- transcript: patched to generate CoRN_notation.ma instead of CoRN.ma

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

diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd.ma
index ba33d14417adceddccdc0f6a8ab35caa035f5862..44ca12dec4b32f01ed1108a85972d18b9f3b8e78 100644 (file)
--- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd.ma
+++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd.ma
@@ -32,7 +32,7 @@ theorem lift_lref_lt:
  \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (lt n 
 d)).(eq_ind bool true (\lambda (b: bool).(eq T (TLRef (match b with [true 
 \Rightarrow n | false \Rightarrow (plus n h)])) (TLRef n))) (refl_equal T 
-(TLRef n)) (blt n d) (sym_equal bool (blt n d) true (lt_blt d n H)))))).
+(TLRef n)) (blt n d) (sym_eq bool (blt n d) true (lt_blt d n H)))))).
 
 theorem lift_lref_ge:
  \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((le d n) \to (eq T 
@@ -41,7 +41,7 @@ theorem lift_lref_ge:
  \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (le d 
 n)).(eq_ind bool false (\lambda (b: bool).(eq T (TLRef (match b with [true 
 \Rightarrow n | false \Rightarrow (plus n h)])) (TLRef (plus n h)))) 
-(refl_equal T (TLRef (plus n h))) (blt n d) (sym_equal bool (blt n d) false 
+(refl_equal T (TLRef (plus n h))) (blt n d) (sym_eq bool (blt n d) false 
 (le_bge d n H)))))).
 
 theorem lift_head:
@@ -78,38 +78,38 @@ theorem lift_gen_sort:
 n0)))).(sym_eq T (TSort n) (TSort n0) H))) (\lambda (n0: nat).(\lambda (H: 
 (eq T (TSort n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TSort 
 n)) (\lambda (H0: (lt n0 d)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) 
-(\lambda (t: T).(eq T (TSort n) t)) H (TLRef n0) (lift_lref_lt n0 h d H0)) in 
-(let H2 \def (match H1 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? 
-t)).((eq T t (TLRef n0)) \to (eq T (TLRef n0) (TSort n))))) with [refl_equal 
-\Rightarrow (\lambda (H1: (eq T (TSort n) (TLRef n0))).(let H2 \def (eq_ind T 
-(TSort n) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with 
-[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) 
-\Rightarrow False])) I (TLRef n0) H1) in (False_ind (eq T (TLRef n0) (TSort 
-n)) H2)))]) in (H2 (refl_equal T (TLRef n0)))))) (\lambda (H0: (le d 
-n0)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T 
-(TSort n) t)) H (TLRef (plus n0 h)) (lift_lref_ge n0 h d H0)) in (let H2 \def 
-(match H1 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t 
-(TLRef (plus n0 h))) \to (eq T (TLRef n0) (TSort n))))) with [refl_equal 
-\Rightarrow (\lambda (H1: (eq T (TSort n) (TLRef (plus n0 h)))).(let H2 \def 
-(eq_ind T (TSort n) (\lambda (e: T).(match e in T return (\lambda (_: 
+(\lambda (t0: T).(eq T (TSort n) t0)) H (TLRef n0) (lift_lref_lt n0 h d H0)) 
+in (let H2 \def (match H1 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? 
+t0)).((eq T t0 (TLRef n0)) \to (eq T (TLRef n0) (TSort n))))) with 
+[refl_equal \Rightarrow (\lambda (H2: (eq T (TSort n) (TLRef n0))).(let H3 
+\def (eq_ind T (TSort n) (\lambda (e: T).(match e in T return (\lambda (_: 
 T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow False])) I (TLRef (plus n0 h)) H1) in (False_ind 
-(eq T (TLRef n0) (TSort n)) H2)))]) in (H2 (refl_equal T (TLRef (plus n0 
-h)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TSort n) 
-(lift h d t0)) \to (eq T t0 (TSort n))))).(\lambda (t1: T).(\lambda (_: (((eq 
-T (TSort n) (lift h d t1)) \to (eq T t1 (TSort n))))).(\lambda (H1: (eq T 
-(TSort n) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d 
-(THead k t0 t1)) (\lambda (t: T).(eq T (TSort n) t)) H1 (THead k (lift h d 
-t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def (match H2 
-in eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead k (lift 
-h d t0) (lift h (s k d) t1))) \to (eq T (THead k t0 t1) (TSort n))))) with 
-[refl_equal \Rightarrow (\lambda (H2: (eq T (TSort n) (THead k (lift h d t0) 
-(lift h (s k d) t1)))).(let H3 \def (eq_ind T (TSort n) (\lambda (e: 
-T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
-True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I 
-(THead k (lift h d t0) (lift h (s k d) t1)) H2) in (False_ind (eq T (THead k 
-t0 t1) (TSort n)) H3)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h 
-(s k d) t1)))))))))))) t)))).
+(THead _ _ _) \Rightarrow False])) I (TLRef n0) H2) in (False_ind (eq T 
+(TLRef n0) (TSort n)) H3)))]) in (H2 (refl_equal T (TLRef n0)))))) (\lambda 
+(H0: (le d n0)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: 
+T).(eq T (TSort n) t0)) H (TLRef (plus n0 h)) (lift_lref_ge n0 h d H0)) in 
+(let H2 \def (match H1 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? 
+t0)).((eq T t0 (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TSort n))))) with 
+[refl_equal \Rightarrow (\lambda (H2: (eq T (TSort n) (TLRef (plus n0 
+h)))).(let H3 \def (eq_ind T (TSort n) (\lambda (e: T).(match e in T return 
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) 
+\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef (plus n0 h)) 
+H2) in (False_ind (eq T (TLRef n0) (TSort n)) H3)))]) in (H2 (refl_equal T 
+(TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: 
+(((eq T (TSort n) (lift h d t0)) \to (eq T t0 (TSort n))))).(\lambda (t1: 
+T).(\lambda (_: (((eq T (TSort n) (lift h d t1)) \to (eq T t1 (TSort 
+n))))).(\lambda (H1: (eq T (TSort n) (lift h d (THead k t0 t1)))).(let H2 
+\def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (TSort n) 
+t2)) H1 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) 
+in (let H3 \def (match H2 in eq return (\lambda (t2: T).(\lambda (_: (eq ? ? 
+t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d) t1))) \to (eq T (THead 
+k t0 t1) (TSort n))))) with [refl_equal \Rightarrow (\lambda (H3: (eq T 
+(TSort n) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H4 \def (eq_ind 
+T (TSort n) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with 
+[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) 
+\Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) t1)) H3) in 
+(False_ind (eq T (THead k t0 t1) (TSort n)) H4)))]) in (H3 (refl_equal T 
+(THead k (lift h d t0) (lift h (s k d) t1)))))))))))) t)))).
 
 theorem lift_gen_lref:
  \forall (t: T).(\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T 
@@ -121,7 +121,7 @@ nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to (or (land (lt i d)
 (eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 (TLRef (minus i 
 h)))))))))) (\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda 
 (i: nat).(\lambda (H: (eq T (TLRef i) (lift h d (TSort n)))).(let H0 \def 
-(eq_ind T (lift h d (TSort n)) (\lambda (t: T).(eq T (TLRef i) t)) H (TSort 
+(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TSort 
 n) (lift_sort n h d)) in (let H1 \def (eq_ind T (TLRef i) (\lambda (ee: 
 T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
 False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I 
@@ -131,40 +131,40 @@ False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I
 nat).(\lambda (H: (eq T (TLRef i) (lift h d (TLRef n)))).(lt_le_e n d (or 
 (land (lt i d) (eq T (TLRef n) (TLRef i))) (land (le (plus d h) i) (eq T 
 (TLRef n) (TLRef (minus i h))))) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind 
-T (lift h d (TLRef n)) (\lambda (t: T).(eq T (TLRef i) t)) H (TLRef n) 
+T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TLRef n) 
 (lift_lref_lt n h d H0)) in (let H2 \def (f_equal T nat (\lambda (e: 
 T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i | 
-(TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) 
-H1) in (eq_ind_r nat n (\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef n) 
-(TLRef n0))) (land (le (plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 
+(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef 
+n) H1) in (eq_ind_r nat n (\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef 
+n) (TLRef n0))) (land (le (plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 
 h)))))) (or_introl (land (lt n d) (eq T (TLRef n) (TLRef n))) (land (le (plus 
 d h) n) (eq T (TLRef n) (TLRef (minus n h)))) (conj (lt n d) (eq T (TLRef n) 
 (TLRef n)) H0 (refl_equal T (TLRef n)))) i H2)))) (\lambda (H0: (le d 
-n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t: T).(eq T (TLRef 
-i) t)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (f_equal 
-T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort 
-_) \Rightarrow i | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i])) 
-(TLRef i) (TLRef (plus n h)) H1) in (eq_ind_r nat (plus n h) (\lambda (n0: 
-nat).(or (land (lt n0 d) (eq T (TLRef n) (TLRef n0))) (land (le (plus d h) 
-n0) (eq T (TLRef n) (TLRef (minus n0 h)))))) (eq_ind_r nat n (\lambda (n0: 
-nat).(or (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n h)))) (land 
-(le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n0))))) (or_intror (land 
-(lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n h)))) (land (le (plus d h) 
-(plus n h)) (eq T (TLRef n) (TLRef n))) (conj (le (plus d h) (plus n h)) (eq 
-T (TLRef n) (TLRef n)) (plus_le_compat d n h h H0 (le_n h)) (refl_equal T 
-(TLRef n)))) (minus (plus n h) h) (minus_plus_r n h)) i H2)))))))))) (\lambda 
-(k: K).(\lambda (t0: T).(\lambda (_: ((\forall (d: nat).(\forall (h: 
-nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to (or (land (lt i d) 
-(eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 (TLRef (minus i 
-h))))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (d: nat).(\forall (h: 
-nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t1)) \to (or (land (lt i d) 
-(eq T t1 (TLRef i))) (land (le (plus d h) i) (eq T t1 (TLRef (minus i 
-h))))))))))).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i: nat).(\lambda 
-(H1: (eq T (TLRef i) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T 
-(lift h d (THead k t0 t1)) (\lambda (t: T).(eq T (TLRef i) t)) H1 (THead k 
-(lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def 
-(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
+n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef 
+i) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def 
+(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with 
+[(TSort _) \Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _) 
+\Rightarrow i])) (TLRef i) (TLRef (plus n h)) H1) in (eq_ind_r nat (plus n h) 
+(\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef n) (TLRef n0))) (land (le 
+(plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 h)))))) (eq_ind_r nat n 
+(\lambda (n0: nat).(or (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n 
+h)))) (land (le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n0))))) 
+(or_intror (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n h)))) (land 
+(le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n))) (conj (le (plus d h) 
+(plus n h)) (eq T (TLRef n) (TLRef n)) (plus_le_compat d n h h H0 (le_n h)) 
+(refl_equal T (TLRef n)))) (minus (plus n h) h) (minus_plus_r n h)) i 
+H2)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: ((\forall (d: 
+nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to 
+(or (land (lt i d) (eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 
+(TLRef (minus i h))))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (d: 
+nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t1)) \to 
+(or (land (lt i d) (eq T t1 (TLRef i))) (land (le (plus d h) i) (eq T t1 
+(TLRef (minus i h))))))))))).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i: 
+nat).(\lambda (H1: (eq T (TLRef i) (lift h d (THead k t0 t1)))).(let H2 \def 
+(eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (TLRef i) t2)) H1 
+(THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let 
+H3 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda 
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
 (THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) 
 t1)) H2) in (False_ind (or (land (lt i d) (eq T (THead k t0 t1) (TLRef i))) 
 (land (le (plus d h) i) (eq T (THead k t0 t1) (TLRef (minus i h))))) 
@@ -180,35 +180,36 @@ d)).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq T (TLRef n) (lift h d t0))
 (lift h d (TSort n0)))).(sym_eq T (TLRef n) (TSort n0) H0))) (\lambda (n0: 
 nat).(\lambda (H0: (eq T (TLRef n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq 
 T (TLRef n0) (TLRef n)) (\lambda (H1: (lt n0 d)).(let H2 \def (eq_ind T (lift 
-h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef n) t)) H0 (TLRef n0) 
+h d (TLRef n0)) (\lambda (t0: T).(eq T (TLRef n) t0)) H0 (TLRef n0) 
 (lift_lref_lt n0 h d H1)) in (sym_eq T (TLRef n) (TLRef n0) H2))) (\lambda 
-(H1: (le d n0)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: 
-T).(eq T (TLRef n) t)) H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in 
-(let H3 \def (match H2 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? 
-t)).((eq T t (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TLRef n))))) with 
-[refl_equal \Rightarrow (\lambda (H2: (eq T (TLRef n) (TLRef (plus n0 
-h)))).(let H3 \def (f_equal T nat (\lambda (e: T).(match e in T return 
-(\lambda (_: T).nat) with [(TSort _) \Rightarrow n | (TLRef n) \Rightarrow n 
-| (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef (plus n0 h)) H2) in (eq_ind 
-nat (plus n0 h) (\lambda (n: nat).(eq T (TLRef n0) (TLRef n))) (let H0 \def 
-(eq_ind nat n (\lambda (n: nat).(lt n d)) H (plus n0 h) H3) in (le_false d n0 
-(eq T (TLRef n0) (TLRef (plus n0 h))) H1 (lt_le_S n0 d (le_lt_trans n0 (plus 
-n0 h) d (le_plus_l n0 h) H0)))) n (sym_eq nat n (plus n0 h) H3))))]) in (H3 
-(refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0: 
-T).(\lambda (_: (((eq T (TLRef n) (lift h d t0)) \to (eq T t0 (TLRef 
-n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef n) (lift h d t1)) \to (eq 
-T t1 (TLRef n))))).(\lambda (H2: (eq T (TLRef n) (lift h d (THead k t0 
-t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t: T).(eq 
-T (TLRef n) t)) H2 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k 
-t0 t1 h d)) in (let H4 \def (match H3 in eq return (\lambda (t: T).(\lambda 
-(_: (eq ? ? t)).((eq T t (THead k (lift h d t0) (lift h (s k d) t1))) \to (eq 
-T (THead k t0 t1) (TLRef n))))) with [refl_equal \Rightarrow (\lambda (H3: 
-(eq T (TLRef n) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H4 \def 
-(eq_ind T (TLRef n) (\lambda (e: T).(match e in T return (\lambda (_: 
-T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | 
-(THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) 
-t1)) H3) in (False_ind (eq T (THead k t0 t1) (TLRef n)) H4)))]) in (H4 
-(refl_equal T (THead k (lift h d t0) (lift h (s k d) t1)))))))))))) t))))).
+(H1: (le d n0)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: 
+T).(eq T (TLRef n) t0)) H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in 
+(let H3 \def (match H2 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? 
+t0)).((eq T t0 (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TLRef n))))) with 
+[refl_equal \Rightarrow (\lambda (H3: (eq T (TLRef n) (TLRef (plus n0 
+h)))).(let H4 \def (f_equal T nat (\lambda (e: T).(match e in T return 
+(\lambda (_: T).nat) with [(TSort _) \Rightarrow n | (TLRef n1) \Rightarrow 
+n1 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef (plus n0 h)) H3) in 
+(eq_ind nat (plus n0 h) (\lambda (n1: nat).(eq T (TLRef n0) (TLRef n1))) (let 
+H5 \def (eq_ind nat n (\lambda (n1: nat).(lt n1 d)) H (plus n0 h) H4) in 
+(le_false d n0 (eq T (TLRef n0) (TLRef (plus n0 h))) H1 (lt_le_S n0 d 
+(le_lt_trans n0 (plus n0 h) d (le_plus_l n0 h) H5)))) n (sym_eq nat n (plus 
+n0 h) H4))))]) in (H3 (refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: 
+K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef n) (lift h d t0)) \to (eq T 
+t0 (TLRef n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef n) (lift h d 
+t1)) \to (eq T t1 (TLRef n))))).(\lambda (H2: (eq T (TLRef n) (lift h d 
+(THead k t0 t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda 
+(t2: T).(eq T (TLRef n) t2)) H2 (THead k (lift h d t0) (lift h (s k d) t1)) 
+(lift_head k t0 t1 h d)) in (let H4 \def (match H3 in eq return (\lambda (t2: 
+T).(\lambda (_: (eq ? ? t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d) 
+t1))) \to (eq T (THead k t0 t1) (TLRef n))))) with [refl_equal \Rightarrow 
+(\lambda (H4: (eq T (TLRef n) (THead k (lift h d t0) (lift h (s k d) 
+t1)))).(let H5 \def (eq_ind T (TLRef n) (\lambda (e: T).(match e in T return 
+(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d 
+t0) (lift h (s k d) t1)) H4) in (False_ind (eq T (THead k t0 t1) (TLRef n)) 
+H5)))]) in (H4 (refl_equal T (THead k (lift h d t0) (lift h (s k d) 
+t1)))))))))))) t))))).
 
 theorem lift_gen_lref_false:
  \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n 
@@ -219,8 +220,8 @@ theorem lift_gen_lref_false:
 n)).(\lambda (H0: (lt n (plus d h))).(\lambda (t: T).(T_ind (\lambda (t0: 
 T).((eq T (TLRef n) (lift h d t0)) \to (\forall (P: Prop).P))) (\lambda (n0: 
 nat).(\lambda (H1: (eq T (TLRef n) (lift h d (TSort n0)))).(\lambda (P: 
-Prop).(let H2 \def (match H1 in eq return (\lambda (t: T).(\lambda (_: (eq ? 
-? t)).((eq T t (lift h d (TSort n0))) \to P))) with [refl_equal \Rightarrow 
+Prop).(let H2 \def (match H1 in eq return (\lambda (t0: T).(\lambda (_: (eq ? 
+? t0)).((eq T t0 (lift h d (TSort n0))) \to P))) with [refl_equal \Rightarrow 
 (\lambda (H2: (eq T (TLRef n) (lift h d (TSort n0)))).(let H3 \def (eq_ind T 
 (TLRef n) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with 
 [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) 
@@ -228,40 +229,41 @@ Prop).(let H2 \def (match H1 in eq return (\lambda (t: T).(\lambda (_: (eq ?
 (H2 (refl_equal T (lift h d (TSort n0)))))))) (\lambda (n0: nat).(\lambda 
 (H1: (eq T (TLRef n) (lift h d (TLRef n0)))).(\lambda (P: Prop).(lt_le_e n0 d 
 P (\lambda (H2: (lt n0 d)).(let H3 \def (eq_ind T (lift h d (TLRef n0)) 
-(\lambda (t: T).(eq T (TLRef n) t)) H1 (TLRef n0) (lift_lref_lt n0 h d H2)) 
-in (let H4 \def (match H3 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? 
-t)).((eq T t (TLRef n0)) \to P))) with [refl_equal \Rightarrow (\lambda (H3: 
-(eq T (TLRef n) (TLRef n0))).(let H4 \def (f_equal T nat (\lambda (e: 
+(\lambda (t0: T).(eq T (TLRef n) t0)) H1 (TLRef n0) (lift_lref_lt n0 h d H2)) 
+in (let H4 \def (match H3 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? 
+t0)).((eq T t0 (TLRef n0)) \to P))) with [refl_equal \Rightarrow (\lambda 
+(H4: (eq T (TLRef n) (TLRef n0))).(let H5 \def (f_equal T nat (\lambda (e: 
 T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n | 
-(TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef n0) 
-H3) in (eq_ind nat n0 (\lambda (_: nat).P) (let H1 \def (eq_ind_r nat n0 
-(\lambda (n: nat).(lt n d)) H2 n H4) in (le_false d n P H H1)) n (sym_eq nat 
-n n0 H4))))]) in (H4 (refl_equal T (TLRef n0)))))) (\lambda (H2: (le d 
-n0)).(let H3 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T 
-(TLRef n) t)) H1 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H2)) in (let H4 
-\def (match H3 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t 
-(TLRef (plus n0 h))) \to P))) with [refl_equal \Rightarrow (\lambda (H3: (eq 
-T (TLRef n) (TLRef (plus n0 h)))).(let H4 \def (f_equal T nat (\lambda (e: 
-T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n | 
-(TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef 
-(plus n0 h)) H3) in (eq_ind nat (plus n0 h) (\lambda (_: nat).P) (let H1 \def 
-(eq_ind nat n (\lambda (n: nat).(lt n (plus d h))) H0 (plus n0 h) H4) in 
-(le_false d n0 P H2 (lt_le_S n0 d (simpl_lt_plus_r h n0 d H1)))) n (sym_eq 
-nat n (plus n0 h) H4))))]) in (H4 (refl_equal T (TLRef (plus n0 h))))))))))) 
-(\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef n) (lift h d 
-t0)) \to (\forall (P: Prop).P)))).(\lambda (t1: T).(\lambda (_: (((eq T 
-(TLRef n) (lift h d t1)) \to (\forall (P: Prop).P)))).(\lambda (H3: (eq T 
-(TLRef n) (lift h d (THead k t0 t1)))).(\lambda (P: Prop).(let H4 \def 
-(eq_ind T (lift h d (THead k t0 t1)) (\lambda (t: T).(eq T (TLRef n) t)) H3 
+(TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef 
+n0) H4) in (eq_ind nat n0 (\lambda (_: nat).P) (let H6 \def (eq_ind_r nat n0 
+(\lambda (n1: nat).(lt n1 d)) H2 n H5) in (le_false d n P H H6)) n (sym_eq 
+nat n n0 H5))))]) in (H4 (refl_equal T (TLRef n0)))))) (\lambda (H2: (le d 
+n0)).(let H3 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: T).(eq T 
+(TLRef n) t0)) H1 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H2)) in (let H4 
+\def (match H3 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T 
+t0 (TLRef (plus n0 h))) \to P))) with [refl_equal \Rightarrow (\lambda (H4: 
+(eq T (TLRef n) (TLRef (plus n0 h)))).(let H5 \def (f_equal T nat (\lambda 
+(e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow 
+n | (TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n])) (TLRef n) 
+(TLRef (plus n0 h)) H4) in (eq_ind nat (plus n0 h) (\lambda (_: nat).P) (let 
+H6 \def (eq_ind nat n (\lambda (n1: nat).(lt n1 (plus d h))) H0 (plus n0 h) 
+H5) in (le_false d n0 P H2 (lt_le_S n0 d (simpl_lt_plus_r h n0 d H6)))) n 
+(sym_eq nat n (plus n0 h) H5))))]) in (H4 (refl_equal T (TLRef (plus n0 
+h))))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef n) 
+(lift h d t0)) \to (\forall (P: Prop).P)))).(\lambda (t1: T).(\lambda (_: 
+(((eq T (TLRef n) (lift h d t1)) \to (\forall (P: Prop).P)))).(\lambda (H3: 
+(eq T (TLRef n) (lift h d (THead k t0 t1)))).(\lambda (P: Prop).(let H4 \def 
+(eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (TLRef n) t2)) H3 
 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let 
-H5 \def (match H4 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq 
-T t (THead k (lift h d t0) (lift h (s k d) t1))) \to P))) with [refl_equal 
-\Rightarrow (\lambda (H4: (eq T (TLRef n) (THead k (lift h d t0) (lift h (s k 
-d) t1)))).(let H5 \def (eq_ind T (TLRef n) (\lambda (e: T).(match e in T 
-return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
-\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d 
-t0) (lift h (s k d) t1)) H4) in (False_ind P H5)))]) in (H5 (refl_equal T 
-(THead k (lift h d t0) (lift h (s k d) t1))))))))))))) t)))))).
+H5 \def (match H4 in eq return (\lambda (t2: T).(\lambda (_: (eq ? ? 
+t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d) t1))) \to P))) with 
+[refl_equal \Rightarrow (\lambda (H5: (eq T (TLRef n) (THead k (lift h d t0) 
+(lift h (s k d) t1)))).(let H6 \def (eq_ind T (TLRef n) (\lambda (e: 
+T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
+False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I 
+(THead k (lift h d t0) (lift h (s k d) t1)) H5) in (False_ind P H6)))]) in 
+(H5 (refl_equal T (THead k (lift h d t0) (lift h (s k d) t1))))))))))))) 
+t)))))).
 
 theorem lift_gen_lref_ge:
  \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to (\forall 
@@ -271,7 +273,7 @@ theorem lift_gen_lref_ge:
 n)).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq T (TLRef (plus n h)) (lift h 
 d t0)) \to (eq T t0 (TLRef n)))) (\lambda (n0: nat).(\lambda (H0: (eq T 
 (TLRef (plus n h)) (lift h d (TSort n0)))).(let H1 \def (match H0 in eq 
-return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (lift h d (TSort 
+return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (lift h d (TSort 
 n0))) \to (eq T (TSort n0) (TLRef n))))) with [refl_equal \Rightarrow 
 (\lambda (H1: (eq T (TLRef (plus n h)) (lift h d (TSort n0)))).(let H2 \def 
 (eq_ind T (TLRef (plus n h)) (\lambda (e: T).(match e in T return (\lambda 
@@ -280,52 +282,53 @@ n0))) \to (eq T (TSort n0) (TLRef n))))) with [refl_equal \Rightarrow
 (eq T (TSort n0) (TLRef n)) H2)))]) in (H1 (refl_equal T (lift h d (TSort 
 n0))))))) (\lambda (n0: nat).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d 
 (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TLRef n)) (\lambda (H1: (lt n0 
-d)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef 
-(plus n h)) t)) H0 (TLRef n0) (lift_lref_lt n0 h d H1)) in (let H3 \def 
-(match H2 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t 
-(TLRef n0)) \to (eq T (TLRef n0) (TLRef n))))) with [refl_equal \Rightarrow 
-(\lambda (H2: (eq T (TLRef (plus n h)) (TLRef n0))).(let H3 \def (f_equal T 
-nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) 
-\Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m: 
-nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in 
-plus) n h) | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow ((let rec 
-plus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O 
-\Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus) n h)])) (TLRef 
-(plus n h)) (TLRef n0) H2) in (eq_ind nat (plus n h) (\lambda (n0: nat).(eq T 
-(TLRef n0) (TLRef n))) (let H0 \def (eq_ind_r nat n0 (\lambda (n: nat).(lt n 
-d)) H1 (plus n h) H3) in (le_false d n (eq T (TLRef (plus n h)) (TLRef n)) H 
-(lt_le_S n d (le_lt_trans n (plus n h) d (le_plus_l n h) H0)))) n0 H3)))]) in 
-(H3 (refl_equal T (TLRef n0)))))) (\lambda (H1: (le d n0)).(let H2 \def 
-(eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef (plus n h)) t)) 
-H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in (let H3 \def (match H2 in 
-eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (TLRef (plus n0 
-h))) \to (eq T (TLRef n0) (TLRef n))))) with [refl_equal \Rightarrow (\lambda 
-(H2: (eq T (TLRef (plus n h)) (TLRef (plus n0 h)))).(let H3 \def (f_equal T 
-nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) 
-\Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m: 
-nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in 
-plus) n h) | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow ((let rec 
-plus (n: nat) on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O 
-\Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in plus) n h)])) (TLRef 
-(plus n h)) (TLRef (plus n0 h)) H2) in (eq_ind nat (plus n h) (\lambda (_: 
-nat).(eq T (TLRef n0) (TLRef n))) (f_equal nat T TLRef n0 n (simpl_plus_r h 
-n0 n (sym_eq nat (plus n h) (plus n0 h) H3))) (plus n0 h) H3)))]) in (H3 
-(refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0: 
-T).(\lambda (_: (((eq T (TLRef (plus n h)) (lift h d t0)) \to (eq T t0 (TLRef 
-n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef (plus n h)) (lift h d 
-t1)) \to (eq T t1 (TLRef n))))).(\lambda (H2: (eq T (TLRef (plus n h)) (lift 
-h d (THead k t0 t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) 
-(\lambda (t: T).(eq T (TLRef (plus n h)) t)) H2 (THead k (lift h d t0) (lift 
-h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H4 \def (match H3 in eq 
-return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead k (lift h d 
-t0) (lift h (s k d) t1))) \to (eq T (THead k t0 t1) (TLRef n))))) with 
-[refl_equal \Rightarrow (\lambda (H3: (eq T (TLRef (plus n h)) (THead k (lift 
-h d t0) (lift h (s k d) t1)))).(let H4 \def (eq_ind T (TLRef (plus n h)) 
-(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) 
-\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow 
-False])) I (THead k (lift h d t0) (lift h (s k d) t1)) H3) in (False_ind (eq 
-T (THead k t0 t1) (TLRef n)) H4)))]) in (H4 (refl_equal T (THead k (lift h d 
-t0) (lift h (s k d) t1)))))))))))) t))))).
+d)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: T).(eq T 
+(TLRef (plus n h)) t0)) H0 (TLRef n0) (lift_lref_lt n0 h d H1)) in (let H3 
+\def (match H2 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T 
+t0 (TLRef n0)) \to (eq T (TLRef n0) (TLRef n))))) with [refl_equal 
+\Rightarrow (\lambda (H3: (eq T (TLRef (plus n h)) (TLRef n0))).(let H4 \def 
+(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with 
+[(TSort _) \Rightarrow ((let rec plus (n1: nat) on n1: (nat \to nat) \def 
+(\lambda (m: nat).(match n1 with [O \Rightarrow m | (S p) \Rightarrow (S 
+(plus p m))])) in plus) n h) | (TLRef n1) \Rightarrow n1 | (THead _ _ _) 
+\Rightarrow ((let rec plus (n1: nat) on n1: (nat \to nat) \def (\lambda (m: 
+nat).(match n1 with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in 
+plus) n h)])) (TLRef (plus n h)) (TLRef n0) H3) in (eq_ind nat (plus n h) 
+(\lambda (n1: nat).(eq T (TLRef n1) (TLRef n))) (let H5 \def (eq_ind_r nat n0 
+(\lambda (n1: nat).(lt n1 d)) H1 (plus n h) H4) in (le_false d n (eq T (TLRef 
+(plus n h)) (TLRef n)) H (lt_le_S n d (le_lt_trans n (plus n h) d (le_plus_l 
+n h) H5)))) n0 H4)))]) in (H3 (refl_equal T (TLRef n0)))))) (\lambda (H1: (le 
+d n0)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: T).(eq T 
+(TLRef (plus n h)) t0)) H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in 
+(let H3 \def (match H2 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? 
+t0)).((eq T t0 (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TLRef n))))) with 
+[refl_equal \Rightarrow (\lambda (H3: (eq T (TLRef (plus n h)) (TLRef (plus 
+n0 h)))).(let H4 \def (f_equal T nat (\lambda (e: T).(match e in T return 
+(\lambda (_: T).nat) with [(TSort _) \Rightarrow ((let rec plus (n1: nat) on 
+n1: (nat \to nat) \def (\lambda (m: nat).(match n1 with [O \Rightarrow m | (S 
+p) \Rightarrow (S (plus p m))])) in plus) n h) | (TLRef n1) \Rightarrow n1 | 
+(THead _ _ _) \Rightarrow ((let rec plus (n1: nat) on n1: (nat \to nat) \def 
+(\lambda (m: nat).(match n1 with [O \Rightarrow m | (S p) \Rightarrow (S 
+(plus p m))])) in plus) n h)])) (TLRef (plus n h)) (TLRef (plus n0 h)) H3) in 
+(eq_ind nat (plus n h) (\lambda (_: nat).(eq T (TLRef n0) (TLRef n))) 
+(f_equal nat T TLRef n0 n (simpl_plus_r h n0 n (sym_eq nat (plus n h) (plus 
+n0 h) H4))) (plus n0 h) H4)))]) in (H3 (refl_equal T (TLRef (plus n0 
+h)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef 
+(plus n h)) (lift h d t0)) \to (eq T t0 (TLRef n))))).(\lambda (t1: 
+T).(\lambda (_: (((eq T (TLRef (plus n h)) (lift h d t1)) \to (eq T t1 (TLRef 
+n))))).(\lambda (H2: (eq T (TLRef (plus n h)) (lift h d (THead k t0 
+t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq 
+T (TLRef (plus n h)) t2)) H2 (THead k (lift h d t0) (lift h (s k d) t1)) 
+(lift_head k t0 t1 h d)) in (let H4 \def (match H3 in eq return (\lambda (t2: 
+T).(\lambda (_: (eq ? ? t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d) 
+t1))) \to (eq T (THead k t0 t1) (TLRef n))))) with [refl_equal \Rightarrow 
+(\lambda (H4: (eq T (TLRef (plus n h)) (THead k (lift h d t0) (lift h (s k d) 
+t1)))).(let H5 \def (eq_ind T (TLRef (plus n h)) (\lambda (e: T).(match e in 
+T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
+\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d 
+t0) (lift h (s k d) t1)) H4) in (False_ind (eq T (THead k t0 t1) (TLRef n)) 
+H5)))]) in (H4 (refl_equal T (THead k (lift h d t0) (lift h (s k d) 
+t1)))))))))))) t))))).
 
 theorem lift_gen_head:
  \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: 
@@ -363,27 +366,27 @@ eq return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (TLRef n)) \to
 (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) 
 (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
 T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))) with [refl_equal 
-\Rightarrow (\lambda (H1: (eq T (THead k u t) (TLRef n))).(let H2 \def 
+\Rightarrow (\lambda (H2: (eq T (THead k u t) (TLRef n))).(let H3 \def 
 (eq_ind T (THead k u t) (\lambda (e: T).(match e in T return (\lambda (_: 
 T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T 
+(THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in (False_ind (ex3_2 T T 
 (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: 
 T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
-T).(eq T t (lift h (s k d) z))))) H2)))]) in (H2 (refl_equal T (TLRef n)))))) 
+T).(eq T t (lift h (s k d) z))))) H3)))]) in (H2 (refl_equal T (TLRef n)))))) 
 (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda 
 (t0: T).(eq T (THead k u t) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d 
 H0)) in (let H2 \def (match H1 in eq return (\lambda (t0: T).(\lambda (_: (eq 
 ? ? t0)).((eq T t0 (TLRef (plus n h))) \to (ex3_2 T T (\lambda (y: 
 T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: T).(\lambda 
 (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift 
-h (s k d) z)))))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (THead k 
-u t) (TLRef (plus n h)))).(let H2 \def (eq_ind T (THead k u t) (\lambda (e: 
+h (s k d) z)))))))) with [refl_equal \Rightarrow (\lambda (H2: (eq T (THead k 
+u t) (TLRef (plus n h)))).(let H3 \def (eq_ind T (THead k u t) (\lambda (e: 
 T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow 
 False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I 
-(TLRef (plus n h)) H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: 
+(TLRef (plus n h)) H2) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: 
 T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u 
 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) 
-z))))) H2)))]) in (H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda 
+z))))) H3)))]) in (H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda 
 (k0: K).(\lambda (t0: T).(\lambda (_: ((\forall (h: nat).(\forall (d: 
 nat).((eq T (THead k u t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: 
 T).(\lambda (z: T).(eq T t0 (THead k y z)))) (\lambda (y: T).(\lambda (_: 
@@ -394,42 +397,42 @@ T).(\lambda (z: T).(eq T t1 (THead k y z)))) (\lambda (y: T).(\lambda (_:
 T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s 
 k d) z)))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T 
 (THead k u t) (lift h d (THead k0 t0 t1)))).(let H2 \def (eq_ind T (lift h d 
-(THead k0 t0 t1)) (\lambda (t0: T).(eq T (THead k u t) t0)) H1 (THead k0 
+(THead k0 t0 t1)) (\lambda (t2: T).(eq T (THead k u t) t2)) H1 (THead k0 
 (lift h d t0) (lift h (s k0 d) t1)) (lift_head k0 t0 t1 h d)) in (let H3 \def 
 (match H2 in eq return (\lambda (t2: T).(\lambda (_: (eq ? ? t2)).((eq T t2 
 (THead k0 (lift h d t0) (lift h (s k0 d) t1))) \to (ex3_2 T T (\lambda (y: 
 T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k y z)))) (\lambda (y: 
 T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
 T).(eq T t (lift h (s k d) z)))))))) with [refl_equal \Rightarrow (\lambda 
-(H2: (eq T (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)))).(let 
-H3 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
-with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t) 
-\Rightarrow t])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) 
-H2) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e in T return 
+(H3: (eq T (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)))).(let 
+H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) 
+with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t2) 
+\Rightarrow t2])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) 
+H3) 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 k u t) (THead k0 (lift h d t0) (lift h 
-(s k0 d) t1)) H2) in ((let H5 \def (f_equal T K (\lambda (e: T).(match e in T 
-return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) 
-\Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u t) (THead k0 (lift 
-h d t0) (lift h (s k0 d) t1)) H2) in (eq_ind K k0 (\lambda (k: K).((eq T u 
-(lift h d t0)) \to ((eq T t (lift h (s k0 d) t1)) \to (ex3_2 T T (\lambda (y: 
-T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k y z)))) (\lambda (y: 
+(THead _ t2 _) \Rightarrow t2])) (THead k u t) (THead k0 (lift h d t0) (lift 
+h (s k0 d) t1)) H3) 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 k1 _ _) \Rightarrow k1])) (THead k u t) (THead k0 
+(lift h d t0) (lift h (s k0 d) t1)) H3) in (eq_ind K k0 (\lambda (k1: K).((eq 
+T u (lift h d t0)) \to ((eq T t (lift h (s k0 d) t1)) \to (ex3_2 T T (\lambda 
+(y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k1 y z)))) (\lambda (y: 
 T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
-T).(eq T t (lift h (s k d) z)))))))) (\lambda (H6: (eq T u (lift h d 
+T).(eq T t (lift h (s k1 d) z)))))))) (\lambda (H7: (eq T u (lift h d 
 t0))).(eq_ind T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s k0 d) t1)) 
 \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead 
 k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda 
-(_: T).(\lambda (z: T).(eq T t (lift h (s k0 d) z))))))) (\lambda (H7: (eq T 
-t (lift h (s k0 d) t1))).(eq_ind T (lift h (s k0 d) t1) (\lambda (t: 
+(_: T).(\lambda (z: T).(eq T t (lift h (s k0 d) z))))))) (\lambda (H8: (eq T 
+t (lift h (s k0 d) t1))).(eq_ind T (lift h (s k0 d) t1) (\lambda (t2: 
 T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead 
 k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d 
-y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k0 d) z)))))) 
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (s k0 d) z)))))) 
 (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) 
 (THead k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h 
 d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k0 d) t1) (lift h (s 
 k0 d) z)))) t0 t1 (refl_equal T (THead k0 t0 t1)) (refl_equal T (lift h d 
 t0)) (refl_equal T (lift h (s k0 d) t1))) t (sym_eq T t (lift h (s k0 d) t1) 
-H7))) u (sym_eq T u (lift h d t0) H6))) k (sym_eq K k k0 H5))) H4)) H3)))]) 
+H8))) u (sym_eq T u (lift h d t0) H7))) k (sym_eq K k k0 H6))) H5)) H4)))]) 
 in (H3 (refl_equal T (THead k0 (lift h d t0) (lift h (s k0 d) 
 t1)))))))))))))) x)))).
 
@@ -470,13 +473,13 @@ T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0)) H
 T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z)))) 
 (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
 T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))) with [refl_equal 
-\Rightarrow (\lambda (H1: (eq T (THead (Bind b) u t) (TLRef n))).(let H2 \def 
+\Rightarrow (\lambda (H2: (eq T (THead (Bind b) u t) (TLRef n))).(let H3 \def 
 (eq_ind T (THead (Bind b) u t) (\lambda (e: T).(match e in T return (\lambda 
 (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T 
+| (THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in (False_ind (ex3_2 T T 
 (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z)))) 
 (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
-T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H2)))]) in (H2 (refl_equal T 
+T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H3)))]) in (H2 (refl_equal T 
 (TLRef n)))))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d 
 (TLRef n)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0)) H (TLRef (plus n 
 h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 in eq return (\lambda 
@@ -484,13 +487,13 @@ h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 in eq return (\lambda
 T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z)))) 
 (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
 T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))) with [refl_equal 
-\Rightarrow (\lambda (H1: (eq T (THead (Bind b) u t) (TLRef (plus n 
-h)))).(let H2 \def (eq_ind T (THead (Bind b) u t) (\lambda (e: T).(match e in 
+\Rightarrow (\lambda (H2: (eq T (THead (Bind b) u t) (TLRef (plus n 
+h)))).(let H3 \def (eq_ind T (THead (Bind b) u t) (\lambda (e: T).(match e in 
 T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) 
 \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef (plus n h)) 
-H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) 
+H2) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) 
 (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d 
-y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H2)))]) in 
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H3)))]) in 
 (H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda (k: K).(\lambda (t0: 
 T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead (Bind b) u 
 t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 
@@ -502,45 +505,45 @@ T).(\lambda (z: T).(eq T t1 (THead (Bind b) y z)))) (\lambda (y: T).(\lambda
 (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift 
 h (S d) z)))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T 
 (THead (Bind b) u t) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T 
-(lift h d (THead k t0 t1)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0)) 
+(lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (THead (Bind b) u t) t2)) 
 H1 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in 
 (let H3 \def (match H2 in eq return (\lambda (t2: T).(\lambda (_: (eq ? ? 
 t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d) t1))) \to (ex3_2 T T 
 (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead (Bind b) y z)))) 
 (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
 T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))) with [refl_equal 
-\Rightarrow (\lambda (H2: (eq T (THead (Bind b) u t) (THead k (lift h d t0) 
-(lift h (s k d) t1)))).(let H3 \def (f_equal T T (\lambda (e: T).(match e in 
+\Rightarrow (\lambda (H3: (eq T (THead (Bind b) u t) (THead k (lift h d t0) 
+(lift h (s k d) t1)))).(let H4 \def (f_equal T T (\lambda (e: T).(match e in 
 T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) 
-\Rightarrow t | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u t) (THead k 
-(lift h d t0) (lift h (s k d) t1)) H2) in ((let H4 \def (f_equal T T (\lambda 
-(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u 
-| (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u 
-t) (THead k (lift h d t0) (lift h (s k d) t1)) H2) in ((let H5 \def (f_equal 
-T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) 
-\Rightarrow (Bind b) | (TLRef _) \Rightarrow (Bind b) | (THead k _ _) 
-\Rightarrow k])) (THead (Bind b) u t) (THead k (lift h d t0) (lift h (s k d) 
-t1)) H2) in (eq_ind K (Bind b) (\lambda (k: K).((eq T u (lift h d t0)) \to 
-((eq T t (lift h (s k d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: 
-T).(eq T (THead k t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: 
-T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S 
-d) z)))))))) (\lambda (H6: (eq T u (lift h d t0))).(eq_ind T (lift h d t0) 
-(\lambda (t2: T).((eq T t (lift h (s (Bind b) d) t1)) \to (ex3_2 T T (\lambda 
-(y: T).(\lambda (z: T).(eq T (THead (Bind b) t0 t1) (THead (Bind b) y z)))) 
-(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda (_: 
-T).(\lambda (z: T).(eq T t (lift h (S d) z))))))) (\lambda (H7: (eq T t (lift 
-h (s (Bind b) d) t1))).(eq_ind T (lift h (s (Bind b) d) t1) (\lambda (t: 
-T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) t0 t1) 
-(THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) 
-(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))) 
-(ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) t0 t1) 
-(THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) 
-(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s (Bind b) d) 
-t1) (lift h (S d) z)))) t0 t1 (refl_equal T (THead (Bind b) t0 t1)) 
-(refl_equal T (lift h d t0)) (refl_equal T (lift h (S d) t1))) t (sym_eq T t 
-(lift h (s (Bind b) d) t1) H7))) u (sym_eq T u (lift h d t0) H6))) k H5)) 
-H4)) H3)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h (s k d) 
-t1)))))))))))))) x)))).
+\Rightarrow t | (THead _ _ t2) \Rightarrow t2])) (THead (Bind b) u t) (THead 
+k (lift h d t0) (lift h (s k d) t1)) H3) 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 _ t2 _) \Rightarrow t2])) 
+(THead (Bind b) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H3) in ((let 
+H6 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) 
+with [(TSort _) \Rightarrow (Bind b) | (TLRef _) \Rightarrow (Bind b) | 
+(THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u t) (THead k (lift h d t0) 
+(lift h (s k d) t1)) H3) in (eq_ind K (Bind b) (\lambda (k0: K).((eq T u 
+(lift h d t0)) \to ((eq T t (lift h (s k0 d) t1)) \to (ex3_2 T T (\lambda (y: 
+T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead (Bind b) y z)))) (\lambda 
+(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T t (lift h (S d) z)))))))) (\lambda (H7: (eq T u (lift h d 
+t0))).(eq_ind T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s (Bind b) 
+d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) 
+t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift 
+h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))))) 
+(\lambda (H8: (eq T t (lift h (s (Bind b) d) t1))).(eq_ind T (lift h (s (Bind 
+b) d) t1) (\lambda (t2: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T 
+(THead (Bind b) t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: 
+T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T 
+t2 (lift h (S d) z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq 
+T (THead (Bind b) t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: 
+T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T 
+(lift h (s (Bind b) d) t1) (lift h (S d) z)))) t0 t1 (refl_equal T (THead 
+(Bind b) t0 t1)) (refl_equal T (lift h d t0)) (refl_equal T (lift h (S d) 
+t1))) t (sym_eq T t (lift h (s (Bind b) d) t1) H8))) u (sym_eq T u (lift h d 
+t0) H7))) k H6)) H5)) H4)))]) in (H3 (refl_equal T (THead k (lift h d t0) 
+(lift h (s k d) t1)))))))))))))) x)))).
 
 theorem lift_gen_flat:
  \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: 
@@ -578,13 +581,13 @@ eq return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (TLRef n)) \to
 (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y 
 z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
 T).(\lambda (z: T).(eq T t (lift h d z)))))))) with [refl_equal \Rightarrow 
-(\lambda (H1: (eq T (THead (Flat f) u t) (TLRef n))).(let H2 \def (eq_ind T 
+(\lambda (H2: (eq T (THead (Flat f) u t) (TLRef n))).(let H3 \def (eq_ind T 
 (THead (Flat f) u t) (\lambda (e: T).(match e in T return (\lambda (_: 
 T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | 
-(THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T 
+(THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in (False_ind (ex3_2 T T 
 (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y z)))) 
 (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
-T).(\lambda (z: T).(eq T t (lift h d z))))) H2)))]) in (H2 (refl_equal T 
+T).(\lambda (z: T).(eq T t (lift h d z))))) H3)))]) in (H2 (refl_equal T 
 (TLRef n)))))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d 
 (TLRef n)) (\lambda (t0: T).(eq T (THead (Flat f) u t) t0)) H (TLRef (plus n 
 h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 in eq return (\lambda 
@@ -592,13 +595,13 @@ h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 in eq return (\lambda
 T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y z)))) 
 (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
 T).(\lambda (z: T).(eq T t (lift h d z)))))))) with [refl_equal \Rightarrow 
-(\lambda (H1: (eq T (THead (Flat f) u t) (TLRef (plus n h)))).(let H2 \def 
+(\lambda (H2: (eq T (THead (Flat f) u t) (TLRef (plus n h)))).(let H3 \def 
 (eq_ind T (THead (Flat f) u t) (\lambda (e: T).(match e in T return (\lambda 
 (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
-| (THead _ _ _) \Rightarrow True])) I (TLRef (plus n h)) H1) in (False_ind 
+| (THead _ _ _) \Rightarrow True])) I (TLRef (plus n h)) H2) in (False_ind 
 (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y 
 z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
-T).(\lambda (z: T).(eq T t (lift h d z))))) H2)))]) in (H2 (refl_equal T 
+T).(\lambda (z: T).(eq T t (lift h d z))))) H3)))]) in (H2 (refl_equal T 
 (TLRef (plus n h)))))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: 
 ((\forall (h: nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d 
 t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Flat f) 
@@ -610,42 +613,42 @@ f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
 (_: T).(\lambda (z: T).(eq T t (lift h d z)))))))))).(\lambda (h: 
 nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead (Flat f) u t) (lift h d 
 (THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda 
-(t0: T).(eq T (THead (Flat f) u t) t0)) H1 (THead k (lift h d t0) (lift h (s 
+(t2: T).(eq T (THead (Flat f) u t) t2)) H1 (THead k (lift h d t0) (lift h (s 
 k d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def (match H2 in eq return 
 (\lambda (t2: T).(\lambda (_: (eq ? ? t2)).((eq T t2 (THead k (lift h d t0) 
 (lift h (s k d) t1))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T 
 (THead k t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T 
 u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z)))))))) 
-with [refl_equal \Rightarrow (\lambda (H2: (eq T (THead (Flat f) u t) (THead 
-k (lift h d t0) (lift h (s k d) t1)))).(let H3 \def (f_equal T T (\lambda (e: 
+with [refl_equal \Rightarrow (\lambda (H3: (eq T (THead (Flat f) u t) (THead 
+k (lift h d t0) (lift h (s k d) t1)))).(let H4 \def (f_equal T T (\lambda (e: 
 T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | 
-(TLRef _) \Rightarrow t | (THead _ _ t) \Rightarrow t])) (THead (Flat f) u t) 
-(THead k (lift h d t0) (lift h (s k d) t1)) H2) in ((let H4 \def (f_equal T T 
-(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) 
-\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) 
-(THead (Flat f) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H2) in ((let 
-H5 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) 
+(TLRef _) \Rightarrow t | (THead _ _ t2) \Rightarrow t2])) (THead (Flat f) u 
+t) (THead k (lift h d t0) (lift h (s k d) t1)) H3) 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 _ t2 _) \Rightarrow t2])) 
+(THead (Flat f) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H3) in ((let 
+H6 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) 
 with [(TSort _) \Rightarrow (Flat f) | (TLRef _) \Rightarrow (Flat f) | 
-(THead k _ _) \Rightarrow k])) (THead (Flat f) u t) (THead k (lift h d t0) 
-(lift h (s k d) t1)) H2) in (eq_ind K (Flat f) (\lambda (k: K).((eq T u (lift 
-h d t0)) \to ((eq T t (lift h (s k d) t1)) \to (ex3_2 T T (\lambda (y: 
-T).(\lambda (z: T).(eq T (THead k t0 t1) (THead (Flat f) y z)))) (\lambda (y: 
-T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
-T).(eq T t (lift h d z)))))))) (\lambda (H6: (eq T u (lift h d t0))).(eq_ind 
+(THead k0 _ _) \Rightarrow k0])) (THead (Flat f) u t) (THead k (lift h d t0) 
+(lift h (s k d) t1)) H3) in (eq_ind K (Flat f) (\lambda (k0: K).((eq T u 
+(lift h d t0)) \to ((eq T t (lift h (s k0 d) t1)) \to (ex3_2 T T (\lambda (y: 
+T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead (Flat f) y z)))) (\lambda 
+(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T t (lift h d z)))))))) (\lambda (H7: (eq T u (lift h d t0))).(eq_ind 
 T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s (Flat f) d) t1)) \to 
 (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) t0 t1) 
 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d 
 y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))))) (\lambda 
-(H7: (eq T t (lift h (s (Flat f) d) t1))).(eq_ind T (lift h (s (Flat f) d) 
-t1) (\lambda (t: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead 
+(H8: (eq T t (lift h (s (Flat f) d) t1))).(eq_ind T (lift h (s (Flat f) d) 
+t1) (\lambda (t2: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead 
+(Flat f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T 
+(lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift 
+h d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead 
 (Flat f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T 
-(lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h 
-d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat 
-f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift 
-h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s 
-(Flat f) d) t1) (lift h d z)))) t0 t1 (refl_equal T (THead (Flat f) t0 t1)) 
-(refl_equal T (lift h d t0)) (refl_equal T (lift h d t1))) t (sym_eq T t 
-(lift h (s (Flat f) d) t1) H7))) u (sym_eq T u (lift h d t0) H6))) k H5)) 
-H4)) H3)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h (s k d) 
+(lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h 
+(s (Flat f) d) t1) (lift h d z)))) t0 t1 (refl_equal T (THead (Flat f) t0 
+t1)) (refl_equal T (lift h d t0)) (refl_equal T (lift h d t1))) t (sym_eq T t 
+(lift h (s (Flat f) d) t1) H8))) u (sym_eq T u (lift h d t0) H7))) k H6)) 
+H5)) H4)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h (s k d) 
 t1)))))))))))))) x)))).