we exported some inversors from coq

[helm.git] / helm / software / matita / contribs / LAMBDA-TYPES / Level-1 / LambdaDelta / pr0 / fwd.ma

diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/fwd.ma
index d24ae097ef3061288e11b4511157e0090ad46036..f0cde41e030695eeb52aa9f2a82870ff21d79ade 100644 (file)
--- a/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/fwd.ma
+++ b/helm/software/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/fwd.ma
@@ -18,22 +18,401 @@ set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/pr0/fwd".
 
 include "pr0/props.ma".
 
-axiom pr0_gen_sort:
+theorem pr0_inv_coq:
+ \forall (t1: T).(\forall (t2: T).(\forall (P: ((T \to (T \to 
+Prop)))).((((pr0 t1 t2) \to (\forall (t: T).((eq T t t1) \to ((eq T t t2) \to 
+(P t1 t2)))))) \to ((((pr0 t1 t2) \to (\forall (u1: T).(\forall (u2: 
+T).(\forall (t0: T).(\forall (t3: T).(\forall (k: K).((eq T (THead k u1 t0) 
+t1) \to ((eq T (THead k u2 t3) t2) \to ((pr0 u1 u2) \to ((pr0 t0 t3) \to (P 
+t1 t2)))))))))))) \to ((((pr0 t1 t2) \to (\forall (u: T).(\forall (v1: 
+T).(\forall (v2: T).(\forall (t0: T).(\forall (t3: T).((eq T (THead (Flat 
+Appl) v1 (THead (Bind Abst) u t0)) t1) \to ((eq T (THead (Bind Abbr) v2 t3) 
+t2) \to ((pr0 v1 v2) \to ((pr0 t0 t3) \to (P t1 t2)))))))))))) \to ((((pr0 t1 
+t2) \to (\forall (b: B).(\forall (v1: T).(\forall (v2: T).(\forall (u1: 
+T).(\forall (u2: T).(\forall (t0: T).(\forall (t3: T).((eq T (THead (Flat 
+Appl) v1 (THead (Bind b) u1 t0)) t1) \to ((eq T (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t3)) t2) \to ((not (eq B b Abst)) \to ((pr0 v1 
+v2) \to ((pr0 u1 u2) \to ((pr0 t0 t3) \to (P t1 t2)))))))))))))))) \to 
+((((pr0 t1 t2) \to (\forall (u1: T).(\forall (u2: T).(\forall (t0: 
+T).(\forall (t3: T).(\forall (w: T).((eq T (THead (Bind Abbr) u1 t0) t1) \to 
+((eq T (THead (Bind Abbr) u2 w) t2) \to ((pr0 u1 u2) \to ((pr0 t0 t3) \to 
+((subst0 O u2 t3 w) \to (P t1 t2))))))))))))) \to ((((pr0 t1 t2) \to (\forall 
+(b: B).(\forall (t0: T).(\forall (t3: T).(\forall (u: T).((eq T (THead (Bind 
+b) u (lift (S O) O t0)) t1) \to ((eq T t3 t2) \to ((not (eq B b Abst)) \to 
+((pr0 t0 t3) \to (P t1 t2))))))))))) \to ((((pr0 t1 t2) \to (\forall (t0: 
+T).(\forall (t3: T).(\forall (u: T).((eq T (THead (Flat Cast) u t0) t1) \to 
+((eq T t3 t2) \to ((pr0 t0 t3) \to (P t1 t2))))))))) \to ((pr0 t1 t2) \to (P 
+t1 t2)))))))))))
+\def
+ \lambda (t1: T).(\lambda (t2: T).(\lambda (P: ((T \to (T \to 
+Prop)))).(\lambda (H: (((pr0 t1 t2) \to (\forall (t: T).((eq T t t1) \to ((eq 
+T t t2) \to (P t1 t2))))))).(\lambda (H0: (((pr0 t1 t2) \to (\forall (u1: 
+T).(\forall (u2: T).(\forall (t0: T).(\forall (t3: T).(\forall (k: K).((eq T 
+(THead k u1 t0) t1) \to ((eq T (THead k u2 t3) t2) \to ((pr0 u1 u2) \to ((pr0 
+t0 t3) \to (P t1 t2))))))))))))).(\lambda (H1: (((pr0 t1 t2) \to (\forall (u: 
+T).(\forall (v1: T).(\forall (v2: T).(\forall (t0: T).(\forall (t3: T).((eq T 
+(THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t1) \to ((eq T (THead (Bind 
+Abbr) v2 t3) t2) \to ((pr0 v1 v2) \to ((pr0 t0 t3) \to (P t1 
+t2))))))))))))).(\lambda (H2: (((pr0 t1 t2) \to (\forall (b: B).(\forall (v1: 
+T).(\forall (v2: T).(\forall (u1: T).(\forall (u2: T).(\forall (t0: 
+T).(\forall (t3: T).((eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t1) 
+\to ((eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t2) 
+\to ((not (eq B b Abst)) \to ((pr0 v1 v2) \to ((pr0 u1 u2) \to ((pr0 t0 t3) 
+\to (P t1 t2))))))))))))))))).(\lambda (H3: (((pr0 t1 t2) \to (\forall (u1: 
+T).(\forall (u2: T).(\forall (t0: T).(\forall (t3: T).(\forall (w: T).((eq T 
+(THead (Bind Abbr) u1 t0) t1) \to ((eq T (THead (Bind Abbr) u2 w) t2) \to 
+((pr0 u1 u2) \to ((pr0 t0 t3) \to ((subst0 O u2 t3 w) \to (P t1 
+t2)))))))))))))).(\lambda (H4: (((pr0 t1 t2) \to (\forall (b: B).(\forall 
+(t0: T).(\forall (t3: T).(\forall (u: T).((eq T (THead (Bind b) u (lift (S O) 
+O t0)) t1) \to ((eq T t3 t2) \to ((not (eq B b Abst)) \to ((pr0 t0 t3) \to (P 
+t1 t2)))))))))))).(\lambda (H5: (((pr0 t1 t2) \to (\forall (t0: T).(\forall 
+(t3: T).(\forall (u: T).((eq T (THead (Flat Cast) u t0) t1) \to ((eq T t3 t2) 
+\to ((pr0 t0 t3) \to (P t1 t2)))))))))).(\lambda (H6: (pr0 t1 t2)).(let H7 
+\def (match H6 in pr0 return (\lambda (t: T).(\lambda (t0: T).(\lambda (_: 
+(pr0 t t0)).((eq T t t1) \to ((eq T t0 t2) \to (P t1 t2)))))) with [(pr0_refl 
+t) \Rightarrow (\lambda (H7: (eq T t t1)).(\lambda (H8: (eq T t t2)).(H H6 t 
+H7 H8))) | (pr0_comp u1 u2 H7 t0 t3 H8 k) \Rightarrow (\lambda (H9: (eq T 
+(THead k u1 t0) t1)).(\lambda (H10: (eq T (THead k u2 t3) t2)).(H0 H6 u1 u2 
+t0 t3 k H9 H10 H7 H8))) | (pr0_beta u v1 v2 H7 t0 t3 H8) \Rightarrow (\lambda 
+(H9: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) t1)).(\lambda 
+(H10: (eq T (THead (Bind Abbr) v2 t3) t2)).(H1 H6 u v1 v2 t0 t3 H9 H10 H7 
+H8))) | (pr0_upsilon b H7 v1 v2 H8 u1 u2 H9 t0 t3 H10) \Rightarrow (\lambda 
+(H11: (eq T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) t1)).(\lambda (H12: 
+(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) t2)).(H2 
+H6 b v1 v2 u1 u2 t0 t3 H11 H12 H7 H8 H9 H10))) | (pr0_delta u1 u2 H7 t0 t3 H8 
+w H9) \Rightarrow (\lambda (H10: (eq T (THead (Bind Abbr) u1 t0) 
+t1)).(\lambda (H11: (eq T (THead (Bind Abbr) u2 w) t2)).(H3 H6 u1 u2 t0 t3 w 
+H10 H11 H7 H8 H9))) | (pr0_zeta b H7 t0 t3 H8 u) \Rightarrow (\lambda (H9: 
+(eq T (THead (Bind b) u (lift (S O) O t0)) t1)).(\lambda (H10: (eq T t3 
+t2)).(H4 H6 b t0 t3 u H9 H10 H7 H8))) | (pr0_epsilon t0 t3 H7 u) \Rightarrow 
+(\lambda (H8: (eq T (THead (Flat Cast) u t0) t1)).(\lambda (H9: (eq T t3 
+t2)).(H5 H6 t0 t3 u H8 H9 H7)))]) in (H7 (refl_equal T t1) (refl_equal T 
+t2))))))))))))).
+
+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)).(pr0_inv_coq (TSort n) x (\lambda (t: T).(\lambda (t0: T).(eq T t0 t))) 
+(\lambda (H0: (pr0 (TSort n) x)).(\lambda (t: T).(\lambda (H1: (eq T t (TSort 
+n))).(\lambda (H2: (eq T t x)).(let H3 \def (eq_ind T t (\lambda (t0: T).(eq 
+T t0 (TSort n))) H1 x H2) in (let H4 \def (eq_ind T x (\lambda (t0: T).(pr0 
+(TSort n) t0)) H0 (TSort n) H3) in (let H5 \def (eq_ind T x (\lambda (t0: 
+T).(pr0 (TSort n) t0)) H (TSort n) H3) in (eq_ind_r T (TSort n) (\lambda (t0: 
+T).(eq T t0 (TSort n))) (refl_equal T (TSort n)) x H3)))))))) (\lambda (H0: 
+(pr0 (TSort n) x)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t0: 
+T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H2: (eq T (THead k u1 t0) 
+(TSort n))).(\lambda (H3: (eq T (THead k u2 t3) x)).(\lambda (_: (pr0 u1 
+u2)).(\lambda (_: (pr0 t0 t3)).(let H5 \def (eq_ind_r T x (\lambda (t: 
+T).(pr0 (TSort n) t)) H0 (THead k u2 t3) H3) in (let H6 \def (eq_ind_r T x 
+(\lambda (t: T).(pr0 (TSort n) t)) H (THead k u2 t3) H3) in (eq_ind T (THead 
+k u2 t3) (\lambda (t: T).(eq T t (TSort n))) (let H7 \def (eq_ind T (THead k 
+u1 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TSort n) H2) in (False_ind (eq T (THead k u2 t3) 
+(TSort n)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (TSort n) x)).(\lambda 
+(u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (TSort 
+n))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t3) x)).(\lambda (_: (pr0 v1 
+v2)).(\lambda (_: (pr0 t0 t3)).(let H5 \def (eq_ind_r T x (\lambda (t: 
+T).(pr0 (TSort n) t)) H0 (THead (Bind Abbr) v2 t3) H3) in (let H6 \def 
+(eq_ind_r T x (\lambda (t: T).(pr0 (TSort n) t)) H (THead (Bind Abbr) v2 t3) 
+H3) in (eq_ind T (THead (Bind Abbr) v2 t3) (\lambda (t: T).(eq T t (TSort 
+n))) (let H7 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) 
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TSort n) H2) in (False_ind (eq T (THead (Bind Abbr) v2 t3) (TSort 
+n)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (TSort n) x)).(\lambda (b: 
+B).(\lambda (v1: T).(\lambda (v2: T).(\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H4: (eq T (THead (Flat Appl) 
+v1 (THead (Bind b) u1 t0)) (TSort n))).(\lambda (H5: (eq T (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t3)) x)).(\lambda (_: (not (eq B b 
+Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 
+t0 t3)).(let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TSort n) t)) H0 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (let H8 
+\def (eq_ind_r T x (\lambda (t: T).(pr0 (TSort n) t)) H (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (eq_ind T (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t3)) (\lambda (t: T).(eq T t (TSort n))) 
+(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (\lambda 
+(ee: T).(match ee in T return (\lambda (_: T).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) t3)) (TSort n)) H9)) x H5))))))))))))))))) (\lambda 
+(H0: (pr0 (TSort n) x)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t0: 
+T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H3: (eq T (THead (Bind Abbr) u1 
+t0) (TSort n))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).(\lambda (_: 
+(pr0 u1 u2)).(\lambda (_: (pr0 t0 t3)).(\lambda (_: (subst0 O u2 t3 w)).(let 
+H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TSort n) t)) H0 (THead (Bind 
+Abbr) u2 w) H4) in (let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TSort n) 
+t)) H (THead (Bind Abbr) u2 w) H4) in (eq_ind T (THead (Bind Abbr) u2 w) 
+(\lambda (t: T).(eq T t (TSort n))) (let H8 \def (eq_ind T (THead (Bind Abbr) 
+u1 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).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) (TSort n)) H8)) x H4)))))))))))))) (\lambda (_: (pr0 (TSort n) 
+x)).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (u: 
+T).(\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (TSort 
+n))).(\lambda (H3: (eq T t3 x)).(\lambda (_: (not (eq B b Abst))).(\lambda 
+(H4: (pr0 t0 t3)).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr0 t0 t)) H4 x 
+H3) in (let H6 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda 
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TSort n) H2) in (False_ind (eq T x (TSort n)) H6)))))))))))) 
+(\lambda (_: (pr0 (TSort n) x)).(\lambda (t0: T).(\lambda (t3: T).(\lambda 
+(u: T).(\lambda (H1: (eq T (THead (Flat Cast) u t0) (TSort n))).(\lambda (H2: 
+(eq T t3 x)).(\lambda (H3: (pr0 t0 t3)).(let H4 \def (eq_ind T t3 (\lambda 
+(t: T).(pr0 t0 t)) H3 x H2) in (let H5 \def (eq_ind T (THead (Flat Cast) u 
+t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort 
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TSort n) H1) in (False_ind (eq T x (TSort n)) 
+H5)))))))))) H))).
 
-axiom pr0_gen_lref:
+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)).(pr0_inv_coq (TLRef n) x (\lambda (t: T).(\lambda (t0: T).(eq T t0 t))) 
+(\lambda (H0: (pr0 (TLRef n) x)).(\lambda (t: T).(\lambda (H1: (eq T t (TLRef 
+n))).(\lambda (H2: (eq T t x)).(let H3 \def (eq_ind T t (\lambda (t0: T).(eq 
+T t0 (TLRef n))) H1 x H2) in (let H4 \def (eq_ind T x (\lambda (t0: T).(pr0 
+(TLRef n) t0)) H0 (TLRef n) H3) in (let H5 \def (eq_ind T x (\lambda (t0: 
+T).(pr0 (TLRef n) t0)) H (TLRef n) H3) in (eq_ind_r T (TLRef n) (\lambda (t0: 
+T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) x H3)))))))) (\lambda (H0: 
+(pr0 (TLRef n) x)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t0: 
+T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H2: (eq T (THead k u1 t0) 
+(TLRef n))).(\lambda (H3: (eq T (THead k u2 t3) x)).(\lambda (_: (pr0 u1 
+u2)).(\lambda (_: (pr0 t0 t3)).(let H5 \def (eq_ind_r T x (\lambda (t: 
+T).(pr0 (TLRef n) t)) H0 (THead k u2 t3) H3) in (let H6 \def (eq_ind_r T x 
+(\lambda (t: T).(pr0 (TLRef n) t)) H (THead k u2 t3) H3) in (eq_ind T (THead 
+k u2 t3) (\lambda (t: T).(eq T t (TLRef n))) (let H7 \def (eq_ind T (THead k 
+u1 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TLRef n) H2) in (False_ind (eq T (THead k u2 t3) 
+(TLRef n)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (TLRef n) x)).(\lambda 
+(u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (H2: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) (TLRef 
+n))).(\lambda (H3: (eq T (THead (Bind Abbr) v2 t3) x)).(\lambda (_: (pr0 v1 
+v2)).(\lambda (_: (pr0 t0 t3)).(let H5 \def (eq_ind_r T x (\lambda (t: 
+T).(pr0 (TLRef n) t)) H0 (THead (Bind Abbr) v2 t3) H3) in (let H6 \def 
+(eq_ind_r T x (\lambda (t: T).(pr0 (TLRef n) t)) H (THead (Bind Abbr) v2 t3) 
+H3) in (eq_ind T (THead (Bind Abbr) v2 t3) (\lambda (t: T).(eq T t (TLRef 
+n))) (let H7 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u t0)) 
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TLRef n) H2) in (False_ind (eq T (THead (Bind Abbr) v2 t3) (TLRef 
+n)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (TLRef n) x)).(\lambda (b: 
+B).(\lambda (v1: T).(\lambda (v2: T).(\lambda (u1: T).(\lambda (u2: 
+T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (H4: (eq T (THead (Flat Appl) 
+v1 (THead (Bind b) u1 t0)) (TLRef n))).(\lambda (H5: (eq T (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t3)) x)).(\lambda (_: (not (eq B b 
+Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u1 u2)).(\lambda (_: (pr0 
+t0 t3)).(let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TLRef n) t)) H0 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (let H8 
+\def (eq_ind_r T x (\lambda (t: T).(pr0 (TLRef n) t)) H (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (eq_ind T (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t3)) (\lambda (t: T).(eq T t (TLRef n))) 
+(let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u1 t0)) (\lambda 
+(ee: T).(match ee in T return (\lambda (_: T).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) t3)) (TLRef n)) H9)) x H5))))))))))))))))) (\lambda 
+(H0: (pr0 (TLRef n) x)).(\lambda (u1: T).(\lambda (u2: T).(\lambda (t0: 
+T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H3: (eq T (THead (Bind Abbr) u1 
+t0) (TLRef n))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) x)).(\lambda (_: 
+(pr0 u1 u2)).(\lambda (_: (pr0 t0 t3)).(\lambda (_: (subst0 O u2 t3 w)).(let 
+H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TLRef n) t)) H0 (THead (Bind 
+Abbr) u2 w) H4) in (let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (TLRef n) 
+t)) H (THead (Bind Abbr) u2 w) H4) in (eq_ind T (THead (Bind Abbr) u2 w) 
+(\lambda (t: T).(eq T t (TLRef n))) (let H8 \def (eq_ind T (THead (Bind Abbr) 
+u1 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).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) (TLRef n)) H8)) x H4)))))))))))))) (\lambda (_: (pr0 (TLRef n) 
+x)).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: T).(\lambda (u: 
+T).(\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) (TLRef 
+n))).(\lambda (H3: (eq T t3 x)).(\lambda (_: (not (eq B b Abst))).(\lambda 
+(H4: (pr0 t0 t3)).(let H5 \def (eq_ind T t3 (\lambda (t: T).(pr0 t0 t)) H4 x 
+H3) in (let H6 \def (eq_ind T (THead (Bind b) u (lift (S O) O t0)) (\lambda 
+(ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow 
+True])) I (TLRef n) H2) in (False_ind (eq T x (TLRef n)) H6)))))))))))) 
+(\lambda (_: (pr0 (TLRef n) x)).(\lambda (t0: T).(\lambda (t3: T).(\lambda 
+(u: T).(\lambda (H1: (eq T (THead (Flat Cast) u t0) (TLRef n))).(\lambda (H2: 
+(eq T t3 x)).(\lambda (H3: (pr0 t0 t3)).(let H4 \def (eq_ind T t3 (\lambda 
+(t: T).(pr0 t0 t)) H3 x H2) in (let H5 \def (eq_ind T (THead (Flat Cast) u 
+t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort 
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow True])) I (TLRef n) H1) in (False_ind (eq T x (TLRef n)) 
+H5)))))))))) H))).
 
-axiom pr0_gen_abst:
+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)).(pr0_inv_coq (THead (Bind Abst) u1 t1) x (\lambda (_: 
+T).(\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)))))) (\lambda (H0: (pr0 (THead 
+(Bind Abst) u1 t1) x)).(\lambda (t: T).(\lambda (H1: (eq T t (THead (Bind 
+Abst) u1 t1))).(\lambda (H2: (eq T t x)).(let H3 \def (eq_ind T t (\lambda 
+(t0: T).(eq T t0 (THead (Bind Abst) u1 t1))) H1 x H2) in (let H4 \def (eq_ind 
+T x (\lambda (t0: T).(pr0 (THead (Bind Abst) u1 t1) t0)) H0 (THead (Bind 
+Abst) u1 t1) H3) in (let H5 \def (eq_ind T x (\lambda (t0: T).(pr0 (THead 
+(Bind Abst) u1 t1) t0)) H (THead (Bind Abst) u1 t1) H3) in (eq_ind_r 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 H3)))))))) (\lambda 
+(H0: (pr0 (THead (Bind Abst) u1 t1) x)).(\lambda (u0: T).(\lambda (u2: 
+T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H2: (eq T 
+(THead k u0 t0) (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq T (THead k u2 
+t3) x)).(\lambda (H1: (pr0 u0 u2)).(\lambda (H4: (pr0 t0 t3)).(let H5 \def 
+(eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t1) t)) H0 (THead k 
+u2 t3) H3) in (let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind 
+Abst) u1 t1) t)) H (THead k u2 t3) H3) in (eq_ind T (THead k u2 t3) (\lambda 
+(t: T).(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Bind 
+Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t2: T).(pr0 t1 t2))))) (let H7 \def (f_equal T K (\lambda (e: 
+T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | 
+(TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) 
+(THead (Bind Abst) u1 t1) H2) in ((let H8 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | 
+(TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) 
+(THead (Bind Abst) u1 t1) H2) in ((let H9 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | 
+(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) 
+(THead (Bind Abst) u1 t1) H2) in (\lambda (H10: (eq T u0 u1)).(\lambda (H11: 
+(eq K k (Bind Abst))).(let H12 \def (eq_ind K k (\lambda (k0: K).(pr0 (THead 
+(Bind Abst) u1 t1) (THead k0 u2 t3))) H6 (Bind Abst) H11) in (let H13 \def 
+(eq_ind K k (\lambda (k0: K).(pr0 (THead (Bind Abst) u1 t1) (THead k0 u2 
+t3))) H5 (Bind Abst) H11) in (eq_ind_r K (Bind Abst) (\lambda (k0: K).(ex3_2 
+T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k0 u2 t3) (THead (Bind 
+Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t2: T).(pr0 t1 t2))))) (let H14 \def (eq_ind T t0 (\lambda (t: 
+T).(pr0 t t3)) H4 t1 H9) in (let H15 \def (eq_ind T u0 (\lambda (t: T).(pr0 t 
+u2)) H1 u1 H10) in (ex3_2_intro T T (\lambda (u3: T).(\lambda (t2: T).(eq T 
+(THead (Bind Abst) u2 t3) (THead (Bind Abst) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
+t2))) u2 t3 (refl_equal T (THead (Bind Abst) u2 t3)) H15 H14))) k H11)))))) 
+H8)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (THead (Bind Abst) u1 t1) 
+x)).(\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t0: 
+T).(\lambda (t3: T).(\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 t3) x)).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t0 t3)).(let H5 
+\def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t1) t)) H0 
+(THead (Bind Abbr) v2 t3) H3) in (let H6 \def (eq_ind_r T x (\lambda (t: 
+T).(pr0 (THead (Bind Abst) u1 t1) t)) H (THead (Bind Abbr) v2 t3) H3) in 
+(eq_ind T (THead (Bind Abbr) v2 t3) (\lambda (t: T).(ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T t (THead (Bind Abst) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
+t2))))) (let H7 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind Abst) u 
+t0)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1 
+t1) H2) in (False_ind (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T 
+(THead (Bind Abbr) v2 t3) (THead (Bind Abst) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
+t2)))) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (THead (Bind Abst) u1 t1) 
+x)).(\lambda (b: B).(\lambda (v1: T).(\lambda (v2: T).(\lambda (u0: 
+T).(\lambda (u2: T).(\lambda (t0: T).(\lambda (t3: T).(\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) t3)) x)).(\lambda (_: (not (eq B b Abst))).(\lambda (_: (pr0 v1 
+v2)).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (pr0 t0 t3)).(let H7 \def 
+(eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t1) t)) H0 (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (let H8 \def 
+(eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t1) t)) H (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) in (eq_ind T (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (\lambda (t: T).(ex3_2 
+T T (\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Bind Abst) u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
+T).(pr0 t1 t2))))) (let H9 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind 
+b) u0 t0)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
+\Rightarrow False | (Flat _) \Rightarrow True])])) I (THead (Bind Abst) u1 
+t1) H4) in (False_ind (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Bind 
+Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t2: T).(pr0 t1 t2)))) H9)) x H5))))))))))))))))) (\lambda (H0: 
+(pr0 (THead (Bind Abst) u1 t1) x)).(\lambda (u0: T).(\lambda (u2: T).(\lambda 
+(t0: T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H3: (eq T (THead (Bind 
+Abbr) u0 t0) (THead (Bind Abst) u1 t1))).(\lambda (H4: (eq T (THead (Bind 
+Abbr) u2 w) x)).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (pr0 t0 t3)).(\lambda 
+(_: (subst0 O u2 t3 w)).(let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 
+(THead (Bind Abst) u1 t1) t)) H0 (THead (Bind Abbr) u2 w) H4) in (let H7 \def 
+(eq_ind_r T x (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t1) t)) H (THead 
+(Bind Abbr) u2 w) H4) in (eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: 
+T).(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Bind Abst) u3 
+t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t2: T).(pr0 t1 t2))))) (let H8 \def (eq_ind T (THead (Bind Abbr) 
+u0 t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind b) 
+\Rightarrow (match b in B return (\lambda (_: B).Prop) with [Abbr \Rightarrow 
+True | Abst \Rightarrow False | Void \Rightarrow False]) | (Flat _) 
+\Rightarrow False])])) I (THead (Bind Abst) u1 t1) H3) in (False_ind (ex3_2 T 
+T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u2 w) (THead 
+(Bind Abst) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda 
+(_: T).(\lambda (t2: T).(pr0 t1 t2)))) H8)) x H4)))))))))))))) (\lambda (H0: 
+(pr0 (THead (Bind Abst) u1 t1) x)).(\lambda (b: B).(\lambda (t0: T).(\lambda 
+(t3: T).(\lambda (u: T).(\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O 
+t0)) (THead (Bind Abst) u1 t1))).(\lambda (H3: (eq T t3 x)).(\lambda (H1: 
+(not (eq B b Abst))).(\lambda (H4: (pr0 t0 t3)).(let H5 \def (eq_ind T t3 
+(\lambda (t: T).(pr0 t0 t)) H4 x H3) in (let H6 \def (f_equal T B (\lambda 
+(e: T).(match e in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b 
+| (TLRef _) \Rightarrow b | (THead k _ _) \Rightarrow (match k in K return 
+(\lambda (_: K).B) with [(Bind b0) \Rightarrow b0 | (Flat _) \Rightarrow 
+b])])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 t1) H2) in 
+((let H7 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: 
+T).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 H8 \def (f_equal T T (\lambda (e: T).(match e in T return 
+(\lambda (_: T).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 u0 t2) \Rightarrow 
+(THead k (lref_map f d u0) (lref_map f (s k d) t2))]) in lref_map) (\lambda 
+(x0: nat).(plus x0 (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 u0 t2) 
+\Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) t2))]) in 
+lref_map) (\lambda (x0: nat).(plus x0 (S O))) O t0) | (THead _ _ t) 
+\Rightarrow t])) (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Abst) u1 
+t1) H2) in (\lambda (_: (eq T u u1)).(\lambda (H10: (eq B b Abst)).(let H11 
+\def (eq_ind B b (\lambda (b0: B).(not (eq B b0 Abst))) H1 Abst H10) in (let 
+H12 \def (eq_ind_r T t1 (\lambda (t: T).(pr0 (THead (Bind Abst) u1 t) x)) H0 
+(lift (S O) O t0) H8) in (let H13 \def (eq_ind_r T t1 (\lambda (t: T).(pr0 
+(THead (Bind Abst) u1 t) x)) H (lift (S O) O t0) H8) in (eq_ind T (lift (S O) 
+O t0) (\lambda (t: T).(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 t t2))))) (let H14 \def (match (H11 
+(refl_equal B Abst)) in False return (\lambda (_: False).(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 (lift 
+(S O) O t0) t2))))) with []) in H14) t1 H8))))))) H7)) H6)))))))))))) 
+(\lambda (_: (pr0 (THead (Bind Abst) u1 t1) x)).(\lambda (t0: T).(\lambda 
+(t3: T).(\lambda (u: T).(\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead 
+(Bind Abst) u1 t1))).(\lambda (H2: (eq T t3 x)).(\lambda (H3: (pr0 t0 
+t3)).(let H4 \def (eq_ind T t3 (\lambda (t: T).(pr0 t0 t)) H3 x H2) in (let 
+H5 \def (eq_ind T (THead (Flat Cast) u t0) (\lambda (ee: T).(match ee in T 
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
+(_: K).Prop) with [(Bind _) \Rightarrow False | (Flat _) \Rightarrow 
+True])])) I (THead (Bind Abst) u1 t1) H1) in (False_ind (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)))) H5)))))))))) H)))).
 
-axiom pr0_gen_appl:
+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 
@@ -54,33 +433,1304 @@ 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)).(pr0_inv_coq (THead (Flat Appl) u1 t1) x (\lambda (_: 
+T).(\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))))))))))) (\lambda (H0: (pr0 
+(THead (Flat Appl) u1 t1) x)).(\lambda (t: T).(\lambda (H1: (eq T t (THead 
+(Flat Appl) u1 t1))).(\lambda (H2: (eq T t x)).(let H3 \def (eq_ind T t 
+(\lambda (t0: T).(eq T t0 (THead (Flat Appl) u1 t1))) H1 x H2) in (let H4 
+\def (eq_ind T x (\lambda (t0: T).(pr0 (THead (Flat Appl) u1 t1) t0)) H0 
+(THead (Flat Appl) u1 t1) H3) in (let H5 \def (eq_ind T x (\lambda (t0: 
+T).(pr0 (THead (Flat Appl) u1 t1) t0)) H (THead (Flat Appl) u1 t1) H3) in 
+(eq_ind_r 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 H3)))))))) (\lambda (H0: (pr0 (THead 
+(Flat Appl) u1 t1) x)).(\lambda (u0: T).(\lambda (u2: T).(\lambda (t0: 
+T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H2: (eq T (THead k u0 t0) 
+(THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T (THead k u2 t3) x)).(\lambda 
+(H1: (pr0 u0 u2)).(\lambda (H4: (pr0 t0 t3)).(let H5 \def (eq_ind_r T x 
+(\lambda (t: T).(pr0 (THead (Flat Appl) u1 t1) t)) H0 (THead k u2 t3) H3) in 
+(let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t1) t)) 
+H (THead k u2 t3) H3) in (eq_ind T (THead k u2 t3) (\lambda (t: T).(or3 
+(ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Flat Appl) u3 
+t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\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 (u3: T).(\lambda (t2: 
+T).(eq T t (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\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 (u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T t (THead (Bind b) 
+v2 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\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 
+(t2: T).(pr0 z1 t2)))))))))) (let H7 \def (f_equal T K (\lambda (e: T).(match 
+e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (THead k u0 t0) (THead (Flat 
+Appl) u1 t1) H2) in ((let H8 \def (f_equal T T (\lambda (e: T).(match e in T 
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow u0 | (TLRef _) 
+\Rightarrow u0 | (THead _ t _) \Rightarrow t])) (THead k u0 t0) (THead (Flat 
+Appl) u1 t1) H2) in ((let H9 \def (f_equal T T (\lambda (e: T).(match e in T 
+return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | (TLRef _) 
+\Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead k u0 t0) (THead (Flat 
+Appl) u1 t1) H2) in (\lambda (H10: (eq T u0 u1)).(\lambda (H11: (eq K k (Flat 
+Appl))).(let H12 \def (eq_ind K k (\lambda (k0: K).(pr0 (THead (Flat Appl) u1 
+t1) (THead k0 u2 t3))) H6 (Flat Appl) H11) in (let H13 \def (eq_ind K k 
+(\lambda (k0: K).(pr0 (THead (Flat Appl) u1 t1) (THead k0 u2 t3))) H5 (Flat 
+Appl) H11) in (eq_ind_r K (Flat Appl) (\lambda (k0: K).(or3 (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T (THead k0 u2 t3) (THead (Flat Appl) 
+u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\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 (u3: T).(\lambda (t2: 
+T).(eq T (THead k0 u2 t3) (THead (Bind Abbr) u3 t2)))))) (\lambda (_: 
+T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\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 (u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T 
+(THead k0 u2 t3) (THead (Bind b) v2 (THead (Flat Appl) (lift (S O) O u3) 
+t2))))))))) (\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 (t2: T).(pr0 z1 t2)))))))))) (let H14 \def 
+(eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H4 t1 H9) in (let H15 \def (eq_ind T 
+u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H10) in (or3_intro0 (ex3_2 T T (\lambda 
+(u3: T).(\lambda (t2: T).(eq T (THead (Flat Appl) u2 t3) (THead (Flat Appl) 
+u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\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 (u3: T).(\lambda (t2: 
+T).(eq T (THead (Flat Appl) u2 t3) (THead (Bind Abbr) u3 t2)))))) (\lambda 
+(_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) 
+(\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 (u3: T).(\lambda (v2: T).(\lambda 
+(t2: T).(eq T (THead (Flat Appl) u2 t3) (THead (Bind b) v2 (THead (Flat Appl) 
+(lift (S O) O u3) t2))))))))) (\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 (t2: T).(pr0 z1 
+t2)))))))) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead 
+(Flat Appl) u2 t3) (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2))) u2 t3 
+(refl_equal T (THead (Flat Appl) u2 t3)) H15 H14)))) k H11)))))) H8)) H7)) x 
+H3))))))))))))) (\lambda (H0: (pr0 (THead (Flat Appl) u1 t1) x)).(\lambda (u: 
+T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t0: T).(\lambda (t3: 
+T).(\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 t3) 
+x)).(\lambda (H1: (pr0 v1 v2)).(\lambda (H4: (pr0 t0 t3)).(let H5 \def 
+(eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t1) t)) H0 (THead 
+(Bind Abbr) v2 t3) H3) in (let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 
+(THead (Flat Appl) u1 t1) t)) H (THead (Bind Abbr) v2 t3) H3) in (eq_ind T 
+(THead (Bind Abbr) v2 t3) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: 
+T).(\lambda (t2: T).(eq T t (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 t (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 (v3: T).(\lambda (t2: T).(eq T t (THead (Bind b) v3 (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 (v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: 
+B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda 
+(t2: T).(pr0 z1 t2)))))))))) (let H7 \def (f_equal T T (\lambda (e: T).(match 
+e in T return (\lambda (_: T).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 ((let H8 \def (f_equal T 
+T (\lambda (e: T).(match e in T return (\lambda (_: T).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 (\lambda (H9: (eq T v1 
+u1)).(let H10 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H1 u1 H9) in (let 
+H11 \def (eq_ind_r T t1 (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t) (THead 
+(Bind Abbr) v2 t3))) H6 (THead (Bind Abst) u t0) H8) in (let H12 \def 
+(eq_ind_r T t1 (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t) (THead (Bind 
+Abbr) v2 t3))) H5 (THead (Bind Abst) u t0) H8) in (eq_ind T (THead (Bind 
+Abst) u t0) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u2: T).(\lambda (t2: 
+T).(eq T (THead (Bind Abbr) v2 t3) (THead (Flat Appl) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t 
+t2)))) (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 (t2: T).(eq T (THead (Bind Abbr) 
+v2 t3) (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 t 
+(THead (Bind b) y1 z1)))))))) (\lambda (b: B).(\lambda (_: T).(\lambda (_: 
+T).(\lambda (u2: T).(\lambda (v3: T).(\lambda (t2: T).(eq T (THead (Bind 
+Abbr) v2 t3) (THead (Bind b) v3 (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 (v3: T).(\lambda (_: T).(pr0 
+y1 v3))))))) (\lambda (_: B).(\lambda (_: T).(\lambda (z1: T).(\lambda (_: 
+T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))))))) (or3_intro1 (ex3_2 T 
+T (\lambda (u2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) v2 t3) (THead 
+(Flat Appl) u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda 
+(_: T).(\lambda (t2: T).(pr0 (THead (Bind Abst) u t0) t2)))) (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 (t2: T).(eq T (THead (Bind Abbr) 
+v2 t3) (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 
+(THead (Bind Abst) u t0) (THead (Bind b) y1 z1)))))))) (\lambda (b: 
+B).(\lambda (_: T).(\lambda (_: T).(\lambda (u2: T).(\lambda (v3: T).(\lambda 
+(t2: T).(eq T (THead (Bind Abbr) v2 t3) (THead (Bind b) v3 (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 
+(v3: T).(\lambda (_: T).(pr0 y1 v3))))))) (\lambda (_: B).(\lambda (_: 
+T).(\lambda (z1: T).(\lambda (_: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 
+t2)))))))) (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 (t2: 
+T).(eq T (THead (Bind Abbr) v2 t3) (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))))) u t0 v2 t3 (refl_equal T (THead (Bind Abst) u t0)) (refl_equal T 
+(THead (Bind Abbr) v2 t3)) H10 H4)) t1 H8)))))) H7)) x H3))))))))))))) 
+(\lambda (H0: (pr0 (THead (Flat Appl) u1 t1) x)).(\lambda (b: B).(\lambda 
+(v1: T).(\lambda (v2: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda (t0: 
+T).(\lambda (t3: T).(\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) t3)) x)).(\lambda (H1: (not (eq B b 
+Abst))).(\lambda (H2: (pr0 v1 v2)).(\lambda (H3: (pr0 u0 u2)).(\lambda (H6: 
+(pr0 t0 t3)).(let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat 
+Appl) u1 t1) t)) H0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t3)) H5) in (let H8 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat 
+Appl) u1 t1) t)) H (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t3)) H5) in (eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) 
+t3)) (\lambda (t: T).(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T 
+t (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
+(\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 (u3: T).(\lambda 
+(t2: T).(eq T t (THead (Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: 
+T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda 
+(z1: T).(\lambda (_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (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 (t2: T).(eq T t (THead (Bind 
+b0) v3 (THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\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 (t2: T).(pr0 z1 t2)))))))))) (let H9 \def (f_equal T T (\lambda 
+(e: T).(match e in T return (\lambda (_: T).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 ((let H10 
+\def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).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 (\lambda (H11: (eq T 
+v1 u1)).(let H12 \def (eq_ind T v1 (\lambda (t: T).(pr0 t v2)) H2 u1 H11) in 
+(let H13 \def (eq_ind_r T t1 (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t) 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)))) H8 (THead 
+(Bind b) u0 t0) H10) in (let H14 \def (eq_ind_r T t1 (\lambda (t: T).(pr0 
+(THead (Flat Appl) u1 t) (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O 
+v2) t3)))) H7 (THead (Bind b) u0 t0) H10) in (eq_ind T (THead (Bind b) u0 t0) 
+(\lambda (t: T).(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Flat 
+Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t2: T).(pr0 t t2)))) (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 (t2: 
+T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead 
+(Bind Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda 
+(_: T).(\lambda (t2: T).(pr0 z1 t2)))))) (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 (t2: T).(eq T (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t3)) (THead (Bind b0) v3 (THead (Flat Appl) 
+(lift (S O) O u3) t2))))))))) (\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 (t2: T).(pr0 z1 
+t2)))))))))) (or3_intro2 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Flat 
+Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t2: T).(pr0 (THead (Bind b) u0 t0) t2)))) (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 (t2: T).(eq T (THead (Bind b) u2 
+(THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Bind Abbr) u3 t2)))))) 
+(\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: T).(pr0 u1 
+u3))))) (\lambda (_: T).(\lambda (z1: T).(\lambda (_: T).(\lambda (t2: 
+T).(pr0 z1 t2)))))) (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 (t2: T).(eq T (THead (Bind b) 
+u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Bind b0) v3 (THead (Flat 
+Appl) (lift (S O) O u3) t2))))))))) (\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 (t2: T).(pr0 z1 
+t2)))))))) (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 (t2: T).(eq T (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) t3)) (THead (Bind b0) v3 (THead (Flat Appl) 
+(lift (S O) O u3) t2))))))))) (\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 (t2: T).(pr0 z1 
+t2))))))) b u0 t0 v2 u2 t3 H1 (refl_equal T (THead (Bind b) u0 t0)) 
+(refl_equal T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3))) 
+H12 H3 H6)) t1 H10)))))) H9)) x H5))))))))))))))))) (\lambda (H0: (pr0 (THead 
+(Flat Appl) u1 t1) x)).(\lambda (u0: T).(\lambda (u2: T).(\lambda (t0: 
+T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H3: (eq T (THead (Bind Abbr) u0 
+t0) (THead (Flat Appl) u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) 
+x)).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (pr0 t0 t3)).(\lambda (_: (subst0 
+O u2 t3 w)).(let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat 
+Appl) u1 t1) t)) H0 (THead (Bind Abbr) u2 w) H4) in (let H7 \def (eq_ind_r T 
+x (\lambda (t: T).(pr0 (THead (Flat Appl) u1 t1) t)) H (THead (Bind Abbr) u2 
+w) H4) in (eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).(or3 (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Flat Appl) u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\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 (u3: T).(\lambda (t2: T).(eq T t (THead (Bind 
+Abbr) u3 t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda 
+(_: T).(pr0 u1 u3))))) (\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 
+(u3: T).(\lambda (v2: T).(\lambda (t2: T).(eq T t (THead (Bind b) v2 (THead 
+(Flat Appl) (lift (S O) O u3) t2))))))))) (\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 
+(t2: T).(pr0 z1 t2)))))))))) (let H8 \def (eq_ind T (THead (Bind Abbr) u0 t0) 
+(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) 
+\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow 
+(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow True | 
+(Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1 t1) H3) in (False_ind 
+(or3 (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u2 
+w) (THead (Flat Appl) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
+u3))) (\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 (u3: 
+T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind Abbr) u3 
+t2)))))) (\lambda (_: T).(\lambda (_: T).(\lambda (u3: T).(\lambda (_: 
+T).(pr0 u1 u3))))) (\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 (u3: T).(\lambda 
+(v2: T).(\lambda (t2: T).(eq T (THead (Bind Abbr) u2 w) (THead (Bind b) v2 
+(THead (Flat Appl) (lift (S O) O u3) t2))))))))) (\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 
+(t2: T).(pr0 z1 t2))))))))) H8)) x H4)))))))))))))) (\lambda (_: (pr0 (THead 
+(Flat Appl) u1 t1) x)).(\lambda (b: B).(\lambda (t0: T).(\lambda (t3: 
+T).(\lambda (u: T).(\lambda (H2: (eq T (THead (Bind b) u (lift (S O) O t0)) 
+(THead (Flat Appl) u1 t1))).(\lambda (H3: (eq T t3 x)).(\lambda (_: (not (eq 
+B b Abst))).(\lambda (H4: (pr0 t0 t3)).(let H5 \def (eq_ind T t3 (\lambda (t: 
+T).(pr0 t0 t)) H4 x H3) in (let H6 \def (eq_ind T (THead (Bind b) u (lift (S 
+O) O t0)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Appl) u1 
+t1) H2) in (False_ind (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 (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 (t2: T).(eq T x (THead (Bind 
+b0) 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))))))))) H6)))))))))))) (\lambda (_: (pr0 
+(THead (Flat Appl) u1 t1) x)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (u: 
+T).(\lambda (H1: (eq T (THead (Flat Cast) u t0) (THead (Flat Appl) u1 
+t1))).(\lambda (H2: (eq T t3 x)).(\lambda (H3: (pr0 t0 t3)).(let H4 \def 
+(eq_ind T t3 (\lambda (t: T).(pr0 t0 t)) H3 x H2) in (let H5 \def (eq_ind T 
+(THead (Flat Cast) u t0) (\lambda (ee: T).(match ee in T return (\lambda (_: 
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).Prop) with 
+[(Bind _) \Rightarrow False | (Flat f) \Rightarrow (match f in F return 
+(\lambda (_: F).Prop) with [Appl \Rightarrow False | Cast \Rightarrow 
+True])])])) I (THead (Flat Appl) u1 t1) H1) in (False_ind (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))))))))) H5)))))))))) H)))).
 
-axiom pr0_gen_cast:
+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)).(pr0_inv_coq (THead (Flat Cast) u1 t1) x (\lambda (_: 
+T).(\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)))) (\lambda (H0: 
+(pr0 (THead (Flat Cast) u1 t1) x)).(\lambda (t: T).(\lambda (H1: (eq T t 
+(THead (Flat Cast) u1 t1))).(\lambda (H2: (eq T t x)).(let H3 \def (eq_ind T 
+t (\lambda (t0: T).(eq T t0 (THead (Flat Cast) u1 t1))) H1 x H2) in (let H4 
+\def (eq_ind T x (\lambda (t0: T).(pr0 (THead (Flat Cast) u1 t1) t0)) H0 
+(THead (Flat Cast) u1 t1) H3) in (let H5 \def (eq_ind T x (\lambda (t0: 
+T).(pr0 (THead (Flat Cast) u1 t1) t0)) H (THead (Flat Cast) u1 t1) H3) in 
+(eq_ind_r 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 H3)))))))) 
+(\lambda (H0: (pr0 (THead (Flat Cast) u1 t1) x)).(\lambda (u0: T).(\lambda 
+(u2: T).(\lambda (t0: T).(\lambda (t3: T).(\lambda (k: K).(\lambda (H2: (eq T 
+(THead k u0 t0) (THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T (THead k u2 
+t3) x)).(\lambda (H1: (pr0 u0 u2)).(\lambda (H4: (pr0 t0 t3)).(let H5 \def 
+(eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Cast) u1 t1) t)) H0 (THead k 
+u2 t3) H3) in (let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat 
+Cast) u1 t1) t)) H (THead k u2 t3) H3) in (eq_ind T (THead k u2 t3) (\lambda 
+(t: T).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Flat 
+Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t))) (let H7 \def (f_equal T K 
+(\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) 
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) 
+(THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in ((let H8 \def (f_equal T T 
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow u0 | (TLRef _) \Rightarrow u0 | (THead _ t _) \Rightarrow t])) 
+(THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in ((let H9 \def (f_equal T T 
+(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
+\Rightarrow t0 | (TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) 
+(THead k u0 t0) (THead (Flat Cast) u1 t1) H2) in (\lambda (H10: (eq T u0 
+u1)).(\lambda (H11: (eq K k (Flat Cast))).(let H12 \def (eq_ind K k (\lambda 
+(k0: K).(pr0 (THead (Flat Cast) u1 t1) (THead k0 u2 t3))) H6 (Flat Cast) H11) 
+in (let H13 \def (eq_ind K k (\lambda (k0: K).(pr0 (THead (Flat Cast) u1 t1) 
+(THead k0 u2 t3))) H5 (Flat Cast) H11) in (eq_ind_r K (Flat Cast) (\lambda 
+(k0: K).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead k0 u2 
+t3) (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 
+u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead k0 u2 
+t3)))) (let H14 \def (eq_ind T t0 (\lambda (t: T).(pr0 t t3)) H4 t1 H9) in 
+(let H15 \def (eq_ind T u0 (\lambda (t: T).(pr0 t u2)) H1 u1 H10) in 
+(or_introl (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead (Flat 
+Cast) u2 t3) (THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: 
+T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 
+(THead (Flat Cast) u2 t3)) (ex3_2_intro T T (\lambda (u3: T).(\lambda (t2: 
+T).(eq T (THead (Flat Cast) u2 t3) (THead (Flat Cast) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
+t2))) u2 t3 (refl_equal T (THead (Flat Cast) u2 t3)) H15 H14)))) k H11)))))) 
+H8)) H7)) x H3))))))))))))) (\lambda (H0: (pr0 (THead (Flat Cast) u1 t1) 
+x)).(\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda (t0: 
+T).(\lambda (t3: T).(\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 t3) x)).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 t0 t3)).(let H5 
+\def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Cast) u1 t1) t)) H0 
+(THead (Bind Abbr) v2 t3) H3) in (let H6 \def (eq_ind_r T x (\lambda (t: 
+T).(pr0 (THead (Flat Cast) u1 t1) t)) H (THead (Bind Abbr) v2 t3) H3) in 
+(eq_ind T (THead (Bind Abbr) v2 t3) (\lambda (t: T).(or (ex3_2 T T (\lambda 
+(u2: T).(\lambda (t2: T).(eq T t (THead (Flat Cast) u2 t2)))) (\lambda (u2: 
+T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
+t2)))) (pr0 t1 t))) (let H7 \def (eq_ind T (THead (Flat Appl) v1 (THead (Bind 
+Abst) u t0)) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) 
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
+_) \Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
+\Rightarrow False | (Flat f) \Rightarrow (match f in F return (\lambda (_: 
+F).Prop) with [Appl \Rightarrow True | Cast \Rightarrow False])])])) I (THead 
+(Flat Cast) u1 t1) H2) in (False_ind (or (ex3_2 T T (\lambda (u2: T).(\lambda 
+(t2: T).(eq T (THead (Bind Abbr) v2 t3) (THead (Flat Cast) u2 t2)))) (\lambda 
+(u2: T).(\lambda (_: T).(pr0 u1 u2))) (\lambda (_: T).(\lambda (t2: T).(pr0 
+t1 t2)))) (pr0 t1 (THead (Bind Abbr) v2 t3))) H7)) x H3))))))))))))) (\lambda 
+(H0: (pr0 (THead (Flat Cast) u1 t1) x)).(\lambda (b: B).(\lambda (v1: 
+T).(\lambda (v2: T).(\lambda (u0: T).(\lambda (u2: T).(\lambda (t0: 
+T).(\lambda (t3: T).(\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) t3)) x)).(\lambda (_: (not (eq B b 
+Abst))).(\lambda (_: (pr0 v1 v2)).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (pr0 
+t0 t3)).(let H7 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Cast) u1 
+t1) t)) H0 (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) 
+in (let H8 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat Cast) u1 t1) 
+t)) H (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) H5) in 
+(eq_ind T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) 
+(\lambda (t: T).(or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T t 
+(THead (Flat Cast) u3 t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) 
+(\lambda (_: T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 t))) (let H9 \def 
+(eq_ind T (THead (Flat Appl) v1 (THead (Bind b) u0 t0)) (\lambda (ee: 
+T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K 
+return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | (Flat f) 
+\Rightarrow (match f in F return (\lambda (_: F).Prop) with [Appl \Rightarrow 
+True | Cast \Rightarrow False])])])) I (THead (Flat Cast) u1 t1) H4) in 
+(False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (THead (Flat Cast) u3 
+t2)))) (\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: 
+T).(\lambda (t2: T).(pr0 t1 t2)))) (pr0 t1 (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t3)))) H9)) x H5))))))))))))))))) (\lambda (H0: (pr0 
+(THead (Flat Cast) u1 t1) x)).(\lambda (u0: T).(\lambda (u2: T).(\lambda (t0: 
+T).(\lambda (t3: T).(\lambda (w: T).(\lambda (H3: (eq T (THead (Bind Abbr) u0 
+t0) (THead (Flat Cast) u1 t1))).(\lambda (H4: (eq T (THead (Bind Abbr) u2 w) 
+x)).(\lambda (_: (pr0 u0 u2)).(\lambda (_: (pr0 t0 t3)).(\lambda (_: (subst0 
+O u2 t3 w)).(let H6 \def (eq_ind_r T x (\lambda (t: T).(pr0 (THead (Flat 
+Cast) u1 t1) t)) H0 (THead (Bind Abbr) u2 w) H4) in (let H7 \def (eq_ind_r T 
+x (\lambda (t: T).(pr0 (THead (Flat Cast) u1 t1) t)) H (THead (Bind Abbr) u2 
+w) H4) in (eq_ind T (THead (Bind Abbr) u2 w) (\lambda (t: T).(or (ex3_2 T T 
+(\lambda (u3: T).(\lambda (t2: T).(eq T t (THead (Flat Cast) u3 t2)))) 
+(\lambda (u3: T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: 
+T).(pr0 t1 t2)))) (pr0 t1 t))) (let H8 \def (eq_ind T (THead (Bind Abbr) u0 
+t0) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort 
+_) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k in K return (\lambda (_: K).Prop) with [(Bind _) 
+\Rightarrow True | (Flat _) \Rightarrow False])])) I (THead (Flat Cast) u1 
+t1) H3) in (False_ind (or (ex3_2 T T (\lambda (u3: T).(\lambda (t2: T).(eq T 
+(THead (Bind Abbr) u2 w) (THead (Flat Cast) u3 t2)))) (\lambda (u3: 
+T).(\lambda (_: T).(pr0 u1 u3))) (\lambda (_: T).(\lambda (t2: T).(pr0 t1 
+t2)))) (pr0 t1 (THead (Bind Abbr) u2 w))) H8)) x H4)))))))))))))) (\lambda 
+(_: (pr0 (THead (Flat Cast) u1 t1) x)).(\lambda (b: B).(\lambda (t0: 
+T).(\lambda (t3: T).(\lambda (u: T).(\lambda (H2: (eq T (THead (Bind b) u 
+(lift (S O) O t0)) (THead (Flat Cast) u1 t1))).(\lambda (H3: (eq T t3 
+x)).(\lambda (_: (not (eq B b Abst))).(\lambda (H4: (pr0 t0 t3)).(let H5 \def 
+(eq_ind T t3 (\lambda (t: T).(pr0 t0 t)) H4 x H3) in (let H6 \def (eq_ind T 
+(THead (Bind b) u (lift (S O) O t0)) (\lambda (ee: T).(match ee in T return 
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
+(_: K).Prop) with [(Bind _) \Rightarrow True | (Flat _) \Rightarrow 
+False])])) I (THead (Flat Cast) u1 t1) H2) in (False_ind (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)) H6)))))))))))) (\lambda (_: (pr0 (THead (Flat 
+Cast) u1 t1) x)).(\lambda (t0: T).(\lambda (t3: T).(\lambda (u: T).(\lambda 
+(H1: (eq T (THead (Flat Cast) u t0) (THead (Flat Cast) u1 t1))).(\lambda (H2: 
+(eq T t3 x)).(\lambda (H3: (pr0 t0 t3)).(let H4 \def (eq_ind T t3 (\lambda 
+(t: T).(pr0 t0 t)) H3 x H2) in (let H5 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | 
+(TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Flat Cast) u 
+t0) (THead (Flat Cast) u1 t1) H1) in ((let H6 \def (f_equal T T (\lambda (e: 
+T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t0 | 
+(TLRef _) \Rightarrow t0 | (THead _ _ t) \Rightarrow t])) (THead (Flat Cast) 
+u t0) (THead (Flat Cast) u1 t1) H1) in (\lambda (_: (eq T u u1)).(let H8 \def 
+(eq_ind T t0 (\lambda (t: T).(pr0 t x)) H4 t1 H6) in (or_intror (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) H8)))) H5)))))))))) H)))).
 
-axiom pr0_gen_abbr:
+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 in pr0 return (\lambda (t: 
+T).(\lambda (t0: T).(\lambda (_: (pr0 t t0)).((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 in T 
+return (\lambda (_: T).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 in T 
+return (\lambda (_: T).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 in T 
+return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
+\Rightarrow k | (THead k0 _ _) \Rightarrow k0])) (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 in T return (\lambda (_: T).Prop) with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k in K return (\lambda (_: K).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 in T return (\lambda (_: 
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).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 in T return (\lambda (_: T).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 in T return (\lambda (_: 
+T).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 in T return (\lambda (_: T).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 u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
+t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (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 u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
+t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (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 in T return 
+(\lambda (_: T).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 
+in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) 
+\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: 
+K).B) with [(Bind b0) \Rightarrow b0 | (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 
+in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef 
+_) \Rightarrow False | (THead k _ _) \Rightarrow (match k in K return 
+(\lambda (_: K).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)))))).
 
-axiom pr0_gen_void:
+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 in pr0 return (\lambda (t: 
+T).(\lambda (t0: T).(\lambda (_: (pr0 t t0)).((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 in T return (\lambda (_: T).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 in T return (\lambda (_: T).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 in T return (\lambda (_: T).K) with [(TSort _) 
+\Rightarrow k | (TLRef _) \Rightarrow k | (THead k0 _ _) \Rightarrow k0])) 
+(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 in T return (\lambda (_: T).Prop) with 
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) 
+\Rightarrow (match k in K return (\lambda (_: K).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 in T return (\lambda (_: 
+T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
+(THead k _ _) \Rightarrow (match k in K return (\lambda (_: K).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 in T return 
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow False | (THead k _ _) \Rightarrow (match k in K return (\lambda 
+(_: K).Prop) with [(Bind b) \Rightarrow (match b in B return (\lambda (_: 
+B).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 in T return (\lambda (_: T).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 u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
+t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (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 u0 t3) \Rightarrow (THead k (lref_map f d u0) (lref_map f (s k d) 
+t3))]) in lref_map) (\lambda (x0: nat).(plus x0 (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 in T return 
+(\lambda (_: T).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 
+in T return (\lambda (_: T).B) with [(TSort _) \Rightarrow b | (TLRef _) 
+\Rightarrow b | (THead k _ _) \Rightarrow (match k in K return (\lambda (_: 
+K).B) with [(Bind b0) \Rightarrow b0 | (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 in T return (\lambda (_: T).Prop) 
+with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ 
+_) \Rightarrow (match k in K return (\lambda (_: K).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)))))).
 
-axiom pr0_gen_lift:
+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)))))))
-.
+\def
+ \lambda (t1: T).(\lambda (x: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda 
+(H: (pr0 (lift h d t1) x)).(insert_eq T (lift h d t1) (\lambda (t: T).(pr0 t 
+x)) (ex2 T (\lambda (t2: T).(eq T x (lift h d t2))) (\lambda (t2: T).(pr0 t1 
+t2))) (\lambda (y: T).(\lambda (H0: (pr0 y x)).(unintro nat d (\lambda (n: 
+nat).((eq T y (lift h n t1)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h n 
+t2))) (\lambda (t2: T).(pr0 t1 t2))))) (unintro T t1 (\lambda (t: T).(\forall 
+(x0: nat).((eq T y (lift h x0 t)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h 
+x0 t2))) (\lambda (t2: T).(pr0 t t2)))))) (pr0_ind (\lambda (t: T).(\lambda 
+(t0: T).(\forall (x0: T).(\forall (x1: nat).((eq T t (lift h x1 x0)) \to (ex2 
+T (\lambda (t2: T).(eq T t0 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 
+t2)))))))) (\lambda (t: T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H1: 
+(eq T t (lift h x1 x0))).(ex_intro2 T (\lambda (t2: T).(eq T t (lift h x1 
+t2))) (\lambda (t2: T).(pr0 x0 t2)) x0 H1 (pr0_refl x0)))))) (\lambda (u1: 
+T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H2: ((\forall (x0: 
+T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T (\lambda (t2: 
+T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda 
+(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H4: ((\forall 
+(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: 
+T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (k: 
+K).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H5: (eq T (THead k u1 t2) 
+(lift h x1 x0))).(K_ind (\lambda (k0: K).((eq T (THead k0 u1 t2) (lift h x1 
+x0)) \to (ex2 T (\lambda (t4: T).(eq T (THead k0 u2 t3) (lift h x1 t4))) 
+(\lambda (t4: T).(pr0 x0 t4))))) (\lambda (b: B).(\lambda (H6: (eq T (THead 
+(Bind b) u1 t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: 
+T).(eq T x0 (THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T 
+u1 (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) 
+z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t3) (lift h x1 t4))) 
+(\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda 
+(H7: (eq T x0 (THead (Bind b) x2 x3))).(\lambda (H8: (eq T u1 (lift h x1 
+x2))).(\lambda (H9: (eq T t2 (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) 
+x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t3) 
+(lift h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (ex2_ind T (\lambda (t4: 
+T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 T 
+(\lambda (t4: T).(eq T (THead (Bind b) u2 t3) (lift h x1 t4))) (\lambda (t4: 
+T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T t3 
+(lift h (S x1) x4))).(\lambda (H10: (pr0 x3 x4)).(eq_ind_r T (lift h (S x1) 
+x4) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 t) (lift 
+h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4)))) (ex2_ind T 
+(\lambda (t4: T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 
+T (\lambda (t4: T).(eq T (THead (Bind b) u2 (lift h (S x1) x4)) (lift h x1 
+t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x5: 
+T).(\lambda (H_x0: (eq T u2 (lift h x1 x5))).(\lambda (H11: (pr0 x2 
+x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T 
+(THead (Bind b) t (lift h (S x1) x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 
+(THead (Bind b) x2 x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (THead 
+(Bind b) (lift h x1 x5) (lift h (S x1) x4)) (lift h x1 t4))) (\lambda (t4: 
+T).(pr0 (THead (Bind b) x2 x3) t4)) (THead (Bind b) x5 x4) (sym_eq T (lift h 
+x1 (THead (Bind b) x5 x4)) (THead (Bind b) (lift h x1 x5) (lift h (S x1) x4)) 
+(lift_bind b x5 x4 h x1)) (pr0_comp x2 x5 H11 x3 x4 H10 (Bind b))) u2 
+H_x0)))) (H2 x2 x1 H8)) t3 H_x)))) (H4 x3 (S x1) H9)) x0 H7)))))) 
+(lift_gen_bind b u1 t2 x0 h x1 H6)))) (\lambda (f: F).(\lambda (H6: (eq T 
+(THead (Flat f) u1 t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: 
+T).(\lambda (z: T).(eq T x0 (THead (Flat f) y0 z)))) (\lambda (y0: 
+T).(\lambda (_: T).(eq T u1 (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T t2 (lift h x1 z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Flat f) u2 
+t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda 
+(x3: T).(\lambda (H7: (eq T x0 (THead (Flat f) x2 x3))).(\lambda (H8: (eq T 
+u1 (lift h x1 x2))).(\lambda (H9: (eq T t2 (lift h x1 x3))).(eq_ind_r T 
+(THead (Flat f) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead 
+(Flat f) u2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) (ex2_ind T 
+(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x3 t4)) (ex2 
+T (\lambda (t4: T).(eq T (THead (Flat f) u2 t3) (lift h x1 t4))) (\lambda 
+(t4: T).(pr0 (THead (Flat f) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq 
+T t3 (lift h x1 x4))).(\lambda (H10: (pr0 x3 x4)).(eq_ind_r T (lift h x1 x4) 
+(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Flat f) u2 t) (lift h 
+x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat f) x2 x3) t4)))) (ex2_ind T 
+(\lambda (t4: T).(eq T u2 (lift h x1 t4))) (\lambda (t4: T).(pr0 x2 t4)) (ex2 
+T (\lambda (t4: T).(eq T (THead (Flat f) u2 (lift h x1 x4)) (lift h x1 t4))) 
+(\lambda (t4: T).(pr0 (THead (Flat f) x2 x3) t4))) (\lambda (x5: T).(\lambda 
+(H_x0: (eq T u2 (lift h x1 x5))).(\lambda (H11: (pr0 x2 x5)).(eq_ind_r T 
+(lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Flat f) 
+t (lift h x1 x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat f) x2 
+x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (THead (Flat f) (lift h x1 x5) 
+(lift h x1 x4)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat f) x2 x3) 
+t4)) (THead (Flat f) x5 x4) (sym_eq T (lift h x1 (THead (Flat f) x5 x4)) 
+(THead (Flat f) (lift h x1 x5) (lift h x1 x4)) (lift_flat f x5 x4 h x1)) 
+(pr0_comp x2 x5 H11 x3 x4 H10 (Flat f))) u2 H_x0)))) (H2 x2 x1 H8)) t3 
+H_x)))) (H4 x3 x1 H9)) x0 H7)))))) (lift_gen_flat f u1 t2 x0 h x1 H6)))) k 
+H5))))))))))))) (\lambda (u: T).(\lambda (v1: T).(\lambda (v2: T).(\lambda 
+(_: (pr0 v1 v2)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T v1 
+(lift h x1 x0)) \to (ex2 T (\lambda (t2: T).(eq T v2 (lift h x1 t2))) 
+(\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda (t2: T).(\lambda (t3: 
+T).(\lambda (_: (pr0 t2 t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: 
+nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h 
+x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (x0: T).(\lambda (x1: 
+nat).(\lambda (H5: (eq T (THead (Flat Appl) v1 (THead (Bind Abst) u t2)) 
+(lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 
+(THead (Flat Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T v1 (lift h 
+x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T (THead (Bind Abst) u t2) 
+(lift h x1 z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) 
+(lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda 
+(x3: T).(\lambda (H6: (eq T x0 (THead (Flat Appl) x2 x3))).(\lambda (H7: (eq 
+T v1 (lift h x1 x2))).(\lambda (H8: (eq T (THead (Bind Abst) u t2) (lift h x1 
+x3))).(eq_ind_r T (THead (Flat Appl) x2 x3) (\lambda (t: T).(ex2 T (\lambda 
+(t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift h x1 t4))) (\lambda (t4: 
+T).(pr0 t t4)))) (ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x3 
+(THead (Bind Abst) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h 
+x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) z)))) (ex2 
+T (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t3) (lift h x1 t4))) (\lambda 
+(t4: T).(pr0 (THead (Flat Appl) x2 x3) t4))) (\lambda (x4: T).(\lambda (x5: 
+T).(\lambda (H9: (eq T x3 (THead (Bind Abst) x4 x5))).(\lambda (_: (eq T u 
+(lift h x1 x4))).(\lambda (H11: (eq T t2 (lift h (S x1) x5))).(eq_ind_r T 
+(THead (Bind Abst) x4 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T 
+(THead (Bind Abbr) v2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat 
+Appl) x2 t) t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h (S x1) t4))) 
+(\lambda (t4: T).(pr0 x5 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Bind 
+Abbr) v2 t3) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 
+(THead (Bind Abst) x4 x5)) t4))) (\lambda (x6: T).(\lambda (H_x: (eq T t3 
+(lift h (S x1) x6))).(\lambda (H12: (pr0 x5 x6)).(eq_ind_r T (lift h (S x1) 
+x6) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) v2 t) 
+(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind 
+Abst) x4 x5)) t4)))) (ex2_ind T (\lambda (t4: T).(eq T v2 (lift h x1 t4))) 
+(\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Bind 
+Abbr) v2 (lift h (S x1) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead 
+(Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4))) (\lambda (x7: T).(\lambda 
+(H_x0: (eq T v2 (lift h x1 x7))).(\lambda (H13: (pr0 x2 x7)).(eq_ind_r T 
+(lift h x1 x7) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind 
+Abbr) t (lift h (S x1) x6)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead 
+(Flat Appl) x2 (THead (Bind Abst) x4 x5)) t4)))) (ex_intro2 T (\lambda (t4: 
+T).(eq T (THead (Bind Abbr) (lift h x1 x7) (lift h (S x1) x6)) (lift h x1 
+t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind Abst) x4 x5)) 
+t4)) (THead (Bind Abbr) x7 x6) (sym_eq T (lift h x1 (THead (Bind Abbr) x7 
+x6)) (THead (Bind Abbr) (lift h x1 x7) (lift h (S x1) x6)) (lift_bind Abbr x7 
+x6 h x1)) (pr0_beta x4 x2 x7 H13 x5 x6 H12)) v2 H_x0)))) (H2 x2 x1 H7)) t3 
+H_x)))) (H4 x5 (S x1) H11)) x3 H9)))))) (lift_gen_bind Abst u t2 x3 h x1 H8)) 
+x0 H6)))))) (lift_gen_flat Appl v1 (THead (Bind Abst) u t2) x0 h x1 
+H5)))))))))))))) (\lambda (b: B).(\lambda (H1: (not (eq B b Abst))).(\lambda 
+(v1: T).(\lambda (v2: T).(\lambda (_: (pr0 v1 v2)).(\lambda (H3: ((\forall 
+(x0: T).(\forall (x1: nat).((eq T v1 (lift h x1 x0)) \to (ex2 T (\lambda (t2: 
+T).(eq T v2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda 
+(u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 u2)).(\lambda (H5: ((\forall 
+(x0: T).(\forall (x1: nat).((eq T u1 (lift h x1 x0)) \to (ex2 T (\lambda (t2: 
+T).(eq T u2 (lift h x1 t2))) (\lambda (t2: T).(pr0 x0 t2)))))))).(\lambda 
+(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H7: ((\forall 
+(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: 
+T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda 
+(x0: T).(\lambda (x1: nat).(\lambda (H8: (eq T (THead (Flat Appl) v1 (THead 
+(Bind b) u1 t2)) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda 
+(z: T).(eq T x0 (THead (Flat Appl) y0 z)))) (\lambda (y0: T).(\lambda (_: 
+T).(eq T v1 (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T (THead 
+(Bind b) u1 t2) (lift h x1 z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind 
+b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h x1 t4))) (\lambda 
+(t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H9: (eq T 
+x0 (THead (Flat Appl) x2 x3))).(\lambda (H10: (eq T v1 (lift h x1 
+x2))).(\lambda (H11: (eq T (THead (Bind b) u1 t2) (lift h x1 x3))).(eq_ind_r 
+T (THead (Flat Appl) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T 
+(THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h x1 t4))) 
+(\lambda (t4: T).(pr0 t t4)))) (ex3_2_ind T T (\lambda (y0: T).(\lambda (z: 
+T).(eq T x3 (THead (Bind b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T 
+u1 (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S x1) 
+z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) 
+(lift (S O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat 
+Appl) x2 x3) t4))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H12: (eq T x3 
+(THead (Bind b) x4 x5))).(\lambda (H13: (eq T u1 (lift h x1 x4))).(\lambda 
+(H14: (eq T t2 (lift h (S x1) x5))).(eq_ind_r T (THead (Bind b) x4 x5) 
+(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead (Flat 
+Appl) (lift (S O) O v2) t3)) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead 
+(Flat Appl) x2 t) t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h (S x1) 
+t4))) (\lambda (t4: T).(pr0 x5 t4)) (ex2 T (\lambda (t4: T).(eq T (THead 
+(Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t3)) (lift h x1 t4))) 
+(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4))) 
+(\lambda (x6: T).(\lambda (H_x: (eq T t3 (lift h (S x1) x6))).(\lambda (H15: 
+(pr0 x5 x6)).(eq_ind_r T (lift h (S x1) x6) (\lambda (t: T).(ex2 T (\lambda 
+(t4: T).(eq T (THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t)) 
+(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) 
+x4 x5)) t4)))) (ex2_ind T (\lambda (t4: T).(eq T u2 (lift h x1 t4))) (\lambda 
+(t4: T).(pr0 x4 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Bind b) u2 (THead 
+(Flat Appl) (lift (S O) O v2) (lift h (S x1) x6))) (lift h x1 t4))) (\lambda 
+(t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4))) (\lambda 
+(x7: T).(\lambda (H_x0: (eq T u2 (lift h x1 x7))).(\lambda (H16: (pr0 x4 
+x7)).(eq_ind_r T (lift h x1 x7) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T 
+(THead (Bind b) t (THead (Flat Appl) (lift (S O) O v2) (lift h (S x1) x6))) 
+(lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) 
+x4 x5)) t4)))) (ex2_ind T (\lambda (t4: T).(eq T v2 (lift h x1 t4))) (\lambda 
+(t4: T).(pr0 x2 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 
+x7) (THead (Flat Appl) (lift (S O) O v2) (lift h (S x1) x6))) (lift h x1 
+t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) 
+t4))) (\lambda (x8: T).(\lambda (H_x1: (eq T v2 (lift h x1 x8))).(\lambda 
+(H17: (pr0 x2 x8)).(eq_ind_r T (lift h x1 x8) (\lambda (t: T).(ex2 T (\lambda 
+(t4: T).(eq T (THead (Bind b) (lift h x1 x7) (THead (Flat Appl) (lift (S O) O 
+t) (lift h (S x1) x6))) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat 
+Appl) x2 (THead (Bind b) x4 x5)) t4)))) (eq_ind T (lift h (plus (S O) x1) 
+(lift (S O) O x8)) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind 
+b) (lift h x1 x7) (THead (Flat Appl) t (lift h (S x1) x6))) (lift h x1 t4))) 
+(\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) t4)))) 
+(eq_ind T (lift h (S x1) (THead (Flat Appl) (lift (S O) O x8) x6)) (\lambda 
+(t: T).(ex2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 x7) t) (lift 
+h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 
+x5)) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (THead (Bind b) (lift h x1 
+x7) (lift h (S x1) (THead (Flat Appl) (lift (S O) O x8) x6))) (lift h x1 
+t4))) (\lambda (t4: T).(pr0 (THead (Flat Appl) x2 (THead (Bind b) x4 x5)) 
+t4)) (THead (Bind b) x7 (THead (Flat Appl) (lift (S O) O x8) x6)) (sym_eq T 
+(lift h x1 (THead (Bind b) x7 (THead (Flat Appl) (lift (S O) O x8) x6))) 
+(THead (Bind b) (lift h x1 x7) (lift h (S x1) (THead (Flat Appl) (lift (S O) 
+O x8) x6))) (lift_bind b x7 (THead (Flat Appl) (lift (S O) O x8) x6) h x1)) 
+(pr0_upsilon b H1 x2 x8 H17 x4 x7 H16 x5 x6 H15)) (THead (Flat Appl) (lift h 
+(S x1) (lift (S O) O x8)) (lift h (S x1) x6)) (lift_flat Appl (lift (S O) O 
+x8) x6 h (S x1))) (lift (S O) O (lift h x1 x8)) (lift_d x8 h (S O) x1 O 
+(le_O_n x1))) v2 H_x1)))) (H3 x2 x1 H10)) u2 H_x0)))) (H5 x4 x1 H13)) t3 
+H_x)))) (H7 x5 (S x1) H14)) x3 H12)))))) (lift_gen_bind b u1 t2 x3 h x1 H11)) 
+x0 H9)))))) (lift_gen_flat Appl v1 (THead (Bind b) u1 t2) x0 h x1 
+H8))))))))))))))))))) (\lambda (u1: T).(\lambda (u2: T).(\lambda (_: (pr0 u1 
+u2)).(\lambda (H2: ((\forall (x0: T).(\forall (x1: nat).((eq T u1 (lift h x1 
+x0)) \to (ex2 T (\lambda (t2: T).(eq T u2 (lift h x1 t2))) (\lambda (t2: 
+T).(pr0 x0 t2)))))))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 
+t3)).(\lambda (H4: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 
+x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: 
+T).(pr0 x0 t4)))))))).(\lambda (w: T).(\lambda (H5: (subst0 O u2 t3 
+w)).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H6: (eq T (THead (Bind 
+Abbr) u1 t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: 
+T).(eq T x0 (THead (Bind Abbr) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq 
+T u1 (lift h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (S 
+x1) z)))) (ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift h x1 
+t4))) (\lambda (t4: T).(pr0 x0 t4))) (\lambda (x2: T).(\lambda (x3: 
+T).(\lambda (H7: (eq T x0 (THead (Bind Abbr) x2 x3))).(\lambda (H8: (eq T u1 
+(lift h x1 x2))).(\lambda (H9: (eq T t2 (lift h (S x1) x3))).(eq_ind_r T 
+(THead (Bind Abbr) x2 x3) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T 
+(THead (Bind Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) 
+(ex2_ind T (\lambda (t4: T).(eq T t3 (lift h (S x1) t4))) (\lambda (t4: 
+T).(pr0 x3 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Bind Abbr) u2 w) (lift 
+h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4))) (\lambda 
+(x4: T).(\lambda (H_x: (eq T t3 (lift h (S x1) x4))).(\lambda (H10: (pr0 x3 
+x4)).(let H11 \def (eq_ind T t3 (\lambda (t: T).(subst0 O u2 t w)) H5 (lift h 
+(S x1) x4) H_x) in (ex2_ind T (\lambda (t4: T).(eq T u2 (lift h x1 t4))) 
+(\lambda (t4: T).(pr0 x2 t4)) (ex2 T (\lambda (t4: T).(eq T (THead (Bind 
+Abbr) u2 w) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind Abbr) x2 x3) 
+t4))) (\lambda (x5: T).(\lambda (H_x0: (eq T u2 (lift h x1 x5))).(\lambda 
+(H12: (pr0 x2 x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda 
+(t4: T).(eq T (THead (Bind Abbr) t w) (lift h x1 t4))) (\lambda (t4: T).(pr0 
+(THead (Bind Abbr) x2 x3) t4)))) (let H13 \def (eq_ind T u2 (\lambda (t: 
+T).(subst0 O t (lift h (S x1) x4) w)) H11 (lift h x1 x5) H_x0) in (let H14 
+\def (refl_equal nat (S (plus O x1))) in (let H15 \def (eq_ind nat (S x1) 
+(\lambda (n: nat).(subst0 O (lift h x1 x5) (lift h n x4) w)) H13 (S (plus O 
+x1)) H14) in (ex2_ind T (\lambda (t4: T).(eq T w (lift h (S (plus O x1)) 
+t4))) (\lambda (t4: T).(subst0 O x5 x4 t4)) (ex2 T (\lambda (t4: T).(eq T 
+(THead (Bind Abbr) (lift h x1 x5) w) (lift h x1 t4))) (\lambda (t4: T).(pr0 
+(THead (Bind Abbr) x2 x3) t4))) (\lambda (x6: T).(\lambda (H16: (eq T w (lift 
+h (S (plus O x1)) x6))).(\lambda (H17: (subst0 O x5 x4 x6)).(eq_ind_r T (lift 
+h (S (plus O x1)) x6) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T (THead 
+(Bind Abbr) (lift h x1 x5) t) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead 
+(Bind Abbr) x2 x3) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (THead (Bind 
+Abbr) (lift h x1 x5) (lift h (S (plus O x1)) x6)) (lift h x1 t4))) (\lambda 
+(t4: T).(pr0 (THead (Bind Abbr) x2 x3) t4)) (THead (Bind Abbr) x5 x6) (sym_eq 
+T (lift h x1 (THead (Bind Abbr) x5 x6)) (THead (Bind Abbr) (lift h x1 x5) 
+(lift h (S (plus O x1)) x6)) (lift_bind Abbr x5 x6 h (plus O x1))) (pr0_delta 
+x2 x5 H12 x3 x4 H10 x6 H17)) w H16)))) (subst0_gen_lift_lt x5 x4 w O h x1 
+H15))))) u2 H_x0)))) (H2 x2 x1 H8)))))) (H4 x3 (S x1) H9)) x0 H7)))))) 
+(lift_gen_bind Abbr u1 t2 x0 h x1 H6))))))))))))))) (\lambda (b: B).(\lambda 
+(H1: (not (eq B b Abst))).(\lambda (t2: T).(\lambda (t3: T).(\lambda (_: (pr0 
+t2 t3)).(\lambda (H3: ((\forall (x0: T).(\forall (x1: nat).((eq T t2 (lift h 
+x1 x0)) \to (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: 
+T).(pr0 x0 t4)))))))).(\lambda (u: T).(\lambda (x0: T).(\lambda (x1: 
+nat).(\lambda (H4: (eq T (THead (Bind b) u (lift (S O) O t2)) (lift h x1 
+x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T x0 (THead (Bind 
+b) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift h x1 y0)))) 
+(\lambda (_: T).(\lambda (z: T).(eq T (lift (S O) O t2) (lift h (S x1) z)))) 
+(ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 
+t4))) (\lambda (x2: T).(\lambda (x3: T).(\lambda (H5: (eq T x0 (THead (Bind 
+b) x2 x3))).(\lambda (_: (eq T u (lift h x1 x2))).(\lambda (H7: (eq T (lift 
+(S O) O t2) (lift h (S x1) x3))).(eq_ind_r T (THead (Bind b) x2 x3) (\lambda 
+(t: T).(ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: 
+T).(pr0 t t4)))) (let H8 \def (eq_ind_r nat (plus (S O) x1) (\lambda (n: 
+nat).(eq nat (S x1) n)) (refl_equal nat (plus (S O) x1)) (plus x1 (S O)) 
+(plus_comm x1 (S O))) in (let H9 \def (eq_ind nat (S x1) (\lambda (n: 
+nat).(eq T (lift (S O) O t2) (lift h n x3))) H7 (plus x1 (S O)) H8) in 
+(ex2_ind T (\lambda (t4: T).(eq T x3 (lift (S O) O t4))) (\lambda (t4: T).(eq 
+T t2 (lift h x1 t4))) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) 
+(\lambda (t4: T).(pr0 (THead (Bind b) x2 x3) t4))) (\lambda (x4: T).(\lambda 
+(H10: (eq T x3 (lift (S O) O x4))).(\lambda (H11: (eq T t2 (lift h x1 
+x4))).(eq_ind_r T (lift (S O) O x4) (\lambda (t: T).(ex2 T (\lambda (t4: 
+T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 t) 
+t4)))) (ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: 
+T).(pr0 x4 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda 
+(t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4))) (\lambda (x5: 
+T).(\lambda (H_x: (eq T t3 (lift h x1 x5))).(\lambda (H12: (pr0 x4 
+x5)).(eq_ind_r T (lift h x1 x5) (\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T 
+t (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O 
+x4)) t4)))) (ex_intro2 T (\lambda (t4: T).(eq T (lift h x1 x5) (lift h x1 
+t4))) (\lambda (t4: T).(pr0 (THead (Bind b) x2 (lift (S O) O x4)) t4)) x5 
+(refl_equal T (lift h x1 x5)) (pr0_zeta b H1 x4 x5 H12 x2)) t3 H_x)))) (H3 x4 
+x1 H11)) x3 H10)))) (lift_gen_lift t2 x3 (S O) h O x1 (le_O_n x1) H9)))) x0 
+H5)))))) (lift_gen_bind b u (lift (S O) O t2) x0 h x1 H4)))))))))))) (\lambda 
+(t2: T).(\lambda (t3: T).(\lambda (_: (pr0 t2 t3)).(\lambda (H2: ((\forall 
+(x0: T).(\forall (x1: nat).((eq T t2 (lift h x1 x0)) \to (ex2 T (\lambda (t4: 
+T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4)))))))).(\lambda (u: 
+T).(\lambda (x0: T).(\lambda (x1: nat).(\lambda (H3: (eq T (THead (Flat Cast) 
+u t2) (lift h x1 x0))).(ex3_2_ind T T (\lambda (y0: T).(\lambda (z: T).(eq T 
+x0 (THead (Flat Cast) y0 z)))) (\lambda (y0: T).(\lambda (_: T).(eq T u (lift 
+h x1 y0)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h x1 z)))) (ex2 T 
+(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 x0 t4))) 
+(\lambda (x2: T).(\lambda (x3: T).(\lambda (H4: (eq T x0 (THead (Flat Cast) 
+x2 x3))).(\lambda (_: (eq T u (lift h x1 x2))).(\lambda (H6: (eq T t2 (lift h 
+x1 x3))).(eq_ind_r T (THead (Flat Cast) x2 x3) (\lambda (t: T).(ex2 T 
+(\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 t t4)))) 
+(ex2_ind T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: T).(pr0 
+x3 t4)) (ex2 T (\lambda (t4: T).(eq T t3 (lift h x1 t4))) (\lambda (t4: 
+T).(pr0 (THead (Flat Cast) x2 x3) t4))) (\lambda (x4: T).(\lambda (H_x: (eq T 
+t3 (lift h x1 x4))).(\lambda (H7: (pr0 x3 x4)).(eq_ind_r T (lift h x1 x4) 
+(\lambda (t: T).(ex2 T (\lambda (t4: T).(eq T t (lift h x1 t4))) (\lambda 
+(t4: T).(pr0 (THead (Flat Cast) x2 x3) t4)))) (ex_intro2 T (\lambda (t4: 
+T).(eq T (lift h x1 x4) (lift h x1 t4))) (\lambda (t4: T).(pr0 (THead (Flat 
+Cast) x2 x3) t4)) x4 (refl_equal T (lift h x1 x4)) (pr0_epsilon x3 x4 H7 x2)) 
+t3 H_x)))) (H2 x3 x1 H6)) x0 H4)))))) (lift_gen_flat Cast u t2 x0 h x1 
+H3)))))))))) y x H0))))) H))))).