 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))) (\lambda (H7: (eq T u u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 (Bind Abst) t) c) \to ((csuba g d1 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) (\lambda (H8: (eq C (CHead c2 (Bind Abst) u1) c)).(eq_ind C (CHead c2 (Bind Abst) u1) (\lambda (c: C).((csuba g d1 c2) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))) (\lambda (H9: (csuba g d1 c2)).(or_introl (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abst) u1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) c2 (refl_equal C (CHead c2 (Bind Abst) u1)) H9))) c H8)) u (sym_eq T u u1 H7))) k (sym_eq K k (Bind Abst) H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u) c)).((let H5 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow t | (CHead _ _ t) \Rightarrow t])) (CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1) H3) in ((let H6 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead d1 (Bind Abst) u1) H3) in (eq_ind C d1 (\lambda (c0: C).((eq T t u1) \to ((eq C (CHead c2 (Bind Abbr) u) c) \to ((csuba g c0 c2) \to ((arity g c0 t (asucc g a)) \to ((arity g c2 u a) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0)))))))))))) (\lambda (H7: (eq T t u1)).(eq_ind T u1 (\lambda (t0: T).((eq C (CHead c2 (Bind Abbr) u) c) \to ((csuba g d1 c2) \to ((arity g d1 t0 (asucc g a)) \to ((arity g c2 u a) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0))))))))))) (\lambda (H8: (eq C (CHead c2 (Bind Abbr) u) c)).(eq_ind C (CHead c2 (Bind Abbr) u) (\lambda (c: C).((csuba g d1 c2) \to ((arity g d1 u1 (asucc g a)) \to ((arity g c2 u a) \to (or (ex2 C (\lambda (d2: C).(eq C c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0)))))))))) (\lambda (H9: (csuba g d1 c2)).(\lambda (H10: (arity g d1 u1 (asucc g a))).(\lambda (H11: (arity g c2 u a)).(or_intror (ex2 C (\lambda (d2: C).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C (CHead c2 (Bind Abbr) u) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a0: A).(arity g d1 u1 (asucc g a0))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a0: A).(arity g d2 u2 a0)))) c2 u a (refl_equal C (CHead c2 (Bind Abbr) u)) H9 H10 H11))))) c H8)) t (sym_eq T t u1 H7))) c1 (sym_eq C c1 d1 H6))) H5)) H4 H0 H1 H2)))]) in (H0 (refl_equal C (CHead d1 (Bind Abst) u1)) (refl_equal C c))))))).
 
 theorem csuba_gen_flat:
  \forall (g: G).(\forall (d1: C).(\forall (c: C).(\forall (u1: T).(\forall (f: F).((csuba g (CHead d1 (Flat f) u1) c) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))))))
 \def
- \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda (f: F).(\lambda (H: (csuba g (CHead d1 (Flat f) u1) c)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (c1: C).((eq C c0 (CHead d1 (Flat f) u1)) \to ((eq C c1 c) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Flat f) u1))).(\lambda (H1: (eq C (CSort n) c)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Flat f) u1) H0) in (False_ind ((eq C (CSort n) c) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))) H2)) H1))) | (csuba_head c1 c2 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) (CHead d1 (Flat f) u1))).(\lambda (H2: (eq C (CHead c2 k u) c)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead d1 (Flat f) u1) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c1 k u) (CHead d1 (Flat f) u1) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead d1 (Flat f) u1) H1) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Flat f)) \to ((eq T u u1) \to ((eq C (CHead c2 k u) c) \to ((csuba g c0 c2) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))))) (\lambda (H6: (eq K k (Flat f))).(eq_ind K (Flat f) (\lambda (k0: K).((eq T u u1) \to ((eq C (CHead c2 k0 u) c) \to ((csuba g d1 c2) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))))) (\lambda (H7: (eq T u u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 (Flat f) t) c) \to ((csuba g d1 c2) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))) (\lambda (H8: (eq C (CHead c2 (Flat f) u1) c)).(eq_ind C (CHead c2 (Flat f) u1) (\lambda (c: C).((csuba g d1 c2) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) (\lambda (H9: (csuba g d1 c2)).(ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 (Flat f) u1) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))) c2 u1 (refl_equal C (CHead c2 (Flat f) u1)) H9)) c H8)) u (sym_eq T u u1 H7))) k (sym_eq K k (Flat f) H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1 (Flat f) u1))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u) c)).((let H5 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H3) in (False_ind ((eq C (CHead c2 (Bind Abbr) u) c) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u a) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))))) H5)) H4 H0 H1 H2)))]) in (H0 (refl_equal C (CHead d1 (Flat f) u1)) (refl_equal C c)))))))).
+ \lambda (g: G).(\lambda (d1: C).(\lambda (c: C).(\lambda (u1: T).(\lambda (f: F).(\lambda (H: (csuba g (CHead d1 (Flat f) u1) c)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c0: C).(\lambda (c1: C).((eq C c0 (CHead d1 (Flat f) u1)) \to ((eq C c1 c) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Flat f) u1))).(\lambda (H1: (eq C (CSort n) c)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Flat f) u1) H0) in (False_ind ((eq C (CSort n) c) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))) H2)) H1))) | (csuba_head c1 c2 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) (CHead d1 (Flat f) u1))).(\lambda (H2: (eq C (CHead c2 k u) c)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead d1 (Flat f) u1) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c1 k u) (CHead d1 (Flat f) u1) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead d1 (Flat f) u1) H1) in (eq_ind C d1 (\lambda (c0: C).((eq K k (Flat f)) \to ((eq T u u1) \to ((eq C (CHead c2 k u) c) \to ((csuba g c0 c2) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))))) (\lambda (H6: (eq K k (Flat f))).(eq_ind K (Flat f) (\lambda (k0: K).((eq T u u1) \to ((eq C (CHead c2 k0 u) c) \to ((csuba g d1 c2) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))))) (\lambda (H7: (eq T u u1)).(eq_ind T u1 (\lambda (t: T).((eq C (CHead c2 (Flat f) t) c) \to ((csuba g d1 c2) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))))))) (\lambda (H8: (eq C (CHead c2 (Flat f) u1) c)).(eq_ind C (CHead c2 (Flat f) u1) (\lambda (c: C).((csuba g d1 c2) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))) (\lambda (H9: (csuba g d1 c2)).(ex2_2_intro C T (\lambda (d2: C).(\lambda (u2: T).(eq C (CHead c2 (Flat f) u1) (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2))) c2 u1 (refl_equal C (CHead c2 (Flat f) u1)) H9)) c H8)) u (sym_eq T u u1 H7))) k (sym_eq K k (Flat f) H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c2 H0 t a H1 u H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead d1 (Flat f) u1))).(\lambda (H4: (eq C (CHead c2 (Bind Abbr) u) c)).((let H5 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 (Flat f) u1) H3) in (False_ind ((eq C (CHead c2 (Bind Abbr) u) c) \to ((csuba g c1 c2) \to ((arity g c1 t (asucc g a)) \to ((arity g c2 u a) \to (ex2_2 C T (\lambda (d2: C).(\lambda (u2: T).(eq C c (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g d1 d2)))))))) H5)) H4 H0 H1 H2)))]) in (H0 (refl_equal C (CHead d1 (Flat f) u1)) (refl_equal C c)))))))).
 
 theorem csuba_gen_bind:
  \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csuba g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))))))
 \def
- \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v1: T).(\lambda (H: (csuba g (CHead e1 (Bind b1) v1) c2)).(let H0 \def (match H return (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind b1) v1)) \to ((eq C c0 c2) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead e1 (Bind b1) v1))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1) v1) H0) in (False_ind ((eq C (CSort n) c2) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))) H2)) H1))) | (csuba_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(\lambda (H2: (eq C (CHead c0 k u) c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in (eq_ind C e1 (\lambda (c: C).((eq K k (Bind b1)) \to ((eq T u v1) \to ((eq C (CHead c0 k u) c2) \to ((csuba g c c0) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))))) (\lambda (H6: (eq K k (Bind b1))).(eq_ind K (Bind b1) (\lambda (k0: K).((eq T u v1) \to ((eq C (CHead c0 k0 u) c2) \to ((csuba g e1 c0) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))))) (\lambda (H7: (eq T u v1)).(eq_ind T v1 (\lambda (t: T).((eq C (CHead c0 (Bind b1) t) c2) \to ((csuba g e1 c0) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))) (\lambda (H8: (eq C (CHead c0 (Bind b1) v1) c2)).(eq_ind C (CHead c0 (Bind b1) v1) (\lambda (c: C).((csuba g e1 c0) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) (\lambda (H9: (csuba g e1 c0)).(let H10 \def (eq_ind_r C c2 (\lambda (c: C).(csuba g (CHead e1 (Bind b1) v1) c)) H (CHead c0 (Bind b1) v1) H8) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))) b1 c0 v1 (refl_equal C (CHead c0 (Bind b1) v1)) H9))) c2 H8)) u (sym_eq T u v1 H7))) k (sym_eq K k (Bind b1) H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c0 H0 t a H1 u H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1))).(\lambda (H4: (eq C (CHead c0 (Bind Abbr) u) c2)).((let H5 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow t | (CHead _ _ t) \Rightarrow t])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H3) in ((let H6 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H3) in ((let H7 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H3) in (eq_ind C e1 (\lambda (c: C).((eq B Abst b1) \to ((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csuba g c c0) \to ((arity g c t (asucc g a)) \to ((arity g c0 u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))))))) (\lambda (H8: (eq B Abst b1)).(eq_ind B Abst (\lambda (_: B).((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csuba g e1 c0) \to ((arity g e1 t (asucc g a)) \to ((arity g c0 u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))))))) (\lambda (H9: (eq T t v1)).(eq_ind T v1 (\lambda (t0: T).((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csuba g e1 c0) \to ((arity g e1 t0 (asucc g a)) \to ((arity g c0 u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))))) (\lambda (H10: (eq C (CHead c0 (Bind Abbr) u) c2)).(eq_ind C (CHead c0 (Bind Abbr) u) (\lambda (c: C).((csuba g e1 c0) \to ((arity g e1 v1 (asucc g a)) \to ((arity g c0 u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))))) (\lambda (H11: (csuba g e1 c0)).(\lambda (_: (arity g e1 v1 (asucc g a))).(\lambda (_: (arity g c0 u a)).(let H14 \def (eq_ind_r C c2 (\lambda (c: C).(csuba g (CHead e1 (Bind b1) v1) c)) H (CHead c0 (Bind Abbr) u) H10) in (let H15 \def (eq_ind_r B b1 (\lambda (b: B).(csuba g (CHead e1 (Bind b) v1) (CHead c0 (Bind Abbr) u))) H14 Abst H8) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind Abbr) u) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))) Abbr c0 u (refl_equal C (CHead c0 (Bind Abbr) u)) H11)))))) c2 H10)) t (sym_eq T t v1 H9))) b1 H8)) c1 (sym_eq C c1 e1 H7))) H6)) H5)) H4 H0 H1 H2)))]) in (H0 (refl_equal C (CHead e1 (Bind b1) v1)) (refl_equal C c2)))))))).
+ \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v1: T).(\lambda (H: (csuba g (CHead e1 (Bind b1) v1) c2)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind b1) v1)) \to ((eq C c0 c2) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))))) with [(csuba_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead e1 (Bind b1) v1))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1) v1) H0) in (False_ind ((eq C (CSort n) c2) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))) H2)) H1))) | (csuba_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(\lambda (H2: (eq C (CHead c0 k u) c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in (eq_ind C e1 (\lambda (c: C).((eq K k (Bind b1)) \to ((eq T u v1) \to ((eq C (CHead c0 k u) c2) \to ((csuba g c c0) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))))) (\lambda (H6: (eq K k (Bind b1))).(eq_ind K (Bind b1) (\lambda (k0: K).((eq T u v1) \to ((eq C (CHead c0 k0 u) c2) \to ((csuba g e1 c0) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))))) (\lambda (H7: (eq T u v1)).(eq_ind T v1 (\lambda (t: T).((eq C (CHead c0 (Bind b1) t) c2) \to ((csuba g e1 c0) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))) (\lambda (H8: (eq C (CHead c0 (Bind b1) v1) c2)).(eq_ind C (CHead c0 (Bind b1) v1) (\lambda (c: C).((csuba g e1 c0) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))) (\lambda (H9: (csuba g e1 c0)).(let H10 \def (eq_ind_r C c2 (\lambda (c: C).(csuba g (CHead e1 (Bind b1) v1) c)) H (CHead c0 (Bind b1) v1) H8) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))) b1 c0 v1 (refl_equal C (CHead c0 (Bind b1) v1)) H9))) c2 H8)) u (sym_eq T u v1 H7))) k (sym_eq K k (Bind b1) H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) | (csuba_abst c1 c0 H0 t a H1 u H2) \Rightarrow (\lambda (H3: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1))).(\lambda (H4: (eq C (CHead c0 (Bind Abbr) u) c2)).((let H5 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow t | (CHead _ _ t) \Rightarrow t])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H3) in ((let H6 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H3) in ((let H7 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H3) in (eq_ind C e1 (\lambda (c: C).((eq B Abst b1) \to ((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csuba g c c0) \to ((arity g c t (asucc g a)) \to ((arity g c0 u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))))))) (\lambda (H8: (eq B Abst b1)).(eq_ind B Abst (\lambda (_: B).((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csuba g e1 c0) \to ((arity g e1 t (asucc g a)) \to ((arity g c0 u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))))))) (\lambda (H9: (eq T t v1)).(eq_ind T v1 (\lambda (t0: T).((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csuba g e1 c0) \to ((arity g e1 t0 (asucc g a)) \to ((arity g c0 u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))))))))) (\lambda (H10: (eq C (CHead c0 (Bind Abbr) u) c2)).(eq_ind C (CHead c0 (Bind Abbr) u) (\lambda (c: C).((csuba g e1 c0) \to ((arity g e1 v1 (asucc g a)) \to ((arity g c0 u a) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2))))))))) (\lambda (H11: (csuba g e1 c0)).(\lambda (_: (arity g e1 v1 (asucc g a))).(\lambda (_: (arity g c0 u a)).(let H14 \def (eq_ind_r C c2 (\lambda (c: C).(csuba g (CHead e1 (Bind b1) v1) c)) H (CHead c0 (Bind Abbr) u) H10) in (let H15 \def (eq_ind_r B b1 (\lambda (b: B).(csuba g (CHead e1 (Bind b) v1) (CHead c0 (Bind Abbr) u))) H14 Abst H8) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind Abbr) u) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g e1 e2)))) Abbr c0 u (refl_equal C (CHead c0 (Bind Abbr) u)) H11)))))) c2 H10)) t (sym_eq T t v1 H9))) b1 H8)) c1 (sym_eq C c1 e1 H7))) H6)) H5)) H4 H0 H1 H2)))]) in (H0 (refl_equal C (CHead e1 (Bind b1) v1)) (refl_equal C c2)))))))).
 
 theorem csuba_refl:
  \forall (g: G).(\forall (c: C).(csuba g c c))
@@ -2308,27 +2308,27 @@ theorem csuba_refl:
 theorem csuba_clear_conf:
  \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csuba g c1 c2) \to (\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c2 e2))))))))
 \def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c1 c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c0 e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c4 e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: (clear (CHead c3 k u) e1)).((match k return (\lambda (k0: K).((clear (CHead c3 k0 u) e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear (CHead c4 k0 u) e2))))) with [(Bind b) \Rightarrow (\lambda (H3: (clear (CHead c3 (Bind b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3 (Bind b) u) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind b) u) (csuba_head g c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1 (clear_gen_bind b c3 e1 u H3))) | (Flat f) \Rightarrow (\lambda (H3: (clear (CHead c3 (Flat f) u) e1)).(let H4 \def (H1 e1 (clear_gen_flat f c3 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x: C).(\lambda (H5: (csuba g e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2)) x H5 (clear_flat c4 x H6 f u))))) H4)))]) H2))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c4 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H2: (arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u a)).(\lambda (e1: C).(\lambda (H4: (clear (CHead c3 (Bind Abst) t) e1)).(eq_ind_r C (CHead c3 (Bind Abst) t) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3 (Bind Abst) t) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)) (CHead c4 (Bind Abbr) u) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abbr c4 u)) e1 (clear_gen_bind Abst c3 e1 t H4))))))))))))) c1 c2 H)))).
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csuba g c1 c2)).(csuba_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c0 e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c4 e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: (clear (CHead c3 k u) e1)).((match k return (\lambda (_: ?).(\lambda (k0: K).((clear (CHead c3 k0 u) e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear (CHead c4 k0 u) e2)))))) with [(Bind b) \Rightarrow (\lambda (H3: (clear (CHead c3 (Bind b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3 (Bind b) u) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind b) u) (csuba_head g c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1 (clear_gen_bind b c3 e1 u H3))) | (Flat f) \Rightarrow (\lambda (H3: (clear (CHead c3 (Flat f) u) e1)).(let H4 \def (H1 e1 (clear_gen_flat f c3 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x: C).(\lambda (H5: (csuba g e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2)) x H5 (clear_flat c4 x H6 f u))))) H4)))]) H2))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csuba g c3 c4)).(\lambda (_: ((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csuba g e1 e2)) (\lambda (e2: C).(clear c4 e2))))))).(\lambda (t: T).(\lambda (a: A).(\lambda (H2: (arity g c3 t (asucc g a))).(\lambda (u: T).(\lambda (H3: (arity g c4 u a)).(\lambda (e1: C).(\lambda (H4: (clear (CHead c3 (Bind Abst) t) e1)).(eq_ind_r C (CHead c3 (Bind Abst) t) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csuba g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)))) (ex_intro2 C (\lambda (e2: C).(csuba g (CHead c3 (Bind Abst) t) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)) (CHead c4 (Bind Abbr) u) (csuba_abst g c3 c4 H0 t a H2 u H3) (clear_bind Abbr c4 u)) e1 (clear_gen_bind Abst c3 e1 t H4))))))))))))) c1 c2 H)))).
 
 theorem csuba_drop_abbr:
  \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop i O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))))
 \def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1 (CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (csuba g c1 c2)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c2)) H0 (CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H)) in (let H2 \def (csuba_gen_abbr g d1 c2 u H1) in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x (Bind Abbr) u))).(\lambda (H4: (csuba g d1 x)).(eq_ind_r C (CHead x (Bind Abbr) u) (\lambda (c: C).(ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (ex_intro2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_refl (CHead x (Bind Abbr) u)) H4) c2 H3)))) H2)))))))))) (\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g c c2) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))))) (\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n) O (CSort n0) (CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (csuba g (CSort n0) c2)).(and3_ind (eq C (CHead d1 (Bind Abbr) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (H2: (eq C (CHead d1 (Bind Abbr) u) (CSort n0))).(\lambda (_: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (match H2 return (\lambda (c: C).((eq C c (CSort n0)) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))) with [refl_equal \Rightarrow (\lambda (H4: (eq C (CHead d1 (Bind Abbr) u) (CSort n0))).(let H5 \def (eq_ind C (CHead d1 (Bind Abbr) u) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n0) H4) in (False_ind (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H5)))]) in (H5 (refl_equal C (CSort n0))))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abbr) u) H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g c c2) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n) O (CHead c k t) (CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2: (csuba g (CHead c k t) c2)).(K_ind (\lambda (k0: K).((csuba g (CHead c k0 t) c2) \to ((drop (r k0 n) O c (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (b: B).(\lambda (H3: (csuba g (CHead c (Bind b) t) c2)).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abbr) u))).(B_ind (\lambda (b0: B).((csuba g (CHead c (Bind b0) t) c2) \to ((drop (r (Bind b0) n) O c (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (H5: (csuba g (CHead c (Bind Abbr) t) c2)).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abbr) u))).(let H7 \def (csuba_gen_abbr g c c2 t H5) in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g c d2)) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind Abbr) t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abbr) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H10 \def (H c d1 u H6 g x H9) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead x0 (Bind Abbr) u))).(\lambda (H12: (csuba g d1 x0)).(let H13 \def (refl_equal nat (r (Bind Abbr) n)) in (let H14 \def (eq_ind nat n (\lambda (n: nat).(drop n O x (CHead x0 (Bind Abbr) u))) H11 (r (Bind Abbr) n) H13) in (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abbr) n x (CHead x0 (Bind Abbr) u) H14 t) H12)))))) H10)) c2 H8)))) H7)))) (\lambda (H5: (csuba g (CHead c (Bind Abst) t) c2)).(\lambda (H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abbr) u))).(let H7 \def (csuba_gen_abst g c c2 t H5) in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H9: (eq C c2 (CHead x (Bind Abst) t))).(\lambda (H10: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H11 \def (H c d1 u H6 g x H10) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abbr) u))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def (refl_equal nat (r (Bind Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda (n: nat).(drop n O x (CHead x0 (Bind Abbr) u))) H12 (r (Bind Abbr) n) H14) in (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abbr) u) H15 t) H13)))))) H11)) c2 H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t (asucc g x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H13 \def (H c d1 u H6 g x0 H10) in (ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H14: (drop n O x0 (CHead x (Bind Abbr) u))).(\lambda (H15: (csuba g d1 x)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def (eq_ind nat n (\lambda (n: nat).(drop n O x0 (CHead x (Bind Abbr) u))) H14 (r (Bind Abbr) n) H16) in (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0 (CHead x (Bind Abbr) u) H17 x1) H15)))))) H13)) c2 H9)))))))) H8)) H7)))) (\lambda (H5: (csuba g (CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abbr) u))).(let H7 \def (csuba_gen_void g c c2 t H5) in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: C).(csuba g c d2)) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Void) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H10 \def (H c d1 u H6 g x H9) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead x0 (Bind Abbr) u))).(\lambda (H12: (csuba g d1 x0)).(let H13 \def (refl_equal nat (r (Bind Abbr) n)) in (let H14 \def (eq_ind nat n (\lambda (n: nat).(drop n O x (CHead x0 (Bind Abbr) u))) H11 (r (Bind Abbr) n) H13) in (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abbr) u) H14 t) H12)))))) H10)) c2 H8)))) H7)))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g (CHead c (Flat f) t) c2)).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr) u))).(let H5 \def (csuba_gen_flat g c c2 t f H3) in (ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g c d2))) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H8 \def (H0 d1 u H4 g x0 H7) in (ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H9: (drop (S n) O x0 (CHead x (Bind Abbr) u))).(\lambda (H10: (csuba g d1 x)).(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abbr) u) H9 x1) H10)))) H8)) c2 H6))))) H5))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u) t n H1)))))))))))) c1)))) i).
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda (H: (drop O O c1 (CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (csuba g c1 c2)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c2)) H0 (CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H)) in (let H2 \def (csuba_gen_abbr g d1 c2 u H1) in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H3: (eq C c2 (CHead x (Bind Abbr) u))).(\lambda (H4: (csuba g d1 x)).(eq_ind_r C (CHead x (Bind Abbr) u) (\lambda (c: C).(ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (ex_intro2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_refl (CHead x (Bind Abbr) u)) H4) c2 H3)))) H2)))))))))) (\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g c c2) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))))) (\lambda (n0: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop (S n) O (CSort n0) (CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (csuba g (CSort n0) c2)).(and3_ind (eq C (CHead d1 (Bind Abbr) u) (CSort n0)) (eq nat (S n) O) (eq nat O O) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (H2: (eq C (CHead d1 (Bind Abbr) u) (CSort n0))).(\lambda (_: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (match H2 return (\lambda (_: ?).(\lambda (c: C).((eq C c (CSort n0)) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))) with [refl_equal \Rightarrow (\lambda (H4: (eq C (CHead d1 (Bind Abbr) u) (CSort n0))).(let H5 \def (eq_ind C (CHead d1 (Bind Abbr) u) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n0) H4) in (False_ind (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) H5)))]) in (H5 (refl_equal C (CSort n0))))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abbr) u) H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u: T).((drop (S n) O c (CHead d1 (Bind Abbr) u)) \to (\forall (g: G).(\forall (c2: C).((csuba g c c2) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n) O (CHead c k t) (CHead d1 (Bind Abbr) u))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2: (csuba g (CHead c k t) c2)).(K_ind (\lambda (k0: K).((csuba g (CHead c k0 t) c2) \to ((drop (r k0 n) O c (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (b: B).(\lambda (H3: (csuba g (CHead c (Bind b) t) c2)).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abbr) u))).(B_ind (\lambda (b0: B).((csuba g (CHead c (Bind b0) t) c2) \to ((drop (r (Bind b0) n) O c (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))) (\lambda (H5: (csuba g (CHead c (Bind Abbr) t) c2)).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abbr) u))).(let H7 \def (csuba_gen_abbr g c c2 t H5) in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g c d2)) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind Abbr) t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abbr) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H10 \def (H c d1 u H6 g x H9) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead x0 (Bind Abbr) u))).(\lambda (H12: (csuba g d1 x0)).(let H13 \def (refl_equal nat (r (Bind Abbr) n)) in (let H14 \def (eq_ind nat n (\lambda (n: nat).(drop n O x (CHead x0 (Bind Abbr) u))) H11 (r (Bind Abbr) n) H13) in (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abbr) n x (CHead x0 (Bind Abbr) u) H14 t) H12)))))) H10)) c2 H8)))) H7)))) (\lambda (H5: (csuba g (CHead c (Bind Abst) t) c2)).(\lambda (H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abbr) u))).(let H7 \def (csuba_gen_abst g c c2 t H5) in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H9: (eq C c2 (CHead x (Bind Abst) t))).(\lambda (H10: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H11 \def (H c d1 u H6 g x H10) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abbr) u))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def (refl_equal nat (r (Bind Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda (n: nat).(drop n O x (CHead x0 (Bind Abbr) u))) H12 (r (Bind Abbr) n) H14) in (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abbr) u) H15 t) H13)))))) H11)) c2 H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t (asucc g x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H13 \def (H c d1 u H6 g x0 H10) in (ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H14: (drop n O x0 (CHead x (Bind Abbr) u))).(\lambda (H15: (csuba g d1 x)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def (eq_ind nat n (\lambda (n: nat).(drop n O x0 (CHead x (Bind Abbr) u))) H14 (r (Bind Abbr) n) H16) in (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0 (CHead x (Bind Abbr) u) H17 x1) H15)))))) H13)) c2 H9)))))))) H8)) H7)))) (\lambda (H5: (csuba g (CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abbr) u))).(let H7 \def (csuba_gen_void g c c2 t H5) in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: C).(csuba g c d2)) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Void) t) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H10 \def (H c d1 u H6 g x H9) in (ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x0: C).(\lambda (H11: (drop n O x (CHead x0 (Bind Abbr) u))).(\lambda (H12: (csuba g d1 x0)).(let H13 \def (refl_equal nat (r (Bind Abbr) n)) in (let H14 \def (eq_ind nat n (\lambda (n: nat).(drop n O x (CHead x0 (Bind Abbr) u))) H11 (r (Bind Abbr) n) H13) in (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abbr) u) H14 t) H12)))))) H10)) c2 H8)))) H7)))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g (CHead c (Flat f) t) c2)).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abbr) u))).(let H5 \def (csuba_gen_flat g c c2 t f H3) in (ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g c d2))) (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c0: C).(ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))) (let H8 \def (H0 d1 u H4 g x0 H7) in (ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x: C).(\lambda (H9: (drop (S n) O x0 (CHead x (Bind Abbr) u))).(\lambda (H10: (csuba g d1 x)).(ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abbr) u) H9 x1) H10)))) H8)) c2 H6))))) H5))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abbr) u) t n H1)))))))))))) c1)))) i).
 
 theorem csuba_drop_abst:
  \forall (i: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop i O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))))))
 \def
- \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H: (drop O O c1 (CHead d1 (Bind Abst) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (csuba g c1 c2)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c2)) H0 (CHead d1 (Bind Abst) u1) (drop_gen_refl c1 (CHead d1 (Bind Abst) u1) H)) in (let H2 \def (csuba_gen_abst g d1 c2 u1 H1) in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H3: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x (Bind Abst) u1))).(\lambda (H5: (csuba g d1 x)).(eq_ind_r C (CHead x (Bind Abst) u1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x (Bind Abst) u1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_refl (CHead x (Bind Abst) u1)) H5)) c2 H4)))) H3)) (\lambda (H3: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H5: (csuba g d1 x0)).(\lambda (H6: (arity g d1 u1 (asucc g x2))).(\lambda (H7: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_refl (CHead x0 (Bind Abbr) x1)) H5 H6 H7)) c2 H4)))))))) H3)) H2)))))))))) (\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c c2) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))))) (\lambda (n0: nat).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) (CHead d1 (Bind Abst) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (csuba g (CSort n0) c2)).(and3_ind (eq C (CHead d1 (Bind Abst) u1) (CSort n0)) (eq nat (S n) O) (eq nat O O) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H2: (eq C (CHead d1 (Bind Abst) u1) (CSort n0))).(\lambda (_: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (match H2 return (\lambda (c: C).((eq C c (CSort n0)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))) with [refl_equal \Rightarrow (\lambda (H4: (eq C (CHead d1 (Bind Abst) u1) (CSort n0))).(let H5 \def (eq_ind C (CHead d1 (Bind Abst) u1) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n0) H4) in (False_ind (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) H5)))]) in (H5 (refl_equal C (CSort n0))))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u1) H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c c2) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H1: (drop (S n) O (CHead c k t) (CHead d1 (Bind Abst) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2: (csuba g (CHead c k t) c2)).(K_ind (\lambda (k0: K).((csuba g (CHead c k0 t) c2) \to ((drop (r k0 n) O c (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) (\lambda (b: B).(\lambda (H3: (csuba g (CHead c (Bind b) t) c2)).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abst) u1))).(B_ind (\lambda (b0: B).((csuba g (CHead c (Bind b0) t) c2) \to ((drop (r (Bind b0) n) O c (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) (\lambda (H5: (csuba g (CHead c (Bind Abbr) t) c2)).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abst) u1))).(let H7 \def (csuba_gen_abbr g c c2 t H5) in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g c d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind Abbr) t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abbr) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H10 \def (H c d1 u1 H6 g x H9) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H11: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abst) u1))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def (refl_equal nat (r (Bind Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda (n: nat).(drop n O x (CHead x0 (Bind Abst) u1))) H12 (r (Bind Abbr) n) H14) in (or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abbr) n x (CHead x0 (Bind Abst) u1) H15 t) H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: (drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H13: (csuba g d1 x0)).(\lambda (H14: (arity g d1 u1 (asucc g x2))).(\lambda (H15: (arity g x0 x1 x2)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def (eq_ind nat n (\lambda (n: nat).(drop n O x (CHead x0 (Bind Abbr) x1))) H12 (r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Abbr) n x (CHead x0 (Bind Abbr) x1) H17 t) H13 H14 H15))))))))))) H11)) H10)) c2 H8)))) H7)))) (\lambda (H5: (csuba g (CHead c (Bind Abst) t) c2)).(\lambda (H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst) u1))).(let H7 \def (csuba_gen_abst g c c2 t H5) in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H9: (eq C c2 (CHead x (Bind Abst) t))).(\lambda (H10: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H11 \def (H c d1 u1 H6 g x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) u1))).(\lambda (H14: (csuba g d1 x0)).(let H15 \def (refl_equal nat (r (Bind Abbr) n)) in (let H16 \def (eq_ind nat n (\lambda (n: nat).(drop n O x (CHead x0 (Bind Abst) u1))) H13 (r (Bind Abbr) n) H15) in (or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abst) u1) H16 t) H14))))))) H12)) (\lambda (H12: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H14: (csuba g d1 x0)).(\lambda (H15: (arity g d1 u1 (asucc g x2))).(\lambda (H16: (arity g x0 x1 x2)).(let H17 \def (refl_equal nat (r (Bind Abbr) n)) in (let H18 \def (eq_ind nat n (\lambda (n: nat).(drop n O x (CHead x0 (Bind Abbr) x1))) H13 (r (Bind Abbr) n) H17) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abbr) x1) H18 t) H14 H15 H16))))))))))) H12)) H11)) c2 H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t (asucc g x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H13 \def (H c d1 u1 H6 g x0 H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H15: (drop n O x0 (CHead x (Bind Abst) u1))).(\lambda (H16: (csuba g d1 x)).(let H17 \def (refl_equal nat (r (Bind Abbr) n)) in (let H18 \def (eq_ind nat n (\lambda (n: nat).(drop n O x0 (CHead x (Bind Abst) u1))) H15 (r (Bind Abbr) n) H17) in (or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0 (CHead x (Bind Abst) u1) H18 x1) H16))))))) H14)) (\lambda (H14: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Abbr) x4))).(\lambda (H16: (csuba g d1 x3)).(\lambda (H17: (arity g d1 u1 (asucc g x5))).(\lambda (H18: (arity g x3 x4 x5)).(let H19 \def (refl_equal nat (r (Bind Abbr) n)) in (let H20 \def (eq_ind nat n (\lambda (n: nat).(drop n O x0 (CHead x3 (Bind Abbr) x4))) H15 (r (Bind Abbr) n) H19) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5 (drop_drop (Bind Abbr) n x0 (CHead x3 (Bind Abbr) x4) H20 x1) H16 H17 H18))))))))))) H14)) H13)) c2 H9)))))))) H8)) H7)))) (\lambda (H5: (csuba g (CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abst) u1))).(let H7 \def (csuba_gen_void g c c2 t H5) in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: C).(csuba g c d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Void) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H10 \def (H c d1 u1 H6 g x H9) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H11: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abst) u1))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def (refl_equal nat (r (Bind Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda (n: nat).(drop n O x (CHead x0 (Bind Abst) u1))) H12 (r (Bind Abbr) n) H14) in (or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abst) u1) H15 t) H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: (drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H13: (csuba g d1 x0)).(\lambda (H14: (arity g d1 u1 (asucc g x2))).(\lambda (H15: (arity g x0 x1 x2)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def (eq_ind nat n (\lambda (n: nat).(drop n O x (CHead x0 (Bind Abbr) x1))) H12 (r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Void) n x (CHead x0 (Bind Abbr) x1) H17 t) H13 H14 H15))))))))))) H11)) H10)) c2 H8)))) H7)))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g (CHead c (Flat f) t) c2)).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abst) u1))).(let H5 \def (csuba_gen_flat g c c2 t f H3) in (ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g c d2))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H8 \def (H0 d1 u1 H4 g x0 H7) in (or_ind (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abst) u1))).(\lambda (H11: (csuba g d1 x)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u1) H10 x1) H11))))) H9)) (\lambda (H9: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: A).(\lambda (H10: (drop (S n) O x0 (CHead x2 (Bind Abbr) x3))).(\lambda (H11: (csuba g d1 x2)).(\lambda (H12: (arity g d1 u1 (asucc g x4))).(\lambda (H13: (arity g x2 x3 x4)).(or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x2 x3 x4 (drop_drop (Flat f) n x0 (CHead x2 (Bind Abbr) x3) H10 x1) H11 H12 H13))))))))) H9)) H8)) c2 H6))))) H5))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abst) u1) t n H1)))))))))))) c1)))) i).
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))))))) (\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H: (drop O O c1 (CHead d1 (Bind Abst) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H0: (csuba g c1 c2)).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csuba g c c2)) H0 (CHead d1 (Bind Abst) u1) (drop_gen_refl c1 (CHead d1 (Bind Abst) u1) H)) in (let H2 \def (csuba_gen_abst g d1 c2 u1 H1) in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H3: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H4: (eq C c2 (CHead x (Bind Abst) u1))).(\lambda (H5: (csuba g d1 x)).(eq_ind_r C (CHead x (Bind Abst) u1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (or_introl (ex2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x (Bind Abst) u1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(drop O O (CHead x (Bind Abst) u1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_refl (CHead x (Bind Abst) u1)) H5)) c2 H4)))) H3)) (\lambda (H3: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H4: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H5: (csuba g d1 x0)).(\lambda (H6: (arity g d1 u1 (asucc g x2))).(\lambda (H7: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c: C).(or (ex2 C (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O c (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (or_intror (ex2 C (\lambda (d2: C).(drop O O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop O O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_refl (CHead x0 (Bind Abbr) x1)) H5 H6 H7)) c2 H4)))))))) H3)) H2)))))))))) (\lambda (n: nat).(\lambda (H: ((\forall (c1: C).(\forall (d1: C).(\forall (u1: T).((drop n O c1 (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (d1: C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c c2) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))))) (\lambda (n0: nat).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H0: (drop (S n) O (CSort n0) (CHead d1 (Bind Abst) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (_: (csuba g (CSort n0) c2)).(and3_ind (eq C (CHead d1 (Bind Abst) u1) (CSort n0)) (eq nat (S n) O) (eq nat O O) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H2: (eq C (CHead d1 (Bind Abst) u1) (CSort n0))).(\lambda (_: (eq nat (S n) O)).(\lambda (_: (eq nat O O)).(let H5 \def (match H2 return (\lambda (_: ?).(\lambda (c: C).((eq C c (CSort n0)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) with [refl_equal \Rightarrow (\lambda (H4: (eq C (CHead d1 (Bind Abst) u1) (CSort n0))).(let H5 \def (eq_ind C (CHead d1 (Bind Abst) u1) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n0) H4) in (False_ind (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) H5)))]) in (H5 (refl_equal C (CSort n0))))))) (drop_gen_sort n0 (S n) O (CHead d1 (Bind Abst) u1) H0))))))))) (\lambda (c: C).(\lambda (H0: ((\forall (d1: C).(\forall (u1: T).((drop (S n) O c (CHead d1 (Bind Abst) u1)) \to (\forall (g: G).(\forall (c2: C).((csuba g c c2) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (d1: C).(\lambda (u1: T).(\lambda (H1: (drop (S n) O (CHead c k t) (CHead d1 (Bind Abst) u1))).(\lambda (g: G).(\lambda (c2: C).(\lambda (H2: (csuba g (CHead c k t) c2)).(K_ind (\lambda (k0: K).((csuba g (CHead c k0 t) c2) \to ((drop (r k0 n) O c (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) (\lambda (b: B).(\lambda (H3: (csuba g (CHead c (Bind b) t) c2)).(\lambda (H4: (drop (r (Bind b) n) O c (CHead d1 (Bind Abst) u1))).(B_ind (\lambda (b0: B).((csuba g (CHead c (Bind b0) t) c2) \to ((drop (r (Bind b0) n) O c (CHead d1 (Bind Abst) u1)) \to (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))) (\lambda (H5: (csuba g (CHead c (Bind Abbr) t) c2)).(\lambda (H6: (drop (r (Bind Abbr) n) O c (CHead d1 (Bind Abst) u1))).(let H7 \def (csuba_gen_abbr g c c2 t H5) in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abbr) t))) (\lambda (d2: C).(csuba g c d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind Abbr) t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abbr) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H10 \def (H c d1 u1 H6 g x H9) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H11: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abst) u1))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def (refl_equal nat (r (Bind Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda (n: nat).(drop n O x (CHead x0 (Bind Abst) u1))) H12 (r (Bind Abbr) n) H14) in (or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abbr) n x (CHead x0 (Bind Abst) u1) H15 t) H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: (drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H13: (csuba g d1 x0)).(\lambda (H14: (arity g d1 u1 (asucc g x2))).(\lambda (H15: (arity g x0 x1 x2)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def (eq_ind nat n (\lambda (n: nat).(drop n O x (CHead x0 (Bind Abbr) x1))) H12 (r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abbr) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Abbr) n x (CHead x0 (Bind Abbr) x1) H17 t) H13 H14 H15))))))))))) H11)) H10)) c2 H8)))) H7)))) (\lambda (H5: (csuba g (CHead c (Bind Abst) t) c2)).(\lambda (H6: (drop (r (Bind Abst) n) O c (CHead d1 (Bind Abst) u1))).(let H7 \def (csuba_gen_abst g c c2 t H5) in (or_ind (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H8: (ex2 C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)))).(ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Abst) t))) (\lambda (d2: C).(csuba g c d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H9: (eq C c2 (CHead x (Bind Abst) t))).(\lambda (H10: (csuba g c x)).(eq_ind_r C (CHead x (Bind Abst) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H11 \def (H c d1 u1 H6 g x H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H12: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (H13: (drop n O x (CHead x0 (Bind Abst) u1))).(\lambda (H14: (csuba g d1 x0)).(let H15 \def (refl_equal nat (r (Bind Abbr) n)) in (let H16 \def (eq_ind nat n (\lambda (n: nat).(drop n O x (CHead x0 (Bind Abst) u1))) H13 (r (Bind Abbr) n) H15) in (or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abst) u1) H16 t) H14))))))) H12)) (\lambda (H12: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H13: (drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H14: (csuba g d1 x0)).(\lambda (H15: (arity g d1 u1 (asucc g x2))).(\lambda (H16: (arity g x0 x1 x2)).(let H17 \def (refl_equal nat (r (Bind Abbr) n)) in (let H18 \def (eq_ind nat n (\lambda (n: nat).(drop n O x (CHead x0 (Bind Abbr) x1))) H13 (r (Bind Abbr) n) H17) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Abst) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Abst) n x (CHead x0 (Bind Abbr) x1) H18 t) H14 H15 H16))))))))))) H12)) H11)) c2 H9)))) H8)) (\lambda (H8: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g c d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g c t (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H9: (eq C c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H10: (csuba g c x0)).(\lambda (_: (arity g c t (asucc g x2))).(\lambda (_: (arity g x0 x1 x2)).(eq_ind_r C (CHead x0 (Bind Abbr) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H13 \def (H c d1 u1 H6 g x0 H10) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H14: (ex2 C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H15: (drop n O x0 (CHead x (Bind Abst) u1))).(\lambda (H16: (csuba g d1 x)).(let H17 \def (refl_equal nat (r (Bind Abbr) n)) in (let H18 \def (eq_ind nat n (\lambda (n: nat).(drop n O x0 (CHead x (Bind Abst) u1))) H15 (r (Bind Abbr) n) H17) in (or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Bind Abbr) n x0 (CHead x (Bind Abst) u1) H18 x1) H16))))))) H14)) (\lambda (H14: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: A).(\lambda (H15: (drop n O x0 (CHead x3 (Bind Abbr) x4))).(\lambda (H16: (csuba g d1 x3)).(\lambda (H17: (arity g d1 u1 (asucc g x5))).(\lambda (H18: (arity g x3 x4 x5)).(let H19 \def (refl_equal nat (r (Bind Abbr) n)) in (let H20 \def (eq_ind nat n (\lambda (n: nat).(drop n O x0 (CHead x3 (Bind Abbr) x4))) H15 (r (Bind Abbr) n) H19) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Bind Abbr) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x3 x4 x5 (drop_drop (Bind Abbr) n x0 (CHead x3 (Bind Abbr) x4) H20 x1) H16 H17 H18))))))))))) H14)) H13)) c2 H9)))))))) H8)) H7)))) (\lambda (H5: (csuba g (CHead c (Bind Void) t) c2)).(\lambda (H6: (drop (r (Bind Void) n) O c (CHead d1 (Bind Abst) u1))).(let H7 \def (csuba_gen_void g c c2 t H5) in (ex2_ind C (\lambda (d2: C).(eq C c2 (CHead d2 (Bind Void) t))) (\lambda (d2: C).(csuba g c d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H8: (eq C c2 (CHead x (Bind Void) t))).(\lambda (H9: (csuba g c x)).(eq_ind_r C (CHead x (Bind Void) t) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H10 \def (H c d1 u1 H6 g x H9) in (or_ind (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H11: (ex2 C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop n O x (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (H12: (drop n O x (CHead x0 (Bind Abst) u1))).(\lambda (H13: (csuba g d1 x0)).(let H14 \def (refl_equal nat (r (Bind Abbr) n)) in (let H15 \def (eq_ind nat n (\lambda (n: nat).(drop n O x (CHead x0 (Bind Abst) u1))) H12 (r (Bind Abbr) n) H14) in (or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x0 (drop_drop (Bind Void) n x (CHead x0 (Bind Abst) u1) H15 t) H13))))))) H11)) (\lambda (H11: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop n O x (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H12: (drop n O x (CHead x0 (Bind Abbr) x1))).(\lambda (H13: (csuba g d1 x0)).(\lambda (H14: (arity g d1 u1 (asucc g x2))).(\lambda (H15: (arity g x0 x1 x2)).(let H16 \def (refl_equal nat (r (Bind Abbr) n)) in (let H17 \def (eq_ind nat n (\lambda (n: nat).(drop n O x (CHead x0 (Bind Abbr) x1))) H12 (r (Bind Abbr) n) H16) in (or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x (Bind Void) t) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (drop_drop (Bind Void) n x (CHead x0 (Bind Abbr) x1) H17 t) H13 H14 H15))))))))))) H11)) H10)) c2 H8)))) H7)))) b H3 H4)))) (\lambda (f: F).(\lambda (H3: (csuba g (CHead c (Flat f) t) c2)).(\lambda (H4: (drop (r (Flat f) n) O c (CHead d1 (Bind Abst) u1))).(let H5 \def (csuba_gen_flat g c c2 t f H3) in (ex2_2_ind C T (\lambda (d2: C).(\lambda (u2: T).(eq C c2 (CHead d2 (Flat f) u2)))) (\lambda (d2: C).(\lambda (_: T).(csuba g c d2))) (or (ex2 C (\lambda (d2: C).(drop (S n) O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C c2 (CHead x0 (Flat f) x1))).(\lambda (H7: (csuba g c x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c0: C).(or (ex2 C (\lambda (d2: C).(drop (S n) O c0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O c0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))) (let H8 \def (H0 d1 u1 H4 g x0 H7) in (or_ind (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H9: (ex2 C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop (S n) O x0 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x: C).(\lambda (H10: (drop (S n) O x0 (CHead x (Bind Abst) u1))).(\lambda (H11: (csuba g d1 x)).(or_introl (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x (drop_drop (Flat f) n x0 (CHead x (Bind Abst) u1) H10 x1) H11))))) H9)) (\lambda (H9: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O x0 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: A).(\lambda (H10: (drop (S n) O x0 (CHead x2 (Bind Abbr) x3))).(\lambda (H11: (csuba g d1 x2)).(\lambda (H12: (arity g d1 u1 (asucc g x4))).(\lambda (H13: (arity g x2 x3 x4)).(or_intror (ex2 C (\lambda (d2: C).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop (S n) O (CHead x0 (Flat f) x1) (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x2 x3 x4 (drop_drop (Flat f) n x0 (CHead x2 (Bind Abbr) x3) H10 x1) H11 H12 H13))))))))) H9)) H8)) c2 H6))))) H5))))) k H2 (drop_gen_drop k c (CHead d1 (Bind Abst) u1) t n H1)))))))))))) c1)))) i).
 
 theorem csuba_getl_abbr:
  \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))))
 \def
- \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u))).(let H0 \def (getl_gen_all c1 (CHead d1 (Bind Abbr) u) i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u))) (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind Abbr) u))).((match x return (\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) with [(CSort n) \Rightarrow (\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n) (CHead d1 (Bind Abbr) u))).(clear_gen_sort (CHead d1 (Bind Abbr) u) n H4 (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) | (CHead c k t) \Rightarrow (\lambda (H3: (drop i O c1 (CHead c k t))).(\lambda (H4: (clear (CHead c k t) (CHead d1 (Bind Abbr) u))).((match k return (\lambda (k0: K).((drop i O c1 (CHead c k0 t)) \to ((clear (CHead c k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) with [(Bind b) \Rightarrow (\lambda (H5: (drop i O c1 (CHead c (Bind b) t))).(\lambda (H6: (clear (CHead c (Bind b) t) (CHead d1 (Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abbr) u) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abbr) u) t H6)) in ((let H8 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abbr) u) t H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind Abbr) u) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abbr) u) t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 c)).(\lambda (c2: C).(\lambda (H12: (csuba g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda (t: T).(drop i O c1 (CHead c (Bind b) t))) H5 u H9) in (let H14 \def (eq_ind_r B b (\lambda (b: B).(drop i O c1 (CHead c (Bind b) u))) H13 Abbr H10) in (let H15 \def (eq_ind_r C c (\lambda (c: C).(drop i O c1 (CHead c (Bind Abbr) u))) H14 d1 H11) in (let H16 \def (csuba_drop_abbr i c1 d1 u H15 g c2 H12) in (ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x0: C).(\lambda (H17: (drop i O c2 (CHead x0 (Bind Abbr) u))).(\lambda (H18: (csuba g d1 x0)).(ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (getl_intro i c2 (CHead x0 (Bind Abbr) u) (CHead x0 (Bind Abbr) u) H17 (clear_bind Abbr x0 u)) H18)))) H16)))))))))) H8)) H7)))) | (Flat f) \Rightarrow (\lambda (H5: (drop i O c1 (CHead c (Flat f) t))).(\lambda (H6: (clear (CHead c (Flat f) t) (CHead d1 (Bind Abbr) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c0: C).((drop i O c0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csuba g c0 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))) (nat_ind (\lambda (n: nat).(\forall (x0: C).((drop n O x0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csuba g x0 c2) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) (\lambda (x0: C).(\lambda (H8: (drop O O x0 (CHead c (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g x0 c2)).(let H10 \def (eq_ind C x0 (\lambda (c: C).(csuba g c c2)) H9 (CHead c (Flat f) t) (drop_gen_refl x0 (CHead c (Flat f) t) H8)) in (let H_y \def (clear_flat c (CHead d1 (Bind Abbr) u) (clear_gen_flat f c (CHead d1 (Bind Abbr) u) t H6) f t) in (let H11 \def (csuba_clear_conf g (CHead c (Flat f) t) c2 H10 (CHead d1 (Bind Abbr) u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1 (Bind Abbr) u) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x1: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abbr) u) x1)).(\lambda (H13: (clear c2 x1)).(let H14 \def (csuba_gen_abbr g d1 x1 u H12) in (ex2_ind C (\lambda (d2: C).(eq C x1 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x2: C).(\lambda (H15: (eq C x1 (CHead x2 (Bind Abbr) u))).(\lambda (H16: (csuba g d1 x2)).(let H17 \def (eq_ind C x1 (\lambda (c: C).(clear c2 c)) H13 (CHead x2 (Bind Abbr) u) H15) in (ex_intro2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x2 (getl_intro O c2 (CHead x2 (Bind Abbr) u) c2 (drop_refl c2) H17) H16))))) H14))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x: C).((drop n O x (CHead c (Flat f) t)) \to (\forall (c2: C).((csuba g x c2) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))).(\lambda (x0: C).(\lambda (H9: (drop (S n) O x0 (CHead c (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g x0 c2)).(let H11 \def (drop_clear x0 (CHead c (Flat f) t) n H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x0 (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead c (Flat f) t))))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (clear x0 (CHead x2 (Bind x1) x3))).(\lambda (H13: (drop n O x2 (CHead c (Flat f) t))).(let H14 \def (csuba_clear_conf g x0 c2 H10 (CHead x2 (Bind x1) x3) H12) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead x2 (Bind x1) x3) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x4: C).(\lambda (H15: (csuba g (CHead x2 (Bind x1) x3) x4)).(\lambda (H16: (clear c2 x4)).(let H17 \def (csuba_gen_bind g x1 x2 x4 x3 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x4 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g x2 e2)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: T).(\lambda (H18: (eq C x4 (CHead x6 (Bind x5) x7))).(\lambda (H19: (csuba g x2 x6)).(let H20 \def (eq_ind C x4 (\lambda (c: C).(clear c2 c)) H16 (CHead x6 (Bind x5) x7) H18) in (let H21 \def (H8 x2 H13 x6 H19) in (ex2_ind C (\lambda (d2: C).(getl n x6 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x8: C).(\lambda (H22: (getl n x6 (CHead x8 (Bind Abbr) u))).(\lambda (H23: (csuba g d1 x8)).(ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x8 (getl_clear_bind x5 c2 x6 x7 H20 (CHead x8 (Bind Abbr) u) n H22) H23)))) H21)))))))) H17))))) H14))))))) H11)))))))) i) H7))))]) H3 H4)))]) H1 H2)))) H0))))))).
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abbr) u))).(let H0 \def (getl_gen_all c1 (CHead d1 (Bind Abbr) u) i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u))) (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))) (\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind Abbr) u))).((match x return (\lambda (_: ?).(\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))) with [(CSort n) \Rightarrow (\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n) (CHead d1 (Bind Abbr) u))).(clear_gen_sort (CHead d1 (Bind Abbr) u) n H4 (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) | (CHead c k t) \Rightarrow (\lambda (H3: (drop i O c1 (CHead c k t))).(\lambda (H4: (clear (CHead c k t) (CHead d1 (Bind Abbr) u))).((match k return (\lambda (_: ?).(\lambda (k0: K).((drop i O c1 (CHead c k0 t)) \to ((clear (CHead c k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csuba g c1 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))) with [(Bind b) \Rightarrow (\lambda (H5: (drop i O c1 (CHead c (Bind b) t))).(\lambda (H6: (clear (CHead c (Bind b) t) (CHead d1 (Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abbr) u) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abbr) u) t H6)) in ((let H8 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abbr) u) t H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind Abbr) u) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abbr) u) t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 c)).(\lambda (c2: C).(\lambda (H12: (csuba g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda (t: T).(drop i O c1 (CHead c (Bind b) t))) H5 u H9) in (let H14 \def (eq_ind_r B b (\lambda (b: B).(drop i O c1 (CHead c (Bind b) u))) H13 Abbr H10) in (let H15 \def (eq_ind_r C c (\lambda (c: C).(drop i O c1 (CHead c (Bind Abbr) u))) H14 d1 H11) in (let H16 \def (csuba_drop_abbr i c1 d1 u H15 g c2 H12) in (ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x0: C).(\lambda (H17: (drop i O c2 (CHead x0 (Bind Abbr) u))).(\lambda (H18: (csuba g d1 x0)).(ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x0 (getl_intro i c2 (CHead x0 (Bind Abbr) u) (CHead x0 (Bind Abbr) u) H17 (clear_bind Abbr x0 u)) H18)))) H16)))))))))) H8)) H7)))) | (Flat f) \Rightarrow (\lambda (H5: (drop i O c1 (CHead c (Flat f) t))).(\lambda (H6: (clear (CHead c (Flat f) t) (CHead d1 (Bind Abbr) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c0: C).((drop i O c0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csuba g c0 c2) \to (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))) (nat_ind (\lambda (n: nat).(\forall (x0: C).((drop n O x0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csuba g x0 c2) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)))))))) (\lambda (x0: C).(\lambda (H8: (drop O O x0 (CHead c (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g x0 c2)).(let H10 \def (eq_ind C x0 (\lambda (c: C).(csuba g c c2)) H9 (CHead c (Flat f) t) (drop_gen_refl x0 (CHead c (Flat f) t) H8)) in (let H_y \def (clear_flat c (CHead d1 (Bind Abbr) u) (clear_gen_flat f c (CHead d1 (Bind Abbr) u) t H6) f t) in (let H11 \def (csuba_clear_conf g (CHead c (Flat f) t) c2 H10 (CHead d1 (Bind Abbr) u) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1 (Bind Abbr) u) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x1: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abbr) u) x1)).(\lambda (H13: (clear c2 x1)).(let H14 \def (csuba_gen_abbr g d1 x1 u H12) in (ex2_ind C (\lambda (d2: C).(eq C x1 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x2: C).(\lambda (H15: (eq C x1 (CHead x2 (Bind Abbr) u))).(\lambda (H16: (csuba g d1 x2)).(let H17 \def (eq_ind C x1 (\lambda (c: C).(clear c2 c)) H13 (CHead x2 (Bind Abbr) u) H15) in (ex_intro2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x2 (getl_intro O c2 (CHead x2 (Bind Abbr) u) c2 (drop_refl c2) H17) H16))))) H14))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x: C).((drop n O x (CHead c (Flat f) t)) \to (\forall (c2: C).((csuba g x c2) \to (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))))))))).(\lambda (x0: C).(\lambda (H9: (drop (S n) O x0 (CHead c (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g x0 c2)).(let H11 \def (drop_clear x0 (CHead c (Flat f) t) n H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x0 (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead c (Flat f) t))))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (clear x0 (CHead x2 (Bind x1) x3))).(\lambda (H13: (drop n O x2 (CHead c (Flat f) t))).(let H14 \def (csuba_clear_conf g x0 c2 H10 (CHead x2 (Bind x1) x3) H12) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead x2 (Bind x1) x3) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x4: C).(\lambda (H15: (csuba g (CHead x2 (Bind x1) x3) x4)).(\lambda (H16: (clear c2 x4)).(let H17 \def (csuba_gen_bind g x1 x2 x4 x3 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x4 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g x2 e2)))) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: T).(\lambda (H18: (eq C x4 (CHead x6 (Bind x5) x7))).(\lambda (H19: (csuba g x2 x6)).(let H20 \def (eq_ind C x4 (\lambda (c: C).(clear c2 c)) H16 (CHead x6 (Bind x5) x7) H18) in (let H21 \def (H8 x2 H13 x6 H19) in (ex2_ind C (\lambda (d2: C).(getl n x6 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2))) (\lambda (x8: C).(\lambda (H22: (getl n x6 (CHead x8 (Bind Abbr) u))).(\lambda (H23: (csuba g d1 x8)).(ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(csuba g d1 d2)) x8 (getl_clear_bind x5 c2 x6 x7 H20 (CHead x8 (Bind Abbr) u) n H22) H23)))) H21)))))))) H17))))) H14))))))) H11)))))))) i) H7))))]) H3 H4)))]) H1 H2)))) H0))))))).
 
 theorem csuba_getl_abst:
  \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u1: T).(\forall (i: nat).((getl i c1 (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))))))
 \def
- \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda (i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u1))).(let H0 \def (getl_gen_all c1 (CHead d1 (Bind Abst) u1) i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u1))) (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))) (\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind Abst) u1))).((match x return (\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) with [(CSort n) \Rightarrow (\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n) (CHead d1 (Bind Abst) u1))).(clear_gen_sort (CHead d1 (Bind Abst) u1) n H4 (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) | (CHead c k t) \Rightarrow (\lambda (H3: (drop i O c1 (CHead c k t))).(\lambda (H4: (clear (CHead c k t) (CHead d1 (Bind Abst) u1))).((match k return (\lambda (k0: K).((drop i O c1 (CHead c k0 t)) \to ((clear (CHead c k0 t) (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) with [(Bind b) \Rightarrow (\lambda (H5: (drop i O c1 (CHead c (Bind b) t))).(\lambda (H6: (clear (CHead c (Bind b) t) (CHead d1 (Bind Abst) u1))).(let H7 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) u1) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abst) u1) t H6)) in ((let H8 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d1 (Bind Abst) u1) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abst) u1) t H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind Abst) u1) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abst) u1) t H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 c)).(\lambda (c2: C).(\lambda (H12: (csuba g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda (t: T).(drop i O c1 (CHead c (Bind b) t))) H5 u1 H9) in (let H14 \def (eq_ind_r B b (\lambda (b: B).(drop i O c1 (CHead c (Bind b) u1))) H13 Abst H10) in (let H15 \def (eq_ind_r C c (\lambda (c: C).(drop i O c1 (CHead c (Bind Abst) u1))) H14 d1 H11) in (let H16 \def (csuba_drop_abst i c1 d1 u1 H15 g c2 H12) in (or_ind (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H17: (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (H18: (drop i O c2 (CHead x0 (Bind Abst) u1))).(\lambda (H19: (csuba g d1 x0)).(or_introl (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x0 (getl_intro i c2 (CHead x0 (Bind Abst) u1) (CHead x0 (Bind Abst) u1) H18 (clear_bind Abst x0 u1)) H19))))) H17)) (\lambda (H17: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H18: (drop i O c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H19: (csuba g d1 x0)).(\lambda (H20: (arity g d1 u1 (asucc g x2))).(\lambda (H21: (arity g x0 x1 x2)).(or_intror (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (getl_intro i c2 (CHead x0 (Bind Abbr) x1) (CHead x0 (Bind Abbr) x1) H18 (clear_bind Abbr x0 x1)) H19 H20 H21))))))))) H17)) H16)))))))))) H8)) H7)))) | (Flat f) \Rightarrow (\lambda (H5: (drop i O c1 (CHead c (Flat f) t))).(\lambda (H6: (clear (CHead c (Flat f) t) (CHead d1 (Bind Abst) u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c0: C).((drop i O c0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csuba g c0 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))) (nat_ind (\lambda (n: nat).(\forall (x0: C).((drop n O x0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csuba g x0 c2) \to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) (\lambda (x0: C).(\lambda (H8: (drop O O x0 (CHead c (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g x0 c2)).(let H10 \def (eq_ind C x0 (\lambda (c: C).(csuba g c c2)) H9 (CHead c (Flat f) t) (drop_gen_refl x0 (CHead c (Flat f) t) H8)) in (let H_y \def (clear_flat c (CHead d1 (Bind Abst) u1) (clear_gen_flat f c (CHead d1 (Bind Abst) u1) t H6) f t) in (let H11 \def (csuba_clear_conf g (CHead c (Flat f) t) c2 H10 (CHead d1 (Bind Abst) u1) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1 (Bind Abst) u1) e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x1: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abst) u1) x1)).(\lambda (H13: (clear c2 x1)).(let H14 \def (csuba_gen_abst g d1 x1 u1 H12) in (or_ind (ex2 C (\lambda (d2: C).(eq C x1 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C x1 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H15: (ex2 C (\lambda (d2: C).(eq C x1 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(eq C x1 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x2: C).(\lambda (H16: (eq C x1 (CHead x2 (Bind Abst) u1))).(\lambda (H17: (csuba g d1 x2)).(let H18 \def (eq_ind C x1 (\lambda (c: C).(clear c2 c)) H13 (CHead x2 (Bind Abst) u1) H16) in (or_introl (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x2 (getl_intro O c2 (CHead x2 (Bind Abst) u1) c2 (drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C x1 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C x1 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: A).(\lambda (H16: (eq C x1 (CHead x2 (Bind Abbr) x3))).(\lambda (H17: (csuba g d1 x2)).(\lambda (H18: (arity g d1 u1 (asucc g x4))).(\lambda (H19: (arity g x2 x3 x4)).(let H20 \def (eq_ind C x1 (\lambda (c: C).(clear c2 c)) H13 (CHead x2 (Bind Abbr) x3) H16) in (or_intror (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x2 x3 x4 (getl_intro O c2 (CHead x2 (Bind Abbr) x3) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15)) H14))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x: C).((drop n O x (CHead c (Flat f) t)) \to (\forall (c2: C).((csuba g x c2) \to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))))).(\lambda (x0: C).(\lambda (H9: (drop (S n) O x0 (CHead c (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g x0 c2)).(let H11 \def (drop_clear x0 (CHead c (Flat f) t) n H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x0 (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead c (Flat f) t))))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (clear x0 (CHead x2 (Bind x1) x3))).(\lambda (H13: (drop n O x2 (CHead c (Flat f) t))).(let H14 \def (csuba_clear_conf g x0 c2 H10 (CHead x2 (Bind x1) x3) H12) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead x2 (Bind x1) x3) e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x4: C).(\lambda (H15: (csuba g (CHead x2 (Bind x1) x3) x4)).(\lambda (H16: (clear c2 x4)).(let H17 \def (csuba_gen_bind g x1 x2 x4 x3 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x4 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g x2 e2)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: T).(\lambda (H18: (eq C x4 (CHead x6 (Bind x5) x7))).(\lambda (H19: (csuba g x2 x6)).(let H20 \def (eq_ind C x4 (\lambda (c: C).(clear c2 c)) H16 (CHead x6 (Bind x5) x7) H18) in (let H21 \def (H8 x2 H13 x6 H19) in (or_ind (ex2 C (\lambda (d2: C).(getl n x6 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x6 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(getl n x6 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(getl n x6 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x8: C).(\lambda (H23: (getl n x6 (CHead x8 (Bind Abst) u1))).(\lambda (H24: (csuba g d1 x8)).(or_introl (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x8 (getl_clear_bind x5 c2 x6 x7 H20 (CHead x8 (Bind Abst) u1) n H23) H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x6 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x6 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x8: C).(\lambda (x9: T).(\lambda (x10: A).(\lambda (H23: (getl n x6 (CHead x8 (Bind Abbr) x9))).(\lambda (H24: (csuba g d1 x8)).(\lambda (H25: (arity g d1 u1 (asucc g x10))).(\lambda (H26: (arity g x8 x9 x10)).(or_intror (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x8 x9 x10 (getl_clear_bind x5 c2 x6 x7 H20 (CHead x8 (Bind Abbr) x9) n H23) H24 H25 H26))))))))) H22)) H21)))))))) H17))))) H14))))))) H11)))))))) i) H7))))]) H3 H4)))]) H1 H2)))) H0))))))).
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u1: T).(\lambda (i: nat).(\lambda (H: (getl i c1 (CHead d1 (Bind Abst) u1))).(let H0 \def (getl_gen_all c1 (CHead d1 (Bind Abst) u1) i H) in (ex2_ind C (\lambda (e: C).(drop i O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) u1))) (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))) (\lambda (x: C).(\lambda (H1: (drop i O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind Abst) u1))).((match x return (\lambda (_: ?).(\lambda (c: C).((drop i O c1 c) \to ((clear c (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))))) with [(CSort n) \Rightarrow (\lambda (_: (drop i O c1 (CSort n))).(\lambda (H4: (clear (CSort n) (CHead d1 (Bind Abst) u1))).(clear_gen_sort (CHead d1 (Bind Abst) u1) n H4 (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) | (CHead c k t) \Rightarrow (\lambda (H3: (drop i O c1 (CHead c k t))).(\lambda (H4: (clear (CHead c k t) (CHead d1 (Bind Abst) u1))).((match k return (\lambda (_: ?).(\lambda (k0: K).((drop i O c1 (CHead c k0 t)) \to ((clear (CHead c k0 t) (CHead d1 (Bind Abst) u1)) \to (\forall (c2: C).((csuba g c1 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))))) with [(Bind b) \Rightarrow (\lambda (H5: (drop i O c1 (CHead c (Bind b) t))).(\lambda (H6: (clear (CHead c (Bind b) t) (CHead d1 (Bind Abst) u1))).(let H7 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) u1) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abst) u1) t H6)) in ((let H8 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d1 (Bind Abst) u1) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abst) u1) t H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind Abst) u1) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abst) u1) t H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 c)).(\lambda (c2: C).(\lambda (H12: (csuba g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda (t: T).(drop i O c1 (CHead c (Bind b) t))) H5 u1 H9) in (let H14 \def (eq_ind_r B b (\lambda (b: B).(drop i O c1 (CHead c (Bind b) u1))) H13 Abst H10) in (let H15 \def (eq_ind_r C c (\lambda (c: C).(drop i O c1 (CHead c (Bind Abst) u1))) H14 d1 H11) in (let H16 \def (csuba_drop_abst i c1 d1 u1 H15 g c2 H12) in (or_ind (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H17: (ex2 C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(drop i O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (H18: (drop i O c2 (CHead x0 (Bind Abst) u1))).(\lambda (H19: (csuba g d1 x0)).(or_introl (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x0 (getl_intro i c2 (CHead x0 (Bind Abst) u1) (CHead x0 (Bind Abst) u1) H18 (clear_bind Abst x0 u1)) H19))))) H17)) (\lambda (H17: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(drop i O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: A).(\lambda (H18: (drop i O c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H19: (csuba g d1 x0)).(\lambda (H20: (arity g d1 u1 (asucc g x2))).(\lambda (H21: (arity g x0 x1 x2)).(or_intror (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x0 x1 x2 (getl_intro i c2 (CHead x0 (Bind Abbr) x1) (CHead x0 (Bind Abbr) x1) H18 (clear_bind Abbr x0 x1)) H19 H20 H21))))))))) H17)) H16)))))))))) H8)) H7)))) | (Flat f) \Rightarrow (\lambda (H5: (drop i O c1 (CHead c (Flat f) t))).(\lambda (H6: (clear (CHead c (Flat f) t) (CHead d1 (Bind Abst) u1))).(let H7 \def H5 in (unintro C c1 (\lambda (c0: C).((drop i O c0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csuba g c0 c2) \to (or (ex2 C (\lambda (d2: C).(getl i c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl i c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))) (nat_ind (\lambda (n: nat).(\forall (x0: C).((drop n O x0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csuba g x0 c2) \to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))))))))) (\lambda (x0: C).(\lambda (H8: (drop O O x0 (CHead c (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csuba g x0 c2)).(let H10 \def (eq_ind C x0 (\lambda (c: C).(csuba g c c2)) H9 (CHead c (Flat f) t) (drop_gen_refl x0 (CHead c (Flat f) t) H8)) in (let H_y \def (clear_flat c (CHead d1 (Bind Abst) u1) (clear_gen_flat f c (CHead d1 (Bind Abst) u1) t H6) f t) in (let H11 \def (csuba_clear_conf g (CHead c (Flat f) t) c2 H10 (CHead d1 (Bind Abst) u1) H_y) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead d1 (Bind Abst) u1) e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x1: C).(\lambda (H12: (csuba g (CHead d1 (Bind Abst) u1) x1)).(\lambda (H13: (clear c2 x1)).(let H14 \def (csuba_gen_abst g d1 x1 u1 H12) in (or_ind (ex2 C (\lambda (d2: C).(eq C x1 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C x1 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H15: (ex2 C (\lambda (d2: C).(eq C x1 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(eq C x1 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x2: C).(\lambda (H16: (eq C x1 (CHead x2 (Bind Abst) u1))).(\lambda (H17: (csuba g d1 x2)).(let H18 \def (eq_ind C x1 (\lambda (c: C).(clear c2 c)) H13 (CHead x2 (Bind Abst) u1) H16) in (or_introl (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x2 (getl_intro O c2 (CHead x2 (Bind Abst) u1) c2 (drop_refl c2) H18) H17)))))) H15)) (\lambda (H15: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C x1 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(eq C x1 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (x4: A).(\lambda (H16: (eq C x1 (CHead x2 (Bind Abbr) x3))).(\lambda (H17: (csuba g d1 x2)).(\lambda (H18: (arity g d1 u1 (asucc g x4))).(\lambda (H19: (arity g x2 x3 x4)).(let H20 \def (eq_ind C x1 (\lambda (c: C).(clear c2 c)) H13 (CHead x2 (Bind Abbr) x3) H16) in (or_intror (ex2 C (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl O c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x2 x3 x4 (getl_intro O c2 (CHead x2 (Bind Abbr) x3) c2 (drop_refl c2) H20) H17 H18 H19)))))))))) H15)) H14))))) H11)))))))) (\lambda (n: nat).(\lambda (H8: ((\forall (x: C).((drop n O x (CHead c (Flat f) t)) \to (\forall (c2: C).((csuba g x c2) \to (or (ex2 C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))))))))).(\lambda (x0: C).(\lambda (H9: (drop (S n) O x0 (CHead c (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csuba g x0 c2)).(let H11 \def (drop_clear x0 (CHead c (Flat f) t) n H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x0 (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n O e (CHead c (Flat f) t))))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (clear x0 (CHead x2 (Bind x1) x3))).(\lambda (H13: (drop n O x2 (CHead c (Flat f) t))).(let H14 \def (csuba_clear_conf g x0 c2 H10 (CHead x2 (Bind x1) x3) H12) in (ex2_ind C (\lambda (e2: C).(csuba g (CHead x2 (Bind x1) x3) e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x4: C).(\lambda (H15: (csuba g (CHead x2 (Bind x1) x3) x4)).(\lambda (H16: (clear c2 x4)).(let H17 \def (csuba_gen_bind g x1 x2 x4 x3 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x4 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csuba g x2 e2)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: T).(\lambda (H18: (eq C x4 (CHead x6 (Bind x5) x7))).(\lambda (H19: (csuba g x2 x6)).(let H20 \def (eq_ind C x4 (\lambda (c: C).(clear c2 c)) H16 (CHead x6 (Bind x5) x7) H18) in (let H21 \def (H8 x2 H13 x6 H19) in (or_ind (ex2 C (\lambda (d2: C).(getl n x6 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x6 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(getl n x6 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)))).(ex2_ind C (\lambda (d2: C).(getl n x6 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x8: C).(\lambda (H23: (getl n x6 (CHead x8 (Bind Abst) u1))).(\lambda (H24: (csuba g d1 x8)).(or_introl (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex_intro2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2)) x8 (getl_clear_bind x5 c2 x6 x7 H20 (CHead x8 (Bind Abst) u1) n H23) H24))))) H22)) (\lambda (H22: (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x6 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))).(ex4_3_ind C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl n x6 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) (or (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))))) (\lambda (x8: C).(\lambda (x9: T).(\lambda (x10: A).(\lambda (H23: (getl n x6 (CHead x8 (Bind Abbr) x9))).(\lambda (H24: (csuba g d1 x8)).(\lambda (H25: (arity g d1 u1 (asucc g x10))).(\lambda (H26: (arity g x8 x9 x10)).(or_intror (ex2 C (\lambda (d2: C).(getl (S n) c2 (CHead d2 (Bind Abst) u1))) (\lambda (d2: C).(csuba g d1 d2))) (ex4_3 C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a))))) (ex4_3_intro C T A (\lambda (d2: C).(\lambda (u2: T).(\lambda (_: A).(getl (S n) c2 (CHead d2 (Bind Abbr) u2))))) (\lambda (d2: C).(\lambda (_: T).(\lambda (_: A).(csuba g d1 d2)))) (\lambda (_: C).(\lambda (_: T).(\lambda (a: A).(arity g d1 u1 (asucc g a))))) (\lambda (d2: C).(\lambda (u2: T).(\lambda (a: A).(arity g d2 u2 a)))) x8 x9 x10 (getl_clear_bind x5 c2 x6 x7 H20 (CHead x8 (Bind Abbr) x9) n H23) H24 H25 H26))))))))) H22)) H21)))))))) H17))))) H14))))))) H11)))))))) i) H7))))]) H3 H4)))]) H1 H2)))) H0))))))).
 
 theorem csuba_arity:
  \forall (g: G).(\forall (c1: C).(\forall (t: T).(\forall (a: A).((arity g c1 t a) \to (\forall (c2: C).((csuba g c1 c2) \to (arity g c2 t a)))))))
@@ -2432,7 +2432,7 @@ theorem nf2_lift:
 theorem nf2_lift1:
  \forall (e: C).(\forall (hds: PList).(\forall (c: C).(\forall (t: T).((drop1 hds c e) \to ((nf2 e t) \to (nf2 c (lift1 hds t)))))))
 \def
- \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c: C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p t))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c e)).(\lambda (H0: (nf2 e t)).(let H1 \def (match H return (\lambda (p: PList).(\lambda (c0: C).(\lambda (c1: C).((eq PList p PNil) \to ((eq C c0 c) \to ((eq C c1 e) \to (nf2 c t))))))) with [(drop1_nil c0) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq C c0 e)).(eq_ind C c (\lambda (c1: C).((eq C c1 e) \to (nf2 c t))) (\lambda (H4: (eq C c e)).(eq_ind C e (\lambda (c: C).(nf2 c t)) H0 c (sym_eq C c e H4))) c0 (sym_eq C c0 c H2) H3)))) | (drop1_cons c1 c2 h d H1 c3 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c1 c)).(\lambda (H5: (eq C c3 e)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e0: PList).(match e0 return (\lambda (_: ?).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c1 c) \to ((eq C c3 e) \to ((drop h d c1 c2) \to ((drop1 hds c2 c3) \to (nf2 c t))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c) (refl_equal C e))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p t)))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (drop1 (PCons n n0 p) c e)).(\lambda (H1: (nf2 e t)).(let H2 \def (match H0 return (\lambda (p0: PList).(\lambda (c0: C).(\lambda (c1: C).((eq PList p0 (PCons n n0 p)) \to ((eq C c0 c) \to ((eq C c1 e) \to (nf2 c (lift n n0 (lift1 p t))))))))) with [(drop1_nil c0) \Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq C c0 e)).((let H5 \def (eq_ind PList PNil (\lambda (e0: PList).(match e0 return (\lambda (_: ?).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c0 c) \to ((eq C c0 e) \to (nf2 c (lift n n0 (lift1 p t))))) H5)) H3 H4)))) | (drop1_cons c1 c2 h d H2 c3 hds H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds) (PCons n n0 p))).(\lambda (H5: (eq C c1 c)).(\lambda (H6: (eq C c3 e)).((let H7 \def (f_equal PList PList (\lambda (e0: PList).(match e0 return (\lambda (_: ?).PList) with [PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat (\lambda (e0: PList).(match e0 return (\lambda (_: ?).nat) with [PNil \Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e0: PList).(match e0 return (\lambda (_: ?).nat) with [PNil \Rightarrow h | (PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n1 d c1 c2) \to ((drop1 hds c2 c3) \to (nf2 c (lift n n0 (lift1 p t)))))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n n1 c1 c2) \to ((drop1 hds c2 c3) \to (nf2 c (lift n n0 (lift1 p t))))))))) (\lambda (H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c1 c) \to ((eq C c3 e) \to ((drop n n0 c1 c2) \to ((drop1 p0 c2 c3) \to (nf2 c (lift n n0 (lift1 p t)))))))) (\lambda (H12: (eq C c1 c)).(eq_ind C c (\lambda (c0: C).((eq C c3 e) \to ((drop n n0 c0 c2) \to ((drop1 p c2 c3) \to (nf2 c (lift n n0 (lift1 p t))))))) (\lambda (H13: (eq C c3 e)).(eq_ind C e (\lambda (c0: C).((drop n n0 c c2) \to ((drop1 p c2 c0) \to (nf2 c (lift n n0 (lift1 p t)))))) (\lambda (H14: (drop n n0 c c2)).(\lambda (H15: (drop1 p c2 e)).(nf2_lift c2 (lift1 p t) (H c2 t H15 H1) c n n0 H14))) c3 (sym_eq C c3 e H13))) c1 (sym_eq C c1 c H12))) hds (sym_eq PList hds p H11))) d (sym_eq nat d n0 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p)) (refl_equal C c) (refl_equal C e))))))))))) hds)).
+ \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c: C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p t))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c e)).(\lambda (H0: (nf2 e t)).(let H1 \def (match H return (\lambda (_: ?).(\lambda (p: PList).(\lambda (c0: C).(\lambda (c1: C).((eq PList p PNil) \to ((eq C c0 c) \to ((eq C c1 e) \to (nf2 c t)))))))) with [(drop1_nil c0) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq C c0 e)).(eq_ind C c (\lambda (c1: C).((eq C c1 e) \to (nf2 c t))) (\lambda (H4: (eq C c e)).(eq_ind C e (\lambda (c: C).(nf2 c t)) H0 c (sym_eq C c e H4))) c0 (sym_eq C c0 c H2) H3)))) | (drop1_cons c1 c2 h d H1 c3 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c1 c)).(\lambda (H5: (eq C c3 e)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e0: PList).(match e0 return (\lambda (_: ?).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c1 c) \to ((eq C c3 e) \to ((drop h d c1 c2) \to ((drop1 hds c2 c3) \to (nf2 c t))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c) (refl_equal C e))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p t)))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (drop1 (PCons n n0 p) c e)).(\lambda (H1: (nf2 e t)).(let H2 \def (match H0 return (\lambda (_: ?).(\lambda (p0: PList).(\lambda (c0: C).(\lambda (c1: C).((eq PList p0 (PCons n n0 p)) \to ((eq C c0 c) \to ((eq C c1 e) \to (nf2 c (lift n n0 (lift1 p t)))))))))) with [(drop1_nil c0) \Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq C c0 e)).((let H5 \def (eq_ind PList PNil (\lambda (e0: PList).(match e0 return (\lambda (_: ?).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c0 c) \to ((eq C c0 e) \to (nf2 c (lift n n0 (lift1 p t))))) H5)) H3 H4)))) | (drop1_cons c1 c2 h d H2 c3 hds H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds) (PCons n n0 p))).(\lambda (H5: (eq C c1 c)).(\lambda (H6: (eq C c3 e)).((let H7 \def (f_equal PList PList (\lambda (e0: PList).(match e0 return (\lambda (_: ?).PList) with [PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat (\lambda (e0: PList).(match e0 return (\lambda (_: ?).nat) with [PNil \Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e0: PList).(match e0 return (\lambda (_: ?).nat) with [PNil \Rightarrow h | (PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n1 d c1 c2) \to ((drop1 hds c2 c3) \to (nf2 c (lift n n0 (lift1 p t)))))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n n1 c1 c2) \to ((drop1 hds c2 c3) \to (nf2 c (lift n n0 (lift1 p t))))))))) (\lambda (H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c1 c) \to ((eq C c3 e) \to ((drop n n0 c1 c2) \to ((drop1 p0 c2 c3) \to (nf2 c (lift n n0 (lift1 p t)))))))) (\lambda (H12: (eq C c1 c)).(eq_ind C c (\lambda (c0: C).((eq C c3 e) \to ((drop n n0 c0 c2) \to ((drop1 p c2 c3) \to (nf2 c (lift n n0 (lift1 p t))))))) (\lambda (H13: (eq C c3 e)).(eq_ind C e (\lambda (c0: C).((drop n n0 c c2) \to ((drop1 p c2 c0) \to (nf2 c (lift n n0 (lift1 p t)))))) (\lambda (H14: (drop n n0 c c2)).(\lambda (H15: (drop1 p c2 e)).(nf2_lift c2 (lift1 p t) (H c2 t H15 H1) c n n0 H14))) c3 (sym_eq C c3 e H13))) c1 (sym_eq C c1 c H12))) hds (sym_eq PList hds p H11))) d (sym_eq nat d n0 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p)) (refl_equal C c) (refl_equal C e))))))))))) hds)).
 
 theorem nf2_iso_appls_lref:
  \forall (c: C).(\forall (i: nat).((nf2 c (TLRef i)) \to (\forall (vs: TList).(\forall (u: T).((pr3 c (THeads (Flat Appl) vs (TLRef i)) u) \to (iso (THeads (Flat Appl) vs (TLRef i)) u))))))
@@ -2442,7 +2442,7 @@ theorem nf2_iso_appls_lref:
 theorem nf2_dec:
  \forall (c: C).(\forall (t1: T).(or (nf2 c t1) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c t1 t2)))))
 \def
- \lambda (c: C).(c_tail_ind (\lambda (c0: C).(\forall (t1: T).(or (\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))))) (\lambda (n: nat).(\lambda (t1: T).(let H_x \def (nf0_dec t1) in (let H \def H_x in (or_ind (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2))) (or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CSort n) t1 t2)))) (\lambda (H0: ((\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))))).(or_introl (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CSort n) t1 t2))) (\lambda (t2: T).(\lambda (H1: (pr2 (CSort n) t1 t2)).(let H_y \def (pr2_gen_csort t1 t2 n H1) in (H0 t2 H_y)))))) (\lambda (H0: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)) (or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CSort n) t1 t2)))) (\lambda (x: T).(\lambda (H1: (((eq T t1 x) \to (\forall (P: Prop).P)))).(\lambda (H2: (pr0 t1 x)).(or_intror (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CSort n) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CSort n) t1 t2)) x H1 (pr2_free (CSort n) t1 x H2)))))) H0)) H))))) (\lambda (c0: C).(\lambda (H: ((\forall (t1: T).(or (\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (t1: T).(let H_x \def (H t1) in (let H0 \def H_x in (or_ind (\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2))) (or (\forall (t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (H1: ((\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))))).(match k return (\lambda (k0: K).(or (\forall (t2: T).((pr2 (CTail k0 t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k0 t c0) t1 t2))))) with [(Bind b) \Rightarrow (match b return (\lambda (b0: B).(or (\forall (t2: T).((pr2 (CTail (Bind b0) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind b0) t c0) t1 t2))))) with [Abbr \Rightarrow (let H_x0 \def (dnf_dec t t1 (clen c0)) in (let H2 \def H_x0 in (ex_ind T (\lambda (v: T).(or (subst0 (clen c0) t t1 (lift (S O) (clen c0) v)) (eq T t1 (lift (S O) (clen c0) v)))) (or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)))) (\lambda (x: T).(\lambda (H3: (or (subst0 (clen c0) t t1 (lift (S O) (clen c0) x)) (eq T t1 (lift (S O) (clen c0) x)))).(or_ind (subst0 (clen c0) t t1 (lift (S O) (clen c0) x)) (eq T t1 (lift (S O) (clen c0) x)) (or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)))) (\lambda (H4: (subst0 (clen c0) t t1 (lift (S O) (clen c0) x))).(let H_x1 \def (getl_ctail_clen Abbr t c0) in (let H5 \def H_x1 in (ex_ind nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind Abbr) t c0) (CHead (CSort n) (Bind Abbr) t))) (or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)))) (\lambda (x0: nat).(\lambda (H6: (getl (clen c0) (CTail (Bind Abbr) t c0) (CHead (CSort x0) (Bind Abbr) t))).(or_intror (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)) (lift (S O) (clen c0) x) (\lambda (H7: (eq T t1 (lift (S O) (clen c0) x))).(\lambda (P: Prop).(let H8 \def (eq_ind T t1 (\lambda (t0: T).(subst0 (clen c0) t t0 (lift (S O) (clen c0) x))) H4 (lift (S O) (clen c0) x) H7) in (subst0_gen_lift_false x t (lift (S O) (clen c0) x) (S O) (clen c0) (clen c0) (le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0)) (\lambda (n: nat).(lt (clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen c0) (S O)) (plus_comm (clen c0) (S O))) H8 P)))) (pr2_delta (CTail (Bind Abbr) t c0) (CSort x0) t (clen c0) H6 t1 t1 (pr0_refl t1) (lift (S O) (clen c0) x) H4))))) H5)))) (\lambda (H4: (eq T t1 (lift (S O) (clen c0) x))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: T).((pr2 c0 t t2) \to (eq T t t2)))) H1 (lift (S O) (clen c0) x) H4) in (eq_ind_r T (lift (S O) (clen c0) x) (\lambda (t0: T).(or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t0 t2))))) (or_introl (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2) \to (eq T (lift (S O) (clen c0) x) t2))) (ex2 T (\lambda (t2: T).((eq T (lift (S O) (clen c0) x) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2))) (\lambda (t2: T).(\lambda (H6: (pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2)).(let H_x1 \def (pr2_gen_ctail (Bind Abbr) c0 t (lift (S O) (clen c0) x) t2 H6) in (let H7 \def H_x1 in (or_ind (pr2 c0 (lift (S O) (clen c0) x) t2) (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T (lift (S O) (clen c0) x) t2) (\lambda (H8: (pr2 c0 (lift (S O) (clen c0) x) t2)).(H5 t2 H8)) (\lambda (H8: (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t: T).(pr0 (lift (S O) (clen c0) x) t)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)) (eq T (lift (S O) (clen c0) x) t2) (\lambda (x0: T).(\lambda (_: (eq K (Bind Abbr) (Bind Abbr))).(\lambda (H10: (pr0 (lift (S O) (clen c0) x) x0)).(\lambda (H11: (subst0 (clen c0) t x0 t2)).(ex2_ind T (\lambda (t3: T).(eq T x0 (lift (S O) (clen c0) t3))) (\lambda (t3: T).(pr0 x t3)) (eq T (lift (S O) (clen c0) x) t2) (\lambda (x1: T).(\lambda (H12: (eq T x0 (lift (S O) (clen c0) x1))).(\lambda (_: (pr0 x x1)).(let H14 \def (eq_ind T x0 (\lambda (t0: T).(subst0 (clen c0) t t0 t2)) H11 (lift (S O) (clen c0) x1) H12) in (subst0_gen_lift_false x1 t t2 (S O) (clen c0) (clen c0) (le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0)) (\lambda (n: nat).(lt (clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen c0) (S O)) (plus_comm (clen c0) (S O))) H14 (eq T (lift (S O) (clen c0) x) t2)))))) (pr0_gen_lift x x0 (S O) (clen c0) H10)))))) H8)) H7)))))) t1 H4))) H3))) H2))) | Abst \Rightarrow (or_introl (\forall (t2: T).((pr2 (CTail (Bind Abst) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abst) t c0) t1 t2))) (\lambda (t2: T).(\lambda (H2: (pr2 (CTail (Bind Abst) t c0) t1 t2)).(let H_x0 \def (pr2_gen_ctail (Bind Abst) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind (pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T (\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: (eq K (Bind Abst) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: (subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Abst) (\lambda (ee: K).(match ee return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3)))))) | Void \Rightarrow (or_introl (\forall (t2: T).((pr2 (CTail (Bind Void) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Void) t c0) t1 t2))) (\lambda (t2: T).(\lambda (H2: (pr2 (CTail (Bind Void) t c0) t1 t2)).(let H_x0 \def (pr2_gen_ctail (Bind Void) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind (pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T (\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: (eq K (Bind Void) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: (subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Void) (\lambda (ee: K).(match ee return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3))))))]) | (Flat f) \Rightarrow (or_introl (\forall (t2: T).((pr2 (CTail (Flat f) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Flat f) t c0) t1 t2))) (\lambda (t2: T).(\lambda (H2: (pr2 (CTail (Flat f) t c0) t1 t2)).(let H_x0 \def (pr2_gen_ctail (Flat f) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind (pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T (\lambda (_: T).(eq K (Flat f) (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: (eq K (Flat f) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: (subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3))))))])) (\lambda (H1: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)) (or (\forall (t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (x: T).(\lambda (H2: (((eq T t1 x) \to (\forall (P: Prop).P)))).(\lambda (H3: (pr2 c0 t1 x)).(or_intror (\forall (t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)) x H2 (pr2_ctail c0 t1 x H3 k t)))))) H1)) H0)))))))) c).
+ \lambda (c: C).(c_tail_ind (\lambda (c0: C).(\forall (t1: T).(or (\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))))) (\lambda (n: nat).(\lambda (t1: T).(let H_x \def (nf0_dec t1) in (let H \def H_x in (or_ind (\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2))) (or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CSort n) t1 t2)))) (\lambda (H0: ((\forall (t2: T).((pr0 t1 t2) \to (eq T t1 t2))))).(or_introl (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CSort n) t1 t2))) (\lambda (t2: T).(\lambda (H1: (pr2 (CSort n) t1 t2)).(let H_y \def (pr2_gen_csort t1 t2 n H1) in (H0 t2 H_y)))))) (\lambda (H0: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr0 t1 t2)) (or (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CSort n) t1 t2)))) (\lambda (x: T).(\lambda (H1: (((eq T t1 x) \to (\forall (P: Prop).P)))).(\lambda (H2: (pr0 t1 x)).(or_intror (\forall (t2: T).((pr2 (CSort n) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CSort n) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CSort n) t1 t2)) x H1 (pr2_free (CSort n) t1 x H2)))))) H0)) H))))) (\lambda (c0: C).(\lambda (H: ((\forall (t1: T).(or (\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (t1: T).(let H_x \def (H t1) in (let H0 \def H_x in (or_ind (\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2))) (or (\forall (t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (H1: ((\forall (t2: T).((pr2 c0 t1 t2) \to (eq T t1 t2))))).(match k return (\lambda (_: ?).(\lambda (k0: K).(or (\forall (t2: T).((pr2 (CTail k0 t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k0 t c0) t1 t2)))))) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).(\lambda (b0: B).(or (\forall (t2: T).((pr2 (CTail (Bind b0) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind b0) t c0) t1 t2)))))) with [Abbr \Rightarrow (let H_x0 \def (dnf_dec t t1 (clen c0)) in (let H2 \def H_x0 in (ex_ind T (\lambda (v: T).(or (subst0 (clen c0) t t1 (lift (S O) (clen c0) v)) (eq T t1 (lift (S O) (clen c0) v)))) (or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)))) (\lambda (x: T).(\lambda (H3: (or (subst0 (clen c0) t t1 (lift (S O) (clen c0) x)) (eq T t1 (lift (S O) (clen c0) x)))).(or_ind (subst0 (clen c0) t t1 (lift (S O) (clen c0) x)) (eq T t1 (lift (S O) (clen c0) x)) (or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)))) (\lambda (H4: (subst0 (clen c0) t t1 (lift (S O) (clen c0) x))).(let H_x1 \def (getl_ctail_clen Abbr t c0) in (let H5 \def H_x1 in (ex_ind nat (\lambda (n: nat).(getl (clen c0) (CTail (Bind Abbr) t c0) (CHead (CSort n) (Bind Abbr) t))) (or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)))) (\lambda (x0: nat).(\lambda (H6: (getl (clen c0) (CTail (Bind Abbr) t c0) (CHead (CSort x0) (Bind Abbr) t))).(or_intror (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t1 t2)) (lift (S O) (clen c0) x) (\lambda (H7: (eq T t1 (lift (S O) (clen c0) x))).(\lambda (P: Prop).(let H8 \def (eq_ind T t1 (\lambda (t0: T).(subst0 (clen c0) t t0 (lift (S O) (clen c0) x))) H4 (lift (S O) (clen c0) x) H7) in (subst0_gen_lift_false x t (lift (S O) (clen c0) x) (S O) (clen c0) (clen c0) (le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0)) (\lambda (n: nat).(lt (clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen c0) (S O)) (plus_comm (clen c0) (S O))) H8 P)))) (pr2_delta (CTail (Bind Abbr) t c0) (CSort x0) t (clen c0) H6 t1 t1 (pr0_refl t1) (lift (S O) (clen c0) x) H4))))) H5)))) (\lambda (H4: (eq T t1 (lift (S O) (clen c0) x))).(let H5 \def (eq_ind T t1 (\lambda (t: T).(\forall (t2: T).((pr2 c0 t t2) \to (eq T t t2)))) H1 (lift (S O) (clen c0) x) H4) in (eq_ind_r T (lift (S O) (clen c0) x) (\lambda (t0: T).(or (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) t0 t2) \to (eq T t0 t2))) (ex2 T (\lambda (t2: T).((eq T t0 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) t0 t2))))) (or_introl (\forall (t2: T).((pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2) \to (eq T (lift (S O) (clen c0) x) t2))) (ex2 T (\lambda (t2: T).((eq T (lift (S O) (clen c0) x) t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2))) (\lambda (t2: T).(\lambda (H6: (pr2 (CTail (Bind Abbr) t c0) (lift (S O) (clen c0) x) t2)).(let H_x1 \def (pr2_gen_ctail (Bind Abbr) c0 t (lift (S O) (clen c0) x) t2 H6) in (let H7 \def H_x1 in (or_ind (pr2 c0 (lift (S O) (clen c0) x) t2) (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T (lift (S O) (clen c0) x) t2) (\lambda (H8: (pr2 c0 (lift (S O) (clen c0) x) t2)).(H5 t2 H8)) (\lambda (H8: (ex3 T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t: T).(pr0 (lift (S O) (clen c0) x) t)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Bind Abbr) (Bind Abbr))) (\lambda (t0: T).(pr0 (lift (S O) (clen c0) x) t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)) (eq T (lift (S O) (clen c0) x) t2) (\lambda (x0: T).(\lambda (_: (eq K (Bind Abbr) (Bind Abbr))).(\lambda (H10: (pr0 (lift (S O) (clen c0) x) x0)).(\lambda (H11: (subst0 (clen c0) t x0 t2)).(ex2_ind T (\lambda (t3: T).(eq T x0 (lift (S O) (clen c0) t3))) (\lambda (t3: T).(pr0 x t3)) (eq T (lift (S O) (clen c0) x) t2) (\lambda (x1: T).(\lambda (H12: (eq T x0 (lift (S O) (clen c0) x1))).(\lambda (_: (pr0 x x1)).(let H14 \def (eq_ind T x0 (\lambda (t0: T).(subst0 (clen c0) t t0 t2)) H11 (lift (S O) (clen c0) x1) H12) in (subst0_gen_lift_false x1 t t2 (S O) (clen c0) (clen c0) (le_n (clen c0)) (eq_ind_r nat (plus (S O) (clen c0)) (\lambda (n: nat).(lt (clen c0) n)) (le_n (plus (S O) (clen c0))) (plus (clen c0) (S O)) (plus_comm (clen c0) (S O))) H14 (eq T (lift (S O) (clen c0) x) t2)))))) (pr0_gen_lift x x0 (S O) (clen c0) H10)))))) H8)) H7)))))) t1 H4))) H3))) H2))) | Abst \Rightarrow (or_introl (\forall (t2: T).((pr2 (CTail (Bind Abst) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Abst) t c0) t1 t2))) (\lambda (t2: T).(\lambda (H2: (pr2 (CTail (Bind Abst) t c0) t1 t2)).(let H_x0 \def (pr2_gen_ctail (Bind Abst) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind (pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T (\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Bind Abst) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: (eq K (Bind Abst) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: (subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Abst) (\lambda (ee: K).(match ee return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3)))))) | Void \Rightarrow (or_introl (\forall (t2: T).((pr2 (CTail (Bind Void) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Bind Void) t c0) t1 t2))) (\lambda (t2: T).(\lambda (H2: (pr2 (CTail (Bind Void) t c0) t1 t2)).(let H_x0 \def (pr2_gen_ctail (Bind Void) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind (pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T (\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Bind Void) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: (eq K (Bind Void) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: (subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Bind Void) (\lambda (ee: K).(match ee return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3))))))]) | (Flat f) \Rightarrow (or_introl (\forall (t2: T).((pr2 (CTail (Flat f) t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail (Flat f) t c0) t1 t2))) (\lambda (t2: T).(\lambda (H2: (pr2 (CTail (Flat f) t c0) t1 t2)).(let H_x0 \def (pr2_gen_ctail (Flat f) c0 t t1 t2 H2) in (let H3 \def H_x0 in (or_ind (pr2 c0 t1 t2) (ex3 T (\lambda (_: T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2))) (eq T t1 t2) (\lambda (H4: (pr2 c0 t1 t2)).(H1 t2 H4)) (\lambda (H4: (ex3 T (\lambda (_: T).(eq K (Flat f) (Bind Abbr))) (\lambda (t: T).(pr0 t1 t)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)))).(ex3_ind T (\lambda (_: T).(eq K (Flat f) (Bind Abbr))) (\lambda (t0: T).(pr0 t1 t0)) (\lambda (t0: T).(subst0 (clen c0) t t0 t2)) (eq T t1 t2) (\lambda (x0: T).(\lambda (H5: (eq K (Flat f) (Bind Abbr))).(\lambda (_: (pr0 t1 x0)).(\lambda (_: (subst0 (clen c0) t x0 t2)).(let H8 \def (eq_ind K (Flat f) (\lambda (ee: K).(match ee return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])) I (Bind Abbr) H5) in (False_ind (eq T t1 t2) H8)))))) H4)) H3))))))])) (\lambda (H1: (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)))).(ex2_ind T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 c0 t1 t2)) (or (\forall (t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)))) (\lambda (x: T).(\lambda (H2: (((eq T t1 x) \to (\forall (P: Prop).P)))).(\lambda (H3: (pr2 c0 t1 x)).(or_intror (\forall (t2: T).((pr2 (CTail k t c0) t1 t2) \to (eq T t1 t2))) (ex2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2))) (ex_intro2 T (\lambda (t2: T).((eq T t1 t2) \to (\forall (P: Prop).P))) (\lambda (t2: T).(pr2 (CTail k t c0) t1 t2)) x H2 (pr2_ctail c0 t1 x H3 k t)))))) H1)) H0)))))))) c).
 
 inductive sn3 (c:C): T \to Prop \def
 | sn3_sing: \forall (t1: T).(((\forall (t2: T).((((eq T t1 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t1 t2) \to (sn3 c t2))))) \to (sn3 c t1)).
@@ -2495,7 +2495,7 @@ theorem sn3_appl_cast:
 theorem sn3_appl_appl:
  \forall (v1: T).(\forall (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in (\forall (c: C).((sn3 c u1) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 u1)))))))))
 \def
- \lambda (v1: T).(\lambda (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in (\lambda (c: C).(\lambda (H: (sn3 c (THead (Flat Appl) v1 t1))).(insert_eq T (THead (Flat Appl) v1 t1) (\lambda (t: T).(sn3 c t)) (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) v1 t1) u2) \to ((((iso (THead (Flat Appl) v1 t1) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) v1 t1)))))) (\lambda (y: T).(\lambda (H0: (sn3 c y)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Flat Appl) v1 t)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) v1 t) u2) \to ((((iso (THead (Flat Appl) v1 t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) v1 t)))))))) (unintro T v1 (\lambda (t: T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) t x) u2) \to ((((iso (THead (Flat Appl) t x) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) t x))))))))) (sn3_ind c (\lambda (t: T).(\forall (x: T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) x x0)))))))))) (\lambda (t2: T).(\lambda (H1: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H2: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x: T).(\forall (x0: T).((eq T t3 (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) x x0)))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t2 (THead (Flat Appl) x x0))).(\lambda (v2: T).(\lambda (H4: (sn3 c v2)).(sn3_ind c (\lambda (t: T).(((\forall (u2: T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t u2)))))) \to (sn3 c (THead (Flat Appl) t (THead (Flat Appl) x x0))))) (\lambda (t0: T).(\lambda (H5: ((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2)))))).(\lambda (H6: ((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t2 u2)))))) \to (sn3 c (THead (Flat Appl) t2 (THead (Flat Appl) x x0))))))))).(\lambda (H7: ((\forall (u2: T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2))))))).(let H8 \def (eq_ind T t2 (\lambda (t: T).(\forall (t2: T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) x x0))))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H9 \def (eq_ind T t2 (\lambda (t: T).(\forall (t2: T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (sn3 c t2))))) H1 (THead (Flat Appl) x x0) H3) in (sn3_pr2_intro c (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (\lambda (t3: T).(\lambda (H10: (((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H11: (pr2 c (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t3)).(let H12 \def (pr2_gen_appl c t0 (THead (Flat Appl) x x0) t3 H11) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Flat Appl) x x0) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (sn3 c t3) (\lambda (H13: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t3 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c (THead (Flat Appl) x x0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Flat Appl) x x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H14: (eq T t3 (THead (Flat Appl) x1 x2))).(\lambda (H15: (pr2 c t0 x1)).(\lambda (H16: (pr2 c (THead (Flat Appl) x x0) x2)).(let H17 \def (eq_ind T t3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P: Prop).P))) H10 (THead (Flat Appl) x1 x2) H14) in (eq_ind_r T (THead (Flat Appl) x1 x2) (\lambda (t: T).(sn3 c t)) (let H18 \def (pr2_gen_appl c x x0 x2 H16) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (H19: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x2 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c x0 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H20: (eq T x2 (THead (Flat Appl) x3 x4))).(\lambda (H21: (pr2 c x x3)).(\lambda (H22: (pr2 c x0 x4)).(let H23 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P: Prop).P))) H17 (THead (Flat Appl) x3 x4) H20) in (eq_ind_r T (THead (Flat Appl) x3 x4) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H_x \def (term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) in (let H24 \def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) ((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 x4))) (\lambda (H25: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))).(let H26 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H25) in ((let H27 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H25) in (\lambda (H28: (eq T x x3)).(let H29 \def (eq_ind_r T x4 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 (THead (Flat Appl) x3 t))) \to (\forall (P: Prop).P))) H23 x0 H27) in (let H30 \def (eq_ind_r T x4 (\lambda (t: T).(pr2 c x0 t)) H22 x0 H27) in (eq_ind T x0 (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 t)))) (let H31 \def (eq_ind_r T x3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 (THead (Flat Appl) t x0))) \to (\forall (P: Prop).P))) H29 x H28) in (let H32 \def (eq_ind_r T x3 (\lambda (t: T).(pr2 c x t)) H21 x H28) in (eq_ind T x (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) t x0)))) (let H_x0 \def (term_dec t0 x1) in (let H33 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x x0))) (\lambda (H34: (eq T t0 x1)).(let H35 \def (eq_ind_r T x1 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) t (THead (Flat Appl) x x0))) \to (\forall (P: Prop).P))) H31 t0 H34) in (let H36 \def (eq_ind_r T x1 (\lambda (t: T).(pr2 c t0 t)) H15 t0 H34) in (eq_ind T t0 (\lambda (t: T).(sn3 c (THead (Flat Appl) t (THead (Flat Appl) x x0)))) (H35 (refl_equal T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))) (sn3 c (THead (Flat Appl) t0 (THead (Flat Appl) x x0)))) x1 H34)))) (\lambda (H34: (((eq T t0 x1) \to (\forall (P: Prop).P)))).(H6 x1 H34 (pr3_pr2 c t0 x1 H15) (\lambda (u2: T).(\lambda (H35: (pr3 c (THead (Flat Appl) x x0) u2)).(\lambda (H36: (((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 u2) (H7 u2 H35 H36) (THead (Flat Appl) x1 u2) (pr3_pr2 c (THead (Flat Appl) t0 u2) (THead (Flat Appl) x1 u2) (pr2_head_1 c t0 x1 H15 (Flat Appl) u2)))))))) H33))) x3 H28))) x4 H27))))) H26))) (\lambda (H25: (((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) \to (\forall (P: Prop).P)))).(H8 (THead (Flat Appl) x3 x4) H25 (pr3_head_12 c x x3 (pr3_pr2 c x x3 H21) (Flat Appl) x0 x4 (pr3_pr2 (CHead c (Flat Appl) x3) x0 x4 (pr2_cflat c x0 x4 H22 Appl x3))) x3 x4 (refl_equal T (THead (Flat Appl) x3 x4)) x1 (sn3_pr3_trans c t0 (sn3_sing c t0 H5) x1 (pr3_pr2 c t0 x1 H15)) (\lambda (u2: T).(\lambda (H26: (pr3 c (THead (Flat Appl) x3 x4) u2)).(\lambda (H27: (((iso (THead (Flat Appl) x3 x4) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 u2) (H7 u2 (pr3_sing c (THead (Flat Appl) x x4) (THead (Flat Appl) x x0) (pr2_thin_dx c x0 x4 H22 x Appl) u2 (pr3_sing c (THead (Flat Appl) x3 x4) (THead (Flat Appl) x x4) (pr2_head_1 c x x3 H21 (Flat Appl) x4) u2 H26)) (\lambda (H28: (iso (THead (Flat Appl) x x0) u2)).(\lambda (P: Prop).(H27 (iso_trans (THead (Flat Appl) x3 x4) (THead (Flat Appl) x x0) (iso_head (Flat Appl) x3 x x4 x0) u2 H28) P)))) (THead (Flat Appl) x1 u2) (pr3_pr2 c (THead (Flat Appl) t0 u2) (THead (Flat Appl) x1 u2) (pr2_head_1 c t0 x1 H15 (Flat Appl) u2)))))))) H24))) x2 H20))))))) H19)) (\lambda (H19: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x2 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H20: (eq T x0 (THead (Bind Abst) x3 x4))).(\lambda (H21: (eq T x2 (THead (Bind Abbr) x5 x6))).(\lambda (H22: (pr2 c x x5)).(\lambda (H23: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x4 x6))))).(let H24 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P: Prop).P))) H17 (THead (Bind Abbr) x5 x6) H21) in (eq_ind_r T (THead (Bind Abbr) x5 x6) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H25 \def (eq_ind T x0 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x t)) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6))) \to (\forall (P: Prop).P))) H24 (THead (Bind Abst) x3 x4) H20) in (let H26 \def (eq_ind T x0 (\lambda (t: T).(\forall (t2: T).((((eq T (THead (Flat Appl) x t) t2) \to (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t2) \to (sn3 c t2))))) H9 (THead (Bind Abst) x3 x4) H20) in (let H27 \def (eq_ind T x0 (\lambda (t: T).(\forall (t2: T).((((eq T (THead (Flat Appl) x t) t2) \to (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) x x0))))))))))))) H8 (THead (Bind Abst) x3 x4) H20) in (let H28 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H7 (THead (Bind Abst) x3 x4) H20) in (let H29 \def (eq_ind T x0 (\lambda (t: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t2 u2)))))) \to (sn3 c (THead (Flat Appl) t2 (THead (Flat Appl) x t)))))))) H6 (THead (Bind Abst) x3 x4) H20) in (sn3_pr3_trans c (THead (Flat Appl) t0 (THead (Bind Abbr) x5 x6)) (H28 (THead (Bind Abbr) x5 x6) (pr3_sing c (THead (Bind Abbr) x x4) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (pr2_free c (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind Abbr) x x4) (pr0_beta x3 x x (pr0_refl x) x4 x4 (pr0_refl x4))) (THead (Bind Abbr) x5 x6) (pr3_head_12 c x x5 (pr3_pr2 c x x5 H22) (Bind Abbr) x4 x6 (pr3_pr2 (CHead c (Bind Abbr) x5) x4 x6 (H23 Abbr x5)))) (\lambda (H30: (iso (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind Abbr) x5 x6))).(\lambda (P: Prop).(let H31 \def (match H30 return (\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl) x (THead (Bind Abst) x3 x4))) \to ((eq T t0 (THead (Bind Abbr) x5 x6)) \to P)))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H1: (eq T (TSort n2) (THead (Bind Abbr) x5 x6))).((let H2 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H0) in (False_ind ((eq T (TSort n2) (THead (Bind Abbr) x5 x6)) \to P) H2)) H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0: (eq T (TLRef i1) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H1: (eq T (TLRef i2) (THead (Bind Abbr) x5 x6))).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H0) in (False_ind ((eq T (TLRef i2) (THead (Bind Abbr) x5 x6)) \to P) H2)) H1))) | (iso_head k v4 v5 t1 t2) \Rightarrow (\lambda (H0: (eq T (THead k v4 t1) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H1: (eq T (THead k v5 t2) (THead (Bind Abbr) x5 x6))).((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t])) (THead k v4 t1) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H0) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4 | (THead _ t _) \Rightarrow t])) (THead k v4 t1) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H0) in ((let H4 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k v4 t1) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H0) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v4 x) \to ((eq T t1 (THead (Bind Abst) x3 x4)) \to ((eq T (THead k0 v5 t2) (THead (Bind Abbr) x5 x6)) \to P)))) (\lambda (H5: (eq T v4 x)).(eq_ind T x (\lambda (_: T).((eq T t1 (THead (Bind Abst) x3 x4)) \to ((eq T (THead (Flat Appl) v5 t2) (THead (Bind Abbr) x5 x6)) \to P))) (\lambda (H6: (eq T t1 (THead (Bind Abst) x3 x4))).(eq_ind T (THead (Bind Abst) x3 x4) (\lambda (_: T).((eq T (THead (Flat Appl) v5 t2) (THead (Bind Abbr) x5 x6)) \to P)) (\lambda (H7: (eq T (THead (Flat Appl) v5 t2) (THead (Bind Abbr) x5 x6))).(let H8 \def (eq_ind T (THead (Flat Appl) v5 t2) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) x5 x6) H7) in (False_ind P H8))) t1 (sym_eq T t1 (THead (Bind Abst) x3 x4) H6))) v4 (sym_eq T v4 x H5))) k (sym_eq K k (Flat Appl) H4))) H3)) H2)) H1)))]) in (H31 (refl_equal T (THead (Flat Appl) x (THead (Bind Abst) x3 x4))) (refl_equal T (THead (Bind Abbr) x5 x6))))))) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6)) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind Abbr) x5 x6)) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6)) (pr2_head_1 c t0 x1 H15 (Flat Appl) (THead (Bind Abbr) x5 x6))))))))) x2 H21)))))))))) H19)) (\lambda (H19: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (x3: B).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H20: (not (eq B x3 Abst))).(\lambda (H21: (eq T x0 (THead (Bind x3) x4 x5))).(\lambda (H22: (eq T x2 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).(\lambda (H23: (pr2 c x x7)).(\lambda (H24: (pr2 c x4 x8)).(\lambda (H25: (pr2 (CHead c (Bind x3) x8) x5 x6)).(let H26 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P: Prop).P))) H17 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) H22) in (eq_ind_r T (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H27 \def (eq_ind T x0 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x t)) (THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))) \to (\forall (P: Prop).P))) H26 (THead (Bind x3) x4 x5) H21) in (let H28 \def (eq_ind T x0 (\lambda (t: T).(\forall (t2: T).((((eq T (THead (Flat Appl) x t) t2) \to (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t2) \to (sn3 c t2))))) H9 (THead (Bind x3) x4 x5) H21) in (let H29 \def (eq_ind T x0 (\lambda (t: T).(\forall (t2: T).((((eq T (THead (Flat Appl) x t) t2) \to (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) x x0))))))))))))) H8 (THead (Bind x3) x4 x5) H21) in (let H30 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H7 (THead (Bind x3) x4 x5) H21) in (let H31 \def (eq_ind T x0 (\lambda (t: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t2 u2)))))) \to (sn3 c (THead (Flat Appl) t2 (THead (Flat Appl) x t)))))))) H6 (THead (Bind x3) x4 x5) H21) in (sn3_pr3_trans c (THead (Flat Appl) t0 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (H30 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (pr3_sing c (THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) x5)) (THead (Flat Appl) x (THead (Bind x3) x4 x5)) (pr2_free c (THead (Flat Appl) x (THead (Bind x3) x4 x5)) (THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) x5)) (pr0_upsilon x3 H20 x x (pr0_refl x) x4 x4 (pr0_refl x4) x5 x5 (pr0_refl x5))) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (pr3_head_12 c x4 x8 (pr3_pr2 c x4 x8 H24) (Bind x3) (THead (Flat Appl) (lift (S O) O x) x5) (THead (Flat Appl) (lift (S O) O x7) x6) (pr3_head_12 (CHead c (Bind x3) x8) (lift (S O) O x) (lift (S O) O x7) (pr3_lift (CHead c (Bind x3) x8) c (S O) O (drop_drop (Bind x3) O c c (drop_refl c) x8) x x7 (pr3_pr2 c x x7 H23)) (Flat Appl) x5 x6 (pr3_pr2 (CHead (CHead c (Bind x3) x8) (Flat Appl) (lift (S O) O x7)) x5 x6 (pr2_cflat (CHead c (Bind x3) x8) x5 x6 H25 Appl (lift (S O) O x7)))))) (\lambda (H32: (iso (THead (Flat Appl) x (THead (Bind x3) x4 x5)) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).(\lambda (P: Prop).(let H33 \def (match H32 return (\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl) x (THead (Bind x3) x4 x5))) \to ((eq T t0 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P)))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H1: (eq T (TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).((let H2 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x (THead (Bind x3) x4 x5)) H0) in (False_ind ((eq T (TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P) H2)) H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0: (eq T (TLRef i1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H1: (eq T (TLRef i2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x (THead (Bind x3) x4 x5)) H0) in (False_ind ((eq T (TLRef i2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P) H2)) H1))) | (iso_head k v4 v5 t1 t2) \Rightarrow (\lambda (H0: (eq T (THead k v4 t1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H1: (eq T (THead k v5 t2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t])) (THead k v4 t1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)) H0) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4 | (THead _ t _) \Rightarrow t])) (THead k v4 t1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)) H0) in ((let H4 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k v4 t1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)) H0) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v4 x) \to ((eq T t1 (THead (Bind x3) x4 x5)) \to ((eq T (THead k0 v5 t2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P)))) (\lambda (H5: (eq T v4 x)).(eq_ind T x (\lambda (_: T).((eq T t1 (THead (Bind x3) x4 x5)) \to ((eq T (THead (Flat Appl) v5 t2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P))) (\lambda (H6: (eq T t1 (THead (Bind x3) x4 x5))).(eq_ind T (THead (Bind x3) x4 x5) (\lambda (_: T).((eq T (THead (Flat Appl) v5 t2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P)) (\lambda (H7: (eq T (THead (Flat Appl) v5 t2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).(let H8 \def (eq_ind T (THead (Flat Appl) v5 t2) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) H7) in (False_ind P H8))) t1 (sym_eq T t1 (THead (Bind x3) x4 x5) H6))) v4 (sym_eq T v4 x H5))) k (sym_eq K k (Flat Appl) H4))) H3)) H2)) H1)))]) in (H33 (refl_equal T (THead (Flat Appl) x (THead (Bind x3) x4 x5))) (refl_equal T (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))))))) (THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (pr2_head_1 c t0 x1 H15 (Flat Appl) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))))))))) x2 H22)))))))))))))) H19)) H18)) t3 H14))))))) H13)) (\lambda (H13: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T t3 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H14: (eq T (THead (Flat Appl) x x0) (THead (Bind Abst) x1 x2))).(\lambda (H15: (eq T t3 (THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c t0 x3)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let H18 \def (eq_ind T t3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P: Prop).P))) H10 (THead (Bind Abbr) x3 x4) H15) in (eq_ind_r T (THead (Bind Abbr) x3 x4) (\lambda (t: T).(sn3 c t)) (let H19 \def (eq_ind T (THead (Flat Appl) x x0) (\lambda (ee: T).(match ee return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) x1 x2) H14) in (False_ind (sn3 c (THead (Bind Abbr) x3 x4)) H19)) t3 H15)))))))))) H13)) (\lambda (H13: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3) (\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H15: (eq T (THead (Flat Appl) x x0) (THead (Bind x1) x2 x3))).(\lambda (H16: (eq T t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda (_: (pr2 c t0 x5)).(\lambda (_: (pr2 c x2 x6)).(\lambda (_: (pr2 (CHead c (Bind x1) x6) x3 x4)).(let H20 \def (eq_ind T t3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P: Prop).P))) H10 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) H16) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) (\lambda (t: T).(sn3 c t)) (let H21 \def (eq_ind T (THead (Flat Appl) x x0) (\lambda (ee: T).(match ee return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x1) x2 x3) H15) in (False_ind (sn3 c (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))) H21)) t3 H16)))))))))))))) H13)) H12)))))))))))) v2 H4))))))))) y H0))))) H))))).
+ \lambda (v1: T).(\lambda (t1: T).(let u1 \def (THead (Flat Appl) v1 t1) in (\lambda (c: C).(\lambda (H: (sn3 c (THead (Flat Appl) v1 t1))).(insert_eq T (THead (Flat Appl) v1 t1) (\lambda (t: T).(sn3 c t)) (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) v1 t1) u2) \to ((((iso (THead (Flat Appl) v1 t1) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) v1 t1)))))) (\lambda (y: T).(\lambda (H0: (sn3 c y)).(unintro T t1 (\lambda (t: T).((eq T y (THead (Flat Appl) v1 t)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) v1 t) u2) \to ((((iso (THead (Flat Appl) v1 t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) v1 t)))))))) (unintro T v1 (\lambda (t: T).(\forall (x: T).((eq T y (THead (Flat Appl) t x)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) t x) u2) \to ((((iso (THead (Flat Appl) t x) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) t x))))))))) (sn3_ind c (\lambda (t: T).(\forall (x: T).(\forall (x0: T).((eq T t (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) x x0)))))))))) (\lambda (t2: T).(\lambda (H1: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (sn3 c t3)))))).(\lambda (H2: ((\forall (t3: T).((((eq T t2 t3) \to (\forall (P: Prop).P))) \to ((pr3 c t2 t3) \to (\forall (x: T).(\forall (x0: T).((eq T t3 (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) x x0)))))))))))))).(\lambda (x: T).(\lambda (x0: T).(\lambda (H3: (eq T t2 (THead (Flat Appl) x x0))).(\lambda (v2: T).(\lambda (H4: (sn3 c v2)).(sn3_ind c (\lambda (t: T).(((\forall (u2: T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t u2)))))) \to (sn3 c (THead (Flat Appl) t (THead (Flat Appl) x x0))))) (\lambda (t0: T).(\lambda (H5: ((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (sn3 c t2)))))).(\lambda (H6: ((\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t2 u2)))))) \to (sn3 c (THead (Flat Appl) t2 (THead (Flat Appl) x x0))))))))).(\lambda (H7: ((\forall (u2: T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2))))))).(let H8 \def (eq_ind T t2 (\lambda (t: T).(\forall (t2: T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) x x0))))))))))))) H2 (THead (Flat Appl) x x0) H3) in (let H9 \def (eq_ind T t2 (\lambda (t: T).(\forall (t2: T).((((eq T t t2) \to (\forall (P: Prop).P))) \to ((pr3 c t t2) \to (sn3 c t2))))) H1 (THead (Flat Appl) x x0) H3) in (sn3_pr2_intro c (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (\lambda (t3: T).(\lambda (H10: (((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t3) \to (\forall (P: Prop).P)))).(\lambda (H11: (pr2 c (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t3)).(let H12 \def (pr2_gen_appl c t0 (THead (Flat Appl) x x0) t3 H11) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Flat Appl) x x0) t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (sn3 c t3) (\lambda (H13: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t3 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c (THead (Flat Appl) x x0) t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c (THead (Flat Appl) x x0) t4))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (H14: (eq T t3 (THead (Flat Appl) x1 x2))).(\lambda (H15: (pr2 c t0 x1)).(\lambda (H16: (pr2 c (THead (Flat Appl) x x0) x2)).(let H17 \def (eq_ind T t3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P: Prop).P))) H10 (THead (Flat Appl) x1 x2) H14) in (eq_ind_r T (THead (Flat Appl) x1 x2) (\lambda (t: T).(sn3 c t)) (let H18 \def (pr2_gen_appl c x x0 x2 H16) in (or3_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4)))) (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4)))))))) (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2)))))))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (H19: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T x2 (THead (Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t2: T).(pr2 c x0 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Flat Appl) u2 t4)))) (\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))) (\lambda (_: T).(\lambda (t4: T).(pr2 c x0 t4))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (H20: (eq T x2 (THead (Flat Appl) x3 x4))).(\lambda (H21: (pr2 c x x3)).(\lambda (H22: (pr2 c x0 x4)).(let H23 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P: Prop).P))) H17 (THead (Flat Appl) x3 x4) H20) in (eq_ind_r T (THead (Flat Appl) x3 x4) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H_x \def (term_dec (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) in (let H24 \def H_x in (or_ind (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) ((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 x4))) (\lambda (H25: (eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4))).(let H26 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow x | (TLRef _) \Rightarrow x | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H25) in ((let H27 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow x0 | (TLRef _) \Rightarrow x0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4) H25) in (\lambda (H28: (eq T x x3)).(let H29 \def (eq_ind_r T x4 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 (THead (Flat Appl) x3 t))) \to (\forall (P: Prop).P))) H23 x0 H27) in (let H30 \def (eq_ind_r T x4 (\lambda (t: T).(pr2 c x0 t)) H22 x0 H27) in (eq_ind T x0 (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x3 t)))) (let H31 \def (eq_ind_r T x3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 (THead (Flat Appl) t x0))) \to (\forall (P: Prop).P))) H29 x H28) in (let H32 \def (eq_ind_r T x3 (\lambda (t: T).(pr2 c x t)) H21 x H28) in (eq_ind T x (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) t x0)))) (let H_x0 \def (term_dec t0 x1) in (let H33 \def H_x0 in (or_ind (eq T t0 x1) ((eq T t0 x1) \to (\forall (P: Prop).P)) (sn3 c (THead (Flat Appl) x1 (THead (Flat Appl) x x0))) (\lambda (H34: (eq T t0 x1)).(let H35 \def (eq_ind_r T x1 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) t (THead (Flat Appl) x x0))) \to (\forall (P: Prop).P))) H31 t0 H34) in (let H36 \def (eq_ind_r T x1 (\lambda (t: T).(pr2 c t0 t)) H15 t0 H34) in (eq_ind T t0 (\lambda (t: T).(sn3 c (THead (Flat Appl) t (THead (Flat Appl) x x0)))) (H35 (refl_equal T (THead (Flat Appl) t0 (THead (Flat Appl) x x0))) (sn3 c (THead (Flat Appl) t0 (THead (Flat Appl) x x0)))) x1 H34)))) (\lambda (H34: (((eq T t0 x1) \to (\forall (P: Prop).P)))).(H6 x1 H34 (pr3_pr2 c t0 x1 H15) (\lambda (u2: T).(\lambda (H35: (pr3 c (THead (Flat Appl) x x0) u2)).(\lambda (H36: (((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 u2) (H7 u2 H35 H36) (THead (Flat Appl) x1 u2) (pr3_pr2 c (THead (Flat Appl) t0 u2) (THead (Flat Appl) x1 u2) (pr2_head_1 c t0 x1 H15 (Flat Appl) u2)))))))) H33))) x3 H28))) x4 H27))))) H26))) (\lambda (H25: (((eq T (THead (Flat Appl) x x0) (THead (Flat Appl) x3 x4)) \to (\forall (P: Prop).P)))).(H8 (THead (Flat Appl) x3 x4) H25 (pr3_head_12 c x x3 (pr3_pr2 c x x3 H21) (Flat Appl) x0 x4 (pr3_pr2 (CHead c (Flat Appl) x3) x0 x4 (pr2_cflat c x0 x4 H22 Appl x3))) x3 x4 (refl_equal T (THead (Flat Appl) x3 x4)) x1 (sn3_pr3_trans c t0 (sn3_sing c t0 H5) x1 (pr3_pr2 c t0 x1 H15)) (\lambda (u2: T).(\lambda (H26: (pr3 c (THead (Flat Appl) x3 x4) u2)).(\lambda (H27: (((iso (THead (Flat Appl) x3 x4) u2) \to (\forall (P: Prop).P)))).(sn3_pr3_trans c (THead (Flat Appl) t0 u2) (H7 u2 (pr3_sing c (THead (Flat Appl) x x4) (THead (Flat Appl) x x0) (pr2_thin_dx c x0 x4 H22 x Appl) u2 (pr3_sing c (THead (Flat Appl) x3 x4) (THead (Flat Appl) x x4) (pr2_head_1 c x x3 H21 (Flat Appl) x4) u2 H26)) (\lambda (H28: (iso (THead (Flat Appl) x x0) u2)).(\lambda (P: Prop).(H27 (iso_trans (THead (Flat Appl) x3 x4) (THead (Flat Appl) x x0) (iso_head (Flat Appl) x3 x x4 x0) u2 H28) P)))) (THead (Flat Appl) x1 u2) (pr3_pr2 c (THead (Flat Appl) t0 u2) (THead (Flat Appl) x1 u2) (pr2_head_1 c t0 x1 H15 (Flat Appl) u2)))))))) H24))) x2 H20))))))) H19)) (\lambda (H19: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T x2 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T x2 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6: T).(\lambda (H20: (eq T x0 (THead (Bind Abst) x3 x4))).(\lambda (H21: (eq T x2 (THead (Bind Abbr) x5 x6))).(\lambda (H22: (pr2 c x x5)).(\lambda (H23: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x4 x6))))).(let H24 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P: Prop).P))) H17 (THead (Bind Abbr) x5 x6) H21) in (eq_ind_r T (THead (Bind Abbr) x5 x6) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H25 \def (eq_ind T x0 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x t)) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6))) \to (\forall (P: Prop).P))) H24 (THead (Bind Abst) x3 x4) H20) in (let H26 \def (eq_ind T x0 (\lambda (t: T).(\forall (t2: T).((((eq T (THead (Flat Appl) x t) t2) \to (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t2) \to (sn3 c t2))))) H9 (THead (Bind Abst) x3 x4) H20) in (let H27 \def (eq_ind T x0 (\lambda (t: T).(\forall (t2: T).((((eq T (THead (Flat Appl) x t) t2) \to (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) x x0))))))))))))) H8 (THead (Bind Abst) x3 x4) H20) in (let H28 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H7 (THead (Bind Abst) x3 x4) H20) in (let H29 \def (eq_ind T x0 (\lambda (t: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t2 u2)))))) \to (sn3 c (THead (Flat Appl) t2 (THead (Flat Appl) x t)))))))) H6 (THead (Bind Abst) x3 x4) H20) in (sn3_pr3_trans c (THead (Flat Appl) t0 (THead (Bind Abbr) x5 x6)) (H28 (THead (Bind Abbr) x5 x6) (pr3_sing c (THead (Bind Abbr) x x4) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (pr2_free c (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind Abbr) x x4) (pr0_beta x3 x x (pr0_refl x) x4 x4 (pr0_refl x4))) (THead (Bind Abbr) x5 x6) (pr3_head_12 c x x5 (pr3_pr2 c x x5 H22) (Bind Abbr) x4 x6 (pr3_pr2 (CHead c (Bind Abbr) x5) x4 x6 (H23 Abbr x5)))) (\lambda (H30: (iso (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) (THead (Bind Abbr) x5 x6))).(\lambda (P: Prop).(let H31 \def (match H30 return (\lambda (_: ?).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl) x (THead (Bind Abst) x3 x4))) \to ((eq T t0 (THead (Bind Abbr) x5 x6)) \to P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H1: (eq T (TSort n2) (THead (Bind Abbr) x5 x6))).((let H2 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H0) in (False_ind ((eq T (TSort n2) (THead (Bind Abbr) x5 x6)) \to P) H2)) H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0: (eq T (TLRef i1) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H1: (eq T (TLRef i2) (THead (Bind Abbr) x5 x6))).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H0) in (False_ind ((eq T (TLRef i2) (THead (Bind Abbr) x5 x6)) \to P) H2)) H1))) | (iso_head k v4 v5 t1 t2) \Rightarrow (\lambda (H0: (eq T (THead k v4 t1) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)))).(\lambda (H1: (eq T (THead k v5 t2) (THead (Bind Abbr) x5 x6))).((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t])) (THead k v4 t1) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H0) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4 | (THead _ t _) \Rightarrow t])) (THead k v4 t1) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H0) in ((let H4 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k v4 t1) (THead (Flat Appl) x (THead (Bind Abst) x3 x4)) H0) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v4 x) \to ((eq T t1 (THead (Bind Abst) x3 x4)) \to ((eq T (THead k0 v5 t2) (THead (Bind Abbr) x5 x6)) \to P)))) (\lambda (H5: (eq T v4 x)).(eq_ind T x (\lambda (_: T).((eq T t1 (THead (Bind Abst) x3 x4)) \to ((eq T (THead (Flat Appl) v5 t2) (THead (Bind Abbr) x5 x6)) \to P))) (\lambda (H6: (eq T t1 (THead (Bind Abst) x3 x4))).(eq_ind T (THead (Bind Abst) x3 x4) (\lambda (_: T).((eq T (THead (Flat Appl) v5 t2) (THead (Bind Abbr) x5 x6)) \to P)) (\lambda (H7: (eq T (THead (Flat Appl) v5 t2) (THead (Bind Abbr) x5 x6))).(let H8 \def (eq_ind T (THead (Flat Appl) v5 t2) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abbr) x5 x6) H7) in (False_ind P H8))) t1 (sym_eq T t1 (THead (Bind Abst) x3 x4) H6))) v4 (sym_eq T v4 x H5))) k (sym_eq K k (Flat Appl) H4))) H3)) H2)) H1)))]) in (H31 (refl_equal T (THead (Flat Appl) x (THead (Bind Abst) x3 x4))) (refl_equal T (THead (Bind Abbr) x5 x6))))))) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6)) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind Abbr) x5 x6)) (THead (Flat Appl) x1 (THead (Bind Abbr) x5 x6)) (pr2_head_1 c t0 x1 H15 (Flat Appl) (THead (Bind Abbr) x5 x6))))))))) x2 H21)))))))))) H19)) (\lambda (H19: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T x0 (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T x2 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c x u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c (THead (Flat Appl) x1 x2)) (\lambda (x3: B).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: T).(\lambda (H20: (not (eq B x3 Abst))).(\lambda (H21: (eq T x0 (THead (Bind x3) x4 x5))).(\lambda (H22: (eq T x2 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).(\lambda (H23: (pr2 c x x7)).(\lambda (H24: (pr2 c x4 x8)).(\lambda (H25: (pr2 (CHead c (Bind x3) x8) x5 x6)).(let H26 \def (eq_ind T x2 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) (THead (Flat Appl) x1 t)) \to (\forall (P: Prop).P))) H17 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) H22) in (eq_ind_r T (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (\lambda (t: T).(sn3 c (THead (Flat Appl) x1 t))) (let H27 \def (eq_ind T x0 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x t)) (THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))) \to (\forall (P: Prop).P))) H26 (THead (Bind x3) x4 x5) H21) in (let H28 \def (eq_ind T x0 (\lambda (t: T).(\forall (t2: T).((((eq T (THead (Flat Appl) x t) t2) \to (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t2) \to (sn3 c t2))))) H9 (THead (Bind x3) x4 x5) H21) in (let H29 \def (eq_ind T x0 (\lambda (t: T).(\forall (t2: T).((((eq T (THead (Flat Appl) x t) t2) \to (\forall (P: Prop).P))) \to ((pr3 c (THead (Flat Appl) x t) t2) \to (\forall (x: T).(\forall (x0: T).((eq T t2 (THead (Flat Appl) x x0)) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x x0) u2) \to ((((iso (THead (Flat Appl) x x0) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 (THead (Flat Appl) x x0))))))))))))) H8 (THead (Bind x3) x4 x5) H21) in (let H30 \def (eq_ind T x0 (\lambda (t: T).(\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t0 u2)))))) H7 (THead (Bind x3) x4 x5) H21) in (let H31 \def (eq_ind T x0 (\lambda (t: T).(\forall (t2: T).((((eq T t0 t2) \to (\forall (P: Prop).P))) \to ((pr3 c t0 t2) \to (((\forall (u2: T).((pr3 c (THead (Flat Appl) x t) u2) \to ((((iso (THead (Flat Appl) x t) u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) t2 u2)))))) \to (sn3 c (THead (Flat Appl) t2 (THead (Flat Appl) x t)))))))) H6 (THead (Bind x3) x4 x5) H21) in (sn3_pr3_trans c (THead (Flat Appl) t0 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (H30 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (pr3_sing c (THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) x5)) (THead (Flat Appl) x (THead (Bind x3) x4 x5)) (pr2_free c (THead (Flat Appl) x (THead (Bind x3) x4 x5)) (THead (Bind x3) x4 (THead (Flat Appl) (lift (S O) O x) x5)) (pr0_upsilon x3 H20 x x (pr0_refl x) x4 x4 (pr0_refl x4) x5 x5 (pr0_refl x5))) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) (pr3_head_12 c x4 x8 (pr3_pr2 c x4 x8 H24) (Bind x3) (THead (Flat Appl) (lift (S O) O x) x5) (THead (Flat Appl) (lift (S O) O x7) x6) (pr3_head_12 (CHead c (Bind x3) x8) (lift (S O) O x) (lift (S O) O x7) (pr3_lift (CHead c (Bind x3) x8) c (S O) O (drop_drop (Bind x3) O c c (drop_refl c) x8) x x7 (pr3_pr2 c x x7 H23)) (Flat Appl) x5 x6 (pr3_pr2 (CHead (CHead c (Bind x3) x8) (Flat Appl) (lift (S O) O x7)) x5 x6 (pr2_cflat (CHead c (Bind x3) x8) x5 x6 H25 Appl (lift (S O) O x7)))))) (\lambda (H32: (iso (THead (Flat Appl) x (THead (Bind x3) x4 x5)) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).(\lambda (P: Prop).(let H33 \def (match H32 return (\lambda (_: ?).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Appl) x (THead (Bind x3) x4 x5))) \to ((eq T t0 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P))))) with [(iso_sort n1 n2) \Rightarrow (\lambda (H0: (eq T (TSort n1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H1: (eq T (TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).((let H2 \def (eq_ind T (TSort n1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x (THead (Bind x3) x4 x5)) H0) in (False_ind ((eq T (TSort n2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P) H2)) H1))) | (iso_lref i1 i2) \Rightarrow (\lambda (H0: (eq T (TLRef i1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H1: (eq T (TLRef i2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).((let H2 \def (eq_ind T (TLRef i1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) x (THead (Bind x3) x4 x5)) H0) in (False_ind ((eq T (TLRef i2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P) H2)) H1))) | (iso_head k v4 v5 t1 t2) \Rightarrow (\lambda (H0: (eq T (THead k v4 t1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)))).(\lambda (H1: (eq T (THead k v5 t2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).((let H2 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t])) (THead k v4 t1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)) H0) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v4 | (TLRef _) \Rightarrow v4 | (THead _ t _) \Rightarrow t])) (THead k v4 t1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)) H0) in ((let H4 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k v4 t1) (THead (Flat Appl) x (THead (Bind x3) x4 x5)) H0) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T v4 x) \to ((eq T t1 (THead (Bind x3) x4 x5)) \to ((eq T (THead k0 v5 t2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P)))) (\lambda (H5: (eq T v4 x)).(eq_ind T x (\lambda (_: T).((eq T t1 (THead (Bind x3) x4 x5)) \to ((eq T (THead (Flat Appl) v5 t2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P))) (\lambda (H6: (eq T t1 (THead (Bind x3) x4 x5))).(eq_ind T (THead (Bind x3) x4 x5) (\lambda (_: T).((eq T (THead (Flat Appl) v5 t2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) \to P)) (\lambda (H7: (eq T (THead (Flat Appl) v5 t2) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))).(let H8 \def (eq_ind T (THead (Flat Appl) v5 t2) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)) H7) in (False_ind P H8))) t1 (sym_eq T t1 (THead (Bind x3) x4 x5) H6))) v4 (sym_eq T v4 x H5))) k (sym_eq K k (Flat Appl) H4))) H3)) H2)) H1)))]) in (H33 (refl_equal T (THead (Flat Appl) x (THead (Bind x3) x4 x5))) (refl_equal T (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))))))) (THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (pr3_pr2 c (THead (Flat Appl) t0 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (THead (Flat Appl) x1 (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6))) (pr2_head_1 c t0 x1 H15 (Flat Appl) (THead (Bind x3) x8 (THead (Flat Appl) (lift (S O) O x7) x6)))))))))) x2 H22)))))))))))))) H19)) H18)) t3 H14))))))) H13)) (\lambda (H13: (ex4_4 T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t2: T).(eq T t3 (THead (Bind Abbr) u2 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t2))))))))).(ex4_4_ind T T T T (\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind Abst) y1 z1)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (t4: T).(eq T t3 (THead (Bind Abbr) u2 t4)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t4: T).(\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) z1 t4))))))) (sn3 c t3) (\lambda (x1: T).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (H14: (eq T (THead (Flat Appl) x x0) (THead (Bind Abst) x1 x2))).(\lambda (H15: (eq T t3 (THead (Bind Abbr) x3 x4))).(\lambda (_: (pr2 c t0 x3)).(\lambda (_: ((\forall (b: B).(\forall (u: T).(pr2 (CHead c (Bind b) u) x2 x4))))).(let H18 \def (eq_ind T t3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P: Prop).P))) H10 (THead (Bind Abbr) x3 x4) H15) in (eq_ind_r T (THead (Bind Abbr) x3 x4) (\lambda (t: T).(sn3 c t)) (let H19 \def (eq_ind T (THead (Flat Appl) x x0) (\lambda (ee: T).(match ee return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) x1 x2) H14) in (False_ind (sn3 c (THead (Bind Abbr) x3 x4)) H19)) t3 H15)))))))))) H13)) (\lambda (H13: (ex6_6 B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))))).(ex6_6_ind B T T T T T (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(not (eq B b Abst)))))))) (\lambda (b: B).(\lambda (y1: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(eq T (THead (Flat Appl) x x0) (THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: T).(\lambda (z2: T).(\lambda (u2: T).(\lambda (y2: T).(eq T t3 (THead (Bind b) y2 (THead (Flat Appl) (lift (S O) O u2) z2))))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (_: T).(pr2 c t0 u2))))))) (\lambda (_: B).(\lambda (y1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (_: T).(\lambda (y2: T).(pr2 c y1 y2))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (z2: T).(\lambda (_: T).(\lambda (y2: T).(pr2 (CHead c (Bind b) y2) z1 z2))))))) (sn3 c t3) (\lambda (x1: B).(\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (x6: T).(\lambda (_: (not (eq B x1 Abst))).(\lambda (H15: (eq T (THead (Flat Appl) x x0) (THead (Bind x1) x2 x3))).(\lambda (H16: (eq T t3 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)))).(\lambda (_: (pr2 c t0 x5)).(\lambda (_: (pr2 c x2 x6)).(\lambda (_: (pr2 (CHead c (Bind x1) x6) x3 x4)).(let H20 \def (eq_ind T t3 (\lambda (t: T).((eq T (THead (Flat Appl) t0 (THead (Flat Appl) x x0)) t) \to (\forall (P: Prop).P))) H10 (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) H16) in (eq_ind_r T (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4)) (\lambda (t: T).(sn3 c t)) (let H21 \def (eq_ind T (THead (Flat Appl) x x0) (\lambda (ee: T).(match ee return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind x1) x2 x3) H15) in (False_ind (sn3 c (THead (Bind x1) x6 (THead (Flat Appl) (lift (S O) O x5) x4))) H21)) t3 H16)))))))))))))) H13)) H12)))))))))))) v2 H4))))))))) y H0))))) H))))).
 
 theorem sn3_appl_appls:
  \forall (v1: T).(\forall (t1: T).(\forall (vs: TList).(let u1 \def (THeads (Flat Appl) (TCons v1 vs) t1) in (\forall (c: C).((sn3 c u1) \to (\forall (v2: T).((sn3 c v2) \to (((\forall (u2: T).((pr3 c u1 u2) \to ((((iso u1 u2) \to (\forall (P: Prop).P))) \to (sn3 c (THead (Flat Appl) v2 u2)))))) \to (sn3 c (THead (Flat Appl) v2 u1))))))))))
@@ -2530,7 +2530,7 @@ theorem sns3_lifts:
 theorem sns3_lifts1:
  \forall (e: C).(\forall (hds: PList).(\forall (c: C).((drop1 hds c e) \to (\forall (ts: TList).((sns3 e ts) \to (sns3 c (lifts1 hds ts)))))))
 \def
- \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to (sns3 c (lifts1 p ts))))))) (\lambda (c: C).(\lambda (H: (drop1 PNil c e)).(\lambda (ts: TList).(\lambda (H0: (sns3 e ts)).(let H1 \def (match H return (\lambda (p: PList).(\lambda (c0: C).(\lambda (c1: C).((eq PList p PNil) \to ((eq C c0 c) \to ((eq C c1 e) \to (sns3 c (lifts1 PNil ts)))))))) with [(drop1_nil c0) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq C c0 e)).(eq_ind C c (\lambda (c1: C).((eq C c1 e) \to (sns3 c (lifts1 PNil ts)))) (\lambda (H4: (eq C c e)).(eq_ind C e (\lambda (c: C).(sns3 c (lifts1 PNil ts))) (eq_ind_r TList ts (\lambda (t: TList).(sns3 e t)) H0 (lifts1 PNil ts) (lifts1_nil ts)) c (sym_eq C c e H4))) c0 (sym_eq C c0 c H2) H3)))) | (drop1_cons c1 c2 h d H1 c3 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c1 c)).(\lambda (H5: (eq C c3 e)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e0: PList).(match e0 return (\lambda (_: ?).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c1 c) \to ((eq C c3 e) \to ((drop h d c1 c2) \to ((drop1 hds c2 c3) \to (sns3 c (lifts1 PNil ts)))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c) (refl_equal C e))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to (sns3 c (lifts1 p ts)))))))).(\lambda (c: C).(\lambda (H0: (drop1 (PCons n n0 p) c e)).(\lambda (ts: TList).(\lambda (H1: (sns3 e ts)).(let H2 \def (match H0 return (\lambda (p0: PList).(\lambda (c0: C).(\lambda (c1: C).((eq PList p0 (PCons n n0 p)) \to ((eq C c0 c) \to ((eq C c1 e) \to (sns3 c (lifts1 (PCons n n0 p) ts)))))))) with [(drop1_nil c0) \Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq C c0 e)).((let H5 \def (eq_ind PList PNil (\lambda (e0: PList).(match e0 return (\lambda (_: ?).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c0 c) \to ((eq C c0 e) \to (sns3 c (lifts1 (PCons n n0 p) ts)))) H5)) H3 H4)))) | (drop1_cons c1 c2 h d H2 c3 hds H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds) (PCons n n0 p))).(\lambda (H5: (eq C c1 c)).(\lambda (H6: (eq C c3 e)).((let H7 \def (f_equal PList PList (\lambda (e0: PList).(match e0 return (\lambda (_: ?).PList) with [PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat (\lambda (e0: PList).(match e0 return (\lambda (_: ?).nat) with [PNil \Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e0: PList).(match e0 return (\lambda (_: ?).nat) with [PNil \Rightarrow h | (PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n1 d c1 c2) \to ((drop1 hds c2 c3) \to (sns3 c (lifts1 (PCons n n0 p) ts))))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n n1 c1 c2) \to ((drop1 hds c2 c3) \to (sns3 c (lifts1 (PCons n n0 p) ts)))))))) (\lambda (H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c1 c) \to ((eq C c3 e) \to ((drop n n0 c1 c2) \to ((drop1 p0 c2 c3) \to (sns3 c (lifts1 (PCons n n0 p) ts))))))) (\lambda (H12: (eq C c1 c)).(eq_ind C c (\lambda (c0: C).((eq C c3 e) \to ((drop n n0 c0 c2) \to ((drop1 p c2 c3) \to (sns3 c (lifts1 (PCons n n0 p) ts)))))) (\lambda (H13: (eq C c3 e)).(eq_ind C e (\lambda (c0: C).((drop n n0 c c2) \to ((drop1 p c2 c0) \to (sns3 c (lifts1 (PCons n n0 p) ts))))) (\lambda (H14: (drop n n0 c c2)).(\lambda (H15: (drop1 p c2 e)).(eq_ind_r TList (lifts n n0 (lifts1 p ts)) (\lambda (t: TList).(sns3 c t)) (sns3_lifts c c2 n n0 H14 (lifts1 p ts) (H c2 H15 ts H1)) (lifts1 (PCons n n0 p) ts) (lifts1_cons n n0 p ts)))) c3 (sym_eq C c3 e H13))) c1 (sym_eq C c1 c H12))) hds (sym_eq PList hds p H11))) d (sym_eq nat d n0 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p)) (refl_equal C c) (refl_equal C e))))))))))) hds)).
+ \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to (sns3 c (lifts1 p ts))))))) (\lambda (c: C).(\lambda (H: (drop1 PNil c e)).(\lambda (ts: TList).(\lambda (H0: (sns3 e ts)).(let H1 \def (match H return (\lambda (_: ?).(\lambda (p: PList).(\lambda (c0: C).(\lambda (c1: C).((eq PList p PNil) \to ((eq C c0 c) \to ((eq C c1 e) \to (sns3 c (lifts1 PNil ts))))))))) with [(drop1_nil c0) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq C c0 e)).(eq_ind C c (\lambda (c1: C).((eq C c1 e) \to (sns3 c (lifts1 PNil ts)))) (\lambda (H4: (eq C c e)).(eq_ind C e (\lambda (c: C).(sns3 c (lifts1 PNil ts))) (eq_ind_r TList ts (\lambda (t: TList).(sns3 e t)) H0 (lifts1 PNil ts) (lifts1_nil ts)) c (sym_eq C c e H4))) c0 (sym_eq C c0 c H2) H3)))) | (drop1_cons c1 c2 h d H1 c3 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c1 c)).(\lambda (H5: (eq C c3 e)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e0: PList).(match e0 return (\lambda (_: ?).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c1 c) \to ((eq C c3 e) \to ((drop h d c1 c2) \to ((drop1 hds c2 c3) \to (sns3 c (lifts1 PNil ts)))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c) (refl_equal C e))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to (sns3 c (lifts1 p ts)))))))).(\lambda (c: C).(\lambda (H0: (drop1 (PCons n n0 p) c e)).(\lambda (ts: TList).(\lambda (H1: (sns3 e ts)).(let H2 \def (match H0 return (\lambda (_: ?).(\lambda (p0: PList).(\lambda (c0: C).(\lambda (c1: C).((eq PList p0 (PCons n n0 p)) \to ((eq C c0 c) \to ((eq C c1 e) \to (sns3 c (lifts1 (PCons n n0 p) ts))))))))) with [(drop1_nil c0) \Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq C c0 e)).((let H5 \def (eq_ind PList PNil (\lambda (e0: PList).(match e0 return (\lambda (_: ?).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c0 c) \to ((eq C c0 e) \to (sns3 c (lifts1 (PCons n n0 p) ts)))) H5)) H3 H4)))) | (drop1_cons c1 c2 h d H2 c3 hds H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds) (PCons n n0 p))).(\lambda (H5: (eq C c1 c)).(\lambda (H6: (eq C c3 e)).((let H7 \def (f_equal PList PList (\lambda (e0: PList).(match e0 return (\lambda (_: ?).PList) with [PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat (\lambda (e0: PList).(match e0 return (\lambda (_: ?).nat) with [PNil \Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e0: PList).(match e0 return (\lambda (_: ?).nat) with [PNil \Rightarrow h | (PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n1 d c1 c2) \to ((drop1 hds c2 c3) \to (sns3 c (lifts1 (PCons n n0 p) ts))))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n n1 c1 c2) \to ((drop1 hds c2 c3) \to (sns3 c (lifts1 (PCons n n0 p) ts)))))))) (\lambda (H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c1 c) \to ((eq C c3 e) \to ((drop n n0 c1 c2) \to ((drop1 p0 c2 c3) \to (sns3 c (lifts1 (PCons n n0 p) ts))))))) (\lambda (H12: (eq C c1 c)).(eq_ind C c (\lambda (c0: C).((eq C c3 e) \to ((drop n n0 c0 c2) \to ((drop1 p c2 c3) \to (sns3 c (lifts1 (PCons n n0 p) ts)))))) (\lambda (H13: (eq C c3 e)).(eq_ind C e (\lambda (c0: C).((drop n n0 c c2) \to ((drop1 p c2 c0) \to (sns3 c (lifts1 (PCons n n0 p) ts))))) (\lambda (H14: (drop n n0 c c2)).(\lambda (H15: (drop1 p c2 e)).(eq_ind_r TList (lifts n n0 (lifts1 p ts)) (\lambda (t: TList).(sns3 c t)) (sns3_lifts c c2 n n0 H14 (lifts1 p ts) (H c2 H15 ts H1)) (lifts1 (PCons n n0 p) ts) (lifts1_cons n n0 p ts)))) c3 (sym_eq C c3 e H13))) c1 (sym_eq C c1 c H12))) hds (sym_eq PList hds p H11))) d (sym_eq nat d n0 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p)) (refl_equal C c) (refl_equal C e))))))))))) hds)).
 
 theorem sn3_gen_lift:
  \forall (c1: C).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((sn3 c1 (lift h d t)) \to (\forall (c2: C).((drop h d c1 c2) \to (sn3 c2 t)))))))
@@ -2560,7 +2560,7 @@ theorem sc3_lift:
 theorem sc3_lift1:
  \forall (g: G).(\forall (e: C).(\forall (a: A).(\forall (hds: PList).(\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 hds c e) \to (sc3 g a c (lift1 hds t)))))))))
 \def
- \lambda (g: G).(\lambda (e: C).(\lambda (a: A).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (H: (sc3 g a e t)).(\lambda (H0: (drop1 PNil c e)).(let H1 \def (match H0 return (\lambda (p: PList).(\lambda (c0: C).(\lambda (c1: C).((eq PList p PNil) \to ((eq C c0 c) \to ((eq C c1 e) \to (sc3 g a c t))))))) with [(drop1_nil c0) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq C c0 e)).(eq_ind C c (\lambda (c1: C).((eq C c1 e) \to (sc3 g a c t))) (\lambda (H4: (eq C c e)).(eq_ind C e (\lambda (c: C).(sc3 g a c t)) H c (sym_eq C c e H4))) c0 (sym_eq C c0 c H2) H3)))) | (drop1_cons c1 c2 h d H1 c3 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c1 c)).(\lambda (H5: (eq C c3 e)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e0: PList).(match e0 return (\lambda (_: ?).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c1 c) \to ((eq C c3 e) \to ((drop h d c1 c2) \to ((drop1 hds c2 c3) \to (sc3 g a c t))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c) (refl_equal C e))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t)))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (sc3 g a e t)).(\lambda (H1: (drop1 (PCons n n0 p) c e)).(let H2 \def (match H1 return (\lambda (p0: PList).(\lambda (c0: C).(\lambda (c1: C).((eq PList p0 (PCons n n0 p)) \to ((eq C c0 c) \to ((eq C c1 e) \to (sc3 g a c (lift n n0 (lift1 p t))))))))) with [(drop1_nil c0) \Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq C c0 e)).((let H5 \def (eq_ind PList PNil (\lambda (e0: PList).(match e0 return (\lambda (_: ?).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c0 c) \to ((eq C c0 e) \to (sc3 g a c (lift n n0 (lift1 p t))))) H5)) H3 H4)))) | (drop1_cons c1 c2 h d H2 c3 hds H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds) (PCons n n0 p))).(\lambda (H5: (eq C c1 c)).(\lambda (H6: (eq C c3 e)).((let H7 \def (f_equal PList PList (\lambda (e0: PList).(match e0 return (\lambda (_: ?).PList) with [PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat (\lambda (e0: PList).(match e0 return (\lambda (_: ?).nat) with [PNil \Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e0: PList).(match e0 return (\lambda (_: ?).nat) with [PNil \Rightarrow h | (PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n1 d c1 c2) \to ((drop1 hds c2 c3) \to (sc3 g a c (lift n n0 (lift1 p t)))))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n n1 c1 c2) \to ((drop1 hds c2 c3) \to (sc3 g a c (lift n n0 (lift1 p t))))))))) (\lambda (H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c1 c) \to ((eq C c3 e) \to ((drop n n0 c1 c2) \to ((drop1 p0 c2 c3) \to (sc3 g a c (lift n n0 (lift1 p t)))))))) (\lambda (H12: (eq C c1 c)).(eq_ind C c (\lambda (c0: C).((eq C c3 e) \to ((drop n n0 c0 c2) \to ((drop1 p c2 c3) \to (sc3 g a c (lift n n0 (lift1 p t))))))) (\lambda (H13: (eq C c3 e)).(eq_ind C e (\lambda (c0: C).((drop n n0 c c2) \to ((drop1 p c2 c0) \to (sc3 g a c (lift n n0 (lift1 p t)))))) (\lambda (H14: (drop n n0 c c2)).(\lambda (H15: (drop1 p c2 e)).(sc3_lift g a c2 (lift1 p t) (H c2 t H0 H15) c n n0 H14))) c3 (sym_eq C c3 e H13))) c1 (sym_eq C c1 c H12))) hds (sym_eq PList hds p H11))) d (sym_eq nat d n0 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p)) (refl_equal C c) (refl_equal C e))))))))))) hds)))).
+ \lambda (g: G).(\lambda (e: C).(\lambda (a: A).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (H: (sc3 g a e t)).(\lambda (H0: (drop1 PNil c e)).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (p: PList).(\lambda (c0: C).(\lambda (c1: C).((eq PList p PNil) \to ((eq C c0 c) \to ((eq C c1 e) \to (sc3 g a c t)))))))) with [(drop1_nil c0) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq C c0 e)).(eq_ind C c (\lambda (c1: C).((eq C c1 e) \to (sc3 g a c t))) (\lambda (H4: (eq C c e)).(eq_ind C e (\lambda (c: C).(sc3 g a c t)) H c (sym_eq C c e H4))) c0 (sym_eq C c0 c H2) H3)))) | (drop1_cons c1 c2 h d H1 c3 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c1 c)).(\lambda (H5: (eq C c3 e)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e0: PList).(match e0 return (\lambda (_: ?).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c1 c) \to ((eq C c3 e) \to ((drop h d c1 c2) \to ((drop1 hds c2 c3) \to (sc3 g a c t))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c) (refl_equal C e))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).(\forall (t: T).((sc3 g a e t) \to ((drop1 p c e) \to (sc3 g a c (lift1 p t)))))))).(\lambda (c: C).(\lambda (t: T).(\lambda (H0: (sc3 g a e t)).(\lambda (H1: (drop1 (PCons n n0 p) c e)).(let H2 \def (match H1 return (\lambda (_: ?).(\lambda (p0: PList).(\lambda (c0: C).(\lambda (c1: C).((eq PList p0 (PCons n n0 p)) \to ((eq C c0 c) \to ((eq C c1 e) \to (sc3 g a c (lift n n0 (lift1 p t)))))))))) with [(drop1_nil c0) \Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq C c0 e)).((let H5 \def (eq_ind PList PNil (\lambda (e0: PList).(match e0 return (\lambda (_: ?).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c0 c) \to ((eq C c0 e) \to (sc3 g a c (lift n n0 (lift1 p t))))) H5)) H3 H4)))) | (drop1_cons c1 c2 h d H2 c3 hds H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds) (PCons n n0 p))).(\lambda (H5: (eq C c1 c)).(\lambda (H6: (eq C c3 e)).((let H7 \def (f_equal PList PList (\lambda (e0: PList).(match e0 return (\lambda (_: ?).PList) with [PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat (\lambda (e0: PList).(match e0 return (\lambda (_: ?).nat) with [PNil \Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e0: PList).(match e0 return (\lambda (_: ?).nat) with [PNil \Rightarrow h | (PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n1 d c1 c2) \to ((drop1 hds c2 c3) \to (sc3 g a c (lift n n0 (lift1 p t)))))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n n1 c1 c2) \to ((drop1 hds c2 c3) \to (sc3 g a c (lift n n0 (lift1 p t))))))))) (\lambda (H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c1 c) \to ((eq C c3 e) \to ((drop n n0 c1 c2) \to ((drop1 p0 c2 c3) \to (sc3 g a c (lift n n0 (lift1 p t)))))))) (\lambda (H12: (eq C c1 c)).(eq_ind C c (\lambda (c0: C).((eq C c3 e) \to ((drop n n0 c0 c2) \to ((drop1 p c2 c3) \to (sc3 g a c (lift n n0 (lift1 p t))))))) (\lambda (H13: (eq C c3 e)).(eq_ind C e (\lambda (c0: C).((drop n n0 c c2) \to ((drop1 p c2 c0) \to (sc3 g a c (lift n n0 (lift1 p t)))))) (\lambda (H14: (drop n n0 c c2)).(\lambda (H15: (drop1 p c2 e)).(sc3_lift g a c2 (lift1 p t) (H c2 t H0 H15) c n n0 H14))) c3 (sym_eq C c3 e H13))) c1 (sym_eq C c1 c H12))) hds (sym_eq PList hds p H11))) d (sym_eq nat d n0 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p)) (refl_equal C c) (refl_equal C e))))))))))) hds)))).
 
 axiom sc3_abbr:
  \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (i: nat).(\forall (d: C).(\forall (v: T).(\forall (c: C).((sc3 g a c (THeads (Flat Appl) vs (lift (S i) O v))) \to ((getl i c (CHead d (Bind Abbr) v)) \to (sc3 g a c (THeads (Flat Appl) vs (TLRef i)))))))))))
@@ -2569,7 +2569,7 @@ axiom sc3_abbr:
 theorem sc3_cast:
  \forall (g: G).(\forall (a: A).(\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a) c (THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a c (THeads (Flat Appl) vs t)) \to (sc3 g a c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))
 \def
- \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to (sc3 g a0 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (sc3 g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) c (THeads (Flat Appl) vs u))).(\lambda (t: T).(\lambda (H0: (land (arity g c (THeads (Flat Appl) vs t) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs t)))).((match n return (\lambda (n1: nat).((sc3 g (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) c (THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads (Flat Appl) vs t) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))) with [O \Rightarrow (\lambda (H1: (sc3 g (ASort O (next g n0)) c (THeads (Flat Appl) vs u))).(\lambda (H2: (land (arity g c (THeads (Flat Appl) vs t) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs t)))).(let H3 \def H1 in (and_ind (arity g c (THeads (Flat Appl) vs u) (ASort O (next g n0))) (sn3 c (THeads (Flat Appl) vs u)) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (ASort O (next g n0)))).(\lambda (H5: (sn3 c (THeads (Flat Appl) vs u))).(let H6 \def H2 in (and_ind (arity g c (THeads (Flat Appl) vs t) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs t)) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (ASort O n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (arity_appls_cast g c u t vs (ASort O n0) H4 H7) (sn3_appls_cast c vs u H5 t H8)))) H6)))) H3)))) | (S n1) \Rightarrow (\lambda (H1: (sc3 g (ASort n1 n0) c (THeads (Flat Appl) vs u))).(\lambda (H2: (land (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs t)))).(let H3 \def H1 in (and_ind (arity g c (THeads (Flat Appl) vs u) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs u)) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (ASort n1 n0))).(\lambda (H5: (sn3 c (THeads (Flat Appl) vs u))).(let H6 \def H2 in (and_ind (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs t)) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (arity_appls_cast g c u t vs (ASort (S n1) n0) H4 H7) (sn3_appls_cast c vs u H5 t H8)))) H6)))) H3))))]) H H0))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to (sc3 g a0 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))).(\lambda (a1: A).(\lambda (H0: ((\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a1) c (THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a1 c (THeads (Flat Appl) vs t)) \to (sc3 g a1 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u: T).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs u))))))))))).(\lambda (t: T).(\lambda (H2: (land (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs t))))))))))).(let H3 \def H1 in (and_ind (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs u))))))))) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g a1)))).(\lambda (H5: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs u))))))))))).(let H6 \def H2 in (and_ind (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs t))))))))) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))) (\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1))).(\lambda (H8: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs t))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (arity_appls_cast g c u t vs (AHead a0 a1) H4 H7) (\lambda (d: C).(\lambda (w: T).(\lambda (H9: (sc3 g a0 d w)).(\lambda (is: PList).(\lambda (H10: (drop1 is d c)).(let H_y \def (H0 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is (THead (Flat Cast) u t))) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w t0))) (eq_ind_r T (THead (Flat Cast) (lift1 is u) (lift1 is t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat Appl) (lifts1 is vs) t0)))) (H_y d (lift1 is u) (eq_ind T (lift1 is (THeads (Flat Appl) vs u)) (\lambda (t0: T).(sc3 g (asucc g a1) d (THead (Flat Appl) w t0))) (H5 d w H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is u)) (lifts1_flat Appl is u vs)) (lift1 is t) (eq_ind T (lift1 is (THeads (Flat Appl) vs t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w t0))) (H8 d w H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is t)) (lifts1_flat Appl is t vs))) (lift1 is (THead (Flat Cast) u t)) (lift1_flat Cast is u t)) (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (lifts1_flat Appl is (THead (Flat Cast) u t) vs))))))))))) H6)))) H3)))))))))))) a)).
+ \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to (sc3 g a0 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u: T).(\lambda (H: (sc3 g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) c (THeads (Flat Appl) vs u))).(\lambda (t: T).(\lambda (H0: (land (arity g c (THeads (Flat Appl) vs t) (ASort n n0)) (sn3 c (THeads (Flat Appl) vs t)))).((match n return (\lambda (_: ?).(\lambda (n1: nat).((sc3 g (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) c (THeads (Flat Appl) vs u)) \to ((land (arity g c (THeads (Flat Appl) vs t) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs t))) \to (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))) with [O \Rightarrow (\lambda (H1: (sc3 g (ASort O (next g n0)) c (THeads (Flat Appl) vs u))).(\lambda (H2: (land (arity g c (THeads (Flat Appl) vs t) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs t)))).(let H3 \def H1 in (and_ind (arity g c (THeads (Flat Appl) vs u) (ASort O (next g n0))) (sn3 c (THeads (Flat Appl) vs u)) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (ASort O (next g n0)))).(\lambda (H5: (sn3 c (THeads (Flat Appl) vs u))).(let H6 \def H2 in (and_ind (arity g c (THeads (Flat Appl) vs t) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs t)) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (ASort O n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort O n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (arity_appls_cast g c u t vs (ASort O n0) H4 H7) (sn3_appls_cast c vs u H5 t H8)))) H6)))) H3)))) | (S n1) \Rightarrow (\lambda (H1: (sc3 g (ASort n1 n0) c (THeads (Flat Appl) vs u))).(\lambda (H2: (land (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs t)))).(let H3 \def H1 in (and_ind (arity g c (THeads (Flat Appl) vs u) (ASort n1 n0)) (sn3 c (THeads (Flat Appl) vs u)) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (ASort n1 n0))).(\lambda (H5: (sn3 c (THeads (Flat Appl) vs u))).(let H6 \def H2 in (and_ind (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs t)) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t)))) (\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (ASort (S n1) n0))).(\lambda (H8: (sn3 c (THeads (Flat Appl) vs t))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (ASort (S n1) n0)) (sn3 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (arity_appls_cast g c u t vs (ASort (S n1) n0) H4 H7) (sn3_appls_cast c vs u H5 t H8)))) H6)))) H3))))]) H H0))))))))) (\lambda (a0: A).(\lambda (_: ((\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a0) c (THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a0 c (THeads (Flat Appl) vs t)) \to (sc3 g a0 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))).(\lambda (a1: A).(\lambda (H0: ((\forall (vs: TList).(\forall (c: C).(\forall (u: T).((sc3 g (asucc g a1) c (THeads (Flat Appl) vs u)) \to (\forall (t: T).((sc3 g a1 c (THeads (Flat Appl) vs t)) \to (sc3 g a1 c (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))).(\lambda (vs: TList).(\lambda (c: C).(\lambda (u: T).(\lambda (H1: (land (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs u))))))))))).(\lambda (t: T).(\lambda (H2: (land (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs t))))))))))).(let H3 \def H1 in (and_ind (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g a1))) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs u))))))))) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))) (\lambda (H4: (arity g c (THeads (Flat Appl) vs u) (AHead a0 (asucc g a1)))).(\lambda (H5: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g (asucc g a1) d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs u))))))))))).(let H6 \def H2 in (and_ind (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs t))))))))) (land (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u t))))))))))) (\lambda (H7: (arity g c (THeads (Flat Appl) vs t) (AHead a0 a1))).(\lambda (H8: ((\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs t))))))))))).(conj (arity g c (THeads (Flat Appl) vs (THead (Flat Cast) u t)) (AHead a0 a1)) (\forall (d: C).(\forall (w: T).((sc3 g a0 d w) \to (\forall (is: PList).((drop1 is d c) \to (sc3 g a1 d (THead (Flat Appl) w (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u t)))))))))) (arity_appls_cast g c u t vs (AHead a0 a1) H4 H7) (\lambda (d: C).(\lambda (w: T).(\lambda (H9: (sc3 g a0 d w)).(\lambda (is: PList).(\lambda (H10: (drop1 is d c)).(let H_y \def (H0 (TCons w (lifts1 is vs))) in (eq_ind_r T (THeads (Flat Appl) (lifts1 is vs) (lift1 is (THead (Flat Cast) u t))) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w t0))) (eq_ind_r T (THead (Flat Cast) (lift1 is u) (lift1 is t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w (THeads (Flat Appl) (lifts1 is vs) t0)))) (H_y d (lift1 is u) (eq_ind T (lift1 is (THeads (Flat Appl) vs u)) (\lambda (t0: T).(sc3 g (asucc g a1) d (THead (Flat Appl) w t0))) (H5 d w H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is u)) (lifts1_flat Appl is u vs)) (lift1 is t) (eq_ind T (lift1 is (THeads (Flat Appl) vs t)) (\lambda (t0: T).(sc3 g a1 d (THead (Flat Appl) w t0))) (H8 d w H9 is H10) (THeads (Flat Appl) (lifts1 is vs) (lift1 is t)) (lifts1_flat Appl is t vs))) (lift1 is (THead (Flat Cast) u t)) (lift1_flat Cast is u t)) (lift1 is (THeads (Flat Appl) vs (THead (Flat Cast) u t))) (lifts1_flat Appl is (THead (Flat Cast) u t) vs))))))))))) H6)))) H3)))))))))))) a)).
 
 axiom sc3_bind:
  \forall (g: G).(\forall (b: B).((not (eq B b Abst)) \to (\forall (a1: A).(\forall (a2: A).(\forall (vs: TList).(\forall (c: C).(\forall (v: T).(\forall (t: T).((sc3 g a2 (CHead c (Bind b) v) (THeads (Flat Appl) (lifts (S O) O vs) t)) \to ((sc3 g a1 c v) \to (sc3 g a2 c (THeads (Flat Appl) vs (THead (Bind b) v t)))))))))))))
@@ -2617,22 +2617,22 @@ theorem ceqc_sym:
 theorem drop_csubc_trans:
  \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))))))))
 \def
- \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c c1)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e2 e1)).(and3_ind (eq C e2 (CSort n)) (eq nat h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: (eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)))) (let H4 \def (eq_ind C e2 (\lambda (c: C).(csubc g c e1)) H0 (CSort n) H1) in (ex_intro2 C (\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)) e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H))))))))) (\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c c1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t) e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) (\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1))))))) (\lambda (H0: (drop O O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H2 \def (eq_ind_r C e2 (\lambda (c: C).(csubc g c e1)) H1 (CHead c k t) (drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)) e1 (drop_refl e1) H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))).(\lambda (H1: (drop (S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 e1)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in (let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1)) (\lambda (c1: C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1))) (\lambda (x: C).(\lambda (H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g c x)).(ex_intro2 C (\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g c x H5 k t))))) H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n (CHead c k t) e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 e1)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n) x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda (c: C).(csubc g c e1)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 (\lambda (c0: C).(\forall (h: nat).((drop h n (CHead c k t) c0) \to (\forall (e1: C).((csubc g c0 e1) \to (ex2 C (\lambda (c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) H0 (CHead x0 k x1) H3) in (let H8 \def (eq_ind T t (\lambda (t: T).(\forall (h: nat).((drop h n (CHead c k t) (CHead x0 k x1)) \to (\forall (e1: C).((csubc g (CHead x0 k x1) e1) \to (ex2 C (\lambda (c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t0) c1)))) (let H9 \def (match H6 return (\lambda (c0: C).(\lambda (c1: C).((eq C c0 (CHead x0 k x1)) \to ((eq C c1 e1) \to (ex2 C (\lambda (c2: C).(drop h (S n) c2 e1)) (\lambda (c2: C).(csubc g (CHead c k (lift h (r k n) x1)) c2))))))) with [(csubc_sort n0) \Rightarrow (\lambda (H1: (eq C (CSort n0) (CHead x0 k x1))).(\lambda (H3: (eq C (CSort n0) e1)).((let H4 \def (eq_ind C (CSort n0) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead x0 k x1) H1) in (False_ind ((eq C (CSort n0) e1) \to (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)))) H4)) H3))) | (csubc_head c1 c2 H1 k0 v) \Rightarrow (\lambda (H3: (eq C (CHead c1 k0 v) (CHead x0 k x1))).(\lambda (H6: (eq C (CHead c2 k0 v) e1)).((let H2 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c1 k0 v) (CHead x0 k x1) H3) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c1 k0 v) (CHead x0 k x1) H3) in ((let H7 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k0 v) (CHead x0 k x1) H3) in (eq_ind C x0 (\lambda (c0: C).((eq K k0 k) \to ((eq T v x1) \to ((eq C (CHead c2 k0 v) e1) \to ((csubc g c0 c2) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) c3)))))))) (\lambda (H8: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T v x1) \to ((eq C (CHead c2 k1 v) e1) \to ((csubc g x0 c2) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) c3))))))) (\lambda (H9: (eq T v x1)).(eq_ind T x1 (\lambda (t: T).((eq C (CHead c2 k t) e1) \to ((csubc g x0 c2) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) c3)))))) (\lambda (H10: (eq C (CHead c2 k x1) e1)).(eq_ind C (CHead c2 k x1) (\lambda (c0: C).((csubc g x0 c2) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 c0)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) c3))))) (\lambda (H11: (csubc g x0 c2)).(let H_x \def (H x0 (r k n) h H5 c2 H11) in (let H5 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r k n) c3 c2)) (\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c2 k x1))) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) c3))) (\lambda (x: C).(\lambda (H12: (drop h (r k n) x c2)).(\lambda (H13: (csubc g c x)).(ex_intro2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c2 k x1))) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) c3)) (CHead x k (lift h (r k n) x1)) (drop_skip k h n x c2 H12 x1) (csubc_head g c x H13 k (lift h (r k n) x1)))))) H5)))) e1 H10)) v (sym_eq T v x1 H9))) k0 (sym_eq K k0 k H8))) c1 (sym_eq C c1 x0 H7))) H4)) H2)) H6 H1))) | (csubc_abst c1 c2 H1 v a H3 w H5) \Rightarrow (\lambda (H6: (eq C (CHead c1 (Bind Abst) v) (CHead x0 k x1))).(\lambda (H7: (eq C (CHead c2 (Bind Abbr) w) e1)).((let H2 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c1 (Bind Abst) v) (CHead x0 k x1) H6) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow (Bind Abst) | (CHead _ k _) \Rightarrow k])) (CHead c1 (Bind Abst) v) (CHead x0 k x1) H6) in ((let H9 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind Abst) v) (CHead x0 k x1) H6) in (eq_ind C x0 (\lambda (c0: C).((eq K (Bind Abst) k) \to ((eq T v x1) \to ((eq C (CHead c2 (Bind Abbr) w) e1) \to ((csubc g c0 c2) \to ((sc3 g (asucc g a) c0 v) \to ((sc3 g a c2 w) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) c3)))))))))) (\lambda (H10: (eq K (Bind Abst) k)).(eq_ind K (Bind Abst) (\lambda (k: K).((eq T v x1) \to ((eq C (CHead c2 (Bind Abbr) w) e1) \to ((csubc g x0 c2) \to ((sc3 g (asucc g a) x0 v) \to ((sc3 g a c2 w) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) c3))))))))) (\lambda (H11: (eq T v x1)).(eq_ind T x1 (\lambda (t: T).((eq C (CHead c2 (Bind Abbr) w) e1) \to ((csubc g x0 c2) \to ((sc3 g (asucc g a) x0 t) \to ((sc3 g a c2 w) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c3)))))))) (\lambda (H12: (eq C (CHead c2 (Bind Abbr) w) e1)).(eq_ind C (CHead c2 (Bind Abbr) w) (\lambda (c0: C).((csubc g x0 c2) \to ((sc3 g (asucc g a) x0 x1) \to ((sc3 g a c2 w) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 c0)) (\lambda (c3: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c3))))))) (\lambda (H13: (csubc g x0 c2)).(\lambda (H14: (sc3 g (asucc g a) x0 x1)).(\lambda (H15: (sc3 g a c2 w)).(let H8 \def (eq_ind_r K k (\lambda (k: K).(\forall (h0: nat).((drop h0 n (CHead c k (lift h (r k n) x1)) (CHead x0 k x1)) \to (\forall (e1: C).((csubc g (CHead x0 k x1) e1) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)))))))) H8 (Bind Abst) H10) in (let H16 \def (eq_ind_r K k (\lambda (k: K).(drop h (r k n) c x0)) H5 (Bind Abst) H10) in (let H_x \def (H x0 (r (Bind Abst) n) h H16 c2 H13) in (let H17 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r (Bind Abst) n) c3 c2)) (\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c2 (Bind Abbr) w))) (\lambda (c3: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c3))) (\lambda (x: C).(\lambda (H18: (drop h (r (Bind Abst) n) x c2)).(\lambda (H19: (csubc g c x)).(ex_intro2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c2 (Bind Abbr) w))) (\lambda (c3: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c3)) (CHead x (Bind Abbr) (lift h n w)) (drop_skip_bind h n x c2 H18 Abbr w) (csubc_abst g c x H19 (lift h (r (Bind Abst) n) x1) a (sc3_lift g (asucc g a) x0 x1 H14 c h (r (Bind Abst) n) H16) (lift h n w) (sc3_lift g a c2 w H15 x h n H18)))))) H17)))))))) e1 H12)) v (sym_eq T v x1 H11))) k H10)) c1 (sym_eq C c1 x0 H9))) H4)) H2)) H7 H1 H3 H5)))]) in (H9 (refl_equal C (CHead x0 k x1)) (refl_equal C e1))) t H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)).
+ \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c c1)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e2 e1)).(and3_ind (eq C e2 (CSort n)) (eq nat h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: (eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)))) (let H4 \def (eq_ind C e2 (\lambda (c: C).(csubc g c e1)) H0 (CSort n) H1) in (ex_intro2 C (\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g (CSort n) c1)) e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H))))))))) (\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c c1))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t) e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) (\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1))))))) (\lambda (H0: (drop O O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H2 \def (eq_ind_r C e2 (\lambda (c: C).(csubc g c e1)) H1 (CHead c k t) (drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)) e1 (drop_refl e1) H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))).(\lambda (H1: (drop (S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 e1)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in (let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1)) (\lambda (c1: C).(csubc g c c1)) (ex2 C (\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1))) (\lambda (x: C).(\lambda (H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g c x)).(ex_intro2 C (\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g c x H5 k t))))) H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n (CHead c k t) e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e2 e1)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n) x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda (c: C).(csubc g c e1)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 (\lambda (c0: C).(\forall (h: nat).((drop h n (CHead c k t) c0) \to (\forall (e1: C).((csubc g c0 e1) \to (ex2 C (\lambda (c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) H0 (CHead x0 k x1) H3) in (let H8 \def (eq_ind T t (\lambda (t: T).(\forall (h: nat).((drop h n (CHead c k t) (CHead x0 k x1)) \to (\forall (e1: C).((csubc g (CHead x0 k x1) e1) \to (ex2 C (\lambda (c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t) c1)))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g (CHead c k t0) c1)))) (let H9 \def (match H6 return (\lambda (_: ?).(\lambda (c0: C).(\lambda (c1: C).((eq C c0 (CHead x0 k x1)) \to ((eq C c1 e1) \to (ex2 C (\lambda (c2: C).(drop h (S n) c2 e1)) (\lambda (c2: C).(csubc g (CHead c k (lift h (r k n) x1)) c2)))))))) with [(csubc_sort n0) \Rightarrow (\lambda (H1: (eq C (CSort n0) (CHead x0 k x1))).(\lambda (H3: (eq C (CSort n0) e1)).((let H4 \def (eq_ind C (CSort n0) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead x0 k x1) H1) in (False_ind ((eq C (CSort n0) e1) \to (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)))) H4)) H3))) | (csubc_head c1 c2 H1 k0 v) \Rightarrow (\lambda (H3: (eq C (CHead c1 k0 v) (CHead x0 k x1))).(\lambda (H6: (eq C (CHead c2 k0 v) e1)).((let H2 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c1 k0 v) (CHead x0 k x1) H3) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c1 k0 v) (CHead x0 k x1) H3) in ((let H7 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k0 v) (CHead x0 k x1) H3) in (eq_ind C x0 (\lambda (c0: C).((eq K k0 k) \to ((eq T v x1) \to ((eq C (CHead c2 k0 v) e1) \to ((csubc g c0 c2) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) c3)))))))) (\lambda (H8: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T v x1) \to ((eq C (CHead c2 k1 v) e1) \to ((csubc g x0 c2) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) c3))))))) (\lambda (H9: (eq T v x1)).(eq_ind T x1 (\lambda (t: T).((eq C (CHead c2 k t) e1) \to ((csubc g x0 c2) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) c3)))))) (\lambda (H10: (eq C (CHead c2 k x1) e1)).(eq_ind C (CHead c2 k x1) (\lambda (c0: C).((csubc g x0 c2) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 c0)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) c3))))) (\lambda (H11: (csubc g x0 c2)).(let H_x \def (H x0 (r k n) h H5 c2 H11) in (let H5 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r k n) c3 c2)) (\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c2 k x1))) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) c3))) (\lambda (x: C).(\lambda (H12: (drop h (r k n) x c2)).(\lambda (H13: (csubc g c x)).(ex_intro2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c2 k x1))) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) c3)) (CHead x k (lift h (r k n) x1)) (drop_skip k h n x c2 H12 x1) (csubc_head g c x H13 k (lift h (r k n) x1)))))) H5)))) e1 H10)) v (sym_eq T v x1 H9))) k0 (sym_eq K k0 k H8))) c1 (sym_eq C c1 x0 H7))) H4)) H2)) H6 H1))) | (csubc_abst c1 c2 H1 v a H3 w H5) \Rightarrow (\lambda (H6: (eq C (CHead c1 (Bind Abst) v) (CHead x0 k x1))).(\lambda (H7: (eq C (CHead c2 (Bind Abbr) w) e1)).((let H2 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c1 (Bind Abst) v) (CHead x0 k x1) H6) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow (Bind Abst) | (CHead _ k _) \Rightarrow k])) (CHead c1 (Bind Abst) v) (CHead x0 k x1) H6) in ((let H9 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind Abst) v) (CHead x0 k x1) H6) in (eq_ind C x0 (\lambda (c0: C).((eq K (Bind Abst) k) \to ((eq T v x1) \to ((eq C (CHead c2 (Bind Abbr) w) e1) \to ((csubc g c0 c2) \to ((sc3 g (asucc g a) c0 v) \to ((sc3 g a c2 w) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) c3)))))))))) (\lambda (H10: (eq K (Bind Abst) k)).(eq_ind K (Bind Abst) (\lambda (k: K).((eq T v x1) \to ((eq C (CHead c2 (Bind Abbr) w) e1) \to ((csubc g x0 c2) \to ((sc3 g (asucc g a) x0 v) \to ((sc3 g a c2 w) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c k (lift h (r k n) x1)) c3))))))))) (\lambda (H11: (eq T v x1)).(eq_ind T x1 (\lambda (t: T).((eq C (CHead c2 (Bind Abbr) w) e1) \to ((csubc g x0 c2) \to ((sc3 g (asucc g a) x0 t) \to ((sc3 g a c2 w) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 e1)) (\lambda (c3: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c3)))))))) (\lambda (H12: (eq C (CHead c2 (Bind Abbr) w) e1)).(eq_ind C (CHead c2 (Bind Abbr) w) (\lambda (c0: C).((csubc g x0 c2) \to ((sc3 g (asucc g a) x0 x1) \to ((sc3 g a c2 w) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 c0)) (\lambda (c3: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c3))))))) (\lambda (H13: (csubc g x0 c2)).(\lambda (H14: (sc3 g (asucc g a) x0 x1)).(\lambda (H15: (sc3 g a c2 w)).(let H8 \def (eq_ind_r K k (\lambda (k: K).(\forall (h0: nat).((drop h0 n (CHead c k (lift h (r k n) x1)) (CHead x0 k x1)) \to (\forall (e1: C).((csubc g (CHead x0 k x1) e1) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e1)) (\lambda (c1: C).(csubc g (CHead c k (lift h (r k n) x1)) c1)))))))) H8 (Bind Abst) H10) in (let H16 \def (eq_ind_r K k (\lambda (k: K).(drop h (r k n) c x0)) H5 (Bind Abst) H10) in (let H_x \def (H x0 (r (Bind Abst) n) h H16 c2 H13) in (let H17 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r (Bind Abst) n) c3 c2)) (\lambda (c3: C).(csubc g c c3)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c2 (Bind Abbr) w))) (\lambda (c3: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c3))) (\lambda (x: C).(\lambda (H18: (drop h (r (Bind Abst) n) x c2)).(\lambda (H19: (csubc g c x)).(ex_intro2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c2 (Bind Abbr) w))) (\lambda (c3: C).(csubc g (CHead c (Bind Abst) (lift h (r (Bind Abst) n) x1)) c3)) (CHead x (Bind Abbr) (lift h n w)) (drop_skip_bind h n x c2 H18 Abbr w) (csubc_abst g c x H19 (lift h (r (Bind Abst) n) x1) a (sc3_lift g (asucc g a) x0 x1 H14 c h (r (Bind Abst) n) H16) (lift h n w) (sc3_lift g a c2 w H15 x h n H18)))))) H17)))))))) e1 H12)) v (sym_eq T v x1 H11))) k H10)) c1 (sym_eq C c1 x0 H9))) H4)) H2)) H7 H1 H3 H5)))]) in (H9 (refl_equal C (CHead x0 k x1)) (refl_equal C e1))) t H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)).
 
 theorem csubc_drop_conf_rev:
  \forall (g: G).(\forall (c2: C).(\forall (e2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))))))))
 \def
- \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 c)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e1 e2)).(and3_ind (eq C e2 (CSort n)) (eq nat h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n)))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: (eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))))) (let H4 \def (eq_ind C e2 (\lambda (c: C).(csubc g e1 c)) H0 (CSort n) H1) in (ex_intro2 C (\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))) e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H))))))))) (\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 c))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t) e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) (\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t)))))))) (\lambda (H0: (drop O O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H2 \def (eq_ind_r C e2 (\lambda (c: C).(csubc g e1 c)) H1 (CHead c k t) (drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))) e1 (drop_refl e1) H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))).(\lambda (H1: (drop (S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 e2)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in (let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t)))) (\lambda (x: C).(\lambda (H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g x c)).(ex_intro2 C (\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g x c H5 k t))))) H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n (CHead c k t) e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t)))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 e2)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n) x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda (c: C).(csubc g e1 c)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 (\lambda (c0: C).(\forall (h: nat).((drop h n (CHead c k t) c0) \to (\forall (e1: C).((csubc g e1 c0) \to (ex2 C (\lambda (c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) H0 (CHead x0 k x1) H3) in (let H8 \def (eq_ind T t (\lambda (t: T).(\forall (h: nat).((drop h n (CHead c k t) (CHead x0 k x1)) \to (\forall (e1: C).((csubc g e1 (CHead x0 k x1)) \to (ex2 C (\lambda (c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t0))))) (let H9 \def (match H6 return (\lambda (c0: C).(\lambda (c1: C).((eq C c0 e1) \to ((eq C c1 (CHead x0 k x1)) \to (ex2 C (\lambda (c2: C).(drop h (S n) c2 e1)) (\lambda (c2: C).(csubc g c2 (CHead c k (lift h (r k n) x1))))))))) with [(csubc_sort n0) \Rightarrow (\lambda (H1: (eq C (CSort n0) e1)).(\lambda (H3: (eq C (CSort n0) (CHead x0 k x1))).(eq_ind C (CSort n0) (\lambda (c0: C).((eq C (CSort n0) (CHead x0 k x1)) \to (ex2 C (\lambda (c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))))) (\lambda (H4: (eq C (CSort n0) (CHead x0 k x1))).(let H5 \def (eq_ind C (CSort n0) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead x0 k x1) H4) in (False_ind (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CSort n0))) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) H5))) e1 H1 H3))) | (csubc_head c1 c2 H1 k0 v) \Rightarrow (\lambda (H3: (eq C (CHead c1 k0 v) e1)).(\lambda (H6: (eq C (CHead c2 k0 v) (CHead x0 k x1))).(eq_ind C (CHead c1 k0 v) (\lambda (c0: C).((eq C (CHead c2 k0 v) (CHead x0 k x1)) \to ((csubc g c1 c2) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 c0)) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1)))))))) (\lambda (H7: (eq C (CHead c2 k0 v) (CHead x0 k x1))).(let H2 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c2 k0 v) (CHead x0 k x1) H7) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c2 k0 v) (CHead x0 k x1) H7) in ((let H8 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 k0 v) (CHead x0 k x1) H7) in (eq_ind C x0 (\lambda (c0: C).((eq K k0 k) \to ((eq T v x1) \to ((csubc g c1 c0) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k0 v))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1))))))))) (\lambda (H9: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T v x1) \to ((csubc g c1 x0) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k1 v))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1)))))))) (\lambda (H10: (eq T v x1)).(eq_ind T x1 (\lambda (t: T).((csubc g c1 x0) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k t))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1))))))) (\lambda (H11: (csubc g c1 x0)).(let H12 \def (eq_ind T v (\lambda (t: T).(eq C (CHead c1 k0 t) e1)) H3 x1 H10) in (let H13 \def (eq_ind K k0 (\lambda (k: K).(eq C (CHead c1 k x1) e1)) H12 k H9) in (let H_x \def (H x0 (r k n) h H5 c1 H11) in (let H5 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r k n) c3 c1)) (\lambda (c3: C).(csubc g c3 c)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k x1))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1))))) (\lambda (x: C).(\lambda (H14: (drop h (r k n) x c1)).(\lambda (H15: (csubc g x c)).(ex_intro2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k x1))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1)))) (CHead x k (lift h (r k n) x1)) (drop_skip k h n x c1 H14 x1) (csubc_head g x c H15 k (lift h (r k n) x1)))))) H5)))))) v (sym_eq T v x1 H10))) k0 (sym_eq K k0 k H9))) c2 (sym_eq C c2 x0 H8))) H4)) H2))) e1 H3 H6 H1))) | (csubc_abst c1 c2 H1 v a H3 w H5) \Rightarrow (\lambda (H6: (eq C (CHead c1 (Bind Abst) v) e1)).(\lambda (H7: (eq C (CHead c2 (Bind Abbr) w) (CHead x0 k x1))).(eq_ind C (CHead c1 (Bind Abst) v) (\lambda (c0: C).((eq C (CHead c2 (Bind Abbr) w) (CHead x0 k x1)) \to ((csubc g c1 c2) \to ((sc3 g (asucc g a) c1 v) \to ((sc3 g a c2 w) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 c0)) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1)))))))))) (\lambda (H9: (eq C (CHead c2 (Bind Abbr) w) (CHead x0 k x1))).(let H2 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind Abbr) w) (CHead x0 k x1) H9) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow (Bind Abbr) | (CHead _ k _) \Rightarrow k])) (CHead c2 (Bind Abbr) w) (CHead x0 k x1) H9) in ((let H10 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 (Bind Abbr) w) (CHead x0 k x1) H9) in (eq_ind C x0 (\lambda (c0: C).((eq K (Bind Abbr) k) \to ((eq T w x1) \to ((csubc g c1 c0) \to ((sc3 g (asucc g a) c1 v) \to ((sc3 g a c0 w) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1))))))))))) (\lambda (H11: (eq K (Bind Abbr) k)).(eq_ind K (Bind Abbr) (\lambda (k: K).((eq T w x1) \to ((csubc g c1 x0) \to ((sc3 g (asucc g a) c1 v) \to ((sc3 g a x0 w) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1)))))))))) (\lambda (H12: (eq T w x1)).(eq_ind T x1 (\lambda (t: T).((csubc g c1 x0) \to ((sc3 g (asucc g a) c1 v) \to ((sc3 g a x0 t) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1))))))))) (\lambda (H13: (csubc g c1 x0)).(\lambda (H14: (sc3 g (asucc g a) c1 v)).(\lambda (H15: (sc3 g a x0 x1)).(let H8 \def (eq_ind_r K k (\lambda (k: K).(\forall (h0: nat).((drop h0 n (CHead c k (lift h (r k n) x1)) (CHead x0 k x1)) \to (\forall (e1: C).((csubc g e1 (CHead x0 k x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))))))) H8 (Bind Abbr) H11) in (let H16 \def (eq_ind_r K k (\lambda (k: K).(drop h (r k n) c x0)) H5 (Bind Abbr) H11) in (let H_x \def (H x0 (r (Bind Abbr) n) h H16 c1 H13) in (let H17 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r (Bind Abbr) n) c3 c1)) (\lambda (c3: C).(csubc g c3 c)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1))))) (\lambda (x: C).(\lambda (H18: (drop h (r (Bind Abbr) n) x c1)).(\lambda (H19: (csubc g x c)).(ex_intro2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1)))) (CHead x (Bind Abst) (lift h n v)) (drop_skip_bind h n x c1 H18 Abst v) (csubc_abst g x c H19 (lift h n v) a (sc3_lift g (asucc g a) c1 v H14 x h n H18) (lift h (r (Bind Abbr) n) x1) (sc3_lift g a x0 x1 H15 c h (r (Bind Abbr) n) H16)))))) H17)))))))) w (sym_eq T w x1 H12))) k H11)) c2 (sym_eq C c2 x0 H10))) H4)) H2))) e1 H6 H7 H1 H3 H5)))]) in (H9 (refl_equal C e1) (refl_equal C (CHead x0 k x1)))) t H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)).
+ \lambda (g: G).(\lambda (c2: C).(C_ind (\lambda (c: C).(\forall (e2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 c)))))))))) (\lambda (n: nat).(\lambda (e2: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e1 e2)).(and3_ind (eq C e2 (CSort n)) (eq nat h O) (eq nat d O) (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n)))) (\lambda (H1: (eq C e2 (CSort n))).(\lambda (H2: (eq nat h O)).(\lambda (H3: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: C).(drop n0 d c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))))) (eq_ind_r nat O (\lambda (n0: nat).(ex2 C (\lambda (c1: C).(drop O n0 c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))))) (let H4 \def (eq_ind C e2 (\lambda (c: C).(csubc g e1 c)) H0 (CSort n) H1) in (ex_intro2 C (\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g c1 (CSort n))) e1 (drop_refl e1) H4)) d H3) h H2)))) (drop_gen_sort n h d e2 H))))))))) (\lambda (c: C).(\lambda (H: ((\forall (e2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h d c1 e1)) (\lambda (c1: C).(csubc g c1 c))))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e2: C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c k t) e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) (\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c k t) e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t)))))))) (\lambda (H0: (drop O O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H2 \def (eq_ind_r C e2 (\lambda (c: C).(csubc g e1 c)) H1 (CHead c k t) (drop_gen_refl (CHead c k t) e2 H0)) in (ex_intro2 C (\lambda (c1: C).(drop O O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))) e1 (drop_refl e1) H2))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c k t) e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop n O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))).(\lambda (H1: (drop (S n) O (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 e2)).(let H_x \def (H e2 O (r k n) (drop_gen_drop k c e2 t n H1) e1 H2) in (let H3 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (r k n) O c1 e1)) (\lambda (c1: C).(csubc g c1 c)) (ex2 C (\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t)))) (\lambda (x: C).(\lambda (H4: (drop (r k n) O x e1)).(\lambda (H5: (csubc g x c)).(ex_intro2 C (\lambda (c1: C).(drop (S n) O c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))) (CHead x k t) (drop_drop k n x e1 H4 t) (csubc_head g x c H5 k t))))) H3)))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n (CHead c k t) e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t)))))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c k t) e2)).(\lambda (e1: C).(\lambda (H2: (csubc g e1 e2)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C e2 (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k n) c e))) (ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3: (eq C e2 (CHead x0 k x1))).(\lambda (H4: (eq T t (lift h (r k n) x1))).(\lambda (H5: (drop h (r k n) c x0)).(let H6 \def (eq_ind C e2 (\lambda (c: C).(csubc g e1 c)) H2 (CHead x0 k x1) H3) in (let H7 \def (eq_ind C e2 (\lambda (c0: C).(\forall (h: nat).((drop h n (CHead c k t) c0) \to (\forall (e1: C).((csubc g e1 c0) \to (ex2 C (\lambda (c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) H0 (CHead x0 k x1) H3) in (let H8 \def (eq_ind T t (\lambda (t: T).(\forall (h: nat).((drop h n (CHead c k t) (CHead x0 k x1)) \to (\forall (e1: C).((csubc g e1 (CHead x0 k x1)) \to (ex2 C (\lambda (c1: C).(drop h n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t))))))))) H7 (lift h (r k n) x1) H4) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0: T).(ex2 C (\lambda (c1: C).(drop h (S n) c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k t0))))) (let H9 \def (match H6 return (\lambda (_: ?).(\lambda (c0: C).(\lambda (c1: C).((eq C c0 e1) \to ((eq C c1 (CHead x0 k x1)) \to (ex2 C (\lambda (c2: C).(drop h (S n) c2 e1)) (\lambda (c2: C).(csubc g c2 (CHead c k (lift h (r k n) x1)))))))))) with [(csubc_sort n0) \Rightarrow (\lambda (H1: (eq C (CSort n0) e1)).(\lambda (H3: (eq C (CSort n0) (CHead x0 k x1))).(eq_ind C (CSort n0) (\lambda (c0: C).((eq C (CSort n0) (CHead x0 k x1)) \to (ex2 C (\lambda (c1: C).(drop h (S n) c1 c0)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))))) (\lambda (H4: (eq C (CSort n0) (CHead x0 k x1))).(let H5 \def (eq_ind C (CSort n0) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead x0 k x1) H4) in (False_ind (ex2 C (\lambda (c1: C).(drop h (S n) c1 (CSort n0))) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1))))) H5))) e1 H1 H3))) | (csubc_head c1 c2 H1 k0 v) \Rightarrow (\lambda (H3: (eq C (CHead c1 k0 v) e1)).(\lambda (H6: (eq C (CHead c2 k0 v) (CHead x0 k x1))).(eq_ind C (CHead c1 k0 v) (\lambda (c0: C).((eq C (CHead c2 k0 v) (CHead x0 k x1)) \to ((csubc g c1 c2) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 c0)) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1)))))))) (\lambda (H7: (eq C (CHead c2 k0 v) (CHead x0 k x1))).(let H2 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow v | (CHead _ _ t) \Rightarrow t])) (CHead c2 k0 v) (CHead x0 k x1) H7) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c2 k0 v) (CHead x0 k x1) H7) in ((let H8 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 k0 v) (CHead x0 k x1) H7) in (eq_ind C x0 (\lambda (c0: C).((eq K k0 k) \to ((eq T v x1) \to ((csubc g c1 c0) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k0 v))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1))))))))) (\lambda (H9: (eq K k0 k)).(eq_ind K k (\lambda (k1: K).((eq T v x1) \to ((csubc g c1 x0) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k1 v))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1)))))))) (\lambda (H10: (eq T v x1)).(eq_ind T x1 (\lambda (t: T).((csubc g c1 x0) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k t))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1))))))) (\lambda (H11: (csubc g c1 x0)).(let H12 \def (eq_ind T v (\lambda (t: T).(eq C (CHead c1 k0 t) e1)) H3 x1 H10) in (let H13 \def (eq_ind K k0 (\lambda (k: K).(eq C (CHead c1 k x1) e1)) H12 k H9) in (let H_x \def (H x0 (r k n) h H5 c1 H11) in (let H5 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r k n) c3 c1)) (\lambda (c3: C).(csubc g c3 c)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k x1))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1))))) (\lambda (x: C).(\lambda (H14: (drop h (r k n) x c1)).(\lambda (H15: (csubc g x c)).(ex_intro2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 k x1))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1)))) (CHead x k (lift h (r k n) x1)) (drop_skip k h n x c1 H14 x1) (csubc_head g x c H15 k (lift h (r k n) x1)))))) H5)))))) v (sym_eq T v x1 H10))) k0 (sym_eq K k0 k H9))) c2 (sym_eq C c2 x0 H8))) H4)) H2))) e1 H3 H6 H1))) | (csubc_abst c1 c2 H1 v a H3 w H5) \Rightarrow (\lambda (H6: (eq C (CHead c1 (Bind Abst) v) e1)).(\lambda (H7: (eq C (CHead c2 (Bind Abbr) w) (CHead x0 k x1))).(eq_ind C (CHead c1 (Bind Abst) v) (\lambda (c0: C).((eq C (CHead c2 (Bind Abbr) w) (CHead x0 k x1)) \to ((csubc g c1 c2) \to ((sc3 g (asucc g a) c1 v) \to ((sc3 g a c2 w) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 c0)) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1)))))))))) (\lambda (H9: (eq C (CHead c2 (Bind Abbr) w) (CHead x0 k x1))).(let H2 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow w | (CHead _ _ t) \Rightarrow t])) (CHead c2 (Bind Abbr) w) (CHead x0 k x1) H9) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow (Bind Abbr) | (CHead _ k _) \Rightarrow k])) (CHead c2 (Bind Abbr) w) (CHead x0 k x1) H9) in ((let H10 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c2 | (CHead c _ _) \Rightarrow c])) (CHead c2 (Bind Abbr) w) (CHead x0 k x1) H9) in (eq_ind C x0 (\lambda (c0: C).((eq K (Bind Abbr) k) \to ((eq T w x1) \to ((csubc g c1 c0) \to ((sc3 g (asucc g a) c1 v) \to ((sc3 g a c0 w) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1))))))))))) (\lambda (H11: (eq K (Bind Abbr) k)).(eq_ind K (Bind Abbr) (\lambda (k: K).((eq T w x1) \to ((csubc g c1 x0) \to ((sc3 g (asucc g a) c1 v) \to ((sc3 g a x0 w) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c k (lift h (r k n) x1)))))))))) (\lambda (H12: (eq T w x1)).(eq_ind T x1 (\lambda (t: T).((csubc g c1 x0) \to ((sc3 g (asucc g a) c1 v) \to ((sc3 g a x0 t) \to (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1))))))))) (\lambda (H13: (csubc g c1 x0)).(\lambda (H14: (sc3 g (asucc g a) c1 v)).(\lambda (H15: (sc3 g a x0 x1)).(let H8 \def (eq_ind_r K k (\lambda (k: K).(\forall (h0: nat).((drop h0 n (CHead c k (lift h (r k n) x1)) (CHead x0 k x1)) \to (\forall (e1: C).((csubc g e1 (CHead x0 k x1)) \to (ex2 C (\lambda (c1: C).(drop h0 n c1 e1)) (\lambda (c1: C).(csubc g c1 (CHead c k (lift h (r k n) x1)))))))))) H8 (Bind Abbr) H11) in (let H16 \def (eq_ind_r K k (\lambda (k: K).(drop h (r k n) c x0)) H5 (Bind Abbr) H11) in (let H_x \def (H x0 (r (Bind Abbr) n) h H16 c1 H13) in (let H17 \def H_x in (ex2_ind C (\lambda (c3: C).(drop h (r (Bind Abbr) n) c3 c1)) (\lambda (c3: C).(csubc g c3 c)) (ex2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1))))) (\lambda (x: C).(\lambda (H18: (drop h (r (Bind Abbr) n) x c1)).(\lambda (H19: (csubc g x c)).(ex_intro2 C (\lambda (c3: C).(drop h (S n) c3 (CHead c1 (Bind Abst) v))) (\lambda (c3: C).(csubc g c3 (CHead c (Bind Abbr) (lift h (r (Bind Abbr) n) x1)))) (CHead x (Bind Abst) (lift h n v)) (drop_skip_bind h n x c1 H18 Abst v) (csubc_abst g x c H19 (lift h n v) a (sc3_lift g (asucc g a) c1 v H14 x h n H18) (lift h (r (Bind Abbr) n) x1) (sc3_lift g a x0 x1 H15 c h (r (Bind Abbr) n) H16)))))) H17)))))))) w (sym_eq T w x1 H12))) k H11)) c2 (sym_eq C c2 x0 H10))) H4)) H2))) e1 H6 H7 H1 H3 H5)))]) in (H9 (refl_equal C e1) (refl_equal C (CHead x0 k x1)))) t H4))))))))) (drop_gen_skip_l c e2 t h n k H1)))))))) d))))))) c2)).
 
 theorem drop1_csubc_trans:
  \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))
 \def
- \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2 e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e2 e1)).(let H1 \def (match H return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c c2)).(\lambda (H3: (eq C c e2)).(eq_ind C c2 (\lambda (c0: C).((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))) (\lambda (H4: (eq C c2 e2)).(eq_ind C e2 (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c0 c1)))) (let H \def (eq_ind_r C e2 (\lambda (c: C).(csubc g c e1)) H0 c2 H4) in (eq_ind C c2 (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c0 c1)))) (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c2 c1)) e1 (drop1_nil e1) H) e2 H4)) c2 (sym_eq C c2 e2 H4))) c (sym_eq C c c2 H2) H3)))) | (drop1_cons c1 c0 h d H1 c3 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c1 c2)).(\lambda (H5: (eq C c3 e2)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e: PList).(match e return (\lambda (_: ?).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop h d c1 c0) \to ((drop1 hds c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 PNil c2 e1)) (\lambda (c4: C).(csubc g c2 c4))))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c2) (refl_equal C e2)))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))).(\lambda (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H2 \def (match H0 return (\lambda (p0: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p0 (PCons n n0 p)) \to ((eq C c c2) \to ((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c e2)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e return (\lambda (_: ?).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c c2) \to ((eq C c e2) \to (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))) H5)) H3 H4)))) | (drop1_cons c1 c0 h d H2 c3 hds H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds) (PCons n n0 p))).(\lambda (H5: (eq C c1 c2)).(\lambda (H6: (eq C c3 e2)).((let H7 \def (f_equal PList PList (\lambda (e: PList).(match e return (\lambda (_: ?).PList) with [PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_: ?).nat) with [PNil \Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_: ?).nat) with [PNil \Rightarrow h | (PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n1 d c1 c0) \to ((drop1 hds c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2 c4)))))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n n1 c1 c0) \to ((drop1 hds c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2 c4))))))))) (\lambda (H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n n0 c1 c0) \to ((drop1 p0 c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2 c4)))))))) (\lambda (H12: (eq C c1 c2)).(eq_ind C c2 (\lambda (c: C).((eq C c3 e2) \to ((drop n n0 c c0) \to ((drop1 p c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2 c4))))))) (\lambda (H13: (eq C c3 e2)).(eq_ind C e2 (\lambda (c: C).((drop n n0 c2 c0) \to ((drop1 p c0 c) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2 c4)))))) (\lambda (H14: (drop n n0 c2 c0)).(\lambda (H15: (drop1 p c0 e2)).(let H_x \def (H c0 e2 H15 e1 H1) in (let H0 \def H_x in (ex2_ind C (\lambda (c2: C).(drop1 p c2 e1)) (\lambda (c2: C).(csubc g c0 c2)) (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2 c4))) (\lambda (x: C).(\lambda (H1: (drop1 p x e1)).(\lambda (H16: (csubc g c0 x)).(let H_x0 \def (drop_csubc_trans g c2 c0 n0 n H14 x H16) in (let H \def H_x0 in (ex2_ind C (\lambda (c2: C).(drop n n0 c2 x)) (\lambda (c4: C).(csubc g c2 c4)) (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2 c4))) (\lambda (x0: C).(\lambda (H17: (drop n n0 x0 x)).(\lambda (H18: (csubc g c2 x0)).(ex_intro2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2 c4)) x0 (drop1_cons x0 x n n0 H17 e1 p H1) H18)))) H)))))) H0))))) c3 (sym_eq C c3 e2 H13))) c1 (sym_eq C c1 c2 H12))) hds (sym_eq PList hds p H11))) d (sym_eq nat d n0 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p)) (refl_equal C c2) (refl_equal C e2)))))))))))) hds)).
+ \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2 e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e2 e1)).(let H1 \def (match H return (\lambda (_: ?).(\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c c2)).(\lambda (H3: (eq C c e2)).(eq_ind C c2 (\lambda (c0: C).((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))) (\lambda (H4: (eq C c2 e2)).(eq_ind C e2 (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c0 c1)))) (let H \def (eq_ind_r C e2 (\lambda (c: C).(csubc g c e1)) H0 c2 H4) in (eq_ind C c2 (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c0 c1)))) (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c2 c1)) e1 (drop1_nil e1) H) e2 H4)) c2 (sym_eq C c2 e2 H4))) c (sym_eq C c c2 H2) H3)))) | (drop1_cons c1 c0 h d H1 c3 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c1 c2)).(\lambda (H5: (eq C c3 e2)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e: PList).(match e return (\lambda (_: ?).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop h d c1 c0) \to ((drop1 hds c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 PNil c2 e1)) (\lambda (c4: C).(csubc g c2 c4))))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c2) (refl_equal C e2)))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))).(\lambda (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e2 e1)).(let H2 \def (match H0 return (\lambda (_: ?).(\lambda (p0: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p0 (PCons n n0 p)) \to ((eq C c c2) \to ((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c e2)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e return (\lambda (_: ?).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c c2) \to ((eq C c e2) \to (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c2 c1))))) H5)) H3 H4)))) | (drop1_cons c1 c0 h d H2 c3 hds H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds) (PCons n n0 p))).(\lambda (H5: (eq C c1 c2)).(\lambda (H6: (eq C c3 e2)).((let H7 \def (f_equal PList PList (\lambda (e: PList).(match e return (\lambda (_: ?).PList) with [PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_: ?).nat) with [PNil \Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_: ?).nat) with [PNil \Rightarrow h | (PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n1 d c1 c0) \to ((drop1 hds c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2 c4)))))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n n1 c1 c0) \to ((drop1 hds c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2 c4))))))))) (\lambda (H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n n0 c1 c0) \to ((drop1 p0 c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2 c4)))))))) (\lambda (H12: (eq C c1 c2)).(eq_ind C c2 (\lambda (c: C).((eq C c3 e2) \to ((drop n n0 c c0) \to ((drop1 p c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2 c4))))))) (\lambda (H13: (eq C c3 e2)).(eq_ind C e2 (\lambda (c: C).((drop n n0 c2 c0) \to ((drop1 p c0 c) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2 c4)))))) (\lambda (H14: (drop n n0 c2 c0)).(\lambda (H15: (drop1 p c0 e2)).(let H_x \def (H c0 e2 H15 e1 H1) in (let H0 \def H_x in (ex2_ind C (\lambda (c2: C).(drop1 p c2 e1)) (\lambda (c2: C).(csubc g c0 c2)) (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2 c4))) (\lambda (x: C).(\lambda (H1: (drop1 p x e1)).(\lambda (H16: (csubc g c0 x)).(let H_x0 \def (drop_csubc_trans g c2 c0 n0 n H14 x H16) in (let H \def H_x0 in (ex2_ind C (\lambda (c2: C).(drop n n0 c2 x)) (\lambda (c4: C).(csubc g c2 c4)) (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2 c4))) (\lambda (x0: C).(\lambda (H17: (drop n n0 x0 x)).(\lambda (H18: (csubc g c2 x0)).(ex_intro2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c2 c4)) x0 (drop1_cons x0 x n n0 H17 e1 p H1) H18)))) H)))))) H0))))) c3 (sym_eq C c3 e2 H13))) c1 (sym_eq C c1 c2 H12))) hds (sym_eq PList hds p H11))) d (sym_eq nat d n0 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p)) (refl_equal C c2) (refl_equal C e2)))))))))))) hds)).
 
 theorem csubc_drop1_conf_rev:
  \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: C).((drop1 hds c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))
 \def
- \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2 e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e1 e2)).(let H1 \def (match H return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c c2)).(\lambda (H3: (eq C c e2)).(eq_ind C c2 (\lambda (c0: C).((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))) (\lambda (H4: (eq C c2 e2)).(eq_ind C e2 (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1 c0)))) (let H \def (eq_ind_r C e2 (\lambda (c: C).(csubc g e1 c)) H0 c2 H4) in (eq_ind C c2 (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1 c0)))) (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1 c2)) e1 (drop1_nil e1) H) e2 H4)) c2 (sym_eq C c2 e2 H4))) c (sym_eq C c c2 H2) H3)))) | (drop1_cons c1 c0 h d H1 c3 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c1 c2)).(\lambda (H5: (eq C c3 e2)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e: PList).(match e return (\lambda (_: ?).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop h d c1 c0) \to ((drop1 hds c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 PNil c2 e1)) (\lambda (c4: C).(csubc g c4 c2))))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c2) (refl_equal C e2)))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))).(\lambda (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H2 \def (match H0 return (\lambda (p0: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p0 (PCons n n0 p)) \to ((eq C c c2) \to ((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c e2)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e return (\lambda (_: ?).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c c2) \to ((eq C c e2) \to (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))) H5)) H3 H4)))) | (drop1_cons c1 c0 h d H2 c3 hds H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds) (PCons n n0 p))).(\lambda (H5: (eq C c1 c2)).(\lambda (H6: (eq C c3 e2)).((let H7 \def (f_equal PList PList (\lambda (e: PList).(match e return (\lambda (_: ?).PList) with [PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_: ?).nat) with [PNil \Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_: ?).nat) with [PNil \Rightarrow h | (PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n1 d c1 c0) \to ((drop1 hds c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4 c2)))))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n n1 c1 c0) \to ((drop1 hds c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4 c2))))))))) (\lambda (H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n n0 c1 c0) \to ((drop1 p0 c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4 c2)))))))) (\lambda (H12: (eq C c1 c2)).(eq_ind C c2 (\lambda (c: C).((eq C c3 e2) \to ((drop n n0 c c0) \to ((drop1 p c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4 c2))))))) (\lambda (H13: (eq C c3 e2)).(eq_ind C e2 (\lambda (c: C).((drop n n0 c2 c0) \to ((drop1 p c0 c) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4 c2)))))) (\lambda (H14: (drop n n0 c2 c0)).(\lambda (H15: (drop1 p c0 e2)).(let H_x \def (H c0 e2 H15 e1 H1) in (let H0 \def H_x in (ex2_ind C (\lambda (c2: C).(drop1 p c2 e1)) (\lambda (c2: C).(csubc g c2 c0)) (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4 c2))) (\lambda (x: C).(\lambda (H1: (drop1 p x e1)).(\lambda (H16: (csubc g x c0)).(let H_x0 \def (csubc_drop_conf_rev g c2 c0 n0 n H14 x H16) in (let H \def H_x0 in (ex2_ind C (\lambda (c2: C).(drop n n0 c2 x)) (\lambda (c4: C).(csubc g c4 c2)) (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4 c2))) (\lambda (x0: C).(\lambda (H17: (drop n n0 x0 x)).(\lambda (H18: (csubc g x0 c2)).(ex_intro2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4 c2)) x0 (drop1_cons x0 x n n0 H17 e1 p H1) H18)))) H)))))) H0))))) c3 (sym_eq C c3 e2 H13))) c1 (sym_eq C c1 c2 H12))) hds (sym_eq PList hds p H11))) d (sym_eq nat d n0 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p)) (refl_equal C c2) (refl_equal C e2)))))))))))) hds)).
+ \lambda (g: G).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))))))) (\lambda (c2: C).(\lambda (e2: C).(\lambda (H: (drop1 PNil c2 e2)).(\lambda (e1: C).(\lambda (H0: (csubc g e1 e2)).(let H1 \def (match H return (\lambda (_: ?).(\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p PNil) \to ((eq C c c2) \to ((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c c2)).(\lambda (H3: (eq C c e2)).(eq_ind C c2 (\lambda (c0: C).((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))) (\lambda (H4: (eq C c2 e2)).(eq_ind C e2 (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1 c0)))) (let H \def (eq_ind_r C e2 (\lambda (c: C).(csubc g e1 c)) H0 c2 H4) in (eq_ind C c2 (\lambda (c0: C).(ex2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1 c0)))) (ex_intro2 C (\lambda (c1: C).(drop1 PNil c1 e1)) (\lambda (c1: C).(csubc g c1 c2)) e1 (drop1_nil e1) H) e2 H4)) c2 (sym_eq C c2 e2 H4))) c (sym_eq C c c2 H2) H3)))) | (drop1_cons c1 c0 h d H1 c3 hds H2) \Rightarrow (\lambda (H3: (eq PList (PCons h d hds) PNil)).(\lambda (H4: (eq C c1 c2)).(\lambda (H5: (eq C c3 e2)).((let H6 \def (eq_ind PList (PCons h d hds) (\lambda (e: PList).(match e return (\lambda (_: ?).Prop) with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop h d c1 c0) \to ((drop1 hds c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 PNil c2 e1)) (\lambda (c4: C).(csubc g c4 c2))))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil) (refl_equal C c2) (refl_equal C e2)))))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c2: C).(\forall (e2: C).((drop1 p c2 e2) \to (\forall (e1: C).((csubc g e1 e2) \to (ex2 C (\lambda (c1: C).(drop1 p c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))).(\lambda (c2: C).(\lambda (e2: C).(\lambda (H0: (drop1 (PCons n n0 p) c2 e2)).(\lambda (e1: C).(\lambda (H1: (csubc g e1 e2)).(let H2 \def (match H0 return (\lambda (_: ?).(\lambda (p0: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p0 (PCons n n0 p)) \to ((eq C c c2) \to ((eq C c0 e2) \to (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2)))))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c e2)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e return (\lambda (_: ?).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c c2) \to ((eq C c e2) \to (ex2 C (\lambda (c1: C).(drop1 (PCons n n0 p) c1 e1)) (\lambda (c1: C).(csubc g c1 c2))))) H5)) H3 H4)))) | (drop1_cons c1 c0 h d H2 c3 hds H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds) (PCons n n0 p))).(\lambda (H5: (eq C c1 c2)).(\lambda (H6: (eq C c3 e2)).((let H7 \def (f_equal PList PList (\lambda (e: PList).(match e return (\lambda (_: ?).PList) with [PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p])) (PCons h d hds) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_: ?).nat) with [PNil \Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e: PList).(match e return (\lambda (_: ?).nat) with [PNil \Rightarrow h | (PCons n _ _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n1 d c1 c0) \to ((drop1 hds c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4 c2)))))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds p) \to ((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n n1 c1 c0) \to ((drop1 hds c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4 c2))))))))) (\lambda (H11: (eq PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c1 c2) \to ((eq C c3 e2) \to ((drop n n0 c1 c0) \to ((drop1 p0 c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4 c2)))))))) (\lambda (H12: (eq C c1 c2)).(eq_ind C c2 (\lambda (c: C).((eq C c3 e2) \to ((drop n n0 c c0) \to ((drop1 p c0 c3) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4 c2))))))) (\lambda (H13: (eq C c3 e2)).(eq_ind C e2 (\lambda (c: C).((drop n n0 c2 c0) \to ((drop1 p c0 c) \to (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4 c2)))))) (\lambda (H14: (drop n n0 c2 c0)).(\lambda (H15: (drop1 p c0 e2)).(let H_x \def (H c0 e2 H15 e1 H1) in (let H0 \def H_x in (ex2_ind C (\lambda (c2: C).(drop1 p c2 e1)) (\lambda (c2: C).(csubc g c2 c0)) (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4 c2))) (\lambda (x: C).(\lambda (H1: (drop1 p x e1)).(\lambda (H16: (csubc g x c0)).(let H_x0 \def (csubc_drop_conf_rev g c2 c0 n0 n H14 x H16) in (let H \def H_x0 in (ex2_ind C (\lambda (c2: C).(drop n n0 c2 x)) (\lambda (c4: C).(csubc g c4 c2)) (ex2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4 c2))) (\lambda (x0: C).(\lambda (H17: (drop n n0 x0 x)).(\lambda (H18: (csubc g x0 c2)).(ex_intro2 C (\lambda (c2: C).(drop1 (PCons n n0 p) c2 e1)) (\lambda (c4: C).(csubc g c4 c2)) x0 (drop1_cons x0 x n n0 H17 e1 p H1) H18)))) H)))))) H0))))) c3 (sym_eq C c3 e2 H13))) c1 (sym_eq C c1 c2 H12))) hds (sym_eq PList hds p H11))) d (sym_eq nat d n0 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p)) (refl_equal C c2) (refl_equal C e2)))))))))))) hds)).
 
 theorem drop1_ceqc_trans:
  \forall (g: G).(\forall (hds: PList).(\forall (c2: C).(\forall (e2: C).((drop1 hds c2 e2) \to (\forall (e1: C).((ceqc g e2 e1) \to (ex2 C (\lambda (c1: C).(drop1 hds c1 e1)) (\lambda (c1: C).(ceqc g c2 c1)))))))))
@@ -2796,12 +2796,12 @@ theorem pc3_head_21:
 theorem pc3_pr0_pr2_t:
  \forall (u1: T).(\forall (u2: T).((pr0 u2 u1) \to (\forall (c: C).(\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3 (CHead c k u1) t1 t2))))))))
 \def
- \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u2 u1)).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2 (CHead c k u2) t1 t2)).(let H1 \def (match H0 return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c k u2)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pc3 (CHead c k u1) t1 t2))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0 (CHead c k u2))).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C (CHead c k u2) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pc3 (CHead c k u1) t1 t2))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pc3 (CHead c k u1) t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pc3 (CHead c k u1) t1 t2))) (\lambda (H7: (pr0 t1 t2)).(pc3_pr2_r (CHead c k u1) t1 t2 (pr2_free (CHead c k u1) t1 t2 H7))) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 (CHead c k u2) H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t0 t3 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T t0 t1)).(\lambda (H6: (eq T t t2)).(eq_ind C (CHead c k u2) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pc3 (CHead c k u1) t1 t2))))))) (\lambda (H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pc3 (CHead c k u1) t1 t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pc3 (CHead c k u1) t1 t2))))) (\lambda (H9: (getl i (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda (H11: (subst0 i u t3 t2)).(nat_ind (\lambda (n: nat).((getl n (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 n u t3 t2) \to (pc3 (CHead c k u1) t1 t2)))) (\lambda (H12: (getl O (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H13: (subst0 O u t3 t2)).(K_ind (\lambda (k: K).((clear (CHead c k u2) (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k u1) t1 t2))) (\lambda (b: B).(\lambda (H14: (clear (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(let H0 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H14)) in ((let H15 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H14)) in ((let H16 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H14)) in (\lambda (H17: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H19 \def (eq_ind T u (\lambda (t: T).(subst0 O t t3 t2)) H13 u2 H16) in (eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c (Bind b0) u1) t1 t2)) (ex2_ind T (\lambda (t1: T).(subst0 O u1 t3 t1)) (\lambda (t1: T).(pr0 t2 t1)) (pc3 (CHead c (Bind Abbr) u1) t1 t2) (\lambda (x: T).(\lambda (H: (subst0 O u1 t3 x)).(\lambda (H20: (pr0 t2 x)).(pc3_pr3_t (CHead c (Bind Abbr) u1) t1 x (pr3_pr2 (CHead c (Bind Abbr) u1) t1 x (pr2_delta (CHead c (Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t1 t3 H10 x H)) t2 (pr3_pr2 (CHead c (Bind Abbr) u1) t2 x (pr2_free (CHead c (Bind Abbr) u1) t2 x H20)))))) (pr0_subst0_fwd u2 t3 t2 O H19 u1 H)) b H17))))) H15)) H0)))) (\lambda (f: F).(\lambda (H14: (clear (CHead c (Flat f) u2) (CHead d (Bind Abbr) u))).(clear_pc3_trans (CHead d (Bind Abbr) u) t1 t2 (pc3_pr2_r (CHead d (Bind Abbr) u) t1 t2 (pr2_delta (CHead d (Bind Abbr) u) d u O (getl_refl Abbr d u) t1 t3 H10 t2 H13)) (CHead c (Flat f) u1) (clear_flat c (CHead d (Bind Abbr) u) (clear_gen_flat f c (CHead d (Bind Abbr) u) u2 H14) f u1)))) k (getl_gen_O (CHead c k u2) (CHead d (Bind Abbr) u) H12)))) (\lambda (i0: nat).(\lambda (IHi: (((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pc3 (CHead c k u1) t1 t2))))).(\lambda (H12: (getl (S i0) (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H13: (subst0 (S i0) u t3 t2)).(K_ind (\lambda (k: K).((((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pc3 (CHead c k u1) t1 t2)))) \to ((getl (r k i0) c (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k u1) t1 t2)))) (\lambda (b: B).(\lambda (_: (((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pc3 (CHead c (Bind b) u1) t1 t2))))).(\lambda (H0: (getl (r (Bind b) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Bind b) u1) t1 t2 (pr2_delta (CHead c (Bind b) u1) d u (S i0) (getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) H0 u1) t1 t3 H10 t2 H13))))) (\lambda (f: F).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pc3 (CHead c (Flat f) u1) t1 t2))))).(\lambda (H0: (getl (r (Flat f) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Flat f) u1) t1 t2 (pr2_cflat c t1 t2 (pr2_delta c d u (r (Flat f) i0) H0 t1 t3 H10 t2 H13) f u1))))) k IHi (getl_gen_S k c (CHead d (Bind Abbr) u) u2 i0 H12)))))) i H9 H11)))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0 (CHead c k u2) H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C (CHead c k u2)) (refl_equal T t1) (refl_equal T t2)))))))))).
+ \lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr0 u2 u1)).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (k: K).(\lambda (H0: (pr2 (CHead c k u2) t1 t2)).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 (CHead c k u2)) \to ((eq T t t1) \to ((eq T t0 t2) \to (pc3 (CHead c k u1) t1 t2)))))))) with [(pr2_free c0 t0 t3 H1) \Rightarrow (\lambda (H2: (eq C c0 (CHead c k u2))).(\lambda (H3: (eq T t0 t1)).(\lambda (H4: (eq T t3 t2)).(eq_ind C (CHead c k u2) (\lambda (_: C).((eq T t0 t1) \to ((eq T t3 t2) \to ((pr0 t0 t3) \to (pc3 (CHead c k u1) t1 t2))))) (\lambda (H5: (eq T t0 t1)).(eq_ind T t1 (\lambda (t: T).((eq T t3 t2) \to ((pr0 t t3) \to (pc3 (CHead c k u1) t1 t2)))) (\lambda (H6: (eq T t3 t2)).(eq_ind T t2 (\lambda (t: T).((pr0 t1 t) \to (pc3 (CHead c k u1) t1 t2))) (\lambda (H7: (pr0 t1 t2)).(pc3_pr2_r (CHead c k u1) t1 t2 (pr2_free (CHead c k u1) t1 t2 H7))) t3 (sym_eq T t3 t2 H6))) t0 (sym_eq T t0 t1 H5))) c0 (sym_eq C c0 (CHead c k u2) H2) H3 H4 H1)))) | (pr2_delta c0 d u i H1 t0 t3 H2 t H3) \Rightarrow (\lambda (H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T t0 t1)).(\lambda (H6: (eq T t t2)).(eq_ind C (CHead c k u2) (\lambda (c1: C).((eq T t0 t1) \to ((eq T t t2) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t0 t3) \to ((subst0 i u t3 t) \to (pc3 (CHead c k u1) t1 t2))))))) (\lambda (H7: (eq T t0 t1)).(eq_ind T t1 (\lambda (t4: T).((eq T t t2) \to ((getl i (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((pr0 t4 t3) \to ((subst0 i u t3 t) \to (pc3 (CHead c k u1) t1 t2)))))) (\lambda (H8: (eq T t t2)).(eq_ind T t2 (\lambda (t4: T).((getl i (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((pr0 t1 t3) \to ((subst0 i u t3 t4) \to (pc3 (CHead c k u1) t1 t2))))) (\lambda (H9: (getl i (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H10: (pr0 t1 t3)).(\lambda (H11: (subst0 i u t3 t2)).(nat_ind (\lambda (n: nat).((getl n (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 n u t3 t2) \to (pc3 (CHead c k u1) t1 t2)))) (\lambda (H12: (getl O (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H13: (subst0 O u t3 t2)).(K_ind (\lambda (k: K).((clear (CHead c k u2) (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k u1) t1 t2))) (\lambda (b: B).(\lambda (H14: (clear (CHead c (Bind b) u2) (CHead d (Bind Abbr) u))).(let H0 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H14)) in ((let H15 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H14)) in ((let H16 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d (Bind Abbr) u) u2 H14)) in (\lambda (H17: (eq B Abbr b)).(\lambda (_: (eq C d c)).(let H19 \def (eq_ind T u (\lambda (t: T).(subst0 O t t3 t2)) H13 u2 H16) in (eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c (Bind b0) u1) t1 t2)) (ex2_ind T (\lambda (t1: T).(subst0 O u1 t3 t1)) (\lambda (t1: T).(pr0 t2 t1)) (pc3 (CHead c (Bind Abbr) u1) t1 t2) (\lambda (x: T).(\lambda (H: (subst0 O u1 t3 x)).(\lambda (H20: (pr0 t2 x)).(pc3_pr3_t (CHead c (Bind Abbr) u1) t1 x (pr3_pr2 (CHead c (Bind Abbr) u1) t1 x (pr2_delta (CHead c (Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t1 t3 H10 x H)) t2 (pr3_pr2 (CHead c (Bind Abbr) u1) t2 x (pr2_free (CHead c (Bind Abbr) u1) t2 x H20)))))) (pr0_subst0_fwd u2 t3 t2 O H19 u1 H)) b H17))))) H15)) H0)))) (\lambda (f: F).(\lambda (H14: (clear (CHead c (Flat f) u2) (CHead d (Bind Abbr) u))).(clear_pc3_trans (CHead d (Bind Abbr) u) t1 t2 (pc3_pr2_r (CHead d (Bind Abbr) u) t1 t2 (pr2_delta (CHead d (Bind Abbr) u) d u O (getl_refl Abbr d u) t1 t3 H10 t2 H13)) (CHead c (Flat f) u1) (clear_flat c (CHead d (Bind Abbr) u) (clear_gen_flat f c (CHead d (Bind Abbr) u) u2 H14) f u1)))) k (getl_gen_O (CHead c k u2) (CHead d (Bind Abbr) u) H12)))) (\lambda (i0: nat).(\lambda (IHi: (((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pc3 (CHead c k u1) t1 t2))))).(\lambda (H12: (getl (S i0) (CHead c k u2) (CHead d (Bind Abbr) u))).(\lambda (H13: (subst0 (S i0) u t3 t2)).(K_ind (\lambda (k: K).((((getl i0 (CHead c k u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pc3 (CHead c k u1) t1 t2)))) \to ((getl (r k i0) c (CHead d (Bind Abbr) u)) \to (pc3 (CHead c k u1) t1 t2)))) (\lambda (b: B).(\lambda (_: (((getl i0 (CHead c (Bind b) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pc3 (CHead c (Bind b) u1) t1 t2))))).(\lambda (H0: (getl (r (Bind b) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Bind b) u1) t1 t2 (pr2_delta (CHead c (Bind b) u1) d u (S i0) (getl_head (Bind b) i0 c (CHead d (Bind Abbr) u) H0 u1) t1 t3 H10 t2 H13))))) (\lambda (f: F).(\lambda (_: (((getl i0 (CHead c (Flat f) u2) (CHead d (Bind Abbr) u)) \to ((subst0 i0 u t3 t2) \to (pc3 (CHead c (Flat f) u1) t1 t2))))).(\lambda (H0: (getl (r (Flat f) i0) c (CHead d (Bind Abbr) u))).(pc3_pr2_r (CHead c (Flat f) u1) t1 t2 (pr2_cflat c t1 t2 (pr2_delta c d u (r (Flat f) i0) H0 t1 t3 H10 t2 H13) f u1))))) k IHi (getl_gen_S k c (CHead d (Bind Abbr) u) u2 i0 H12)))))) i H9 H11)))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0 (CHead c k u2) H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C (CHead c k u2)) (refl_equal T t1) (refl_equal T t2)))))))))).
 
 theorem pc3_pr2_pr2_t:
  \forall (c: C).(\forall (u1: T).(\forall (u2: T).((pr2 c u2 u1) \to (\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3 (CHead c k u1) t1 t2))))))))
 \def
- \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u2 u1)).(let H0 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to ((eq T t u2) \to ((eq T t0 u1) \to (\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3 (CHead c k u1) t1 t2))))))))))) with [(pr2_free c0 t1 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1 u2)).(\lambda (H3: (eq T t2 u1)).(eq_ind C c (\lambda (_: C).((eq T t1 u2) \to ((eq T t2 u1) \to ((pr0 t1 t2) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 t4))))))))) (\lambda (H4: (eq T t1 u2)).(eq_ind T u2 (\lambda (t: T).((eq T t2 u1) \to ((pr0 t t2) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 t4)))))))) (\lambda (H5: (eq T t2 u1)).(eq_ind T u1 (\lambda (t: T).((pr0 u2 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 t4))))))) (\lambda (H6: (pr0 u2 u1)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H: (pr2 (CHead c k u2) t0 t3)).(pc3_pr0_pr2_t u1 u2 H6 c t0 t3 k H)))))) t2 (sym_eq T t2 u1 H5))) t1 (sym_eq T t1 u2 H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t1 u2)).(\lambda (H5: (eq T t u1)).(eq_ind C c (\lambda (c1: C).((eq T t1 u2) \to ((eq T t u1) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1 t2) \to ((subst0 i u t2 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 t4))))))))))) (\lambda (H6: (eq T t1 u2)).(eq_ind T u2 (\lambda (t0: T).((eq T t u1) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 t4)))))))))) (\lambda (H7: (eq T t u1)).(eq_ind T u1 (\lambda (t0: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 u2 t2) \to ((subst0 i u t2 t0) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 t4))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 u2 t2)).(\lambda (H10: (subst0 i u t2 u1)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H: (pr2 (CHead c k u2) t0 t3)).(let H11 \def (match H return (\lambda (c0: C).(\lambda (t: T).(\lambda (t1: T).((eq C c0 (CHead c k u2)) \to ((eq T t t0) \to ((eq T t1 t3) \to (pc3 (CHead c k u1) t0 t3))))))) with [(pr2_free c0 t1 t4 H3) \Rightarrow (\lambda (H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T t1 t0)).(\lambda (H6: (eq T t4 t3)).(eq_ind C (CHead c k u2) (\lambda (_: C).((eq T t1 t0) \to ((eq T t4 t3) \to ((pr0 t1 t4) \to (pc3 (CHead c k u1) t0 t3))))) (\lambda (H7: (eq T t1 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t4 t3) \to ((pr0 t t4) \to (pc3 (CHead c k u1) t0 t3)))) (\lambda (H8: (eq T t4 t3)).(eq_ind T t3 (\lambda (t: T).((pr0 t0 t) \to (pc3 (CHead c k u1) t0 t3))) (\lambda (H9: (pr0 t0 t3)).(pc3_pr2_r (CHead c k u1) t0 t3 (pr2_free (CHead c k u1) t0 t3 H9))) t4 (sym_eq T t4 t3 H8))) t1 (sym_eq T t1 t0 H7))) c0 (sym_eq C c0 (CHead c k u2) H4) H5 H6 H3)))) | (pr2_delta c0 d0 u0 i0 H3 t1 t4 H4 t H5) \Rightarrow (\lambda (H6: (eq C c0 (CHead c k u2))).(\lambda (H7: (eq T t1 t0)).(\lambda (H11: (eq T t t3)).(eq_ind C (CHead c k u2) (\lambda (c1: C).((eq T t1 t0) \to ((eq T t t3) \to ((getl i0 c1 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t1 t4) \to ((subst0 i0 u0 t4 t) \to (pc3 (CHead c k u1) t0 t3))))))) (\lambda (H12: (eq T t1 t0)).(eq_ind T t0 (\lambda (t2: T).((eq T t t3) \to ((getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to ((pr0 t2 t4) \to ((subst0 i0 u0 t4 t) \to (pc3 (CHead c k u1) t0 t3)))))) (\lambda (H13: (eq T t t3)).(eq_ind T t3 (\lambda (t2: T).((getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t4) \to ((subst0 i0 u0 t4 t2) \to (pc3 (CHead c k u1) t0 t3))))) (\lambda (H14: (getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0))).(\lambda (H15: (pr0 t0 t4)).(\lambda (H16: (subst0 i0 u0 t4 t3)).(nat_ind (\lambda (n: nat).((getl n (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t4 t3) \to (pc3 (CHead c k u1) t0 t3)))) (\lambda (H17: (getl O (CHead c k u2) (CHead d0 (Bind Abbr) u0))).(\lambda (H18: (subst0 O u0 t4 t3)).((match k return (\lambda (k: K).((clear (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to (pc3 (CHead c k u1) t0 t3))) with [(Bind b) \Rightarrow (\lambda (H19: (clear (CHead c (Bind b) u2) (CHead d0 (Bind Abbr) u0))).(let H \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d0 | (CHead c _ _) \Rightarrow c])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H19)) in ((let H0 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H19)) in ((let H1 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H19)) in (\lambda (H20: (eq B Abbr b)).(\lambda (_: (eq C d0 c)).(let H22 \def (eq_ind T u0 (\lambda (t: T).(subst0 O t t4 t3)) H18 u2 H1) in (eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c (Bind b0) u1) t0 t3)) (ex2_ind T (\lambda (t0: T).(subst0 O t2 t4 t0)) (\lambda (t0: T).(pr0 t3 t0)) (pc3 (CHead c (Bind Abbr) u1) t0 t3) (\lambda (x: T).(\lambda (H2: (subst0 O t2 t4 x)).(\lambda (H9: (pr0 t3 x)).(ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(subst0 (S (plus i O)) u x t0)) (pc3 (CHead c (Bind Abbr) u1) t0 t3) (\lambda (x0: T).(\lambda (H10: (subst0 O u1 t4 x0)).(\lambda (H23: (subst0 (S (plus i O)) u x x0)).(let H24 \def (f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H25 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n u x x0)) H23 (S i) H24) in (pc3_pr2_u (CHead c (Bind Abbr) u1) x0 t0 (pr2_delta (CHead c (Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t0 t4 H15 x0 H10) t3 (pc3_pr2_x (CHead c (Bind Abbr) u1) x0 t3 (pr2_delta (CHead c (Bind Abbr) u1) d u (S i) (getl_head (Bind Abbr) i c (CHead d (Bind Abbr) u) H8 u1) t3 x H9 x0 H25)))))))) (subst0_subst0_back t4 x t2 O H2 u1 u i H10))))) (pr0_subst0_fwd u2 t4 t3 O H22 t2 H9)) b H20))))) H0)) H))) | (Flat f) \Rightarrow (\lambda (H8: (clear (CHead c (Flat f) u2) (CHead d0 (Bind Abbr) u0))).(clear_pc3_trans (CHead d0 (Bind Abbr) u0) t0 t3 (pc3_pr2_r (CHead d0 (Bind Abbr) u0) t0 t3 (pr2_delta (CHead d0 (Bind Abbr) u0) d0 u0 O (getl_refl Abbr d0 u0) t0 t4 H15 t3 H18)) (CHead c (Flat f) u1) (clear_flat c (CHead d0 (Bind Abbr) u0) (clear_gen_flat f c (CHead d0 (Bind Abbr) u0) u2 H8) f u1)))]) (getl_gen_O (CHead c k u2) (CHead d0 (Bind Abbr) u0) H17)))) (\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to ((subst0 i1 u0 t4 t3) \to (pc3 (CHead c k u1) t0 t3))))).(\lambda (H8: (getl (S i1) (CHead c k u2) (CHead d0 (Bind Abbr) u0))).(\lambda (H9: (subst0 (S i1) u0 t4 t3)).(K_ind (\lambda (k: K).((getl (r k i1) c (CHead d0 (Bind Abbr) u0)) \to (pc3 (CHead c k u1) t0 t3))) (\lambda (b: B).(\lambda (H: (getl (r (Bind b) i1) c (CHead d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead c (Bind b) u1) t0 t3 (pr2_delta (CHead c (Bind b) u1) d0 u0 (S i1) (getl_head (Bind b) i1 c (CHead d0 (Bind Abbr) u0) H u1) t0 t4 H15 t3 H9)))) (\lambda (f: F).(\lambda (H: (getl (r (Flat f) i1) c (CHead d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead c (Flat f) u1) t0 t3 (pr2_cflat c t0 t3 (pr2_delta c d0 u0 (r (Flat f) i1) H t0 t4 H15 t3 H9) f u1)))) k (getl_gen_S k c (CHead d0 (Bind Abbr) u0) u2 i1 H8)))))) i0 H14 H16)))) t (sym_eq T t t3 H13))) t1 (sym_eq T t1 t0 H12))) c0 (sym_eq C c0 (CHead c k u2) H6) H7 H11 H3 H4 H5))))]) in (H11 (refl_equal C (CHead c k u2)) (refl_equal T t0) (refl_equal T t3)))))))))) t (sym_eq T t u1 H7))) t1 (sym_eq T t1 u2 H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T u2) (refl_equal T u1)))))).
+ \lambda (c: C).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H: (pr2 c u2 u1)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).((eq C c0 c) \to ((eq T t u2) \to ((eq T t0 u1) \to (\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr2 (CHead c k u2) t1 t2) \to (pc3 (CHead c k u1) t1 t2)))))))))))) with [(pr2_free c0 t1 t2 H0) \Rightarrow (\lambda (H1: (eq C c0 c)).(\lambda (H2: (eq T t1 u2)).(\lambda (H3: (eq T t2 u1)).(eq_ind C c (\lambda (_: C).((eq T t1 u2) \to ((eq T t2 u1) \to ((pr0 t1 t2) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 t4))))))))) (\lambda (H4: (eq T t1 u2)).(eq_ind T u2 (\lambda (t: T).((eq T t2 u1) \to ((pr0 t t2) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 t4)))))))) (\lambda (H5: (eq T t2 u1)).(eq_ind T u1 (\lambda (t: T).((pr0 u2 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 t4))))))) (\lambda (H6: (pr0 u2 u1)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H: (pr2 (CHead c k u2) t0 t3)).(pc3_pr0_pr2_t u1 u2 H6 c t0 t3 k H)))))) t2 (sym_eq T t2 u1 H5))) t1 (sym_eq T t1 u2 H4))) c0 (sym_eq C c0 c H1) H2 H3 H0)))) | (pr2_delta c0 d u i H0 t1 t2 H1 t H2) \Rightarrow (\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq T t1 u2)).(\lambda (H5: (eq T t u1)).(eq_ind C c (\lambda (c1: C).((eq T t1 u2) \to ((eq T t u1) \to ((getl i c1 (CHead d (Bind Abbr) u)) \to ((pr0 t1 t2) \to ((subst0 i u t2 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 t4))))))))))) (\lambda (H6: (eq T t1 u2)).(eq_ind T u2 (\lambda (t0: T).((eq T t u1) \to ((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 t0 t2) \to ((subst0 i u t2 t) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 t4)))))))))) (\lambda (H7: (eq T t u1)).(eq_ind T u1 (\lambda (t0: T).((getl i c (CHead d (Bind Abbr) u)) \to ((pr0 u2 t2) \to ((subst0 i u t2 t0) \to (\forall (t3: T).(\forall (t4: T).(\forall (k: K).((pr2 (CHead c k u2) t3 t4) \to (pc3 (CHead c k u1) t3 t4))))))))) (\lambda (H8: (getl i c (CHead d (Bind Abbr) u))).(\lambda (H9: (pr0 u2 t2)).(\lambda (H10: (subst0 i u t2 u1)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H: (pr2 (CHead c k u2) t0 t3)).(let H11 \def (match H return (\lambda (_: ?).(\lambda (c0: C).(\lambda (t: T).(\lambda (t1: T).((eq C c0 (CHead c k u2)) \to ((eq T t t0) \to ((eq T t1 t3) \to (pc3 (CHead c k u1) t0 t3)))))))) with [(pr2_free c0 t1 t4 H3) \Rightarrow (\lambda (H4: (eq C c0 (CHead c k u2))).(\lambda (H5: (eq T t1 t0)).(\lambda (H6: (eq T t4 t3)).(eq_ind C (CHead c k u2) (\lambda (_: C).((eq T t1 t0) \to ((eq T t4 t3) \to ((pr0 t1 t4) \to (pc3 (CHead c k u1) t0 t3))))) (\lambda (H7: (eq T t1 t0)).(eq_ind T t0 (\lambda (t: T).((eq T t4 t3) \to ((pr0 t t4) \to (pc3 (CHead c k u1) t0 t3)))) (\lambda (H8: (eq T t4 t3)).(eq_ind T t3 (\lambda (t: T).((pr0 t0 t) \to (pc3 (CHead c k u1) t0 t3))) (\lambda (H9: (pr0 t0 t3)).(pc3_pr2_r (CHead c k u1) t0 t3 (pr2_free (CHead c k u1) t0 t3 H9))) t4 (sym_eq T t4 t3 H8))) t1 (sym_eq T t1 t0 H7))) c0 (sym_eq C c0 (CHead c k u2) H4) H5 H6 H3)))) | (pr2_delta c0 d0 u0 i0 H3 t1 t4 H4 t H5) \Rightarrow (\lambda (H6: (eq C c0 (CHead c k u2))).(\lambda (H7: (eq T t1 t0)).(\lambda (H11: (eq T t t3)).(eq_ind C (CHead c k u2) (\lambda (c1: C).((eq T t1 t0) \to ((eq T t t3) \to ((getl i0 c1 (CHead d0 (Bind Abbr) u0)) \to ((pr0 t1 t4) \to ((subst0 i0 u0 t4 t) \to (pc3 (CHead c k u1) t0 t3))))))) (\lambda (H12: (eq T t1 t0)).(eq_ind T t0 (\lambda (t2: T).((eq T t t3) \to ((getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to ((pr0 t2 t4) \to ((subst0 i0 u0 t4 t) \to (pc3 (CHead c k u1) t0 t3)))))) (\lambda (H13: (eq T t t3)).(eq_ind T t3 (\lambda (t2: T).((getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to ((pr0 t0 t4) \to ((subst0 i0 u0 t4 t2) \to (pc3 (CHead c k u1) t0 t3))))) (\lambda (H14: (getl i0 (CHead c k u2) (CHead d0 (Bind Abbr) u0))).(\lambda (H15: (pr0 t0 t4)).(\lambda (H16: (subst0 i0 u0 t4 t3)).(nat_ind (\lambda (n: nat).((getl n (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to ((subst0 n u0 t4 t3) \to (pc3 (CHead c k u1) t0 t3)))) (\lambda (H17: (getl O (CHead c k u2) (CHead d0 (Bind Abbr) u0))).(\lambda (H18: (subst0 O u0 t4 t3)).((match k return (\lambda (_: ?).(\lambda (k: K).((clear (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to (pc3 (CHead c k u1) t0 t3)))) with [(Bind b) \Rightarrow (\lambda (H19: (clear (CHead c (Bind b) u2) (CHead d0 (Bind Abbr) u0))).(let H \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d0 | (CHead c _ _) \Rightarrow c])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H19)) in ((let H0 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H19)) in ((let H1 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead d0 (Bind Abbr) u0) (CHead c (Bind b) u2) (clear_gen_bind b c (CHead d0 (Bind Abbr) u0) u2 H19)) in (\lambda (H20: (eq B Abbr b)).(\lambda (_: (eq C d0 c)).(let H22 \def (eq_ind T u0 (\lambda (t: T).(subst0 O t t4 t3)) H18 u2 H1) in (eq_ind B Abbr (\lambda (b0: B).(pc3 (CHead c (Bind b0) u1) t0 t3)) (ex2_ind T (\lambda (t0: T).(subst0 O t2 t4 t0)) (\lambda (t0: T).(pr0 t3 t0)) (pc3 (CHead c (Bind Abbr) u1) t0 t3) (\lambda (x: T).(\lambda (H2: (subst0 O t2 t4 x)).(\lambda (H9: (pr0 t3 x)).(ex2_ind T (\lambda (t0: T).(subst0 O u1 t4 t0)) (\lambda (t0: T).(subst0 (S (plus i O)) u x t0)) (pc3 (CHead c (Bind Abbr) u1) t0 t3) (\lambda (x0: T).(\lambda (H10: (subst0 O u1 t4 x0)).(\lambda (H23: (subst0 (S (plus i O)) u x x0)).(let H24 \def (f_equal nat nat S (plus i O) i (sym_eq nat i (plus i O) (plus_n_O i))) in (let H25 \def (eq_ind nat (S (plus i O)) (\lambda (n: nat).(subst0 n u x x0)) H23 (S i) H24) in (pc3_pr2_u (CHead c (Bind Abbr) u1) x0 t0 (pr2_delta (CHead c (Bind Abbr) u1) c u1 O (getl_refl Abbr c u1) t0 t4 H15 x0 H10) t3 (pc3_pr2_x (CHead c (Bind Abbr) u1) x0 t3 (pr2_delta (CHead c (Bind Abbr) u1) d u (S i) (getl_head (Bind Abbr) i c (CHead d (Bind Abbr) u) H8 u1) t3 x H9 x0 H25)))))))) (subst0_subst0_back t4 x t2 O H2 u1 u i H10))))) (pr0_subst0_fwd u2 t4 t3 O H22 t2 H9)) b H20))))) H0)) H))) | (Flat f) \Rightarrow (\lambda (H8: (clear (CHead c (Flat f) u2) (CHead d0 (Bind Abbr) u0))).(clear_pc3_trans (CHead d0 (Bind Abbr) u0) t0 t3 (pc3_pr2_r (CHead d0 (Bind Abbr) u0) t0 t3 (pr2_delta (CHead d0 (Bind Abbr) u0) d0 u0 O (getl_refl Abbr d0 u0) t0 t4 H15 t3 H18)) (CHead c (Flat f) u1) (clear_flat c (CHead d0 (Bind Abbr) u0) (clear_gen_flat f c (CHead d0 (Bind Abbr) u0) u2 H8) f u1)))]) (getl_gen_O (CHead c k u2) (CHead d0 (Bind Abbr) u0) H17)))) (\lambda (i1: nat).(\lambda (_: (((getl i1 (CHead c k u2) (CHead d0 (Bind Abbr) u0)) \to ((subst0 i1 u0 t4 t3) \to (pc3 (CHead c k u1) t0 t3))))).(\lambda (H8: (getl (S i1) (CHead c k u2) (CHead d0 (Bind Abbr) u0))).(\lambda (H9: (subst0 (S i1) u0 t4 t3)).(K_ind (\lambda (k: K).((getl (r k i1) c (CHead d0 (Bind Abbr) u0)) \to (pc3 (CHead c k u1) t0 t3))) (\lambda (b: B).(\lambda (H: (getl (r (Bind b) i1) c (CHead d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead c (Bind b) u1) t0 t3 (pr2_delta (CHead c (Bind b) u1) d0 u0 (S i1) (getl_head (Bind b) i1 c (CHead d0 (Bind Abbr) u0) H u1) t0 t4 H15 t3 H9)))) (\lambda (f: F).(\lambda (H: (getl (r (Flat f) i1) c (CHead d0 (Bind Abbr) u0))).(pc3_pr2_r (CHead c (Flat f) u1) t0 t3 (pr2_cflat c t0 t3 (pr2_delta c d0 u0 (r (Flat f) i1) H t0 t4 H15 t3 H9) f u1)))) k (getl_gen_S k c (CHead d0 (Bind Abbr) u0) u2 i1 H8)))))) i0 H14 H16)))) t (sym_eq T t t3 H13))) t1 (sym_eq T t1 t0 H12))) c0 (sym_eq C c0 (CHead c k u2) H6) H7 H11 H3 H4 H5))))]) in (H11 (refl_equal C (CHead c k u2)) (refl_equal T t0) (refl_equal T t3)))))))))) t (sym_eq T t u1 H7))) t1 (sym_eq T t1 u2 H6))) c0 (sym_eq C c0 c H3) H4 H5 H0 H1 H2))))]) in (H0 (refl_equal C c) (refl_equal T u2) (refl_equal T u1)))))).
 
 theorem pc3_pr2_pr3_t:
  \forall (c: C).(\forall (u2: T).(\forall (t1: T).(\forall (t2: T).(\forall (k: K).((pr3 (CHead c k u2) t1 t2) \to (\forall (u1: T).((pr2 c u2 u1) \to (pc3 (CHead c k u1) t1 t2))))))))
@@ -2821,7 +2821,7 @@ theorem pc3_lift:
 theorem pc3_wcpr0__pc3_wcpr0_t_aux:
  \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (k: K).(\forall (u: T).(\forall (t1: T).(\forall (t2: T).((pr3 (CHead c1 k u) t1 t2) \to (pc3 (CHead c2 k u) t1 t2))))))))
 \def
- \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (k: K).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 (CHead c1 k u) t1 t2)).(pr3_ind (CHead c1 k u) (\lambda (t: T).(\lambda (t0: T).(pc3 (CHead c2 k u) t t0))) (\lambda (t: T).(pc3_refl (CHead c2 k u) t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (pr2 (CHead c1 k u) t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead c1 k u) t0 t4)).(\lambda (H3: (pc3 (CHead c2 k u) t0 t4)).(pc3_t t0 (CHead c2 k u) t3 (let H4 \def (match H1 return (\lambda (c: C).(\lambda (t: T).(\lambda (t1: T).((eq C c (CHead c1 k u)) \to ((eq T t t3) \to ((eq T t1 t0) \to (pc3 (CHead c2 k u) t3 t0))))))) with [(pr2_free c t1 t2 H2) \Rightarrow (\lambda (H3: (eq C c (CHead c1 k u))).(\lambda (H4: (eq T t1 t3)).(\lambda (H5: (eq T t2 t0)).(eq_ind C (CHead c1 k u) (\lambda (_: C).((eq T t1 t3) \to ((eq T t2 t0) \to ((pr0 t1 t2) \to (pc3 (CHead c2 k u) t3 t0))))) (\lambda (H6: (eq T t1 t3)).(eq_ind T t3 (\lambda (t: T).((eq T t2 t0) \to ((pr0 t t2) \to (pc3 (CHead c2 k u) t3 t0)))) (\lambda (H7: (eq T t2 t0)).(eq_ind T t0 (\lambda (t: T).((pr0 t3 t) \to (pc3 (CHead c2 k u) t3 t0))) (\lambda (H8: (pr0 t3 t0)).(pc3_pr2_r (CHead c2 k u) t3 t0 (pr2_free (CHead c2 k u) t3 t0 H8))) t2 (sym_eq T t2 t0 H7))) t1 (sym_eq T t1 t3 H6))) c (sym_eq C c (CHead c1 k u) H3) H4 H5 H2)))) | (pr2_delta c d u0 i H2 t1 t2 H3 t H4) \Rightarrow (\lambda (H5: (eq C c (CHead c1 k u))).(\lambda (H6: (eq T t1 t3)).(\lambda (H7: (eq T t t0)).(eq_ind C (CHead c1 k u) (\lambda (c0: C).((eq T t1 t3) \to ((eq T t t0) \to ((getl i c0 (CHead d (Bind Abbr) u0)) \to ((pr0 t1 t2) \to ((subst0 i u0 t2 t) \to (pc3 (CHead c2 k u) t3 t0))))))) (\lambda (H8: (eq T t1 t3)).(eq_ind T t3 (\lambda (t4: T).((eq T t t0) \to ((getl i (CHead c1 k u) (CHead d (Bind Abbr) u0)) \to ((pr0 t4 t2) \to ((subst0 i u0 t2 t) \to (pc3 (CHead c2 k u) t3 t0)))))) (\lambda (H9: (eq T t t0)).(eq_ind T t0 (\lambda (t4: T).((getl i (CHead c1 k u) (CHead d (Bind Abbr) u0)) \to ((pr0 t3 t2) \to ((subst0 i u0 t2 t4) \to (pc3 (CHead c2 k u) t3 t0))))) (\lambda (H10: (getl i (CHead c1 k u) (CHead d (Bind Abbr) u0))).(\lambda (H11: (pr0 t3 t2)).(\lambda (H12: (subst0 i u0 t2 t0)).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u2: T).(getl i (CHead c2 k u) (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u0 u2))) (pc3 (CHead c2 k u) t3 t0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H0: (getl i (CHead c2 k u) (CHead x0 (Bind Abbr) x1))).(\lambda (_: (wcpr0 d x0)).(\lambda (H14: (pr0 u0 x1)).(ex2_ind T (\lambda (t0: T).(subst0 i x1 t2 t0)) (\lambda (t3: T).(pr0 t0 t3)) (pc3 (CHead c2 k u) t3 t0) (\lambda (x: T).(\lambda (H15: (subst0 i x1 t2 x)).(\lambda (H16: (pr0 t0 x)).(pc3_pr2_u (CHead c2 k u) x t3 (pr2_delta (CHead c2 k u) x0 x1 i H0 t3 t2 H11 x H15) t0 (pc3_pr2_x (CHead c2 k u) x t0 (pr2_free (CHead c2 k u) t0 x H16)))))) (pr0_subst0_fwd u0 t2 t0 i H12 x1 H14))))))) (wcpr0_getl (CHead c1 k u) (CHead c2 k u) (wcpr0_comp c1 c2 H u u (pr0_refl u) k) i d u0 (Bind Abbr) H10))))) t (sym_eq T t t0 H9))) t1 (sym_eq T t1 t3 H8))) c (sym_eq C c (CHead c1 k u) H5) H6 H7 H2 H3 H4))))]) in (H4 (refl_equal C (CHead c1 k u)) (refl_equal T t3) (refl_equal T t0))) t4 H3))))))) t1 t2 H0)))))))).
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (wcpr0 c1 c2)).(\lambda (k: K).(\lambda (u: T).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H0: (pr3 (CHead c1 k u) t1 t2)).(pr3_ind (CHead c1 k u) (\lambda (t: T).(\lambda (t0: T).(pc3 (CHead c2 k u) t t0))) (\lambda (t: T).(pc3_refl (CHead c2 k u) t)) (\lambda (t0: T).(\lambda (t3: T).(\lambda (H1: (pr2 (CHead c1 k u) t3 t0)).(\lambda (t4: T).(\lambda (_: (pr3 (CHead c1 k u) t0 t4)).(\lambda (H3: (pc3 (CHead c2 k u) t0 t4)).(pc3_t t0 (CHead c2 k u) t3 (let H4 \def (match H1 return (\lambda (_: ?).(\lambda (c: C).(\lambda (t: T).(\lambda (t1: T).((eq C c (CHead c1 k u)) \to ((eq T t t3) \to ((eq T t1 t0) \to (pc3 (CHead c2 k u) t3 t0)))))))) with [(pr2_free c t1 t2 H2) \Rightarrow (\lambda (H3: (eq C c (CHead c1 k u))).(\lambda (H4: (eq T t1 t3)).(\lambda (H5: (eq T t2 t0)).(eq_ind C (CHead c1 k u) (\lambda (_: C).((eq T t1 t3) \to ((eq T t2 t0) \to ((pr0 t1 t2) \to (pc3 (CHead c2 k u) t3 t0))))) (\lambda (H6: (eq T t1 t3)).(eq_ind T t3 (\lambda (t: T).((eq T t2 t0) \to ((pr0 t t2) \to (pc3 (CHead c2 k u) t3 t0)))) (\lambda (H7: (eq T t2 t0)).(eq_ind T t0 (\lambda (t: T).((pr0 t3 t) \to (pc3 (CHead c2 k u) t3 t0))) (\lambda (H8: (pr0 t3 t0)).(pc3_pr2_r (CHead c2 k u) t3 t0 (pr2_free (CHead c2 k u) t3 t0 H8))) t2 (sym_eq T t2 t0 H7))) t1 (sym_eq T t1 t3 H6))) c (sym_eq C c (CHead c1 k u) H3) H4 H5 H2)))) | (pr2_delta c d u0 i H2 t1 t2 H3 t H4) \Rightarrow (\lambda (H5: (eq C c (CHead c1 k u))).(\lambda (H6: (eq T t1 t3)).(\lambda (H7: (eq T t t0)).(eq_ind C (CHead c1 k u) (\lambda (c0: C).((eq T t1 t3) \to ((eq T t t0) \to ((getl i c0 (CHead d (Bind Abbr) u0)) \to ((pr0 t1 t2) \to ((subst0 i u0 t2 t) \to (pc3 (CHead c2 k u) t3 t0))))))) (\lambda (H8: (eq T t1 t3)).(eq_ind T t3 (\lambda (t4: T).((eq T t t0) \to ((getl i (CHead c1 k u) (CHead d (Bind Abbr) u0)) \to ((pr0 t4 t2) \to ((subst0 i u0 t2 t) \to (pc3 (CHead c2 k u) t3 t0)))))) (\lambda (H9: (eq T t t0)).(eq_ind T t0 (\lambda (t4: T).((getl i (CHead c1 k u) (CHead d (Bind Abbr) u0)) \to ((pr0 t3 t2) \to ((subst0 i u0 t2 t4) \to (pc3 (CHead c2 k u) t3 t0))))) (\lambda (H10: (getl i (CHead c1 k u) (CHead d (Bind Abbr) u0))).(\lambda (H11: (pr0 t3 t2)).(\lambda (H12: (subst0 i u0 t2 t0)).(ex3_2_ind C T (\lambda (e2: C).(\lambda (u2: T).(getl i (CHead c2 k u) (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u0 u2))) (pc3 (CHead c2 k u) t3 t0) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H0: (getl i (CHead c2 k u) (CHead x0 (Bind Abbr) x1))).(\lambda (_: (wcpr0 d x0)).(\lambda (H14: (pr0 u0 x1)).(ex2_ind T (\lambda (t0: T).(subst0 i x1 t2 t0)) (\lambda (t3: T).(pr0 t0 t3)) (pc3 (CHead c2 k u) t3 t0) (\lambda (x: T).(\lambda (H15: (subst0 i x1 t2 x)).(\lambda (H16: (pr0 t0 x)).(pc3_pr2_u (CHead c2 k u) x t3 (pr2_delta (CHead c2 k u) x0 x1 i H0 t3 t2 H11 x H15) t0 (pc3_pr2_x (CHead c2 k u) x t0 (pr2_free (CHead c2 k u) t0 x H16)))))) (pr0_subst0_fwd u0 t2 t0 i H12 x1 H14))))))) (wcpr0_getl (CHead c1 k u) (CHead c2 k u) (wcpr0_comp c1 c2 H u u (pr0_refl u) k) i d u0 (Bind Abbr) H10))))) t (sym_eq T t t0 H9))) t1 (sym_eq T t1 t3 H8))) c (sym_eq C c (CHead c1 k u) H5) H6 H7 H2 H3 H4))))]) in (H4 (refl_equal C (CHead c1 k u)) (refl_equal T t3) (refl_equal T t0))) t4 H3))))))) t1 t2 H0)))))))).
 
 theorem pc3_wcpr0_t:
  \forall (c1: C).(\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t1: T).(\forall (t2: T).((pr3 c1 t1 t2) \to (pc3 c2 t1 t2))))))
@@ -2996,17 +2996,17 @@ inductive csub3 (g:G): C \to (C \to Prop) \def
 theorem csub3_gen_abbr:
  \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v: T).((csub3 g (CHead e1 (Bind Abbr) v) c2) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))))
 \def
- \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: (csub3 g (CHead e1 (Bind Abbr) v) c2)).(let H0 \def (match H return (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abbr) v)) \to ((eq C c0 c2) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2))))))) with [(csub3_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead e1 (Bind Abbr) v))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind Abbr) v) H0) in (False_ind ((eq C (CSort n) c2) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))) H2)) H1))) | (csub3_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) (CHead e1 (Bind Abbr) v))).(\lambda (H2: (eq C (CHead c0 k u) c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind Abbr) v) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c1 k u) (CHead e1 (Bind Abbr) v) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Bind Abbr) v) H1) in (eq_ind C e1 (\lambda (c: C).((eq K k (Bind Abbr)) \to ((eq T u v) \to ((eq C (CHead c0 k u) c2) \to ((csub3 g c c0) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))))) (\lambda (H6: (eq K k (Bind Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T u v) \to ((eq C (CHead c0 k0 u) c2) \to ((csub3 g e1 c0) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2))))))) (\lambda (H7: (eq T u v)).(eq_ind T v (\lambda (t: T).((eq C (CHead c0 (Bind Abbr) t) c2) \to ((csub3 g e1 c0) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))) (\lambda (H8: (eq C (CHead c0 (Bind Abbr) v) c2)).(eq_ind C (CHead c0 (Bind Abbr) v) (\lambda (c: C).((csub3 g e1 c0) \to (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2))))) (\lambda (H9: (csub3 g e1 c0)).(let H10 \def (eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind Abbr) v) c)) H (CHead c0 (Bind Abbr) v) H8) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead c0 (Bind Abbr) v) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)) c0 (refl_equal C (CHead c0 (Bind Abbr) v)) H9))) c2 H8)) u (sym_eq T u v H7))) k (sym_eq K k (Bind Abbr) H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) | (csub3_void c1 c0 H0 b H1 u1 u2) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind Abbr) v))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2) c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abbr) v) H2) in (False_ind ((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g c1 c0) \to ((not (eq B b Void)) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))) H4)) H3 H0 H1))) | (csub3_abst c1 c0 H0 u t H1) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abbr) v))).(\lambda (H3: (eq C (CHead c0 (Bind Abbr) u) c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abbr) v) H2) in (False_ind ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g c1 c0) \to ((ty3 g c0 u t) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CHead e1 (Bind Abbr) v)) (refl_equal C c2))))))).
+ \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v: T).(\lambda (H: (csub3 g (CHead e1 (Bind Abbr) v) c2)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abbr) v)) \to ((eq C c0 c2) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))))) with [(csub3_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead e1 (Bind Abbr) v))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind Abbr) v) H0) in (False_ind ((eq C (CSort n) c2) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))) H2)) H1))) | (csub3_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) (CHead e1 (Bind Abbr) v))).(\lambda (H2: (eq C (CHead c0 k u) c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind Abbr) v) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c1 k u) (CHead e1 (Bind Abbr) v) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Bind Abbr) v) H1) in (eq_ind C e1 (\lambda (c: C).((eq K k (Bind Abbr)) \to ((eq T u v) \to ((eq C (CHead c0 k u) c2) \to ((csub3 g c c0) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))))) (\lambda (H6: (eq K k (Bind Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T u v) \to ((eq C (CHead c0 k0 u) c2) \to ((csub3 g e1 c0) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2))))))) (\lambda (H7: (eq T u v)).(eq_ind T v (\lambda (t: T).((eq C (CHead c0 (Bind Abbr) t) c2) \to ((csub3 g e1 c0) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))) (\lambda (H8: (eq C (CHead c0 (Bind Abbr) v) c2)).(eq_ind C (CHead c0 (Bind Abbr) v) (\lambda (c: C).((csub3 g e1 c0) \to (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2))))) (\lambda (H9: (csub3 g e1 c0)).(let H10 \def (eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind Abbr) v) c)) H (CHead c0 (Bind Abbr) v) H8) in (ex_intro2 C (\lambda (e2: C).(eq C (CHead c0 (Bind Abbr) v) (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)) c0 (refl_equal C (CHead c0 (Bind Abbr) v)) H9))) c2 H8)) u (sym_eq T u v H7))) k (sym_eq K k (Bind Abbr) H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) | (csub3_void c1 c0 H0 b H1 u1 u2) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind Abbr) v))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2) c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abbr) v) H2) in (False_ind ((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g c1 c0) \to ((not (eq B b Void)) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))) H4)) H3 H0 H1))) | (csub3_abst c1 c0 H0 u t H1) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abbr) v))).(\lambda (H3: (eq C (CHead c0 (Bind Abbr) u) c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abbr) v) H2) in (False_ind ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g c1 c0) \to ((ty3 g c0 u t) \to (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abbr) v))) (\lambda (e2: C).(csub3 g e1 e2)))))) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CHead e1 (Bind Abbr) v)) (refl_equal C c2))))))).
 
 theorem csub3_gen_abst:
  \forall (g: G).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csub3 g (CHead e1 (Bind Abst) v1) c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))))
 \def
- \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v1: T).(\lambda (H: (csub3 g (CHead e1 (Bind Abst) v1) c2)).(let H0 \def (match H return (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abst) v1)) \to ((eq C c0 c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))))) with [(csub3_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead e1 (Bind Abst) v1))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind Abst) v1) H0) in (False_ind ((eq C (CSort n) c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) H2)) H1))) | (csub3_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) (CHead e1 (Bind Abst) v1))).(\lambda (H2: (eq C (CHead c0 k u) c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind Abst) v1) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c1 k u) (CHead e1 (Bind Abst) v1) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Bind Abst) v1) H1) in (eq_ind C e1 (\lambda (c: C).((eq K k (Bind Abst)) \to ((eq T u v1) \to ((eq C (CHead c0 k u) c2) \to ((csub3 g c c0) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))))) (\lambda (H6: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) (\lambda (k0: K).((eq T u v1) \to ((eq C (CHead c0 k0 u) c2) \to ((csub3 g e1 c0) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))))) (\lambda (H7: (eq T u v1)).(eq_ind T v1 (\lambda (t: T).((eq C (CHead c0 (Bind Abst) t) c2) \to ((csub3 g e1 c0) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) (\lambda (H8: (eq C (CHead c0 (Bind Abst) v1) c2)).(eq_ind C (CHead c0 (Bind Abst) v1) (\lambda (c: C).((csub3 g e1 c0) \to (or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) (\lambda (H9: (csub3 g e1 c0)).(let H10 \def (eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind Abst) v1) c)) H (CHead c0 (Bind Abst) v1) H8) in (or_introl (ex2 C (\lambda (e2: C).(eq C (CHead c0 (Bind Abst) v1) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind Abst) v1) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) (ex_intro2 C (\lambda (e2: C).(eq C (CHead c0 (Bind Abst) v1) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2)) c0 (refl_equal C (CHead c0 (Bind Abst) v1)) H9)))) c2 H8)) u (sym_eq T u v1 H7))) k (sym_eq K k (Bind Abst) H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) | (csub3_void c1 c0 H0 b H1 u1 u2) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind Abst) v1))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2) c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abst) v1) H2) in (False_ind ((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g c1 c0) \to ((not (eq B b Void)) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) H4)) H3 H0 H1))) | (csub3_abst c1 c0 H0 u t H1) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1))).(\lambda (H3: (eq C (CHead c0 (Bind Abbr) u) c2)).((let H4 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow t | (CHead _ _ t) \Rightarrow t])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) H2) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) H2) in (eq_ind C e1 (\lambda (c: C).((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g c c0) \to ((ty3 g c0 u t) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))))) (\lambda (H6: (eq T t v1)).(eq_ind T v1 (\lambda (t0: T).((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g e1 c0) \to ((ty3 g c0 u t0) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))))) (\lambda (H7: (eq C (CHead c0 (Bind Abbr) u) c2)).(eq_ind C (CHead c0 (Bind Abbr) u) (\lambda (c: C).((csub3 g e1 c0) \to ((ty3 g c0 u v1) \to (or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) (\lambda (H8: (csub3 g e1 c0)).(\lambda (H9: (ty3 g c0 u v1)).(let H10 \def (eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind Abst) v1) c)) H (CHead c0 (Bind Abbr) u) H7) in (or_intror (ex2 C (\lambda (e2: C).(eq C (CHead c0 (Bind Abbr) u) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))) c0 u (refl_equal C (CHead c0 (Bind Abbr) u)) H8 H9))))) c2 H7)) t (sym_eq T t v1 H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CHead e1 (Bind Abst) v1)) (refl_equal C c2))))))).
+ \lambda (g: G).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v1: T).(\lambda (H: (csub3 g (CHead e1 (Bind Abst) v1) c2)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind Abst) v1)) \to ((eq C c0 c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))))) with [(csub3_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead e1 (Bind Abst) v1))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind Abst) v1) H0) in (False_ind ((eq C (CSort n) c2) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))) H2)) H1))) | (csub3_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) (CHead e1 (Bind Abst) v1))).(\lambda (H2: (eq C (CHead c0 k u) c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind Abst) v1) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c1 k u) (CHead e1 (Bind Abst) v1) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Bind Abst) v1) H1) in (eq_ind C e1 (\lambda (c: C).((eq K k (Bind Abst)) \to ((eq T u v1) \to ((eq C (CHead c0 k u) c2) \to ((csub3 g c c0) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))))) (\lambda (H6: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) (\lambda (k0: K).((eq T u v1) \to ((eq C (CHead c0 k0 u) c2) \to ((csub3 g e1 c0) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))))) (\lambda (H7: (eq T u v1)).(eq_ind T v1 (\lambda (t: T).((eq C (CHead c0 (Bind Abst) t) c2) \to ((csub3 g e1 c0) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) (\lambda (H8: (eq C (CHead c0 (Bind Abst) v1) c2)).(eq_ind C (CHead c0 (Bind Abst) v1) (\lambda (c: C).((csub3 g e1 c0) \to (or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))) (\lambda (H9: (csub3 g e1 c0)).(let H10 \def (eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind Abst) v1) c)) H (CHead c0 (Bind Abst) v1) H8) in (or_introl (ex2 C (\lambda (e2: C).(eq C (CHead c0 (Bind Abst) v1) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind Abst) v1) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) (ex_intro2 C (\lambda (e2: C).(eq C (CHead c0 (Bind Abst) v1) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2)) c0 (refl_equal C (CHead c0 (Bind Abst) v1)) H9)))) c2 H8)) u (sym_eq T u v1 H7))) k (sym_eq K k (Bind Abst) H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) | (csub3_void c1 c0 H0 b H1 u1 u2) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind Abst) v1))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2) c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead e1 (Bind Abst) v1) H2) in (False_ind ((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g c1 c0) \to ((not (eq B b Void)) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) H4)) H3 H0 H1))) | (csub3_abst c1 c0 H0 u t H1) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1))).(\lambda (H3: (eq C (CHead c0 (Bind Abbr) u) c2)).((let H4 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow t | (CHead _ _ t) \Rightarrow t])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) H2) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind Abst) v1) H2) in (eq_ind C e1 (\lambda (c: C).((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g c c0) \to ((ty3 g c0 u t) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))))) (\lambda (H6: (eq T t v1)).(eq_ind T v1 (\lambda (t0: T).((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g e1 c0) \to ((ty3 g c0 u t0) \to (or (ex2 C (\lambda (e2: C).(eq C c2 (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))))))))) (\lambda (H7: (eq C (CHead c0 (Bind Abbr) u) c2)).(eq_ind C (CHead c0 (Bind Abbr) u) (\lambda (c: C).((csub3 g e1 c0) \to ((ty3 g c0 u v1) \to (or (ex2 C (\lambda (e2: C).(eq C c (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))))))) (\lambda (H8: (csub3 g e1 c0)).(\lambda (H9: (ty3 g c0 u v1)).(let H10 \def (eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind Abst) v1) c)) H (CHead c0 (Bind Abbr) u) H7) in (or_intror (ex2 C (\lambda (e2: C).(eq C (CHead c0 (Bind Abbr) u) (CHead e2 (Bind Abst) v1))) (\lambda (e2: C).(csub3 g e1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1)))) (ex3_2_intro C T (\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind Abbr) u) (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 v1))) c0 u (refl_equal C (CHead c0 (Bind Abbr) u)) H8 H9))))) c2 H7)) t (sym_eq T t v1 H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CHead e1 (Bind Abst) v1)) (refl_equal C c2))))))).
 
 theorem csub3_gen_bind:
  \forall (g: G).(\forall (b1: B).(\forall (e1: C).(\forall (c2: C).(\forall (v1: T).((csub3 g (CHead e1 (Bind b1) v1) c2) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))))))))))
 \def
- \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v1: T).(\lambda (H: (csub3 g (CHead e1 (Bind b1) v1) c2)).(let H0 \def (match H return (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind b1) v1)) \to ((eq C c0 c2) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))))))))) with [(csub3_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead e1 (Bind b1) v1))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1) v1) H0) in (False_ind ((eq C (CSort n) c2) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))))) H2)) H1))) | (csub3_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(\lambda (H2: (eq C (CHead c0 k u) c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in (eq_ind C e1 (\lambda (c: C).((eq K k (Bind b1)) \to ((eq T u v1) \to ((eq C (CHead c0 k u) c2) \to ((csub3 g c c0) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))))))))) (\lambda (H6: (eq K k (Bind b1))).(eq_ind K (Bind b1) (\lambda (k0: K).((eq T u v1) \to ((eq C (CHead c0 k0 u) c2) \to ((csub3 g e1 c0) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))))))))) (\lambda (H7: (eq T u v1)).(eq_ind T v1 (\lambda (t: T).((eq C (CHead c0 (Bind b1) t) c2) \to ((csub3 g e1 c0) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))))))) (\lambda (H8: (eq C (CHead c0 (Bind b1) v1) c2)).(eq_ind C (CHead c0 (Bind b1) v1) (\lambda (c: C).((csub3 g e1 c0) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))))))) (\lambda (H9: (csub3 g e1 c0)).(let H10 \def (eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind b1) v1) c)) H (CHead c0 (Bind b1) v1) H8) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))) b1 c0 v1 (refl_equal C (CHead c0 (Bind b1) v1)) H9))) c2 H8)) u (sym_eq T u v1 H7))) k (sym_eq K k (Bind b1) H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) | (csub3_void c1 c0 H0 b H1 u1 u2) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2) c2)).((let H4 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H2) in ((let H5 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Void])])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H2) in ((let H6 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H2) in (eq_ind C e1 (\lambda (c: C).((eq B Void b1) \to ((eq T u1 v1) \to ((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g c c0) \to ((not (eq B b Void)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))))))))))) (\lambda (H7: (eq B Void b1)).(eq_ind B Void (\lambda (_: B).((eq T u1 v1) \to ((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g e1 c0) \to ((not (eq B b Void)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))))))))) (\lambda (H8: (eq T u1 v1)).(eq_ind T v1 (\lambda (_: T).((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g e1 c0) \to ((not (eq B b Void)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))))))))) (\lambda (H9: (eq C (CHead c0 (Bind b) u2) c2)).(eq_ind C (CHead c0 (Bind b) u2) (\lambda (c: C).((csub3 g e1 c0) \to ((not (eq B b Void)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))))))) (\lambda (H10: (csub3 g e1 c0)).(\lambda (_: (not (eq B b Void))).(let H12 \def (eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind b1) v1) c)) H (CHead c0 (Bind b) u2) H9) in (let H13 \def (eq_ind_r B b1 (\lambda (b0: B).(csub3 g (CHead e1 (Bind b0) v1) (CHead c0 (Bind b) u2))) H12 Void H7) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind b) u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))) b c0 u2 (refl_equal C (CHead c0 (Bind b) u2)) H10))))) c2 H9)) u1 (sym_eq T u1 v1 H8))) b1 H7)) c1 (sym_eq C c1 e1 H6))) H5)) H4)) H3 H0 H1))) | (csub3_abst c1 c0 H0 u t H1) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1))).(\lambda (H3: (eq C (CHead c0 (Bind Abbr) u) c2)).((let H4 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow t | (CHead _ _ t) \Rightarrow t])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H2) in ((let H5 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H2) in ((let H6 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H2) in (eq_ind C e1 (\lambda (c: C).((eq B Abst b1) \to ((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g c c0) \to ((ty3 g c0 u t) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))))))))))) (\lambda (H7: (eq B Abst b1)).(eq_ind B Abst (\lambda (_: B).((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g e1 c0) \to ((ty3 g c0 u t) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))))))))) (\lambda (H8: (eq T t v1)).(eq_ind T v1 (\lambda (t0: T).((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g e1 c0) \to ((ty3 g c0 u t0) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))))))))) (\lambda (H9: (eq C (CHead c0 (Bind Abbr) u) c2)).(eq_ind C (CHead c0 (Bind Abbr) u) (\lambda (c: C).((csub3 g e1 c0) \to ((ty3 g c0 u v1) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))))))) (\lambda (H10: (csub3 g e1 c0)).(\lambda (_: (ty3 g c0 u v1)).(let H12 \def (eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind b1) v1) c)) H (CHead c0 (Bind Abbr) u) H9) in (let H13 \def (eq_ind_r B b1 (\lambda (b: B).(csub3 g (CHead e1 (Bind b) v1) (CHead c0 (Bind Abbr) u))) H12 Abst H7) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind Abbr) u) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))) Abbr c0 u (refl_equal C (CHead c0 (Bind Abbr) u)) H10))))) c2 H9)) t (sym_eq T t v1 H8))) b1 H7)) c1 (sym_eq C c1 e1 H6))) H5)) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CHead e1 (Bind b1) v1)) (refl_equal C c2)))))))).
+ \lambda (g: G).(\lambda (b1: B).(\lambda (e1: C).(\lambda (c2: C).(\lambda (v1: T).(\lambda (H: (csub3 g (CHead e1 (Bind b1) v1) c2)).(let H0 \def (match H return (\lambda (_: ?).(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead e1 (Bind b1) v1)) \to ((eq C c0 c2) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))))))))) with [(csub3_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead e1 (Bind b1) v1))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead e1 (Bind b1) v1) H0) in (False_ind ((eq C (CSort n) c2) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))))) H2)) H1))) | (csub3_head c1 c0 H0 k u) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u) (CHead e1 (Bind b1) v1))).(\lambda (H2: (eq C (CHead c0 k u) c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u) (CHead e1 (Bind b1) v1) H1) in (eq_ind C e1 (\lambda (c: C).((eq K k (Bind b1)) \to ((eq T u v1) \to ((eq C (CHead c0 k u) c2) \to ((csub3 g c c0) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))))))))) (\lambda (H6: (eq K k (Bind b1))).(eq_ind K (Bind b1) (\lambda (k0: K).((eq T u v1) \to ((eq C (CHead c0 k0 u) c2) \to ((csub3 g e1 c0) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))))))))) (\lambda (H7: (eq T u v1)).(eq_ind T v1 (\lambda (t: T).((eq C (CHead c0 (Bind b1) t) c2) \to ((csub3 g e1 c0) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))))))) (\lambda (H8: (eq C (CHead c0 (Bind b1) v1) c2)).(eq_ind C (CHead c0 (Bind b1) v1) (\lambda (c: C).((csub3 g e1 c0) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))))))) (\lambda (H9: (csub3 g e1 c0)).(let H10 \def (eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind b1) v1) c)) H (CHead c0 (Bind b1) v1) H8) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind b1) v1) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))) b1 c0 v1 (refl_equal C (CHead c0 (Bind b1) v1)) H9))) c2 H8)) u (sym_eq T u v1 H7))) k (sym_eq K k (Bind b1) H6))) c1 (sym_eq C c1 e1 H5))) H4)) H3)) H2 H0))) | (csub3_void c1 c0 H0 b H1 u1 u2) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2) c2)).((let H4 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u1 | (CHead _ _ t) \Rightarrow t])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H2) in ((let H5 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Void | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Void])])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H2) in ((let H6 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind Void) u1) (CHead e1 (Bind b1) v1) H2) in (eq_ind C e1 (\lambda (c: C).((eq B Void b1) \to ((eq T u1 v1) \to ((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g c c0) \to ((not (eq B b Void)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))))))))))) (\lambda (H7: (eq B Void b1)).(eq_ind B Void (\lambda (_: B).((eq T u1 v1) \to ((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g e1 c0) \to ((not (eq B b Void)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))))))))) (\lambda (H8: (eq T u1 v1)).(eq_ind T v1 (\lambda (_: T).((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g e1 c0) \to ((not (eq B b Void)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))))))))) (\lambda (H9: (eq C (CHead c0 (Bind b) u2) c2)).(eq_ind C (CHead c0 (Bind b) u2) (\lambda (c: C).((csub3 g e1 c0) \to ((not (eq B b Void)) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))))))) (\lambda (H10: (csub3 g e1 c0)).(\lambda (_: (not (eq B b Void))).(let H12 \def (eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind b1) v1) c)) H (CHead c0 (Bind b) u2) H9) in (let H13 \def (eq_ind_r B b1 (\lambda (b0: B).(csub3 g (CHead e1 (Bind b0) v1) (CHead c0 (Bind b) u2))) H12 Void H7) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind b) u2) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))) b c0 u2 (refl_equal C (CHead c0 (Bind b) u2)) H10))))) c2 H9)) u1 (sym_eq T u1 v1 H8))) b1 H7)) c1 (sym_eq C c1 e1 H6))) H5)) H4)) H3 H0 H1))) | (csub3_abst c1 c0 H0 u t H1) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1))).(\lambda (H3: (eq C (CHead c0 (Bind Abbr) u) c2)).((let H4 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow t | (CHead _ _ t) \Rightarrow t])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H2) in ((let H5 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H2) in ((let H6 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 (Bind Abst) t) (CHead e1 (Bind b1) v1) H2) in (eq_ind C e1 (\lambda (c: C).((eq B Abst b1) \to ((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g c c0) \to ((ty3 g c0 u t) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))))))))))) (\lambda (H7: (eq B Abst b1)).(eq_ind B Abst (\lambda (_: B).((eq T t v1) \to ((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g e1 c0) \to ((ty3 g c0 u t) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))))))))) (\lambda (H8: (eq T t v1)).(eq_ind T v1 (\lambda (t0: T).((eq C (CHead c0 (Bind Abbr) u) c2) \to ((csub3 g e1 c0) \to ((ty3 g c0 u t0) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c2 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2))))))))) (\lambda (H9: (eq C (CHead c0 (Bind Abbr) u) c2)).(eq_ind C (CHead c0 (Bind Abbr) u) (\lambda (c: C).((csub3 g e1 c0) \to ((ty3 g c0 u v1) \to (ex2_3 B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C c (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))))))) (\lambda (H10: (csub3 g e1 c0)).(\lambda (_: (ty3 g c0 u v1)).(let H12 \def (eq_ind_r C c2 (\lambda (c: C).(csub3 g (CHead e1 (Bind b1) v1) c)) H (CHead c0 (Bind Abbr) u) H9) in (let H13 \def (eq_ind_r B b1 (\lambda (b: B).(csub3 g (CHead e1 (Bind b) v1) (CHead c0 (Bind Abbr) u))) H12 Abst H7) in (ex2_3_intro B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C (CHead c0 (Bind Abbr) u) (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g e1 e2)))) Abbr c0 u (refl_equal C (CHead c0 (Bind Abbr) u)) H10))))) c2 H9)) t (sym_eq T t v1 H8))) b1 H7)) c1 (sym_eq C c1 e1 H6))) H5)) H4)) H3 H0 H1)))]) in (H0 (refl_equal C (CHead e1 (Bind b1) v1)) (refl_equal C c2)))))))).
 
 theorem csub3_refl:
  \forall (g: G).(\forall (c: C).(csub3 g c c))
@@ -3016,32 +3016,32 @@ theorem csub3_refl:
 theorem csub3_clear_conf:
  \forall (g: G).(\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (e1: C).((clear c1 e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c2 e2))))))))
 \def
- \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csub3 g c1 c2)).(csub3_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c0 e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csub3 g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c4 e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: (clear (CHead c3 k u) e1)).((match k return (\lambda (k0: K).((clear (CHead c3 k0 u) e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear (CHead c4 k0 u) e2))))) with [(Bind b) \Rightarrow (\lambda (H3: (clear (CHead c3 (Bind b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csub3 g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)))) (ex_intro2 C (\lambda (e2: C).(csub3 g (CHead c3 (Bind b) u) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind b) u) (csub3_head g c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1 (clear_gen_bind b c3 e1 u H3))) | (Flat f) \Rightarrow (\lambda (H3: (clear (CHead c3 (Flat f) u) e1)).(let H4 \def (H1 e1 (clear_gen_flat f c3 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x: C).(\lambda (H5: (csub3 g e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2)) x H5 (clear_flat c4 x H6 f u))))) H4)))]) H2))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csub3 g c3 c4)).(\lambda (_: ((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c4 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: (clear (CHead c3 (Bind Void) u1) e1)).(eq_ind_r C (CHead c3 (Bind Void) u1) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csub3 g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u2) e2)))) (ex_intro2 C (\lambda (e2: C).(csub3 g (CHead c3 (Bind Void) u1) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u2) e2)) (CHead c4 (Bind b) u2) (csub3_void g c3 c4 H0 b H2 u1 u2) (clear_bind b c4 u2)) e1 (clear_gen_bind Void c3 e1 u1 H3)))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csub3 g c3 c4)).(\lambda (_: ((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c4 e2))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (ty3 g c4 u t)).(\lambda (e1: C).(\lambda (H3: (clear (CHead c3 (Bind Abst) t) e1)).(eq_ind_r C (CHead c3 (Bind Abst) t) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csub3 g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)))) (ex_intro2 C (\lambda (e2: C).(csub3 g (CHead c3 (Bind Abst) t) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)) (CHead c4 (Bind Abbr) u) (csub3_abst g c3 c4 H0 u t H2) (clear_bind Abbr c4 u)) e1 (clear_gen_bind Abst c3 e1 t H3))))))))))) c1 c2 H)))).
+ \lambda (g: G).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csub3 g c1 c2)).(csub3_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (e1: C).((clear c e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c0 e2))))))) (\lambda (n: nat).(\lambda (e1: C).(\lambda (H0: (clear (CSort n) e1)).(clear_gen_sort e1 n H0 (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear (CSort n) e2))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csub3 g c3 c4)).(\lambda (H1: ((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c4 e2))))))).(\lambda (k: K).(\lambda (u: T).(\lambda (e1: C).(\lambda (H2: (clear (CHead c3 k u) e1)).((match k return (\lambda (_: ?).(\lambda (k0: K).((clear (CHead c3 k0 u) e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear (CHead c4 k0 u) e2)))))) with [(Bind b) \Rightarrow (\lambda (H3: (clear (CHead c3 (Bind b) u) e1)).(eq_ind_r C (CHead c3 (Bind b) u) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csub3 g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)))) (ex_intro2 C (\lambda (e2: C).(csub3 g (CHead c3 (Bind b) u) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u) e2)) (CHead c4 (Bind b) u) (csub3_head g c3 c4 H0 (Bind b) u) (clear_bind b c4 u)) e1 (clear_gen_bind b c3 e1 u H3))) | (Flat f) \Rightarrow (\lambda (H3: (clear (CHead c3 (Flat f) u) e1)).(let H4 \def (H1 e1 (clear_gen_flat f c3 e1 u H3)) in (ex2_ind C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c4 e2)) (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2))) (\lambda (x: C).(\lambda (H5: (csub3 g e1 x)).(\lambda (H6: (clear c4 x)).(ex_intro2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear (CHead c4 (Flat f) u) e2)) x H5 (clear_flat c4 x H6 f u))))) H4)))]) H2))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csub3 g c3 c4)).(\lambda (_: ((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c4 e2))))))).(\lambda (b: B).(\lambda (H2: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (e1: C).(\lambda (H3: (clear (CHead c3 (Bind Void) u1) e1)).(eq_ind_r C (CHead c3 (Bind Void) u1) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csub3 g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u2) e2)))) (ex_intro2 C (\lambda (e2: C).(csub3 g (CHead c3 (Bind Void) u1) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind b) u2) e2)) (CHead c4 (Bind b) u2) (csub3_void g c3 c4 H0 b H2 u1 u2) (clear_bind b c4 u2)) e1 (clear_gen_bind Void c3 e1 u1 H3)))))))))))) (\lambda (c3: C).(\lambda (c4: C).(\lambda (H0: (csub3 g c3 c4)).(\lambda (_: ((\forall (e1: C).((clear c3 e1) \to (ex2 C (\lambda (e2: C).(csub3 g e1 e2)) (\lambda (e2: C).(clear c4 e2))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (H2: (ty3 g c4 u t)).(\lambda (e1: C).(\lambda (H3: (clear (CHead c3 (Bind Abst) t) e1)).(eq_ind_r C (CHead c3 (Bind Abst) t) (\lambda (c: C).(ex2 C (\lambda (e2: C).(csub3 g c e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)))) (ex_intro2 C (\lambda (e2: C).(csub3 g (CHead c3 (Bind Abst) t) e2)) (\lambda (e2: C).(clear (CHead c4 (Bind Abbr) u) e2)) (CHead c4 (Bind Abbr) u) (csub3_abst g c3 c4 H0 u t H2) (clear_bind Abbr c4 u)) e1 (clear_gen_bind Abst c3 e1 t H3))))))))))) c1 c2 H)))).
 
 theorem csub3_drop_flat:
  \forall (g: G).(\forall (f: F).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Flat f) u))))))))))))
 \def
- \lambda (g: G).(\lambda (f: F).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Flat f) u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csub3 g c1 c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 (Flat f) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csub3 g c c2)) H (CHead d1 (Flat f) u) (drop_gen_refl c1 (CHead d1 (Flat f) u) H0)) in (let H2 \def (match H1 return (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead d1 (Flat f) u)) \to ((eq C c0 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u)))))))) with [(csub3_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Flat f) u))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Flat f) u) H0) in (False_ind ((eq C (CSort n) c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u))))) H2)) H1))) | (csub3_head c1 c0 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) (CHead d1 (Flat f) u))).(\lambda (H2: (eq C (CHead c0 k u0) c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u0) (CHead d1 (Flat f) u) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c1 k u0) (CHead d1 (Flat f) u) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u0) (CHead d1 (Flat f) u) H1) in (eq_ind C d1 (\lambda (c: C).((eq K k (Flat f)) \to ((eq T u0 u) \to ((eq C (CHead c0 k u0) c2) \to ((csub3 g c c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))))) (\lambda (H6: (eq K k (Flat f))).(eq_ind K (Flat f) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead c0 k0 u0) c2) \to ((csub3 g d1 c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u)))))))) (\lambda (H7: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c0 (Flat f) t) c2) \to ((csub3 g d1 c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))) (\lambda (H8: (eq C (CHead c0 (Flat f) u) c2)).(eq_ind C (CHead c0 (Flat f) u) (\lambda (c: C).((csub3 g d1 c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c (CHead d2 (Flat f) u)))))) (\lambda (H9: (csub3 g d1 c0)).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O (CHead c0 (Flat f) u) (CHead d2 (Flat f) u))) c0 H9 (drop_refl (CHead c0 (Flat f) u)))) c2 H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Flat f) H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csub3_void c1 c0 H0 b H1 u1 u2) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Void) u1) (CHead d1 (Flat f) u))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2) c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 (Flat f) u) H2) in (False_ind ((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g c1 c0) \to ((not (eq B b Void)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))) H4)) H3 H0 H1))) | (csub3_abst c1 c0 H0 u0 t H1) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Abst) t) (CHead d1 (Flat f) u))).(\lambda (H3: (eq C (CHead c0 (Bind Abbr) u0) c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 (Flat f) u) H2) in (False_ind ((eq C (CHead c0 (Bind Abbr) u0) c2) \to ((csub3 g c1 c0) \to ((ty3 g c0 u0 t) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal C (CHead d1 (Flat f) u)) (refl_equal C c2)))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Flat f) u)))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H0: (csub3 g c1 c2)).(csub3_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (u: T).((drop (S n0) O c (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c0 (CHead d2 (Flat f) u))))))))) (\lambda (n1: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n0) O (CSort n1) (CHead d1 (Flat f) u))).(let H2 \def (match H1 return (\lambda (n: nat).(\lambda (n2: nat).(\lambda (c: C).(\lambda (c0: C).((eq nat n (S n0)) \to ((eq nat n2 O) \to ((eq C c (CSort n1)) \to ((eq C c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u)))))))))))) with [(drop_refl c) \Rightarrow (\lambda (H1: (eq nat O (S n0))).(\lambda (H2: (eq nat O O)).(\lambda (H3: (eq C c (CSort n1))).(\lambda (H4: (eq C c (CHead d1 (Flat f) u))).((let H5 \def (eq_ind nat O (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S n0) H1) in (False_ind ((eq nat O O) \to ((eq C c (CSort n1)) \to ((eq C c (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u))))))) H5)) H2 H3 H4))))) | (drop_drop k h c e H1 u0) \Rightarrow (\lambda (H2: (eq nat (S h) (S n0))).(\lambda (H3: (eq nat O O)).(\lambda (H4: (eq C (CHead c k u0) (CSort n1))).(\lambda (H5: (eq C e (CHead d1 (Flat f) u))).((let H6 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow h | (S n) \Rightarrow n])) (S h) (S n0) H2) in (eq_ind nat n0 (\lambda (n: nat).((eq nat O O) \to ((eq C (CHead c k u0) (CSort n1)) \to ((eq C e (CHead d1 (Flat f) u)) \to ((drop (r k n) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u))))))))) (\lambda (_: (eq nat O O)).(\lambda (H8: (eq C (CHead c k u0) (CSort n1))).(let H9 \def (eq_ind C (CHead c k u0) (\lambda (e0: C).(match e0 return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n1) H8) in (False_ind ((eq C e (CHead d1 (Flat f) u)) \to ((drop (r k n0) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u)))))) H9)))) h (sym_eq nat h n0 H6))) H3 H4 H5 H1))))) | (drop_skip k h d c e H1 u0) \Rightarrow (\lambda (H2: (eq nat h (S n0))).(\lambda (H3: (eq nat (S d) O)).(\lambda (H4: (eq C (CHead c k (lift h (r k d) u0)) (CSort n1))).(\lambda (H5: (eq C (CHead e k u0) (CHead d1 (Flat f) u))).(eq_ind nat (S n0) (\lambda (n: nat).((eq nat (S d) O) \to ((eq C (CHead c k (lift n (r k d) u0)) (CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 (Flat f) u)) \to ((drop n (r k d) c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u))))))))) (\lambda (H6: (eq nat (S d) O)).(let H7 \def (eq_ind nat (S d) (\lambda (e0: nat).(match e0 return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H6) in (False_ind ((eq C (CHead c k (lift (S n0) (r k d) u0)) (CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 (Flat f) u)) \to ((drop (S n0) (r k d) c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u))))))) H7))) h (sym_eq nat h (S n0) H2) H3 H4 H5 H1)))))]) in (H2 (refl_equal nat (S n0)) (refl_equal nat O) (refl_equal C (CSort n1)) (refl_equal C (CHead d1 (Flat f) u)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Flat f) u0)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Flat f) u0))))))))) (\lambda (b: B).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Bind b) u) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csub3 g d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x (Flat f) u0))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Bind b) n0 c3 (CHead x (Flat f) u0) H5 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind b) c0 (CHead d1 (Flat f) u0) u n0 H3)))))))) (\lambda (f0: F).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Flat f0) u) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f0) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csub3 g d1 x)).(\lambda (H5: (drop (S n0) O c3 (CHead x (Flat f) u0))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f0) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Flat f0) n0 c3 (CHead x (Flat f) u0) H5 u))))) (H2 d1 u0 (drop_gen_drop (Flat f0) c0 (CHead d1 (Flat f) u0) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Flat f) u))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H5: (csub3 g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Flat f) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead x (Flat f) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 (CHead d1 (Flat f) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H4: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H5: (csub3 g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u0))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Flat f) u0))) x H5 (drop_drop (Bind Abbr) n0 c3 (CHead x (Flat f) u0) H6 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 (Flat f) u0) t n0 H4))))))))))))) c1 c2 H0)))))) n))).
+ \lambda (g: G).(\lambda (f: F).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Flat f) u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csub3 g c1 c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 (Flat f) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csub3 g c c2)) H (CHead d1 (Flat f) u) (drop_gen_refl c1 (CHead d1 (Flat f) u) H0)) in (let H2 \def (match H1 return (\lambda (_: ?).(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead d1 (Flat f) u)) \to ((eq C c0 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))))) with [(csub3_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Flat f) u))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Flat f) u) H0) in (False_ind ((eq C (CSort n) c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u))))) H2)) H1))) | (csub3_head c1 c0 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) (CHead d1 (Flat f) u))).(\lambda (H2: (eq C (CHead c0 k u0) c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u0) (CHead d1 (Flat f) u) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c1 k u0) (CHead d1 (Flat f) u) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u0) (CHead d1 (Flat f) u) H1) in (eq_ind C d1 (\lambda (c: C).((eq K k (Flat f)) \to ((eq T u0 u) \to ((eq C (CHead c0 k u0) c2) \to ((csub3 g c c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))))) (\lambda (H6: (eq K k (Flat f))).(eq_ind K (Flat f) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead c0 k0 u0) c2) \to ((csub3 g d1 c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u)))))))) (\lambda (H7: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c0 (Flat f) t) c2) \to ((csub3 g d1 c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))) (\lambda (H8: (eq C (CHead c0 (Flat f) u) c2)).(eq_ind C (CHead c0 (Flat f) u) (\lambda (c: C).((csub3 g d1 c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c (CHead d2 (Flat f) u)))))) (\lambda (H9: (csub3 g d1 c0)).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O (CHead c0 (Flat f) u) (CHead d2 (Flat f) u))) c0 H9 (drop_refl (CHead c0 (Flat f) u)))) c2 H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Flat f) H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csub3_void c1 c0 H0 b H1 u1 u2) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Void) u1) (CHead d1 (Flat f) u))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2) c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 (Flat f) u) H2) in (False_ind ((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g c1 c0) \to ((not (eq B b Void)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))) H4)) H3 H0 H1))) | (csub3_abst c1 c0 H0 u0 t H1) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Abst) t) (CHead d1 (Flat f) u))).(\lambda (H3: (eq C (CHead c0 (Bind Abbr) u0) c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (CHead d1 (Flat f) u) H2) in (False_ind ((eq C (CHead c0 (Bind Abbr) u0) c2) \to ((csub3 g c1 c0) \to ((ty3 g c0 u0 t) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Flat f) u))))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal C (CHead d1 (Flat f) u)) (refl_equal C c2)))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Flat f) u)))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H0: (csub3 g c1 c2)).(csub3_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (u: T).((drop (S n0) O c (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c0 (CHead d2 (Flat f) u))))))))) (\lambda (n1: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n0) O (CSort n1) (CHead d1 (Flat f) u))).(let H2 \def (match H1 return (\lambda (_: ?).(\lambda (n: nat).(\lambda (n2: nat).(\lambda (c: C).(\lambda (c0: C).((eq nat n (S n0)) \to ((eq nat n2 O) \to ((eq C c (CSort n1)) \to ((eq C c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u))))))))))))) with [(drop_refl c) \Rightarrow (\lambda (H1: (eq nat O (S n0))).(\lambda (H2: (eq nat O O)).(\lambda (H3: (eq C c (CSort n1))).(\lambda (H4: (eq C c (CHead d1 (Flat f) u))).((let H5 \def (eq_ind nat O (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S n0) H1) in (False_ind ((eq nat O O) \to ((eq C c (CSort n1)) \to ((eq C c (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u))))))) H5)) H2 H3 H4))))) | (drop_drop k h c e H1 u0) \Rightarrow (\lambda (H2: (eq nat (S h) (S n0))).(\lambda (H3: (eq nat O O)).(\lambda (H4: (eq C (CHead c k u0) (CSort n1))).(\lambda (H5: (eq C e (CHead d1 (Flat f) u))).((let H6 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow h | (S n) \Rightarrow n])) (S h) (S n0) H2) in (eq_ind nat n0 (\lambda (n: nat).((eq nat O O) \to ((eq C (CHead c k u0) (CSort n1)) \to ((eq C e (CHead d1 (Flat f) u)) \to ((drop (r k n) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u))))))))) (\lambda (_: (eq nat O O)).(\lambda (H8: (eq C (CHead c k u0) (CSort n1))).(let H9 \def (eq_ind C (CHead c k u0) (\lambda (e0: C).(match e0 return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n1) H8) in (False_ind ((eq C e (CHead d1 (Flat f) u)) \to ((drop (r k n0) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u)))))) H9)))) h (sym_eq nat h n0 H6))) H3 H4 H5 H1))))) | (drop_skip k h d c e H1 u0) \Rightarrow (\lambda (H2: (eq nat h (S n0))).(\lambda (H3: (eq nat (S d) O)).(\lambda (H4: (eq C (CHead c k (lift h (r k d) u0)) (CSort n1))).(\lambda (H5: (eq C (CHead e k u0) (CHead d1 (Flat f) u))).(eq_ind nat (S n0) (\lambda (n: nat).((eq nat (S d) O) \to ((eq C (CHead c k (lift n (r k d) u0)) (CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 (Flat f) u)) \to ((drop n (r k d) c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u))))))))) (\lambda (H6: (eq nat (S d) O)).(let H7 \def (eq_ind nat (S d) (\lambda (e0: nat).(match e0 return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H6) in (False_ind ((eq C (CHead c k (lift (S n0) (r k d) u0)) (CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 (Flat f) u)) \to ((drop (S n0) (r k d) c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Flat f) u))))))) H7))) h (sym_eq nat h (S n0) H2) H3 H4 H5 H1)))))]) in (H2 (refl_equal nat (S n0)) (refl_equal nat O) (refl_equal C (CSort n1)) (refl_equal C (CHead d1 (Flat f) u)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Flat f) u0)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Flat f) u0))))))))) (\lambda (b: B).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Bind b) u) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csub3 g d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x (Flat f) u0))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Bind b) n0 c3 (CHead x (Flat f) u0) H5 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind b) c0 (CHead d1 (Flat f) u0) u n0 H3)))))))) (\lambda (f0: F).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Flat f0) u) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f0) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H4: (csub3 g d1 x)).(\lambda (H5: (drop (S n0) O c3 (CHead x (Flat f) u0))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f0) u) (CHead d2 (Flat f) u0))) x H4 (drop_drop (Flat f0) n0 c3 (CHead x (Flat f) u0) H5 u))))) (H2 d1 u0 (drop_gen_drop (Flat f0) c0 (CHead d1 (Flat f) u0) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Flat f) u))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Flat f) u)))) (\lambda (x: C).(\lambda (H5: (csub3 g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Flat f) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead x (Flat f) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 (CHead d1 (Flat f) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Flat f) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Flat f) u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H4: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Flat f) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Flat f) u0))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Flat f) u0)))) (\lambda (x: C).(\lambda (H5: (csub3 g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Flat f) u0))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Flat f) u0))) x H5 (drop_drop (Bind Abbr) n0 c3 (CHead x (Flat f) u0) H6 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 (Flat f) u0) t n0 H4))))))))))))) c1 c2 H0)))))) n))).
 
 theorem csub3_drop_abbr:
  \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n O c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u)))))))))))
 \def
- \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csub3 g c1 c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 (Bind Abbr) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csub3 g c c2)) H (CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H0)) in (let H2 \def (match H1 return (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead d1 (Bind Abbr) u)) \to ((eq C c0 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u)))))))) with [(csub3_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Bind Abbr) u))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abbr) u) H0) in (False_ind ((eq C (CSort n) c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))))) H2)) H1))) | (csub3_head c1 c0 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) (CHead d1 (Bind Abbr) u))).(\lambda (H2: (eq C (CHead c0 k u0) c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H1) in (eq_ind C d1 (\lambda (c: C).((eq K k (Bind Abbr)) \to ((eq T u0 u) \to ((eq C (CHead c0 k u0) c2) \to ((csub3 g c c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))))))))) (\lambda (H6: (eq K k (Bind Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead c0 k0 u0) c2) \to ((csub3 g d1 c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u)))))))) (\lambda (H7: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c0 (Bind Abbr) t) c2) \to ((csub3 g d1 c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))))))) (\lambda (H8: (eq C (CHead c0 (Bind Abbr) u) c2)).(eq_ind C (CHead c0 (Bind Abbr) u) (\lambda (c: C).((csub3 g d1 c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u)))))) (\lambda (H9: (csub3 g d1 c0)).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O (CHead c0 (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) c0 H9 (drop_refl (CHead c0 (Bind Abbr) u)))) c2 H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Bind Abbr) H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csub3_void c1 c0 H0 b H1 u1 u2) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Void) u1) (CHead d1 (Bind Abbr) u))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2) c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) u) H2) in (False_ind ((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g c1 c0) \to ((not (eq B b Void)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))))))) H4)) H3 H0 H1))) | (csub3_abst c1 c0 H0 u0 t H1) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Abst) t) (CHead d1 (Bind Abbr) u))).(\lambda (H3: (eq C (CHead c0 (Bind Abbr) u0) c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) u) H2) in (False_ind ((eq C (CHead c0 (Bind Abbr) u0) c2) \to ((csub3 g c1 c0) \to ((ty3 g c0 u0 t) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal C (CHead d1 (Bind Abbr) u)) (refl_equal C c2)))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H0: (csub3 g c1 c2)).(csub3_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (u: T).((drop (S n0) O c (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c0 (CHead d2 (Bind Abbr) u))))))))) (\lambda (n1: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n0) O (CSort n1) (CHead d1 (Bind Abbr) u))).(let H2 \def (match H1 return (\lambda (n: nat).(\lambda (n2: nat).(\lambda (c: C).(\lambda (c0: C).((eq nat n (S n0)) \to ((eq nat n2 O) \to ((eq C c (CSort n1)) \to ((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))))))))))) with [(drop_refl c) \Rightarrow (\lambda (H1: (eq nat O (S n0))).(\lambda (H2: (eq nat O O)).(\lambda (H3: (eq C c (CSort n1))).(\lambda (H4: (eq C c (CHead d1 (Bind Abbr) u))).((let H5 \def (eq_ind nat O (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S n0) H1) in (False_ind ((eq nat O O) \to ((eq C c (CSort n1)) \to ((eq C c (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))))) H5)) H2 H3 H4))))) | (drop_drop k h c e H1 u0) \Rightarrow (\lambda (H2: (eq nat (S h) (S n0))).(\lambda (H3: (eq nat O O)).(\lambda (H4: (eq C (CHead c k u0) (CSort n1))).(\lambda (H5: (eq C e (CHead d1 (Bind Abbr) u))).((let H6 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow h | (S n) \Rightarrow n])) (S h) (S n0) H2) in (eq_ind nat n0 (\lambda (n: nat).((eq nat O O) \to ((eq C (CHead c k u0) (CSort n1)) \to ((eq C e (CHead d1 (Bind Abbr) u)) \to ((drop (r k n) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))))))) (\lambda (_: (eq nat O O)).(\lambda (H8: (eq C (CHead c k u0) (CSort n1))).(let H9 \def (eq_ind C (CHead c k u0) (\lambda (e0: C).(match e0 return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n1) H8) in (False_ind ((eq C e (CHead d1 (Bind Abbr) u)) \to ((drop (r k n0) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))))) H9)))) h (sym_eq nat h n0 H6))) H3 H4 H5 H1))))) | (drop_skip k h d c e H1 u0) \Rightarrow (\lambda (H2: (eq nat h (S n0))).(\lambda (H3: (eq nat (S d) O)).(\lambda (H4: (eq C (CHead c k (lift h (r k d) u0)) (CSort n1))).(\lambda (H5: (eq C (CHead e k u0) (CHead d1 (Bind Abbr) u))).(eq_ind nat (S n0) (\lambda (n: nat).((eq nat (S d) O) \to ((eq C (CHead c k (lift n (r k d) u0)) (CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 (Bind Abbr) u)) \to ((drop n (r k d) c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))))))) (\lambda (H6: (eq nat (S d) O)).(let H7 \def (eq_ind nat (S d) (\lambda (e0: nat).(match e0 return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H6) in (False_ind ((eq C (CHead c k (lift (S n0) (r k d) u0)) (CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 (Bind Abbr) u)) \to ((drop (S n0) (r k d) c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))))) H7))) h (sym_eq nat h (S n0) H2) H3 H4 H5 H1)))))]) in (H2 (refl_equal nat (S n0)) (refl_equal nat O) (refl_equal C (CSort n1)) (refl_equal C (CHead d1 (Bind Abbr) u)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Bind Abbr) u0)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abbr) u0))))))))) (\lambda (b: B).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Bind b) u) (CHead d1 (Bind Abbr) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (x: C).(\lambda (H4: (csub3 g d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0))) x H4 (drop_drop (Bind b) n0 c3 (CHead x (Bind Abbr) u0) H5 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind b) c0 (CHead d1 (Bind Abbr) u0) u n0 H3)))))))) (\lambda (f: F).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Flat f) u) (CHead d1 (Bind Abbr) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (x: C).(\lambda (H4: (csub3 g d1 x)).(\lambda (H5: (drop (S n0) O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0))) x H4 (drop_drop (Flat f) n0 c3 (CHead x (Bind Abbr) u0) H5 u))))) (H2 d1 u0 (drop_gen_drop (Flat f) c0 (CHead d1 (Bind Abbr) u0) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Bind Abbr) u))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (x: C).(\lambda (H5: (csub3 g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead x (Bind Abbr) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 (CHead d1 (Bind Abbr) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H4: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind Abbr) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (x: C).(\lambda (H5: (csub3 g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0))) x H5 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind Abbr) u0) H6 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 (Bind Abbr) u0) t n0 H4))))))))))))) c1 c2 H0)))))) n)).
+ \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) u))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csub3 g c1 c2)).(\lambda (d1: C).(\lambda (u: T).(\lambda (H0: (drop O O c1 (CHead d1 (Bind Abbr) u))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csub3 g c c2)) H (CHead d1 (Bind Abbr) u) (drop_gen_refl c1 (CHead d1 (Bind Abbr) u) H0)) in (let H2 \def (match H1 return (\lambda (_: ?).(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead d1 (Bind Abbr) u)) \to ((eq C c0 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))))))))) with [(csub3_sort n) \Rightarrow (\lambda (H0: (eq C (CSort n) (CHead d1 (Bind Abbr) u))).(\lambda (H1: (eq C (CSort n) c2)).((let H2 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abbr) u) H0) in (False_ind ((eq C (CSort n) c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))))) H2)) H1))) | (csub3_head c1 c0 H0 k u0) \Rightarrow (\lambda (H1: (eq C (CHead c1 k u0) (CHead d1 (Bind Abbr) u))).(\lambda (H2: (eq C (CHead c0 k u0) c2)).((let H3 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H1) in ((let H4 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H1) in ((let H5 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c1 | (CHead c _ _) \Rightarrow c])) (CHead c1 k u0) (CHead d1 (Bind Abbr) u) H1) in (eq_ind C d1 (\lambda (c: C).((eq K k (Bind Abbr)) \to ((eq T u0 u) \to ((eq C (CHead c0 k u0) c2) \to ((csub3 g c c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))))))))) (\lambda (H6: (eq K k (Bind Abbr))).(eq_ind K (Bind Abbr) (\lambda (k0: K).((eq T u0 u) \to ((eq C (CHead c0 k0 u0) c2) \to ((csub3 g d1 c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u)))))))) (\lambda (H7: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((eq C (CHead c0 (Bind Abbr) t) c2) \to ((csub3 g d1 c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))))))) (\lambda (H8: (eq C (CHead c0 (Bind Abbr) u) c2)).(eq_ind C (CHead c0 (Bind Abbr) u) (\lambda (c: C).((csub3 g d1 c0) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abbr) u)))))) (\lambda (H9: (csub3 g d1 c0)).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O (CHead c0 (Bind Abbr) u) (CHead d2 (Bind Abbr) u))) c0 H9 (drop_refl (CHead c0 (Bind Abbr) u)))) c2 H8)) u0 (sym_eq T u0 u H7))) k (sym_eq K k (Bind Abbr) H6))) c1 (sym_eq C c1 d1 H5))) H4)) H3)) H2 H0))) | (csub3_void c1 c0 H0 b H1 u1 u2) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Void) u1) (CHead d1 (Bind Abbr) u))).(\lambda (H3: (eq C (CHead c0 (Bind b) u2) c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Void) u1) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) u) H2) in (False_ind ((eq C (CHead c0 (Bind b) u2) c2) \to ((csub3 g c1 c0) \to ((not (eq B b Void)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))))))) H4)) H3 H0 H1))) | (csub3_abst c1 c0 H0 u0 t H1) \Rightarrow (\lambda (H2: (eq C (CHead c1 (Bind Abst) t) (CHead d1 (Bind Abbr) u))).(\lambda (H3: (eq C (CHead c0 (Bind Abbr) u0) c2)).((let H4 \def (eq_ind C (CHead c1 (Bind Abst) t) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abbr) u) H2) in (False_ind ((eq C (CHead c0 (Bind Abbr) u0) c2) \to ((csub3 g c1 c0) \to ((ty3 g c0 u0 t) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abbr) u))))))) H4)) H3 H0 H1)))]) in (H2 (refl_equal C (CHead d1 (Bind Abbr) u)) (refl_equal C c2)))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (u: T).((drop n0 O c1 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H0: (csub3 g c1 c2)).(csub3_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (u: T).((drop (S n0) O c (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c0 (CHead d2 (Bind Abbr) u))))))))) (\lambda (n1: nat).(\lambda (d1: C).(\lambda (u: T).(\lambda (H1: (drop (S n0) O (CSort n1) (CHead d1 (Bind Abbr) u))).(let H2 \def (match H1 return (\lambda (_: ?).(\lambda (n: nat).(\lambda (n2: nat).(\lambda (c: C).(\lambda (c0: C).((eq nat n (S n0)) \to ((eq nat n2 O) \to ((eq C c (CSort n1)) \to ((eq C c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))))))))))) with [(drop_refl c) \Rightarrow (\lambda (H1: (eq nat O (S n0))).(\lambda (H2: (eq nat O O)).(\lambda (H3: (eq C c (CSort n1))).(\lambda (H4: (eq C c (CHead d1 (Bind Abbr) u))).((let H5 \def (eq_ind nat O (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S n0) H1) in (False_ind ((eq nat O O) \to ((eq C c (CSort n1)) \to ((eq C c (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))))) H5)) H2 H3 H4))))) | (drop_drop k h c e H1 u0) \Rightarrow (\lambda (H2: (eq nat (S h) (S n0))).(\lambda (H3: (eq nat O O)).(\lambda (H4: (eq C (CHead c k u0) (CSort n1))).(\lambda (H5: (eq C e (CHead d1 (Bind Abbr) u))).((let H6 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow h | (S n) \Rightarrow n])) (S h) (S n0) H2) in (eq_ind nat n0 (\lambda (n: nat).((eq nat O O) \to ((eq C (CHead c k u0) (CSort n1)) \to ((eq C e (CHead d1 (Bind Abbr) u)) \to ((drop (r k n) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))))))) (\lambda (_: (eq nat O O)).(\lambda (H8: (eq C (CHead c k u0) (CSort n1))).(let H9 \def (eq_ind C (CHead c k u0) (\lambda (e0: C).(match e0 return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n1) H8) in (False_ind ((eq C e (CHead d1 (Bind Abbr) u)) \to ((drop (r k n0) O c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))))) H9)))) h (sym_eq nat h n0 H6))) H3 H4 H5 H1))))) | (drop_skip k h d c e H1 u0) \Rightarrow (\lambda (H2: (eq nat h (S n0))).(\lambda (H3: (eq nat (S d) O)).(\lambda (H4: (eq C (CHead c k (lift h (r k d) u0)) (CSort n1))).(\lambda (H5: (eq C (CHead e k u0) (CHead d1 (Bind Abbr) u))).(eq_ind nat (S n0) (\lambda (n: nat).((eq nat (S d) O) \to ((eq C (CHead c k (lift n (r k d) u0)) (CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 (Bind Abbr) u)) \to ((drop n (r k d) c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))))))) (\lambda (H6: (eq nat (S d) O)).(let H7 \def (eq_ind nat (S d) (\lambda (e0: nat).(match e0 return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H6) in (False_ind ((eq C (CHead c k (lift (S n0) (r k d) u0)) (CSort n1)) \to ((eq C (CHead e k u0) (CHead d1 (Bind Abbr) u)) \to ((drop (S n0) (r k d) c e) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u))))))) H7))) h (sym_eq nat h (S n0) H2) H3 H4 H5 H1)))))]) in (H2 (refl_equal nat (S n0)) (refl_equal nat O) (refl_equal C (CSort n1)) (refl_equal C (CHead d1 (Bind Abbr) u)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (d1: C).(\forall (u0: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Bind Abbr) u0)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abbr) u0))))))))) (\lambda (b: B).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Bind b) u) (CHead d1 (Bind Abbr) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (x: C).(\lambda (H4: (csub3 g d1 x)).(\lambda (H5: (drop n0 O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0))) x H4 (drop_drop (Bind b) n0 c3 (CHead x (Bind Abbr) u0) H5 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind b) c0 (CHead d1 (Bind Abbr) u0) u n0 H3)))))))) (\lambda (f: F).(\lambda (u: T).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Flat f) u) (CHead d1 (Bind Abbr) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (x: C).(\lambda (H4: (csub3 g d1 x)).(\lambda (H5: (drop (S n0) O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0))) x H4 (drop_drop (Flat f) n0 c3 (CHead x (Bind Abbr) u0) H5 u))))) (H2 d1 u0 (drop_gen_drop (Flat f) c0 (CHead d1 (Bind Abbr) u0) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (u: T).(\lambda (H4: (drop (S n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Bind Abbr) u))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (x: C).(\lambda (H5: (csub3 g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u))) x H5 (drop_drop (Bind b) n0 c3 (CHead x (Bind Abbr) u) H6 u2))))) (H c0 c3 H1 d1 u (drop_gen_drop (Bind Void) c0 (CHead d1 (Bind Abbr) u) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (u: T).((drop (S n0) O c0 (CHead d1 (Bind Abbr) u)) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (u0: T).(\lambda (H4: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind Abbr) u0))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abbr) u0))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (x: C).(\lambda (H5: (csub3 g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind Abbr) u0))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0))) x H5 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind Abbr) u0) H6 u))))) (H c0 c3 H1 d1 u0 (drop_gen_drop (Bind Abst) c0 (CHead d1 (Bind Abbr) u0) t n0 H4))))))))))))) c1 c2 H0)))))) n)).
 
 theorem csub3_drop_abst:
  \forall (g: G).(\forall (n: nat).(\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))))
 \def
- \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csub3 g c1 c2)).(\lambda (d1: C).(\lambda (t: T).(\lambda (H0: (drop O O c1 (CHead d1 (Bind Abst) t))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csub3 g c c2)) H (CHead d1 (Bind Abst) t) (drop_gen_refl c1 (CHead d1 (Bind Abst) t) H0)) in (let H2 \def (match H1 return (\lambda (c: C).(\lambda (c0: C).((eq C c (CHead d1 (Bind Abst) t)) \to ((eq C c0 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))) with [(csub3_sort n) \Rightarrow (\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) t))).(\lambda (H2: (eq C (CSort n) c2)).((let H3 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abst) t) H1) in (False_ind ((eq C (CSort n) c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))) H3)) H2))) | (csub3_head c0 c3 H1 k u) \Rightarrow (\lambda (H2: (eq C (CHead c0 k u) (CHead d1 (Bind Abst) t))).(\lambda (H3: (eq C (CHead c3 k u) c2)).((let H4 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c0 k u) (CHead d1 (Bind Abst) t) H2) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c0 k u) (CHead d1 (Bind Abst) t) H2) in ((let H6 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k u) (CHead d1 (Bind Abst) t) H2) in (eq_ind C d1 (\lambda (c: C).((eq K k (Bind Abst)) \to ((eq T u t) \to ((eq C (CHead c3 k u) c2) \to ((csub3 g c c3) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) (\lambda (H7: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) (\lambda (k0: K).((eq T u t) \to ((eq C (CHead c3 k0 u) c2) \to ((csub3 g d1 c3) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))))))) (\lambda (H8: (eq T u t)).(eq_ind T t (\lambda (t0: T).((eq C (CHead c3 (Bind Abst) t0) c2) \to ((csub3 g d1 c3) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))) (\lambda (H9: (eq C (CHead c3 (Bind Abst) t) c2)).(eq_ind C (CHead c3 (Bind Abst) t) (\lambda (c: C).((csub3 g d1 c3) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))))) (\lambda (H10: (csub3 g d1 c3)).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O (CHead c3 (Bind Abst) t) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abst) t) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O (CHead c3 (Bind Abst) t) (CHead d2 (Bind Abst) t))) c3 H10 (drop_refl (CHead c3 (Bind Abst) t))))) c2 H9)) u (sym_eq T u t H8))) k (sym_eq K k (Bind Abst) H7))) c0 (sym_eq C c0 d1 H6))) H5)) H4)) H3 H1))) | (csub3_void c0 c3 H1 b H2 u1 u2) \Rightarrow (\lambda (H3: (eq C (CHead c0 (Bind Void) u1) (CHead d1 (Bind Abst) t))).(\lambda (H4: (eq C (CHead c3 (Bind b) u2) c2)).((let H5 \def (eq_ind C (CHead c0 (Bind Void) u1) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abst) t) H3) in (False_ind ((eq C (CHead c3 (Bind b) u2) c2) \to ((csub3 g c0 c3) \to ((not (eq B b Void)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))) H5)) H4 H1 H2))) | (csub3_abst c0 c3 H1 u t0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c0 (Bind Abst) t0) (CHead d1 (Bind Abst) t))).(\lambda (H4: (eq C (CHead c3 (Bind Abbr) u) c2)).((let H5 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow t0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 (Bind Abst) t0) (CHead d1 (Bind Abst) t) H3) in ((let H6 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Bind Abst) t0) (CHead d1 (Bind Abst) t) H3) in (eq_ind C d1 (\lambda (c: C).((eq T t0 t) \to ((eq C (CHead c3 (Bind Abbr) u) c2) \to ((csub3 g c c3) \to ((ty3 g c3 u t0) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) (\lambda (H7: (eq T t0 t)).(eq_ind T t (\lambda (t1: T).((eq C (CHead c3 (Bind Abbr) u) c2) \to ((csub3 g d1 c3) \to ((ty3 g c3 u t1) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))))))) (\lambda (H8: (eq C (CHead c3 (Bind Abbr) u) c2)).(eq_ind C (CHead c3 (Bind Abbr) u) (\lambda (c: C).((csub3 g d1 c3) \to ((ty3 g c3 u t) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))) (\lambda (H9: (csub3 g d1 c3)).(\lambda (H10: (ty3 g c3 u t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) c3 u H9 (drop_refl (CHead c3 (Bind Abbr) u)) H10)))) c2 H8)) t0 (sym_eq T t0 t H7))) c0 (sym_eq C c0 d1 H6))) H5)) H4 H1 H2)))]) in (H2 (refl_equal C (CHead d1 (Bind Abst) t)) (refl_equal C c2)))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H0: (csub3 g c1 c2)).(csub3_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (t: T).((drop (S n0) O c (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c0 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (n1: nat).(\lambda (d1: C).(\lambda (t: T).(\lambda (H1: (drop (S n0) O (CSort n1) (CHead d1 (Bind Abst) t))).(let H2 \def (match H1 return (\lambda (n: nat).(\lambda (n2: nat).(\lambda (c: C).(\lambda (c0: C).((eq nat n (S n0)) \to ((eq nat n2 O) \to ((eq C c (CSort n1)) \to ((eq C c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))))) with [(drop_refl c) \Rightarrow (\lambda (H1: (eq nat O (S n0))).(\lambda (H2: (eq nat O O)).(\lambda (H3: (eq C c (CSort n1))).(\lambda (H4: (eq C c (CHead d1 (Bind Abst) t))).((let H5 \def (eq_ind nat O (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S n0) H1) in (False_ind ((eq nat O O) \to ((eq C c (CSort n1)) \to ((eq C c (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))) H5)) H2 H3 H4))))) | (drop_drop k h c e H1 u) \Rightarrow (\lambda (H2: (eq nat (S h) (S n0))).(\lambda (H3: (eq nat O O)).(\lambda (H4: (eq C (CHead c k u) (CSort n1))).(\lambda (H5: (eq C e (CHead d1 (Bind Abst) t))).((let H6 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow h | (S n) \Rightarrow n])) (S h) (S n0) H2) in (eq_ind nat n0 (\lambda (n: nat).((eq nat O O) \to ((eq C (CHead c k u) (CSort n1)) \to ((eq C e (CHead d1 (Bind Abst) t)) \to ((drop (r k n) O c e) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) (\lambda (_: (eq nat O O)).(\lambda (H8: (eq C (CHead c k u) (CSort n1))).(let H9 \def (eq_ind C (CHead c k u) (\lambda (e0: C).(match e0 return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n1) H8) in (False_ind ((eq C e (CHead d1 (Bind Abst) t)) \to ((drop (r k n0) O c e) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))))) H9)))) h (sym_eq nat h n0 H6))) H3 H4 H5 H1))))) | (drop_skip k h d c e H1 u) \Rightarrow (\lambda (H2: (eq nat h (S n0))).(\lambda (H3: (eq nat (S d) O)).(\lambda (H4: (eq C (CHead c k (lift h (r k d) u)) (CSort n1))).(\lambda (H5: (eq C (CHead e k u) (CHead d1 (Bind Abst) t))).(eq_ind nat (S n0) (\lambda (n: nat).((eq nat (S d) O) \to ((eq C (CHead c k (lift n (r k d) u)) (CSort n1)) \to ((eq C (CHead e k u) (CHead d1 (Bind Abst) t)) \to ((drop n (r k d) c e) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) (\lambda (H6: (eq nat (S d) O)).(let H7 \def (eq_ind nat (S d) (\lambda (e0: nat).(match e0 return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H6) in (False_ind ((eq C (CHead c k (lift (S n0) (r k d) u)) (CSort n1)) \to ((eq C (CHead e k u) (CHead d1 (Bind Abst) t)) \to ((drop (S n0) (r k d) c e) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))) H7))) h (sym_eq nat h (S n0) H2) H3 H4 H5 H1)))))]) in (H2 (refl_equal nat (S n0)) (refl_equal nat O) (refl_equal C (CSort n1)) (refl_equal C (CHead d1 (Bind Abst) t)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (t: T).((drop (S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (d1: C).(\forall (t: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) (\lambda (b: B).(\lambda (u: T).(\lambda (d1: C).(\lambda (t: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Bind b) u) (CHead d1 (Bind Abst) t))).(or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda (H4: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda (x: C).(\lambda (H5: (csub3 g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t))) x H5 (drop_drop (Bind b) n0 c3 (CHead x (Bind Abst) t) H6 u)))))) H4)) (\lambda (H4: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (csub3 g d1 x0)).(\lambda (H6: (drop n0 O c3 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) x0 x1 H5 (drop_drop (Bind b) n0 c3 (CHead x0 (Bind Abbr) x1) H6 u) H7))))))) H4)) (H c0 c3 H1 d1 t (drop_gen_drop (Bind b) c0 (CHead d1 (Bind Abst) t) u n0 H3)))))))) (\lambda (f: F).(\lambda (u: T).(\lambda (d1: C).(\lambda (t: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Flat f) u) (CHead d1 (Bind Abst) t))).(or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda (H4: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda (x: C).(\lambda (H5: (csub3 g d1 x)).(\lambda (H6: (drop (S n0) O c3 (CHead x (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t))) x H5 (drop_drop (Flat f) n0 c3 (CHead x (Bind Abst) t) H6 u)))))) H4)) (\lambda (H4: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (csub3 g d1 x0)).(\lambda (H6: (drop (S n0) O c3 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) x0 x1 H5 (drop_drop (Flat f) n0 c3 (CHead x0 (Bind Abbr) x1) H6 u) H7))))))) H4)) (H2 d1 t (drop_gen_drop (Flat f) c0 (CHead d1 (Bind Abst) t) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (t: T).((drop (S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (t: T).(\lambda (H4: (drop (S n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Bind Abst) t))).(or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H5: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x: C).(\lambda (H6: (csub3 g d1 x)).(\lambda (H7: (drop n0 O c3 (CHead x (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) t))) x H6 (drop_drop (Bind b) n0 c3 (CHead x (Bind Abst) t) H7 u2)))))) H5)) (\lambda (H5: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (csub3 g d1 x0)).(\lambda (H7: (drop n0 O c3 (CHead x0 (Bind Abbr) x1))).(\lambda (H8: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x0 x1 H6 (drop_drop (Bind b) n0 c3 (CHead x0 (Bind Abbr) x1) H7 u2) H8))))))) H5)) (H c0 c3 H1 d1 t (drop_gen_drop (Bind Void) c0 (CHead d1 (Bind Abst) t) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (t: T).((drop (S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (t0: T).(\lambda (H4: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind Abst) t0))).(or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))) (\lambda (H5: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t0))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t0))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))) (\lambda (x: C).(\lambda (H6: (csub3 g d1 x)).(\lambda (H7: (drop n0 O c3 (CHead x (Bind Abst) t0))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind Abst) t0) H7 u)))))) H5)) (\lambda (H5: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t0))))).(ex3_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (csub3 g d1 x0)).(\lambda (H7: (drop n0 O c3 (CHead x0 (Bind Abbr) x1))).(\lambda (H8: (ty3 g x0 x1 t0)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))) x0 x1 H6 (drop_drop (Bind Abbr) n0 c3 (CHead x0 (Bind Abbr) x1) H7 u) H8))))))) H5)) (H c0 c3 H1 d1 t0 (drop_gen_drop (Bind Abst) c0 (CHead d1 (Bind Abst) t0) t n0 H4))))))))))))) c1 c2 H0)))))) n)).
+ \lambda (g: G).(\lambda (n: nat).(nat_ind (\lambda (n0: nat).(\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))) (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (csub3 g c1 c2)).(\lambda (d1: C).(\lambda (t: T).(\lambda (H0: (drop O O c1 (CHead d1 (Bind Abst) t))).(let H1 \def (eq_ind C c1 (\lambda (c: C).(csub3 g c c2)) H (CHead d1 (Bind Abst) t) (drop_gen_refl c1 (CHead d1 (Bind Abst) t) H0)) in (let H2 \def (match H1 return (\lambda (_: ?).(\lambda (c: C).(\lambda (c0: C).((eq C c (CHead d1 (Bind Abst) t)) \to ((eq C c0 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) with [(csub3_sort n) \Rightarrow (\lambda (H1: (eq C (CSort n) (CHead d1 (Bind Abst) t))).(\lambda (H2: (eq C (CSort n) c2)).((let H3 \def (eq_ind C (CSort n) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow True | (CHead _ _ _) \Rightarrow False])) I (CHead d1 (Bind Abst) t) H1) in (False_ind ((eq C (CSort n) c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))) H3)) H2))) | (csub3_head c0 c3 H1 k u) \Rightarrow (\lambda (H2: (eq C (CHead c0 k u) (CHead d1 (Bind Abst) t))).(\lambda (H3: (eq C (CHead c3 k u) c2)).((let H4 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead c0 k u) (CHead d1 (Bind Abst) t) H2) in ((let H5 \def (f_equal C K (\lambda (e: C).(match e return (\lambda (_: ?).K) with [(CSort _) \Rightarrow k | (CHead _ k _) \Rightarrow k])) (CHead c0 k u) (CHead d1 (Bind Abst) t) H2) in ((let H6 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 k u) (CHead d1 (Bind Abst) t) H2) in (eq_ind C d1 (\lambda (c: C).((eq K k (Bind Abst)) \to ((eq T u t) \to ((eq C (CHead c3 k u) c2) \to ((csub3 g c c3) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) (\lambda (H7: (eq K k (Bind Abst))).(eq_ind K (Bind Abst) (\lambda (k0: K).((eq T u t) \to ((eq C (CHead c3 k0 u) c2) \to ((csub3 g d1 c3) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))))))) (\lambda (H8: (eq T u t)).(eq_ind T t (\lambda (t0: T).((eq C (CHead c3 (Bind Abst) t0) c2) \to ((csub3 g d1 c3) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))) (\lambda (H9: (eq C (CHead c3 (Bind Abst) t) c2)).(eq_ind C (CHead c3 (Bind Abst) t) (\lambda (c: C).((csub3 g d1 c3) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))))) (\lambda (H10: (csub3 g d1 c3)).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O (CHead c3 (Bind Abst) t) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abst) t) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O (CHead c3 (Bind Abst) t) (CHead d2 (Bind Abst) t))) c3 H10 (drop_refl (CHead c3 (Bind Abst) t))))) c2 H9)) u (sym_eq T u t H8))) k (sym_eq K k (Bind Abst) H7))) c0 (sym_eq C c0 d1 H6))) H5)) H4)) H3 H1))) | (csub3_void c0 c3 H1 b H2 u1 u2) \Rightarrow (\lambda (H3: (eq C (CHead c0 (Bind Void) u1) (CHead d1 (Bind Abst) t))).(\lambda (H4: (eq C (CHead c3 (Bind b) u2) c2)).((let H5 \def (eq_ind C (CHead c0 (Bind Void) u1) (\lambda (e: C).(match e return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead d1 (Bind Abst) t) H3) in (False_ind ((eq C (CHead c3 (Bind b) u2) c2) \to ((csub3 g c0 c3) \to ((not (eq B b Void)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop O O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))) H5)) H4 H1 H2))) | (csub3_abst c0 c3 H1 u t0 H2) \Rightarrow (\lambda (H3: (eq C (CHead c0 (Bind Abst) t0) (CHead d1 (Bind Abst) t))).(\lambda (H4: (eq C (CHead c3 (Bind Abbr) u) c2)).((let H5 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow t0 | (CHead _ _ t) \Rightarrow t])) (CHead c0 (Bind Abst) t0) (CHead d1 (Bind Abst) t) H3) in ((let H6 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c])) (CHead c0 (Bind Abst) t0) (CHead d1 (Bind Abst) t) H3) in (eq_ind C d1 (\lambda (c: C).((eq T t0 t) \to ((eq C (CHead c3 (Bind Abbr) u) c2) \to ((csub3 g c c3) \to ((ty3 g c3 u t0) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) (\lambda (H7: (eq T t0 t)).(eq_ind T t (\lambda (t1: T).((eq C (CHead c3 (Bind Abbr) u) c2) \to ((csub3 g d1 c3) \to ((ty3 g c3 u t1) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c2 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))))))) (\lambda (H8: (eq C (CHead c3 (Bind Abbr) u) c2)).(eq_ind C (CHead c3 (Bind Abbr) u) (\lambda (c: C).((csub3 g d1 c3) \to ((ty3 g c3 u t) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O c (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O c (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))) (\lambda (H9: (csub3 g d1 c3)).(\lambda (H10: (ty3 g c3 u t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop O O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop O O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) c3 u H9 (drop_refl (CHead c3 (Bind Abbr) u)) H10)))) c2 H8)) t0 (sym_eq T t0 t H7))) c0 (sym_eq C c0 d1 H6))) H5)) H4 H1 H2)))]) in (H2 (refl_equal C (CHead d1 (Bind Abst) t)) (refl_equal C c2)))))))))) (\lambda (n0: nat).(\lambda (H: ((\forall (c1: C).(\forall (c2: C).((csub3 g c1 c2) \to (\forall (d1: C).(\forall (t: T).((drop n0 O c1 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H0: (csub3 g c1 c2)).(csub3_ind g (\lambda (c: C).(\lambda (c0: C).(\forall (d1: C).(\forall (t: T).((drop (S n0) O c (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c0 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O c0 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (n1: nat).(\lambda (d1: C).(\lambda (t: T).(\lambda (H1: (drop (S n0) O (CSort n1) (CHead d1 (Bind Abst) t))).(let H2 \def (match H1 return (\lambda (_: ?).(\lambda (n: nat).(\lambda (n2: nat).(\lambda (c: C).(\lambda (c0: C).((eq nat n (S n0)) \to ((eq nat n2 O) \to ((eq C c (CSort n1)) \to ((eq C c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))))))) with [(drop_refl c) \Rightarrow (\lambda (H1: (eq nat O (S n0))).(\lambda (H2: (eq nat O O)).(\lambda (H3: (eq C c (CSort n1))).(\lambda (H4: (eq C c (CHead d1 (Bind Abst) t))).((let H5 \def (eq_ind nat O (\lambda (e: nat).(match e return (\lambda (_: ?).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S n0) H1) in (False_ind ((eq nat O O) \to ((eq C c (CSort n1)) \to ((eq C c (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))) H5)) H2 H3 H4))))) | (drop_drop k h c e H1 u) \Rightarrow (\lambda (H2: (eq nat (S h) (S n0))).(\lambda (H3: (eq nat O O)).(\lambda (H4: (eq C (CHead c k u) (CSort n1))).(\lambda (H5: (eq C e (CHead d1 (Bind Abst) t))).((let H6 \def (f_equal nat nat (\lambda (e0: nat).(match e0 return (\lambda (_: ?).nat) with [O \Rightarrow h | (S n) \Rightarrow n])) (S h) (S n0) H2) in (eq_ind nat n0 (\lambda (n: nat).((eq nat O O) \to ((eq C (CHead c k u) (CSort n1)) \to ((eq C e (CHead d1 (Bind Abst) t)) \to ((drop (r k n) O c e) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) (\lambda (_: (eq nat O O)).(\lambda (H8: (eq C (CHead c k u) (CSort n1))).(let H9 \def (eq_ind C (CHead c k u) (\lambda (e0: C).(match e0 return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I (CSort n1) H8) in (False_ind ((eq C e (CHead d1 (Bind Abst) t)) \to ((drop (r k n0) O c e) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))))) H9)))) h (sym_eq nat h n0 H6))) H3 H4 H5 H1))))) | (drop_skip k h d c e H1 u) \Rightarrow (\lambda (H2: (eq nat h (S n0))).(\lambda (H3: (eq nat (S d) O)).(\lambda (H4: (eq C (CHead c k (lift h (r k d) u)) (CSort n1))).(\lambda (H5: (eq C (CHead e k u) (CHead d1 (Bind Abst) t))).(eq_ind nat (S n0) (\lambda (n: nat).((eq nat (S d) O) \to ((eq C (CHead c k (lift n (r k d) u)) (CSort n1)) \to ((eq C (CHead e k u) (CHead d1 (Bind Abst) t)) \to ((drop n (r k d) c e) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) (\lambda (H6: (eq nat (S d) O)).(let H7 \def (eq_ind nat (S d) (\lambda (e0: nat).(match e0 return (\lambda (_: ?).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H6) in (False_ind ((eq C (CHead c k (lift (S n0) (r k d) u)) (CSort n1)) \to ((eq C (CHead e k u) (CHead d1 (Bind Abst) t)) \to ((drop (S n0) (r k d) c e) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CSort n1) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))) H7))) h (sym_eq nat h (S n0) H2) H3 H4 H5 H1)))))]) in (H2 (refl_equal nat (S n0)) (refl_equal nat O) (refl_equal C (CSort n1)) (refl_equal C (CHead d1 (Bind Abst) t)))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (H2: ((\forall (d1: C).(\forall (t: T).((drop (S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (u: T).(\forall (d1: C).(\forall (t: T).((drop (S n0) O (CHead c0 k0 u) (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 k0 u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))))))))) (\lambda (b: B).(\lambda (u: T).(\lambda (d1: C).(\lambda (t: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Bind b) u) (CHead d1 (Bind Abst) t))).(or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda (H4: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda (x: C).(\lambda (H5: (csub3 g d1 x)).(\lambda (H6: (drop n0 O c3 (CHead x (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t))) x H5 (drop_drop (Bind b) n0 c3 (CHead x (Bind Abst) t) H6 u)))))) H4)) (\lambda (H4: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (csub3 g d1 x0)).(\lambda (H6: (drop n0 O c3 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind b) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) x0 x1 H5 (drop_drop (Bind b) n0 c3 (CHead x0 (Bind Abbr) x1) H6 u) H7))))))) H4)) (H c0 c3 H1 d1 t (drop_gen_drop (Bind b) c0 (CHead d1 (Bind Abst) t) u n0 H3)))))))) (\lambda (f: F).(\lambda (u: T).(\lambda (d1: C).(\lambda (t: T).(\lambda (H3: (drop (S n0) O (CHead c0 (Flat f) u) (CHead d1 (Bind Abst) t))).(or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda (H4: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda (x: C).(\lambda (H5: (csub3 g d1 x)).(\lambda (H6: (drop (S n0) O c3 (CHead x (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t))) x H5 (drop_drop (Flat f) n0 c3 (CHead x (Bind Abst) t) H6 u)))))) H4)) (\lambda (H4: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (csub3 g d1 x0)).(\lambda (H6: (drop (S n0) O c3 (CHead x0 (Bind Abbr) x1))).(\lambda (H7: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Flat f) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t))) x0 x1 H5 (drop_drop (Flat f) n0 c3 (CHead x0 (Bind Abbr) x1) H6 u) H7))))))) H4)) (H2 d1 t (drop_gen_drop (Flat f) c0 (CHead d1 (Bind Abst) t) u n0 H3)))))))) k)))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (t: T).((drop (S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))).(\lambda (b: B).(\lambda (_: (not (eq B b Void))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (d1: C).(\lambda (t: T).(\lambda (H4: (drop (S n0) O (CHead c0 (Bind Void) u1) (CHead d1 (Bind Abst) t))).(or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H5: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x: C).(\lambda (H6: (csub3 g d1 x)).(\lambda (H7: (drop n0 O c3 (CHead x (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) t))) x H6 (drop_drop (Bind b) n0 c3 (CHead x (Bind Abst) t) H7 u2)))))) H5)) (\lambda (H5: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (csub3 g d1 x0)).(\lambda (H7: (drop n0 O c3 (CHead x0 (Bind Abbr) x1))).(\lambda (H8: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O (CHead c3 (Bind b) u2) (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x0 x1 H6 (drop_drop (Bind b) n0 c3 (CHead x0 (Bind Abbr) x1) H7 u2) H8))))))) H5)) (H c0 c3 H1 d1 t (drop_gen_drop (Bind Void) c0 (CHead d1 (Bind Abst) t) u1 n0 H4)))))))))))))) (\lambda (c0: C).(\lambda (c3: C).(\lambda (H1: (csub3 g c0 c3)).(\lambda (_: ((\forall (d1: C).(\forall (t: T).((drop (S n0) O c0 (CHead d1 (Bind Abst) t)) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O c3 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop (S n0) O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c3 u t)).(\lambda (d1: C).(\lambda (t0: T).(\lambda (H4: (drop (S n0) O (CHead c0 (Bind Abst) t) (CHead d1 (Bind Abst) t0))).(or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))) (\lambda (H5: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t0))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n0 O c3 (CHead d2 (Bind Abst) t0))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))) (\lambda (x: C).(\lambda (H6: (csub3 g d1 x)).(\lambda (H7: (drop n0 O c3 (CHead x (Bind Abst) t0))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0))) x H6 (drop_drop (Bind Abbr) n0 c3 (CHead x (Bind Abst) t0) H7 u)))))) H5)) (\lambda (H5: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t0))))).(ex3_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop n0 O c3 (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (csub3 g d1 x0)).(\lambda (H7: (drop n0 O c3 (CHead x0 (Bind Abbr) x1))).(\lambda (H8: (ty3 g x0 x1 t0)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abst) t0)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u0: T).(drop (S n0) O (CHead c3 (Bind Abbr) u) (CHead d2 (Bind Abbr) u0)))) (\lambda (d2: C).(\lambda (u0: T).(ty3 g d2 u0 t0))) x0 x1 H6 (drop_drop (Bind Abbr) n0 c3 (CHead x0 (Bind Abbr) x1) H7 u) H8))))))) H5)) (H c0 c3 H1 d1 t0 (drop_gen_drop (Bind Abst) c0 (CHead d1 (Bind Abst) t0) t n0 H4))))))))))))) c1 c2 H0)))))) n)).
 
 theorem csub3_getl_abbr:
  \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (u: T).(\forall (n: nat).((getl n c1 (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csub3 g c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))))))))
 \def
- \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda (n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abbr) u))).(let H0 \def (getl_gen_all c1 (CHead d1 (Bind Abbr) u) n H) in (ex2_ind C (\lambda (e: C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u))) (\forall (c2: C).((csub3 g c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))) (\lambda (x: C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind Abbr) u))).((match x return (\lambda (c: C).((drop n O c1 c) \to ((clear c (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csub3 g c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))))))))) with [(CSort n0) \Rightarrow (\lambda (_: (drop n O c1 (CSort n0))).(\lambda (H4: (clear (CSort n0) (CHead d1 (Bind Abbr) u))).(clear_gen_sort (CHead d1 (Bind Abbr) u) n0 H4 (\forall (c2: C).((csub3 g c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))))))))) | (CHead c k t) \Rightarrow (\lambda (H3: (drop n O c1 (CHead c k t))).(\lambda (H4: (clear (CHead c k t) (CHead d1 (Bind Abbr) u))).((match k return (\lambda (k0: K).((drop n O c1 (CHead c k0 t)) \to ((clear (CHead c k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csub3 g c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))))))))) with [(Bind b) \Rightarrow (\lambda (H5: (drop n O c1 (CHead c (Bind b) t))).(\lambda (H6: (clear (CHead c (Bind b) t) (CHead d1 (Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abbr) u) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abbr) u) t H6)) in ((let H8 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abbr) u) t H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind Abbr) u) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abbr) u) t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 c)).(\lambda (c2: C).(\lambda (H12: (csub3 g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda (t: T).(drop n O c1 (CHead c (Bind b) t))) H5 u H9) in (let H14 \def (eq_ind_r B b (\lambda (b: B).(drop n O c1 (CHead c (Bind b) u))) H13 Abbr H10) in (let H15 \def (eq_ind_r C c (\lambda (c: C).(drop n O c1 (CHead c (Bind Abbr) u))) H14 d1 H11) in (ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x0: C).(\lambda (H16: (csub3 g d1 x0)).(\lambda (H17: (drop n O c2 (CHead x0 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) x0 H16 (getl_intro n c2 (CHead x0 (Bind Abbr) u) (CHead x0 (Bind Abbr) u) H17 (clear_bind Abbr x0 u)))))) (csub3_drop_abbr g n c1 c2 H12 d1 u H15)))))))))) H8)) H7)))) | (Flat f) \Rightarrow (\lambda (H5: (drop n O c1 (CHead c (Flat f) t))).(\lambda (H6: (clear (CHead c (Flat f) t) (CHead d1 (Bind Abbr) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c0: C).((drop n O c0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csub3 g c0 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))))) (nat_ind (\lambda (n0: nat).(\forall (x0: C).((drop n0 O x0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csub3 g x0 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u))))))))) (\lambda (x0: C).(\lambda (H8: (drop O O x0 (CHead c (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csub3 g x0 c2)).(let H10 \def (eq_ind C x0 (\lambda (c: C).(csub3 g c c2)) H9 (CHead c (Flat f) t) (drop_gen_refl x0 (CHead c (Flat f) t) H8)) in (let H_y \def (clear_flat c (CHead d1 (Bind Abbr) u) (clear_gen_flat f c (CHead d1 (Bind Abbr) u) t H6) f t) in (let H11 \def (csub3_clear_conf g (CHead c (Flat f) t) c2 H10 (CHead d1 (Bind Abbr) u) H_y) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead d1 (Bind Abbr) u) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x1: C).(\lambda (H12: (csub3 g (CHead d1 (Bind Abbr) u) x1)).(\lambda (H13: (clear c2 x1)).(let H14 \def (csub3_gen_abbr g d1 x1 u H12) in (ex2_ind C (\lambda (e2: C).(eq C x1 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csub3 g d1 e2)) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x2: C).(\lambda (H15: (eq C x1 (CHead x2 (Bind Abbr) u))).(\lambda (H16: (csub3 g d1 x2)).(let H17 \def (eq_ind C x1 (\lambda (c: C).(clear c2 c)) H13 (CHead x2 (Bind Abbr) u) H15) in (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u))) x2 H16 (getl_intro O c2 (CHead x2 (Bind Abbr) u) c2 (drop_refl c2) H17)))))) H14))))) H11)))))))) (\lambda (n0: nat).(\lambda (H8: ((\forall (x: C).((drop n0 O x (CHead c (Flat f) t)) \to (\forall (c2: C).((csub3 g x c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u)))))))))).(\lambda (x0: C).(\lambda (H9: (drop (S n0) O x0 (CHead c (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csub3 g x0 c2)).(let H11 \def (drop_clear x0 (CHead c (Flat f) t) n0 H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x0 (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead c (Flat f) t))))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (clear x0 (CHead x2 (Bind x1) x3))).(\lambda (H13: (drop n0 O x2 (CHead c (Flat f) t))).(let H14 \def (csub3_clear_conf g x0 c2 H10 (CHead x2 (Bind x1) x3) H12) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead x2 (Bind x1) x3) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x4: C).(\lambda (H15: (csub3 g (CHead x2 (Bind x1) x3) x4)).(\lambda (H16: (clear c2 x4)).(let H17 \def (csub3_gen_bind g x1 x2 x4 x3 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x4 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g x2 e2)))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: T).(\lambda (H18: (eq C x4 (CHead x6 (Bind x5) x7))).(\lambda (H19: (csub3 g x2 x6)).(let H20 \def (eq_ind C x4 (\lambda (c: C).(clear c2 c)) H16 (CHead x6 (Bind x5) x7) H18) in (let H21 \def (H8 x2 H13 x6 H19) in (ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 x6 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x8: C).(\lambda (H22: (csub3 g d1 x8)).(\lambda (H23: (getl n0 x6 (CHead x8 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u))) x8 H22 (getl_clear_bind x5 c2 x6 x7 H20 (CHead x8 (Bind Abbr) u) n0 H23))))) H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))]) H3 H4)))]) H1 H2)))) H0))))))).
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (u: T).(\lambda (n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abbr) u))).(let H0 \def (getl_gen_all c1 (CHead d1 (Bind Abbr) u) n H) in (ex2_ind C (\lambda (e: C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abbr) u))) (\forall (c2: C).((csub3 g c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))) (\lambda (x: C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind Abbr) u))).((match x return (\lambda (_: ?).(\lambda (c: C).((drop n O c1 c) \to ((clear c (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csub3 g c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))))))) with [(CSort n0) \Rightarrow (\lambda (_: (drop n O c1 (CSort n0))).(\lambda (H4: (clear (CSort n0) (CHead d1 (Bind Abbr) u))).(clear_gen_sort (CHead d1 (Bind Abbr) u) n0 H4 (\forall (c2: C).((csub3 g c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))))))))) | (CHead c k t) \Rightarrow (\lambda (H3: (drop n O c1 (CHead c k t))).(\lambda (H4: (clear (CHead c k t) (CHead d1 (Bind Abbr) u))).((match k return (\lambda (_: ?).(\lambda (k0: K).((drop n O c1 (CHead c k0 t)) \to ((clear (CHead c k0 t) (CHead d1 (Bind Abbr) u)) \to (\forall (c2: C).((csub3 g c1 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))))))) with [(Bind b) \Rightarrow (\lambda (H5: (drop n O c1 (CHead c (Bind b) t))).(\lambda (H6: (clear (CHead c (Bind b) t) (CHead d1 (Bind Abbr) u))).(let H7 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abbr) u) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abbr) u) t H6)) in ((let H8 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abbr | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (CHead d1 (Bind Abbr) u) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abbr) u) t H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind Abbr) u) (CHead c (Bind b) t) (clear_gen_bind b c (CHead d1 (Bind Abbr) u) t H6)) in (\lambda (H10: (eq B Abbr b)).(\lambda (H11: (eq C d1 c)).(\lambda (c2: C).(\lambda (H12: (csub3 g c1 c2)).(let H13 \def (eq_ind_r T t (\lambda (t: T).(drop n O c1 (CHead c (Bind b) t))) H5 u H9) in (let H14 \def (eq_ind_r B b (\lambda (b: B).(drop n O c1 (CHead c (Bind b) u))) H13 Abbr H10) in (let H15 \def (eq_ind_r C c (\lambda (c: C).(drop n O c1 (CHead c (Bind Abbr) u))) H14 d1 H11) in (ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x0: C).(\lambda (H16: (csub3 g d1 x0)).(\lambda (H17: (drop n O c2 (CHead x0 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) x0 H16 (getl_intro n c2 (CHead x0 (Bind Abbr) u) (CHead x0 (Bind Abbr) u) H17 (clear_bind Abbr x0 u)))))) (csub3_drop_abbr g n c1 c2 H12 d1 u H15)))))))))) H8)) H7)))) | (Flat f) \Rightarrow (\lambda (H5: (drop n O c1 (CHead c (Flat f) t))).(\lambda (H6: (clear (CHead c (Flat f) t) (CHead d1 (Bind Abbr) u))).(let H7 \def H5 in (unintro C c1 (\lambda (c0: C).((drop n O c0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csub3 g c0 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u)))))))) (nat_ind (\lambda (n0: nat).(\forall (x0: C).((drop n0 O x0 (CHead c (Flat f) t)) \to (\forall (c2: C).((csub3 g x0 c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u))))))))) (\lambda (x0: C).(\lambda (H8: (drop O O x0 (CHead c (Flat f) t))).(\lambda (c2: C).(\lambda (H9: (csub3 g x0 c2)).(let H10 \def (eq_ind C x0 (\lambda (c: C).(csub3 g c c2)) H9 (CHead c (Flat f) t) (drop_gen_refl x0 (CHead c (Flat f) t) H8)) in (let H_y \def (clear_flat c (CHead d1 (Bind Abbr) u) (clear_gen_flat f c (CHead d1 (Bind Abbr) u) t H6) f t) in (let H11 \def (csub3_clear_conf g (CHead c (Flat f) t) c2 H10 (CHead d1 (Bind Abbr) u) H_y) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead d1 (Bind Abbr) u) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x1: C).(\lambda (H12: (csub3 g (CHead d1 (Bind Abbr) u) x1)).(\lambda (H13: (clear c2 x1)).(let H14 \def (csub3_gen_abbr g d1 x1 u H12) in (ex2_ind C (\lambda (e2: C).(eq C x1 (CHead e2 (Bind Abbr) u))) (\lambda (e2: C).(csub3 g d1 e2)) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x2: C).(\lambda (H15: (eq C x1 (CHead x2 (Bind Abbr) u))).(\lambda (H16: (csub3 g d1 x2)).(let H17 \def (eq_ind C x1 (\lambda (c: C).(clear c2 c)) H13 (CHead x2 (Bind Abbr) u) H15) in (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abbr) u))) x2 H16 (getl_intro O c2 (CHead x2 (Bind Abbr) u) c2 (drop_refl c2) H17)))))) H14))))) H11)))))))) (\lambda (n0: nat).(\lambda (H8: ((\forall (x: C).((drop n0 O x (CHead c (Flat f) t)) \to (\forall (c2: C).((csub3 g x c2) \to (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abbr) u)))))))))).(\lambda (x0: C).(\lambda (H9: (drop (S n0) O x0 (CHead c (Flat f) t))).(\lambda (c2: C).(\lambda (H10: (csub3 g x0 c2)).(let H11 \def (drop_clear x0 (CHead c (Flat f) t) n0 H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x0 (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead c (Flat f) t))))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (clear x0 (CHead x2 (Bind x1) x3))).(\lambda (H13: (drop n0 O x2 (CHead c (Flat f) t))).(let H14 \def (csub3_clear_conf g x0 c2 H10 (CHead x2 (Bind x1) x3) H12) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead x2 (Bind x1) x3) e2)) (\lambda (e2: C).(clear c2 e2)) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x4: C).(\lambda (H15: (csub3 g (CHead x2 (Bind x1) x3) x4)).(\lambda (H16: (clear c2 x4)).(let H17 \def (csub3_gen_bind g x1 x2 x4 x3 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x4 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g x2 e2)))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: T).(\lambda (H18: (eq C x4 (CHead x6 (Bind x5) x7))).(\lambda (H19: (csub3 g x2 x6)).(let H20 \def (eq_ind C x4 (\lambda (c: C).(clear c2 c)) H16 (CHead x6 (Bind x5) x7) H18) in (let H21 \def (H8 x2 H13 x6 H19) in (ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 x6 (CHead d2 (Bind Abbr) u))) (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (x8: C).(\lambda (H22: (csub3 g d1 x8)).(\lambda (H23: (getl n0 x6 (CHead x8 (Bind Abbr) u))).(ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abbr) u))) x8 H22 (getl_clear_bind x5 c2 x6 x7 H20 (CHead x8 (Bind Abbr) u) n0 H23))))) H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))]) H3 H4)))]) H1 H2)))) H0))))))).
 
 theorem csub3_getl_abst:
  \forall (g: G).(\forall (c1: C).(\forall (d1: C).(\forall (t: T).(\forall (n: nat).((getl n c1 (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))))
 \def
- \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (t: T).(\lambda (n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abst) t))).(let H0 \def (getl_gen_all c1 (CHead d1 (Bind Abst) t) n H) in (ex2_ind C (\lambda (e: C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) t))) (\forall (c2: C).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))) (\lambda (x: C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind Abst) t))).((match x return (\lambda (c: C).((drop n O c1 c) \to ((clear c (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) with [(CSort n0) \Rightarrow (\lambda (_: (drop n O c1 (CSort n0))).(\lambda (H4: (clear (CSort n0) (CHead d1 (Bind Abst) t))).(clear_gen_sort (CHead d1 (Bind Abst) t) n0 H4 (\forall (c2: C).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) | (CHead c k t0) \Rightarrow (\lambda (H3: (drop n O c1 (CHead c k t0))).(\lambda (H4: (clear (CHead c k t0) (CHead d1 (Bind Abst) t))).((match k return (\lambda (k0: K).((drop n O c1 (CHead c k0 t0)) \to ((clear (CHead c k0 t0) (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) with [(Bind b) \Rightarrow (\lambda (H5: (drop n O c1 (CHead c (Bind b) t0))).(\lambda (H6: (clear (CHead c (Bind b) t0) (CHead d1 (Bind Abst) t))).(let H7 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) t) (CHead c (Bind b) t0) (clear_gen_bind b c (CHead d1 (Bind Abst) t) t0 H6)) in ((let H8 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d1 (Bind Abst) t) (CHead c (Bind b) t0) (clear_gen_bind b c (CHead d1 (Bind Abst) t) t0 H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow t | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind Abst) t) (CHead c (Bind b) t0) (clear_gen_bind b c (CHead d1 (Bind Abst) t) t0 H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 c)).(\lambda (c2: C).(\lambda (H12: (csub3 g c1 c2)).(let H13 \def (eq_ind_r T t0 (\lambda (t: T).(drop n O c1 (CHead c (Bind b) t))) H5 t H9) in (let H14 \def (eq_ind_r B b (\lambda (b: B).(drop n O c1 (CHead c (Bind b) t))) H13 Abst H10) in (let H15 \def (eq_ind_r C c (\lambda (c: C).(drop n O c1 (CHead c (Bind Abst) t))) H14 d1 H11) in (or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H16: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x0: C).(\lambda (H17: (csub3 g d1 x0)).(\lambda (H18: (drop n O c2 (CHead x0 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t))) x0 H17 (getl_intro n c2 (CHead x0 (Bind Abst) t) (CHead x0 (Bind Abst) t) H18 (clear_bind Abst x0 t))))))) H16)) (\lambda (H16: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H17: (csub3 g d1 x0)).(\lambda (H18: (drop n O c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H19: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x0 x1 H17 (getl_intro n c2 (CHead x0 (Bind Abbr) x1) (CHead x0 (Bind Abbr) x1) H18 (clear_bind Abbr x0 x1)) H19))))))) H16)) (csub3_drop_abst g n c1 c2 H12 d1 t H15)))))))))) H8)) H7)))) | (Flat f) \Rightarrow (\lambda (H5: (drop n O c1 (CHead c (Flat f) t0))).(\lambda (H6: (clear (CHead c (Flat f) t0) (CHead d1 (Bind Abst) t))).(let H7 \def H5 in (unintro C c1 (\lambda (c0: C).((drop n O c0 (CHead c (Flat f) t0)) \to (\forall (c2: C).((csub3 g c0 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))) (nat_ind (\lambda (n0: nat).(\forall (x0: C).((drop n0 O x0 (CHead c (Flat f) t0)) \to (\forall (c2: C).((csub3 g x0 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (x0: C).(\lambda (H8: (drop O O x0 (CHead c (Flat f) t0))).(\lambda (c2: C).(\lambda (H9: (csub3 g x0 c2)).(let H10 \def (eq_ind C x0 (\lambda (c: C).(csub3 g c c2)) H9 (CHead c (Flat f) t0) (drop_gen_refl x0 (CHead c (Flat f) t0) H8)) in (let H_y \def (clear_flat c (CHead d1 (Bind Abst) t) (clear_gen_flat f c (CHead d1 (Bind Abst) t) t0 H6) f t0) in (let H11 \def (csub3_clear_conf g (CHead c (Flat f) t0) c2 H10 (CHead d1 (Bind Abst) t) H_y) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead d1 (Bind Abst) t) e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x1: C).(\lambda (H12: (csub3 g (CHead d1 (Bind Abst) t) x1)).(\lambda (H13: (clear c2 x1)).(let H14 \def (csub3_gen_abst g d1 x1 t H12) in (or_ind (ex2 C (\lambda (e2: C).(eq C x1 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csub3 g d1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x1 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g d1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H15: (ex2 C (\lambda (e2: C).(eq C x1 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csub3 g d1 e2)))).(ex2_ind C (\lambda (e2: C).(eq C x1 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csub3 g d1 e2)) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x2: C).(\lambda (H16: (eq C x1 (CHead x2 (Bind Abst) t))).(\lambda (H17: (csub3 g d1 x2)).(let H18 \def (eq_ind C x1 (\lambda (c: C).(clear c2 c)) H13 (CHead x2 (Bind Abst) t) H16) in (or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t))) x2 H17 (getl_intro O c2 (CHead x2 (Bind Abst) t) c2 (drop_refl c2) H18))))))) H15)) (\lambda (H15: (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x1 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g d1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t))))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (v2: T).(eq C x1 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g d1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (H16: (eq C x1 (CHead x2 (Bind Abbr) x3))).(\lambda (H17: (csub3 g d1 x2)).(\lambda (H18: (ty3 g x2 x3 t)).(let H19 \def (eq_ind C x1 (\lambda (c: C).(clear c2 c)) H13 (CHead x2 (Bind Abbr) x3) H16) in (or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x2 x3 H17 (getl_intro O c2 (CHead x2 (Bind Abbr) x3) c2 (drop_refl c2) H19) H18)))))))) H15)) H14))))) H11)))))))) (\lambda (n0: nat).(\lambda (H8: ((\forall (x: C).((drop n0 O x (CHead c (Flat f) t0)) \to (\forall (c2: C).((csub3 g x c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))).(\lambda (x0: C).(\lambda (H9: (drop (S n0) O x0 (CHead c (Flat f) t0))).(\lambda (c2: C).(\lambda (H10: (csub3 g x0 c2)).(let H11 \def (drop_clear x0 (CHead c (Flat f) t0) n0 H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x0 (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead c (Flat f) t0))))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (clear x0 (CHead x2 (Bind x1) x3))).(\lambda (H13: (drop n0 O x2 (CHead c (Flat f) t0))).(let H14 \def (csub3_clear_conf g x0 c2 H10 (CHead x2 (Bind x1) x3) H12) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead x2 (Bind x1) x3) e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x4: C).(\lambda (H15: (csub3 g (CHead x2 (Bind x1) x3) x4)).(\lambda (H16: (clear c2 x4)).(let H17 \def (csub3_gen_bind g x1 x2 x4 x3 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x4 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g x2 e2)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: T).(\lambda (H18: (eq C x4 (CHead x6 (Bind x5) x7))).(\lambda (H19: (csub3 g x2 x6)).(let H20 \def (eq_ind C x4 (\lambda (c: C).(clear c2 c)) H16 (CHead x6 (Bind x5) x7) H18) in (let H21 \def (H8 x2 H13 x6 H19) in (or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 x6 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 x6 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 x6 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 x6 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x8: C).(\lambda (H23: (csub3 g d1 x8)).(\lambda (H24: (getl n0 x6 (CHead x8 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t))) x8 H23 (getl_clear_bind x5 c2 x6 x7 H20 (CHead x8 (Bind Abst) t) n0 H24)))))) H22)) (\lambda (H22: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 x6 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 x6 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x8: C).(\lambda (x9: T).(\lambda (H23: (csub3 g d1 x8)).(\lambda (H24: (getl n0 x6 (CHead x8 (Bind Abbr) x9))).(\lambda (H25: (ty3 g x8 x9 t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x8 x9 H23 (getl_clear_bind x5 c2 x6 x7 H20 (CHead x8 (Bind Abbr) x9) n0 H24) H25))))))) H22)) H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))]) H3 H4)))]) H1 H2)))) H0))))))).
+ \lambda (g: G).(\lambda (c1: C).(\lambda (d1: C).(\lambda (t: T).(\lambda (n: nat).(\lambda (H: (getl n c1 (CHead d1 (Bind Abst) t))).(let H0 \def (getl_gen_all c1 (CHead d1 (Bind Abst) t) n H) in (ex2_ind C (\lambda (e: C).(drop n O c1 e)) (\lambda (e: C).(clear e (CHead d1 (Bind Abst) t))) (\forall (c2: C).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))) (\lambda (x: C).(\lambda (H1: (drop n O c1 x)).(\lambda (H2: (clear x (CHead d1 (Bind Abst) t))).((match x return (\lambda (_: ?).(\lambda (c: C).((drop n O c1 c) \to ((clear c (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))) with [(CSort n0) \Rightarrow (\lambda (_: (drop n O c1 (CSort n0))).(\lambda (H4: (clear (CSort n0) (CHead d1 (Bind Abst) t))).(clear_gen_sort (CHead d1 (Bind Abst) t) n0 H4 (\forall (c2: C).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) | (CHead c k t0) \Rightarrow (\lambda (H3: (drop n O c1 (CHead c k t0))).(\lambda (H4: (clear (CHead c k t0) (CHead d1 (Bind Abst) t))).((match k return (\lambda (_: ?).(\lambda (k0: K).((drop n O c1 (CHead c k0 t0)) \to ((clear (CHead c k0 t0) (CHead d1 (Bind Abst) t)) \to (\forall (c2: C).((csub3 g c1 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))) with [(Bind b) \Rightarrow (\lambda (H5: (drop n O c1 (CHead c (Bind b) t0))).(\lambda (H6: (clear (CHead c (Bind b) t0) (CHead d1 (Bind Abst) t))).(let H7 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d1 | (CHead c _ _) \Rightarrow c])) (CHead d1 (Bind Abst) t) (CHead c (Bind b) t0) (clear_gen_bind b c (CHead d1 (Bind Abst) t) t0 H6)) in ((let H8 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow Abst | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abst])])) (CHead d1 (Bind Abst) t) (CHead c (Bind b) t0) (clear_gen_bind b c (CHead d1 (Bind Abst) t) t0 H6)) in ((let H9 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow t | (CHead _ _ t) \Rightarrow t])) (CHead d1 (Bind Abst) t) (CHead c (Bind b) t0) (clear_gen_bind b c (CHead d1 (Bind Abst) t) t0 H6)) in (\lambda (H10: (eq B Abst b)).(\lambda (H11: (eq C d1 c)).(\lambda (c2: C).(\lambda (H12: (csub3 g c1 c2)).(let H13 \def (eq_ind_r T t0 (\lambda (t: T).(drop n O c1 (CHead c (Bind b) t))) H5 t H9) in (let H14 \def (eq_ind_r B b (\lambda (b: B).(drop n O c1 (CHead c (Bind b) t))) H13 Abst H10) in (let H15 \def (eq_ind_r C c (\lambda (c: C).(drop n O c1 (CHead c (Bind Abst) t))) H14 d1 H11) in (or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H16: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(drop n O c2 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x0: C).(\lambda (H17: (csub3 g d1 x0)).(\lambda (H18: (drop n O c2 (CHead x0 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t))) x0 H17 (getl_intro n c2 (CHead x0 (Bind Abst) t) (CHead x0 (Bind Abst) t) H18 (clear_bind Abst x0 t))))))) H16)) (\lambda (H16: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(drop n O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H17: (csub3 g d1 x0)).(\lambda (H18: (drop n O c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H19: (ty3 g x0 x1 t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x0 x1 H17 (getl_intro n c2 (CHead x0 (Bind Abbr) x1) (CHead x0 (Bind Abbr) x1) H18 (clear_bind Abbr x0 x1)) H19))))))) H16)) (csub3_drop_abst g n c1 c2 H12 d1 t H15)))))))))) H8)) H7)))) | (Flat f) \Rightarrow (\lambda (H5: (drop n O c1 (CHead c (Flat f) t0))).(\lambda (H6: (clear (CHead c (Flat f) t0) (CHead d1 (Bind Abst) t))).(let H7 \def H5 in (unintro C c1 (\lambda (c0: C).((drop n O c0 (CHead c (Flat f) t0)) \to (\forall (c2: C).((csub3 g c0 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))) (nat_ind (\lambda (n0: nat).(\forall (x0: C).((drop n0 O x0 (CHead c (Flat f) t0)) \to (\forall (c2: C).((csub3 g x0 c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))))))))) (\lambda (x0: C).(\lambda (H8: (drop O O x0 (CHead c (Flat f) t0))).(\lambda (c2: C).(\lambda (H9: (csub3 g x0 c2)).(let H10 \def (eq_ind C x0 (\lambda (c: C).(csub3 g c c2)) H9 (CHead c (Flat f) t0) (drop_gen_refl x0 (CHead c (Flat f) t0) H8)) in (let H_y \def (clear_flat c (CHead d1 (Bind Abst) t) (clear_gen_flat f c (CHead d1 (Bind Abst) t) t0 H6) f t0) in (let H11 \def (csub3_clear_conf g (CHead c (Flat f) t0) c2 H10 (CHead d1 (Bind Abst) t) H_y) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead d1 (Bind Abst) t) e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x1: C).(\lambda (H12: (csub3 g (CHead d1 (Bind Abst) t) x1)).(\lambda (H13: (clear c2 x1)).(let H14 \def (csub3_gen_abst g d1 x1 t H12) in (or_ind (ex2 C (\lambda (e2: C).(eq C x1 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csub3 g d1 e2))) (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x1 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g d1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H15: (ex2 C (\lambda (e2: C).(eq C x1 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csub3 g d1 e2)))).(ex2_ind C (\lambda (e2: C).(eq C x1 (CHead e2 (Bind Abst) t))) (\lambda (e2: C).(csub3 g d1 e2)) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x2: C).(\lambda (H16: (eq C x1 (CHead x2 (Bind Abst) t))).(\lambda (H17: (csub3 g d1 x2)).(let H18 \def (eq_ind C x1 (\lambda (c: C).(clear c2 c)) H13 (CHead x2 (Bind Abst) t) H16) in (or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t))) x2 H17 (getl_intro O c2 (CHead x2 (Bind Abst) t) c2 (drop_refl c2) H18))))))) H15)) (\lambda (H15: (ex3_2 C T (\lambda (e2: C).(\lambda (v2: T).(eq C x1 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g d1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t))))).(ex3_2_ind C T (\lambda (e2: C).(\lambda (v2: T).(eq C x1 (CHead e2 (Bind Abbr) v2)))) (\lambda (e2: C).(\lambda (_: T).(csub3 g d1 e2))) (\lambda (e2: C).(\lambda (v2: T).(ty3 g e2 v2 t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x2: C).(\lambda (x3: T).(\lambda (H16: (eq C x1 (CHead x2 (Bind Abbr) x3))).(\lambda (H17: (csub3 g d1 x2)).(\lambda (H18: (ty3 g x2 x3 t)).(let H19 \def (eq_ind C x1 (\lambda (c: C).(clear c2 c)) H13 (CHead x2 (Bind Abbr) x3) H16) in (or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl O c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl O c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x2 x3 H17 (getl_intro O c2 (CHead x2 (Bind Abbr) x3) c2 (drop_refl c2) H19) H18)))))))) H15)) H14))))) H11)))))))) (\lambda (n0: nat).(\lambda (H8: ((\forall (x: C).((drop n0 O x (CHead c (Flat f) t0)) \to (\forall (c2: C).((csub3 g x c2) \to (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))))))))).(\lambda (x0: C).(\lambda (H9: (drop (S n0) O x0 (CHead c (Flat f) t0))).(\lambda (c2: C).(\lambda (H10: (csub3 g x0 c2)).(let H11 \def (drop_clear x0 (CHead c (Flat f) t0) n0 H9) in (ex2_3_ind B C T (\lambda (b: B).(\lambda (e: C).(\lambda (v: T).(clear x0 (CHead e (Bind b) v))))) (\lambda (_: B).(\lambda (e: C).(\lambda (_: T).(drop n0 O e (CHead c (Flat f) t0))))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x1: B).(\lambda (x2: C).(\lambda (x3: T).(\lambda (H12: (clear x0 (CHead x2 (Bind x1) x3))).(\lambda (H13: (drop n0 O x2 (CHead c (Flat f) t0))).(let H14 \def (csub3_clear_conf g x0 c2 H10 (CHead x2 (Bind x1) x3) H12) in (ex2_ind C (\lambda (e2: C).(csub3 g (CHead x2 (Bind x1) x3) e2)) (\lambda (e2: C).(clear c2 e2)) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x4: C).(\lambda (H15: (csub3 g (CHead x2 (Bind x1) x3) x4)).(\lambda (H16: (clear c2 x4)).(let H17 \def (csub3_gen_bind g x1 x2 x4 x3 H15) in (ex2_3_ind B C T (\lambda (b2: B).(\lambda (e2: C).(\lambda (v2: T).(eq C x4 (CHead e2 (Bind b2) v2))))) (\lambda (_: B).(\lambda (e2: C).(\lambda (_: T).(csub3 g x2 e2)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x5: B).(\lambda (x6: C).(\lambda (x7: T).(\lambda (H18: (eq C x4 (CHead x6 (Bind x5) x7))).(\lambda (H19: (csub3 g x2 x6)).(let H20 \def (eq_ind C x4 (\lambda (c: C).(clear c2 c)) H16 (CHead x6 (Bind x5) x7) H18) in (let H21 \def (H8 x2 H13 x6 H19) in (or_ind (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 x6 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 x6 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (H22: (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 x6 (CHead d2 (Bind Abst) t))))).(ex2_ind C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl n0 x6 (CHead d2 (Bind Abst) t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x8: C).(\lambda (H23: (csub3 g d1 x8)).(\lambda (H24: (getl n0 x6 (CHead x8 (Bind Abst) t))).(or_introl (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex_intro2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t))) x8 H23 (getl_clear_bind x5 c2 x6 x7 H20 (CHead x8 (Bind Abst) t) n0 H24)))))) H22)) (\lambda (H22: (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 x6 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))).(ex3_2_ind C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl n0 x6 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) (or (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))))) (\lambda (x8: C).(\lambda (x9: T).(\lambda (H23: (csub3 g d1 x8)).(\lambda (H24: (getl n0 x6 (CHead x8 (Bind Abbr) x9))).(\lambda (H25: (ty3 g x8 x9 t)).(or_intror (ex2 C (\lambda (d2: C).(csub3 g d1 d2)) (\lambda (d2: C).(getl (S n0) c2 (CHead d2 (Bind Abst) t)))) (ex3_2 C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t)))) (ex3_2_intro C T (\lambda (d2: C).(\lambda (_: T).(csub3 g d1 d2))) (\lambda (d2: C).(\lambda (u: T).(getl (S n0) c2 (CHead d2 (Bind Abbr) u)))) (\lambda (d2: C).(\lambda (u: T).(ty3 g d2 u t))) x8 x9 H23 (getl_clear_bind x5 c2 x6 x7 H20 (CHead x8 (Bind Abbr) x9) n0 H24) H25))))))) H22)) H21)))))))) H17))))) H14))))))) H11)))))))) n) H7))))]) H3 H4)))]) H1 H2)))) H0))))))).
 
 theorem csub3_pr2:
  \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1 t1 t2) \to (\forall (c2: C).((csub3 g c1 c2) \to (pr2 c2 t1 t2)))))))
@@ -3066,7 +3066,7 @@ theorem csub3_ty3_ld:
 theorem ty3_sred_wcpr0_pr0:
  \forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t: T).((ty3 g c1 t1 t) \to (\forall (c2: C).((wcpr0 c1 c2) \to (\forall (t2: T).((pr0 t1 t2) \to (ty3 g c2 t2 t)))))))))
 \def
- \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (t2: T).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t3: T).((pr0 t0 t3) \to (ty3 g c2 t3 t2)))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t0: T).(\lambda (_: (ty3 g c t2 t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t3: T).((pr0 t2 t3) \to (ty3 g c2 t3 t0))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t3))))))).(\lambda (H4: (pc3 c t3 t2)).(\lambda (c2: C).(\lambda (H5: (wcpr0 c c2)).(\lambda (t4: T).(\lambda (H6: (pr0 u t4)).(ty3_conv g c2 t2 t0 (H1 c2 H5 t2 (pr0_refl t2)) t4 t3 (H3 c2 H5 t4 H6) (pc3_wcpr0 c c2 H5 t3 t2 H4)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort m) t2)).(eq_ind_r T (TSort m) (\lambda (t0: T).(ty3 g c2 t0 (TSort (next g m)))) (ty3_sort g c2 m) t2 (pr0_gen_sort t2 m H1)))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef n) t2)).(eq_ind_r T (TLRef n) (\lambda (t3: T).(ty3 g c2 t3 (lift (S n) O t0))) (ex3_2_ind C T (\lambda (e2: C).(\lambda (u2: T).(getl n c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2 (TLRef n) (lift (S n) O t0)) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u x1)).(ty3_abbr g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7))))))) (wcpr0_getl c c2 H3 n d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 n H4)))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda (t0: T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef n) t2)).(eq_ind_r T (TLRef n) (\lambda (t3: T).(ty3 g c2 t3 (lift (S n) O u))) (ex3_2_ind C T (\lambda (e2: C).(\lambda (u2: T).(getl n c2 (CHead e2 (Bind Abst) u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abst) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u x1)).(ty3_conv g c2 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g x0 u t0 (H2 x0 H6 u (pr0_refl u)) c2 O (S n) (getl_drop Abst c2 x0 x1 n H5)) (TLRef n) (lift (S n) O x1) (ty3_abst g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7)) (pc3_lift c2 x0 (S n) O (getl_drop Abst c2 x0 x1 n H5) x1 u (pc3_pr2_x x0 x1 u (pr2_free x0 u x1 H7))))))))) (wcpr0_getl c c2 H3 n d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 n H4)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t0: T).(\lambda (_: (ty3 g c u t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (b: B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: (ty3 g (CHead c (Bind b) u) t2 t3)).(\lambda (H3: ((\forall (c2: C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (t4: T).(\lambda (H4: (ty3 g (CHead c (Bind b) u) t3 t4)).(\lambda (H5: ((\forall (c2: C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t2: T).((pr0 t3 t2) \to (ty3 g c2 t2 t4))))))).(\lambda (c2: C).(\lambda (H6: (wcpr0 c c2)).(\lambda (t5: T).(\lambda (H7: (pr0 (THead (Bind b) u t2) t5)).(let H8 \def (match H7 return (\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Bind b) u t2)) \to ((eq T t0 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3)))))) with [(pr0_refl t4) \Rightarrow (\lambda (H7: (eq T t4 (THead (Bind b) u t2))).(\lambda (H8: (eq T t4 t5)).(eq_ind T (THead (Bind b) u t2) (\lambda (t: T).((eq T t t5) \to (ty3 g c2 t5 (THead (Bind b) u t3)))) (\lambda (H9: (eq T (THead (Bind b) u t2) t5)).(eq_ind T (THead (Bind b) u t2) (\lambda (t: T).(ty3 g c2 t (THead (Bind b) u t3))) (ty3_bind g c2 u t0 (H1 c2 H6 u (pr0_refl u)) b t2 t3 (H3 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t2 (pr0_refl t2)) t4 (H5 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t3 (pr0_refl t3))) t5 H9)) t4 (sym_eq T t4 (THead (Bind b) u t2) H7) H8))) | (pr0_comp u1 u2 H7 t4 t5 H8 k) \Rightarrow (\lambda (H9: (eq T (THead k u1 t4) (THead (Bind b) u t2))).(\lambda (H10: (eq T (THead k u2 t5) t5)).((let H11 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) \Rightarrow t])) (THead k u1 t4) (THead (Bind b) u t2) H9) in ((let H12 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead k u1 t4) (THead (Bind b) u t2) H9) in ((let H13 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u1 t4) (THead (Bind b) u t2) H9) in (eq_ind K (Bind b) (\lambda (k0: K).((eq T u1 u) \to ((eq T t4 t2) \to ((eq T (THead k0 u2 t5) t5) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3)))))))) (\lambda (H14: (eq T u1 u)).(eq_ind T u (\lambda (t: T).((eq T t4 t2) \to ((eq T (THead (Bind b) u2 t5) t5) \to ((pr0 t u2) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))) (\lambda (H15: (eq T t4 t2)).(eq_ind T t2 (\lambda (t: T).((eq T (THead (Bind b) u2 t5) t5) \to ((pr0 u u2) \to ((pr0 t t5) \to (ty3 g c2 t5 (THead (Bind b) u t3)))))) (\lambda (H16: (eq T (THead (Bind b) u2 t5) t5)).(eq_ind T (THead (Bind b) u2 t5) (\lambda (t: T).((pr0 u u2) \to ((pr0 t2 t5) \to (ty3 g c2 t (THead (Bind b) u t3))))) (\lambda (H17: (pr0 u u2)).(\lambda (H18: (pr0 t2 t5)).(ex_ind T (\lambda (t: T).(ty3 g (CHead c2 (Bind b) u) t4 t)) (ty3 g c2 (THead (Bind b) u2 t5) (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H19: (ty3 g (CHead c2 (Bind b) u) t4 x)).(ex_ind T (\lambda (t: T).(ty3 g (CHead c2 (Bind b) u2) t3 t)) (ty3 g c2 (THead (Bind b) u2 t5) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H20: (ty3 g (CHead c2 (Bind b) u2) t3 x0)).(ty3_conv g c2 (THead (Bind b) u t3) (THead (Bind b) u t4) (ty3_bind g c2 u t0 (H1 c2 H6 u (pr0_refl u)) b t3 t4 (H5 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t3 (pr0_refl t3)) x H19) (THead (Bind b) u2 t5) (THead (Bind b) u2 t3) (ty3_bind g c2 u2 t0 (H1 c2 H6 u2 H17) b t5 t3 (H3 (CHead c2 (Bind b) u2) (wcpr0_comp c c2 H6 u u2 H17 (Bind b)) t5 H18) x0 H20) (pc3_pr2_x c2 (THead (Bind b) u2 t3) (THead (Bind b) u t3) (pr2_head_1 c2 u u2 (pr2_free c2 u u2 H17) (Bind b) t3))))) (ty3_correct g (CHead c2 (Bind b) u2) t5 t3 (H3 (CHead c2 (Bind b) u2) (wcpr0_comp c c2 H6 u u2 H17 (Bind b)) t5 H18))))) (ty3_correct g (CHead c2 (Bind b) u) t3 t4 (H5 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t3 (pr0_refl t3)))))) t5 H16)) t4 (sym_eq T t4 t2 H15))) u1 (sym_eq T u1 u H14))) k (sym_eq K k (Bind b) H13))) H12)) H11)) H10 H7 H8))) | (pr0_beta u0 v1 v2 H7 t4 t5 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) (THead (Bind b) u t2))).(\lambda (H10: (eq T (THead (Bind Abbr) v2 t5) t5)).((let H11 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2) H9) in (False_ind ((eq T (THead (Bind Abbr) v2 t5) t5) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3))))) H11)) H10 H7 H8))) | (pr0_upsilon b0 H7 v1 v2 H8 u1 u2 H9 t4 t5 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 (THead (Bind b0) u1 t4)) (THead (Bind b) u t2))).(\lambda (H12: (eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t5)).((let H13 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b0) u1 t4)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2) H11) in (False_ind ((eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t5) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))) H13)) H12 H7 H8 H9 H10))) | (pr0_delta u1 u2 H7 t4 t5 H8 w H9) \Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr) u1 t4) (THead (Bind b) u t2))).(\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t5)).((let H12 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abbr) u1 t4) (THead (Bind b) u t2) H10) in ((let H13 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead (Bind Abbr) u1 t4) (THead (Bind b) u t2) H10) in ((let H14 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u1 t4) (THead (Bind b) u t2) H10) in (eq_ind B Abbr (\lambda (b: B).((eq T u1 u) \to ((eq T t4 t2) \to ((eq T (THead (Bind Abbr) u2 w) t5) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))))) (\lambda (H15: (eq T u1 u)).(eq_ind T u (\lambda (t: T).((eq T t4 t2) \to ((eq T (THead (Bind Abbr) u2 w) t5) \to ((pr0 t u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ty3 g c2 t5 (THead (Bind Abbr) u t3)))))))) (\lambda (H16: (eq T t4 t2)).(eq_ind T t2 (\lambda (t: T).((eq T (THead (Bind Abbr) u2 w) t5) \to ((pr0 u u2) \to ((pr0 t t5) \to ((subst0 O u2 t5 w) \to (ty3 g c2 t5 (THead (Bind Abbr) u t3))))))) (\lambda (H17: (eq T (THead (Bind Abbr) u2 w) t5)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).((pr0 u u2) \to ((pr0 t2 t5) \to ((subst0 O u2 t5 w) \to (ty3 g c2 t (THead (Bind Abbr) u t3)))))) (\lambda (H18: (pr0 u u2)).(\lambda (H19: (pr0 t2 t5)).(\lambda (H20: (subst0 O u2 t5 w)).(let H21 \def (eq_ind_r B b (\lambda (b: B).(\forall (c2: C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t2: T).((pr0 t3 t2) \to (ty3 g c2 t2 t4)))))) H5 Abbr H14) in (let H22 \def (eq_ind_r B b (\lambda (b: B).(ty3 g (CHead c (Bind b) u) t3 t4)) H4 Abbr H14) in (let H23 \def (eq_ind_r B b (\lambda (b: B).(\forall (c2: C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g c2 t4 t3)))))) H3 Abbr H14) in (let H24 \def (eq_ind_r B b (\lambda (b: B).(ty3 g (CHead c (Bind b) u) t2 t3)) H2 Abbr H14) in (ex_ind T (\lambda (t: T).(ty3 g (CHead c2 (Bind Abbr) u) t4 t)) (ty3 g c2 (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u t3)) (\lambda (x: T).(\lambda (H25: (ty3 g (CHead c2 (Bind Abbr) u) t4 x)).(ex_ind T (\lambda (t: T).(ty3 g (CHead c2 (Bind Abbr) u2) t3 t)) (ty3 g c2 (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (H26: (ty3 g (CHead c2 (Bind Abbr) u2) t3 x0)).(ty3_conv g c2 (THead (Bind Abbr) u t3) (THead (Bind Abbr) u t4) (ty3_bind g c2 u t0 (H1 c2 H6 u (pr0_refl u)) Abbr t3 t4 (H21 (CHead c2 (Bind Abbr) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind Abbr)) t3 (pr0_refl t3)) x H25) (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u2 t3) (ty3_bind g c2 u2 t0 (H1 c2 H6 u2 H18) Abbr w t3 (ty3_subst0 g (CHead c2 (Bind Abbr) u2) t5 t3 (H23 (CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H6 u u2 H18 (Bind Abbr)) t5 H19) c2 u2 O (getl_refl Abbr c2 u2) w H20) x0 H26) (pc3_pr2_x c2 (THead (Bind Abbr) u2 t3) (THead (Bind Abbr) u t3) (pr2_head_1 c2 u u2 (pr2_free c2 u u2 H18) (Bind Abbr) t3))))) (ty3_correct g (CHead c2 (Bind Abbr) u2) t5 t3 (H23 (CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H6 u u2 H18 (Bind Abbr)) t5 H19))))) (ty3_correct g (CHead c2 (Bind Abbr) u) t3 t4 (H21 (CHead c2 (Bind Abbr) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind Abbr)) t3 (pr0_refl t3))))))))))) t5 H17)) t4 (sym_eq T t4 t2 H16))) u1 (sym_eq T u1 u H15))) b H14)) H13)) H12)) H11 H7 H8 H9))) | (pr0_zeta b0 H7 t4 t5 H8 u0) \Rightarrow (\lambda (H9: (eq T (THead (Bind b0) u0 (lift (S O) O t4)) (THead (Bind b) u t2))).(\lambda (H10: (eq T t5 t5)).((let H11 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t4) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t4) | (THead _ _ t) \Rightarrow t])) (THead (Bind b0) u0 (lift (S O) O t4)) (THead (Bind b) u t2) H9) in ((let H12 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead (Bind b0) u0 (lift (S O) O t4)) (THead (Bind b) u t2) H9) in ((let H13 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 (lift (S O) O t4)) (THead (Bind b) u t2) H9) in (eq_ind B b (\lambda (b1: B).((eq T u0 u) \to ((eq T (lift (S O) O t4) t2) \to ((eq T t5 t5) \to ((not (eq B b1 Abst)) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3)))))))) (\lambda (H14: (eq T u0 u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t4) t2) \to ((eq T t5 t5) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))) (\lambda (H15: (eq T (lift (S O) O t4) t2)).(eq_ind T (lift (S O) O t4) (\lambda (_: T).((eq T t5 t5) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3)))))) (\lambda (H16: (eq T t5 t5)).(eq_ind T t5 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t4 t) \to (ty3 g c2 t5 (THead (Bind b) u t3))))) (\lambda (H17: (not (eq B b Abst))).(\lambda (H18: (pr0 t4 t5)).(let H19 \def (eq_ind_r T t2 (\lambda (t: T).(\forall (c2: C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t2: T).((pr0 t t2) \to (ty3 g c2 t2 t3)))))) H3 (lift (S O) O t4) H15) in (let H20 \def (eq_ind_r T t2 (\lambda (t: T).(ty3 g (CHead c (Bind b) u) t t3)) H2 (lift (S O) O t4) H15) in (ex_ind T (\lambda (t: T).(ty3 g (CHead c2 (Bind b) u) t4 t)) (ty3 g c2 t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H4: (ty3 g (CHead c2 (Bind b) u) t4 x)).(B_ind (\lambda (b: B).((not (eq B b Abst)) \to ((ty3 g (CHead c2 (Bind b) u) t3 t4) \to ((ty3 g (CHead c2 (Bind b) u) t4 x) \to ((ty3 g (CHead c2 (Bind b) u) (lift (S O) O t5) t3) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))) (\lambda (H21: (not (eq B Abbr Abst))).(\lambda (H2: (ty3 g (CHead c2 (Bind Abbr) u) t3 t4)).(\lambda (H5: (ty3 g (CHead c2 (Bind Abbr) u) t4 x)).(\lambda (H22: (ty3 g (CHead c2 (Bind Abbr) u) (lift (S O) O t5) t3)).(let H \def (ty3_gen_cabbr g (CHead c2 (Bind Abbr) u) (lift (S O) O t5) t3 H22 c2 u O (getl_refl Abbr c2 u) (CHead c2 (Bind Abbr) u) (csubst1_refl O u (CHead c2 (Bind Abbr) u)) c2 (drop_drop (Bind Abbr) O c2 c2 (drop_refl c2) u)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 O u (lift (S O) O t5) (lift (S O) O y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 O u t3 (lift (S O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g c2 y1 y2))) (ty3 g c2 t5 (THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H0: (subst1 O u (lift (S O) O t5) (lift (S O) O x0))).(\lambda (H3: (subst1 O u t3 (lift (S O) O x1))).(\lambda (H23: (ty3 g c2 x0 x1)).(let H24 \def (eq_ind T x0 (\lambda (t: T).(ty3 g c2 t x1)) H23 t5 (lift_inj x0 t5 (S O) O (subst1_gen_lift_eq t5 u (lift (S O) O x0) (S O) O O (le_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n)) (le_n (plus (S O) O)) (plus O (S O)) (plus_comm O (S O))) H0))) in (ty3_conv g c2 (THead (Bind Abbr) u t3) (THead (Bind Abbr) u t4) (ty3_bind g c2 u t0 (H1 c2 H6 u (pr0_refl u)) Abbr t3 t4 H2 x H5) t5 x1 H24 (pc3_pr3_x c2 x1 (THead (Bind Abbr) u t3) (pr3_t (THead (Bind Abbr) u (lift (S O) O x1)) (THead (Bind Abbr) u t3) c2 (pr3_pr2 c2 (THead (Bind Abbr) u t3) (THead (Bind Abbr) u (lift (S O) O x1)) (pr2_free c2 (THead (Bind Abbr) u t3) (THead (Bind Abbr) u (lift (S O) O x1)) (pr0_delta1 u u (pr0_refl u) t3 t3 (pr0_refl t3) (lift (S O) O x1) H3))) x1 (pr3_pr2 c2 (THead (Bind Abbr) u (lift (S O) O x1)) x1 (pr2_free c2 (THead (Bind Abbr) u (lift (S O) O x1)) x1 (pr0_zeta Abbr H21 x1 x1 (pr0_refl x1) u)))))))))))) H)))))) (\lambda (H21: (not (eq B Abst Abst))).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) t3 t4)).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) t4 x)).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) (lift (S O) O t5) t3)).(let H \def (match (H21 (refl_equal B Abst)) return (\lambda (_: ?).(ty3 g c2 t5 (THead (Bind Abst) u t3))) with []) in H))))) (\lambda (H21: (not (eq B Void Abst))).(\lambda (H2: (ty3 g (CHead c2 (Bind Void) u) t3 t4)).(\lambda (H5: (ty3 g (CHead c2 (Bind Void) u) t4 x)).(\lambda (H22: (ty3 g (CHead c2 (Bind Void) u) (lift (S O) O t5) t3)).(let H \def (ty3_gen_cvoid g (CHead c2 (Bind Void) u) (lift (S O) O t5) t3 H22 c2 u O (getl_refl Void c2 u) c2 (drop_drop (Bind Void) O c2 c2 (drop_refl c2) u)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) O t5) (lift (S O) O y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g c2 y1 y2))) (ty3 g c2 t5 (THead (Bind Void) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H0: (eq T (lift (S O) O t5) (lift (S O) O x0))).(\lambda (H3: (eq T t3 (lift (S O) O x1))).(\lambda (H23: (ty3 g c2 x0 x1)).(let H24 \def (eq_ind T t3 (\lambda (t: T).(ty3 g (CHead c2 (Bind Void) u) t t4)) H2 (lift (S O) O x1) H3) in (eq_ind_r T (lift (S O) O x1) (\lambda (t: T).(ty3 g c2 t5 (THead (Bind Void) u t))) (let H25 \def (eq_ind_r T x0 (\lambda (t: T).(ty3 g c2 t x1)) H23 t5 (lift_inj t5 x0 (S O) O H0)) in (ty3_conv g c2 (THead (Bind Void) u (lift (S O) O x1)) (THead (Bind Void) u t4) (ty3_bind g c2 u t0 (H1 c2 H6 u (pr0_refl u)) Void (lift (S O) O x1) t4 H24 x H5) t5 x1 H25 (pc3_pr2_x c2 x1 (THead (Bind Void) u (lift (S O) O x1)) (pr2_free c2 (THead (Bind Void) u (lift (S O) O x1)) x1 (pr0_zeta Void H21 x1 x1 (pr0_refl x1) u))))) t3 H3))))))) H)))))) b H17 (H5 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t3 (pr0_refl t3)) H4 (H19 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) (lift (S O) O t5) (pr0_lift t4 t5 H18 (S O) O))))) (ty3_correct g (CHead c2 (Bind b) u) t3 t4 (H5 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t3 (pr0_refl t3)))))))) t5 (sym_eq T t5 t5 H16))) t2 H15)) u0 (sym_eq T u0 u H14))) b0 (sym_eq B b0 b H13))) H12)) H11)) H10 H7 H8))) | (pr0_epsilon t4 t5 H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Flat Cast) u0 t4) (THead (Bind b) u t2))).(\lambda (H9: (eq T t5 t5)).((let H10 \def (eq_ind T (THead (Flat Cast) u0 t4) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2) H8) in (False_ind ((eq T t5 t5) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3)))) H10)) H9 H7)))]) in (H8 (refl_equal T (THead (Bind b) u t2)) (refl_equal T t5)))))))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 w t2) \to (ty3 g c2 t2 u))))))).(\lambda (v: T).(\lambda (t0: T).(\lambda (H2: (ty3 g c v (THead (Bind Abst) u t0))).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 v t2) \to (ty3 g c2 t2 (THead (Bind Abst) u t0)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Flat Appl) w v) t2)).(let H6 \def (match H5 return (\lambda (t: T).(\lambda (t1: T).((eq T t (THead (Flat Appl) w v)) \to ((eq T t1 t2) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))))) with [(pr0_refl t0) \Rightarrow (\lambda (H5: (eq T t0 (THead (Flat Appl) w v))).(\lambda (H6: (eq T t0 t2)).(eq_ind T (THead (Flat Appl) w v) (\lambda (t: T).((eq T t t2) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))) (\lambda (H7: (eq T (THead (Flat Appl) w v) t2)).(eq_ind T (THead (Flat Appl) w v) (\lambda (t: T).(ty3 g c2 t (THead (Flat Appl) w (THead (Bind Abst) u t0)))) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) v t0 (H3 c2 H4 v (pr0_refl v))) t2 H7)) t0 (sym_eq T t0 (THead (Flat Appl) w v) H5) H6))) | (pr0_comp u1 u2 H5 t1 t0 H6 k) \Rightarrow (\lambda (H7: (eq T (THead k u1 t1) (THead (Flat Appl) w v))).(\lambda (H8: (eq T (THead k u2 t0) t2)).((let H9 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t])) (THead k u1 t1) (THead (Flat Appl) w v) H7) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead k u1 t1) (THead (Flat Appl) w v) H7) in ((let H11 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u1 t1) (THead (Flat Appl) w v) H7) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T u1 w) \to ((eq T t1 v) \to ((eq T (THead k0 u2 t0) t2) \to ((pr0 u1 u2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H12: (eq T u1 w)).(eq_ind T w (\lambda (t: T).((eq T t1 v) \to ((eq T (THead (Flat Appl) u2 t0) t2) \to ((pr0 t u2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) (\lambda (H13: (eq T t1 v)).(eq_ind T v (\lambda (t: T).((eq T (THead (Flat Appl) u2 t0) t2) \to ((pr0 w u2) \to ((pr0 t t0) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))))) (\lambda (H14: (eq T (THead (Flat Appl) u2 t0) t2)).(eq_ind T (THead (Flat Appl) u2 t0) (\lambda (t: T).((pr0 w u2) \to ((pr0 v t0) \to (ty3 g c2 t (THead (Flat Appl) w (THead (Bind Abst) u t0)))))) (\lambda (H15: (pr0 w u2)).(\lambda (H16: (pr0 v t0)).(ex_ind T (\lambda (t: T).(ty3 g c2 (THead (Bind Abst) u t0) t)) (ty3 g c2 (THead (Flat Appl) u2 t0) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H17: (ty3 g c2 (THead (Bind Abst) u t0) x)).(ex4_3_ind T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c2 (THead (Bind Abst) u t2) x)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c2 u t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c2 (Bind Abst) u) t2 t3)))) (ty3 g c2 (THead (Flat Appl) u2 t0) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H19: (ty3 g c2 u x1)).(\lambda (H20: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(\lambda (H21: (ty3 g (CHead c2 (Bind Abst) u) x0 x2)).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H19 Abst t0 x0 H20 x2 H21)) (THead (Flat Appl) u2 t0) (THead (Flat Appl) u2 (THead (Bind Abst) u t0)) (ty3_appl g c2 u2 u (H1 c2 H4 u2 H15) t0 t0 (H3 c2 H4 t0 H16)) (pc3_pr2_x c2 (THead (Flat Appl) u2 (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pr2_head_1 c2 w u2 (pr2_free c2 w u2 H15) (Flat Appl) (THead (Bind Abst) u t0))))))))))) (ty3_gen_bind g Abst c2 u t0 x H17)))) (ty3_correct g c2 v (THead (Bind Abst) u t0) (H3 c2 H4 v (pr0_refl v)))))) t2 H14)) t1 (sym_eq T t1 v H13))) u1 (sym_eq T u1 w H12))) k (sym_eq K k (Flat Appl) H11))) H10)) H9)) H8 H5 H6))) | (pr0_beta u0 v1 v2 H5 t1 t0 H6) \Rightarrow (\lambda (H7: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t1)) (THead (Flat Appl) w v))).(\lambda (H8: (eq T (THead (Bind Abbr) v2 t0) t2)).((let H9 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow (THead (Bind Abst) u0 t1) | (TLRef _) \Rightarrow (THead (Bind Abst) u0 t1) | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t1)) (THead (Flat Appl) w v) H7) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t1)) (THead (Flat Appl) w v) H7) in (eq_ind T w (\lambda (t: T).((eq T (THead (Bind Abst) u0 t1) v) \to ((eq T (THead (Bind Abbr) v2 t0) t2) \to ((pr0 t v2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) (\lambda (H11: (eq T (THead (Bind Abst) u0 t1) v)).(eq_ind T (THead (Bind Abst) u0 t1) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t0) t2) \to ((pr0 w v2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))))) (\lambda (H12: (eq T (THead (Bind Abbr) v2 t0) t2)).(eq_ind T (THead (Bind Abbr) v2 t0) (\lambda (t: T).((pr0 w v2) \to ((pr0 t1 t0) \to (ty3 g c2 t (THead (Flat Appl) w (THead (Bind Abst) u t0)))))) (\lambda (H13: (pr0 w v2)).(\lambda (H14: (pr0 t1 t0)).(let H15 \def (eq_ind_r T v (\lambda (t: T).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to (ty3 g c2 t2 (THead (Bind Abst) u t0))))))) H3 (THead (Bind Abst) u0 t1) H11) in (let H16 \def (eq_ind_r T v (\lambda (t: T).(ty3 g c t (THead (Bind Abst) u t0))) H2 (THead (Bind Abst) u0 t1) H11) in (ex_ind T (\lambda (t: T).(ty3 g c2 (THead (Bind Abst) u t0) t)) (ty3 g c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H2: (ty3 g c2 (THead (Bind Abst) u t0) x)).(ex4_3_ind T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c2 (THead (Bind Abst) u t2) x)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c2 u t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c2 (Bind Abst) u) t2 t3)))) (ty3 g c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H17: (ty3 g c2 u x1)).(\lambda (H18: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(\lambda (H19: (ty3 g (CHead c2 (Bind Abst) u) x0 x2)).(ex4_3_ind T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c2 (THead (Bind Abst) u0 t2) (THead (Bind Abst) u t0))))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c2 u0 t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u0) t0 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c2 (Bind Abst) u0) t2 t3)))) (ty3 g c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H0: (pc3 c2 (THead (Bind Abst) u0 x3) (THead (Bind Abst) u t0))).(\lambda (H20: (ty3 g c2 u0 x4)).(\lambda (H21: (ty3 g (CHead c2 (Bind Abst) u0) t0 x3)).(\lambda (H22: (ty3 g (CHead c2 (Bind Abst) u0) x3 x5)).(and_ind (pc3 c2 u0 u) (\forall (b: B).(\forall (u: T).(pc3 (CHead c2 (Bind b) u) x3 t0))) (ty3 g c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H23: (pc3 c2 u0 u)).(\lambda (H24: ((\forall (b: B).(\forall (u: T).(pc3 (CHead c2 (Bind b) u) x3 t0))))).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H17 Abst t0 x0 H18 x2 H19)) (THead (Bind Abbr) v2 t0) (THead (Bind Abbr) v2 x3) (ty3_bind g c2 v2 u (H1 c2 H4 v2 H13) Abbr t0 x3 (csub3_ty3_ld g c2 v2 u0 (ty3_conv g c2 u0 x4 H20 v2 u (H1 c2 H4 v2 H13) (pc3_s c2 u u0 H23)) t0 x3 H21) x5 (csub3_ty3_ld g c2 v2 u0 (ty3_conv g c2 u0 x4 H20 v2 u (H1 c2 H4 v2 H13) (pc3_s c2 u u0 H23)) x3 x5 H22)) (pc3_t (THead (Bind Abbr) v2 t0) c2 (THead (Bind Abbr) v2 x3) (pc3_head_2 c2 v2 x3 t0 (Bind Abbr) (H24 Abbr v2)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc3_pr2_x c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pr2_free c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Bind Abbr) v2 t0) (pr0_beta u w v2 H13 t0 t0 (pr0_refl t0)))))))) (pc3_gen_abst c2 u0 u x3 t0 H0))))))))) (ty3_gen_bind g Abst c2 u0 t0 (THead (Bind Abst) u t0) (H15 c2 H4 (THead (Bind Abst) u0 t0) (pr0_comp u0 u0 (pr0_refl u0) t1 t0 H14 (Bind Abst)))))))))))) (ty3_gen_bind g Abst c2 u t0 x H2)))) (ty3_correct g c2 (THead (Bind Abst) u0 t1) (THead (Bind Abst) u t0) (H15 c2 H4 (THead (Bind Abst) u0 t1) (pr0_refl (THead (Bind Abst) u0 t1))))))))) t2 H12)) v H11)) v1 (sym_eq T v1 w H10))) H9)) H8 H5 H6))) | (pr0_upsilon b H5 v1 v2 H6 u1 u2 H7 t1 t0 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (THead (Flat Appl) w v))).(\lambda (H10: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) t2)).((let H11 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow (THead (Bind b) u1 t1) | (TLRef _) \Rightarrow (THead (Bind b) u1 t1) | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (THead (Flat Appl) w v) H9) in ((let H12 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (THead (Flat Appl) w v) H9) in (eq_ind T w (\lambda (t: T).((eq T (THead (Bind b) u1 t1) v) \to ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) t2) \to ((not (eq B b Abst)) \to ((pr0 t v2) \to ((pr0 u1 u2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))))))) (\lambda (H13: (eq T (THead (Bind b) u1 t1) v)).(eq_ind T (THead (Bind b) u1 t1) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) t2) \to ((not (eq B b Abst)) \to ((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H14: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t1 t0) \to (ty3 g c2 t (THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) (\lambda (H15: (not (eq B b Abst))).(\lambda (H16: (pr0 w v2)).(\lambda (H17: (pr0 u1 u2)).(\lambda (H18: (pr0 t1 t0)).(let H19 \def (eq_ind_r T v (\lambda (t: T).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to (ty3 g c2 t2 (THead (Bind Abst) u t0))))))) H3 (THead (Bind b) u1 t1) H13) in (let H20 \def (eq_ind_r T v (\lambda (t: T).(ty3 g c t (THead (Bind Abst) u t0))) H2 (THead (Bind b) u1 t1) H13) in (ex_ind T (\lambda (t: T).(ty3 g c2 (THead (Bind Abst) u t0) t)) (ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H2: (ty3 g c2 (THead (Bind Abst) u t0) x)).(let H3 \def H2 in (ex4_3_ind T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c2 (THead (Bind Abst) u t2) x)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c2 u t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c2 (Bind Abst) u) t2 t3)))) (ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H22: (ty3 g c2 u x1)).(\lambda (H23: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(\lambda (H24: (ty3 g (CHead c2 (Bind Abst) u) x0 x2)).(ex4_3_ind T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c2 (THead (Bind b) u2 t2) (THead (Bind Abst) u t0))))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c2 u2 t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) u2) t0 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c2 (Bind b) u2) t2 t3)))) (ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H0: (pc3 c2 (THead (Bind b) u2 x3) (THead (Bind Abst) u t0))).(\lambda (H25: (ty3 g c2 u2 x4)).(\lambda (H26: (ty3 g (CHead c2 (Bind b) u2) t0 x3)).(\lambda (_: (ty3 g (CHead c2 (Bind b) u2) x3 x5)).(let H28 \def (eq_ind T (lift (S O) O (THead (Bind Abst) u t0)) (\lambda (t: T).(pc3 (CHead c2 (Bind b) u2) x3 t)) (pc3_gen_not_abst b H15 c2 x3 t0 u2 u H0) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u t0 (S O) O)) in (let H29 \def (eq_ind T (lift (S O) O (THead (Bind Abst) u t0)) (\lambda (t: T).(ty3 g (CHead c2 (Bind b) u2) t (lift (S O) O x))) (ty3_lift g c2 (THead (Bind Abst) u t0) x H2 (CHead c2 (Bind b) u2) O (S O) (drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u t0 (S O) O)) in (ex4_3_ind T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c2 (Bind b) u2) (THead (Bind Abst) (lift (S O) O u) t2) (lift (S O) O x))))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S O) O u)) (lift (S O) (S O) t0) t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S O) O u)) t2 t3)))) (ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: T).(\lambda (_: (pc3 (CHead c2 (Bind b) u2) (THead (Bind Abst) (lift (S O) O u) x6) (lift (S O) O x))).(\lambda (H31: (ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) x7)).(\lambda (H32: (ty3 g (CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S O) O u)) (lift (S O) (S O) t0) x6)).(\lambda (H33: (ty3 g (CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S O) O u)) x6 x8)).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H22 Abst t0 x0 H23 x2 H24)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)))) (ty3_bind g c2 u2 x4 H25 b (THead (Flat Appl) (lift (S O) O v2) t0) (THead (Flat Appl) (lift (S O) O v2) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0))) (ty3_appl g (CHead c2 (Bind b) u2) (lift (S O) O v2) (lift (S O) O u) (ty3_lift g c2 v2 u (H1 c2 H4 v2 H16) (CHead c2 (Bind b) u2) O (S O) (drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) t0 (lift (S O) (S O) t0) (ty3_conv g (CHead c2 (Bind b) u2) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (THead (Bind Abst) (lift (S O) O u) x6) (ty3_bind g (CHead c2 (Bind b) u2) (lift (S O) O u) x7 H31 Abst (lift (S O) (S O) t0) x6 H32 x8 H33) t0 x3 H26 H28)) (THead (Flat Appl) (lift (S O) O v2) (THead (Bind Abst) (lift (S O) O u) x6)) (ty3_appl g (CHead c2 (Bind b) u2) (lift (S O) O v2) (lift (S O) O u) (ty3_lift g c2 v2 u (H1 c2 H4 v2 H16) (CHead c2 (Bind b) u2) O (S O) (drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) x6 (ty3_bind g (CHead c2 (Bind b) u2) (lift (S O) O u) x7 H31 Abst (lift (S O) (S O) t0) x6 H32 x8 H33))) (eq_ind T (lift (S O) O (THead (Bind Abst) u t0)) (\lambda (t: T).(pc3 c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t)) (THead (Flat Appl) w (THead (Bind Abst) u t0)))) (pc3_pc1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O (THead (Bind Abst) u t0)))) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc1_pr0_u2 (THead (Flat Appl) v2 (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u t0)))) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O (THead (Bind Abst) u t0)))) (pr0_upsilon b H15 v2 v2 (pr0_refl v2) u2 u2 (pr0_refl u2) (lift (S O) O (THead (Bind Abst) u t0)) (lift (S O) O (THead (Bind Abst) u t0)) (pr0_refl (lift (S O) O (THead (Bind Abst) u t0)))) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc1_s (THead (Flat Appl) v2 (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u t0)))) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc1_head w v2 (pc1_pr0_r w v2 H16) (THead (Bind Abst) u t0) (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u t0))) (pc1_pr0_x (THead (Bind Abst) u t0) (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u t0))) (pr0_zeta b H15 (THead (Bind Abst) u t0) (THead (Bind Abst) u t0) (pr0_refl (THead (Bind Abst) u t0)) u2)) (Flat Appl)))) c2) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u t0 (S O) O)))))))))) (ty3_gen_bind g Abst (CHead c2 (Bind b) u2) (lift (S O) O u) (lift (S O) (S O) t0) (lift (S O) O x) H29))))))))))) (ty3_gen_bind g b c2 u2 t0 (THead (Bind Abst) u t0) (H19 c2 H4 (THead (Bind b) u2 t0) (pr0_comp u1 u2 H17 t1 t0 H18 (Bind b)))))))))))) (ty3_gen_bind g Abst c2 u t0 x H3))))) (ty3_correct g c2 (THead (Bind b) u2 t0) (THead (Bind Abst) u t0) (H19 c2 H4 (THead (Bind b) u2 t0) (pr0_comp u1 u2 H17 t1 t0 H18 (Bind b))))))))))) t2 H14)) v H13)) v1 (sym_eq T v1 w H12))) H11)) H10 H5 H6 H7 H8))) | (pr0_delta u1 u2 H5 t1 t0 H6 w0 H7) \Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u1 t1) (THead (Flat Appl) w v))).(\lambda (H9: (eq T (THead (Bind Abbr) u2 w0) t2)).((let H10 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) w v) H8) in (False_ind ((eq T (THead (Bind Abbr) u2 w0) t2) \to ((pr0 u1 u2) \to ((pr0 t1 t0) \to ((subst0 O u2 t0 w0) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))))) H10)) H9 H5 H6 H7))) | (pr0_zeta b H5 t1 t0 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead (Bind b) u0 (lift (S O) O t1)) (THead (Flat Appl) w v))).(\lambda (H8: (eq T t0 t2)).((let H9 \def (eq_ind T (THead (Bind b) u0 (lift (S O) O t1)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) w v) H7) in (False_ind ((eq T t0 t2) \to ((not (eq B b Abst)) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))) H9)) H8 H5 H6))) | (pr0_epsilon t1 t0 H5 u0) \Rightarrow (\lambda (H6: (eq T (THead (Flat Cast) u0 t1) (THead (Flat Appl) w v))).(\lambda (H7: (eq T t0 t2)).((let H8 \def (eq_ind T (THead (Flat Cast) u0 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) w v) H6) in (False_ind ((eq T t0 t2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))) H8)) H7 H5)))]) in (H6 (refl_equal T (THead (Flat Appl) w v)) (refl_equal T t2)))))))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t2 t3)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c t3 t0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t3 t2) \to (ty3 g c2 t2 t0))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead (Flat Cast) t3 t2) t4)).(let H6 \def (match H5 return (\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Cast) t3 t2)) \to ((eq T t0 t4) \to (ty3 g c2 t4 t3))))) with [(pr0_refl t) \Rightarrow (\lambda (H5: (eq T t (THead (Flat Cast) t3 t2))).(\lambda (H6: (eq T t t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t0: T).((eq T t0 t4) \to (ty3 g c2 t4 t3))) (\lambda (H7: (eq T (THead (Flat Cast) t3 t2) t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t0: T).(ty3 g c2 t0 t3)) (ty3_cast g c2 t2 t3 (H1 c2 H4 t2 (pr0_refl t2)) t0 (H3 c2 H4 t3 (pr0_refl t3))) t4 H7)) t (sym_eq T t (THead (Flat Cast) t3 t2) H5) H6))) | (pr0_comp u1 u2 H5 t4 t5 H6 k) \Rightarrow (\lambda (H7: (eq T (THead k u1 t4) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T (THead k u2 t5) t4)).((let H9 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) \Rightarrow t])) (THead k u1 t4) (THead (Flat Cast) t3 t2) H7) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead k u1 t4) (THead (Flat Cast) t3 t2) H7) in ((let H11 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u1 t4) (THead (Flat Cast) t3 t2) H7) in (eq_ind K (Flat Cast) (\lambda (k0: K).((eq T u1 t3) \to ((eq T t4 t2) \to ((eq T (THead k0 u2 t5) t4) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ty3 g c2 t4 t3))))))) (\lambda (H12: (eq T u1 t3)).(eq_ind T t3 (\lambda (t: T).((eq T t4 t2) \to ((eq T (THead (Flat Cast) u2 t5) t4) \to ((pr0 t u2) \to ((pr0 t4 t5) \to (ty3 g c2 t4 t3)))))) (\lambda (H13: (eq T t4 t2)).(eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Cast) u2 t5) t4) \to ((pr0 t3 u2) \to ((pr0 t t5) \to (ty3 g c2 t4 t3))))) (\lambda (H14: (eq T (THead (Flat Cast) u2 t5) t4)).(eq_ind T (THead (Flat Cast) u2 t5) (\lambda (t: T).((pr0 t3 u2) \to ((pr0 t2 t5) \to (ty3 g c2 t t3)))) (\lambda (H15: (pr0 t3 u2)).(\lambda (H16: (pr0 t2 t5)).(ty3_conv g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)) (THead (Flat Cast) u2 t5) u2 (ty3_cast g c2 t5 u2 (ty3_conv g c2 u2 t0 (H3 c2 H4 u2 H15) t5 t3 (H1 c2 H4 t5 H16) (pc3_pr2_r c2 t3 u2 (pr2_free c2 t3 u2 H15))) t0 (H3 c2 H4 u2 H15)) (pc3_pr2_x c2 u2 t3 (pr2_free c2 t3 u2 H15))))) t4 H14)) t4 (sym_eq T t4 t2 H13))) u1 (sym_eq T u1 t3 H12))) k (sym_eq K k (Flat Cast) H11))) H10)) H9)) H8 H5 H6))) | (pr0_beta u v1 v2 H5 t4 t5 H6) \Rightarrow (\lambda (H7: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T (THead (Bind Abbr) v2 t5) t4)).((let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H7) in (False_ind ((eq T (THead (Bind Abbr) v2 t5) t4) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ty3 g c2 t4 t3)))) H9)) H8 H5 H6))) | (pr0_upsilon b H5 v1 v2 H6 u1 u2 H7 t4 t5 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) (THead (Flat Cast) t3 t2))).(\lambda (H10: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t4)).((let H11 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H9) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t4) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ty3 g c2 t4 t3)))))) H11)) H10 H5 H6 H7 H8))) | (pr0_delta u1 u2 H5 t4 t5 H6 w H7) \Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u1 t4) (THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T (THead (Bind Abbr) u2 w) t4)).((let H10 \def (eq_ind T (THead (Bind Abbr) u1 t4) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ty3 g c2 t4 t3))))) H10)) H9 H5 H6 H7))) | (pr0_zeta b H5 t4 t5 H6 u) \Rightarrow (\lambda (H7: (eq T (THead (Bind b) u (lift (S O) O t4)) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T t5 t4)).((let H9 \def (eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) t3 t2) H7) in (False_ind ((eq T t5 t4) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ty3 g c2 t4 t3)))) H9)) H8 H5 H6))) | (pr0_epsilon t4 t5 H5 u) \Rightarrow (\lambda (H6: (eq T (THead (Flat Cast) u t4) (THead (Flat Cast) t3 t2))).(\lambda (H7: (eq T t5 t4)).((let H8 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) u t4) (THead (Flat Cast) t3 t2) H6) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u t4) (THead (Flat Cast) t3 t2) H6) in (eq_ind T t3 (\lambda (_: T).((eq T t4 t2) \to ((eq T t5 t4) \to ((pr0 t4 t5) \to (ty3 g c2 t4 t3))))) (\lambda (H10: (eq T t4 t2)).(eq_ind T t2 (\lambda (t: T).((eq T t5 t4) \to ((pr0 t t5) \to (ty3 g c2 t4 t3)))) (\lambda (H11: (eq T t5 t4)).(eq_ind T t4 (\lambda (t: T).((pr0 t2 t) \to (ty3 g c2 t4 t3))) (\lambda (H12: (pr0 t2 t4)).(H1 c2 H4 t4 H12)) t5 (sym_eq T t5 t4 H11))) t4 (sym_eq T t4 t2 H10))) u (sym_eq T u t3 H9))) H8)) H7 H5)))]) in (H6 (refl_equal T (THead (Flat Cast) t3 t2)) (refl_equal T t4))))))))))))))) c1 t1 t H))))).
+ \lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t: T).(\lambda (H: (ty3 g c1 t1 t)).(ty3_ind g (\lambda (c: C).(\lambda (t0: T).(\lambda (t2: T).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t3: T).((pr0 t0 t3) \to (ty3 g c2 t3 t2)))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t0: T).(\lambda (_: (ty3 g c t2 t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t3: T).((pr0 t2 t3) \to (ty3 g c2 t3 t0))))))).(\lambda (u: T).(\lambda (t3: T).(\lambda (_: (ty3 g c u t3)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t3))))))).(\lambda (H4: (pc3 c t3 t2)).(\lambda (c2: C).(\lambda (H5: (wcpr0 c c2)).(\lambda (t4: T).(\lambda (H6: (pr0 u t4)).(ty3_conv g c2 t2 t0 (H1 c2 H5 t2 (pr0_refl t2)) t4 t3 (H3 c2 H5 t4 H6) (pc3_wcpr0 c c2 H5 t3 t2 H4)))))))))))))))) (\lambda (c: C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H1: (pr0 (TSort m) t2)).(eq_ind_r T (TSort m) (\lambda (t0: T).(ty3 g c2 t0 (TSort (next g m)))) (ty3_sort g c2 m) t2 (pr0_gen_sort t2 m H1)))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda (t0: T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef n) t2)).(eq_ind_r T (TLRef n) (\lambda (t3: T).(ty3 g c2 t3 (lift (S n) O t0))) (ex3_2_ind C T (\lambda (e2: C).(\lambda (u2: T).(getl n c2 (CHead e2 (Bind Abbr) u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2 (TLRef n) (lift (S n) O t0)) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abbr) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u x1)).(ty3_abbr g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7))))))) (wcpr0_getl c c2 H3 n d u (Bind Abbr) H0)) t2 (pr0_gen_lref t2 n H4)))))))))))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda (t0: T).(\lambda (_: (ty3 g d u t0)).(\lambda (H2: ((\forall (c2: C).((wcpr0 d c2) \to (\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (c2: C).(\lambda (H3: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H4: (pr0 (TLRef n) t2)).(eq_ind_r T (TLRef n) (\lambda (t3: T).(ty3 g c2 t3 (lift (S n) O u))) (ex3_2_ind C T (\lambda (e2: C).(\lambda (u2: T).(getl n c2 (CHead e2 (Bind Abst) u2)))) (\lambda (e2: C).(\lambda (_: T).(wcpr0 d e2))) (\lambda (_: C).(\lambda (u2: T).(pr0 u u2))) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abst) x1))).(\lambda (H6: (wcpr0 d x0)).(\lambda (H7: (pr0 u x1)).(ty3_conv g c2 (lift (S n) O u) (lift (S n) O t0) (ty3_lift g x0 u t0 (H2 x0 H6 u (pr0_refl u)) c2 O (S n) (getl_drop Abst c2 x0 x1 n H5)) (TLRef n) (lift (S n) O x1) (ty3_abst g n c2 x0 x1 H5 t0 (H2 x0 H6 x1 H7)) (pc3_lift c2 x0 (S n) O (getl_drop Abst c2 x0 x1 n H5) x1 u (pc3_pr2_x x0 x1 u (pr2_free x0 u x1 H7))))))))) (wcpr0_getl c c2 H3 n d u (Bind Abst) H0)) t2 (pr0_gen_lref t2 n H4)))))))))))))) (\lambda (c: C).(\lambda (u: T).(\lambda (t0: T).(\lambda (_: (ty3 g c u t0)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 u t2) \to (ty3 g c2 t2 t0))))))).(\lambda (b: B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: (ty3 g (CHead c (Bind b) u) t2 t3)).(\lambda (H3: ((\forall (c2: C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (t4: T).(\lambda (H4: (ty3 g (CHead c (Bind b) u) t3 t4)).(\lambda (H5: ((\forall (c2: C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t2: T).((pr0 t3 t2) \to (ty3 g c2 t2 t4))))))).(\lambda (c2: C).(\lambda (H6: (wcpr0 c c2)).(\lambda (t5: T).(\lambda (H7: (pr0 (THead (Bind b) u t2) t5)).(let H8 \def (match H7 return (\lambda (_: ?).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Bind b) u t2)) \to ((eq T t0 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))) with [(pr0_refl t4) \Rightarrow (\lambda (H7: (eq T t4 (THead (Bind b) u t2))).(\lambda (H8: (eq T t4 t5)).(eq_ind T (THead (Bind b) u t2) (\lambda (t: T).((eq T t t5) \to (ty3 g c2 t5 (THead (Bind b) u t3)))) (\lambda (H9: (eq T (THead (Bind b) u t2) t5)).(eq_ind T (THead (Bind b) u t2) (\lambda (t: T).(ty3 g c2 t (THead (Bind b) u t3))) (ty3_bind g c2 u t0 (H1 c2 H6 u (pr0_refl u)) b t2 t3 (H3 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t2 (pr0_refl t2)) t4 (H5 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t3 (pr0_refl t3))) t5 H9)) t4 (sym_eq T t4 (THead (Bind b) u t2) H7) H8))) | (pr0_comp u1 u2 H7 t4 t5 H8 k) \Rightarrow (\lambda (H9: (eq T (THead k u1 t4) (THead (Bind b) u t2))).(\lambda (H10: (eq T (THead k u2 t5) t5)).((let H11 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) \Rightarrow t])) (THead k u1 t4) (THead (Bind b) u t2) H9) in ((let H12 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead k u1 t4) (THead (Bind b) u t2) H9) in ((let H13 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u1 t4) (THead (Bind b) u t2) H9) in (eq_ind K (Bind b) (\lambda (k0: K).((eq T u1 u) \to ((eq T t4 t2) \to ((eq T (THead k0 u2 t5) t5) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3)))))))) (\lambda (H14: (eq T u1 u)).(eq_ind T u (\lambda (t: T).((eq T t4 t2) \to ((eq T (THead (Bind b) u2 t5) t5) \to ((pr0 t u2) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))) (\lambda (H15: (eq T t4 t2)).(eq_ind T t2 (\lambda (t: T).((eq T (THead (Bind b) u2 t5) t5) \to ((pr0 u u2) \to ((pr0 t t5) \to (ty3 g c2 t5 (THead (Bind b) u t3)))))) (\lambda (H16: (eq T (THead (Bind b) u2 t5) t5)).(eq_ind T (THead (Bind b) u2 t5) (\lambda (t: T).((pr0 u u2) \to ((pr0 t2 t5) \to (ty3 g c2 t (THead (Bind b) u t3))))) (\lambda (H17: (pr0 u u2)).(\lambda (H18: (pr0 t2 t5)).(ex_ind T (\lambda (t: T).(ty3 g (CHead c2 (Bind b) u) t4 t)) (ty3 g c2 (THead (Bind b) u2 t5) (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H19: (ty3 g (CHead c2 (Bind b) u) t4 x)).(ex_ind T (\lambda (t: T).(ty3 g (CHead c2 (Bind b) u2) t3 t)) (ty3 g c2 (THead (Bind b) u2 t5) (THead (Bind b) u t3)) (\lambda (x0: T).(\lambda (H20: (ty3 g (CHead c2 (Bind b) u2) t3 x0)).(ty3_conv g c2 (THead (Bind b) u t3) (THead (Bind b) u t4) (ty3_bind g c2 u t0 (H1 c2 H6 u (pr0_refl u)) b t3 t4 (H5 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t3 (pr0_refl t3)) x H19) (THead (Bind b) u2 t5) (THead (Bind b) u2 t3) (ty3_bind g c2 u2 t0 (H1 c2 H6 u2 H17) b t5 t3 (H3 (CHead c2 (Bind b) u2) (wcpr0_comp c c2 H6 u u2 H17 (Bind b)) t5 H18) x0 H20) (pc3_pr2_x c2 (THead (Bind b) u2 t3) (THead (Bind b) u t3) (pr2_head_1 c2 u u2 (pr2_free c2 u u2 H17) (Bind b) t3))))) (ty3_correct g (CHead c2 (Bind b) u2) t5 t3 (H3 (CHead c2 (Bind b) u2) (wcpr0_comp c c2 H6 u u2 H17 (Bind b)) t5 H18))))) (ty3_correct g (CHead c2 (Bind b) u) t3 t4 (H5 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t3 (pr0_refl t3)))))) t5 H16)) t4 (sym_eq T t4 t2 H15))) u1 (sym_eq T u1 u H14))) k (sym_eq K k (Bind b) H13))) H12)) H11)) H10 H7 H8))) | (pr0_beta u0 v1 v2 H7 t4 t5 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) (THead (Bind b) u t2))).(\lambda (H10: (eq T (THead (Bind Abbr) v2 t5) t5)).((let H11 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t4)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2) H9) in (False_ind ((eq T (THead (Bind Abbr) v2 t5) t5) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3))))) H11)) H10 H7 H8))) | (pr0_upsilon b0 H7 v1 v2 H8 u1 u2 H9 t4 t5 H10) \Rightarrow (\lambda (H11: (eq T (THead (Flat Appl) v1 (THead (Bind b0) u1 t4)) (THead (Bind b) u t2))).(\lambda (H12: (eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t5)).((let H13 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b0) u1 t4)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2) H11) in (False_ind ((eq T (THead (Bind b0) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t5) \to ((not (eq B b0 Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))) H13)) H12 H7 H8 H9 H10))) | (pr0_delta u1 u2 H7 t4 t5 H8 w H9) \Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr) u1 t4) (THead (Bind b) u t2))).(\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t5)).((let H12 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) \Rightarrow t])) (THead (Bind Abbr) u1 t4) (THead (Bind b) u t2) H10) in ((let H13 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead (Bind Abbr) u1 t4) (THead (Bind b) u t2) H10) in ((let H14 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow Abbr | (TLRef _) \Rightarrow Abbr | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow Abbr])])) (THead (Bind Abbr) u1 t4) (THead (Bind b) u t2) H10) in (eq_ind B Abbr (\lambda (b: B).((eq T u1 u) \to ((eq T t4 t2) \to ((eq T (THead (Bind Abbr) u2 w) t5) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))))) (\lambda (H15: (eq T u1 u)).(eq_ind T u (\lambda (t: T).((eq T t4 t2) \to ((eq T (THead (Bind Abbr) u2 w) t5) \to ((pr0 t u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ty3 g c2 t5 (THead (Bind Abbr) u t3)))))))) (\lambda (H16: (eq T t4 t2)).(eq_ind T t2 (\lambda (t: T).((eq T (THead (Bind Abbr) u2 w) t5) \to ((pr0 u u2) \to ((pr0 t t5) \to ((subst0 O u2 t5 w) \to (ty3 g c2 t5 (THead (Bind Abbr) u t3))))))) (\lambda (H17: (eq T (THead (Bind Abbr) u2 w) t5)).(eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).((pr0 u u2) \to ((pr0 t2 t5) \to ((subst0 O u2 t5 w) \to (ty3 g c2 t (THead (Bind Abbr) u t3)))))) (\lambda (H18: (pr0 u u2)).(\lambda (H19: (pr0 t2 t5)).(\lambda (H20: (subst0 O u2 t5 w)).(let H21 \def (eq_ind_r B b (\lambda (b: B).(\forall (c2: C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t2: T).((pr0 t3 t2) \to (ty3 g c2 t2 t4)))))) H5 Abbr H14) in (let H22 \def (eq_ind_r B b (\lambda (b: B).(ty3 g (CHead c (Bind b) u) t3 t4)) H4 Abbr H14) in (let H23 \def (eq_ind_r B b (\lambda (b: B).(\forall (c2: C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g c2 t4 t3)))))) H3 Abbr H14) in (let H24 \def (eq_ind_r B b (\lambda (b: B).(ty3 g (CHead c (Bind b) u) t2 t3)) H2 Abbr H14) in (ex_ind T (\lambda (t: T).(ty3 g (CHead c2 (Bind Abbr) u) t4 t)) (ty3 g c2 (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u t3)) (\lambda (x: T).(\lambda (H25: (ty3 g (CHead c2 (Bind Abbr) u) t4 x)).(ex_ind T (\lambda (t: T).(ty3 g (CHead c2 (Bind Abbr) u2) t3 t)) (ty3 g c2 (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (H26: (ty3 g (CHead c2 (Bind Abbr) u2) t3 x0)).(ty3_conv g c2 (THead (Bind Abbr) u t3) (THead (Bind Abbr) u t4) (ty3_bind g c2 u t0 (H1 c2 H6 u (pr0_refl u)) Abbr t3 t4 (H21 (CHead c2 (Bind Abbr) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind Abbr)) t3 (pr0_refl t3)) x H25) (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u2 t3) (ty3_bind g c2 u2 t0 (H1 c2 H6 u2 H18) Abbr w t3 (ty3_subst0 g (CHead c2 (Bind Abbr) u2) t5 t3 (H23 (CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H6 u u2 H18 (Bind Abbr)) t5 H19) c2 u2 O (getl_refl Abbr c2 u2) w H20) x0 H26) (pc3_pr2_x c2 (THead (Bind Abbr) u2 t3) (THead (Bind Abbr) u t3) (pr2_head_1 c2 u u2 (pr2_free c2 u u2 H18) (Bind Abbr) t3))))) (ty3_correct g (CHead c2 (Bind Abbr) u2) t5 t3 (H23 (CHead c2 (Bind Abbr) u2) (wcpr0_comp c c2 H6 u u2 H18 (Bind Abbr)) t5 H19))))) (ty3_correct g (CHead c2 (Bind Abbr) u) t3 t4 (H21 (CHead c2 (Bind Abbr) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind Abbr)) t3 (pr0_refl t3))))))))))) t5 H17)) t4 (sym_eq T t4 t2 H16))) u1 (sym_eq T u1 u H15))) b H14)) H13)) H12)) H11 H7 H8 H9))) | (pr0_zeta b0 H7 t4 t5 H8 u0) \Rightarrow (\lambda (H9: (eq T (THead (Bind b0) u0 (lift (S O) O t4)) (THead (Bind b) u t2))).(\lambda (H10: (eq T t5 t5)).((let H11 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t4) | (TLRef _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x (S O))) O t4) | (THead _ _ t) \Rightarrow t])) (THead (Bind b0) u0 (lift (S O) O t4)) (THead (Bind b) u t2) H9) in ((let H12 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead (Bind b0) u0 (lift (S O) O t4)) (THead (Bind b) u t2) H9) in ((let H13 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b0])])) (THead (Bind b0) u0 (lift (S O) O t4)) (THead (Bind b) u t2) H9) in (eq_ind B b (\lambda (b1: B).((eq T u0 u) \to ((eq T (lift (S O) O t4) t2) \to ((eq T t5 t5) \to ((not (eq B b1 Abst)) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3)))))))) (\lambda (H14: (eq T u0 u)).(eq_ind T u (\lambda (_: T).((eq T (lift (S O) O t4) t2) \to ((eq T t5 t5) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))) (\lambda (H15: (eq T (lift (S O) O t4) t2)).(eq_ind T (lift (S O) O t4) (\lambda (_: T).((eq T t5 t5) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3)))))) (\lambda (H16: (eq T t5 t5)).(eq_ind T t5 (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 t4 t) \to (ty3 g c2 t5 (THead (Bind b) u t3))))) (\lambda (H17: (not (eq B b Abst))).(\lambda (H18: (pr0 t4 t5)).(let H19 \def (eq_ind_r T t2 (\lambda (t: T).(\forall (c2: C).((wcpr0 (CHead c (Bind b) u) c2) \to (\forall (t2: T).((pr0 t t2) \to (ty3 g c2 t2 t3)))))) H3 (lift (S O) O t4) H15) in (let H20 \def (eq_ind_r T t2 (\lambda (t: T).(ty3 g (CHead c (Bind b) u) t t3)) H2 (lift (S O) O t4) H15) in (ex_ind T (\lambda (t: T).(ty3 g (CHead c2 (Bind b) u) t4 t)) (ty3 g c2 t5 (THead (Bind b) u t3)) (\lambda (x: T).(\lambda (H4: (ty3 g (CHead c2 (Bind b) u) t4 x)).(B_ind (\lambda (b: B).((not (eq B b Abst)) \to ((ty3 g (CHead c2 (Bind b) u) t3 t4) \to ((ty3 g (CHead c2 (Bind b) u) t4 x) \to ((ty3 g (CHead c2 (Bind b) u) (lift (S O) O t5) t3) \to (ty3 g c2 t5 (THead (Bind b) u t3))))))) (\lambda (H21: (not (eq B Abbr Abst))).(\lambda (H2: (ty3 g (CHead c2 (Bind Abbr) u) t3 t4)).(\lambda (H5: (ty3 g (CHead c2 (Bind Abbr) u) t4 x)).(\lambda (H22: (ty3 g (CHead c2 (Bind Abbr) u) (lift (S O) O t5) t3)).(let H \def (ty3_gen_cabbr g (CHead c2 (Bind Abbr) u) (lift (S O) O t5) t3 H22 c2 u O (getl_refl Abbr c2 u) (CHead c2 (Bind Abbr) u) (csubst1_refl O u (CHead c2 (Bind Abbr) u)) c2 (drop_drop (Bind Abbr) O c2 c2 (drop_refl c2) u)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(subst1 O u (lift (S O) O t5) (lift (S O) O y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 O u t3 (lift (S O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g c2 y1 y2))) (ty3 g c2 t5 (THead (Bind Abbr) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H0: (subst1 O u (lift (S O) O t5) (lift (S O) O x0))).(\lambda (H3: (subst1 O u t3 (lift (S O) O x1))).(\lambda (H23: (ty3 g c2 x0 x1)).(let H24 \def (eq_ind T x0 (\lambda (t: T).(ty3 g c2 t x1)) H23 t5 (lift_inj x0 t5 (S O) O (subst1_gen_lift_eq t5 u (lift (S O) O x0) (S O) O O (le_n O) (eq_ind_r nat (plus (S O) O) (\lambda (n: nat).(lt O n)) (le_n (plus (S O) O)) (plus O (S O)) (plus_comm O (S O))) H0))) in (ty3_conv g c2 (THead (Bind Abbr) u t3) (THead (Bind Abbr) u t4) (ty3_bind g c2 u t0 (H1 c2 H6 u (pr0_refl u)) Abbr t3 t4 H2 x H5) t5 x1 H24 (pc3_pr3_x c2 x1 (THead (Bind Abbr) u t3) (pr3_t (THead (Bind Abbr) u (lift (S O) O x1)) (THead (Bind Abbr) u t3) c2 (pr3_pr2 c2 (THead (Bind Abbr) u t3) (THead (Bind Abbr) u (lift (S O) O x1)) (pr2_free c2 (THead (Bind Abbr) u t3) (THead (Bind Abbr) u (lift (S O) O x1)) (pr0_delta1 u u (pr0_refl u) t3 t3 (pr0_refl t3) (lift (S O) O x1) H3))) x1 (pr3_pr2 c2 (THead (Bind Abbr) u (lift (S O) O x1)) x1 (pr2_free c2 (THead (Bind Abbr) u (lift (S O) O x1)) x1 (pr0_zeta Abbr H21 x1 x1 (pr0_refl x1) u)))))))))))) H)))))) (\lambda (H21: (not (eq B Abst Abst))).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) t3 t4)).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) t4 x)).(\lambda (_: (ty3 g (CHead c2 (Bind Abst) u) (lift (S O) O t5) t3)).(let H \def (match (H21 (refl_equal B Abst)) return (\lambda (_: ?).(ty3 g c2 t5 (THead (Bind Abst) u t3))) with []) in H))))) (\lambda (H21: (not (eq B Void Abst))).(\lambda (H2: (ty3 g (CHead c2 (Bind Void) u) t3 t4)).(\lambda (H5: (ty3 g (CHead c2 (Bind Void) u) t4 x)).(\lambda (H22: (ty3 g (CHead c2 (Bind Void) u) (lift (S O) O t5) t3)).(let H \def (ty3_gen_cvoid g (CHead c2 (Bind Void) u) (lift (S O) O t5) t3 H22 c2 u O (getl_refl Void c2 u) c2 (drop_drop (Bind Void) O c2 c2 (drop_refl c2) u)) in (ex3_2_ind T T (\lambda (y1: T).(\lambda (_: T).(eq T (lift (S O) O t5) (lift (S O) O y1)))) (\lambda (_: T).(\lambda (y2: T).(eq T t3 (lift (S O) O y2)))) (\lambda (y1: T).(\lambda (y2: T).(ty3 g c2 y1 y2))) (ty3 g c2 t5 (THead (Bind Void) u t3)) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H0: (eq T (lift (S O) O t5) (lift (S O) O x0))).(\lambda (H3: (eq T t3 (lift (S O) O x1))).(\lambda (H23: (ty3 g c2 x0 x1)).(let H24 \def (eq_ind T t3 (\lambda (t: T).(ty3 g (CHead c2 (Bind Void) u) t t4)) H2 (lift (S O) O x1) H3) in (eq_ind_r T (lift (S O) O x1) (\lambda (t: T).(ty3 g c2 t5 (THead (Bind Void) u t))) (let H25 \def (eq_ind_r T x0 (\lambda (t: T).(ty3 g c2 t x1)) H23 t5 (lift_inj t5 x0 (S O) O H0)) in (ty3_conv g c2 (THead (Bind Void) u (lift (S O) O x1)) (THead (Bind Void) u t4) (ty3_bind g c2 u t0 (H1 c2 H6 u (pr0_refl u)) Void (lift (S O) O x1) t4 H24 x H5) t5 x1 H25 (pc3_pr2_x c2 x1 (THead (Bind Void) u (lift (S O) O x1)) (pr2_free c2 (THead (Bind Void) u (lift (S O) O x1)) x1 (pr0_zeta Void H21 x1 x1 (pr0_refl x1) u))))) t3 H3))))))) H)))))) b H17 (H5 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t3 (pr0_refl t3)) H4 (H19 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) (lift (S O) O t5) (pr0_lift t4 t5 H18 (S O) O))))) (ty3_correct g (CHead c2 (Bind b) u) t3 t4 (H5 (CHead c2 (Bind b) u) (wcpr0_comp c c2 H6 u u (pr0_refl u) (Bind b)) t3 (pr0_refl t3)))))))) t5 (sym_eq T t5 t5 H16))) t2 H15)) u0 (sym_eq T u0 u H14))) b0 (sym_eq B b0 b H13))) H12)) H11)) H10 H7 H8))) | (pr0_epsilon t4 t5 H7 u0) \Rightarrow (\lambda (H8: (eq T (THead (Flat Cast) u0 t4) (THead (Bind b) u t2))).(\lambda (H9: (eq T t5 t5)).((let H10 \def (eq_ind T (THead (Flat Cast) u0 t4) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u t2) H8) in (False_ind ((eq T t5 t5) \to ((pr0 t4 t5) \to (ty3 g c2 t5 (THead (Bind b) u t3)))) H10)) H9 H7)))]) in (H8 (refl_equal T (THead (Bind b) u t2)) (refl_equal T t5)))))))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 w t2) \to (ty3 g c2 t2 u))))))).(\lambda (v: T).(\lambda (t0: T).(\lambda (H2: (ty3 g c v (THead (Bind Abst) u t0))).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 v t2) \to (ty3 g c2 t2 (THead (Bind Abst) u t0)))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t2: T).(\lambda (H5: (pr0 (THead (Flat Appl) w v) t2)).(let H6 \def (match H5 return (\lambda (_: ?).(\lambda (t: T).(\lambda (t1: T).((eq T t (THead (Flat Appl) w v)) \to ((eq T t1 t2) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) with [(pr0_refl t0) \Rightarrow (\lambda (H5: (eq T t0 (THead (Flat Appl) w v))).(\lambda (H6: (eq T t0 t2)).(eq_ind T (THead (Flat Appl) w v) (\lambda (t: T).((eq T t t2) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))) (\lambda (H7: (eq T (THead (Flat Appl) w v) t2)).(eq_ind T (THead (Flat Appl) w v) (\lambda (t: T).(ty3 g c2 t (THead (Flat Appl) w (THead (Bind Abst) u t0)))) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) v t0 (H3 c2 H4 v (pr0_refl v))) t2 H7)) t0 (sym_eq T t0 (THead (Flat Appl) w v) H5) H6))) | (pr0_comp u1 u2 H5 t1 t0 H6 k) \Rightarrow (\lambda (H7: (eq T (THead k u1 t1) (THead (Flat Appl) w v))).(\lambda (H8: (eq T (THead k u2 t0) t2)).((let H9 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t])) (THead k u1 t1) (THead (Flat Appl) w v) H7) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead k u1 t1) (THead (Flat Appl) w v) H7) in ((let H11 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u1 t1) (THead (Flat Appl) w v) H7) in (eq_ind K (Flat Appl) (\lambda (k0: K).((eq T u1 w) \to ((eq T t1 v) \to ((eq T (THead k0 u2 t0) t2) \to ((pr0 u1 u2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H12: (eq T u1 w)).(eq_ind T w (\lambda (t: T).((eq T t1 v) \to ((eq T (THead (Flat Appl) u2 t0) t2) \to ((pr0 t u2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) (\lambda (H13: (eq T t1 v)).(eq_ind T v (\lambda (t: T).((eq T (THead (Flat Appl) u2 t0) t2) \to ((pr0 w u2) \to ((pr0 t t0) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))))) (\lambda (H14: (eq T (THead (Flat Appl) u2 t0) t2)).(eq_ind T (THead (Flat Appl) u2 t0) (\lambda (t: T).((pr0 w u2) \to ((pr0 v t0) \to (ty3 g c2 t (THead (Flat Appl) w (THead (Bind Abst) u t0)))))) (\lambda (H15: (pr0 w u2)).(\lambda (H16: (pr0 v t0)).(ex_ind T (\lambda (t: T).(ty3 g c2 (THead (Bind Abst) u t0) t)) (ty3 g c2 (THead (Flat Appl) u2 t0) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H17: (ty3 g c2 (THead (Bind Abst) u t0) x)).(ex4_3_ind T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c2 (THead (Bind Abst) u t2) x)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c2 u t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c2 (Bind Abst) u) t2 t3)))) (ty3 g c2 (THead (Flat Appl) u2 t0) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H19: (ty3 g c2 u x1)).(\lambda (H20: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(\lambda (H21: (ty3 g (CHead c2 (Bind Abst) u) x0 x2)).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H19 Abst t0 x0 H20 x2 H21)) (THead (Flat Appl) u2 t0) (THead (Flat Appl) u2 (THead (Bind Abst) u t0)) (ty3_appl g c2 u2 u (H1 c2 H4 u2 H15) t0 t0 (H3 c2 H4 t0 H16)) (pc3_pr2_x c2 (THead (Flat Appl) u2 (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pr2_head_1 c2 w u2 (pr2_free c2 w u2 H15) (Flat Appl) (THead (Bind Abst) u t0))))))))))) (ty3_gen_bind g Abst c2 u t0 x H17)))) (ty3_correct g c2 v (THead (Bind Abst) u t0) (H3 c2 H4 v (pr0_refl v)))))) t2 H14)) t1 (sym_eq T t1 v H13))) u1 (sym_eq T u1 w H12))) k (sym_eq K k (Flat Appl) H11))) H10)) H9)) H8 H5 H6))) | (pr0_beta u0 v1 v2 H5 t1 t0 H6) \Rightarrow (\lambda (H7: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u0 t1)) (THead (Flat Appl) w v))).(\lambda (H8: (eq T (THead (Bind Abbr) v2 t0) t2)).((let H9 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow (THead (Bind Abst) u0 t1) | (TLRef _) \Rightarrow (THead (Bind Abst) u0 t1) | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t1)) (THead (Flat Appl) w v) H7) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind Abst) u0 t1)) (THead (Flat Appl) w v) H7) in (eq_ind T w (\lambda (t: T).((eq T (THead (Bind Abst) u0 t1) v) \to ((eq T (THead (Bind Abbr) v2 t0) t2) \to ((pr0 t v2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) (\lambda (H11: (eq T (THead (Bind Abst) u0 t1) v)).(eq_ind T (THead (Bind Abst) u0 t1) (\lambda (_: T).((eq T (THead (Bind Abbr) v2 t0) t2) \to ((pr0 w v2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))))) (\lambda (H12: (eq T (THead (Bind Abbr) v2 t0) t2)).(eq_ind T (THead (Bind Abbr) v2 t0) (\lambda (t: T).((pr0 w v2) \to ((pr0 t1 t0) \to (ty3 g c2 t (THead (Flat Appl) w (THead (Bind Abst) u t0)))))) (\lambda (H13: (pr0 w v2)).(\lambda (H14: (pr0 t1 t0)).(let H15 \def (eq_ind_r T v (\lambda (t: T).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to (ty3 g c2 t2 (THead (Bind Abst) u t0))))))) H3 (THead (Bind Abst) u0 t1) H11) in (let H16 \def (eq_ind_r T v (\lambda (t: T).(ty3 g c t (THead (Bind Abst) u t0))) H2 (THead (Bind Abst) u0 t1) H11) in (ex_ind T (\lambda (t: T).(ty3 g c2 (THead (Bind Abst) u t0) t)) (ty3 g c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H2: (ty3 g c2 (THead (Bind Abst) u t0) x)).(ex4_3_ind T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c2 (THead (Bind Abst) u t2) x)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c2 u t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c2 (Bind Abst) u) t2 t3)))) (ty3 g c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H17: (ty3 g c2 u x1)).(\lambda (H18: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(\lambda (H19: (ty3 g (CHead c2 (Bind Abst) u) x0 x2)).(ex4_3_ind T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c2 (THead (Bind Abst) u0 t2) (THead (Bind Abst) u t0))))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c2 u0 t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u0) t0 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c2 (Bind Abst) u0) t2 t3)))) (ty3 g c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H0: (pc3 c2 (THead (Bind Abst) u0 x3) (THead (Bind Abst) u t0))).(\lambda (H20: (ty3 g c2 u0 x4)).(\lambda (H21: (ty3 g (CHead c2 (Bind Abst) u0) t0 x3)).(\lambda (H22: (ty3 g (CHead c2 (Bind Abst) u0) x3 x5)).(and_ind (pc3 c2 u0 u) (\forall (b: B).(\forall (u: T).(pc3 (CHead c2 (Bind b) u) x3 t0))) (ty3 g c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (H23: (pc3 c2 u0 u)).(\lambda (H24: ((\forall (b: B).(\forall (u: T).(pc3 (CHead c2 (Bind b) u) x3 t0))))).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H17 Abst t0 x0 H18 x2 H19)) (THead (Bind Abbr) v2 t0) (THead (Bind Abbr) v2 x3) (ty3_bind g c2 v2 u (H1 c2 H4 v2 H13) Abbr t0 x3 (csub3_ty3_ld g c2 v2 u0 (ty3_conv g c2 u0 x4 H20 v2 u (H1 c2 H4 v2 H13) (pc3_s c2 u u0 H23)) t0 x3 H21) x5 (csub3_ty3_ld g c2 v2 u0 (ty3_conv g c2 u0 x4 H20 v2 u (H1 c2 H4 v2 H13) (pc3_s c2 u u0 H23)) x3 x5 H22)) (pc3_t (THead (Bind Abbr) v2 t0) c2 (THead (Bind Abbr) v2 x3) (pc3_head_2 c2 v2 x3 t0 (Bind Abbr) (H24 Abbr v2)) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc3_pr2_x c2 (THead (Bind Abbr) v2 t0) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pr2_free c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Bind Abbr) v2 t0) (pr0_beta u w v2 H13 t0 t0 (pr0_refl t0)))))))) (pc3_gen_abst c2 u0 u x3 t0 H0))))))))) (ty3_gen_bind g Abst c2 u0 t0 (THead (Bind Abst) u t0) (H15 c2 H4 (THead (Bind Abst) u0 t0) (pr0_comp u0 u0 (pr0_refl u0) t1 t0 H14 (Bind Abst)))))))))))) (ty3_gen_bind g Abst c2 u t0 x H2)))) (ty3_correct g c2 (THead (Bind Abst) u0 t1) (THead (Bind Abst) u t0) (H15 c2 H4 (THead (Bind Abst) u0 t1) (pr0_refl (THead (Bind Abst) u0 t1))))))))) t2 H12)) v H11)) v1 (sym_eq T v1 w H10))) H9)) H8 H5 H6))) | (pr0_upsilon b H5 v1 v2 H6 u1 u2 H7 t1 t0 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (THead (Flat Appl) w v))).(\lambda (H10: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) t2)).((let H11 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow (THead (Bind b) u1 t1) | (TLRef _) \Rightarrow (THead (Bind b) u1 t1) | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (THead (Flat Appl) w v) H9) in ((let H12 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v1 (THead (Bind b) u1 t1)) (THead (Flat Appl) w v) H9) in (eq_ind T w (\lambda (t: T).((eq T (THead (Bind b) u1 t1) v) \to ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) t2) \to ((not (eq B b Abst)) \to ((pr0 t v2) \to ((pr0 u1 u2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))))))) (\lambda (H13: (eq T (THead (Bind b) u1 t1) v)).(eq_ind T (THead (Bind b) u1 t1) (\lambda (_: T).((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) t2) \to ((not (eq B b Abst)) \to ((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))))))) (\lambda (H14: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) t2)).(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) (\lambda (t: T).((not (eq B b Abst)) \to ((pr0 w v2) \to ((pr0 u1 u2) \to ((pr0 t1 t0) \to (ty3 g c2 t (THead (Flat Appl) w (THead (Bind Abst) u t0)))))))) (\lambda (H15: (not (eq B b Abst))).(\lambda (H16: (pr0 w v2)).(\lambda (H17: (pr0 u1 u2)).(\lambda (H18: (pr0 t1 t0)).(let H19 \def (eq_ind_r T v (\lambda (t: T).(\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t t2) \to (ty3 g c2 t2 (THead (Bind Abst) u t0))))))) H3 (THead (Bind b) u1 t1) H13) in (let H20 \def (eq_ind_r T v (\lambda (t: T).(ty3 g c t (THead (Bind Abst) u t0))) H2 (THead (Bind b) u1 t1) H13) in (ex_ind T (\lambda (t: T).(ty3 g c2 (THead (Bind Abst) u t0) t)) (ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x: T).(\lambda (H2: (ty3 g c2 (THead (Bind Abst) u t0) x)).(let H3 \def H2 in (ex4_3_ind T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c2 (THead (Bind Abst) u t2) x)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c2 u t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind Abst) u) t0 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c2 (Bind Abst) u) t2 t3)))) (ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Bind Abst) u x0) x)).(\lambda (H22: (ty3 g c2 u x1)).(\lambda (H23: (ty3 g (CHead c2 (Bind Abst) u) t0 x0)).(\lambda (H24: (ty3 g (CHead c2 (Bind Abst) u) x0 x2)).(ex4_3_ind T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c2 (THead (Bind b) u2 t2) (THead (Bind Abst) u t0))))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c2 u2 t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) u2) t0 t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c2 (Bind b) u2) t2 t3)))) (ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x3: T).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H0: (pc3 c2 (THead (Bind b) u2 x3) (THead (Bind Abst) u t0))).(\lambda (H25: (ty3 g c2 u2 x4)).(\lambda (H26: (ty3 g (CHead c2 (Bind b) u2) t0 x3)).(\lambda (_: (ty3 g (CHead c2 (Bind b) u2) x3 x5)).(let H28 \def (eq_ind T (lift (S O) O (THead (Bind Abst) u t0)) (\lambda (t: T).(pc3 (CHead c2 (Bind b) u2) x3 t)) (pc3_gen_not_abst b H15 c2 x3 t0 u2 u H0) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u t0 (S O) O)) in (let H29 \def (eq_ind T (lift (S O) O (THead (Bind Abst) u t0)) (\lambda (t: T).(ty3 g (CHead c2 (Bind b) u2) t (lift (S O) O x))) (ty3_lift g c2 (THead (Bind Abst) u t0) x H2 (CHead c2 (Bind b) u2) O (S O) (drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u t0 (S O) O)) in (ex4_3_ind T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 (CHead c2 (Bind b) u2) (THead (Bind Abst) (lift (S O) O u) t2) (lift (S O) O x))))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S O) O u)) (lift (S O) (S O) t0) t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S O) O u)) t2 t3)))) (ty3 g c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) (THead (Flat Appl) w (THead (Bind Abst) u t0))) (\lambda (x6: T).(\lambda (x7: T).(\lambda (x8: T).(\lambda (_: (pc3 (CHead c2 (Bind b) u2) (THead (Bind Abst) (lift (S O) O u) x6) (lift (S O) O x))).(\lambda (H31: (ty3 g (CHead c2 (Bind b) u2) (lift (S O) O u) x7)).(\lambda (H32: (ty3 g (CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S O) O u)) (lift (S O) (S O) t0) x6)).(\lambda (H33: (ty3 g (CHead (CHead c2 (Bind b) u2) (Bind Abst) (lift (S O) O u)) x6 x8)).(ty3_conv g c2 (THead (Flat Appl) w (THead (Bind Abst) u t0)) (THead (Flat Appl) w (THead (Bind Abst) u x0)) (ty3_appl g c2 w u (H1 c2 H4 w (pr0_refl w)) (THead (Bind Abst) u t0) x0 (ty3_bind g c2 u x1 H22 Abst t0 x0 H23 x2 H24)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t0)) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)))) (ty3_bind g c2 u2 x4 H25 b (THead (Flat Appl) (lift (S O) O v2) t0) (THead (Flat Appl) (lift (S O) O v2) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0))) (ty3_appl g (CHead c2 (Bind b) u2) (lift (S O) O v2) (lift (S O) O u) (ty3_lift g c2 v2 u (H1 c2 H4 v2 H16) (CHead c2 (Bind b) u2) O (S O) (drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) t0 (lift (S O) (S O) t0) (ty3_conv g (CHead c2 (Bind b) u2) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (THead (Bind Abst) (lift (S O) O u) x6) (ty3_bind g (CHead c2 (Bind b) u2) (lift (S O) O u) x7 H31 Abst (lift (S O) (S O) t0) x6 H32 x8 H33) t0 x3 H26 H28)) (THead (Flat Appl) (lift (S O) O v2) (THead (Bind Abst) (lift (S O) O u) x6)) (ty3_appl g (CHead c2 (Bind b) u2) (lift (S O) O v2) (lift (S O) O u) (ty3_lift g c2 v2 u (H1 c2 H4 v2 H16) (CHead c2 (Bind b) u2) O (S O) (drop_drop (Bind b) O c2 c2 (drop_refl c2) u2)) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) x6 (ty3_bind g (CHead c2 (Bind b) u2) (lift (S O) O u) x7 H31 Abst (lift (S O) (S O) t0) x6 H32 x8 H33))) (eq_ind T (lift (S O) O (THead (Bind Abst) u t0)) (\lambda (t: T).(pc3 c2 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t)) (THead (Flat Appl) w (THead (Bind Abst) u t0)))) (pc3_pc1 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O (THead (Bind Abst) u t0)))) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc1_pr0_u2 (THead (Flat Appl) v2 (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u t0)))) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) (lift (S O) O (THead (Bind Abst) u t0)))) (pr0_upsilon b H15 v2 v2 (pr0_refl v2) u2 u2 (pr0_refl u2) (lift (S O) O (THead (Bind Abst) u t0)) (lift (S O) O (THead (Bind Abst) u t0)) (pr0_refl (lift (S O) O (THead (Bind Abst) u t0)))) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc1_s (THead (Flat Appl) v2 (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u t0)))) (THead (Flat Appl) w (THead (Bind Abst) u t0)) (pc1_head w v2 (pc1_pr0_r w v2 H16) (THead (Bind Abst) u t0) (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u t0))) (pc1_pr0_x (THead (Bind Abst) u t0) (THead (Bind b) u2 (lift (S O) O (THead (Bind Abst) u t0))) (pr0_zeta b H15 (THead (Bind Abst) u t0) (THead (Bind Abst) u t0) (pr0_refl (THead (Bind Abst) u t0)) u2)) (Flat Appl)))) c2) (THead (Bind Abst) (lift (S O) O u) (lift (S O) (S O) t0)) (lift_bind Abst u t0 (S O) O)))))))))) (ty3_gen_bind g Abst (CHead c2 (Bind b) u2) (lift (S O) O u) (lift (S O) (S O) t0) (lift (S O) O x) H29))))))))))) (ty3_gen_bind g b c2 u2 t0 (THead (Bind Abst) u t0) (H19 c2 H4 (THead (Bind b) u2 t0) (pr0_comp u1 u2 H17 t1 t0 H18 (Bind b)))))))))))) (ty3_gen_bind g Abst c2 u t0 x H3))))) (ty3_correct g c2 (THead (Bind b) u2 t0) (THead (Bind Abst) u t0) (H19 c2 H4 (THead (Bind b) u2 t0) (pr0_comp u1 u2 H17 t1 t0 H18 (Bind b))))))))))) t2 H14)) v H13)) v1 (sym_eq T v1 w H12))) H11)) H10 H5 H6 H7 H8))) | (pr0_delta u1 u2 H5 t1 t0 H6 w0 H7) \Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u1 t1) (THead (Flat Appl) w v))).(\lambda (H9: (eq T (THead (Bind Abbr) u2 w0) t2)).((let H10 \def (eq_ind T (THead (Bind Abbr) u1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) w v) H8) in (False_ind ((eq T (THead (Bind Abbr) u2 w0) t2) \to ((pr0 u1 u2) \to ((pr0 t1 t0) \to ((subst0 O u2 t0 w0) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))))) H10)) H9 H5 H6 H7))) | (pr0_zeta b H5 t1 t0 H6 u0) \Rightarrow (\lambda (H7: (eq T (THead (Bind b) u0 (lift (S O) O t1)) (THead (Flat Appl) w v))).(\lambda (H8: (eq T t0 t2)).((let H9 \def (eq_ind T (THead (Bind b) u0 (lift (S O) O t1)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) w v) H7) in (False_ind ((eq T t0 t2) \to ((not (eq B b Abst)) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0)))))) H9)) H8 H5 H6))) | (pr0_epsilon t1 t0 H5 u0) \Rightarrow (\lambda (H6: (eq T (THead (Flat Cast) u0 t1) (THead (Flat Appl) w v))).(\lambda (H7: (eq T t0 t2)).((let H8 \def (eq_ind T (THead (Flat Cast) u0 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) w v) H6) in (False_ind ((eq T t0 t2) \to ((pr0 t1 t0) \to (ty3 g c2 t2 (THead (Flat Appl) w (THead (Bind Abst) u t0))))) H8)) H7 H5)))]) in (H6 (refl_equal T (THead (Flat Appl) w v)) (refl_equal T t2)))))))))))))))) (\lambda (c: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g c t2 t3)).(\lambda (H1: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t4: T).((pr0 t2 t4) \to (ty3 g c2 t4 t3))))))).(\lambda (t0: T).(\lambda (_: (ty3 g c t3 t0)).(\lambda (H3: ((\forall (c2: C).((wcpr0 c c2) \to (\forall (t2: T).((pr0 t3 t2) \to (ty3 g c2 t2 t0))))))).(\lambda (c2: C).(\lambda (H4: (wcpr0 c c2)).(\lambda (t4: T).(\lambda (H5: (pr0 (THead (Flat Cast) t3 t2) t4)).(let H6 \def (match H5 return (\lambda (_: ?).(\lambda (t: T).(\lambda (t0: T).((eq T t (THead (Flat Cast) t3 t2)) \to ((eq T t0 t4) \to (ty3 g c2 t4 t3)))))) with [(pr0_refl t) \Rightarrow (\lambda (H5: (eq T t (THead (Flat Cast) t3 t2))).(\lambda (H6: (eq T t t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t0: T).((eq T t0 t4) \to (ty3 g c2 t4 t3))) (\lambda (H7: (eq T (THead (Flat Cast) t3 t2) t4)).(eq_ind T (THead (Flat Cast) t3 t2) (\lambda (t0: T).(ty3 g c2 t0 t3)) (ty3_cast g c2 t2 t3 (H1 c2 H4 t2 (pr0_refl t2)) t0 (H3 c2 H4 t3 (pr0_refl t3))) t4 H7)) t (sym_eq T t (THead (Flat Cast) t3 t2) H5) H6))) | (pr0_comp u1 u2 H5 t4 t5 H6 k) \Rightarrow (\lambda (H7: (eq T (THead k u1 t4) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T (THead k u2 t5) t4)).((let H9 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) \Rightarrow t])) (THead k u1 t4) (THead (Flat Cast) t3 t2) H7) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u1 | (TLRef _) \Rightarrow u1 | (THead _ t _) \Rightarrow t])) (THead k u1 t4) (THead (Flat Cast) t3 t2) H7) in ((let H11 \def (f_equal T K (\lambda (e: T).(match e return (\lambda (_: ?).K) with [(TSort _) \Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u1 t4) (THead (Flat Cast) t3 t2) H7) in (eq_ind K (Flat Cast) (\lambda (k0: K).((eq T u1 t3) \to ((eq T t4 t2) \to ((eq T (THead k0 u2 t5) t4) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ty3 g c2 t4 t3))))))) (\lambda (H12: (eq T u1 t3)).(eq_ind T t3 (\lambda (t: T).((eq T t4 t2) \to ((eq T (THead (Flat Cast) u2 t5) t4) \to ((pr0 t u2) \to ((pr0 t4 t5) \to (ty3 g c2 t4 t3)))))) (\lambda (H13: (eq T t4 t2)).(eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Cast) u2 t5) t4) \to ((pr0 t3 u2) \to ((pr0 t t5) \to (ty3 g c2 t4 t3))))) (\lambda (H14: (eq T (THead (Flat Cast) u2 t5) t4)).(eq_ind T (THead (Flat Cast) u2 t5) (\lambda (t: T).((pr0 t3 u2) \to ((pr0 t2 t5) \to (ty3 g c2 t t3)))) (\lambda (H15: (pr0 t3 u2)).(\lambda (H16: (pr0 t2 t5)).(ty3_conv g c2 t3 t0 (H3 c2 H4 t3 (pr0_refl t3)) (THead (Flat Cast) u2 t5) u2 (ty3_cast g c2 t5 u2 (ty3_conv g c2 u2 t0 (H3 c2 H4 u2 H15) t5 t3 (H1 c2 H4 t5 H16) (pc3_pr2_r c2 t3 u2 (pr2_free c2 t3 u2 H15))) t0 (H3 c2 H4 u2 H15)) (pc3_pr2_x c2 u2 t3 (pr2_free c2 t3 u2 H15))))) t4 H14)) t4 (sym_eq T t4 t2 H13))) u1 (sym_eq T u1 t3 H12))) k (sym_eq K k (Flat Cast) H11))) H10)) H9)) H8 H5 H6))) | (pr0_beta u v1 v2 H5 t4 t5 H6) \Rightarrow (\lambda (H7: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T (THead (Bind Abbr) v2 t5) t4)).((let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t4)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H7) in (False_ind ((eq T (THead (Bind Abbr) v2 t5) t4) \to ((pr0 v1 v2) \to ((pr0 t4 t5) \to (ty3 g c2 t4 t3)))) H9)) H8 H5 H6))) | (pr0_upsilon b H5 v1 v2 H6 u1 u2 H7 t4 t5 H8) \Rightarrow (\lambda (H9: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) (THead (Flat Cast) t3 t2))).(\lambda (H10: (eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t4)).((let H11 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t4)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H9) in (False_ind ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t5)) t4) \to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to (ty3 g c2 t4 t3)))))) H11)) H10 H5 H6 H7 H8))) | (pr0_delta u1 u2 H5 t4 t5 H6 w H7) \Rightarrow (\lambda (H8: (eq T (THead (Bind Abbr) u1 t4) (THead (Flat Cast) t3 t2))).(\lambda (H9: (eq T (THead (Bind Abbr) u2 w) t4)).((let H10 \def (eq_ind T (THead (Bind Abbr) u1 t4) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T (THead (Bind Abbr) u2 w) t4) \to ((pr0 u1 u2) \to ((pr0 t4 t5) \to ((subst0 O u2 t5 w) \to (ty3 g c2 t4 t3))))) H10)) H9 H5 H6 H7))) | (pr0_zeta b H5 t4 t5 H6 u) \Rightarrow (\lambda (H7: (eq T (THead (Bind b) u (lift (S O) O t4)) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T t5 t4)).((let H9 \def (eq_ind T (THead (Bind b) u (lift (S O) O t4)) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) t3 t2) H7) in (False_ind ((eq T t5 t4) \to ((not (eq B b Abst)) \to ((pr0 t4 t5) \to (ty3 g c2 t4 t3)))) H9)) H8 H5 H6))) | (pr0_epsilon t4 t5 H5 u) \Rightarrow (\lambda (H6: (eq T (THead (Flat Cast) u t4) (THead (Flat Cast) t3 t2))).(\lambda (H7: (eq T t5 t4)).((let H8 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) u t4) (THead (Flat Cast) t3 t2) H6) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u t4) (THead (Flat Cast) t3 t2) H6) in (eq_ind T t3 (\lambda (_: T).((eq T t4 t2) \to ((eq T t5 t4) \to ((pr0 t4 t5) \to (ty3 g c2 t4 t3))))) (\lambda (H10: (eq T t4 t2)).(eq_ind T t2 (\lambda (t: T).((eq T t5 t4) \to ((pr0 t t5) \to (ty3 g c2 t4 t3)))) (\lambda (H11: (eq T t5 t4)).(eq_ind T t4 (\lambda (t: T).((pr0 t2 t) \to (ty3 g c2 t4 t3))) (\lambda (H12: (pr0 t2 t4)).(H1 c2 H4 t4 H12)) t5 (sym_eq T t5 t4 H11))) t4 (sym_eq T t4 t2 H10))) u (sym_eq T u t3 H9))) H8)) H7 H5)))]) in (H6 (refl_equal T (THead (Flat Cast) t3 t2)) (refl_equal T t4))))))))))))))) c1 t1 t H))))).
 
 theorem ty3_sred_pr1:
  \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr1 t1 t2) \to (\forall (g: G).(\forall (t: T).((ty3 g c t1 t) \to (ty3 g c t2 t)))))))
@@ -3126,12 +3126,12 @@ theorem ty3_sconv:
 theorem ty3_tau0:
  \forall (g: G).(\forall (c: C).(\forall (u: T).(\forall (t1: T).((ty3 g c u t1) \to (\forall (t2: T).((tau0 g c u t2) \to (ty3 g c u t2)))))))
 \def
- \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H: (ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (_: T).(\forall (t2: T).((tau0 g c0 t t2) \to (ty3 g c0 t t2)))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: ((\forall (t3: T).((tau0 g c0 t2 t3) \to (ty3 g c0 t2 t3))))).(\lambda (u0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c0 u0 t3)).(\lambda (H3: ((\forall (t2: T).((tau0 g c0 u0 t2) \to (ty3 g c0 u0 t2))))).(\lambda (_: (pc3 c0 t3 t2)).(\lambda (t0: T).(\lambda (H5: (tau0 g c0 u0 t0)).(H3 t0 H5))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (t2: T).(\lambda (H0: (tau0 g c0 (TSort m) t2)).(let H1 \def (match H0 return (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c c0) \to ((eq T t (TSort m)) \to ((eq T t0 t2) \to (ty3 g c0 (TSort m) t2))))))) with [(tau0_sort c0 n) \Rightarrow (\lambda (H0: (eq C c0 c0)).(\lambda (H1: (eq T (TSort n) (TSort m))).(\lambda (H2: (eq T (TSort (next g n)) t2)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n) (TSort m)) \to ((eq T (TSort (next g n)) t2) \to (ty3 g c0 (TSort m) t2)))) (\lambda (H3: (eq T (TSort n) (TSort m))).(let H4 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort n) \Rightarrow n | (TLRef _) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TSort n) (TSort m) H3) in (eq_ind nat m (\lambda (n0: nat).((eq T (TSort (next g n0)) t2) \to (ty3 g c0 (TSort m) t2))) (\lambda (H5: (eq T (TSort (next g m)) t2)).(eq_ind T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 (TSort m) t)) (ty3_sort g c0 m) t2 H5)) n (sym_eq nat n m H4)))) c0 (sym_eq C c0 c0 H0) H1 H2)))) | (tau0_abbr c0 d v i H0 w H1) \Rightarrow (\lambda (H2: (eq C c0 c0)).(\lambda (H3: (eq T (TLRef i) (TSort m))).(\lambda (H4: (eq T (lift (S i) O w) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (TSort m)) \to ((eq T (lift (S i) O w) t2) \to ((getl i c (CHead d (Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (TSort m) t2)))))) (\lambda (H5: (eq T (TLRef i) (TSort m))).(let H6 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort m) H5) in (False_ind ((eq T (lift (S i) O w) t2) \to ((getl i c0 (CHead d (Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (TSort m) t2)))) H6))) c0 (sym_eq C c0 c0 H2) H3 H4 H0 H1)))) | (tau0_abst c0 d v i H0 w H1) \Rightarrow (\lambda (H2: (eq C c0 c0)).(\lambda (H3: (eq T (TLRef i) (TSort m))).(\lambda (H4: (eq T (lift (S i) O v) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (TSort m)) \to ((eq T (lift (S i) O v) t2) \to ((getl i c (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (TSort m) t2)))))) (\lambda (H5: (eq T (TLRef i) (TSort m))).(let H6 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort m) H5) in (False_ind ((eq T (lift (S i) O v) t2) \to ((getl i c0 (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (TSort m) t2)))) H6))) c0 (sym_eq C c0 c0 H2) H3 H4 H0 H1)))) | (tau0_bind b c0 v t1 t0 H0) \Rightarrow (\lambda (H1: (eq C c0 c0)).(\lambda (H2: (eq T (THead (Bind b) v t1) (TSort m))).(\lambda (H3: (eq T (THead (Bind b) v t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Bind b) v t1) (TSort m)) \to ((eq T (THead (Bind b) v t0) t2) \to ((tau0 g (CHead c (Bind b) v) t1 t0) \to (ty3 g c0 (TSort m) t2))))) (\lambda (H4: (eq T (THead (Bind b) v t1) (TSort m))).(let H5 \def (eq_ind T (THead (Bind b) v t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort m) H4) in (False_ind ((eq T (THead (Bind b) v t0) t2) \to ((tau0 g (CHead c0 (Bind b) v) t1 t0) \to (ty3 g c0 (TSort m) t2))) H5))) c0 (sym_eq C c0 c0 H1) H2 H3 H0)))) | (tau0_appl c0 v t1 t0 H0) \Rightarrow (\lambda (H1: (eq C c0 c0)).(\lambda (H2: (eq T (THead (Flat Appl) v t1) (TSort m))).(\lambda (H3: (eq T (THead (Flat Appl) v t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Appl) v t1) (TSort m)) \to ((eq T (THead (Flat Appl) v t0) t2) \to ((tau0 g c t1 t0) \to (ty3 g c0 (TSort m) t2))))) (\lambda (H4: (eq T (THead (Flat Appl) v t1) (TSort m))).(let H5 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort m) H4) in (False_ind ((eq T (THead (Flat Appl) v t0) t2) \to ((tau0 g c0 t1 t0) \to (ty3 g c0 (TSort m) t2))) H5))) c0 (sym_eq C c0 c0 H1) H2 H3 H0)))) | (tau0_cast c0 v1 v2 H0 t1 t0 H1) \Rightarrow (\lambda (H2: (eq C c0 c0)).(\lambda (H3: (eq T (THead (Flat Cast) v1 t1) (TSort m))).(\lambda (H4: (eq T (THead (Flat Cast) v2 t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Cast) v1 t1) (TSort m)) \to ((eq T (THead (Flat Cast) v2 t0) t2) \to ((tau0 g c v1 v2) \to ((tau0 g c t1 t0) \to (ty3 g c0 (TSort m) t2)))))) (\lambda (H5: (eq T (THead (Flat Cast) v1 t1) (TSort m))).(let H6 \def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort m) H5) in (False_ind ((eq T (THead (Flat Cast) v2 t0) t2) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0 t1 t0) \to (ty3 g c0 (TSort m) t2)))) H6))) c0 (sym_eq C c0 c0 H2) H3 H4 H0 H1))))]) in (H1 (refl_equal C c0) (refl_equal T (TSort m)) (refl_equal T t2))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall (t2: T).((tau0 g d u0 t2) \to (ty3 g d u0 t2))))).(\lambda (t2: T).(\lambda (H3: (tau0 g c0 (TLRef n) t2)).(let H4 \def (match H3 return (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c c0) \to ((eq T t (TLRef n)) \to ((eq T t0 t2) \to (ty3 g c0 (TLRef n) t2))))))) with [(tau0_sort c0 n0) \Rightarrow (\lambda (H3: (eq C c0 c0)).(\lambda (H4: (eq T (TSort n0) (TLRef n))).(\lambda (H5: (eq T (TSort (next g n0)) t2)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n0) (TLRef n)) \to ((eq T (TSort (next g n0)) t2) \to (ty3 g c0 (TLRef n) t2)))) (\lambda (H6: (eq T (TSort n0) (TLRef n))).(let H7 \def (eq_ind T (TSort n0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n) H6) in (False_ind ((eq T (TSort (next g n0)) t2) \to (ty3 g c0 (TLRef n) t2)) H7))) c0 (sym_eq C c0 c0 H3) H4 H5)))) | (tau0_abbr c0 d0 v i H3 w H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (TLRef i) (TLRef n))).(\lambda (H7: (eq T (lift (S i) O w) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O w) t2) \to ((getl i c (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2)))))) (\lambda (H8: (eq T (TLRef i) (TLRef n))).(let H9 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow i | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H8) in (eq_ind nat n (\lambda (n0: nat).((eq T (lift (S n0) O w) t2) \to ((getl n0 c0 (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H10: (eq T (lift (S n) O w) t2)).(eq_ind T (lift (S n) O w) (\lambda (t: T).((getl n c0 (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t)))) (\lambda (H11: (getl n c0 (CHead d0 (Bind Abbr) v))).(\lambda (H12: (tau0 g d0 v w)).(let H13 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c: C).(getl n c0 c)) H0 (CHead d0 (Bind Abbr) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind Abbr) v) H11)) in (let H14 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind Abbr) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind Abbr) v) H11)) in ((let H15 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind Abbr) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind Abbr) v) H11)) in (\lambda (H16: (eq C d d0)).(let H17 \def (eq_ind_r T v (\lambda (t: T).(getl n c0 (CHead d0 (Bind Abbr) t))) H13 u0 H15) in (let H18 \def (eq_ind_r T v (\lambda (t: T).(tau0 g d0 t w)) H12 u0 H15) in (let H19 \def (eq_ind_r C d0 (\lambda (c: C).(getl n c0 (CHead c (Bind Abbr) u0))) H17 d H16) in (let H20 \def (eq_ind_r C d0 (\lambda (c: C).(tau0 g c u0 w)) H18 d H16) in (ty3_abbr g n c0 d u0 H19 w (H2 w H20)))))))) H14))))) t2 H10)) i (sym_eq nat i n H9)))) c0 (sym_eq C c0 c0 H5) H6 H7 H3 H4)))) | (tau0_abst c0 d0 v i H3 w H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (TLRef i) (TLRef n))).(\lambda (H7: (eq T (lift (S i) O v) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O v) t2) \to ((getl i c (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2)))))) (\lambda (H8: (eq T (TLRef i) (TLRef n))).(let H9 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow i | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H8) in (eq_ind nat n (\lambda (n0: nat).((eq T (lift (S n0) O v) t2) \to ((getl n0 c0 (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H10: (eq T (lift (S n) O v) t2)).(eq_ind T (lift (S n) O v) (\lambda (t: T).((getl n c0 (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t)))) (\lambda (H11: (getl n c0 (CHead d0 (Bind Abst) v))).(\lambda (_: (tau0 g d0 v w)).(let H2 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c: C).(getl n c0 c)) H0 (CHead d0 (Bind Abst) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind Abst) v) H11)) in (let H13 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (ee: C).(match ee return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d0 (Bind Abst) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind Abst) v) H11)) in (False_ind (ty3 g c0 (TLRef n) (lift (S n) O v)) H13))))) t2 H10)) i (sym_eq nat i n H9)))) c0 (sym_eq C c0 c0 H5) H6 H7 H3 H4)))) | (tau0_bind b c0 v t1 t0 H3) \Rightarrow (\lambda (H4: (eq C c0 c0)).(\lambda (H5: (eq T (THead (Bind b) v t1) (TLRef n))).(\lambda (H6: (eq T (THead (Bind b) v t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Bind b) v t1) (TLRef n)) \to ((eq T (THead (Bind b) v t0) t2) \to ((tau0 g (CHead c (Bind b) v) t1 t0) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H7: (eq T (THead (Bind b) v t1) (TLRef n))).(let H8 \def (eq_ind T (THead (Bind b) v t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H7) in (False_ind ((eq T (THead (Bind b) v t0) t2) \to ((tau0 g (CHead c0 (Bind b) v) t1 t0) \to (ty3 g c0 (TLRef n) t2))) H8))) c0 (sym_eq C c0 c0 H4) H5 H6 H3)))) | (tau0_appl c0 v t1 t0 H3) \Rightarrow (\lambda (H4: (eq C c0 c0)).(\lambda (H5: (eq T (THead (Flat Appl) v t1) (TLRef n))).(\lambda (H6: (eq T (THead (Flat Appl) v t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Appl) v t1) (TLRef n)) \to ((eq T (THead (Flat Appl) v t0) t2) \to ((tau0 g c t1 t0) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H7: (eq T (THead (Flat Appl) v t1) (TLRef n))).(let H8 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H7) in (False_ind ((eq T (THead (Flat Appl) v t0) t2) \to ((tau0 g c0 t1 t0) \to (ty3 g c0 (TLRef n) t2))) H8))) c0 (sym_eq C c0 c0 H4) H5 H6 H3)))) | (tau0_cast c0 v1 v2 H3 t1 t0 H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (THead (Flat Cast) v1 t1) (TLRef n))).(\lambda (H7: (eq T (THead (Flat Cast) v2 t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Cast) v1 t1) (TLRef n)) \to ((eq T (THead (Flat Cast) v2 t0) t2) \to ((tau0 g c v1 v2) \to ((tau0 g c t1 t0) \to (ty3 g c0 (TLRef n) t2)))))) (\lambda (H8: (eq T (THead (Flat Cast) v1 t1) (TLRef n))).(let H9 \def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H8) in (False_ind ((eq T (THead (Flat Cast) v2 t0) t2) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0 t1 t0) \to (ty3 g c0 (TLRef n) t2)))) H9))) c0 (sym_eq C c0 c0 H5) H6 H7 H3 H4))))]) in (H4 (refl_equal C c0) (refl_equal T (TLRef n)) (refl_equal T t2))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) u0))).(\lambda (t: T).(\lambda (H1: (ty3 g d u0 t)).(\lambda (_: ((\forall (t2: T).((tau0 g d u0 t2) \to (ty3 g d u0 t2))))).(\lambda (t2: T).(\lambda (H3: (tau0 g c0 (TLRef n) t2)).(let H4 \def (match H3 return (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c c0) \to ((eq T t (TLRef n)) \to ((eq T t0 t2) \to (ty3 g c0 (TLRef n) t2))))))) with [(tau0_sort c0 n0) \Rightarrow (\lambda (H3: (eq C c0 c0)).(\lambda (H4: (eq T (TSort n0) (TLRef n))).(\lambda (H5: (eq T (TSort (next g n0)) t2)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n0) (TLRef n)) \to ((eq T (TSort (next g n0)) t2) \to (ty3 g c0 (TLRef n) t2)))) (\lambda (H6: (eq T (TSort n0) (TLRef n))).(let H7 \def (eq_ind T (TSort n0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n) H6) in (False_ind ((eq T (TSort (next g n0)) t2) \to (ty3 g c0 (TLRef n) t2)) H7))) c0 (sym_eq C c0 c0 H3) H4 H5)))) | (tau0_abbr c0 d0 v i H3 w H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (TLRef i) (TLRef n))).(\lambda (H7: (eq T (lift (S i) O w) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O w) t2) \to ((getl i c (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2)))))) (\lambda (H8: (eq T (TLRef i) (TLRef n))).(let H9 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow i | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H8) in (eq_ind nat n (\lambda (n0: nat).((eq T (lift (S n0) O w) t2) \to ((getl n0 c0 (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H10: (eq T (lift (S n) O w) t2)).(eq_ind T (lift (S n) O w) (\lambda (t: T).((getl n c0 (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t)))) (\lambda (H11: (getl n c0 (CHead d0 (Bind Abbr) v))).(\lambda (_: (tau0 g d0 v w)).(let H2 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (c: C).(getl n c0 c)) H0 (CHead d0 (Bind Abbr) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind Abbr) v) H11)) in (let H13 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (ee: C).(match ee return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d0 (Bind Abbr) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind Abbr) v) H11)) in (False_ind (ty3 g c0 (TLRef n) (lift (S n) O w)) H13))))) t2 H10)) i (sym_eq nat i n H9)))) c0 (sym_eq C c0 c0 H5) H6 H7 H3 H4)))) | (tau0_abst c0 d0 v i H3 w H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (TLRef i) (TLRef n))).(\lambda (H7: (eq T (lift (S i) O v) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O v) t2) \to ((getl i c (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2)))))) (\lambda (H8: (eq T (TLRef i) (TLRef n))).(let H9 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow i | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H8) in (eq_ind nat n (\lambda (n0: nat).((eq T (lift (S n0) O v) t2) \to ((getl n0 c0 (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H10: (eq T (lift (S n) O v) t2)).(eq_ind T (lift (S n) O v) (\lambda (t: T).((getl n c0 (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t)))) (\lambda (H11: (getl n c0 (CHead d0 (Bind Abst) v))).(\lambda (H12: (tau0 g d0 v w)).(let H2 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (c: C).(getl n c0 c)) H0 (CHead d0 (Bind Abst) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind Abst) v) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abst) u0) (CHead d0 (Bind Abst) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind Abst) v) H11)) in ((let H14 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u0) (CHead d0 (Bind Abst) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind Abst) v) H11)) in (\lambda (H15: (eq C d d0)).(let H16 \def (eq_ind_r T v (\lambda (t: T).(getl n c0 (CHead d0 (Bind Abst) t))) H2 u0 H14) in (let H17 \def (eq_ind_r T v (\lambda (t: T).(tau0 g d0 t w)) H12 u0 H14) in (eq_ind T u0 (\lambda (t: T).(ty3 g c0 (TLRef n) (lift (S n) O t))) (let H18 \def (eq_ind_r C d0 (\lambda (c: C).(getl n c0 (CHead c (Bind Abst) u0))) H16 d H15) in (let H19 \def (eq_ind_r C d0 (\lambda (c: C).(tau0 g c u0 w)) H17 d H15) in (ty3_abst g n c0 d u0 H18 t H1))) v H14))))) H13))))) t2 H10)) i (sym_eq nat i n H9)))) c0 (sym_eq C c0 c0 H5) H6 H7 H3 H4)))) | (tau0_bind b c0 v t1 t0 H3) \Rightarrow (\lambda (H4: (eq C c0 c0)).(\lambda (H5: (eq T (THead (Bind b) v t1) (TLRef n))).(\lambda (H6: (eq T (THead (Bind b) v t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Bind b) v t1) (TLRef n)) \to ((eq T (THead (Bind b) v t0) t2) \to ((tau0 g (CHead c (Bind b) v) t1 t0) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H7: (eq T (THead (Bind b) v t1) (TLRef n))).(let H8 \def (eq_ind T (THead (Bind b) v t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H7) in (False_ind ((eq T (THead (Bind b) v t0) t2) \to ((tau0 g (CHead c0 (Bind b) v) t1 t0) \to (ty3 g c0 (TLRef n) t2))) H8))) c0 (sym_eq C c0 c0 H4) H5 H6 H3)))) | (tau0_appl c0 v t1 t0 H3) \Rightarrow (\lambda (H4: (eq C c0 c0)).(\lambda (H5: (eq T (THead (Flat Appl) v t1) (TLRef n))).(\lambda (H6: (eq T (THead (Flat Appl) v t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Appl) v t1) (TLRef n)) \to ((eq T (THead (Flat Appl) v t0) t2) \to ((tau0 g c t1 t0) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H7: (eq T (THead (Flat Appl) v t1) (TLRef n))).(let H8 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H7) in (False_ind ((eq T (THead (Flat Appl) v t0) t2) \to ((tau0 g c0 t1 t0) \to (ty3 g c0 (TLRef n) t2))) H8))) c0 (sym_eq C c0 c0 H4) H5 H6 H3)))) | (tau0_cast c0 v1 v2 H3 t1 t0 H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (THead (Flat Cast) v1 t1) (TLRef n))).(\lambda (H7: (eq T (THead (Flat Cast) v2 t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Cast) v1 t1) (TLRef n)) \to ((eq T (THead (Flat Cast) v2 t0) t2) \to ((tau0 g c v1 v2) \to ((tau0 g c t1 t0) \to (ty3 g c0 (TLRef n) t2)))))) (\lambda (H8: (eq T (THead (Flat Cast) v1 t1) (TLRef n))).(let H9 \def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H8) in (False_ind ((eq T (THead (Flat Cast) v2 t0) t2) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0 t1 t0) \to (ty3 g c0 (TLRef n) t2)))) H9))) c0 (sym_eq C c0 c0 H5) H6 H7 H3 H4))))]) in (H4 (refl_equal C c0) (refl_equal T (TLRef n)) (refl_equal T t2))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u0 t)).(\lambda (_: ((\forall (t2: T).((tau0 g c0 u0 t2) \to (ty3 g c0 u0 t2))))).(\lambda (b: B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t2 t3)).(\lambda (H3: ((\forall (t3: T).((tau0 g (CHead c0 (Bind b) u0) t2 t3) \to (ty3 g (CHead c0 (Bind b) u0) t2 t3))))).(\lambda (t0: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t3 t0)).(\lambda (_: ((\forall (t2: T).((tau0 g (CHead c0 (Bind b) u0) t3 t2) \to (ty3 g (CHead c0 (Bind b) u0) t3 t2))))).(\lambda (t4: T).(\lambda (H6: (tau0 g c0 (THead (Bind b) u0 t2) t4)).(let H7 \def (match H6 return (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c c0) \to ((eq T t (THead (Bind b) u0 t2)) \to ((eq T t0 t4) \to (ty3 g c0 (THead (Bind b) u0 t2) t4))))))) with [(tau0_sort c0 n) \Rightarrow (\lambda (H6: (eq C c0 c0)).(\lambda (H7: (eq T (TSort n) (THead (Bind b) u0 t2))).(\lambda (H8: (eq T (TSort (next g n)) t4)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n) (THead (Bind b) u0 t2)) \to ((eq T (TSort (next g n)) t4) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))) (\lambda (H9: (eq T (TSort n) (THead (Bind b) u0 t2))).(let H10 \def (eq_ind T (TSort n) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u0 t2) H9) in (False_ind ((eq T (TSort (next g n)) t4) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)) H10))) c0 (sym_eq C c0 c0 H6) H7 H8)))) | (tau0_abbr c0 d v i H6 w H7) \Rightarrow (\lambda (H8: (eq C c0 c0)).(\lambda (H9: (eq T (TLRef i) (THead (Bind b) u0 t2))).(\lambda (H10: (eq T (lift (S i) O w) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (THead (Bind b) u0 t2)) \to ((eq T (lift (S i) O w) t4) \to ((getl i c (CHead d (Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))))) (\lambda (H11: (eq T (TLRef i) (THead (Bind b) u0 t2))).(let H12 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u0 t2) H11) in (False_ind ((eq T (lift (S i) O w) t4) \to ((getl i c0 (CHead d (Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))) H12))) c0 (sym_eq C c0 c0 H8) H9 H10 H6 H7)))) | (tau0_abst c0 d v i H6 w H7) \Rightarrow (\lambda (H8: (eq C c0 c0)).(\lambda (H9: (eq T (TLRef i) (THead (Bind b) u0 t2))).(\lambda (H10: (eq T (lift (S i) O v) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (THead (Bind b) u0 t2)) \to ((eq T (lift (S i) O v) t4) \to ((getl i c (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))))) (\lambda (H11: (eq T (TLRef i) (THead (Bind b) u0 t2))).(let H12 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u0 t2) H11) in (False_ind ((eq T (lift (S i) O v) t4) \to ((getl i c0 (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))) H12))) c0 (sym_eq C c0 c0 H8) H9 H10 H6 H7)))) | (tau0_bind b0 c0 v t4 t5 H6) \Rightarrow (\lambda (H7: (eq C c0 c0)).(\lambda (H8: (eq T (THead (Bind b0) v t4) (THead (Bind b) u0 t2))).(\lambda (H9: (eq T (THead (Bind b0) v t5) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Bind b0) v t4) (THead (Bind b) u0 t2)) \to ((eq T (THead (Bind b0) v t5) t4) \to ((tau0 g (CHead c (Bind b0) v) t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4))))) (\lambda (H10: (eq T (THead (Bind b0) v t4) (THead (Bind b) u0 t2))).(let H11 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) \Rightarrow t])) (THead (Bind b0) v t4) (THead (Bind b) u0 t2) H10) in ((let H12 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t _) \Rightarrow t])) (THead (Bind b0) v t4) (THead (Bind b) u0 t2) H10) in ((let H13 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b0])])) (THead (Bind b0) v t4) (THead (Bind b) u0 t2) H10) in (eq_ind B b (\lambda (b1: B).((eq T v u0) \to ((eq T t4 t2) \to ((eq T (THead (Bind b1) v t5) t4) \to ((tau0 g (CHead c0 (Bind b1) v) t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))))) (\lambda (H14: (eq T v u0)).(eq_ind T u0 (\lambda (t: T).((eq T t4 t2) \to ((eq T (THead (Bind b) t t5) t4) \to ((tau0 g (CHead c0 (Bind b) t) t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4))))) (\lambda (H15: (eq T t4 t2)).(eq_ind T t2 (\lambda (t: T).((eq T (THead (Bind b) u0 t5) t4) \to ((tau0 g (CHead c0 (Bind b) u0) t t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))) (\lambda (H16: (eq T (THead (Bind b) u0 t5) t4)).(eq_ind T (THead (Bind b) u0 t5) (\lambda (t: T).((tau0 g (CHead c0 (Bind b) u0) t2 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t))) (\lambda (H17: (tau0 g (CHead c0 (Bind b) u0) t2 t5)).(let H_y \def (H3 t5 H17) in (ex_ind T (\lambda (t: T).(ty3 g (CHead c0 (Bind b) u0) t5 t)) (ty3 g c0 (THead (Bind b) u0 t2) (THead (Bind b) u0 t5)) (\lambda (x: T).(\lambda (H1: (ty3 g (CHead c0 (Bind b) u0) t5 x)).(ty3_bind g c0 u0 t H0 b t2 t5 H_y x H1))) (ty3_correct g (CHead c0 (Bind b) u0) t2 t5 H_y)))) t4 H16)) t4 (sym_eq T t4 t2 H15))) v (sym_eq T v u0 H14))) b0 (sym_eq B b0 b H13))) H12)) H11))) c0 (sym_eq C c0 c0 H7) H8 H9 H6)))) | (tau0_appl c0 v t4 t5 H6) \Rightarrow (\lambda (H7: (eq C c0 c0)).(\lambda (H8: (eq T (THead (Flat Appl) v t4) (THead (Bind b) u0 t2))).(\lambda (H9: (eq T (THead (Flat Appl) v t5) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Appl) v t4) (THead (Bind b) u0 t2)) \to ((eq T (THead (Flat Appl) v t5) t4) \to ((tau0 g c t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4))))) (\lambda (H10: (eq T (THead (Flat Appl) v t4) (THead (Bind b) u0 t2))).(let H11 \def (eq_ind T (THead (Flat Appl) v t4) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 t2) H10) in (False_ind ((eq T (THead (Flat Appl) v t5) t4) \to ((tau0 g c0 t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4))) H11))) c0 (sym_eq C c0 c0 H7) H8 H9 H6)))) | (tau0_cast c0 v1 v2 H6 t4 t5 H7) \Rightarrow (\lambda (H8: (eq C c0 c0)).(\lambda (H9: (eq T (THead (Flat Cast) v1 t4) (THead (Bind b) u0 t2))).(\lambda (H10: (eq T (THead (Flat Cast) v2 t5) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Cast) v1 t4) (THead (Bind b) u0 t2)) \to ((eq T (THead (Flat Cast) v2 t5) t4) \to ((tau0 g c v1 v2) \to ((tau0 g c t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))))) (\lambda (H11: (eq T (THead (Flat Cast) v1 t4) (THead (Bind b) u0 t2))).(let H12 \def (eq_ind T (THead (Flat Cast) v1 t4) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 t2) H11) in (False_ind ((eq T (THead (Flat Cast) v2 t5) t4) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0 t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))) H12))) c0 (sym_eq C c0 c0 H8) H9 H10 H6 H7))))]) in (H7 (refl_equal C c0) (refl_equal T (THead (Bind b) u0 t2)) (refl_equal T t4)))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda (H0: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2: T).((tau0 g c0 w t2) \to (ty3 g c0 w t2))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v (THead (Bind Abst) u0 t))).(\lambda (H3: ((\forall (t2: T).((tau0 g c0 v t2) \to (ty3 g c0 v t2))))).(\lambda (t2: T).(\lambda (H4: (tau0 g c0 (THead (Flat Appl) w v) t2)).(let H5 \def (match H4 return (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c c0) \to ((eq T t (THead (Flat Appl) w v)) \to ((eq T t0 t2) \to (ty3 g c0 (THead (Flat Appl) w v) t2))))))) with [(tau0_sort c0 n) \Rightarrow (\lambda (H4: (eq C c0 c0)).(\lambda (H5: (eq T (TSort n) (THead (Flat Appl) w v))).(\lambda (H6: (eq T (TSort (next g n)) t2)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n) (THead (Flat Appl) w v)) \to ((eq T (TSort (next g n)) t2) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))) (\lambda (H7: (eq T (TSort n) (THead (Flat Appl) w v))).(let H8 \def (eq_ind T (TSort n) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w v) H7) in (False_ind ((eq T (TSort (next g n)) t2) \to (ty3 g c0 (THead (Flat Appl) w v) t2)) H8))) c0 (sym_eq C c0 c0 H4) H5 H6)))) | (tau0_abbr c0 d v0 i H4 w0 H5) \Rightarrow (\lambda (H6: (eq C c0 c0)).(\lambda (H7: (eq T (TLRef i) (THead (Flat Appl) w v))).(\lambda (H8: (eq T (lift (S i) O w0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (THead (Flat Appl) w v)) \to ((eq T (lift (S i) O w0) t2) \to ((getl i c (CHead d (Bind Abbr) v0)) \to ((tau0 g d v0 w0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))))) (\lambda (H9: (eq T (TLRef i) (THead (Flat Appl) w v))).(let H10 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w v) H9) in (False_ind ((eq T (lift (S i) O w0) t2) \to ((getl i c0 (CHead d (Bind Abbr) v0)) \to ((tau0 g d v0 w0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))) H10))) c0 (sym_eq C c0 c0 H6) H7 H8 H4 H5)))) | (tau0_abst c0 d v0 i H4 w0 H5) \Rightarrow (\lambda (H6: (eq C c0 c0)).(\lambda (H7: (eq T (TLRef i) (THead (Flat Appl) w v))).(\lambda (H8: (eq T (lift (S i) O v0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (THead (Flat Appl) w v)) \to ((eq T (lift (S i) O v0) t2) \to ((getl i c (CHead d (Bind Abst) v0)) \to ((tau0 g d v0 w0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))))) (\lambda (H9: (eq T (TLRef i) (THead (Flat Appl) w v))).(let H10 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w v) H9) in (False_ind ((eq T (lift (S i) O v0) t2) \to ((getl i c0 (CHead d (Bind Abst) v0)) \to ((tau0 g d v0 w0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))) H10))) c0 (sym_eq C c0 c0 H6) H7 H8 H4 H5)))) | (tau0_bind b c0 v0 t1 t0 H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (THead (Bind b) v0 t1) (THead (Flat Appl) w v))).(\lambda (H7: (eq T (THead (Bind b) v0 t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Bind b) v0 t1) (THead (Flat Appl) w v)) \to ((eq T (THead (Bind b) v0 t0) t2) \to ((tau0 g (CHead c (Bind b) v0) t1 t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2))))) (\lambda (H8: (eq T (THead (Bind b) v0 t1) (THead (Flat Appl) w v))).(let H9 \def (eq_ind T (THead (Bind b) v0 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) w v) H8) in (False_ind ((eq T (THead (Bind b) v0 t0) t2) \to ((tau0 g (CHead c0 (Bind b) v0) t1 t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2))) H9))) c0 (sym_eq C c0 c0 H5) H6 H7 H4)))) | (tau0_appl c0 v0 t1 t0 H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (THead (Flat Appl) v0 t1) (THead (Flat Appl) w v))).(\lambda (H7: (eq T (THead (Flat Appl) v0 t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Appl) v0 t1) (THead (Flat Appl) w v)) \to ((eq T (THead (Flat Appl) v0 t0) t2) \to ((tau0 g c t1 t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2))))) (\lambda (H8: (eq T (THead (Flat Appl) v0 t1) (THead (Flat Appl) w v))).(let H9 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v0 t1) (THead (Flat Appl) w v) H8) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v0 t1) (THead (Flat Appl) w v) H8) in (eq_ind T w (\lambda (t: T).((eq T t1 v) \to ((eq T (THead (Flat Appl) t t0) t2) \to ((tau0 g c0 t1 t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2))))) (\lambda (H11: (eq T t1 v)).(eq_ind T v (\lambda (t: T).((eq T (THead (Flat Appl) w t0) t2) \to ((tau0 g c0 t t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))) (\lambda (H12: (eq T (THead (Flat Appl) w t0) t2)).(eq_ind T (THead (Flat Appl) w t0) (\lambda (t: T).((tau0 g c0 v t0) \to (ty3 g c0 (THead (Flat Appl) w v) t))) (\lambda (H13: (tau0 g c0 v t0)).(let H_y \def (H3 t0 H13) in (let H1 \def (ty3_unique g c0 v t0 H_y (THead (Bind Abst) u0 t) H2) in (ex_ind T (\lambda (t: T).(ty3 g c0 t0 t)) (ty3 g c0 (THead (Flat Appl) w v) (THead (Flat Appl) w t0)) (\lambda (x: T).(\lambda (H3: (ty3 g c0 t0 x)).(ex_ind T (\lambda (t: T).(ty3 g c0 u0 t)) (ty3 g c0 (THead (Flat Appl) w v) (THead (Flat Appl) w t0)) (\lambda (x0: T).(\lambda (_: (ty3 g c0 u0 x0)).(ex_ind T (\lambda (t2: T).(ty3 g c0 (THead (Bind Abst) u0 t) t2)) (ty3 g c0 (THead (Flat Appl) w v) (THead (Flat Appl) w t0)) (\lambda (x1: T).(\lambda (H15: (ty3 g c0 (THead (Bind Abst) u0 t) x1)).(ex4_3_ind T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind Abst) u0 t2) x1)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u0 t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind Abst) u0) t t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c0 (Bind Abst) u0) t2 t3)))) (ty3 g c0 (THead (Flat Appl) w v) (THead (Flat Appl) w t0)) (\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (_: (pc3 c0 (THead (Bind Abst) u0 x2) x1)).(\lambda (H17: (ty3 g c0 u0 x3)).(\lambda (H18: (ty3 g (CHead c0 (Bind Abst) u0) t x2)).(\lambda (H19: (ty3 g (CHead c0 (Bind Abst) u0) x2 x4)).(ty3_conv g c0 (THead (Flat Appl) w t0) (THead (Flat Appl) w (THead (Bind Abst) u0 x2)) (ty3_appl g c0 w u0 H0 t0 x2 (ty3_sconv g c0 t0 x H3 (THead (Bind Abst) u0 t) (THead (Bind Abst) u0 x2) (ty3_bind g c0 u0 x3 H17 Abst t x2 H18 x4 H19) H1)) (THead (Flat Appl) w v) (THead (Flat Appl) w (THead (Bind Abst) u0 t)) (ty3_appl g c0 w u0 H0 v t H2) (pc3_s c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t)) (THead (Flat Appl) w t0) (pc3_thin_dx c0 t0 (THead (Bind Abst) u0 t) H1 w Appl)))))))))) (ty3_gen_bind g Abst c0 u0 t x1 H15)))) (ty3_correct g c0 v (THead (Bind Abst) u0 t) H2)))) (ty3_correct g c0 w u0 H0)))) (ty3_correct g c0 v t0 H_y))))) t2 H12)) t1 (sym_eq T t1 v H11))) v0 (sym_eq T v0 w H10))) H9))) c0 (sym_eq C c0 c0 H5) H6 H7 H4)))) | (tau0_cast c0 v1 v2 H4 t1 t0 H5) \Rightarrow (\lambda (H6: (eq C c0 c0)).(\lambda (H7: (eq T (THead (Flat Cast) v1 t1) (THead (Flat Appl) w v))).(\lambda (H8: (eq T (THead (Flat Cast) v2 t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Cast) v1 t1) (THead (Flat Appl) w v)) \to ((eq T (THead (Flat Cast) v2 t0) t2) \to ((tau0 g c v1 v2) \to ((tau0 g c t1 t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))))) (\lambda (H9: (eq T (THead (Flat Cast) v1 t1) (THead (Flat Appl) w v))).(let H10 \def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) w v) H9) in (False_ind ((eq T (THead (Flat Cast) v2 t0) t2) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0 t1 t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))) H10))) c0 (sym_eq C c0 c0 H6) H7 H8 H4 H5))))]) in (H5 (refl_equal C c0) (refl_equal T (THead (Flat Appl) w v)) (refl_equal T t2)))))))))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: (ty3 g c0 t2 t3)).(\lambda (H1: ((\forall (t3: T).((tau0 g c0 t2 t3) \to (ty3 g c0 t2 t3))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t3 t0)).(\lambda (H3: ((\forall (t2: T).((tau0 g c0 t3 t2) \to (ty3 g c0 t3 t2))))).(\lambda (t4: T).(\lambda (H4: (tau0 g c0 (THead (Flat Cast) t3 t2) t4)).(let H5 \def (match H4 return (\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c c0) \to ((eq T t (THead (Flat Cast) t3 t2)) \to ((eq T t0 t4) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4))))))) with [(tau0_sort c0 n) \Rightarrow (\lambda (H4: (eq C c0 c0)).(\lambda (H5: (eq T (TSort n) (THead (Flat Cast) t3 t2))).(\lambda (H6: (eq T (TSort (next g n)) t4)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n) (THead (Flat Cast) t3 t2)) \to ((eq T (TSort (next g n)) t4) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))) (\lambda (H7: (eq T (TSort n) (THead (Flat Cast) t3 t2))).(let H8 \def (eq_ind T (TSort n) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t3 t2) H7) in (False_ind ((eq T (TSort (next g n)) t4) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)) H8))) c0 (sym_eq C c0 c0 H4) H5 H6)))) | (tau0_abbr c0 d v i H4 w H5) \Rightarrow (\lambda (H6: (eq C c0 c0)).(\lambda (H7: (eq T (TLRef i) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T (lift (S i) O w) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (THead (Flat Cast) t3 t2)) \to ((eq T (lift (S i) O w) t4) \to ((getl i c (CHead d (Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))))) (\lambda (H9: (eq T (TLRef i) (THead (Flat Cast) t3 t2))).(let H10 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t3 t2) H9) in (False_ind ((eq T (lift (S i) O w) t4) \to ((getl i c0 (CHead d (Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))) H10))) c0 (sym_eq C c0 c0 H6) H7 H8 H4 H5)))) | (tau0_abst c0 d v i H4 w H5) \Rightarrow (\lambda (H6: (eq C c0 c0)).(\lambda (H7: (eq T (TLRef i) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T (lift (S i) O v) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (THead (Flat Cast) t3 t2)) \to ((eq T (lift (S i) O v) t4) \to ((getl i c (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))))) (\lambda (H9: (eq T (TLRef i) (THead (Flat Cast) t3 t2))).(let H10 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t3 t2) H9) in (False_ind ((eq T (lift (S i) O v) t4) \to ((getl i c0 (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))) H10))) c0 (sym_eq C c0 c0 H6) H7 H8 H4 H5)))) | (tau0_bind b c0 v t4 t5 H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (THead (Bind b) v t4) (THead (Flat Cast) t3 t2))).(\lambda (H7: (eq T (THead (Bind b) v t5) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Bind b) v t4) (THead (Flat Cast) t3 t2)) \to ((eq T (THead (Bind b) v t5) t4) \to ((tau0 g (CHead c (Bind b) v) t4 t5) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4))))) (\lambda (H8: (eq T (THead (Bind b) v t4) (THead (Flat Cast) t3 t2))).(let H9 \def (eq_ind T (THead (Bind b) v t4) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T (THead (Bind b) v t5) t4) \to ((tau0 g (CHead c0 (Bind b) v) t4 t5) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4))) H9))) c0 (sym_eq C c0 c0 H5) H6 H7 H4)))) | (tau0_appl c0 v t4 t5 H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (THead (Flat Appl) v t4) (THead (Flat Cast) t3 t2))).(\lambda (H7: (eq T (THead (Flat Appl) v t5) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Appl) v t4) (THead (Flat Cast) t3 t2)) \to ((eq T (THead (Flat Appl) v t5) t4) \to ((tau0 g c t4 t5) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4))))) (\lambda (H8: (eq T (THead (Flat Appl) v t4) (THead (Flat Cast) t3 t2))).(let H9 \def (eq_ind T (THead (Flat Appl) v t4) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T (THead (Flat Appl) v t5) t4) \to ((tau0 g c0 t4 t5) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4))) H9))) c0 (sym_eq C c0 c0 H5) H6 H7 H4)))) | (tau0_cast c0 v1 v2 H4 t4 t5 H5) \Rightarrow (\lambda (H6: (eq C c0 c0)).(\lambda (H7: (eq T (THead (Flat Cast) v1 t4) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T (THead (Flat Cast) v2 t5) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Cast) v1 t4) (THead (Flat Cast) t3 t2)) \to ((eq T (THead (Flat Cast) v2 t5) t4) \to ((tau0 g c v1 v2) \to ((tau0 g c t4 t5) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))))) (\lambda (H9: (eq T (THead (Flat Cast) v1 t4) (THead (Flat Cast) t3 t2))).(let H10 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) v1 t4) (THead (Flat Cast) t3 t2) H9) in ((let H11 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) v1 t4) (THead (Flat Cast) t3 t2) H9) in (eq_ind T t3 (\lambda (t: T).((eq T t4 t2) \to ((eq T (THead (Flat Cast) v2 t5) t4) \to ((tau0 g c0 t v2) \to ((tau0 g c0 t4 t5) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))))) (\lambda (H12: (eq T t4 t2)).(eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Cast) v2 t5) t4) \to ((tau0 g c0 t3 v2) \to ((tau0 g c0 t t5) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4))))) (\lambda (H13: (eq T (THead (Flat Cast) v2 t5) t4)).(eq_ind T (THead (Flat Cast) v2 t5) (\lambda (t: T).((tau0 g c0 t3 v2) \to ((tau0 g c0 t2 t5) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t)))) (\lambda (H14: (tau0 g c0 t3 v2)).(\lambda (H15: (tau0 g c0 t2 t5)).(let H_y \def (H1 t5 H15) in (let H_y0 \def (H3 v2 H14) in (let H3 \def (ty3_unique g c0 t2 t5 H_y t3 H0) in (ex_ind T (\lambda (t: T).(ty3 g c0 v2 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) v2 t5)) (\lambda (x: T).(\lambda (H16: (ty3 g c0 v2 x)).(ex_ind T (\lambda (t: T).(ty3 g c0 t5 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) v2 t5)) (\lambda (x0: T).(\lambda (H17: (ty3 g c0 t5 x0)).(ty3_conv g c0 (THead (Flat Cast) v2 t5) v2 (ty3_cast g c0 t5 v2 (ty3_sconv g c0 t5 x0 H17 t3 v2 H_y0 H3) x H16) (THead (Flat Cast) t3 t2) t3 (ty3_cast g c0 t2 t3 H0 v2 H_y0) (pc3_s c0 t3 (THead (Flat Cast) v2 t5) (pc3_pr2_u c0 t5 (THead (Flat Cast) v2 t5) (pr2_free c0 (THead (Flat Cast) v2 t5) t5 (pr0_epsilon t5 t5 (pr0_refl t5) v2)) t3 H3))))) (ty3_correct g c0 t2 t5 H_y)))) (ty3_correct g c0 t3 v2 H_y0))))))) t4 H13)) t4 (sym_eq T t4 t2 H12))) v1 (sym_eq T v1 t3 H11))) H10))) c0 (sym_eq C c0 c0 H6) H7 H8 H4 H5))))]) in (H5 (refl_equal C c0) (refl_equal T (THead (Flat Cast) t3 t2)) (refl_equal T t4))))))))))))) c u t1 H))))).
+ \lambda (g: G).(\lambda (c: C).(\lambda (u: T).(\lambda (t1: T).(\lambda (H: (ty3 g c u t1)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (_: T).(\forall (t2: T).((tau0 g c0 t t2) \to (ty3 g c0 t t2)))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t2 t)).(\lambda (_: ((\forall (t3: T).((tau0 g c0 t2 t3) \to (ty3 g c0 t2 t3))))).(\lambda (u0: T).(\lambda (t3: T).(\lambda (_: (ty3 g c0 u0 t3)).(\lambda (H3: ((\forall (t2: T).((tau0 g c0 u0 t2) \to (ty3 g c0 u0 t2))))).(\lambda (_: (pc3 c0 t3 t2)).(\lambda (t0: T).(\lambda (H5: (tau0 g c0 u0 t0)).(H3 t0 H5))))))))))))) (\lambda (c0: C).(\lambda (m: nat).(\lambda (t2: T).(\lambda (H0: (tau0 g c0 (TSort m) t2)).(let H1 \def (match H0 return (\lambda (_: ?).(\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c c0) \to ((eq T t (TSort m)) \to ((eq T t0 t2) \to (ty3 g c0 (TSort m) t2)))))))) with [(tau0_sort c0 n) \Rightarrow (\lambda (H0: (eq C c0 c0)).(\lambda (H1: (eq T (TSort n) (TSort m))).(\lambda (H2: (eq T (TSort (next g n)) t2)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n) (TSort m)) \to ((eq T (TSort (next g n)) t2) \to (ty3 g c0 (TSort m) t2)))) (\lambda (H3: (eq T (TSort n) (TSort m))).(let H4 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort n) \Rightarrow n | (TLRef _) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TSort n) (TSort m) H3) in (eq_ind nat m (\lambda (n0: nat).((eq T (TSort (next g n0)) t2) \to (ty3 g c0 (TSort m) t2))) (\lambda (H5: (eq T (TSort (next g m)) t2)).(eq_ind T (TSort (next g m)) (\lambda (t: T).(ty3 g c0 (TSort m) t)) (ty3_sort g c0 m) t2 H5)) n (sym_eq nat n m H4)))) c0 (sym_eq C c0 c0 H0) H1 H2)))) | (tau0_abbr c0 d v i H0 w H1) \Rightarrow (\lambda (H2: (eq C c0 c0)).(\lambda (H3: (eq T (TLRef i) (TSort m))).(\lambda (H4: (eq T (lift (S i) O w) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (TSort m)) \to ((eq T (lift (S i) O w) t2) \to ((getl i c (CHead d (Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (TSort m) t2)))))) (\lambda (H5: (eq T (TLRef i) (TSort m))).(let H6 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort m) H5) in (False_ind ((eq T (lift (S i) O w) t2) \to ((getl i c0 (CHead d (Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (TSort m) t2)))) H6))) c0 (sym_eq C c0 c0 H2) H3 H4 H0 H1)))) | (tau0_abst c0 d v i H0 w H1) \Rightarrow (\lambda (H2: (eq C c0 c0)).(\lambda (H3: (eq T (TLRef i) (TSort m))).(\lambda (H4: (eq T (lift (S i) O v) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (TSort m)) \to ((eq T (lift (S i) O v) t2) \to ((getl i c (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (TSort m) t2)))))) (\lambda (H5: (eq T (TLRef i) (TSort m))).(let H6 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (TSort m) H5) in (False_ind ((eq T (lift (S i) O v) t2) \to ((getl i c0 (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (TSort m) t2)))) H6))) c0 (sym_eq C c0 c0 H2) H3 H4 H0 H1)))) | (tau0_bind b c0 v t1 t0 H0) \Rightarrow (\lambda (H1: (eq C c0 c0)).(\lambda (H2: (eq T (THead (Bind b) v t1) (TSort m))).(\lambda (H3: (eq T (THead (Bind b) v t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Bind b) v t1) (TSort m)) \to ((eq T (THead (Bind b) v t0) t2) \to ((tau0 g (CHead c (Bind b) v) t1 t0) \to (ty3 g c0 (TSort m) t2))))) (\lambda (H4: (eq T (THead (Bind b) v t1) (TSort m))).(let H5 \def (eq_ind T (THead (Bind b) v t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort m) H4) in (False_ind ((eq T (THead (Bind b) v t0) t2) \to ((tau0 g (CHead c0 (Bind b) v) t1 t0) \to (ty3 g c0 (TSort m) t2))) H5))) c0 (sym_eq C c0 c0 H1) H2 H3 H0)))) | (tau0_appl c0 v t1 t0 H0) \Rightarrow (\lambda (H1: (eq C c0 c0)).(\lambda (H2: (eq T (THead (Flat Appl) v t1) (TSort m))).(\lambda (H3: (eq T (THead (Flat Appl) v t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Appl) v t1) (TSort m)) \to ((eq T (THead (Flat Appl) v t0) t2) \to ((tau0 g c t1 t0) \to (ty3 g c0 (TSort m) t2))))) (\lambda (H4: (eq T (THead (Flat Appl) v t1) (TSort m))).(let H5 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort m) H4) in (False_ind ((eq T (THead (Flat Appl) v t0) t2) \to ((tau0 g c0 t1 t0) \to (ty3 g c0 (TSort m) t2))) H5))) c0 (sym_eq C c0 c0 H1) H2 H3 H0)))) | (tau0_cast c0 v1 v2 H0 t1 t0 H1) \Rightarrow (\lambda (H2: (eq C c0 c0)).(\lambda (H3: (eq T (THead (Flat Cast) v1 t1) (TSort m))).(\lambda (H4: (eq T (THead (Flat Cast) v2 t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Cast) v1 t1) (TSort m)) \to ((eq T (THead (Flat Cast) v2 t0) t2) \to ((tau0 g c v1 v2) \to ((tau0 g c t1 t0) \to (ty3 g c0 (TSort m) t2)))))) (\lambda (H5: (eq T (THead (Flat Cast) v1 t1) (TSort m))).(let H6 \def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TSort m) H5) in (False_ind ((eq T (THead (Flat Cast) v2 t0) t2) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0 t1 t0) \to (ty3 g c0 (TSort m) t2)))) H6))) c0 (sym_eq C c0 c0 H2) H3 H4 H0 H1))))]) in (H1 (refl_equal C c0) (refl_equal T (TSort m)) (refl_equal T t2))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u0))).(\lambda (t: T).(\lambda (_: (ty3 g d u0 t)).(\lambda (H2: ((\forall (t2: T).((tau0 g d u0 t2) \to (ty3 g d u0 t2))))).(\lambda (t2: T).(\lambda (H3: (tau0 g c0 (TLRef n) t2)).(let H4 \def (match H3 return (\lambda (_: ?).(\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c c0) \to ((eq T t (TLRef n)) \to ((eq T t0 t2) \to (ty3 g c0 (TLRef n) t2)))))))) with [(tau0_sort c0 n0) \Rightarrow (\lambda (H3: (eq C c0 c0)).(\lambda (H4: (eq T (TSort n0) (TLRef n))).(\lambda (H5: (eq T (TSort (next g n0)) t2)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n0) (TLRef n)) \to ((eq T (TSort (next g n0)) t2) \to (ty3 g c0 (TLRef n) t2)))) (\lambda (H6: (eq T (TSort n0) (TLRef n))).(let H7 \def (eq_ind T (TSort n0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n) H6) in (False_ind ((eq T (TSort (next g n0)) t2) \to (ty3 g c0 (TLRef n) t2)) H7))) c0 (sym_eq C c0 c0 H3) H4 H5)))) | (tau0_abbr c0 d0 v i H3 w H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (TLRef i) (TLRef n))).(\lambda (H7: (eq T (lift (S i) O w) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O w) t2) \to ((getl i c (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2)))))) (\lambda (H8: (eq T (TLRef i) (TLRef n))).(let H9 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow i | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H8) in (eq_ind nat n (\lambda (n0: nat).((eq T (lift (S n0) O w) t2) \to ((getl n0 c0 (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H10: (eq T (lift (S n) O w) t2)).(eq_ind T (lift (S n) O w) (\lambda (t: T).((getl n c0 (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t)))) (\lambda (H11: (getl n c0 (CHead d0 (Bind Abbr) v))).(\lambda (H12: (tau0 g d0 v w)).(let H13 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c: C).(getl n c0 c)) H0 (CHead d0 (Bind Abbr) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind Abbr) v) H11)) in (let H14 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind Abbr) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind Abbr) v) H11)) in ((let H15 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abbr) u0) (CHead d0 (Bind Abbr) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind Abbr) v) H11)) in (\lambda (H16: (eq C d d0)).(let H17 \def (eq_ind_r T v (\lambda (t: T).(getl n c0 (CHead d0 (Bind Abbr) t))) H13 u0 H15) in (let H18 \def (eq_ind_r T v (\lambda (t: T).(tau0 g d0 t w)) H12 u0 H15) in (let H19 \def (eq_ind_r C d0 (\lambda (c: C).(getl n c0 (CHead c (Bind Abbr) u0))) H17 d H16) in (let H20 \def (eq_ind_r C d0 (\lambda (c: C).(tau0 g c u0 w)) H18 d H16) in (ty3_abbr g n c0 d u0 H19 w (H2 w H20)))))))) H14))))) t2 H10)) i (sym_eq nat i n H9)))) c0 (sym_eq C c0 c0 H5) H6 H7 H3 H4)))) | (tau0_abst c0 d0 v i H3 w H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (TLRef i) (TLRef n))).(\lambda (H7: (eq T (lift (S i) O v) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O v) t2) \to ((getl i c (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2)))))) (\lambda (H8: (eq T (TLRef i) (TLRef n))).(let H9 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow i | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H8) in (eq_ind nat n (\lambda (n0: nat).((eq T (lift (S n0) O v) t2) \to ((getl n0 c0 (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H10: (eq T (lift (S n) O v) t2)).(eq_ind T (lift (S n) O v) (\lambda (t: T).((getl n c0 (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t)))) (\lambda (H11: (getl n c0 (CHead d0 (Bind Abst) v))).(\lambda (_: (tau0 g d0 v w)).(let H2 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (c: C).(getl n c0 c)) H0 (CHead d0 (Bind Abst) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind Abst) v) H11)) in (let H13 \def (eq_ind C (CHead d (Bind Abbr) u0) (\lambda (ee: C).(match ee return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d0 (Bind Abst) v) (getl_mono c0 (CHead d (Bind Abbr) u0) n H0 (CHead d0 (Bind Abst) v) H11)) in (False_ind (ty3 g c0 (TLRef n) (lift (S n) O v)) H13))))) t2 H10)) i (sym_eq nat i n H9)))) c0 (sym_eq C c0 c0 H5) H6 H7 H3 H4)))) | (tau0_bind b c0 v t1 t0 H3) \Rightarrow (\lambda (H4: (eq C c0 c0)).(\lambda (H5: (eq T (THead (Bind b) v t1) (TLRef n))).(\lambda (H6: (eq T (THead (Bind b) v t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Bind b) v t1) (TLRef n)) \to ((eq T (THead (Bind b) v t0) t2) \to ((tau0 g (CHead c (Bind b) v) t1 t0) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H7: (eq T (THead (Bind b) v t1) (TLRef n))).(let H8 \def (eq_ind T (THead (Bind b) v t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H7) in (False_ind ((eq T (THead (Bind b) v t0) t2) \to ((tau0 g (CHead c0 (Bind b) v) t1 t0) \to (ty3 g c0 (TLRef n) t2))) H8))) c0 (sym_eq C c0 c0 H4) H5 H6 H3)))) | (tau0_appl c0 v t1 t0 H3) \Rightarrow (\lambda (H4: (eq C c0 c0)).(\lambda (H5: (eq T (THead (Flat Appl) v t1) (TLRef n))).(\lambda (H6: (eq T (THead (Flat Appl) v t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Appl) v t1) (TLRef n)) \to ((eq T (THead (Flat Appl) v t0) t2) \to ((tau0 g c t1 t0) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H7: (eq T (THead (Flat Appl) v t1) (TLRef n))).(let H8 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H7) in (False_ind ((eq T (THead (Flat Appl) v t0) t2) \to ((tau0 g c0 t1 t0) \to (ty3 g c0 (TLRef n) t2))) H8))) c0 (sym_eq C c0 c0 H4) H5 H6 H3)))) | (tau0_cast c0 v1 v2 H3 t1 t0 H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (THead (Flat Cast) v1 t1) (TLRef n))).(\lambda (H7: (eq T (THead (Flat Cast) v2 t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Cast) v1 t1) (TLRef n)) \to ((eq T (THead (Flat Cast) v2 t0) t2) \to ((tau0 g c v1 v2) \to ((tau0 g c t1 t0) \to (ty3 g c0 (TLRef n) t2)))))) (\lambda (H8: (eq T (THead (Flat Cast) v1 t1) (TLRef n))).(let H9 \def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H8) in (False_ind ((eq T (THead (Flat Cast) v2 t0) t2) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0 t1 t0) \to (ty3 g c0 (TLRef n) t2)))) H9))) c0 (sym_eq C c0 c0 H5) H6 H7 H3 H4))))]) in (H4 (refl_equal C c0) (refl_equal T (TLRef n)) (refl_equal T t2))))))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u0: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) u0))).(\lambda (t: T).(\lambda (H1: (ty3 g d u0 t)).(\lambda (_: ((\forall (t2: T).((tau0 g d u0 t2) \to (ty3 g d u0 t2))))).(\lambda (t2: T).(\lambda (H3: (tau0 g c0 (TLRef n) t2)).(let H4 \def (match H3 return (\lambda (_: ?).(\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c c0) \to ((eq T t (TLRef n)) \to ((eq T t0 t2) \to (ty3 g c0 (TLRef n) t2)))))))) with [(tau0_sort c0 n0) \Rightarrow (\lambda (H3: (eq C c0 c0)).(\lambda (H4: (eq T (TSort n0) (TLRef n))).(\lambda (H5: (eq T (TSort (next g n0)) t2)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n0) (TLRef n)) \to ((eq T (TSort (next g n0)) t2) \to (ty3 g c0 (TLRef n) t2)))) (\lambda (H6: (eq T (TSort n0) (TLRef n))).(let H7 \def (eq_ind T (TSort n0) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef n) H6) in (False_ind ((eq T (TSort (next g n0)) t2) \to (ty3 g c0 (TLRef n) t2)) H7))) c0 (sym_eq C c0 c0 H3) H4 H5)))) | (tau0_abbr c0 d0 v i H3 w H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (TLRef i) (TLRef n))).(\lambda (H7: (eq T (lift (S i) O w) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O w) t2) \to ((getl i c (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2)))))) (\lambda (H8: (eq T (TLRef i) (TLRef n))).(let H9 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow i | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H8) in (eq_ind nat n (\lambda (n0: nat).((eq T (lift (S n0) O w) t2) \to ((getl n0 c0 (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H10: (eq T (lift (S n) O w) t2)).(eq_ind T (lift (S n) O w) (\lambda (t: T).((getl n c0 (CHead d0 (Bind Abbr) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t)))) (\lambda (H11: (getl n c0 (CHead d0 (Bind Abbr) v))).(\lambda (_: (tau0 g d0 v w)).(let H2 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (c: C).(getl n c0 c)) H0 (CHead d0 (Bind Abbr) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind Abbr) v) H11)) in (let H13 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (ee: C).(match ee return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow True | Void \Rightarrow False]) | (Flat _) \Rightarrow False])])) I (CHead d0 (Bind Abbr) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind Abbr) v) H11)) in (False_ind (ty3 g c0 (TLRef n) (lift (S n) O w)) H13))))) t2 H10)) i (sym_eq nat i n H9)))) c0 (sym_eq C c0 c0 H5) H6 H7 H3 H4)))) | (tau0_abst c0 d0 v i H3 w H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (TLRef i) (TLRef n))).(\lambda (H7: (eq T (lift (S i) O v) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (TLRef n)) \to ((eq T (lift (S i) O v) t2) \to ((getl i c (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2)))))) (\lambda (H8: (eq T (TLRef i) (TLRef n))).(let H9 \def (f_equal T nat (\lambda (e: T).(match e return (\lambda (_: ?).nat) with [(TSort _) \Rightarrow i | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) H8) in (eq_ind nat n (\lambda (n0: nat).((eq T (lift (S n0) O v) t2) \to ((getl n0 c0 (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H10: (eq T (lift (S n) O v) t2)).(eq_ind T (lift (S n) O v) (\lambda (t: T).((getl n c0 (CHead d0 (Bind Abst) v)) \to ((tau0 g d0 v w) \to (ty3 g c0 (TLRef n) t)))) (\lambda (H11: (getl n c0 (CHead d0 (Bind Abst) v))).(\lambda (H12: (tau0 g d0 v w)).(let H2 \def (eq_ind C (CHead d (Bind Abst) u0) (\lambda (c: C).(getl n c0 c)) H0 (CHead d0 (Bind Abst) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind Abst) v) H11)) in (let H13 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow d | (CHead c _ _) \Rightarrow c])) (CHead d (Bind Abst) u0) (CHead d0 (Bind Abst) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind Abst) v) H11)) in ((let H14 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t) \Rightarrow t])) (CHead d (Bind Abst) u0) (CHead d0 (Bind Abst) v) (getl_mono c0 (CHead d (Bind Abst) u0) n H0 (CHead d0 (Bind Abst) v) H11)) in (\lambda (H15: (eq C d d0)).(let H16 \def (eq_ind_r T v (\lambda (t: T).(getl n c0 (CHead d0 (Bind Abst) t))) H2 u0 H14) in (let H17 \def (eq_ind_r T v (\lambda (t: T).(tau0 g d0 t w)) H12 u0 H14) in (eq_ind T u0 (\lambda (t: T).(ty3 g c0 (TLRef n) (lift (S n) O t))) (let H18 \def (eq_ind_r C d0 (\lambda (c: C).(getl n c0 (CHead c (Bind Abst) u0))) H16 d H15) in (let H19 \def (eq_ind_r C d0 (\lambda (c: C).(tau0 g c u0 w)) H17 d H15) in (ty3_abst g n c0 d u0 H18 t H1))) v H14))))) H13))))) t2 H10)) i (sym_eq nat i n H9)))) c0 (sym_eq C c0 c0 H5) H6 H7 H3 H4)))) | (tau0_bind b c0 v t1 t0 H3) \Rightarrow (\lambda (H4: (eq C c0 c0)).(\lambda (H5: (eq T (THead (Bind b) v t1) (TLRef n))).(\lambda (H6: (eq T (THead (Bind b) v t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Bind b) v t1) (TLRef n)) \to ((eq T (THead (Bind b) v t0) t2) \to ((tau0 g (CHead c (Bind b) v) t1 t0) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H7: (eq T (THead (Bind b) v t1) (TLRef n))).(let H8 \def (eq_ind T (THead (Bind b) v t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H7) in (False_ind ((eq T (THead (Bind b) v t0) t2) \to ((tau0 g (CHead c0 (Bind b) v) t1 t0) \to (ty3 g c0 (TLRef n) t2))) H8))) c0 (sym_eq C c0 c0 H4) H5 H6 H3)))) | (tau0_appl c0 v t1 t0 H3) \Rightarrow (\lambda (H4: (eq C c0 c0)).(\lambda (H5: (eq T (THead (Flat Appl) v t1) (TLRef n))).(\lambda (H6: (eq T (THead (Flat Appl) v t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Appl) v t1) (TLRef n)) \to ((eq T (THead (Flat Appl) v t0) t2) \to ((tau0 g c t1 t0) \to (ty3 g c0 (TLRef n) t2))))) (\lambda (H7: (eq T (THead (Flat Appl) v t1) (TLRef n))).(let H8 \def (eq_ind T (THead (Flat Appl) v t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H7) in (False_ind ((eq T (THead (Flat Appl) v t0) t2) \to ((tau0 g c0 t1 t0) \to (ty3 g c0 (TLRef n) t2))) H8))) c0 (sym_eq C c0 c0 H4) H5 H6 H3)))) | (tau0_cast c0 v1 v2 H3 t1 t0 H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (THead (Flat Cast) v1 t1) (TLRef n))).(\lambda (H7: (eq T (THead (Flat Cast) v2 t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Cast) v1 t1) (TLRef n)) \to ((eq T (THead (Flat Cast) v2 t0) t2) \to ((tau0 g c v1 v2) \to ((tau0 g c t1 t0) \to (ty3 g c0 (TLRef n) t2)))))) (\lambda (H8: (eq T (THead (Flat Cast) v1 t1) (TLRef n))).(let H9 \def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H8) in (False_ind ((eq T (THead (Flat Cast) v2 t0) t2) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0 t1 t0) \to (ty3 g c0 (TLRef n) t2)))) H9))) c0 (sym_eq C c0 c0 H5) H6 H7 H3 H4))))]) in (H4 (refl_equal C c0) (refl_equal T (TLRef n)) (refl_equal T t2))))))))))))) (\lambda (c0: C).(\lambda (u0: T).(\lambda (t: T).(\lambda (H0: (ty3 g c0 u0 t)).(\lambda (_: ((\forall (t2: T).((tau0 g c0 u0 t2) \to (ty3 g c0 u0 t2))))).(\lambda (b: B).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t2 t3)).(\lambda (H3: ((\forall (t3: T).((tau0 g (CHead c0 (Bind b) u0) t2 t3) \to (ty3 g (CHead c0 (Bind b) u0) t2 t3))))).(\lambda (t0: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u0) t3 t0)).(\lambda (_: ((\forall (t2: T).((tau0 g (CHead c0 (Bind b) u0) t3 t2) \to (ty3 g (CHead c0 (Bind b) u0) t3 t2))))).(\lambda (t4: T).(\lambda (H6: (tau0 g c0 (THead (Bind b) u0 t2) t4)).(let H7 \def (match H6 return (\lambda (_: ?).(\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c c0) \to ((eq T t (THead (Bind b) u0 t2)) \to ((eq T t0 t4) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))))))) with [(tau0_sort c0 n) \Rightarrow (\lambda (H6: (eq C c0 c0)).(\lambda (H7: (eq T (TSort n) (THead (Bind b) u0 t2))).(\lambda (H8: (eq T (TSort (next g n)) t4)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n) (THead (Bind b) u0 t2)) \to ((eq T (TSort (next g n)) t4) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))) (\lambda (H9: (eq T (TSort n) (THead (Bind b) u0 t2))).(let H10 \def (eq_ind T (TSort n) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u0 t2) H9) in (False_ind ((eq T (TSort (next g n)) t4) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)) H10))) c0 (sym_eq C c0 c0 H6) H7 H8)))) | (tau0_abbr c0 d v i H6 w H7) \Rightarrow (\lambda (H8: (eq C c0 c0)).(\lambda (H9: (eq T (TLRef i) (THead (Bind b) u0 t2))).(\lambda (H10: (eq T (lift (S i) O w) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (THead (Bind b) u0 t2)) \to ((eq T (lift (S i) O w) t4) \to ((getl i c (CHead d (Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))))) (\lambda (H11: (eq T (TLRef i) (THead (Bind b) u0 t2))).(let H12 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u0 t2) H11) in (False_ind ((eq T (lift (S i) O w) t4) \to ((getl i c0 (CHead d (Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))) H12))) c0 (sym_eq C c0 c0 H8) H9 H10 H6 H7)))) | (tau0_abst c0 d v i H6 w H7) \Rightarrow (\lambda (H8: (eq C c0 c0)).(\lambda (H9: (eq T (TLRef i) (THead (Bind b) u0 t2))).(\lambda (H10: (eq T (lift (S i) O v) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (THead (Bind b) u0 t2)) \to ((eq T (lift (S i) O v) t4) \to ((getl i c (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))))) (\lambda (H11: (eq T (TLRef i) (THead (Bind b) u0 t2))).(let H12 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Bind b) u0 t2) H11) in (False_ind ((eq T (lift (S i) O v) t4) \to ((getl i c0 (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))) H12))) c0 (sym_eq C c0 c0 H8) H9 H10 H6 H7)))) | (tau0_bind b0 c0 v t4 t5 H6) \Rightarrow (\lambda (H7: (eq C c0 c0)).(\lambda (H8: (eq T (THead (Bind b0) v t4) (THead (Bind b) u0 t2))).(\lambda (H9: (eq T (THead (Bind b0) v t5) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Bind b0) v t4) (THead (Bind b) u0 t2)) \to ((eq T (THead (Bind b0) v t5) t4) \to ((tau0 g (CHead c (Bind b0) v) t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4))))) (\lambda (H10: (eq T (THead (Bind b0) v t4) (THead (Bind b) u0 t2))).(let H11 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) \Rightarrow t])) (THead (Bind b0) v t4) (THead (Bind b) u0 t2) H10) in ((let H12 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t _) \Rightarrow t])) (THead (Bind b0) v t4) (THead (Bind b) u0 t2) H10) in ((let H13 \def (f_equal T B (\lambda (e: T).(match e return (\lambda (_: ?).B) with [(TSort _) \Rightarrow b0 | (TLRef _) \Rightarrow b0 | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow b0])])) (THead (Bind b0) v t4) (THead (Bind b) u0 t2) H10) in (eq_ind B b (\lambda (b1: B).((eq T v u0) \to ((eq T t4 t2) \to ((eq T (THead (Bind b1) v t5) t4) \to ((tau0 g (CHead c0 (Bind b1) v) t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))))) (\lambda (H14: (eq T v u0)).(eq_ind T u0 (\lambda (t: T).((eq T t4 t2) \to ((eq T (THead (Bind b) t t5) t4) \to ((tau0 g (CHead c0 (Bind b) t) t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4))))) (\lambda (H15: (eq T t4 t2)).(eq_ind T t2 (\lambda (t: T).((eq T (THead (Bind b) u0 t5) t4) \to ((tau0 g (CHead c0 (Bind b) u0) t t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))) (\lambda (H16: (eq T (THead (Bind b) u0 t5) t4)).(eq_ind T (THead (Bind b) u0 t5) (\lambda (t: T).((tau0 g (CHead c0 (Bind b) u0) t2 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t))) (\lambda (H17: (tau0 g (CHead c0 (Bind b) u0) t2 t5)).(let H_y \def (H3 t5 H17) in (ex_ind T (\lambda (t: T).(ty3 g (CHead c0 (Bind b) u0) t5 t)) (ty3 g c0 (THead (Bind b) u0 t2) (THead (Bind b) u0 t5)) (\lambda (x: T).(\lambda (H1: (ty3 g (CHead c0 (Bind b) u0) t5 x)).(ty3_bind g c0 u0 t H0 b t2 t5 H_y x H1))) (ty3_correct g (CHead c0 (Bind b) u0) t2 t5 H_y)))) t4 H16)) t4 (sym_eq T t4 t2 H15))) v (sym_eq T v u0 H14))) b0 (sym_eq B b0 b H13))) H12)) H11))) c0 (sym_eq C c0 c0 H7) H8 H9 H6)))) | (tau0_appl c0 v t4 t5 H6) \Rightarrow (\lambda (H7: (eq C c0 c0)).(\lambda (H8: (eq T (THead (Flat Appl) v t4) (THead (Bind b) u0 t2))).(\lambda (H9: (eq T (THead (Flat Appl) v t5) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Appl) v t4) (THead (Bind b) u0 t2)) \to ((eq T (THead (Flat Appl) v t5) t4) \to ((tau0 g c t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4))))) (\lambda (H10: (eq T (THead (Flat Appl) v t4) (THead (Bind b) u0 t2))).(let H11 \def (eq_ind T (THead (Flat Appl) v t4) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 t2) H10) in (False_ind ((eq T (THead (Flat Appl) v t5) t4) \to ((tau0 g c0 t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4))) H11))) c0 (sym_eq C c0 c0 H7) H8 H9 H6)))) | (tau0_cast c0 v1 v2 H6 t4 t5 H7) \Rightarrow (\lambda (H8: (eq C c0 c0)).(\lambda (H9: (eq T (THead (Flat Cast) v1 t4) (THead (Bind b) u0 t2))).(\lambda (H10: (eq T (THead (Flat Cast) v2 t5) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Cast) v1 t4) (THead (Bind b) u0 t2)) \to ((eq T (THead (Flat Cast) v2 t5) t4) \to ((tau0 g c v1 v2) \to ((tau0 g c t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))))) (\lambda (H11: (eq T (THead (Flat Cast) v1 t4) (THead (Bind b) u0 t2))).(let H12 \def (eq_ind T (THead (Flat Cast) v1 t4) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind b) u0 t2) H11) in (False_ind ((eq T (THead (Flat Cast) v2 t5) t4) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0 t4 t5) \to (ty3 g c0 (THead (Bind b) u0 t2) t4)))) H12))) c0 (sym_eq C c0 c0 H8) H9 H10 H6 H7))))]) in (H7 (refl_equal C c0) (refl_equal T (THead (Bind b) u0 t2)) (refl_equal T t4)))))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u0: T).(\lambda (H0: (ty3 g c0 w u0)).(\lambda (_: ((\forall (t2: T).((tau0 g c0 w t2) \to (ty3 g c0 w t2))))).(\lambda (v: T).(\lambda (t: T).(\lambda (H2: (ty3 g c0 v (THead (Bind Abst) u0 t))).(\lambda (H3: ((\forall (t2: T).((tau0 g c0 v t2) \to (ty3 g c0 v t2))))).(\lambda (t2: T).(\lambda (H4: (tau0 g c0 (THead (Flat Appl) w v) t2)).(let H5 \def (match H4 return (\lambda (_: ?).(\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c c0) \to ((eq T t (THead (Flat Appl) w v)) \to ((eq T t0 t2) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))))))) with [(tau0_sort c0 n) \Rightarrow (\lambda (H4: (eq C c0 c0)).(\lambda (H5: (eq T (TSort n) (THead (Flat Appl) w v))).(\lambda (H6: (eq T (TSort (next g n)) t2)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n) (THead (Flat Appl) w v)) \to ((eq T (TSort (next g n)) t2) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))) (\lambda (H7: (eq T (TSort n) (THead (Flat Appl) w v))).(let H8 \def (eq_ind T (TSort n) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w v) H7) in (False_ind ((eq T (TSort (next g n)) t2) \to (ty3 g c0 (THead (Flat Appl) w v) t2)) H8))) c0 (sym_eq C c0 c0 H4) H5 H6)))) | (tau0_abbr c0 d v0 i H4 w0 H5) \Rightarrow (\lambda (H6: (eq C c0 c0)).(\lambda (H7: (eq T (TLRef i) (THead (Flat Appl) w v))).(\lambda (H8: (eq T (lift (S i) O w0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (THead (Flat Appl) w v)) \to ((eq T (lift (S i) O w0) t2) \to ((getl i c (CHead d (Bind Abbr) v0)) \to ((tau0 g d v0 w0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))))) (\lambda (H9: (eq T (TLRef i) (THead (Flat Appl) w v))).(let H10 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w v) H9) in (False_ind ((eq T (lift (S i) O w0) t2) \to ((getl i c0 (CHead d (Bind Abbr) v0)) \to ((tau0 g d v0 w0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))) H10))) c0 (sym_eq C c0 c0 H6) H7 H8 H4 H5)))) | (tau0_abst c0 d v0 i H4 w0 H5) \Rightarrow (\lambda (H6: (eq C c0 c0)).(\lambda (H7: (eq T (TLRef i) (THead (Flat Appl) w v))).(\lambda (H8: (eq T (lift (S i) O v0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (THead (Flat Appl) w v)) \to ((eq T (lift (S i) O v0) t2) \to ((getl i c (CHead d (Bind Abst) v0)) \to ((tau0 g d v0 w0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))))) (\lambda (H9: (eq T (TLRef i) (THead (Flat Appl) w v))).(let H10 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Appl) w v) H9) in (False_ind ((eq T (lift (S i) O v0) t2) \to ((getl i c0 (CHead d (Bind Abst) v0)) \to ((tau0 g d v0 w0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))) H10))) c0 (sym_eq C c0 c0 H6) H7 H8 H4 H5)))) | (tau0_bind b c0 v0 t1 t0 H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (THead (Bind b) v0 t1) (THead (Flat Appl) w v))).(\lambda (H7: (eq T (THead (Bind b) v0 t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Bind b) v0 t1) (THead (Flat Appl) w v)) \to ((eq T (THead (Bind b) v0 t0) t2) \to ((tau0 g (CHead c (Bind b) v0) t1 t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2))))) (\lambda (H8: (eq T (THead (Bind b) v0 t1) (THead (Flat Appl) w v))).(let H9 \def (eq_ind T (THead (Bind b) v0 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) w v) H8) in (False_ind ((eq T (THead (Bind b) v0 t0) t2) \to ((tau0 g (CHead c0 (Bind b) v0) t1 t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2))) H9))) c0 (sym_eq C c0 c0 H5) H6 H7 H4)))) | (tau0_appl c0 v0 t1 t0 H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (THead (Flat Appl) v0 t1) (THead (Flat Appl) w v))).(\lambda (H7: (eq T (THead (Flat Appl) v0 t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Appl) v0 t1) (THead (Flat Appl) w v)) \to ((eq T (THead (Flat Appl) v0 t0) t2) \to ((tau0 g c t1 t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2))))) (\lambda (H8: (eq T (THead (Flat Appl) v0 t1) (THead (Flat Appl) w v))).(let H9 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t1 | (TLRef _) \Rightarrow t1 | (THead _ _ t) \Rightarrow t])) (THead (Flat Appl) v0 t1) (THead (Flat Appl) w v) H8) in ((let H10 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v0 | (TLRef _) \Rightarrow v0 | (THead _ t _) \Rightarrow t])) (THead (Flat Appl) v0 t1) (THead (Flat Appl) w v) H8) in (eq_ind T w (\lambda (t: T).((eq T t1 v) \to ((eq T (THead (Flat Appl) t t0) t2) \to ((tau0 g c0 t1 t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2))))) (\lambda (H11: (eq T t1 v)).(eq_ind T v (\lambda (t: T).((eq T (THead (Flat Appl) w t0) t2) \to ((tau0 g c0 t t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))) (\lambda (H12: (eq T (THead (Flat Appl) w t0) t2)).(eq_ind T (THead (Flat Appl) w t0) (\lambda (t: T).((tau0 g c0 v t0) \to (ty3 g c0 (THead (Flat Appl) w v) t))) (\lambda (H13: (tau0 g c0 v t0)).(let H_y \def (H3 t0 H13) in (let H1 \def (ty3_unique g c0 v t0 H_y (THead (Bind Abst) u0 t) H2) in (ex_ind T (\lambda (t: T).(ty3 g c0 t0 t)) (ty3 g c0 (THead (Flat Appl) w v) (THead (Flat Appl) w t0)) (\lambda (x: T).(\lambda (H3: (ty3 g c0 t0 x)).(ex_ind T (\lambda (t: T).(ty3 g c0 u0 t)) (ty3 g c0 (THead (Flat Appl) w v) (THead (Flat Appl) w t0)) (\lambda (x0: T).(\lambda (_: (ty3 g c0 u0 x0)).(ex_ind T (\lambda (t2: T).(ty3 g c0 (THead (Bind Abst) u0 t) t2)) (ty3 g c0 (THead (Flat Appl) w v) (THead (Flat Appl) w t0)) (\lambda (x1: T).(\lambda (H15: (ty3 g c0 (THead (Bind Abst) u0 t) x1)).(ex4_3_ind T T T (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(pc3 c0 (THead (Bind Abst) u0 t2) x1)))) (\lambda (_: T).(\lambda (t: T).(\lambda (_: T).(ty3 g c0 u0 t)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c0 (Bind Abst) u0) t t2)))) (\lambda (t2: T).(\lambda (_: T).(\lambda (t3: T).(ty3 g (CHead c0 (Bind Abst) u0) t2 t3)))) (ty3 g c0 (THead (Flat Appl) w v) (THead (Flat Appl) w t0)) (\lambda (x2: T).(\lambda (x3: T).(\lambda (x4: T).(\lambda (_: (pc3 c0 (THead (Bind Abst) u0 x2) x1)).(\lambda (H17: (ty3 g c0 u0 x3)).(\lambda (H18: (ty3 g (CHead c0 (Bind Abst) u0) t x2)).(\lambda (H19: (ty3 g (CHead c0 (Bind Abst) u0) x2 x4)).(ty3_conv g c0 (THead (Flat Appl) w t0) (THead (Flat Appl) w (THead (Bind Abst) u0 x2)) (ty3_appl g c0 w u0 H0 t0 x2 (ty3_sconv g c0 t0 x H3 (THead (Bind Abst) u0 t) (THead (Bind Abst) u0 x2) (ty3_bind g c0 u0 x3 H17 Abst t x2 H18 x4 H19) H1)) (THead (Flat Appl) w v) (THead (Flat Appl) w (THead (Bind Abst) u0 t)) (ty3_appl g c0 w u0 H0 v t H2) (pc3_s c0 (THead (Flat Appl) w (THead (Bind Abst) u0 t)) (THead (Flat Appl) w t0) (pc3_thin_dx c0 t0 (THead (Bind Abst) u0 t) H1 w Appl)))))))))) (ty3_gen_bind g Abst c0 u0 t x1 H15)))) (ty3_correct g c0 v (THead (Bind Abst) u0 t) H2)))) (ty3_correct g c0 w u0 H0)))) (ty3_correct g c0 v t0 H_y))))) t2 H12)) t1 (sym_eq T t1 v H11))) v0 (sym_eq T v0 w H10))) H9))) c0 (sym_eq C c0 c0 H5) H6 H7 H4)))) | (tau0_cast c0 v1 v2 H4 t1 t0 H5) \Rightarrow (\lambda (H6: (eq C c0 c0)).(\lambda (H7: (eq T (THead (Flat Cast) v1 t1) (THead (Flat Appl) w v))).(\lambda (H8: (eq T (THead (Flat Cast) v2 t0) t2)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Cast) v1 t1) (THead (Flat Appl) w v)) \to ((eq T (THead (Flat Cast) v2 t0) t2) \to ((tau0 g c v1 v2) \to ((tau0 g c t1 t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))))) (\lambda (H9: (eq T (THead (Flat Cast) v1 t1) (THead (Flat Appl) w v))).(let H10 \def (eq_ind T (THead (Flat Cast) v1 t1) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow False | Cast \Rightarrow True])])])) I (THead (Flat Appl) w v) H9) in (False_ind ((eq T (THead (Flat Cast) v2 t0) t2) \to ((tau0 g c0 v1 v2) \to ((tau0 g c0 t1 t0) \to (ty3 g c0 (THead (Flat Appl) w v) t2)))) H10))) c0 (sym_eq C c0 c0 H6) H7 H8 H4 H5))))]) in (H5 (refl_equal C c0) (refl_equal T (THead (Flat Appl) w v)) (refl_equal T t2)))))))))))))) (\lambda (c0: C).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H0: (ty3 g c0 t2 t3)).(\lambda (H1: ((\forall (t3: T).((tau0 g c0 t2 t3) \to (ty3 g c0 t2 t3))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t3 t0)).(\lambda (H3: ((\forall (t2: T).((tau0 g c0 t3 t2) \to (ty3 g c0 t3 t2))))).(\lambda (t4: T).(\lambda (H4: (tau0 g c0 (THead (Flat Cast) t3 t2) t4)).(let H5 \def (match H4 return (\lambda (_: ?).(\lambda (c: C).(\lambda (t: T).(\lambda (t0: T).((eq C c c0) \to ((eq T t (THead (Flat Cast) t3 t2)) \to ((eq T t0 t4) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))))))) with [(tau0_sort c0 n) \Rightarrow (\lambda (H4: (eq C c0 c0)).(\lambda (H5: (eq T (TSort n) (THead (Flat Cast) t3 t2))).(\lambda (H6: (eq T (TSort (next g n)) t4)).(eq_ind C c0 (\lambda (_: C).((eq T (TSort n) (THead (Flat Cast) t3 t2)) \to ((eq T (TSort (next g n)) t4) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))) (\lambda (H7: (eq T (TSort n) (THead (Flat Cast) t3 t2))).(let H8 \def (eq_ind T (TSort n) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t3 t2) H7) in (False_ind ((eq T (TSort (next g n)) t4) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)) H8))) c0 (sym_eq C c0 c0 H4) H5 H6)))) | (tau0_abbr c0 d v i H4 w H5) \Rightarrow (\lambda (H6: (eq C c0 c0)).(\lambda (H7: (eq T (TLRef i) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T (lift (S i) O w) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (THead (Flat Cast) t3 t2)) \to ((eq T (lift (S i) O w) t4) \to ((getl i c (CHead d (Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))))) (\lambda (H9: (eq T (TLRef i) (THead (Flat Cast) t3 t2))).(let H10 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t3 t2) H9) in (False_ind ((eq T (lift (S i) O w) t4) \to ((getl i c0 (CHead d (Bind Abbr) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))) H10))) c0 (sym_eq C c0 c0 H6) H7 H8 H4 H5)))) | (tau0_abst c0 d v i H4 w H5) \Rightarrow (\lambda (H6: (eq C c0 c0)).(\lambda (H7: (eq T (TLRef i) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T (lift (S i) O v) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (TLRef i) (THead (Flat Cast) t3 t2)) \to ((eq T (lift (S i) O v) t4) \to ((getl i c (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))))) (\lambda (H9: (eq T (TLRef i) (THead (Flat Cast) t3 t2))).(let H10 \def (eq_ind T (TLRef i) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead (Flat Cast) t3 t2) H9) in (False_ind ((eq T (lift (S i) O v) t4) \to ((getl i c0 (CHead d (Bind Abst) v)) \to ((tau0 g d v w) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))) H10))) c0 (sym_eq C c0 c0 H6) H7 H8 H4 H5)))) | (tau0_bind b c0 v t4 t5 H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (THead (Bind b) v t4) (THead (Flat Cast) t3 t2))).(\lambda (H7: (eq T (THead (Bind b) v t5) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Bind b) v t4) (THead (Flat Cast) t3 t2)) \to ((eq T (THead (Bind b) v t5) t4) \to ((tau0 g (CHead c (Bind b) v) t4 t5) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4))))) (\lambda (H8: (eq T (THead (Bind b) v t4) (THead (Flat Cast) t3 t2))).(let H9 \def (eq_ind T (THead (Bind b) v t4) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T (THead (Bind b) v t5) t4) \to ((tau0 g (CHead c0 (Bind b) v) t4 t5) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4))) H9))) c0 (sym_eq C c0 c0 H5) H6 H7 H4)))) | (tau0_appl c0 v t4 t5 H4) \Rightarrow (\lambda (H5: (eq C c0 c0)).(\lambda (H6: (eq T (THead (Flat Appl) v t4) (THead (Flat Cast) t3 t2))).(\lambda (H7: (eq T (THead (Flat Appl) v t5) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Appl) v t4) (THead (Flat Cast) t3 t2)) \to ((eq T (THead (Flat Appl) v t5) t4) \to ((tau0 g c t4 t5) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4))))) (\lambda (H8: (eq T (THead (Flat Appl) v t4) (THead (Flat Cast) t3 t2))).(let H9 \def (eq_ind T (THead (Flat Appl) v t4) (\lambda (e: T).(match e return (\lambda (_: ?).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f return (\lambda (_: ?).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead (Flat Cast) t3 t2) H8) in (False_ind ((eq T (THead (Flat Appl) v t5) t4) \to ((tau0 g c0 t4 t5) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4))) H9))) c0 (sym_eq C c0 c0 H5) H6 H7 H4)))) | (tau0_cast c0 v1 v2 H4 t4 t5 H5) \Rightarrow (\lambda (H6: (eq C c0 c0)).(\lambda (H7: (eq T (THead (Flat Cast) v1 t4) (THead (Flat Cast) t3 t2))).(\lambda (H8: (eq T (THead (Flat Cast) v2 t5) t4)).(eq_ind C c0 (\lambda (c: C).((eq T (THead (Flat Cast) v1 t4) (THead (Flat Cast) t3 t2)) \to ((eq T (THead (Flat Cast) v2 t5) t4) \to ((tau0 g c v1 v2) \to ((tau0 g c t4 t5) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))))) (\lambda (H9: (eq T (THead (Flat Cast) v1 t4) (THead (Flat Cast) t3 t2))).(let H10 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow t4 | (TLRef _) \Rightarrow t4 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) v1 t4) (THead (Flat Cast) t3 t2) H9) in ((let H11 \def (f_equal T T (\lambda (e: T).(match e return (\lambda (_: ?).T) with [(TSort _) \Rightarrow v1 | (TLRef _) \Rightarrow v1 | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) v1 t4) (THead (Flat Cast) t3 t2) H9) in (eq_ind T t3 (\lambda (t: T).((eq T t4 t2) \to ((eq T (THead (Flat Cast) v2 t5) t4) \to ((tau0 g c0 t v2) \to ((tau0 g c0 t4 t5) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4)))))) (\lambda (H12: (eq T t4 t2)).(eq_ind T t2 (\lambda (t: T).((eq T (THead (Flat Cast) v2 t5) t4) \to ((tau0 g c0 t3 v2) \to ((tau0 g c0 t t5) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t4))))) (\lambda (H13: (eq T (THead (Flat Cast) v2 t5) t4)).(eq_ind T (THead (Flat Cast) v2 t5) (\lambda (t: T).((tau0 g c0 t3 v2) \to ((tau0 g c0 t2 t5) \to (ty3 g c0 (THead (Flat Cast) t3 t2) t)))) (\lambda (H14: (tau0 g c0 t3 v2)).(\lambda (H15: (tau0 g c0 t2 t5)).(let H_y \def (H1 t5 H15) in (let H_y0 \def (H3 v2 H14) in (let H3 \def (ty3_unique g c0 t2 t5 H_y t3 H0) in (ex_ind T (\lambda (t: T).(ty3 g c0 v2 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) v2 t5)) (\lambda (x: T).(\lambda (H16: (ty3 g c0 v2 x)).(ex_ind T (\lambda (t: T).(ty3 g c0 t5 t)) (ty3 g c0 (THead (Flat Cast) t3 t2) (THead (Flat Cast) v2 t5)) (\lambda (x0: T).(\lambda (H17: (ty3 g c0 t5 x0)).(ty3_conv g c0 (THead (Flat Cast) v2 t5) v2 (ty3_cast g c0 t5 v2 (ty3_sconv g c0 t5 x0 H17 t3 v2 H_y0 H3) x H16) (THead (Flat Cast) t3 t2) t3 (ty3_cast g c0 t2 t3 H0 v2 H_y0) (pc3_s c0 t3 (THead (Flat Cast) v2 t5) (pc3_pr2_u c0 t5 (THead (Flat Cast) v2 t5) (pr2_free c0 (THead (Flat Cast) v2 t5) t5 (pr0_epsilon t5 t5 (pr0_refl t5) v2)) t3 H3))))) (ty3_correct g c0 t2 t5 H_y)))) (ty3_correct g c0 t3 v2 H_y0))))))) t4 H13)) t4 (sym_eq T t4 t2 H12))) v1 (sym_eq T v1 t3 H11))) H10))) c0 (sym_eq C c0 c0 H6) H7 H8 H4 H5))))]) in (H5 (refl_equal C c0) (refl_equal T (THead (Flat Cast) t3 t2)) (refl_equal T t4))))))))))))) c u t1 H))))).
 
 theorem ty3_arity:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c t1 t2) \to (ex2 A (\lambda (a1: A).(arity g c t1 a1)) (\lambda (a1: A).(arity g c t2 (asucc g a1))))))))
 \def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(ex2 A (\lambda (a1: A).(arity g c0 t a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g a1))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))))).(\lambda (H4: (pc3 c0 t4 t3)).(let H5 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g a1)))) (\lambda (x: A).(\lambda (H6: (arity g c0 t3 x)).(\lambda (_: (arity g c0 t (asucc g x))).(let H8 \def H3 in (ex2_ind A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g a1)))) (\lambda (x0: A).(\lambda (H9: (arity g c0 u x0)).(\lambda (H10: (arity g c0 t4 (asucc g x0))).(let H11 \def H4 in (ex2_ind T (\lambda (t0: T).(pr3 c0 t4 t0)) (\lambda (t0: T).(pr3 c0 t3 t0)) (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g a1)))) (\lambda (x1: T).(\lambda (H12: (pr3 c0 t4 x1)).(\lambda (H13: (pr3 c0 t3 x1)).(ex_intro2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g a1))) x0 H9 (arity_repl g c0 t3 x H6 (asucc g x0) (leq_sym g (asucc g x0) x (arity_mono g c0 x1 (asucc g x0) (arity_sred_pr3 c0 t4 x1 H12 g (asucc g x0) H10) x (arity_sred_pr3 c0 t3 x1 H13 g x H6)))))))) H11))))) H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro2 A (\lambda (a1: A).(arity g c0 (TSort m) a1)) (\lambda (a1: A).(arity g c0 (TSort (next g m)) (asucc g a1))) (ASort O m) (arity_sort g c0 m) (arity_sort g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A (\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (H5: (arity g d t (asucc g x))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g a1))) x (arity_abbr g c0 d u n H0 x H4) (arity_lift g d t (asucc g x) H5 c0 (S n) O (getl_drop Abbr c0 d u n H0)))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A (\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (_: (arity g d t (asucc g x))).(let H_x \def (leq_asucc g x) in (let H6 \def H_x in (ex_ind A (\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g a1)))) (\lambda (x0: A).(\lambda (H7: (leq g x (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g a1))) x0 (arity_abst g c0 d u n H0 x0 (arity_repl g d u x H4 (asucc g x0) H7)) (arity_lift g d u (asucc g x0) (arity_repl g d u x H4 (asucc g x0) H7) c0 (S n) O (getl_drop Abst c0 d u n H0))))) H6)))))) H3)))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (b: B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t3 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4 (asucc g a1))))).(\lambda (t0: T).(\lambda (H4: (ty3 g (CHead c0 (Bind b) u) t4 t0)).(\lambda (H5: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t0 (asucc g a1))))).(let H6 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) (asucc g a1)))) (\lambda (x: A).(\lambda (H7: (arity g c0 u x)).(\lambda (_: (arity g c0 t (asucc g x))).(let H9 \def H3 in (ex2_ind A (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t3 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4 (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) (asucc g a1)))) (\lambda (x0: A).(\lambda (H10: (arity g (CHead c0 (Bind b) u) t3 x0)).(\lambda (H11: (arity g (CHead c0 (Bind b) u) t4 (asucc g x0))).(let H_x \def (leq_asucc g x) in (let H12 \def H_x in (ex_ind A (\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) (asucc g a1)))) (\lambda (x1: A).(\lambda (H13: (leq g x (asucc g x1))).((match b return (\lambda (b0: B).((ty3 g (CHead c0 (Bind b0) u) t4 t0) \to ((ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b0) u) t4 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b0) u) t0 (asucc g a1)))) \to ((arity g (CHead c0 (Bind b0) u) t3 x0) \to ((arity g (CHead c0 (Bind b0) u) t4 (asucc g x0)) \to (ex2 A (\lambda (a1: A).(arity g c0 (THead (Bind b0) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b0) u t4) (asucc g a1))))))))) with [Abbr \Rightarrow (\lambda (_: (ty3 g (CHead c0 (Bind Abbr) u) t4 t0)).(\lambda (_: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind Abbr) u) t4 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind Abbr) u) t0 (asucc g a1))))).(\lambda (H16: (arity g (CHead c0 (Bind Abbr) u) t3 x0)).(\lambda (H17: (arity g (CHead c0 (Bind Abbr) u) t4 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u t4) (asucc g a1))) x0 (arity_bind g Abbr not_abbr_abst c0 u x H7 t3 x0 H16) (arity_bind g Abbr not_abbr_abst c0 u x H7 t4 (asucc g x0) H17)))))) | Abst \Rightarrow (\lambda (_: (ty3 g (CHead c0 (Bind Abst) u) t4 t0)).(\lambda (_: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind Abst) u) t4 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind Abst) u) t0 (asucc g a1))))).(\lambda (H16: (arity g (CHead c0 (Bind Abst) u) t3 x0)).(\lambda (H17: (arity g (CHead c0 (Bind Abst) u) t4 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t4) (asucc g a1))) (AHead x1 x0) (arity_head g c0 u x1 (arity_repl g c0 u x H7 (asucc g x1) H13) t3 x0 H16) (arity_repl g c0 (THead (Bind Abst) u t4) (AHead x1 (asucc g x0)) (arity_head g c0 u x1 (arity_repl g c0 u x H7 (asucc g x1) H13) t4 (asucc g x0) H17) (asucc g (AHead x1 x0)) (leq_refl g (asucc g (AHead x1 x0))))))))) | Void \Rightarrow (\lambda (_: (ty3 g (CHead c0 (Bind Void) u) t4 t0)).(\lambda (_: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind Void) u) t4 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind Void) u) t0 (asucc g a1))))).(\lambda (H16: (arity g (CHead c0 (Bind Void) u) t3 x0)).(\lambda (H17: (arity g (CHead c0 (Bind Void) u) t4 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Void) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Void) u t4) (asucc g a1))) x0 (arity_bind g Void not_void_abst c0 u x H7 t3 x0 H16) (arity_bind g Void not_void_abst c0 u x H7 t4 (asucc g x0) H17))))))]) H4 H5 H10 H11))) H12)))))) H9))))) H6))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity g c0 w a1)) (\lambda (a1: A).(arity g c0 u (asucc g a1))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u t))).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g c0 v a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t) (asucc g a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 w a1)) (\lambda (a1: A).(arity g c0 u (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 w x)).(\lambda (H6: (arity g c0 u (asucc g x))).(let H7 \def H3 in (ex2_ind A (\lambda (a1: A).(arity g c0 v a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t) (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x0: A).(\lambda (H8: (arity g c0 v x0)).(\lambda (H9: (arity g c0 (THead (Bind Abst) u t) (asucc g x0))).(let H10 \def (arity_gen_abst g c0 u t (asucc g x0) H9) in (ex3_2_ind A A (\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g x0) (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x1: A).(\lambda (x2: A).(\lambda (H11: (eq A (asucc g x0) (AHead x1 x2))).(\lambda (H12: (arity g c0 u (asucc g x1))).(\lambda (H13: (arity g (CHead c0 (Bind Abst) u) t x2)).(let H14 \def (sym_equal A (asucc g x0) (AHead x1 x2) H11) in (let H15 \def (asucc_gen_head g x1 x2 x0 H14) in (ex2_ind A (\lambda (a0: A).(eq A x0 (AHead x1 a0))) (\lambda (a0: A).(eq A x2 (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x3: A).(\lambda (H16: (eq A x0 (AHead x1 x3))).(\lambda (H17: (eq A x2 (asucc g x3))).(let H18 \def (eq_ind A x2 (\lambda (a: A).(arity g (CHead c0 (Bind Abst) u) t a)) H13 (asucc g x3) H17) in (let H19 \def (eq_ind A x0 (\lambda (a: A).(arity g c0 v a)) H8 (AHead x1 x3) H16) in (ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g a1))) x3 (arity_appl g c0 w x1 (arity_repl g c0 w x H5 x1 (leq_sym g x1 x (asucc_inj g x1 x (arity_mono g c0 u (asucc g x1) H12 (asucc g x) H6)))) v x3 H19) (arity_appl g c0 w x H5 (THead (Bind Abst) u t) (asucc g x3) (arity_head g c0 u x H6 t (asucc g x3) H18)))))))) H15)))))))) H10))))) H7))))) H4))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 t0)).(\lambda (_: (ex2 A (\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 t3 x)).(\lambda (H6: (arity g c0 t4 (asucc g x))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) x (arity_cast g c0 t4 x H6 t3 H5) H6)))) H4)))))))))) c t1 t2 H))))).
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (ty3 g c t1 t2)).(ty3_ind g (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(ex2 A (\lambda (a1: A).(arity g c0 t a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g a1))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 t3 t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 u t4)).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))))).(\lambda (H4: (pc3 c0 t4 t3)).(let H5 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g a1)))) (\lambda (x: A).(\lambda (H6: (arity g c0 t3 x)).(\lambda (_: (arity g c0 t (asucc g x))).(let H8 \def H3 in (ex2_ind A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g a1)))) (\lambda (x0: A).(\lambda (H9: (arity g c0 u x0)).(\lambda (H10: (arity g c0 t4 (asucc g x0))).(let H11 \def H4 in (ex2_ind T (\lambda (t0: T).(pr3 c0 t4 t0)) (\lambda (t0: T).(pr3 c0 t3 t0)) (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g a1)))) (\lambda (x1: T).(\lambda (H12: (pr3 c0 t4 x1)).(\lambda (H13: (pr3 c0 t3 x1)).(ex_intro2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t3 (asucc g a1))) x0 H9 (arity_repl g c0 t3 x H6 (asucc g x0) (leq_sym g (asucc g x0) x (arity_mono g c0 x1 (asucc g x0) (arity_sred_pr3 c0 t4 x1 H12 g (asucc g x0) H10) x (arity_sred_pr3 c0 t3 x1 H13 g x H6)))))))) H11))))) H8))))) H5)))))))))))) (\lambda (c0: C).(\lambda (m: nat).(ex_intro2 A (\lambda (a1: A).(arity g c0 (TSort m) a1)) (\lambda (a1: A).(arity g c0 (TSort (next g m)) (asucc g a1))) (ASort O m) (arity_sort g c0 m) (arity_sort g c0 (next g m))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A (\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (H5: (arity g d t (asucc g x))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O t) (asucc g a1))) x (arity_abbr g c0 d u n H0 x H4) (arity_lift g d t (asucc g x) H5 c0 (S n) O (getl_drop Abbr c0 d u n H0)))))) H3)))))))))) (\lambda (n: nat).(\lambda (c0: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c0 (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (_: (ty3 g d u t)).(\lambda (H2: (ex2 A (\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g a1))))).(let H3 \def H2 in (ex2_ind A (\lambda (a1: A).(arity g d u a1)) (\lambda (a1: A).(arity g d t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g a1)))) (\lambda (x: A).(\lambda (H4: (arity g d u x)).(\lambda (_: (arity g d t (asucc g x))).(let H_x \def (leq_asucc g x) in (let H6 \def H_x in (ex_ind A (\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g a1)))) (\lambda (x0: A).(\lambda (H7: (leq g x (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (TLRef n) a1)) (\lambda (a1: A).(arity g c0 (lift (S n) O u) (asucc g a1))) x0 (arity_abst g c0 d u n H0 x0 (arity_repl g d u x H4 (asucc g x0) H7)) (arity_lift g d u (asucc g x0) (arity_repl g d u x H4 (asucc g x0) H7) c0 (S n) O (getl_drop Abst c0 d u n H0))))) H6)))))) H3)))))))))) (\lambda (c0: C).(\lambda (u: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 u t)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))))).(\lambda (b: B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c0 (Bind b) u) t3 t4)).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t3 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4 (asucc g a1))))).(\lambda (t0: T).(\lambda (H4: (ty3 g (CHead c0 (Bind b) u) t4 t0)).(\lambda (H5: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t0 (asucc g a1))))).(let H6 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 u a1)) (\lambda (a1: A).(arity g c0 t (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) (asucc g a1)))) (\lambda (x: A).(\lambda (H7: (arity g c0 u x)).(\lambda (_: (arity g c0 t (asucc g x))).(let H9 \def H3 in (ex2_ind A (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t3 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b) u) t4 (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) (asucc g a1)))) (\lambda (x0: A).(\lambda (H10: (arity g (CHead c0 (Bind b) u) t3 x0)).(\lambda (H11: (arity g (CHead c0 (Bind b) u) t4 (asucc g x0))).(let H_x \def (leq_asucc g x) in (let H12 \def H_x in (ex_ind A (\lambda (a0: A).(leq g x (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Bind b) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b) u t4) (asucc g a1)))) (\lambda (x1: A).(\lambda (H13: (leq g x (asucc g x1))).((match b return (\lambda (_: ?).(\lambda (b0: B).((ty3 g (CHead c0 (Bind b0) u) t4 t0) \to ((ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind b0) u) t4 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind b0) u) t0 (asucc g a1)))) \to ((arity g (CHead c0 (Bind b0) u) t3 x0) \to ((arity g (CHead c0 (Bind b0) u) t4 (asucc g x0)) \to (ex2 A (\lambda (a1: A).(arity g c0 (THead (Bind b0) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind b0) u t4) (asucc g a1)))))))))) with [Abbr \Rightarrow (\lambda (_: (ty3 g (CHead c0 (Bind Abbr) u) t4 t0)).(\lambda (_: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind Abbr) u) t4 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind Abbr) u) t0 (asucc g a1))))).(\lambda (H16: (arity g (CHead c0 (Bind Abbr) u) t3 x0)).(\lambda (H17: (arity g (CHead c0 (Bind Abbr) u) t4 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abbr) u t4) (asucc g a1))) x0 (arity_bind g Abbr not_abbr_abst c0 u x H7 t3 x0 H16) (arity_bind g Abbr not_abbr_abst c0 u x H7 t4 (asucc g x0) H17)))))) | Abst \Rightarrow (\lambda (_: (ty3 g (CHead c0 (Bind Abst) u) t4 t0)).(\lambda (_: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind Abst) u) t4 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind Abst) u) t0 (asucc g a1))))).(\lambda (H16: (arity g (CHead c0 (Bind Abst) u) t3 x0)).(\lambda (H17: (arity g (CHead c0 (Bind Abst) u) t4 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t4) (asucc g a1))) (AHead x1 x0) (arity_head g c0 u x1 (arity_repl g c0 u x H7 (asucc g x1) H13) t3 x0 H16) (arity_repl g c0 (THead (Bind Abst) u t4) (AHead x1 (asucc g x0)) (arity_head g c0 u x1 (arity_repl g c0 u x H7 (asucc g x1) H13) t4 (asucc g x0) H17) (asucc g (AHead x1 x0)) (leq_refl g (asucc g (AHead x1 x0))))))))) | Void \Rightarrow (\lambda (_: (ty3 g (CHead c0 (Bind Void) u) t4 t0)).(\lambda (_: (ex2 A (\lambda (a1: A).(arity g (CHead c0 (Bind Void) u) t4 a1)) (\lambda (a1: A).(arity g (CHead c0 (Bind Void) u) t0 (asucc g a1))))).(\lambda (H16: (arity g (CHead c0 (Bind Void) u) t3 x0)).(\lambda (H17: (arity g (CHead c0 (Bind Void) u) t4 (asucc g x0))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Bind Void) u t3) a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Void) u t4) (asucc g a1))) x0 (arity_bind g Void not_void_abst c0 u x H7 t3 x0 H16) (arity_bind g Void not_void_abst c0 u x H7 t4 (asucc g x0) H17))))))]) H4 H5 H10 H11))) H12)))))) H9))))) H6))))))))))))))) (\lambda (c0: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3 g c0 w u)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity g c0 w a1)) (\lambda (a1: A).(arity g c0 u (asucc g a1))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c0 v (THead (Bind Abst) u t))).(\lambda (H3: (ex2 A (\lambda (a1: A).(arity g c0 v a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t) (asucc g a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 w a1)) (\lambda (a1: A).(arity g c0 u (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 w x)).(\lambda (H6: (arity g c0 u (asucc g x))).(let H7 \def H3 in (ex2_ind A (\lambda (a1: A).(arity g c0 v a1)) (\lambda (a1: A).(arity g c0 (THead (Bind Abst) u t) (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x0: A).(\lambda (H8: (arity g c0 v x0)).(\lambda (H9: (arity g c0 (THead (Bind Abst) u t) (asucc g x0))).(let H10 \def (arity_gen_abst g c0 u t (asucc g x0) H9) in (ex3_2_ind A A (\lambda (a1: A).(\lambda (a2: A).(eq A (asucc g x0) (AHead a1 a2)))) (\lambda (a1: A).(\lambda (_: A).(arity g c0 u (asucc g a1)))) (\lambda (_: A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x1: A).(\lambda (x2: A).(\lambda (H11: (eq A (asucc g x0) (AHead x1 x2))).(\lambda (H12: (arity g c0 u (asucc g x1))).(\lambda (H13: (arity g (CHead c0 (Bind Abst) u) t x2)).(let H14 \def (sym_equal A (asucc g x0) (AHead x1 x2) H11) in (let H15 \def (asucc_gen_head g x1 x2 x0 H14) in (ex2_ind A (\lambda (a0: A).(eq A x0 (AHead x1 a0))) (\lambda (a0: A).(eq A x2 (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x3: A).(\lambda (H16: (eq A x0 (AHead x1 x3))).(\lambda (H17: (eq A x2 (asucc g x3))).(let H18 \def (eq_ind A x2 (\lambda (a: A).(arity g (CHead c0 (Bind Abst) u) t a)) H13 (asucc g x3) H17) in (let H19 \def (eq_ind A x0 (\lambda (a: A).(arity g c0 v a)) H8 (AHead x1 x3) H16) in (ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g a1))) x3 (arity_appl g c0 w x1 (arity_repl g c0 w x H5 x1 (leq_sym g x1 x (asucc_inj g x1 x (arity_mono g c0 u (asucc g x1) H12 (asucc g x) H6)))) v x3 H19) (arity_appl g c0 w x H5 (THead (Bind Abst) u t) (asucc g x3) (arity_head g c0 u x H6 t (asucc g x3) H18)))))))) H15)))))))) H10))))) H7))))) H4))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c0 t3 t4)).(\lambda (H1: (ex2 A (\lambda (a1: A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))))).(\lambda (t0: T).(\lambda (_: (ty3 g c0 t4 t0)).(\lambda (_: (ex2 A (\lambda (a1: A).(arity g c0 t4 a1)) (\lambda (a1: A).(arity g c0 t0 (asucc g a1))))).(let H4 \def H1 in (ex2_ind A (\lambda (a1: A).(arity g c0 t3 a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1)))) (\lambda (x: A).(\lambda (H5: (arity g c0 t3 x)).(\lambda (H6: (arity g c0 t4 (asucc g x))).(ex_intro2 A (\lambda (a1: A).(arity g c0 (THead (Flat Cast) t4 t3) a1)) (\lambda (a1: A).(arity g c0 t4 (asucc g a1))) x (arity_cast g c0 t4 x H6 t3 H5) H6)))) H4)))))))))) c t1 t2 H))))).
 
 theorem ty3_predicative:
  \forall (g: G).(\forall (c: C).(\forall (v: T).(\forall (t: T).(\forall (u: T).((ty3 g c (THead (Bind Abst) v t) u) \to ((pc3 c u v) \to (\forall (P: Prop).P)))))))
@@ -3161,5 +3161,5 @@ theorem pc3_abst_dec:
 theorem ty3_inference:
  \forall (g: G).(\forall (c: C).(\forall (t1: T).(or (ex T (\lambda (t2: T).(ty3 g c t1 t2))) (\forall (t2: T).((ty3 g c t1 t2) \to (\forall (P: Prop).P))))))
 \def
- \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(flt_wf_ind (\lambda (c0: C).(\lambda (t: T).(or (ex T (\lambda (t2: T).(ty3 g c0 t t2))) (\forall (t2: T).((ty3 g c0 t t2) \to (\forall (P: Prop).P)))))) (\lambda (c2: C).(\lambda (t2: T).(match t2 return (\lambda (t: T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P: Prop).P)))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to (\forall (P: Prop).P)))))) with [(TSort n) \Rightarrow (\lambda (_: ((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (TSort n)) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P: Prop).P))))))))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TSort n) t3))) (\forall (t3: T).((ty3 g c2 (TSort n) t3) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 (TSort n) t3)) (TSort (next g n)) (ty3_sort g c2 n)))) | (TLRef n) \Rightarrow (\lambda (H: ((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (TLRef n)) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P: Prop).P))))))))).(let H_x \def (getl_dec c2 n) in (let H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl n c2 (CHead e (Bind b) v)))))) (\forall (d: C).((getl n c2 d) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P)))) (\lambda (H1: (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl n c2 (CHead e (Bind b) v))))))).(ex_3_ind C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl n c2 (CHead e (Bind b) v))))) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P)))) (\lambda (x0: C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl n c2 (CHead x0 (Bind x1) x2))).(let H3 \def (H x0 x2 (getl_flt x1 c2 x0 x2 n H2)) in (or_ind (ex T (\lambda (t3: T).(ty3 g x0 x2 t3))) (\forall (t3: T).((ty3 g x0 x2 t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P)))) (\lambda (H4: (ex T (\lambda (t2: T).(ty3 g x0 x2 t2)))).(ex_ind T (\lambda (t3: T).(ty3 g x0 x2 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P)))) (\lambda (x: T).(\lambda (H5: (ty3 g x0 x2 x)).((match x1 return (\lambda (b: B).((getl n c2 (CHead x0 (Bind b) x2)) \to (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P)))))) with [Abbr \Rightarrow (\lambda (H6: (getl n c2 (CHead x0 (Bind Abbr) x2))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3)) (lift (S n) O x) (ty3_abbr g n c2 x0 x2 H6 x H5)))) | Abst \Rightarrow (\lambda (H6: (getl n c2 (CHead x0 (Bind Abst) x2))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3)) (lift (S n) O x2) (ty3_abst g n c2 x0 x2 H6 x H5)))) | Void \Rightarrow (\lambda (H6: (getl n c2 (CHead x0 (Bind Void) x2))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H7: (ty3 g c2 (TLRef n) t3)).(\lambda (P: Prop).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) P (\lambda (H8: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) P (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (lift (S n) O x5) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind Abbr) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (c: C).(getl n c2 c)) H6 (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10)) in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match ee return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10)) in (False_ind P H13))))))))) H8)) (\lambda (H8: (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) P (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (lift (S n) O x4) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind Abst) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (c: C).(getl n c2 c)) H6 (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10)) in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match ee return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10)) in (False_ind P H13))))))))) H8)) (ty3_gen_lref g c2 t3 n H7)))))))]) H2))) H4)) (\lambda (H4: ((\forall (t2: T).((ty3 g x0 x2 t2) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H5: (ty3 g c2 (TLRef n) t3)).(\lambda (P: Prop).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) P (\lambda (H6: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) P (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (lift (S n) O x5) t3)).(\lambda (H8: (getl n c2 (CHead x3 (Bind Abbr) x4))).(\lambda (H9: (ty3 g x3 x4 x5)).(let H10 \def (eq_ind C (CHead x0 (Bind x1) x2) (\lambda (c: C).(getl n c2 c)) H2 (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in (let H11 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow x0 | (CHead c _ _) \Rightarrow c])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in ((let H12 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow x1 | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in ((let H13 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow x2 | (CHead _ _ t) \Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in (\lambda (_: (eq B x1 Abbr)).(\lambda (H15: (eq C x0 x3)).(let H16 \def (eq_ind_r T x4 (\lambda (t: T).(getl n c2 (CHead x3 (Bind Abbr) t))) H10 x2 H13) in (let H17 \def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5)) H9 x2 H13) in (let H18 \def (eq_ind_r C x3 (\lambda (c: C).(getl n c2 (CHead c (Bind Abbr) x2))) H16 x0 H15) in (let H19 \def (eq_ind_r C x3 (\lambda (c: C).(ty3 g c x2 x5)) H17 x0 H15) in (H4 x5 H19 P)))))))) H12)) H11))))))))) H6)) (\lambda (H6: (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) P (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H7: (pc3 c2 (lift (S n) O x4) t3)).(\lambda (H8: (getl n c2 (CHead x3 (Bind Abst) x4))).(\lambda (H9: (ty3 g x3 x4 x5)).(let H10 \def (eq_ind C (CHead x0 (Bind x1) x2) (\lambda (c: C).(getl n c2 c)) H2 (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in (let H11 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow x0 | (CHead c _ _) \Rightarrow c])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in ((let H12 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow x1 | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in ((let H13 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow x2 | (CHead _ _ t) \Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in (\lambda (_: (eq B x1 Abst)).(\lambda (H15: (eq C x0 x3)).(let H16 \def (eq_ind_r T x4 (\lambda (t: T).(getl n c2 (CHead x3 (Bind Abst) t))) H10 x2 H13) in (let H17 \def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5)) H9 x2 H13) in (let H18 \def (eq_ind_r T x4 (\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)) H7 x2 H13) in (let H19 \def (eq_ind_r C x3 (\lambda (c: C).(getl n c2 (CHead c (Bind Abst) x2))) H16 x0 H15) in (let H20 \def (eq_ind_r C x3 (\lambda (c: C).(ty3 g c x2 x5)) H17 x0 H15) in (H4 x5 H20 P))))))))) H12)) H11))))))))) H6)) (ty3_gen_lref g c2 t3 n H5))))))) H3)))))) H1)) (\lambda (H1: ((\forall (d: C).((getl n c2 d) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H2: (ty3 g c2 (TLRef n) t3)).(\lambda (P: Prop).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) P (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) P (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2 (lift (S n) O x2) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abbr) x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abbr) x1) H5 P))))))) H3)) (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) P (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2 (lift (S n) O x1) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abst) x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abst) x1) H5 P))))))) H3)) (ty3_gen_lref g c2 t3 n H2))))))) H0)))) | (THead k t t0) \Rightarrow (\lambda (H: ((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead k t t0)) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P: Prop).P))))))))).((match k return (\lambda (k0: K).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead k0 t t0)) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P: Prop).P)))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 (THead k0 t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead k0 t t0) t3) \to (\forall (P: Prop).P)))))) with [(Bind b) \Rightarrow (\lambda (H0: ((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead (Bind b) t t0)) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P: Prop).P))))))))).(let H1 \def (H0 c2 t (flt_thead_sx (Bind b) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H2: (ex T (\lambda (t2: T).(ty3 g c2 t t2)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x: T).(\lambda (H3: (ty3 g c2 t x)).(let H4 \def (H0 (CHead c2 (Bind b) t) t0 (flt_shift (Bind b) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) t0 t3))) (\forall (t3: T).((ty3 g (CHead c2 (Bind b) t) t0 t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H5: (ex T (\lambda (t2: T).(ty3 g (CHead c2 (Bind b) t) t0 t2)))).(ex_ind T (\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) t0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x0: T).(\lambda (H6: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) x0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x1: T).(\lambda (H7: (ty3 g (CHead c2 (Bind b) t) x0 x1)).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3)) (THead (Bind b) t x0) (ty3_bind g c2 t x H3 b t0 x0 H6 x1 H7))))) (ty3_correct g (CHead c2 (Bind b) t) t0 x0 H6)))) H5)) (\lambda (H5: ((\forall (t2: T).((ty3 g (CHead c2 (Bind b) t) t0 t2) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H6: (ty3 g c2 (THead (Bind b) t t0) t3)).(\lambda (P: Prop).(ex4_3_ind T T T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 c2 (THead (Bind b) t t4) t3)))) (\lambda (_: T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c2 t t5)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead c2 (Bind b) t) t4 t6)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Bind b) t x0) t3)).(\lambda (_: (ty3 g c2 t x1)).(\lambda (H9: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(\lambda (_: (ty3 g (CHead c2 (Bind b) t) x0 x2)).(H5 x0 H9 P)))))))) (ty3_gen_bind g b c2 t t0 t3 H6))))))) H4)))) H2)) (\lambda (H2: ((\forall (t2: T).((ty3 g c2 t t2) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H3: (ty3 g c2 (THead (Bind b) t t0) t3)).(\lambda (P: Prop).(ex4_3_ind T T T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 c2 (THead (Bind b) t t4) t3)))) (\lambda (_: T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c2 t t5)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead c2 (Bind b) t) t4 t6)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Bind b) t x0) t3)).(\lambda (H5: (ty3 g c2 t x1)).(\lambda (_: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(\lambda (_: (ty3 g (CHead c2 (Bind b) t) x0 x2)).(H2 x1 H5 P)))))))) (ty3_gen_bind g b c2 t t0 t3 H3))))))) H1))) | (Flat f) \Rightarrow (\lambda (H0: ((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead (Flat f) t t0)) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P: Prop).P))))))))).((match f return (\lambda (f0: F).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead (Flat f0) t t0)) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P: Prop).P)))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat f0) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat f0) t t0) t3) \to (\forall (P: Prop).P)))))) with [Appl \Rightarrow (\lambda (H1: ((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead (Flat Appl) t t0)) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P: Prop).P))))))))).(let H2 \def (H1 c2 t (flt_thead_sx (Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H3: (ex T (\lambda (t2: T).(ty3 g c2 t t2)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x: T).(\lambda (H4: (ty3 g c2 t x)).(let H5 \def (H1 c2 t0 (flt_thead_dx (Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H6: (ex T (\lambda (t2: T).(ty3 g c2 t0 t2)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x0: T).(\lambda (H7: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x1: T).(\lambda (H8: (ty3 g c2 x0 x1)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x2: T).(\lambda (H9: (ty3 g c2 x x2)).(let H10 \def (ty3_sn3 g c2 x x2 H9) in (let H_x \def (nf2_sn3 c2 x H10) in (let H11 \def H_x in (ex2_ind T (\lambda (u: T).(pr3 c2 x u)) (\lambda (u: T).(nf2 c2 u)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x3: T).(\lambda (H12: (pr3 c2 x x3)).(\lambda (H13: (nf2 c2 x3)).(let H14 \def (ty3_sred_pr3 c2 x x3 H12 g x2 H9) in (let H_x0 \def (pc3_abst_dec g c2 x0 x1 H8 x3 x2 H14) in (let H15 \def H_x0 in (or_ind (ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2 v2)))) (\forall (u: T).((pc3 c2 x0 (THead (Bind Abst) x3 u)) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H16: (ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2 v2))))).(ex4_2_ind T T (\lambda (u: T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2 v2))) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H17: (pc3 c2 x0 (THead (Bind Abst) x3 x4))).(\lambda (H18: (ty3 g c2 (THead (Bind Abst) x5 x4) x1)).(\lambda (H19: (pr3 c2 x3 x5)).(\lambda (_: (nf2 c2 x5)).(let H_y \def (nf2_pr3_unfold c2 x3 x5 H19 H13) in (let H21 \def (eq_ind_r T x5 (\lambda (t: T).(pr3 c2 x3 t)) H19 x3 H_y) in (let H22 \def (eq_ind_r T x5 (\lambda (t: T).(ty3 g c2 (THead (Bind Abst) t x4) x1)) H18 x3 H_y) in (or_introl (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3)) (THead (Flat Appl) t (THead (Bind Abst) x3 x4)) (ty3_appl g c2 t x3 (ty3_tred g c2 t x H4 x3 H12) t0 x4 (ty3_conv g c2 (THead (Bind Abst) x3 x4) x1 H22 t0 x0 H7 H17))))))))))))) H16)) (\lambda (H16: ((\forall (u: T).((pc3 c2 x0 (THead (Bind Abst) x3 u)) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H17: (ty3 g c2 (THead (Flat Appl) t t0) t3)).(\lambda (P: Prop).(ex3_2_ind T T (\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 t u))) P (\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) x4 x5)) t3)).(\lambda (H19: (ty3 g c2 t0 (THead (Bind Abst) x4 x5))).(\lambda (H20: (ty3 g c2 t x4)).(let H_y \def (ty3_unique g c2 t x4 H20 x H4) in (let H_y0 \def (ty3_unique g c2 t0 (THead (Bind Abst) x4 x5) H19 x0 H7) in (H16 x5 (pc3_t (THead (Bind Abst) x4 x5) c2 x0 (pc3_s c2 x0 (THead (Bind Abst) x4 x5) H_y0) (THead (Bind Abst) x3 x5) (pc3_head_1 c2 x4 x3 (pc3_t x c2 x4 H_y x3 (pc3_pr3_r c2 x x3 H12)) (Bind Abst) x5)) P)))))))) (ty3_gen_appl g c2 t t0 t3 H17))))))) H15))))))) H11)))))) (ty3_correct g c2 t x H4)))) (ty3_correct g c2 t0 x0 H7)))) H6)) (\lambda (H6: ((\forall (t2: T).((ty3 g c2 t0 t2) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H7: (ty3 g c2 (THead (Flat Appl) t t0) t3)).(\lambda (P: Prop).(ex3_2_ind T T (\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 t u))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) t3)).(\lambda (H9: (ty3 g c2 t0 (THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c2 t x0)).(H6 (THead (Bind Abst) x0 x1) H9 P)))))) (ty3_gen_appl g c2 t t0 t3 H7))))))) H5)))) H3)) (\lambda (H3: ((\forall (t2: T).((ty3 g c2 t t2) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H4: (ty3 g c2 (THead (Flat Appl) t t0) t3)).(\lambda (P: Prop).(ex3_2_ind T T (\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 t u))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) t3)).(\lambda (_: (ty3 g c2 t0 (THead (Bind Abst) x0 x1))).(\lambda (H7: (ty3 g c2 t x0)).(H3 x0 H7 P)))))) (ty3_gen_appl g c2 t t0 t3 H4))))))) H2))) | Cast \Rightarrow (\lambda (H1: ((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead (Flat Cast) t t0)) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P: Prop).P))))))))).(let H2 \def (H1 c2 t (flt_thead_sx (Flat Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H3: (ex T (\lambda (t2: T).(ty3 g c2 t t2)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x: T).(\lambda (H4: (ty3 g c2 t x)).(let H5 \def (H1 c2 t0 (flt_thead_dx (Flat Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H6: (ex T (\lambda (t2: T).(ty3 g c2 t0 t2)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x0: T).(\lambda (H7: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x1: T).(\lambda (H8: (ty3 g c2 x0 x1)).(let H_x \def (pc3_dec g c2 x0 x1 H8 t x H4) in (let H9 \def H_x in (or_ind (pc3 c2 x0 t) ((pc3 c2 x0 t) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H10: (pc3 c2 x0 t)).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3)) t (ty3_cast g c2 t0 t (ty3_conv g c2 t x H4 t0 x0 H7 H10) x H4)))) (\lambda (H10: (((pc3 c2 x0 t) \to (\forall (P: Prop).P)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H11: (ty3 g c2 (THead (Flat Cast) t t0) t3)).(\lambda (P: Prop).(and_ind (pc3 c2 t t3) (ty3 g c2 t0 t) P (\lambda (_: (pc3 c2 t t3)).(\lambda (H13: (ty3 g c2 t0 t)).(let H_y \def (ty3_unique g c2 t0 t H13 x0 H7) in (H10 (pc3_s c2 x0 t H_y) P)))) (ty3_gen_cast g c2 t0 t t3 H11))))))) H9))))) (ty3_correct g c2 t0 x0 H7)))) H6)) (\lambda (H6: ((\forall (t2: T).((ty3 g c2 t0 t2) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H7: (ty3 g c2 (THead (Flat Cast) t t0) t3)).(\lambda (P: Prop).(and_ind (pc3 c2 t t3) (ty3 g c2 t0 t) P (\lambda (_: (pc3 c2 t t3)).(\lambda (H9: (ty3 g c2 t0 t)).(H6 t H9 P))) (ty3_gen_cast g c2 t0 t t3 H7))))))) H5)))) H3)) (\lambda (H3: ((\forall (t2: T).((ty3 g c2 t t2) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H4: (ty3 g c2 (THead (Flat Cast) t t0) t3)).(\lambda (P: Prop).(and_ind (pc3 c2 t t3) (ty3 g c2 t0 t) P (\lambda (_: (pc3 c2 t t3)).(\lambda (H6: (ty3 g c2 t0 t)).(ex_ind T (\lambda (t4: T).(ty3 g c2 t t4)) P (\lambda (x: T).(\lambda (H7: (ty3 g c2 t x)).(H3 x H7 P))) (ty3_correct g c2 t0 t H6)))) (ty3_gen_cast g c2 t0 t t3 H4))))))) H2)))]) H0))]) H))]))) c t1))).
+ \lambda (g: G).(\lambda (c: C).(\lambda (t1: T).(flt_wf_ind (\lambda (c0: C).(\lambda (t: T).(or (ex T (\lambda (t2: T).(ty3 g c0 t t2))) (\forall (t2: T).((ty3 g c0 t t2) \to (\forall (P: Prop).P)))))) (\lambda (c2: C).(\lambda (t2: T).(match t2 return (\lambda (_: ?).(\lambda (t: T).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 t) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P: Prop).P)))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to (\forall (P: Prop).P))))))) with [(TSort n) \Rightarrow (\lambda (_: ((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (TSort n)) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P: Prop).P))))))))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TSort n) t3))) (\forall (t3: T).((ty3 g c2 (TSort n) t3) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 (TSort n) t3)) (TSort (next g n)) (ty3_sort g c2 n)))) | (TLRef n) \Rightarrow (\lambda (H: ((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (TLRef n)) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P: Prop).P))))))))).(let H_x \def (getl_dec c2 n) in (let H0 \def H_x in (or_ind (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl n c2 (CHead e (Bind b) v)))))) (\forall (d: C).((getl n c2 d) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P)))) (\lambda (H1: (ex_3 C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl n c2 (CHead e (Bind b) v))))))).(ex_3_ind C B T (\lambda (e: C).(\lambda (b: B).(\lambda (v: T).(getl n c2 (CHead e (Bind b) v))))) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P)))) (\lambda (x0: C).(\lambda (x1: B).(\lambda (x2: T).(\lambda (H2: (getl n c2 (CHead x0 (Bind x1) x2))).(let H3 \def (H x0 x2 (getl_flt x1 c2 x0 x2 n H2)) in (or_ind (ex T (\lambda (t3: T).(ty3 g x0 x2 t3))) (\forall (t3: T).((ty3 g x0 x2 t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P)))) (\lambda (H4: (ex T (\lambda (t2: T).(ty3 g x0 x2 t2)))).(ex_ind T (\lambda (t3: T).(ty3 g x0 x2 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P)))) (\lambda (x: T).(\lambda (H5: (ty3 g x0 x2 x)).((match x1 return (\lambda (_: ?).(\lambda (b: B).((getl n c2 (CHead x0 (Bind b) x2)) \to (or (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P))))))) with [Abbr \Rightarrow (\lambda (H6: (getl n c2 (CHead x0 (Bind Abbr) x2))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3)) (lift (S n) O x) (ty3_abbr g n c2 x0 x2 H6 x H5)))) | Abst \Rightarrow (\lambda (H6: (getl n c2 (CHead x0 (Bind Abst) x2))).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3)) (lift (S n) O x2) (ty3_abst g n c2 x0 x2 H6 x H5)))) | Void \Rightarrow (\lambda (H6: (getl n c2 (CHead x0 (Bind Void) x2))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H7: (ty3 g c2 (TLRef n) t3)).(\lambda (P: Prop).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) P (\lambda (H8: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) P (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (lift (S n) O x5) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind Abbr) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (c: C).(getl n c2 c)) H6 (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10)) in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match ee return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abbr) x4) H10)) in (False_ind P H13))))))))) H8)) (\lambda (H8: (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) P (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (lift (S n) O x4) t3)).(\lambda (H10: (getl n c2 (CHead x3 (Bind Abst) x4))).(\lambda (_: (ty3 g x3 x4 x5)).(let H12 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (c: C).(getl n c2 c)) H6 (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10)) in (let H13 \def (eq_ind C (CHead x0 (Bind Void) x2) (\lambda (ee: C).(match ee return (\lambda (_: ?).Prop) with [(CSort _) \Rightarrow False | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).Prop) with [(Bind b) \Rightarrow (match b return (\lambda (_: ?).Prop) with [Abbr \Rightarrow False | Abst \Rightarrow False | Void \Rightarrow True]) | (Flat _) \Rightarrow False])])) I (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind Void) x2) n H6 (CHead x3 (Bind Abst) x4) H10)) in (False_ind P H13))))))))) H8)) (ty3_gen_lref g c2 t3 n H7)))))))]) H2))) H4)) (\lambda (H4: ((\forall (t2: T).((ty3 g x0 x2 t2) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H5: (ty3 g c2 (TLRef n) t3)).(\lambda (P: Prop).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) P (\lambda (H6: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) P (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (lift (S n) O x5) t3)).(\lambda (H8: (getl n c2 (CHead x3 (Bind Abbr) x4))).(\lambda (H9: (ty3 g x3 x4 x5)).(let H10 \def (eq_ind C (CHead x0 (Bind x1) x2) (\lambda (c: C).(getl n c2 c)) H2 (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in (let H11 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow x0 | (CHead c _ _) \Rightarrow c])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in ((let H12 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow x1 | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in ((let H13 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow x2 | (CHead _ _ t) \Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind Abbr) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abbr) x4) H8)) in (\lambda (_: (eq B x1 Abbr)).(\lambda (H15: (eq C x0 x3)).(let H16 \def (eq_ind_r T x4 (\lambda (t: T).(getl n c2 (CHead x3 (Bind Abbr) t))) H10 x2 H13) in (let H17 \def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5)) H9 x2 H13) in (let H18 \def (eq_ind_r C x3 (\lambda (c: C).(getl n c2 (CHead c (Bind Abbr) x2))) H16 x0 H15) in (let H19 \def (eq_ind_r C x3 (\lambda (c: C).(ty3 g c x2 x5)) H17 x0 H15) in (H4 x5 H19 P)))))))) H12)) H11))))))))) H6)) (\lambda (H6: (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) P (\lambda (x3: C).(\lambda (x4: T).(\lambda (x5: T).(\lambda (H7: (pc3 c2 (lift (S n) O x4) t3)).(\lambda (H8: (getl n c2 (CHead x3 (Bind Abst) x4))).(\lambda (H9: (ty3 g x3 x4 x5)).(let H10 \def (eq_ind C (CHead x0 (Bind x1) x2) (\lambda (c: C).(getl n c2 c)) H2 (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in (let H11 \def (f_equal C C (\lambda (e: C).(match e return (\lambda (_: ?).C) with [(CSort _) \Rightarrow x0 | (CHead c _ _) \Rightarrow c])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in ((let H12 \def (f_equal C B (\lambda (e: C).(match e return (\lambda (_: ?).B) with [(CSort _) \Rightarrow x1 | (CHead _ k _) \Rightarrow (match k return (\lambda (_: ?).B) with [(Bind b) \Rightarrow b | (Flat _) \Rightarrow x1])])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in ((let H13 \def (f_equal C T (\lambda (e: C).(match e return (\lambda (_: ?).T) with [(CSort _) \Rightarrow x2 | (CHead _ _ t) \Rightarrow t])) (CHead x0 (Bind x1) x2) (CHead x3 (Bind Abst) x4) (getl_mono c2 (CHead x0 (Bind x1) x2) n H2 (CHead x3 (Bind Abst) x4) H8)) in (\lambda (_: (eq B x1 Abst)).(\lambda (H15: (eq C x0 x3)).(let H16 \def (eq_ind_r T x4 (\lambda (t: T).(getl n c2 (CHead x3 (Bind Abst) t))) H10 x2 H13) in (let H17 \def (eq_ind_r T x4 (\lambda (t: T).(ty3 g x3 t x5)) H9 x2 H13) in (let H18 \def (eq_ind_r T x4 (\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)) H7 x2 H13) in (let H19 \def (eq_ind_r C x3 (\lambda (c: C).(getl n c2 (CHead c (Bind Abst) x2))) H16 x0 H15) in (let H20 \def (eq_ind_r C x3 (\lambda (c: C).(ty3 g c x2 x5)) H17 x0 H15) in (H4 x5 H20 P))))))))) H12)) H11))))))))) H6)) (ty3_gen_lref g c2 t3 n H5))))))) H3)))))) H1)) (\lambda (H1: ((\forall (d: C).((getl n c2 d) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (TLRef n) t3))) (\forall (t3: T).((ty3 g c2 (TLRef n) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H2: (ty3 g c2 (TLRef n) t3)).(\lambda (P: Prop).(or_ind (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t))))) P (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (_: T).(\lambda (t: T).(pc3 c2 (lift (S n) O t) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abbr) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) P (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2 (lift (S n) O x2) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abbr) x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abbr) x1) H5 P))))))) H3)) (\lambda (H3: (ex3_3 C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))))).(ex3_3_ind C T T (\lambda (_: C).(\lambda (u: T).(\lambda (_: T).(pc3 c2 (lift (S n) O u) t3)))) (\lambda (e: C).(\lambda (u: T).(\lambda (_: T).(getl n c2 (CHead e (Bind Abst) u))))) (\lambda (e: C).(\lambda (u: T).(\lambda (t: T).(ty3 g e u t)))) P (\lambda (x0: C).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2 (lift (S n) O x1) t3)).(\lambda (H5: (getl n c2 (CHead x0 (Bind Abst) x1))).(\lambda (_: (ty3 g x0 x1 x2)).(H1 (CHead x0 (Bind Abst) x1) H5 P))))))) H3)) (ty3_gen_lref g c2 t3 n H2))))))) H0)))) | (THead k t t0) \Rightarrow (\lambda (H: ((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead k t t0)) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P: Prop).P))))))))).((match k return (\lambda (_: ?).(\lambda (k0: K).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead k0 t t0)) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P: Prop).P)))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 (THead k0 t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead k0 t t0) t3) \to (\forall (P: Prop).P))))))) with [(Bind b) \Rightarrow (\lambda (H0: ((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead (Bind b) t t0)) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P: Prop).P))))))))).(let H1 \def (H0 c2 t (flt_thead_sx (Bind b) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H2: (ex T (\lambda (t2: T).(ty3 g c2 t t2)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x: T).(\lambda (H3: (ty3 g c2 t x)).(let H4 \def (H0 (CHead c2 (Bind b) t) t0 (flt_shift (Bind b) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) t0 t3))) (\forall (t3: T).((ty3 g (CHead c2 (Bind b) t) t0 t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H5: (ex T (\lambda (t2: T).(ty3 g (CHead c2 (Bind b) t) t0 t2)))).(ex_ind T (\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) t0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x0: T).(\lambda (H6: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g (CHead c2 (Bind b) t) x0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x1: T).(\lambda (H7: (ty3 g (CHead c2 (Bind b) t) x0 x1)).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3)) (THead (Bind b) t x0) (ty3_bind g c2 t x H3 b t0 x0 H6 x1 H7))))) (ty3_correct g (CHead c2 (Bind b) t) t0 x0 H6)))) H5)) (\lambda (H5: ((\forall (t2: T).((ty3 g (CHead c2 (Bind b) t) t0 t2) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H6: (ty3 g c2 (THead (Bind b) t t0) t3)).(\lambda (P: Prop).(ex4_3_ind T T T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 c2 (THead (Bind b) t t4) t3)))) (\lambda (_: T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c2 t t5)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead c2 (Bind b) t) t4 t6)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Bind b) t x0) t3)).(\lambda (_: (ty3 g c2 t x1)).(\lambda (H9: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(\lambda (_: (ty3 g (CHead c2 (Bind b) t) x0 x2)).(H5 x0 H9 P)))))))) (ty3_gen_bind g b c2 t t0 t3 H6))))))) H4)))) H2)) (\lambda (H2: ((\forall (t2: T).((ty3 g c2 t t2) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Bind b) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Bind b) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H3: (ty3 g c2 (THead (Bind b) t t0) t3)).(\lambda (P: Prop).(ex4_3_ind T T T (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(pc3 c2 (THead (Bind b) t t4) t3)))) (\lambda (_: T).(\lambda (t5: T).(\lambda (_: T).(ty3 g c2 t t5)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (_: T).(ty3 g (CHead c2 (Bind b) t) t0 t4)))) (\lambda (t4: T).(\lambda (_: T).(\lambda (t6: T).(ty3 g (CHead c2 (Bind b) t) t4 t6)))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (x2: T).(\lambda (_: (pc3 c2 (THead (Bind b) t x0) t3)).(\lambda (H5: (ty3 g c2 t x1)).(\lambda (_: (ty3 g (CHead c2 (Bind b) t) t0 x0)).(\lambda (_: (ty3 g (CHead c2 (Bind b) t) x0 x2)).(H2 x1 H5 P)))))))) (ty3_gen_bind g b c2 t t0 t3 H3))))))) H1))) | (Flat f) \Rightarrow (\lambda (H0: ((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead (Flat f) t t0)) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P: Prop).P))))))))).((match f return (\lambda (_: ?).(\lambda (f0: F).(((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead (Flat f0) t t0)) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P: Prop).P)))))))) \to (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat f0) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat f0) t t0) t3) \to (\forall (P: Prop).P))))))) with [Appl \Rightarrow (\lambda (H1: ((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead (Flat Appl) t t0)) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P: Prop).P))))))))).(let H2 \def (H1 c2 t (flt_thead_sx (Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H3: (ex T (\lambda (t2: T).(ty3 g c2 t t2)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x: T).(\lambda (H4: (ty3 g c2 t x)).(let H5 \def (H1 c2 t0 (flt_thead_dx (Flat Appl) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H6: (ex T (\lambda (t2: T).(ty3 g c2 t0 t2)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x0: T).(\lambda (H7: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x1: T).(\lambda (H8: (ty3 g c2 x0 x1)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x2: T).(\lambda (H9: (ty3 g c2 x x2)).(let H10 \def (ty3_sn3 g c2 x x2 H9) in (let H_x \def (nf2_sn3 c2 x H10) in (let H11 \def H_x in (ex2_ind T (\lambda (u: T).(pr3 c2 x u)) (\lambda (u: T).(nf2 c2 u)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x3: T).(\lambda (H12: (pr3 c2 x x3)).(\lambda (H13: (nf2 c2 x3)).(let H14 \def (ty3_sred_pr3 c2 x x3 H12 g x2 H9) in (let H_x0 \def (pc3_abst_dec g c2 x0 x1 H8 x3 x2 H14) in (let H15 \def H_x0 in (or_ind (ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2 v2)))) (\forall (u: T).((pc3 c2 x0 (THead (Bind Abst) x3 u)) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H16: (ex4_2 T T (\lambda (u: T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2 v2))))).(ex4_2_ind T T (\lambda (u: T).(\lambda (_: T).(pc3 c2 x0 (THead (Bind Abst) x3 u)))) (\lambda (u: T).(\lambda (v2: T).(ty3 g c2 (THead (Bind Abst) v2 u) x1))) (\lambda (_: T).(\lambda (v2: T).(pr3 c2 x3 v2))) (\lambda (_: T).(\lambda (v2: T).(nf2 c2 v2))) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H17: (pc3 c2 x0 (THead (Bind Abst) x3 x4))).(\lambda (H18: (ty3 g c2 (THead (Bind Abst) x5 x4) x1)).(\lambda (H19: (pr3 c2 x3 x5)).(\lambda (_: (nf2 c2 x5)).(let H_y \def (nf2_pr3_unfold c2 x3 x5 H19 H13) in (let H21 \def (eq_ind_r T x5 (\lambda (t: T).(pr3 c2 x3 t)) H19 x3 H_y) in (let H22 \def (eq_ind_r T x5 (\lambda (t: T).(ty3 g c2 (THead (Bind Abst) t x4) x1)) H18 x3 H_y) in (or_introl (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3)) (THead (Flat Appl) t (THead (Bind Abst) x3 x4)) (ty3_appl g c2 t x3 (ty3_tred g c2 t x H4 x3 H12) t0 x4 (ty3_conv g c2 (THead (Bind Abst) x3 x4) x1 H22 t0 x0 H7 H17))))))))))))) H16)) (\lambda (H16: ((\forall (u: T).((pc3 c2 x0 (THead (Bind Abst) x3 u)) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H17: (ty3 g c2 (THead (Flat Appl) t t0) t3)).(\lambda (P: Prop).(ex3_2_ind T T (\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 t u))) P (\lambda (x4: T).(\lambda (x5: T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) x4 x5)) t3)).(\lambda (H19: (ty3 g c2 t0 (THead (Bind Abst) x4 x5))).(\lambda (H20: (ty3 g c2 t x4)).(let H_y \def (ty3_unique g c2 t x4 H20 x H4) in (let H_y0 \def (ty3_unique g c2 t0 (THead (Bind Abst) x4 x5) H19 x0 H7) in (H16 x5 (pc3_t (THead (Bind Abst) x4 x5) c2 x0 (pc3_s c2 x0 (THead (Bind Abst) x4 x5) H_y0) (THead (Bind Abst) x3 x5) (pc3_head_1 c2 x4 x3 (pc3_t x c2 x4 H_y x3 (pc3_pr3_r c2 x x3 H12)) (Bind Abst) x5)) P)))))))) (ty3_gen_appl g c2 t t0 t3 H17))))))) H15))))))) H11)))))) (ty3_correct g c2 t x H4)))) (ty3_correct g c2 t0 x0 H7)))) H6)) (\lambda (H6: ((\forall (t2: T).((ty3 g c2 t0 t2) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H7: (ty3 g c2 (THead (Flat Appl) t t0) t3)).(\lambda (P: Prop).(ex3_2_ind T T (\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 t u))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) t3)).(\lambda (H9: (ty3 g c2 t0 (THead (Bind Abst) x0 x1))).(\lambda (_: (ty3 g c2 t x0)).(H6 (THead (Bind Abst) x0 x1) H9 P)))))) (ty3_gen_appl g c2 t t0 t3 H7))))))) H5)))) H3)) (\lambda (H3: ((\forall (t2: T).((ty3 g c2 t t2) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Appl) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Appl) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H4: (ty3 g c2 (THead (Flat Appl) t t0) t3)).(\lambda (P: Prop).(ex3_2_ind T T (\lambda (u: T).(\lambda (t4: T).(pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) u t4)) t3))) (\lambda (u: T).(\lambda (t4: T).(ty3 g c2 t0 (THead (Bind Abst) u t4)))) (\lambda (u: T).(\lambda (_: T).(ty3 g c2 t u))) P (\lambda (x0: T).(\lambda (x1: T).(\lambda (_: (pc3 c2 (THead (Flat Appl) t (THead (Bind Abst) x0 x1)) t3)).(\lambda (_: (ty3 g c2 t0 (THead (Bind Abst) x0 x1))).(\lambda (H7: (ty3 g c2 t x0)).(H3 x0 H7 P)))))) (ty3_gen_appl g c2 t t0 t3 H4))))))) H2))) | Cast \Rightarrow (\lambda (H1: ((\forall (c1: C).(\forall (t1: T).((flt c1 t1 c2 (THead (Flat Cast) t t0)) \to (or (ex T (\lambda (t2: T).(ty3 g c1 t1 t2))) (\forall (t2: T).((ty3 g c1 t1 t2) \to (\forall (P: Prop).P))))))))).(let H2 \def (H1 c2 t (flt_thead_sx (Flat Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g c2 t t3))) (\forall (t3: T).((ty3 g c2 t t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H3: (ex T (\lambda (t2: T).(ty3 g c2 t t2)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x: T).(\lambda (H4: (ty3 g c2 t x)).(let H5 \def (H1 c2 t0 (flt_thead_dx (Flat Cast) c2 t t0)) in (or_ind (ex T (\lambda (t3: T).(ty3 g c2 t0 t3))) (\forall (t3: T).((ty3 g c2 t0 t3) \to (\forall (P: Prop).P))) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H6: (ex T (\lambda (t2: T).(ty3 g c2 t0 t2)))).(ex_ind T (\lambda (t3: T).(ty3 g c2 t0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x0: T).(\lambda (H7: (ty3 g c2 t0 x0)).(ex_ind T (\lambda (t3: T).(ty3 g c2 x0 t3)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (x1: T).(\lambda (H8: (ty3 g c2 x0 x1)).(let H_x \def (pc3_dec g c2 x0 x1 H8 t x H4) in (let H9 \def H_x in (or_ind (pc3 c2 x0 t) ((pc3 c2 x0 t) \to (\forall (P: Prop).P)) (or (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P)))) (\lambda (H10: (pc3 c2 x0 t)).(or_introl (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P))) (ex_intro T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3)) t (ty3_cast g c2 t0 t (ty3_conv g c2 t x H4 t0 x0 H7 H10) x H4)))) (\lambda (H10: (((pc3 c2 x0 t) \to (\forall (P: Prop).P)))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H11: (ty3 g c2 (THead (Flat Cast) t t0) t3)).(\lambda (P: Prop).(and_ind (pc3 c2 t t3) (ty3 g c2 t0 t) P (\lambda (_: (pc3 c2 t t3)).(\lambda (H13: (ty3 g c2 t0 t)).(let H_y \def (ty3_unique g c2 t0 t H13 x0 H7) in (H10 (pc3_s c2 x0 t H_y) P)))) (ty3_gen_cast g c2 t0 t t3 H11))))))) H9))))) (ty3_correct g c2 t0 x0 H7)))) H6)) (\lambda (H6: ((\forall (t2: T).((ty3 g c2 t0 t2) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H7: (ty3 g c2 (THead (Flat Cast) t t0) t3)).(\lambda (P: Prop).(and_ind (pc3 c2 t t3) (ty3 g c2 t0 t) P (\lambda (_: (pc3 c2 t t3)).(\lambda (H9: (ty3 g c2 t0 t)).(H6 t H9 P))) (ty3_gen_cast g c2 t0 t t3 H7))))))) H5)))) H3)) (\lambda (H3: ((\forall (t2: T).((ty3 g c2 t t2) \to (\forall (P: Prop).P))))).(or_intror (ex T (\lambda (t3: T).(ty3 g c2 (THead (Flat Cast) t t0) t3))) (\forall (t3: T).((ty3 g c2 (THead (Flat Cast) t t0) t3) \to (\forall (P: Prop).P))) (\lambda (t3: T).(\lambda (H4: (ty3 g c2 (THead (Flat Cast) t t0) t3)).(\lambda (P: Prop).(and_ind (pc3 c2 t t3) (ty3 g c2 t0 t) P (\lambda (_: (pc3 c2 t t3)).(\lambda (H6: (ty3 g c2 t0 t)).(ex_ind T (\lambda (t4: T).(ty3 g c2 t t4)) P (\lambda (x: T).(\lambda (H7: (ty3 g c2 t x)).(H3 x H7 P))) (ty3_correct g c2 t0 t H6)))) (ty3_gen_cast g c2 t0 t t3 H4))))))) H2)))]) H0))]) H))]))) c t1))).
 

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Preamble.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Preamble.ma
index 1e73acd7a361b99b88e060a92e04cad6f47f0a3f..2055f18959992b360c9f7364e8476d7fcbbf2918 100644 (file)
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Preamble.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Preamble.ma
@@ -14,8 +14,32 @@
 
 set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/Preamble".
 
-include "legacy/coq.ma".
+(* FG: We should include legacy/coq.ma bit it is not working *)
+(* include "legacy/coq.ma".                                  *)
 
+default "equality"
+ cic:/Coq/Init/Logic/eq.ind
+ cic:/Coq/Init/Logic/sym_eq.con
+ cic:/Coq/Init/Logic/trans_eq.con
+ cic:/Coq/Init/Logic/eq_ind.con
+ cic:/Coq/Init/Logic/eq_ind_r.con
+ cic:/Coq/Init/Logic/f_equal.con
+ cic:/Coq/Init/Logic/f_equal1.con.
+       
+default "true"
+ cic:/Coq/Init/Logic/True.ind.
+
+default "false"
+ cic:/Coq/Init/Logic/False.ind.
+
+default "absurd"
+ cic:/Coq/Init/Logic/absurd.con.
+         
+interpretation "Coq's natural plus" 'plus x y = (cic:/Coq/Init/Peano/plus.con x y).
+interpretation "Coq's natural 'less or equal to'" 'leq x y = (cic:/Coq/Init/Peano/le.ind#xpointer(1/1) x y).
+
+alias id "land" = "cic:/Coq/Init/Logic/and.ind#xpointer(1/1)".
+         
 (* FG/CSC: These aliases should disappear: we would like to write something
  *         like: "disambiguate in cic:/Coq/*"
  *)
@@ -54,3 +78,11 @@ qed.
 theorem sym_not_eq: \forall A:Type. \forall x,y:A. x \neq y \to y \neq x.
  unfold not. intros. apply H. symmetry. assumption.
 qed.
+
+theorem plus_reg_l: \forall (n,m,p:nat). n + m = n + p \to m = p.
+ intros. apply plus_reg_l; auto.
+qed.
+
+theorem plus_le_reg_l: \forall p,n,m. p + n <= p + m \to n <= m.
+ intros. apply plus_le_reg_l; auto.
+qed.