X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Flift%2Ffwd.ma;fp=matita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Flift%2Ffwd.ma;h=824deac30887756bc8077bff678a180a47ab0b97;hp=0000000000000000000000000000000000000000;hb=f61af501fb4608cc4fb062a0864c774e677f0d76;hpb=58ae1809c352e71e7b5530dc41e2bfc834e1aef1

diff --git a/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/fwd.ma b/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/lift/fwd.ma
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@@ -0,0 +1,392 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+(* This file was automatically generated: do not edit *********************)
+
+include "LambdaDelta-1/lift/defs.ma".
+
+theorem lift_sort:
+ \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(eq T (lift h d (TSort 
+n)) (TSort n))))
+\def
+ \lambda (n: nat).(\lambda (_: nat).(\lambda (_: nat).(refl_equal T (TSort 
+n)))).
+
+theorem lift_lref_lt:
+ \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((lt n d) \to (eq T 
+(lift h d (TLRef n)) (TLRef n)))))
+\def
+ \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_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 
+(lift h d (TLRef n)) (TLRef (plus n h))))))
+\def
+ \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_eq bool (blt n d) false 
+(le_bge d n H)))))).
+
+theorem lift_head:
+ \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall 
+(d: nat).(eq T (lift h d (THead k u t)) (THead k (lift h d u) (lift h (s k d) 
+t)))))))
+\def
+ \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda 
+(d: nat).(refl_equal T (THead k (lift h d u) (lift h (s k d) t))))))).
+
+theorem lift_bind:
+ \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall 
+(d: nat).(eq T (lift h d (THead (Bind b) u t)) (THead (Bind b) (lift h d u) 
+(lift h (S d) t)))))))
+\def
+ \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda 
+(d: nat).(refl_equal T (THead (Bind b) (lift h d u) (lift h (S d) t))))))).
+
+theorem lift_flat:
+ \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall 
+(d: nat).(eq T (lift h d (THead (Flat f) u t)) (THead (Flat f) (lift h d u) 
+(lift h d t)))))))
+\def
+ \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda 
+(d: nat).(refl_equal T (THead (Flat f) (lift h d u) (lift h d t))))))).
+
+theorem lift_gen_sort:
+ \forall (h: nat).(\forall (d: nat).(\forall (n: nat).(\forall (t: T).((eq T 
+(TSort n) (lift h d t)) \to (eq T t (TSort n))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (t: T).(T_ind 
+(\lambda (t0: T).((eq T (TSort n) (lift h d t0)) \to (eq T t0 (TSort n)))) 
+(\lambda (n0: nat).(\lambda (H: (eq T (TSort n) (lift h d (TSort 
+n0)))).(sym_eq T (TSort n) (TSort n0) H))) (\lambda (n0: nat).(\lambda (H: 
+(eq T (TSort n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TSort 
+n)) (\lambda (_: (lt n0 d)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) 
+(\lambda (t0: T).(eq T (TSort n) t0)) H (TLRef n0) (lift_lref_lt n0 h d (let 
+H1 \def (eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda 
+(_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | 
+(THead _ _ _) \Rightarrow False])) I (lift h d (TLRef n0)) H) in (False_ind 
+(lt n0 d) H1)))) in (let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match 
+ee 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)))) (\lambda (_: (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 (let H1 \def 
+(eq_ind T (TSort n) (\lambda (ee: T).(match ee in T return (\lambda (_: 
+T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | 
+(THead _ _ _) \Rightarrow False])) I (lift h d (TLRef n0)) H) in (False_ind 
+(le d n0) H1)))) in (let H2 \def (eq_ind T (TSort n) (\lambda (ee: T).(match 
+ee 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))))))) (\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 (eq_ind T (TSort n) (\lambda (ee: 
+T).(match ee 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))))))))) t)))).
+
+theorem lift_gen_lref:
+ \forall (t: T).(\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T 
+(TLRef i) (lift h d t)) \to (or (land (lt i d) (eq T t (TLRef i))) (land (le 
+(plus d h) i) (eq T t (TLRef (minus i h)))))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\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 (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 (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 
+(TSort n) H0) in (False_ind (or (land (lt i d) (eq T (TSort n) (TLRef i))) 
+(land (le (plus d h) i) (eq T (TSort n) (TLRef (minus i h))))) H1)))))))) 
+(\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda (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 (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 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 (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)) (le_plus_plus 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))))) 
+H3)))))))))))) t).
+
+theorem lift_gen_lref_lt:
+ \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((lt n d) \to (\forall 
+(t: T).((eq T (TLRef n) (lift h d t)) \to (eq T t (TLRef n)))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt n 
+d)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef n) (lift h d t))).(let H_x 
+\def (lift_gen_lref t d h n H0) in (let H1 \def H_x in (or_ind (land (lt n d) 
+(eq T t (TLRef n))) (land (le (plus d h) n) (eq T t (TLRef (minus n h)))) (eq 
+T t (TLRef n)) (\lambda (H2: (land (lt n d) (eq T t (TLRef n)))).(land_ind 
+(lt n d) (eq T t (TLRef n)) (eq T t (TLRef n)) (\lambda (_: (lt n 
+d)).(\lambda (H4: (eq T t (TLRef n))).(eq_ind_r T (TLRef n) (\lambda (t0: 
+T).(eq T t0 (TLRef n))) (refl_equal T (TLRef n)) t H4))) H2)) (\lambda (H2: 
+(land (le (plus d h) n) (eq T t (TLRef (minus n h))))).(land_ind (le (plus d 
+h) n) (eq T t (TLRef (minus n h))) (eq T t (TLRef n)) (\lambda (H3: (le (plus 
+d h) n)).(\lambda (H4: (eq T t (TLRef (minus n h)))).(eq_ind_r T (TLRef 
+(minus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false (plus d h) n (eq 
+T (TLRef (minus n h)) (TLRef n)) H3 (lt_le_S n (plus d h) (le_plus_trans (S 
+n) d h H))) t H4))) H2)) H1)))))))).
+
+theorem lift_gen_lref_false:
+ \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n 
+(plus d h)) \to (\forall (t: T).((eq T (TLRef n) (lift h d t)) \to (\forall 
+(P: Prop).P)))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d 
+n)).(\lambda (H0: (lt n (plus d h))).(\lambda (t: T).(\lambda (H1: (eq T 
+(TLRef n) (lift h d t))).(\lambda (P: Prop).(let H_x \def (lift_gen_lref t d 
+h n H1) in (let H2 \def H_x in (or_ind (land (lt n d) (eq T t (TLRef n))) 
+(land (le (plus d h) n) (eq T t (TLRef (minus n h)))) P (\lambda (H3: (land 
+(lt n d) (eq T t (TLRef n)))).(land_ind (lt n d) (eq T t (TLRef n)) P 
+(\lambda (H4: (lt n d)).(\lambda (_: (eq T t (TLRef n))).(le_false d n P H 
+H4))) H3)) (\lambda (H3: (land (le (plus d h) n) (eq T t (TLRef (minus n 
+h))))).(land_ind (le (plus d h) n) (eq T t (TLRef (minus n h))) P (\lambda 
+(H4: (le (plus d h) n)).(\lambda (_: (eq T t (TLRef (minus n h)))).(le_false 
+(plus d h) n P H4 H0))) H3)) H2)))))))))).
+
+theorem lift_gen_lref_ge:
+ \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to (\forall 
+(t: T).((eq T (TLRef (plus n h)) (lift h d t)) \to (eq T t (TLRef n)))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d 
+n)).(\lambda (t: T).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d 
+t))).(let H_x \def (lift_gen_lref t d h (plus n h) H0) in (let H1 \def H_x in 
+(or_ind (land (lt (plus n h) d) (eq T t (TLRef (plus n h)))) (land (le (plus 
+d h) (plus n h)) (eq T t (TLRef (minus (plus n h) h)))) (eq T t (TLRef n)) 
+(\lambda (H2: (land (lt (plus n h) d) (eq T t (TLRef (plus n h))))).(land_ind 
+(lt (plus n h) d) (eq T t (TLRef (plus n h))) (eq T t (TLRef n)) (\lambda 
+(H3: (lt (plus n h) d)).(\lambda (H4: (eq T t (TLRef (plus n h)))).(eq_ind_r 
+T (TLRef (plus n h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (le_false d n (eq 
+T (TLRef (plus n h)) (TLRef n)) H (lt_le_S n d (simpl_lt_plus_r h n d 
+(lt_le_trans (plus n h) d (plus d h) H3 (le_plus_l d h))))) t H4))) H2)) 
+(\lambda (H2: (land (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n 
+h) h))))).(land_ind (le (plus d h) (plus n h)) (eq T t (TLRef (minus (plus n 
+h) h))) (eq T t (TLRef n)) (\lambda (_: (le (plus d h) (plus n h))).(\lambda 
+(H4: (eq T t (TLRef (minus (plus n h) h)))).(eq_ind_r T (TLRef (minus (plus n 
+h) h)) (\lambda (t0: T).(eq T t0 (TLRef n))) (f_equal nat T TLRef (minus 
+(plus n h) h) n (minus_plus_r n h)) t H4))) H2)) H1)))))))).
+
+theorem lift_gen_head:
+ \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: 
+nat).(\forall (d: nat).((eq T (THead k u t) (lift h d x)) \to (ex3_2 T T 
+(\lambda (y: T).(\lambda (z: T).(eq T x (THead k y z)))) (\lambda (y: 
+T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T t (lift h (s k d) z)))))))))))
+\def
+ \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind 
+(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) 
+(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead 
+k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda 
+(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))))))) (\lambda (n: 
+nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) 
+(lift h d (TSort n)))).(let H0 \def (eq_ind T (lift h d (TSort n)) (\lambda 
+(t0: T).(eq T (THead k u t) t0)) H (TSort n) (lift_sort n h d)) in (let H1 
+\def (eq_ind T (THead k u t) (\lambda (ee: T).(match ee in T return (\lambda 
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False 
+| (THead _ _ _) \Rightarrow True])) I (TSort n) H0) in (False_ind (ex3_2 T T 
+(\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead k y z)))) (\lambda (y: 
+T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T t (lift h (s k d) z))))) H1))))))) (\lambda (n: nat).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t) (lift h d (TLRef 
+n)))).(lt_le_e n d (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) 
+(THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) 
+(\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) (\lambda (H0: 
+(lt n d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T 
+(THead k u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def 
+(eq_ind T (THead k u t) (\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 (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)))) (\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 (eq_ind T (THead 
+k u t) (\lambda (ee: T).(match ee 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) (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))))))))) (\lambda (k0: K).(\lambda (t0: 
+T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) 
+(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead 
+k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda 
+(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (t1: 
+T).(\lambda (H0: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) 
+(lift h d t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead 
+k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda 
+(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k u t) (lift h d (THead k0 
+t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k0 t0 t1)) (\lambda (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 (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)) 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 _ t2 _) \Rightarrow t2])) 
+(THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) H2) in ((let H5 
+\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)) 
+H2) in (\lambda (H6: (eq T u (lift h d t0))).(\lambda (H7: (eq K k k0)).(let 
+H8 \def (eq_ind_r K k0 (\lambda (k1: K).(eq T t (lift h (s k1 d) t1))) H5 k 
+H7) in (eq_ind K k (\lambda (k1: K).(ex3_2 T T (\lambda (y: T).(\lambda (z: 
+T).(eq T (THead k1 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)))))) (let H9 \def (eq_ind T t (\lambda (t2: T).(\forall (h0: 
+nat).(\forall (d0: nat).((eq T (THead k u t2) (lift h0 d0 t1)) \to (ex3_2 T T 
+(\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z)))) (\lambda (y: 
+T).(\lambda (_: T).(eq T u (lift h0 d0 y)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T t2 (lift h0 (s k d0) z))))))))) H0 (lift h (s k d) t1) H8) in (let 
+H10 \def (eq_ind T t (\lambda (t2: T).(\forall (h0: nat).(\forall (d0: 
+nat).((eq T (THead k u t2) (lift h0 d0 t0)) \to (ex3_2 T T (\lambda (y: 
+T).(\lambda (z: T).(eq T t0 (THead k y z)))) (\lambda (y: T).(\lambda (_: 
+T).(eq T u (lift h0 d0 y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift 
+h0 (s k d0) z))))))))) H (lift h (s k d) t1) H8) in (eq_ind_r T (lift h (s k 
+d) t1) (\lambda (t2: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T 
+(THead k 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 t2 (lift h (s k d) 
+z)))))) (let H11 \def (eq_ind T u (\lambda (t2: T).(\forall (h0: 
+nat).(\forall (d0: nat).((eq T (THead k t2 (lift h (s k d) t1)) (lift h0 d0 
+t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead k y z)))) 
+(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h0 d0 y)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h0 (s k d0) z))))))))) H10 
+(lift h d t0) H6) in (let H12 \def (eq_ind T u (\lambda (t2: T).(\forall (h0: 
+nat).(\forall (d0: nat).((eq T (THead k t2 (lift h (s k d) t1)) (lift h0 d0 
+t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead k y z)))) 
+(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h0 d0 y)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h0 (s k d0) z))))))))) H9 
+(lift h d t0) H6) in (eq_ind_r T (lift h d t0) (\lambda (t2: T).(ex3_2 T T 
+(\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead k y z)))) 
+(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T (lift h (s k d) t1) (lift h (s k d) z)))))) 
+(ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead 
+k 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 k d) t1) (lift h (s k d) 
+z)))) t0 t1 (refl_equal T (THead k t0 t1)) (refl_equal T (lift h d t0)) 
+(refl_equal T (lift h (s k d) t1))) u H6))) t H8))) k0 H7))))) H4)) 
+H3))))))))))) x)))).
+
+theorem lift_gen_bind:
+ \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: 
+nat).(\forall (d: nat).((eq T (THead (Bind b) u t) (lift h d x)) \to (ex3_2 T 
+T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda 
+(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T t (lift h (S d) z)))))))))))
+\def
+ \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u t) (lift h d 
+x))).(let H_x \def (lift_gen_head (Bind b) u t x h d H) in (let H0 \def H_x 
+in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y 
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T t (lift h (S d) z)))) (ex3_2 T T (\lambda (y: 
+T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda (y: T).(\lambda 
+(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift 
+h (S d) z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead 
+(Bind b) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t 
+(lift h (S d) x1))).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t0: 
+T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Bind b) y 
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T t (lift h (S d) z)))))) (eq_ind_r T (lift h (S d) 
+x1) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead 
+(Bind b) x0 x1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T 
+u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h (S d) 
+z)))))) (eq_ind_r T (lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y: 
+T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind b) y z)))) 
+(\lambda (y: T).(\lambda (_: T).(eq T t0 (lift h d y)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d) z)))))) (ex3_2_intro 
+T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) x0 x1) (THead (Bind 
+b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d x0) (lift h d 
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) x1) (lift h (S d) 
+z)))) x0 x1 (refl_equal T (THead (Bind b) x0 x1)) (refl_equal T (lift h d 
+x0)) (refl_equal T (lift h (S d) x1))) u H2) t H3) x H1)))))) H0))))))))).
+
+theorem lift_gen_flat:
+ \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: 
+nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d x)) \to (ex3_2 T 
+T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda 
+(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T t (lift h d z)))))))))))
+\def
+ \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(\lambda (h: 
+nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Flat f) u t) (lift h d 
+x))).(let H_x \def (lift_gen_head (Flat f) u t x h d H) in (let H0 \def H_x 
+in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y 
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: 
+T).(\lambda (z: T).(eq T t (lift h d z)))) (ex3_2 T T (\lambda (y: 
+T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda 
+(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift 
+h d z))))) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H1: (eq T x (THead 
+(Flat f) x0 x1))).(\lambda (H2: (eq T u (lift h d x0))).(\lambda (H3: (eq T t 
+(lift h d x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t0: T).(ex3_2 T 
+T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Flat f) y z)))) (\lambda 
+(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: 
+T).(eq T t (lift h d z)))))) (eq_ind_r T (lift h d x1) (\lambda (t0: 
+T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) x0 x1) 
+(THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d 
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t0 (lift h d z)))))) (eq_ind_r T 
+(lift h d x0) (\lambda (t0: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq 
+T (THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: 
+T).(eq T t0 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h d 
+x1) (lift h d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T 
+(THead (Flat f) x0 x1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: 
+T).(eq T (lift h d x0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T 
+(lift h d x1) (lift h d z)))) x0 x1 (refl_equal T (THead (Flat f) x0 x1)) 
+(refl_equal T (lift h d x0)) (refl_equal T (lift h d x1))) u H2) t H3) x 
+H1)))))) H0))))))))).
+