alias id "S" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1/2)".
alias id "true" = "cic:/Coq/Init/Datatypes/bool.ind#xpointer(1/1/1)".
alias id "True" = "cic:/Coq/Init/Logic/True.ind#xpointer(1/1)".
+alias id "plus_lt_compat_r" = "cic:/Coq/Arith/Plus/plus_lt_compat_r.con".
+alias id "plus_n_O" = "cic:/Coq/Init/Peano/plus_n_O.con".
+alias id "plus_le_lt_compat" = "cic:/Coq/Arith/Plus/plus_le_lt_compat.con".
+alias id "lt_wf_ind" = "cic:/Coq/Arith/Wf_nat/lt_wf_ind.con".
+alias id "minus_Sn_m" = "cic:/Coq/Arith/Minus/minus_Sn_m.con".
+alias id "and_ind" = "cic:/Coq/Init/Logic/and_ind.con".
+alias id "le_lt_trans" = "cic:/Coq/Arith/Lt/le_lt_trans.con".
+alias id "lt_le_trans" = "cic:/Coq/Arith/Lt/lt_le_trans.con".
+alias id "le_lt_trans" = "cic:/Coq/Arith/Lt/le_lt_trans.con".
+alias id "plus_n_O" = "cic:/Coq/Init/Peano/plus_n_O.con".
+alias id "f_equal3" = "cic:/Coq/Init/Logic/f_equal3.con".
+alias id "S_pred" = "cic:/Coq/Arith/Lt/S_pred.con".
+alias id "lt_le_trans" = "cic:/Coq/Arith/Lt/lt_le_trans.con".
+alias id "plus_lt_compat_r" = "cic:/Coq/Arith/Plus/plus_lt_compat_r.con".
+alias id "le_plus_trans" = "cic:/Coq/Arith/Plus/le_plus_trans.con".
+alias id "f_equal2" = "cic:/Coq/Init/Logic/f_equal2.con".
+alias id "le_plus_trans" = "cic:/Coq/Arith/Plus/le_plus_trans.con".
+alias id "f_equal2" = "cic:/Coq/Init/Logic/f_equal2.con".
+alias id "plus_n_O" = "cic:/Coq/Init/Peano/plus_n_O.con".
+alias id "plus_n_O" = "cic:/Coq/Init/Peano/plus_n_O.con".
+alias id "lt_trans" = "cic:/Coq/Arith/Lt/lt_trans.con".
+alias id "minus_Sn_m" = "cic:/Coq/Arith/Minus/minus_Sn_m.con".
+alias id "ex_intro" = "cic:/Coq/Init/Logic/ex.ind#xpointer(1/1/1)".
+alias id "lt_trans" = "cic:/Coq/Arith/Lt/lt_trans.con".
+alias id "lt_n_Sn" = "cic:/Coq/Arith/Lt/lt_n_Sn.con".
+alias id "lt_le_trans" = "cic:/Coq/Arith/Lt/lt_le_trans.con".
+alias id "lt_wf_ind" = "cic:/Coq/Arith/Wf_nat/lt_wf_ind.con".
theorem f_equal: \forall A,B:Type. \forall f:A \to B.
\forall x,y:A. x = y \to f x = f y.
include "blt/props.ma".
+include "ext/plist.ma".
+
set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/G/defs".
-include "../Base/theory.ma".
+include "preamble.ma".
record G : Set \def {
next: (nat \to nat);
set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/T/defs".
-include "../Base/theory.ma".
+include "preamble.ma".
inductive B: Set \def
| Abbr: B
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/cnt/defs".
+
+include "T/defs.ma".
+
+inductive cnt: T \to Prop \def
+| cnt_sort: \forall (n: nat).(cnt (TSort n))
+| cnt_head: \forall (t: T).((cnt t) \to (\forall (k: K).(\forall (v: T).(cnt
+(THead k v t))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/cnt/props".
+
+include "cnt/defs.ma".
+
+include "lift/defs.ma".
+
+theorem cnt_lift:
+ \forall (t: T).((cnt t) \to (\forall (i: nat).(\forall (d: nat).(cnt (lift i
+d t)))))
+\def
+ \lambda (t: T).(\lambda (H: (cnt t)).(cnt_ind (\lambda (t0: T).(\forall (i:
+nat).(\forall (d: nat).(cnt (lift i d t0))))) (\lambda (n: nat).(\lambda (i:
+nat).(\lambda (d: nat).(eq_ind_r T (TSort n) (\lambda (t0: T).(cnt t0))
+(cnt_sort n) (lift i d (TSort n)) (lift_sort n i d))))) (\lambda (t0:
+T).(\lambda (_: (cnt t0)).(\lambda (H1: ((\forall (i: nat).(\forall (d:
+nat).(cnt (lift i d t0)))))).(\lambda (k: K).(\lambda (v: T).(\lambda (i:
+nat).(\lambda (d: nat).(eq_ind_r T (THead k (lift i d v) (lift i (s k d) t0))
+(\lambda (t1: T).(cnt t1)) (cnt_head (lift i (s k d) t0) (H1 i (s k d)) k
+(lift i d v)) (lift i d (THead k v t0)) (lift_head k v t0 i d))))))))) t H)).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/drop/defs".
+
+include "C/defs.ma".
+
+include "lift/defs.ma".
+
+inductive drop: nat \to (nat \to (C \to (C \to Prop))) \def
+| drop_refl: \forall (c: C).(drop O O c c)
+| drop_drop: \forall (k: K).(\forall (h: nat).(\forall (c: C).(\forall (e:
+C).((drop (r k h) O c e) \to (\forall (u: T).(drop (S h) O (CHead c k u)
+e))))))
+| drop_skip: \forall (k: K).(\forall (h: nat).(\forall (d: nat).(\forall (c:
+C).(\forall (e: C).((drop h (r k d) c e) \to (\forall (u: T).(drop h (S d)
+(CHead c k (lift h (r k d) u)) (CHead e k u)))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/drop/fwd".
+
+include "drop/defs.ma".
+
+theorem drop_gen_sort:
+ \forall (n: nat).(\forall (h: nat).(\forall (d: nat).(\forall (x: C).((drop
+h d (CSort n) x) \to (and3 (eq C x (CSort n)) (eq nat h O) (eq nat d O))))))
+\def
+ \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (x:
+C).(\lambda (H: (drop h d (CSort n) x)).(insert_eq C (CSort n) (\lambda (c:
+C).(drop h d c x)) (and3 (eq C x (CSort n)) (eq nat h O) (eq nat d O))
+(\lambda (y: C).(\lambda (H0: (drop h d y x)).(drop_ind (\lambda (n0:
+nat).(\lambda (n1: nat).(\lambda (c: C).(\lambda (c0: C).((eq C c (CSort n))
+\to (and3 (eq C c0 (CSort n)) (eq nat n0 O) (eq nat n1 O))))))) (\lambda (c:
+C).(\lambda (H1: (eq C c (CSort n))).(let H2 \def (f_equal C C (\lambda (e:
+C).e) c (CSort n) H1) in (eq_ind_r C (CSort n) (\lambda (c0: C).(and3 (eq C
+c0 (CSort n)) (eq nat O O) (eq nat O O))) (and3_intro (eq C (CSort n) (CSort
+n)) (eq nat O O) (eq nat O O) (refl_equal C (CSort n)) (refl_equal nat O)
+(refl_equal nat O)) c H2)))) (\lambda (k: K).(\lambda (h0: nat).(\lambda (c:
+C).(\lambda (e: C).(\lambda (_: (drop (r k h0) O c e)).(\lambda (_: (((eq C c
+(CSort n)) \to (and3 (eq C e (CSort n)) (eq nat (r k h0) O) (eq nat O
+O))))).(\lambda (u: T).(\lambda (H3: (eq C (CHead c k u) (CSort n))).(let H4
+\def (eq_ind C (CHead c k u) (\lambda (ee: C).(match ee in C return (\lambda
+(_: C).Prop) with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow
+True])) I (CSort n) H3) in (False_ind (and3 (eq C e (CSort n)) (eq nat (S h0)
+O) (eq nat O O)) H4)))))))))) (\lambda (k: K).(\lambda (h0: nat).(\lambda
+(d0: nat).(\lambda (c: C).(\lambda (e: C).(\lambda (_: (drop h0 (r k d0) c
+e)).(\lambda (_: (((eq C c (CSort n)) \to (and3 (eq C e (CSort n)) (eq nat h0
+O) (eq nat (r k d0) O))))).(\lambda (u: T).(\lambda (H3: (eq C (CHead c k
+(lift h0 (r k d0) u)) (CSort n))).(let H4 \def (eq_ind C (CHead c k (lift h0
+(r k d0) u)) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
+with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I
+(CSort n) H3) in (False_ind (and3 (eq C (CHead e k u) (CSort n)) (eq nat h0
+O) (eq nat (S d0) O)) H4))))))))))) h d y x H0))) H))))).
+
+theorem drop_gen_refl:
+ \forall (x: C).(\forall (e: C).((drop O O x e) \to (eq C x e)))
+\def
+ \lambda (x: C).(\lambda (e: C).(\lambda (H: (drop O O x e)).(insert_eq nat O
+(\lambda (n: nat).(drop n O x e)) (eq C x e) (\lambda (y: nat).(\lambda (H0:
+(drop y O x e)).(insert_eq nat O (\lambda (n: nat).(drop y n x e)) ((eq nat y
+O) \to (eq C x e)) (\lambda (y0: nat).(\lambda (H1: (drop y y0 x
+e)).(drop_ind (\lambda (n: nat).(\lambda (n0: nat).(\lambda (c: C).(\lambda
+(c0: C).((eq nat n0 O) \to ((eq nat n O) \to (eq C c c0))))))) (\lambda (c:
+C).(\lambda (_: (eq nat O O)).(\lambda (_: (eq nat O O)).(refl_equal C c))))
+(\lambda (k: K).(\lambda (h: nat).(\lambda (c: C).(\lambda (e0: C).(\lambda
+(_: (drop (r k h) O c e0)).(\lambda (_: (((eq nat O O) \to ((eq nat (r k h)
+O) \to (eq C c e0))))).(\lambda (u: T).(\lambda (_: (eq nat O O)).(\lambda
+(H5: (eq nat (S h) O)).(let H6 \def (eq_ind nat (S h) (\lambda (ee:
+nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow
+False | (S _) \Rightarrow True])) I O H5) in (False_ind (eq C (CHead c k u)
+e0) H6))))))))))) (\lambda (k: K).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (c: C).(\lambda (e0: C).(\lambda (H2: (drop h (r k d) c
+e0)).(\lambda (H3: (((eq nat (r k d) O) \to ((eq nat h O) \to (eq C c
+e0))))).(\lambda (u: T).(\lambda (H4: (eq nat (S d) O)).(\lambda (H5: (eq nat
+h O)).(let H6 \def (f_equal nat nat (\lambda (e1: nat).e1) h O H5) in (let H7
+\def (eq_ind nat h (\lambda (n: nat).((eq nat (r k d) O) \to ((eq nat n O)
+\to (eq C c e0)))) H3 O H6) in (let H8 \def (eq_ind nat h (\lambda (n:
+nat).(drop n (r k d) c e0)) H2 O H6) in (eq_ind_r nat O (\lambda (n: nat).(eq
+C (CHead c k (lift n (r k d) u)) (CHead e0 k u))) (let H9 \def (eq_ind nat (S
+d) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (eq C
+(CHead c k (lift O (r k d) u)) (CHead e0 k u)) H9)) h H6)))))))))))))) y y0 x
+e H1))) H0))) H))).
+
+theorem drop_gen_drop:
+ \forall (k: K).(\forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h:
+nat).((drop (S h) O (CHead c k u) x) \to (drop (r k h) O c x))))))
+\def
+ \lambda (k: K).(\lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h:
+nat).(\lambda (H: (drop (S h) O (CHead c k u) x)).(insert_eq C (CHead c k u)
+(\lambda (c0: C).(drop (S h) O c0 x)) (drop (r k h) O c x) (\lambda (y:
+C).(\lambda (H0: (drop (S h) O y x)).(insert_eq nat O (\lambda (n: nat).(drop
+(S h) n y x)) ((eq C y (CHead c k u)) \to (drop (r k h) O c x)) (\lambda (y0:
+nat).(\lambda (H1: (drop (S h) y0 y x)).(insert_eq nat (S h) (\lambda (n:
+nat).(drop n y0 y x)) ((eq nat y0 O) \to ((eq C y (CHead c k u)) \to (drop (r
+k h) O c x))) (\lambda (y1: nat).(\lambda (H2: (drop y1 y0 y x)).(drop_ind
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq
+nat n (S h)) \to ((eq nat n0 O) \to ((eq C c0 (CHead c k u)) \to (drop (r k
+h) O c c1)))))))) (\lambda (c0: C).(\lambda (H3: (eq nat O (S h))).(\lambda
+(_: (eq nat O O)).(\lambda (_: (eq C c0 (CHead c k u))).(let H6 \def (match
+H3 in eq return (\lambda (n: nat).(\lambda (_: (eq ? ? n)).((eq nat n (S h))
+\to (drop (r k h) O c c0)))) with [refl_equal \Rightarrow (\lambda (H2: (eq
+nat O (S h))).(let H3 \def (eq_ind nat O (\lambda (e: nat).(match e in nat
+return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow
+False])) I (S h) H2) in (False_ind (drop (r k h) O c c0) H3)))]) in (H6
+(refl_equal nat (S h)))))))) (\lambda (k0: K).(\lambda (h0: nat).(\lambda
+(c0: C).(\lambda (e: C).(\lambda (H3: (drop (r k0 h0) O c0 e)).(\lambda (_:
+(((eq nat (r k0 h0) (S h)) \to ((eq nat O O) \to ((eq C c0 (CHead c k u)) \to
+(drop (r k h) O c e)))))).(\lambda (u0: T).(\lambda (H5: (eq nat (S h0) (S
+h))).(\lambda (_: (eq nat O O)).(\lambda (H7: (eq C (CHead c0 k0 u0) (CHead c
+k u))).(let H8 \def (match H5 in eq return (\lambda (n: nat).(\lambda (_: (eq
+? ? n)).((eq nat n (S h)) \to (drop (r k h) O c e)))) with [refl_equal
+\Rightarrow (\lambda (H4: (eq nat (S h0) (S h))).(let H5 \def (f_equal nat
+nat (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).nat) with [O
+\Rightarrow h0 | (S n) \Rightarrow n])) (S h0) (S h) H4) in (eq_ind nat h
+(\lambda (_: nat).(drop (r k h) O c e)) (let H6 \def (match H7 in eq return
+(\lambda (c0: C).(\lambda (_: (eq ? ? c0)).((eq C c0 (CHead c k u)) \to (drop
+(r k h) O c e)))) with [refl_equal \Rightarrow (\lambda (H4: (eq C (CHead c0
+k0 u0) (CHead c k u))).(let H6 \def (f_equal C T (\lambda (e0: C).(match e0
+in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t)
+\Rightarrow t])) (CHead c0 k0 u0) (CHead c k u) H4) in ((let H7 \def (f_equal
+C K (\lambda (e0: C).(match e0 in C return (\lambda (_: C).K) with [(CSort _)
+\Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead c0 k0 u0) (CHead c k
+u) H4) in ((let H8 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow
+c])) (CHead c0 k0 u0) (CHead c k u) H4) in (eq_ind C c (\lambda (_: C).((eq K
+k0 k) \to ((eq T u0 u) \to (drop (r k h) O c e)))) (\lambda (H9: (eq K k0
+k)).(eq_ind K k (\lambda (_: K).((eq T u0 u) \to (drop (r k h) O c e)))
+(\lambda (H10: (eq T u0 u)).(eq_ind T u (\lambda (_: T).(drop (r k h) O c e))
+(eq_ind nat h0 (\lambda (n: nat).(drop (r k n) O c e)) (eq_ind C c0 (\lambda
+(c: C).(drop (r k h0) O c e)) (eq_ind K k0 (\lambda (k: K).(drop (r k h0) O
+c0 e)) H3 k H9) c H8) h H5) u0 (sym_eq T u0 u H10))) k0 (sym_eq K k0 k H9)))
+c0 (sym_eq C c0 c H8))) H7)) H6)))]) in (H6 (refl_equal C (CHead c k u)))) h0
+(sym_eq nat h0 h H5))))]) in (H8 (refl_equal nat (S h)))))))))))))) (\lambda
+(k0: K).(\lambda (h0: nat).(\lambda (d: nat).(\lambda (c0: C).(\lambda (e:
+C).(\lambda (_: (drop h0 (r k0 d) c0 e)).(\lambda (_: (((eq nat h0 (S h)) \to
+((eq nat (r k0 d) O) \to ((eq C c0 (CHead c k u)) \to (drop (r k h) O c
+e)))))).(\lambda (u0: T).(\lambda (_: (eq nat h0 (S h))).(\lambda (H6: (eq
+nat (S d) O)).(\lambda (_: (eq C (CHead c0 k0 (lift h0 (r k0 d) u0)) (CHead c
+k u))).(let H8 \def (match H6 in eq return (\lambda (n: nat).(\lambda (_: (eq
+? ? n)).((eq nat n O) \to (drop (r k h) O c (CHead e k0 u0))))) with
+[refl_equal \Rightarrow (\lambda (H4: (eq nat (S d) O)).(let H5 \def (eq_ind
+nat (S d) (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).Prop)
+with [O \Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind
+(drop (r k h) O c (CHead e k0 u0)) H5)))]) in (H8 (refl_equal nat
+O)))))))))))))) y1 y0 y x H2))) H1))) H0))) H)))))).
+
+theorem drop_gen_skip_r:
+ \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall
+(d: nat).(\forall (k: K).((drop h (S d) x (CHead c k u)) \to (ex2 C (\lambda
+(e: C).(eq C x (CHead e k (lift h (r k d) u)))) (\lambda (e: C).(drop h (r k
+d) e c)))))))))
+\def
+ \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) x (CHead c k u))).(let H0
+\def (match H in drop return (\lambda (n: nat).(\lambda (n0: nat).(\lambda
+(c0: C).(\lambda (c1: C).(\lambda (_: (drop n n0 c0 c1)).((eq nat n h) \to
+((eq nat n0 (S d)) \to ((eq C c0 x) \to ((eq C c1 (CHead c k u)) \to (ex2 C
+(\lambda (e: C).(eq C x (CHead e k (lift h (r k d) u)))) (\lambda (e:
+C).(drop h (r k d) e c)))))))))))) with [(drop_refl c0) \Rightarrow (\lambda
+(H0: (eq nat O h)).(\lambda (H1: (eq nat O (S d))).(\lambda (H2: (eq C c0
+x)).(\lambda (H3: (eq C c0 (CHead c k u))).(eq_ind nat O (\lambda (n:
+nat).((eq nat O (S d)) \to ((eq C c0 x) \to ((eq C c0 (CHead c k u)) \to (ex2
+C (\lambda (e: C).(eq C x (CHead e k (lift n (r k d) u)))) (\lambda (e:
+C).(drop n (r k d) e c))))))) (\lambda (H4: (eq nat O (S d))).(let H5 \def
+(eq_ind nat O (\lambda (e: nat).(match e in nat return (\lambda (_:
+nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S d) H4)
+in (False_ind ((eq C c0 x) \to ((eq C c0 (CHead c k u)) \to (ex2 C (\lambda
+(e: C).(eq C x (CHead e k (lift O (r k d) u)))) (\lambda (e: C).(drop O (r k
+d) e c))))) H5))) h H0 H1 H2 H3))))) | (drop_drop k0 h0 c0 e H0 u0)
+\Rightarrow (\lambda (H1: (eq nat (S h0) h)).(\lambda (H2: (eq nat O (S
+d))).(\lambda (H3: (eq C (CHead c0 k0 u0) x)).(\lambda (H4: (eq C e (CHead c
+k u))).(eq_ind nat (S h0) (\lambda (n: nat).((eq nat O (S d)) \to ((eq C
+(CHead c0 k0 u0) x) \to ((eq C e (CHead c k u)) \to ((drop (r k0 h0) O c0 e)
+\to (ex2 C (\lambda (e0: C).(eq C x (CHead e0 k (lift n (r k d) u))))
+(\lambda (e0: C).(drop n (r k d) e0 c)))))))) (\lambda (H5: (eq nat O (S
+d))).(let H6 \def (eq_ind nat O (\lambda (e0: nat).(match e0 in nat return
+(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
+I (S d) H5) in (False_ind ((eq C (CHead c0 k0 u0) x) \to ((eq C e (CHead c k
+u)) \to ((drop (r k0 h0) O c0 e) \to (ex2 C (\lambda (e0: C).(eq C x (CHead
+e0 k (lift (S h0) (r k d) u)))) (\lambda (e0: C).(drop (S h0) (r k d) e0
+c)))))) H6))) h H1 H2 H3 H4 H0))))) | (drop_skip k0 h0 d0 c0 e H0 u0)
+\Rightarrow (\lambda (H1: (eq nat h0 h)).(\lambda (H2: (eq nat (S d0) (S
+d))).(\lambda (H3: (eq C (CHead c0 k0 (lift h0 (r k0 d0) u0)) x)).(\lambda
+(H4: (eq C (CHead e k0 u0) (CHead c k u))).(eq_ind nat h (\lambda (n:
+nat).((eq nat (S d0) (S d)) \to ((eq C (CHead c0 k0 (lift n (r k0 d0) u0)) x)
+\to ((eq C (CHead e k0 u0) (CHead c k u)) \to ((drop n (r k0 d0) c0 e) \to
+(ex2 C (\lambda (e0: C).(eq C x (CHead e0 k (lift h (r k d) u)))) (\lambda
+(e0: C).(drop h (r k d) e0 c)))))))) (\lambda (H5: (eq nat (S d0) (S
+d))).(let H6 \def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat return
+(\lambda (_: nat).nat) with [O \Rightarrow d0 | (S n) \Rightarrow n])) (S d0)
+(S d) H5) in (eq_ind nat d (\lambda (n: nat).((eq C (CHead c0 k0 (lift h (r
+k0 n) u0)) x) \to ((eq C (CHead e k0 u0) (CHead c k u)) \to ((drop h (r k0 n)
+c0 e) \to (ex2 C (\lambda (e0: C).(eq C x (CHead e0 k (lift h (r k d) u))))
+(\lambda (e0: C).(drop h (r k d) e0 c))))))) (\lambda (H7: (eq C (CHead c0 k0
+(lift h (r k0 d) u0)) x)).(eq_ind C (CHead c0 k0 (lift h (r k0 d) u0))
+(\lambda (c1: C).((eq C (CHead e k0 u0) (CHead c k u)) \to ((drop h (r k0 d)
+c0 e) \to (ex2 C (\lambda (e0: C).(eq C c1 (CHead e0 k (lift h (r k d) u))))
+(\lambda (e0: C).(drop h (r k d) e0 c)))))) (\lambda (H8: (eq C (CHead e k0
+u0) (CHead c k u))).(let H9 \def (f_equal C T (\lambda (e0: C).(match e0 in C
+return (\lambda (_: C).T) with [(CSort _) \Rightarrow u0 | (CHead _ _ t)
+\Rightarrow t])) (CHead e k0 u0) (CHead c k u) H8) in ((let H10 \def (f_equal
+C K (\lambda (e0: C).(match e0 in C return (\lambda (_: C).K) with [(CSort _)
+\Rightarrow k0 | (CHead _ k _) \Rightarrow k])) (CHead e k0 u0) (CHead c k u)
+H8) in ((let H11 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow e | (CHead c _ _) \Rightarrow
+c])) (CHead e k0 u0) (CHead c k u) H8) in (eq_ind C c (\lambda (c1: C).((eq K
+k0 k) \to ((eq T u0 u) \to ((drop h (r k0 d) c0 c1) \to (ex2 C (\lambda (e0:
+C).(eq C (CHead c0 k0 (lift h (r k0 d) u0)) (CHead e0 k (lift h (r k d) u))))
+(\lambda (e0: C).(drop h (r k d) e0 c))))))) (\lambda (H12: (eq K k0
+k)).(eq_ind K k (\lambda (k1: K).((eq T u0 u) \to ((drop h (r k1 d) c0 c) \to
+(ex2 C (\lambda (e0: C).(eq C (CHead c0 k1 (lift h (r k1 d) u0)) (CHead e0 k
+(lift h (r k d) u)))) (\lambda (e0: C).(drop h (r k d) e0 c)))))) (\lambda
+(H13: (eq T u0 u)).(eq_ind T u (\lambda (t: T).((drop h (r k d) c0 c) \to
+(ex2 C (\lambda (e0: C).(eq C (CHead c0 k (lift h (r k d) t)) (CHead e0 k
+(lift h (r k d) u)))) (\lambda (e0: C).(drop h (r k d) e0 c))))) (\lambda
+(H14: (drop h (r k d) c0 c)).(let H15 \def (eq_ind T u0 (\lambda (t: T).(eq C
+(CHead c0 k0 (lift h (r k0 d) t)) x)) H7 u H13) in (let H16 \def (eq_ind K k0
+(\lambda (k: K).(eq C (CHead c0 k (lift h (r k d) u)) x)) H15 k H12) in (let
+H17 \def (eq_ind_r C x (\lambda (c0: C).(drop h (S d) c0 (CHead c k u))) H
+(CHead c0 k (lift h (r k d) u)) H16) in (ex_intro2 C (\lambda (e0: C).(eq C
+(CHead c0 k (lift h (r k d) u)) (CHead e0 k (lift h (r k d) u)))) (\lambda
+(e0: C).(drop h (r k d) e0 c)) c0 (refl_equal C (CHead c0 k (lift h (r k d)
+u))) H14))))) u0 (sym_eq T u0 u H13))) k0 (sym_eq K k0 k H12))) e (sym_eq C e
+c H11))) H10)) H9))) x H7)) d0 (sym_eq nat d0 d H6)))) h0 (sym_eq nat h0 h
+H1) H2 H3 H4 H0)))))]) in (H0 (refl_equal nat h) (refl_equal nat (S d))
+(refl_equal C x) (refl_equal C (CHead c k u)))))))))).
+
+theorem drop_gen_skip_l:
+ \forall (c: C).(\forall (x: C).(\forall (u: T).(\forall (h: nat).(\forall
+(d: nat).(\forall (k: K).((drop h (S d) (CHead c k u) x) \to (ex3_2 C T
+(\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_:
+C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_:
+T).(drop h (r k d) c e))))))))))
+\def
+ \lambda (c: C).(\lambda (x: C).(\lambda (u: T).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (k: K).(\lambda (H: (drop h (S d) (CHead c k u) x)).(let H0
+\def (match H in drop return (\lambda (n: nat).(\lambda (n0: nat).(\lambda
+(c0: C).(\lambda (c1: C).(\lambda (_: (drop n n0 c0 c1)).((eq nat n h) \to
+((eq nat n0 (S d)) \to ((eq C c0 (CHead c k u)) \to ((eq C c1 x) \to (ex3_2 C
+T (\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_:
+C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e: C).(\lambda (_:
+T).(drop h (r k d) c e))))))))))))) with [(drop_refl c0) \Rightarrow (\lambda
+(H0: (eq nat O h)).(\lambda (H1: (eq nat O (S d))).(\lambda (H2: (eq C c0
+(CHead c k u))).(\lambda (H3: (eq C c0 x)).(eq_ind nat O (\lambda (n:
+nat).((eq nat O (S d)) \to ((eq C c0 (CHead c k u)) \to ((eq C c0 x) \to
+(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda
+(_: C).(\lambda (v: T).(eq T u (lift n (r k d) v)))) (\lambda (e: C).(\lambda
+(_: T).(drop n (r k d) c e)))))))) (\lambda (H4: (eq nat O (S d))).(let H5
+\def (eq_ind nat O (\lambda (e: nat).(match e in nat return (\lambda (_:
+nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False])) I (S d) H4)
+in (False_ind ((eq C c0 (CHead c k u)) \to ((eq C c0 x) \to (ex3_2 C T
+(\lambda (e: C).(\lambda (v: T).(eq C x (CHead e k v)))) (\lambda (_:
+C).(\lambda (v: T).(eq T u (lift O (r k d) v)))) (\lambda (e: C).(\lambda (_:
+T).(drop O (r k d) c e)))))) H5))) h H0 H1 H2 H3))))) | (drop_drop k0 h0 c0 e
+H0 u0) \Rightarrow (\lambda (H1: (eq nat (S h0) h)).(\lambda (H2: (eq nat O
+(S d))).(\lambda (H3: (eq C (CHead c0 k0 u0) (CHead c k u))).(\lambda (H4:
+(eq C e x)).(eq_ind nat (S h0) (\lambda (n: nat).((eq nat O (S d)) \to ((eq C
+(CHead c0 k0 u0) (CHead c k u)) \to ((eq C e x) \to ((drop (r k0 h0) O c0 e)
+\to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k v))))
+(\lambda (_: C).(\lambda (v: T).(eq T u (lift n (r k d) v)))) (\lambda (e0:
+C).(\lambda (_: T).(drop n (r k d) c e0))))))))) (\lambda (H5: (eq nat O (S
+d))).(let H6 \def (eq_ind nat O (\lambda (e0: nat).(match e0 in nat return
+(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
+I (S d) H5) in (False_ind ((eq C (CHead c0 k0 u0) (CHead c k u)) \to ((eq C e
+x) \to ((drop (r k0 h0) O c0 e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v:
+T).(eq C x (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift (S
+h0) (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop (S h0) (r k d) c
+e0))))))) H6))) h H1 H2 H3 H4 H0))))) | (drop_skip k0 h0 d0 c0 e H0 u0)
+\Rightarrow (\lambda (H1: (eq nat h0 h)).(\lambda (H2: (eq nat (S d0) (S
+d))).(\lambda (H3: (eq C (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k
+u))).(\lambda (H4: (eq C (CHead e k0 u0) x)).(eq_ind nat h (\lambda (n:
+nat).((eq nat (S d0) (S d)) \to ((eq C (CHead c0 k0 (lift n (r k0 d0) u0))
+(CHead c k u)) \to ((eq C (CHead e k0 u0) x) \to ((drop n (r k0 d0) c0 e) \to
+(ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0 k v))))
+(\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v)))) (\lambda (e0:
+C).(\lambda (_: T).(drop h (r k d) c e0))))))))) (\lambda (H5: (eq nat (S d0)
+(S d))).(let H6 \def (f_equal nat nat (\lambda (e0: nat).(match e0 in nat
+return (\lambda (_: nat).nat) with [O \Rightarrow d0 | (S n) \Rightarrow n]))
+(S d0) (S d) H5) in (eq_ind nat d (\lambda (n: nat).((eq C (CHead c0 k0 (lift
+h (r k0 n) u0)) (CHead c k u)) \to ((eq C (CHead e k0 u0) x) \to ((drop h (r
+k0 n) c0 e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead e0
+k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v))))
+(\lambda (e0: C).(\lambda (_: T).(drop h (r k d) c e0)))))))) (\lambda (H7:
+(eq C (CHead c0 k0 (lift h (r k0 d) u0)) (CHead c k u))).(let H8 \def
+(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with
+[(CSort _) \Rightarrow ((let rec lref_map (f: ((nat \to nat))) (d: nat) (t:
+T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i)
+\Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false
+\Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u)
+(lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x h)) (r k0
+d) u0) | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 (lift h (r k0 d) u0))
+(CHead c k u) H7) in ((let H9 \def (f_equal C K (\lambda (e0: C).(match e0 in
+C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k _)
+\Rightarrow k])) (CHead c0 k0 (lift h (r k0 d) u0)) (CHead c k u) H7) in
+((let H10 \def (f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda
+(_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow c]))
+(CHead c0 k0 (lift h (r k0 d) u0)) (CHead c k u) H7) in (eq_ind C c (\lambda
+(c1: C).((eq K k0 k) \to ((eq T (lift h (r k0 d) u0) u) \to ((eq C (CHead e
+k0 u0) x) \to ((drop h (r k0 d) c1 e) \to (ex3_2 C T (\lambda (e0:
+C).(\lambda (v: T).(eq C x (CHead e0 k v)))) (\lambda (_: C).(\lambda (v:
+T).(eq T u (lift h (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r
+k d) c e0))))))))) (\lambda (H11: (eq K k0 k)).(eq_ind K k (\lambda (k1:
+K).((eq T (lift h (r k1 d) u0) u) \to ((eq C (CHead e k1 u0) x) \to ((drop h
+(r k1 d) c e) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C x (CHead
+e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k d) v))))
+(\lambda (e0: C).(\lambda (_: T).(drop h (r k d) c e0)))))))) (\lambda (H12:
+(eq T (lift h (r k d) u0) u)).(eq_ind T (lift h (r k d) u0) (\lambda (t:
+T).((eq C (CHead e k u0) x) \to ((drop h (r k d) c e) \to (ex3_2 C T (\lambda
+(e0: C).(\lambda (v: T).(eq C x (CHead e0 k v)))) (\lambda (_: C).(\lambda
+(v: T).(eq T t (lift h (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h
+(r k d) c e0))))))) (\lambda (H13: (eq C (CHead e k u0) x)).(eq_ind C (CHead
+e k u0) (\lambda (c1: C).((drop h (r k d) c e) \to (ex3_2 C T (\lambda (e0:
+C).(\lambda (v: T).(eq C c1 (CHead e0 k v)))) (\lambda (_: C).(\lambda (v:
+T).(eq T (lift h (r k d) u0) (lift h (r k d) v)))) (\lambda (e0: C).(\lambda
+(_: T).(drop h (r k d) c e0)))))) (\lambda (H14: (drop h (r k d) c e)).(let
+H15 \def (eq_ind_r T u (\lambda (t: T).(drop h (S d) (CHead c k t) x)) H
+(lift h (r k d) u0) H12) in (let H16 \def (eq_ind_r C x (\lambda (c0:
+C).(drop h (S d) (CHead c k (lift h (r k d) u0)) c0)) H15 (CHead e k u0) H13)
+in (ex3_2_intro C T (\lambda (e0: C).(\lambda (v: T).(eq C (CHead e k u0)
+(CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T (lift h (r k d) u0)
+(lift h (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k d) c
+e0))) e u0 (refl_equal C (CHead e k u0)) (refl_equal T (lift h (r k d) u0))
+H14)))) x H13)) u H12)) k0 (sym_eq K k0 k H11))) c0 (sym_eq C c0 c H10)))
+H9)) H8))) d0 (sym_eq nat d0 d H6)))) h0 (sym_eq nat h0 h H1) H2 H3 H4
+H0)))))]) in (H0 (refl_equal nat h) (refl_equal nat (S d)) (refl_equal C
+(CHead c k u)) (refl_equal C x))))))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/drop/props".
+
+include "drop/defs.ma".
+
+include "lift/props.ma".
+
+theorem drop_skip_bind:
+ \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h
+d c e) \to (\forall (b: B).(\forall (u: T).(drop h (S d) (CHead c (Bind b)
+(lift h d u)) (CHead e (Bind b) u))))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda
+(H: (drop h d c e)).(\lambda (b: B).(\lambda (u: T).(eq_ind nat (r (Bind b)
+d) (\lambda (n: nat).(drop h (S d) (CHead c (Bind b) (lift h n u)) (CHead e
+(Bind b) u))) (drop_skip (Bind b) h d c e H u) d (refl_equal nat d)))))))).
+
+theorem drop_skip_flat:
+ \forall (h: nat).(\forall (d: nat).(\forall (c: C).(\forall (e: C).((drop h
+(S d) c e) \to (\forall (f: F).(\forall (u: T).(drop h (S d) (CHead c (Flat
+f) (lift h (S d) u)) (CHead e (Flat f) u))))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (c: C).(\lambda (e: C).(\lambda
+(H: (drop h (S d) c e)).(\lambda (f: F).(\lambda (u: T).(eq_ind nat (r (Flat
+f) d) (\lambda (n: nat).(drop h (S d) (CHead c (Flat f) (lift h n u)) (CHead
+e (Flat f) u))) (drop_skip (Flat f) h d c e H u) (S d) (refl_equal nat (S
+d))))))))).
+
+theorem drop_S:
+ \forall (b: B).(\forall (c: C).(\forall (e: C).(\forall (u: T).(\forall (h:
+nat).((drop h O c (CHead e (Bind b) u)) \to (drop (S h) O c e))))))
+\def
+ \lambda (b: B).(\lambda (c: C).(C_ind (\lambda (c0: C).(\forall (e:
+C).(\forall (u: T).(\forall (h: nat).((drop h O c0 (CHead e (Bind b) u)) \to
+(drop (S h) O c0 e)))))) (\lambda (n: nat).(\lambda (e: C).(\lambda (u:
+T).(\lambda (h: nat).(\lambda (H: (drop h O (CSort n) (CHead e (Bind b)
+u))).(and3_ind (eq C (CHead e (Bind b) u) (CSort n)) (eq nat h O) (eq nat O
+O) (drop (S h) O (CSort n) e) (\lambda (H0: (eq C (CHead e (Bind b) u) (CSort
+n))).(\lambda (H1: (eq nat h O)).(\lambda (_: (eq nat O O)).(eq_ind_r nat O
+(\lambda (n0: nat).(drop (S n0) O (CSort n) e)) (let H3 \def (eq_ind C (CHead
+e (Bind b) u) (\lambda (ee: C).(match ee in C return (\lambda (_: C).Prop)
+with [(CSort _) \Rightarrow False | (CHead _ _ _) \Rightarrow True])) I
+(CSort n) H0) in (False_ind (drop (S O) O (CSort n) e) H3)) h H1))))
+(drop_gen_sort n h O (CHead e (Bind b) u) H))))))) (\lambda (c0: C).(\lambda
+(H: ((\forall (e: C).(\forall (u: T).(\forall (h: nat).((drop h O c0 (CHead e
+(Bind b) u)) \to (drop (S h) O c0 e))))))).(\lambda (k: K).(\lambda (t:
+T).(\lambda (e: C).(\lambda (u: T).(\lambda (h: nat).(nat_ind (\lambda (n:
+nat).((drop n O (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S n) O (CHead
+c0 k t) e))) (\lambda (H0: (drop O O (CHead c0 k t) (CHead e (Bind b)
+u))).(let H1 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow
+c])) (CHead c0 k t) (CHead e (Bind b) u) (drop_gen_refl (CHead c0 k t) (CHead
+e (Bind b) u) H0)) in ((let H2 \def (f_equal C K (\lambda (e0: C).(match e0
+in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k _)
+\Rightarrow k])) (CHead c0 k t) (CHead e (Bind b) u) (drop_gen_refl (CHead c0
+k t) (CHead e (Bind b) u) H0)) in ((let H3 \def (f_equal C T (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t |
+(CHead _ _ t) \Rightarrow t])) (CHead c0 k t) (CHead e (Bind b) u)
+(drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) H0)) in (\lambda (H4: (eq
+K k (Bind b))).(\lambda (H5: (eq C c0 e)).(eq_ind C c0 (\lambda (c1: C).(drop
+(S O) O (CHead c0 k t) c1)) (eq_ind_r K (Bind b) (\lambda (k0: K).(drop (S O)
+O (CHead c0 k0 t) c0)) (drop_drop (Bind b) O c0 c0 (drop_refl c0) t) k H4) e
+H5)))) H2)) H1))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c0 k t)
+(CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 k t) e)))).(\lambda (H1:
+(drop (S n) O (CHead c0 k t) (CHead e (Bind b) u))).(drop_drop k (S n) c0 e
+(eq_ind_r nat (S (r k n)) (\lambda (n0: nat).(drop n0 O c0 e)) (H e u (r k n)
+(drop_gen_drop k c0 (CHead e (Bind b) u) t n H1)) (r k (S n)) (r_S k n))
+t)))) h)))))))) c)).
+
+theorem drop_ctail:
+ \forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop
+h d c1 c2) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k u c1)
+(CTail k u c2))))))))
+\def
+ \lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c c2) \to (\forall (k: K).(\forall (u:
+T).(drop h d (CTail k u c) (CTail k u c2))))))))) (\lambda (n: nat).(\lambda
+(c2: C).(\lambda (d: nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n)
+c2)).(\lambda (k: K).(\lambda (u: T).(and3_ind (eq C c2 (CSort n)) (eq nat h
+O) (eq nat d O) (drop h d (CTail k u (CSort n)) (CTail k u c2)) (\lambda (H0:
+(eq C c2 (CSort n))).(\lambda (H1: (eq nat h O)).(\lambda (H2: (eq nat d
+O)).(eq_ind_r nat O (\lambda (n0: nat).(drop n0 d (CTail k u (CSort n))
+(CTail k u c2))) (eq_ind_r nat O (\lambda (n0: nat).(drop O n0 (CTail k u
+(CSort n)) (CTail k u c2))) (eq_ind_r C (CSort n) (\lambda (c: C).(drop O O
+(CTail k u (CSort n)) (CTail k u c))) (drop_refl (CTail k u (CSort n))) c2
+H0) d H2) h H1)))) (drop_gen_sort n h d c2 H))))))))) (\lambda (c2:
+C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h:
+nat).((drop h d c2 c3) \to (\forall (k: K).(\forall (u: T).(drop h d (CTail k
+u c2) (CTail k u c3)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (c3:
+C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n
+(CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u
+(CHead c2 k t)) (CTail k0 u c3))))))) (\lambda (h: nat).(nat_ind (\lambda (n:
+nat).((drop n O (CHead c2 k t) c3) \to (\forall (k0: K).(\forall (u: T).(drop
+n O (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)))))) (\lambda (H: (drop O O
+(CHead c2 k t) c3)).(\lambda (k0: K).(\lambda (u: T).(eq_ind C (CHead c2 k t)
+(\lambda (c: C).(drop O O (CTail k0 u (CHead c2 k t)) (CTail k0 u c)))
+(drop_refl (CTail k0 u (CHead c2 k t))) c3 (drop_gen_refl (CHead c2 k t) c3
+H))))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to
+(\forall (k0: K).(\forall (u: T).(drop n O (CTail k0 u (CHead c2 k t)) (CTail
+k0 u c3))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t) c3)).(\lambda (k0:
+K).(\lambda (u: T).(drop_drop k n (CTail k0 u c2) (CTail k0 u c3) (IHc c3 O
+(r k n) (drop_gen_drop k c2 c3 t n H0) k0 u) t)))))) h)) (\lambda (n:
+nat).(\lambda (H: ((\forall (h: nat).((drop h n (CHead c2 k t) c3) \to
+(\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u (CHead c2 k t)) (CTail
+k0 u c3)))))))).(\lambda (h: nat).(\lambda (H0: (drop h (S n) (CHead c2 k t)
+c3)).(\lambda (k0: K).(\lambda (u: T).(ex3_2_ind C T (\lambda (e: C).(\lambda
+(v: T).(eq C c3 (CHead e k v)))) (\lambda (_: C).(\lambda (v: T).(eq T t
+(lift h (r k n) v)))) (\lambda (e: C).(\lambda (_: T).(drop h (r k n) c2 e)))
+(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)) (\lambda (x0:
+C).(\lambda (x1: T).(\lambda (H1: (eq C c3 (CHead x0 k x1))).(\lambda (H2:
+(eq T t (lift h (r k n) x1))).(\lambda (H3: (drop h (r k n) c2 x0)).(let H4
+\def (eq_ind C c3 (\lambda (c: C).(\forall (h: nat).((drop h n (CHead c2 k t)
+c) \to (\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u (CHead c2 k t))
+(CTail k0 u c))))))) H (CHead x0 k x1) H1) in (eq_ind_r C (CHead x0 k x1)
+(\lambda (c: C).(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u c)))
+(let H5 \def (eq_ind T t (\lambda (t: T).(\forall (h: nat).((drop h n (CHead
+c2 k t) (CHead x0 k x1)) \to (\forall (k0: K).(\forall (u: T).(drop h n
+(CTail k0 u (CHead c2 k t)) (CTail k0 u (CHead x0 k x1)))))))) H4 (lift h (r
+k n) x1) H2) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0: T).(drop h (S
+n) (CTail k0 u (CHead c2 k t0)) (CTail k0 u (CHead x0 k x1)))) (drop_skip k h
+n (CTail k0 u c2) (CTail k0 u x0) (IHc x0 (r k n) h H3 k0 u) x1) t H2)) c3
+H1))))))) (drop_gen_skip_l c2 c3 t h n k H0)))))))) d))))))) c1).
+
+theorem drop_mono:
+ \forall (c: C).(\forall (x1: C).(\forall (d: nat).(\forall (h: nat).((drop h
+d c x1) \to (\forall (x2: C).((drop h d c x2) \to (eq C x1 x2)))))))
+\def
+ \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (x1: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0
+x2) \to (eq C x1 x2)))))))) (\lambda (n: nat).(\lambda (x1: C).(\lambda (d:
+nat).(\lambda (h: nat).(\lambda (H: (drop h d (CSort n) x1)).(\lambda (x2:
+C).(\lambda (H0: (drop h d (CSort n) x2)).(and3_ind (eq C x2 (CSort n)) (eq
+nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H1: (eq C x2 (CSort
+n))).(\lambda (H2: (eq nat h O)).(\lambda (H3: (eq nat d O)).(and3_ind (eq C
+x1 (CSort n)) (eq nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H4: (eq C x1
+(CSort n))).(\lambda (H5: (eq nat h O)).(\lambda (H6: (eq nat d O)).(eq_ind_r
+C (CSort n) (\lambda (c0: C).(eq C x1 c0)) (let H7 \def (eq_ind nat h
+(\lambda (n: nat).(eq nat n O)) H2 O H5) in (let H8 \def (eq_ind nat d
+(\lambda (n: nat).(eq nat n O)) H3 O H6) in (eq_ind_r C (CSort n) (\lambda
+(c0: C).(eq C c0 (CSort n))) (refl_equal C (CSort n)) x1 H4))) x2 H1))))
+(drop_gen_sort n h d x1 H))))) (drop_gen_sort n h d x2 H0))))))))) (\lambda
+(c0: C).(\lambda (H: ((\forall (x1: C).(\forall (d: nat).(\forall (h:
+nat).((drop h d c0 x1) \to (\forall (x2: C).((drop h d c0 x2) \to (eq C x1
+x2))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (x1: C).(\lambda (d:
+nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).((drop h n (CHead c0 k t)
+x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq C x1 x2))))))
+(\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 k t) x1)
+\to (\forall (x2: C).((drop n O (CHead c0 k t) x2) \to (eq C x1 x2)))))
+(\lambda (H0: (drop O O (CHead c0 k t) x1)).(\lambda (x2: C).(\lambda (H1:
+(drop O O (CHead c0 k t) x2)).(eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C
+x1 c1)) (eq_ind C (CHead c0 k t) (\lambda (c1: C).(eq C c1 (CHead c0 k t)))
+(refl_equal C (CHead c0 k t)) x1 (drop_gen_refl (CHead c0 k t) x1 H0)) x2
+(drop_gen_refl (CHead c0 k t) x2 H1))))) (\lambda (n: nat).(\lambda (_:
+(((drop n O (CHead c0 k t) x1) \to (\forall (x2: C).((drop n O (CHead c0 k t)
+x2) \to (eq C x1 x2)))))).(\lambda (H1: (drop (S n) O (CHead c0 k t)
+x1)).(\lambda (x2: C).(\lambda (H2: (drop (S n) O (CHead c0 k t) x2)).(H x1 O
+(r k n) (drop_gen_drop k c0 x1 t n H1) x2 (drop_gen_drop k c0 x2 t n
+H2))))))) h)) (\lambda (n: nat).(\lambda (H0: ((\forall (h: nat).((drop h n
+(CHead c0 k t) x1) \to (\forall (x2: C).((drop h n (CHead c0 k t) x2) \to (eq
+C x1 x2))))))).(\lambda (h: nat).(\lambda (H1: (drop h (S n) (CHead c0 k t)
+x1)).(\lambda (x2: C).(\lambda (H2: (drop h (S n) (CHead c0 k t)
+x2)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x2 (CHead e k v))))
+(\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e:
+C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x0:
+C).(\lambda (x3: T).(\lambda (H3: (eq C x2 (CHead x0 k x3))).(\lambda (H4:
+(eq T t (lift h (r k n) x3))).(\lambda (H5: (drop h (r k n) c0
+x0)).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C x1 (CHead e k v))))
+(\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k n) v)))) (\lambda (e:
+C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x4:
+C).(\lambda (x5: T).(\lambda (H6: (eq C x1 (CHead x4 k x5))).(\lambda (H7:
+(eq T t (lift h (r k n) x5))).(\lambda (H8: (drop h (r k n) c0 x4)).(eq_ind_r
+C (CHead x0 k x3) (\lambda (c1: C).(eq C x1 c1)) (let H9 \def (eq_ind C x1
+(\lambda (c: C).(\forall (h: nat).((drop h n (CHead c0 k t) c) \to (\forall
+(x2: C).((drop h n (CHead c0 k t) x2) \to (eq C c x2)))))) H0 (CHead x4 k x5)
+H6) in (eq_ind_r C (CHead x4 k x5) (\lambda (c1: C).(eq C c1 (CHead x0 k
+x3))) (let H10 \def (eq_ind T t (\lambda (t: T).(\forall (h: nat).((drop h n
+(CHead c0 k t) (CHead x4 k x5)) \to (\forall (x2: C).((drop h n (CHead c0 k
+t) x2) \to (eq C (CHead x4 k x5) x2)))))) H9 (lift h (r k n) x5) H7) in (let
+H11 \def (eq_ind T t (\lambda (t: T).(eq T t (lift h (r k n) x3))) H4 (lift h
+(r k n) x5) H7) in (let H12 \def (eq_ind T x5 (\lambda (t: T).(\forall (h0:
+nat).((drop h0 n (CHead c0 k (lift h (r k n) t)) (CHead x4 k t)) \to (\forall
+(x2: C).((drop h0 n (CHead c0 k (lift h (r k n) t)) x2) \to (eq C (CHead x4 k
+t) x2)))))) H10 x3 (lift_inj x5 x3 h (r k n) H11)) in (eq_ind_r T x3 (\lambda
+(t0: T).(eq C (CHead x4 k t0) (CHead x0 k x3))) (sym_equal C (CHead x0 k x3)
+(CHead x4 k x3) (sym_equal C (CHead x4 k x3) (CHead x0 k x3) (sym_equal C
+(CHead x0 k x3) (CHead x4 k x3) (f_equal3 C K T C CHead x0 x4 k k x3 x3 (H x0
+(r k n) h H5 x4 H8) (refl_equal K k) (refl_equal T x3))))) x5 (lift_inj x5 x3
+h (r k n) H11))))) x1 H6)) x2 H3)))))) (drop_gen_skip_l c0 x1 t h n k
+H1))))))) (drop_gen_skip_l c0 x2 t h n k H2)))))))) d))))))) c).
+
+theorem drop_conf_lt:
+ \forall (k: K).(\forall (i: nat).(\forall (u: T).(\forall (c0: C).(\forall
+(c: C).((drop i O c (CHead c0 k u)) \to (\forall (e: C).(\forall (h:
+nat).(\forall (d: nat).((drop h (S (plus i d)) c e) \to (ex3_2 T C (\lambda
+(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop i O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop
+h (r k d) c0 e0)))))))))))))
+\def
+ \lambda (k: K).(\lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (u:
+T).(\forall (c0: C).(\forall (c: C).((drop n O c (CHead c0 k u)) \to (\forall
+(e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus n d)) c e) \to
+(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v))))
+(\lambda (v: T).(\lambda (e0: C).(drop n O e (CHead e0 k v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))))) (\lambda (u:
+T).(\lambda (c0: C).(\lambda (c: C).(\lambda (H: (drop O O c (CHead c0 k
+u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop
+h (S (plus O d)) c e)).(let H1 \def (eq_ind C c (\lambda (c: C).(drop h (S
+(plus O d)) c e)) H0 (CHead c0 k u) (drop_gen_refl c (CHead c0 k u) H)) in
+(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v))))
+(\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k (plus O d)) v))))
+(\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus O d)) c0 e0))) (ex3_2 T C
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v:
+T).(\lambda (e0: C).(drop O O e (CHead e0 k v)))) (\lambda (_: T).(\lambda
+(e0: C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda
+(H2: (eq C e (CHead x0 k x1))).(\lambda (H3: (eq T u (lift h (r k (plus O d))
+x1))).(\lambda (H4: (drop h (r k (plus O d)) c0 x0)).(eq_ind_r C (CHead x0 k
+x1) (\lambda (c1: C).(ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift
+h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop O O c1 (CHead e0 k
+v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))) (eq_ind_r T
+(lift h (r k (plus O d)) x1) (\lambda (t: T).(ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T t (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda
+(e0: C).(drop h (r k d) c0 e0))))) (ex3_2_intro T C (\lambda (v: T).(\lambda
+(_: C).(eq T (lift h (r k (plus O d)) x1) (lift h (r k d) v)))) (\lambda (v:
+T).(\lambda (e0: C).(drop O O (CHead x0 k x1) (CHead e0 k v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h (r k d) c0 e0))) x1 x0 (refl_equal T (lift h (r k
+d) x1)) (drop_refl (CHead x0 k x1)) H4) u H3) e H2)))))) (drop_gen_skip_l c0
+e u h (plus O d) k H1))))))))))) (\lambda (i0: nat).(\lambda (H: ((\forall
+(u: T).(\forall (c0: C).(\forall (c: C).((drop i0 O c (CHead c0 k u)) \to
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus i0 d))
+c e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d)
+v)))) (\lambda (v: T).(\lambda (e0: C).(drop i0 O e (CHead e0 k v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0)))))))))))))).(\lambda
+(u: T).(\lambda (c0: C).(\lambda (c: C).(C_ind (\lambda (c1: C).((drop (S i0)
+O c1 (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d:
+nat).((drop h (S (plus (S i0) d)) c1 e) \to (ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0)))))))))) (\lambda (n: nat).(\lambda (_: (drop (S
+i0) O (CSort n) (CHead c0 k u))).(\lambda (e: C).(\lambda (h: nat).(\lambda
+(d: nat).(\lambda (H1: (drop h (S (plus (S i0) d)) (CSort n) e)).(and3_ind
+(eq C e (CSort n)) (eq nat h O) (eq nat (S (plus (S i0) d)) O) (ex3_2 T C
+(\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v:
+T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h (r k d) c0 e0)))) (\lambda (_: (eq C e (CSort
+n))).(\lambda (_: (eq nat h O)).(\lambda (H4: (eq nat (S (plus (S i0) d))
+O)).(let H5 \def (eq_ind nat (S (plus (S i0) d)) (\lambda (ee: nat).(match ee
+in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
+\Rightarrow True])) I O H4) in (False_ind (ex3_2 T C (\lambda (v: T).(\lambda
+(_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop
+(S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d)
+c0 e0)))) H5))))) (drop_gen_sort n h (S (plus (S i0) d)) e H1))))))))
+(\lambda (c1: C).(\lambda (H0: (((drop (S i0) O c1 (CHead c0 k u)) \to
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h (S (plus (S i0)
+d)) c1 e) \to (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k
+d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O e (CHead e0 k v))))
+(\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))).(\lambda
+(k0: K).(K_ind (\lambda (k1: K).(\forall (t: T).((drop (S i0) O (CHead c1 k1
+t) (CHead c0 k u)) \to (\forall (e: C).(\forall (h: nat).(\forall (d:
+nat).((drop h (S (plus (S i0) d)) (CHead c1 k1 t) e) \to (ex3_2 T C (\lambda
+(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0))))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda
+(H1: (drop (S i0) O (CHead c1 (Bind b) t) (CHead c0 k u))).(\lambda (e:
+C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h (S (plus (S i0)
+d)) (CHead c1 (Bind b) t) e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v:
+T).(eq C e (CHead e0 (Bind b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t
+(lift h (r (Bind b) (plus (S i0) d)) v)))) (\lambda (e0: C).(\lambda (_:
+T).(drop h (r (Bind b) (plus (S i0) d)) c1 e0))) (ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3:
+(eq C e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b)
+(plus (S i0) d)) x1))).(\lambda (H5: (drop h (r (Bind b) (plus (S i0) d)) c1
+x0)).(eq_ind_r C (CHead x0 (Bind b) x1) (\lambda (c2: C).(ex3_2 T C (\lambda
+(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop (S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0))))) (let H6 \def (H u c0 c1 (drop_gen_drop (Bind b)
+c1 (CHead c0 k u) t i0 H1) x0 h d H5) in (ex3_2_ind T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop i0 O x0 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T
+u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O
+(CHead x0 (Bind b) x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0)))) (\lambda (x2: T).(\lambda (x3: C).(\lambda (H7:
+(eq T u (lift h (r k d) x2))).(\lambda (H8: (drop i0 O x0 (CHead x3 k
+x2))).(\lambda (H9: (drop h (r k d) c0 x3)).(ex3_2_intro T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop (S i0) O (CHead x0 (Bind b) x1) (CHead e0 k v)))) (\lambda (_:
+T).(\lambda (e0: C).(drop h (r k d) c0 e0))) x2 x3 H7 (drop_drop (Bind b) i0
+x0 (CHead x3 k x2) H8 x1) H9)))))) H6)) e H3)))))) (drop_gen_skip_l c1 e t h
+(plus (S i0) d) (Bind b) H2))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda
+(H1: (drop (S i0) O (CHead c1 (Flat f) t) (CHead c0 k u))).(\lambda (e:
+C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: (drop h (S (plus (S i0)
+d)) (CHead c1 (Flat f) t) e)).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v:
+T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t
+(lift h (r (Flat f) (plus (S i0) d)) v)))) (\lambda (e0: C).(\lambda (_:
+T).(drop h (r (Flat f) (plus (S i0) d)) c1 e0))) (ex3_2 T C (\lambda (v:
+T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop (S i0) O e (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0)))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H3:
+(eq C e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t (lift h (r (Flat f)
+(plus (S i0) d)) x1))).(\lambda (H5: (drop h (r (Flat f) (plus (S i0) d)) c1
+x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c2: C).(ex3_2 T C (\lambda
+(v: T).(\lambda (_: C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda
+(e0: C).(drop (S i0) O c2 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0:
+C).(drop h (r k d) c0 e0))))) (ex3_2_ind T C (\lambda (v: T).(\lambda (_:
+C).(eq T u (lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S
+i0) O x0 (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d)
+c0 e0))) (ex3_2 T C (\lambda (v: T).(\lambda (_: C).(eq T u (lift h (r k d)
+v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead x0 (Flat f) x1)
+(CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r k d) c0 e0))))
+(\lambda (x2: T).(\lambda (x3: C).(\lambda (H6: (eq T u (lift h (r k d)
+x2))).(\lambda (H7: (drop (S i0) O x0 (CHead x3 k x2))).(\lambda (H8: (drop h
+(r k d) c0 x3)).(ex3_2_intro T C (\lambda (v: T).(\lambda (_: C).(eq T u
+(lift h (r k d) v)))) (\lambda (v: T).(\lambda (e0: C).(drop (S i0) O (CHead
+x0 (Flat f) x1) (CHead e0 k v)))) (\lambda (_: T).(\lambda (e0: C).(drop h (r
+k d) c0 e0))) x2 x3 H6 (drop_drop (Flat f) i0 x0 (CHead x3 k x2) H7 x1)
+H8)))))) (H0 (drop_gen_drop (Flat f) c1 (CHead c0 k u) t i0 H1) x0 h d H5)) e
+H3)))))) (drop_gen_skip_l c1 e t h (plus (S i0) d) (Flat f) H2)))))))))
+k0)))) c)))))) i)).
+
+theorem drop_conf_ge:
+ \forall (i: nat).(\forall (a: C).(\forall (c: C).((drop i O c a) \to
+(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) \to ((le
+(plus d h) i) \to (drop (minus i h) O e a)))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (a: C).(\forall (c:
+C).((drop n O c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d c e) \to ((le (plus d h) n) \to (drop (minus n h) O e
+a)))))))))) (\lambda (a: C).(\lambda (c: C).(\lambda (H: (drop O O c
+a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h
+d c e)).(\lambda (H1: (le (plus d h) O)).(let H2 \def (eq_ind C c (\lambda
+(c: C).(drop h d c e)) H0 a (drop_gen_refl c a H)) in (let H3 \def (match H1
+in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O) \to (drop
+(minus O h) O e a)))) with [le_n \Rightarrow (\lambda (H: (eq nat (plus d h)
+O)).(let H3 \def (f_equal nat nat (\lambda (e0: nat).e0) (plus d h) O H) in
+(eq_ind nat (plus d h) (\lambda (n: nat).(drop (minus n h) n e a)) (eq_ind_r
+nat O (\lambda (n: nat).(drop (minus n h) n e a)) (and_ind (eq nat d O) (eq
+nat h O) (drop O O e a) (\lambda (H0: (eq nat d O)).(\lambda (H1: (eq nat h
+O)).(let H2 \def (eq_ind nat d (\lambda (n: nat).(drop h n a e)) H2 O H0) in
+(let H4 \def (eq_ind nat h (\lambda (n: nat).(drop n O a e)) H2 O H1) in
+(eq_ind C a (\lambda (c: C).(drop O O c a)) (drop_refl a) e (drop_gen_refl a
+e H4)))))) (plus_O d h H3)) (plus d h) H3) O H3))) | (le_S m H) \Rightarrow
+(\lambda (H2: (eq nat (S m) O)).((let H0 \def (eq_ind nat (S m) (\lambda (e0:
+nat).(match e0 in nat return (\lambda (_: nat).Prop) with [O \Rightarrow
+False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (plus d h) m)
+\to (drop (minus O h) O e a)) H0)) H))]) in (H3 (refl_equal nat O))))))))))))
+(\lambda (i0: nat).(\lambda (H: ((\forall (a: C).(\forall (c: C).((drop i0 O
+c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e)
+\to ((le (plus d h) i0) \to (drop (minus i0 h) O e a))))))))))).(\lambda (a:
+C).(\lambda (c: C).(C_ind (\lambda (c0: C).((drop (S i0) O c0 a) \to (\forall
+(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le (plus d
+h) (S i0)) \to (drop (minus (S i0) h) O e a)))))))) (\lambda (n:
+nat).(\lambda (H0: (drop (S i0) O (CSort n) a)).(\lambda (e: C).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H1: (drop h d (CSort n) e)).(\lambda (H2:
+(le (plus d h) (S i0))).(and3_ind (eq C e (CSort n)) (eq nat h O) (eq nat d
+O) (drop (minus (S i0) h) O e a) (\lambda (H3: (eq C e (CSort n))).(\lambda
+(H4: (eq nat h O)).(\lambda (H5: (eq nat d O)).(and3_ind (eq C a (CSort n))
+(eq nat (S i0) O) (eq nat O O) (drop (minus (S i0) h) O e a) (\lambda (H6:
+(eq C a (CSort n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O
+O)).(let H9 \def (eq_ind nat d (\lambda (n: nat).(le (plus n h) (S i0))) H2 O
+H5) in (let H10 \def (eq_ind nat h (\lambda (n: nat).(le (plus O n) (S i0)))
+H9 O H4) in (eq_ind_r nat O (\lambda (n0: nat).(drop (minus (S i0) n0) O e
+a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O c0 a))
+(eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O (CSort n)
+c0)) (let H11 \def (eq_ind nat (S i0) (\lambda (ee: nat).(match ee in nat
+return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
+True])) I O H7) in (False_ind (drop (minus (S i0) O) O (CSort n) (CSort n))
+H11)) a H6) e H3) h H4)))))) (drop_gen_sort n (S i0) O a H0)))))
+(drop_gen_sort n h d e H1))))))))) (\lambda (c0: C).(\lambda (H0: (((drop (S
+i0) O c0 a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h
+d c0 e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e
+a))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).((drop (S
+i0) O (CHead c0 k0 t) a) \to (\forall (e: C).(\forall (h: nat).(\forall (d:
+nat).((drop h d (CHead c0 k0 t) e) \to ((le (plus d h) (S i0)) \to (drop
+(minus (S i0) h) O e a))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H1:
+(drop (S i0) O (CHead c0 (Bind b) t) a)).(\lambda (e: C).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H2: (drop h d (CHead c0 (Bind b) t)
+e)).(\lambda (H3: (le (plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h
+n (CHead c0 (Bind b) t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S
+i0) h) O e a)))) (\lambda (H4: (drop h O (CHead c0 (Bind b) t) e)).(\lambda
+(H5: (le (plus O h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0
+(Bind b) t) e) \to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e
+a)))) (\lambda (H6: (drop O O (CHead c0 (Bind b) t) e)).(\lambda (_: (le
+(plus O O) (S i0))).(eq_ind C (CHead c0 (Bind b) t) (\lambda (c1: C).(drop
+(minus (S i0) O) O c1 a)) (drop_drop (Bind b) i0 c0 a (drop_gen_drop (Bind b)
+c0 a t i0 H1) t) e (drop_gen_refl (CHead c0 (Bind b) t) e H6)))) (\lambda
+(h0: nat).(\lambda (_: (((drop h0 O (CHead c0 (Bind b) t) e) \to ((le (plus O
+h0) (S i0)) \to (drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0)
+O (CHead c0 (Bind b) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H a
+c0 (drop_gen_drop (Bind b) c0 a t i0 H1) e h0 O (drop_gen_drop (Bind b) c0 e
+t h0 H6) (le_S_n (plus O h0) i0 H7)))))) h H4 H5))) (\lambda (d0:
+nat).(\lambda (_: (((drop h d0 (CHead c0 (Bind b) t) e) \to ((le (plus d0 h)
+(S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0)
+(CHead c0 (Bind b) t) e)).(\lambda (H5: (le (plus (S d0) h) (S
+i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Bind
+b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Bind b) d0)
+v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Bind b) d0) c0 e0))) (drop
+(minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C
+e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) d0)
+x1))).(\lambda (H8: (drop h (r (Bind b) d0) c0 x0)).(eq_ind_r C (CHead x0
+(Bind b) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (eq_ind nat (S
+(minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Bind b) x1) a))
+(drop_drop (Bind b) (minus i0 h) x0 a (H a c0 (drop_gen_drop (Bind b) c0 a t
+i0 H1) x0 h d0 H8 (le_S_n (plus d0 h) i0 H5)) x1) (minus (S i0) h)
+(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) e
+H6)))))) (drop_gen_skip_l c0 e t h d0 (Bind b) H4)))))) d H2 H3)))))))))
+(\lambda (f: F).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c0 (Flat
+f) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2:
+(drop h d (CHead c0 (Flat f) t) e)).(\lambda (H3: (le (plus d h) (S
+i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Flat f) t) e) \to ((le
+(plus n h) (S i0)) \to (drop (minus (S i0) h) O e a)))) (\lambda (H4: (drop h
+O (CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus O h) (S i0))).(nat_ind
+(\lambda (n: nat).((drop n O (CHead c0 (Flat f) t) e) \to ((le (plus O n) (S
+i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda (H6: (drop O O (CHead c0
+(Flat f) t) e)).(\lambda (_: (le (plus O O) (S i0))).(eq_ind C (CHead c0
+(Flat f) t) (\lambda (c1: C).(drop (minus (S i0) O) O c1 a)) (drop_drop (Flat
+f) i0 c0 a (drop_gen_drop (Flat f) c0 a t i0 H1) t) e (drop_gen_refl (CHead
+c0 (Flat f) t) e H6)))) (\lambda (h0: nat).(\lambda (_: (((drop h0 O (CHead
+c0 (Flat f) t) e) \to ((le (plus O h0) (S i0)) \to (drop (minus (S i0) h0) O
+e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 (Flat f) t) e)).(\lambda (H7:
+(le (plus O (S h0)) (S i0))).(H0 (drop_gen_drop (Flat f) c0 a t i0 H1) e (S
+h0) O (drop_gen_drop (Flat f) c0 e t h0 H6) H7))))) h H4 H5))) (\lambda (d0:
+nat).(\lambda (_: (((drop h d0 (CHead c0 (Flat f) t) e) \to ((le (plus d0 h)
+(S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0)
+(CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus (S d0) h) (S
+i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Flat
+f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0)
+v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Flat f) d0) c0 e0))) (drop
+(minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C
+e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t (lift h (r (Flat f) d0)
+x1))).(\lambda (H8: (drop h (r (Flat f) d0) c0 x0)).(eq_ind_r C (CHead x0
+(Flat f) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (let H9 \def
+(eq_ind_r nat (minus (S i0) h) (\lambda (n: nat).(drop n O x0 a)) (H0
+(drop_gen_drop (Flat f) c0 a t i0 H1) x0 h (S d0) H8 H5) (S (minus i0 h))
+(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) in
+(eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Flat f)
+x1) a)) (drop_drop (Flat f) (minus i0 h) x0 a H9 x1) (minus (S i0) h)
+(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5))))) e
+H6)))))) (drop_gen_skip_l c0 e t h d0 (Flat f) H4)))))) d H2 H3)))))))))
+k)))) c))))) i).
+
+theorem drop_conf_rev:
+ \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to
+(\forall (c2: C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1:
+C).(drop j O c1 c2)) (\lambda (c1: C).(drop i j c1 e1)))))))))
+\def
+ \lambda (j: nat).(nat_ind (\lambda (n: nat).(\forall (e1: C).(\forall (e2:
+C).((drop n O e1 e2) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e2)
+\to (ex2 C (\lambda (c1: C).(drop n O c1 c2)) (\lambda (c1: C).(drop i n c1
+e1)))))))))) (\lambda (e1: C).(\lambda (e2: C).(\lambda (H: (drop O O e1
+e2)).(\lambda (c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(let
+H1 \def (eq_ind_r C e2 (\lambda (c: C).(drop i O c2 c)) H0 e1 (drop_gen_refl
+e1 e2 H)) in (ex_intro2 C (\lambda (c1: C).(drop O O c1 c2)) (\lambda (c1:
+C).(drop i O c1 e1)) c2 (drop_refl c2) H1)))))))) (\lambda (j0: nat).(\lambda
+(IHj: ((\forall (e1: C).(\forall (e2: C).((drop j0 O e1 e2) \to (\forall (c2:
+C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop j0 O
+c1 c2)) (\lambda (c1: C).(drop i j0 c1 e1))))))))))).(\lambda (e1: C).(C_ind
+(\lambda (c: C).(\forall (e2: C).((drop (S j0) O c e2) \to (\forall (c2:
+C).(\forall (i: nat).((drop i O c2 e2) \to (ex2 C (\lambda (c1: C).(drop (S
+j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 c))))))))) (\lambda (n:
+nat).(\lambda (e2: C).(\lambda (H: (drop (S j0) O (CSort n) e2)).(\lambda
+(c2: C).(\lambda (i: nat).(\lambda (H0: (drop i O c2 e2)).(and3_ind (eq C e2
+(CSort n)) (eq nat (S j0) O) (eq nat O O) (ex2 C (\lambda (c1: C).(drop (S
+j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) (\lambda (H1:
+(eq C e2 (CSort n))).(\lambda (H2: (eq nat (S j0) O)).(\lambda (_: (eq nat O
+O)).(let H4 \def (eq_ind C e2 (\lambda (c: C).(drop i O c2 c)) H0 (CSort n)
+H1) in (let H5 \def (eq_ind nat (S j0) (\lambda (ee: nat).(match ee in nat
+return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
+True])) I O H2) in (False_ind (ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2))
+(\lambda (c1: C).(drop i (S j0) c1 (CSort n)))) H5)))))) (drop_gen_sort n (S
+j0) O e2 H)))))))) (\lambda (e2: C).(\lambda (IHe1: ((\forall (e3: C).((drop
+(S j0) O e2 e3) \to (\forall (c2: C).(\forall (i: nat).((drop i O c2 e3) \to
+(ex2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1: C).(drop i (S
+j0) c1 e2)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (e3: C).(\lambda
+(H: (drop (S j0) O (CHead e2 k t) e3)).(\lambda (c2: C).(\lambda (i:
+nat).(\lambda (H0: (drop i O c2 e3)).((match k in K return (\lambda (k0:
+K).((drop (r k0 j0) O e2 e3) \to (ex2 C (\lambda (c1: C).(drop (S j0) O c1
+c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 k0 t)))))) with [(Bind b)
+\Rightarrow (\lambda (H1: (drop (r (Bind b) j0) O e2 e3)).(let H_x \def (IHj
+e2 e3 H1 c2 i H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c1: C).(drop j0
+O c1 c2)) (\lambda (c1: C).(drop i j0 c1 e2)) (ex2 C (\lambda (c1: C).(drop
+(S j0) O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Bind b) t))))
+(\lambda (x: C).(\lambda (H3: (drop j0 O x c2)).(\lambda (H4: (drop i j0 x
+e2)).(ex_intro2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1:
+C).(drop i (S j0) c1 (CHead e2 (Bind b) t))) (CHead x (Bind b) (lift i (r
+(Bind b) j0) t)) (drop_drop (Bind b) j0 x c2 H3 (lift i (r (Bind b) j0) t))
+(drop_skip (Bind b) i j0 x e2 H4 t))))) H2)))) | (Flat f) \Rightarrow
+(\lambda (H1: (drop (r (Flat f) j0) O e2 e3)).(let H_x \def (IHe1 e3 H1 c2 i
+H0) in (let H2 \def H_x in (ex2_ind C (\lambda (c1: C).(drop (S j0) O c1 c2))
+(\lambda (c1: C).(drop i (S j0) c1 e2)) (ex2 C (\lambda (c1: C).(drop (S j0)
+O c1 c2)) (\lambda (c1: C).(drop i (S j0) c1 (CHead e2 (Flat f) t))))
+(\lambda (x: C).(\lambda (H3: (drop (S j0) O x c2)).(\lambda (H4: (drop i (S
+j0) x e2)).(ex_intro2 C (\lambda (c1: C).(drop (S j0) O c1 c2)) (\lambda (c1:
+C).(drop i (S j0) c1 (CHead e2 (Flat f) t))) (CHead x (Flat f) (lift i (r
+(Flat f) j0) t)) (drop_drop (Flat f) j0 x c2 H3 (lift i (r (Flat f) j0) t))
+(drop_skip (Flat f) i j0 x e2 H4 t))))) H2))))]) (drop_gen_drop k e2 e3 t j0
+H))))))))))) e1)))) j).
+
+theorem drop_trans_le:
+ \forall (i: nat).(\forall (d: nat).((le i d) \to (\forall (c1: C).(\forall
+(c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O
+c2 e2) \to (ex2 C (\lambda (e1: C).(drop i O c1 e1)) (\lambda (e1: C).(drop h
+(minus d i) e1 e2)))))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (d: nat).((le n d) \to
+(\forall (c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to
+(\forall (e2: C).((drop n O c2 e2) \to (ex2 C (\lambda (e1: C).(drop n O c1
+e1)) (\lambda (e1: C).(drop h (minus d n) e1 e2)))))))))))) (\lambda (d:
+nat).(\lambda (_: (le O d)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h:
+nat).(\lambda (H0: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H1: (drop O O
+c2 e2)).(let H2 \def (eq_ind C c2 (\lambda (c: C).(drop h d c1 c)) H0 e2
+(drop_gen_refl c2 e2 H1)) in (eq_ind nat d (\lambda (n: nat).(ex2 C (\lambda
+(e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h n e1 e2)))) (ex_intro2 C
+(\lambda (e1: C).(drop O O c1 e1)) (\lambda (e1: C).(drop h d e1 e2)) c1
+(drop_refl c1) H2) (minus d O) (minus_n_O d))))))))))) (\lambda (i0:
+nat).(\lambda (IHi: ((\forall (d: nat).((le i0 d) \to (\forall (c1:
+C).(\forall (c2: C).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2:
+C).((drop i0 O c2 e2) \to (ex2 C (\lambda (e1: C).(drop i0 O c1 e1)) (\lambda
+(e1: C).(drop h (minus d i0) e1 e2))))))))))))).(\lambda (d: nat).(nat_ind
+(\lambda (n: nat).((le (S i0) n) \to (\forall (c1: C).(\forall (c2:
+C).(\forall (h: nat).((drop h n c1 c2) \to (\forall (e2: C).((drop (S i0) O
+c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1:
+C).(drop h (minus n (S i0)) e1 e2))))))))))) (\lambda (H: (le (S i0)
+O)).(\lambda (c1: C).(\lambda (c2: C).(\lambda (h: nat).(\lambda (_: (drop h
+O c1 c2)).(\lambda (e2: C).(\lambda (_: (drop (S i0) O c2 e2)).(let H2 \def
+(match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O)
+\to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1: C).(drop h
+(minus O (S i0)) e1 e2)))))) with [le_n \Rightarrow (\lambda (H2: (eq nat (S
+i0) O)).(let H3 \def (eq_ind nat (S i0) (\lambda (e: nat).(match e in nat
+return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow
+True])) I O H2) in (False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1))
+(\lambda (e1: C).(drop h (minus O (S i0)) e1 e2))) H3))) | (le_S m H2)
+\Rightarrow (\lambda (H3: (eq nat (S m) O)).((let H4 \def (eq_ind nat (S m)
+(\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow False | (S _) \Rightarrow True])) I O H3) in (False_ind ((le (S
+i0) m) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1 e1)) (\lambda (e1:
+C).(drop h (minus O (S i0)) e1 e2)))) H4)) H2))]) in (H2 (refl_equal nat
+O)))))))))) (\lambda (d0: nat).(\lambda (_: (((le (S i0) d0) \to (\forall
+(c1: C).(\forall (c2: C).(\forall (h: nat).((drop h d0 c1 c2) \to (\forall
+(e2: C).((drop (S i0) O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c1
+e1)) (\lambda (e1: C).(drop h (minus d0 (S i0)) e1 e2)))))))))))).(\lambda
+(H: (le (S i0) (S d0))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2:
+C).(\forall (h: nat).((drop h (S d0) c c2) \to (\forall (e2: C).((drop (S i0)
+O c2 e2) \to (ex2 C (\lambda (e1: C).(drop (S i0) O c e1)) (\lambda (e1:
+C).(drop h (minus (S d0) (S i0)) e1 e2))))))))) (\lambda (n: nat).(\lambda
+(c2: C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CSort n)
+c2)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c2 e2)).(and3_ind (eq C c2
+(CSort n)) (eq nat h O) (eq nat (S d0) O) (ex2 C (\lambda (e1: C).(drop (S
+i0) O (CSort n) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)))
+(\lambda (H2: (eq C c2 (CSort n))).(\lambda (_: (eq nat h O)).(\lambda (_:
+(eq nat (S d0) O)).(let H5 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c
+e2)) H1 (CSort n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq
+nat O O) (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda (e1:
+C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (H6: (eq C e2 (CSort
+n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C
+(CSort n) (\lambda (c: C).(ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n)
+e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 c)))) (let H9 \def
+(eq_ind nat (S i0) (\lambda (ee: nat).(match ee in nat return (\lambda (_:
+nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H7) in
+(False_ind (ex2 C (\lambda (e1: C).(drop (S i0) O (CSort n) e1)) (\lambda
+(e1: C).(drop h (minus (S d0) (S i0)) e1 (CSort n)))) H9)) e2 H6))))
+(drop_gen_sort n (S i0) O e2 H5)))))) (drop_gen_sort n h (S d0) c2 H0))))))))
+(\lambda (c2: C).(\lambda (IHc: ((\forall (c3: C).(\forall (h: nat).((drop h
+(S d0) c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda
+(e1: C).(drop (S i0) O c2 e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0))
+e1 e2)))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t:
+T).(\forall (c3: C).(\forall (h: nat).((drop h (S d0) (CHead c2 k0 t) c3) \to
+(\forall (e2: C).((drop (S i0) O c3 e2) \to (ex2 C (\lambda (e1: C).(drop (S
+i0) O (CHead c2 k0 t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1
+e2)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (c3: C).(\lambda (h:
+nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Bind b) t) c3)).(\lambda (e2:
+C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T (\lambda (e:
+C).(\lambda (v: T).(eq C c3 (CHead e (Bind b) v)))) (\lambda (_: C).(\lambda
+(v: T).(eq T t (lift h (r (Bind b) d0) v)))) (\lambda (e: C).(\lambda (_:
+T).(drop h (r (Bind b) d0) c2 e))) (ex2 C (\lambda (e1: C).(drop (S i0) O
+(CHead c2 (Bind b) t) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1
+e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0
+(Bind b) x1))).(\lambda (H3: (eq T t (lift h (r (Bind b) d0) x1))).(\lambda
+(H4: (drop h (r (Bind b) d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c:
+C).(drop (S i0) O c e2)) H1 (CHead x0 (Bind b) x1) H2) in (eq_ind_r T (lift h
+(r (Bind b) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1: C).(drop (S i0) O
+(CHead c2 (Bind b) t0) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1
+e2)))) (ex2_ind C (\lambda (e1: C).(drop i0 O c2 e1)) (\lambda (e1: C).(drop
+h (minus d0 i0) e1 e2)) (ex2 C (\lambda (e1: C).(drop (S i0) O (CHead c2
+(Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S
+d0) (S i0)) e1 e2))) (\lambda (x: C).(\lambda (H6: (drop i0 O c2 x)).(\lambda
+(H7: (drop h (minus d0 i0) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0)
+O (CHead c2 (Bind b) (lift h (r (Bind b) d0) x1)) e1)) (\lambda (e1: C).(drop
+h (minus (S d0) (S i0)) e1 e2)) x (drop_drop (Bind b) i0 c2 x H6 (lift h (r
+(Bind b) d0) x1)) H7)))) (IHi d0 (le_S_n i0 d0 H) c2 x0 h H4 e2
+(drop_gen_drop (Bind b) x0 e2 x1 i0 H5))) t H3))))))) (drop_gen_skip_l c2 c3
+t h d0 (Bind b) H0))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (c3:
+C).(\lambda (h: nat).(\lambda (H0: (drop h (S d0) (CHead c2 (Flat f) t)
+c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(ex3_2_ind C T
+(\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e (Flat f) v)))) (\lambda (_:
+C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0) v)))) (\lambda (e:
+C).(\lambda (_: T).(drop h (r (Flat f) d0) c2 e))) (ex2 C (\lambda (e1:
+C).(drop (S i0) O (CHead c2 (Flat f) t) e1)) (\lambda (e1: C).(drop h (minus
+(S d0) (S i0)) e1 e2))) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C
+c3 (CHead x0 (Flat f) x1))).(\lambda (H3: (eq T t (lift h (r (Flat f) d0)
+x1))).(\lambda (H4: (drop h (r (Flat f) d0) c2 x0)).(let H5 \def (eq_ind C c3
+(\lambda (c: C).(drop (S i0) O c e2)) H1 (CHead x0 (Flat f) x1) H2) in
+(eq_ind_r T (lift h (r (Flat f) d0) x1) (\lambda (t0: T).(ex2 C (\lambda (e1:
+C).(drop (S i0) O (CHead c2 (Flat f) t0) e1)) (\lambda (e1: C).(drop h (minus
+(S d0) (S i0)) e1 e2)))) (ex2_ind C (\lambda (e1: C).(drop (S i0) O c2 e1))
+(\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2)) (ex2 C (\lambda (e1:
+C).(drop (S i0) O (CHead c2 (Flat f) (lift h (r (Flat f) d0) x1)) e1))
+(\lambda (e1: C).(drop h (minus (S d0) (S i0)) e1 e2))) (\lambda (x:
+C).(\lambda (H6: (drop (S i0) O c2 x)).(\lambda (H7: (drop h (minus (S d0) (S
+i0)) x e2)).(ex_intro2 C (\lambda (e1: C).(drop (S i0) O (CHead c2 (Flat f)
+(lift h (r (Flat f) d0) x1)) e1)) (\lambda (e1: C).(drop h (minus (S d0) (S
+i0)) e1 e2)) x (drop_drop (Flat f) i0 c2 x H6 (lift h (r (Flat f) d0) x1))
+H7)))) (IHc x0 h H4 e2 (drop_gen_drop (Flat f) x0 e2 x1 i0 H5))) t H3)))))))
+(drop_gen_skip_l c2 c3 t h d0 (Flat f) H0))))))))) k)))) c1))))) d)))) i).
+
+theorem drop_trans_ge:
+ \forall (i: nat).(\forall (c1: C).(\forall (c2: C).(\forall (d:
+nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i O c2
+e2) \to ((le d i) \to (drop (plus i h) O c1 e2)))))))))
+\def
+ \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (c1: C).(\forall (c2:
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2:
+C).((drop n O c2 e2) \to ((le d n) \to (drop (plus n h) O c1 e2))))))))))
+(\lambda (c1: C).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h:
+nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H0: (drop O O
+c2 e2)).(\lambda (H1: (le d O)).(eq_ind C c2 (\lambda (c: C).(drop (plus O h)
+O c1 c)) (let H2 \def (match H1 in le return (\lambda (n: nat).(\lambda (_:
+(le ? n)).((eq nat n O) \to (drop (plus O h) O c1 c2)))) with [le_n
+\Rightarrow (\lambda (H0: (eq nat d O)).(eq_ind nat O (\lambda (_: nat).(drop
+(plus O h) O c1 c2)) (let H2 \def (eq_ind nat d (\lambda (n: nat).(le n O))
+H1 O H0) in (let H3 \def (eq_ind nat d (\lambda (n: nat).(drop h n c1 c2)) H
+O H0) in H3)) d (sym_eq nat d O H0))) | (le_S m H0) \Rightarrow (\lambda (H2:
+(eq nat (S m) O)).((let H1 \def (eq_ind nat (S m) (\lambda (e: nat).(match e
+in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
+\Rightarrow True])) I O H2) in (False_ind ((le d m) \to (drop (plus O h) O c1
+c2)) H1)) H0))]) in (H2 (refl_equal nat O))) e2 (drop_gen_refl c2 e2
+H0)))))))))) (\lambda (i0: nat).(\lambda (IHi: ((\forall (c1: C).(\forall
+(c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall
+(e2: C).((drop i0 O c2 e2) \to ((le d i0) \to (drop (plus i0 h) O c1
+e2))))))))))).(\lambda (c1: C).(C_ind (\lambda (c: C).(\forall (c2:
+C).(\forall (d: nat).(\forall (h: nat).((drop h d c c2) \to (\forall (e2:
+C).((drop (S i0) O c2 e2) \to ((le d (S i0)) \to (drop (plus (S i0) h) O c
+e2))))))))) (\lambda (n: nat).(\lambda (c2: C).(\lambda (d: nat).(\lambda (h:
+nat).(\lambda (H: (drop h d (CSort n) c2)).(\lambda (e2: C).(\lambda (H0:
+(drop (S i0) O c2 e2)).(\lambda (H1: (le d (S i0))).(and3_ind (eq C c2 (CSort
+n)) (eq nat h O) (eq nat d O) (drop (S (plus i0 h)) O (CSort n) e2) (\lambda
+(H2: (eq C c2 (CSort n))).(\lambda (H3: (eq nat h O)).(\lambda (H4: (eq nat d
+O)).(eq_ind_r nat O (\lambda (n0: nat).(drop (S (plus i0 n0)) O (CSort n)
+e2)) (let H5 \def (eq_ind nat d (\lambda (n: nat).(le n (S i0))) H1 O H4) in
+(let H6 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CSort n)
+H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop (S
+(plus i0 O)) O (CSort n) e2) (\lambda (H7: (eq C e2 (CSort n))).(\lambda (H8:
+(eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n) (\lambda
+(c: C).(drop (S (plus i0 O)) O (CSort n) c)) (let H10 \def (eq_ind nat (S i0)
+(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow False | (S _) \Rightarrow True])) I O H8) in (False_ind (drop (S
+(plus i0 O)) O (CSort n) (CSort n)) H10)) e2 H7)))) (drop_gen_sort n (S i0) O
+e2 H6)))) h H3)))) (drop_gen_sort n h d c2 H)))))))))) (\lambda (c2:
+C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h:
+nat).((drop h d c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le d
+(S i0)) \to (drop (S (plus i0 h)) O c2 e2)))))))))).(\lambda (k: K).(\lambda
+(t: T).(\lambda (c3: C).(\lambda (d: nat).(nat_ind (\lambda (n: nat).(\forall
+(h: nat).((drop h n (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O
+c3 e2) \to ((le n (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t)
+e2))))))) (\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c2 k
+t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le O (S i0)) \to
+(drop (S (plus i0 n)) O (CHead c2 k t) e2)))))) (\lambda (H: (drop O O (CHead
+c2 k t) c3)).(\lambda (e2: C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda
+(_: (le O (S i0))).(let H2 \def (eq_ind_r C c3 (\lambda (c: C).(drop (S i0) O
+c e2)) H0 (CHead c2 k t) (drop_gen_refl (CHead c2 k t) c3 H)) in (eq_ind nat
+i0 (\lambda (n: nat).(drop (S n) O (CHead c2 k t) e2)) (drop_drop k i0 c2 e2
+(drop_gen_drop k c2 e2 t i0 H2) t) (plus i0 O) (plus_n_O i0))))))) (\lambda
+(n: nat).(\lambda (_: (((drop n O (CHead c2 k t) c3) \to (\forall (e2:
+C).((drop (S i0) O c3 e2) \to ((le O (S i0)) \to (drop (S (plus i0 n)) O
+(CHead c2 k t) e2))))))).(\lambda (H0: (drop (S n) O (CHead c2 k t)
+c3)).(\lambda (e2: C).(\lambda (H1: (drop (S i0) O c3 e2)).(\lambda (H2: (le
+O (S i0))).(eq_ind nat (S (plus i0 n)) (\lambda (n0: nat).(drop (S n0) O
+(CHead c2 k t) e2)) (drop_drop k (S (plus i0 n)) c2 e2 (eq_ind_r nat (S (r k
+(plus i0 n))) (\lambda (n0: nat).(drop n0 O c2 e2)) (eq_ind_r nat (plus i0 (r
+k n)) (\lambda (n0: nat).(drop (S n0) O c2 e2)) (IHc c3 O (r k n)
+(drop_gen_drop k c2 c3 t n H0) e2 H1 H2) (r k (plus i0 n)) (r_plus_sym k i0
+n)) (r k (S (plus i0 n))) (r_S k (plus i0 n))) t) (plus i0 (S n)) (plus_n_Sm
+i0 n)))))))) h)) (\lambda (d0: nat).(\lambda (IHd: ((\forall (h: nat).((drop
+h d0 (CHead c2 k t) c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le
+d0 (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) e2)))))))).(\lambda (h:
+nat).(\lambda (H: (drop h (S d0) (CHead c2 k t) c3)).(\lambda (e2:
+C).(\lambda (H0: (drop (S i0) O c3 e2)).(\lambda (H1: (le (S d0) (S
+i0))).(ex3_2_ind C T (\lambda (e: C).(\lambda (v: T).(eq C c3 (CHead e k
+v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k d0) v)))) (\lambda
+(e: C).(\lambda (_: T).(drop h (r k d0) c2 e))) (drop (S (plus i0 h)) O
+(CHead c2 k t) e2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3
+(CHead x0 k x1))).(\lambda (H3: (eq T t (lift h (r k d0) x1))).(\lambda (H4:
+(drop h (r k d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: C).(\forall
+(h: nat).((drop h d0 (CHead c2 k t) c) \to (\forall (e2: C).((drop (S i0) O c
+e2) \to ((le d0 (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) e2)))))))
+IHd (CHead x0 k x1) H2) in (let H6 \def (eq_ind C c3 (\lambda (c: C).(drop (S
+i0) O c e2)) H0 (CHead x0 k x1) H2) in (let H7 \def (eq_ind T t (\lambda (t:
+T).(\forall (h: nat).((drop h d0 (CHead c2 k t) (CHead x0 k x1)) \to (\forall
+(e2: C).((drop (S i0) O (CHead x0 k x1) e2) \to ((le d0 (S i0)) \to (drop (S
+(plus i0 h)) O (CHead c2 k t) e2))))))) H5 (lift h (r k d0) x1) H3) in
+(eq_ind_r T (lift h (r k d0) x1) (\lambda (t0: T).(drop (S (plus i0 h)) O
+(CHead c2 k t0) e2)) (drop_drop k (plus i0 h) c2 e2 (K_ind (\lambda (k0:
+K).((drop h (r k0 d0) c2 x0) \to ((drop (r k0 i0) O x0 e2) \to (drop (r k0
+(plus i0 h)) O c2 e2)))) (\lambda (b: B).(\lambda (H8: (drop h (r (Bind b)
+d0) c2 x0)).(\lambda (H9: (drop (r (Bind b) i0) O x0 e2)).(IHi c2 x0 (r (Bind
+b) d0) h H8 e2 H9 (le_S_n (r (Bind b) d0) i0 H1))))) (\lambda (f: F).(\lambda
+(H8: (drop h (r (Flat f) d0) c2 x0)).(\lambda (H9: (drop (r (Flat f) i0) O x0
+e2)).(IHc x0 (r (Flat f) d0) h H8 e2 H9 H1)))) k H4 (drop_gen_drop k x0 e2 x1
+i0 H6)) (lift h (r k d0) x1)) t H3))))))))) (drop_gen_skip_l c2 c3 t h d0 k
+H))))))))) d))))))) c1)))) i).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift/defs".
+
+include "T/defs.ma".
+
+definition lref_map:
+ ((nat \to nat)) \to (nat \to (T \to T))
+\def
+ let rec lref_map (f: ((nat \to nat))) (d: nat) (t: T) on t: T \def (match t
+with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match
+(blt i d) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k u
+t0) \Rightarrow (THead k (lref_map f d u) (lref_map f (s k d) t0))]) in
+lref_map.
+
+definition lift:
+ nat \to (nat \to (T \to T))
+\def
+ \lambda (h: nat).(\lambda (i: nat).(\lambda (t: T).(lref_map (\lambda (x:
+nat).(plus x h)) i t))).
+
+definition lifts:
+ nat \to (nat \to (TList \to TList))
+\def
+ let rec lifts (h: nat) (d: nat) (ts: TList) on ts: TList \def (match ts with
+[TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (TCons (lift h d t) (lifts
+h d ts0))]) in lifts.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd".
+
+include "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_equal 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_equal 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 (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 (_:
+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)))).
+
+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 (t: T).(eq T (TLRef i) t)) 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 (t: T).(eq T (TLRef i) t)) 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
+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 |
+(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).(T_ind (\lambda (t0: T).((eq T (TLRef n) (lift h d t0))
+\to (eq T t0 (TLRef n)))) (\lambda (n0: nat).(\lambda (H0: (eq T (TLRef n)
+(lift h d (TSort n0)))).(sym_eq T (TLRef n) (TSort n0) H0))) (\lambda (n0:
+nat).(\lambda (H0: (eq T (TLRef n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq
+T (TLRef n0) (TLRef n)) (\lambda (H1: (lt n0 d)).(let H2 \def (eq_ind T (lift
+h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef n) t)) H0 (TLRef n0)
+(lift_lref_lt n0 h d H1)) in (sym_eq T (TLRef n) (TLRef n0) H2))) (\lambda
+(H1: (le d n0)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t:
+T).(eq T (TLRef n) t)) H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in
+(let H3 \def (match H2 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))))).
+
+theorem lift_gen_lref_false:
+ \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n
+(plus d h)) \to (\forall (t: T).((eq T (TLRef n) (lift h d t)) \to (\forall
+(P: Prop).P)))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d
+n)).(\lambda (H0: (lt n (plus d h))).(\lambda (t: T).(T_ind (\lambda (t0:
+T).((eq T (TLRef n) (lift h d t0)) \to (\forall (P: Prop).P))) (\lambda (n0:
+nat).(\lambda (H1: (eq T (TLRef n) (lift h d (TSort n0)))).(\lambda (P:
+Prop).(let H2 \def (match H1 in eq return (\lambda (t: T).(\lambda (_: (eq ?
+? t)).((eq T t (lift h d (TSort n0))) \to P))) with [refl_equal \Rightarrow
+(\lambda (H2: (eq T (TLRef n) (lift h d (TSort n0)))).(let H3 \def (eq_ind T
+(TLRef n) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _)
+\Rightarrow False])) I (lift h d (TSort n0)) H2) in (False_ind P H3)))]) in
+(H2 (refl_equal T (lift h d (TSort n0)))))))) (\lambda (n0: nat).(\lambda
+(H1: (eq T (TLRef n) (lift h d (TLRef n0)))).(\lambda (P: Prop).(lt_le_e n0 d
+P (\lambda (H2: (lt n0 d)).(let H3 \def (eq_ind T (lift h d (TLRef n0))
+(\lambda (t: T).(eq T (TLRef n) t)) H1 (TLRef n0) (lift_lref_lt n0 h d H2))
+in (let H4 \def (match H3 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:
+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
+(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)))))).
+
+theorem lift_gen_lref_ge:
+ \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to (\forall
+(t: T).((eq T (TLRef (plus n h)) (lift h d t)) \to (eq T t (TLRef n)))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (n: nat).(\lambda (H: (le d
+n)).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq T (TLRef (plus n h)) (lift h
+d t0)) \to (eq T t0 (TLRef n)))) (\lambda (n0: nat).(\lambda (H0: (eq T
+(TLRef (plus n h)) (lift h d (TSort n0)))).(let H1 \def (match H0 in eq
+return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (lift h d (TSort
+n0))) \to (eq T (TSort n0) (TLRef n))))) with [refl_equal \Rightarrow
+(\lambda (H1: (eq T (TLRef (plus n h)) (lift h d (TSort n0)))).(let H2 \def
+(eq_ind T (TLRef (plus n h)) (\lambda (e: T).(match e in T return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True |
+(THead _ _ _) \Rightarrow False])) I (lift h d (TSort n0)) H1) in (False_ind
+(eq T (TSort n0) (TLRef n)) H2)))]) in (H1 (refl_equal T (lift h d (TSort
+n0))))))) (\lambda (n0: nat).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d
+(TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TLRef n)) (\lambda (H1: (lt n0
+d)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef
+(plus n h)) t)) H0 (TLRef n0) (lift_lref_lt n0 h d H1)) in (let H3 \def
+(match H2 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))))).
+
+theorem lift_gen_head:
+ \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
+nat).(\forall (d: nat).((eq T (THead k u t) (lift h d x)) \to (ex3_2 T T
+(\lambda (y: T).(\lambda (z: T).(eq T x (THead k y z)))) (\lambda (y:
+T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h (s k d) z)))))))))))
+\def
+ \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind
+(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k u t)
+(lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead
+k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
+(_: T).(\lambda (z: T).(eq T t (lift h (s k d) z))))))))) (\lambda (n:
+nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k u t)
+(lift h d (TSort n)))).(let H0 \def (match H in eq return (\lambda (t0:
+T).(\lambda (_: (eq ? ? t0)).((eq T t0 (lift h d (TSort n))) \to (ex3_2 T T
+(\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead k y z)))) (\lambda (y:
+T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h (s k d) z)))))))) with [refl_equal \Rightarrow (\lambda
+(H0: (eq T (THead k u t) (lift h d (TSort n)))).(let H1 \def (eq_ind T (THead
+k u t) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (lift h d (TSort n)) H0) in (False_ind (ex3_2 T T
+(\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead k y z)))) (\lambda (y:
+T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h (s k d) z))))) H1)))]) in (H0 (refl_equal T (lift h d
+(TSort n))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H: (eq T (THead k u t) (lift h d (TLRef n)))).(lt_le_e n d
+(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z))))
+(\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T t (lift h (s k d) z))))) (\lambda (H0: (lt n
+d)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead
+k u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def (match H1 in
+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
+(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
+(\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y:
+T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h (s k d) z))))) H2)))]) in (H2 (refl_equal T (TLRef n))))))
+(\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda
+(t0: T).(eq T (THead k u t) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d
+H0)) in (let H2 \def (match H1 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:
+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) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u
+(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d)
+z))))) H2)))]) in (H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda
+(k0: K).(\lambda (t0: T).(\lambda (_: ((\forall (h: nat).(\forall (d:
+nat).((eq T (THead k u t) (lift h d t0)) \to (ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T t0 (THead k y z)))) (\lambda (y: T).(\lambda (_:
+T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s
+k d) z)))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (h: nat).(\forall
+(d: nat).((eq T (THead k u t) (lift h d t1)) \to (ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T t1 (THead k y z)))) (\lambda (y: T).(\lambda (_:
+T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s
+k d) z)))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T
+(THead k u t) (lift h d (THead k0 t0 t1)))).(let H2 \def (eq_ind T (lift h d
+(THead k0 t0 t1)) (\lambda (t0: T).(eq T (THead k u t) t0)) H1 (THead k0
+(lift h d t0) (lift h (s k0 d) t1)) (lift_head k0 t0 t1 h d)) in (let H3 \def
+(match H2 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
+(\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:
+T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h (s k d) z)))))))) (\lambda (H6: (eq T u (lift h d
+t0))).(eq_ind T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s k0 d) t1))
+\to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead
+k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda
+(_: T).(\lambda (z: T).(eq T t (lift h (s k0 d) z))))))) (\lambda (H7: (eq T
+t (lift h (s k0 d) t1))).(eq_ind T (lift h (s k0 d) t1) (\lambda (t:
+T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead
+k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k0 d) z))))))
+(ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1)
+(THead k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h
+d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k0 d) t1) (lift h (s
+k0 d) z)))) t0 t1 (refl_equal T (THead k0 t0 t1)) (refl_equal T (lift h d
+t0)) (refl_equal T (lift h (s k0 d) t1))) t (sym_eq T t (lift h (s k0 d) t1)
+H7))) u (sym_eq T u (lift h d t0) H6))) k (sym_eq K k k0 H5))) H4)) H3)))])
+in (H3 (refl_equal T (THead k0 (lift h d t0) (lift h (s k0 d)
+t1)))))))))))))) x)))).
+
+theorem lift_gen_bind:
+ \forall (b: B).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
+nat).(\forall (d: nat).((eq T (THead (Bind b) u t) (lift h d x)) \to (ex3_2 T
+T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda
+(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h (S d) z)))))))))))
+\def
+ \lambda (b: B).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind
+(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead (Bind b) u
+t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0
+(THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))))
+(\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T
+(THead (Bind b) u t) (lift h d (TSort n)))).(let H0 \def (match H in eq
+return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (lift h d (TSort
+n))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead
+(Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y))))
+(\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))) with
+[refl_equal \Rightarrow (\lambda (H0: (eq T (THead (Bind b) u t) (lift h d
+(TSort n)))).(let H1 \def (eq_ind T (THead (Bind b) u t) (\lambda (e:
+T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow
+False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I
+(lift h d (TSort n)) H0) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda
+(z: T).(eq T (TSort n) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_:
+T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S
+d) z))))) H1)))]) in (H0 (refl_equal T (lift h d (TSort n))))))))) (\lambda
+(n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind
+b) u t) (lift h d (TLRef n)))).(lt_le_e n d (ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z)))) (\lambda (y:
+T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h (S d) z))))) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind
+T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0)) H
+(TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def (match H1 in 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 (Bind b) y z))))
+(\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))) with [refl_equal
+\Rightarrow (\lambda (H1: (eq T (THead (Bind b) u t) (TLRef n))).(let H2 \def
+(eq_ind T (THead (Bind b) u t) (\lambda (e: T).(match e 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 (Bind b) y z))))
+(\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H2)))]) in (H2 (refl_equal T
+(TLRef n)))))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d
+(TLRef n)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0)) H (TLRef (plus n
+h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 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 (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
+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 (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H2)))]) in
+(H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda (k: K).(\lambda (t0:
+T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead (Bind b) u
+t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0
+(THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d)
+z)))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (h: nat).(\forall (d:
+nat).((eq T (THead (Bind b) u t) (lift h d t1)) \to (ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T t1 (THead (Bind b) y z)))) (\lambda (y: T).(\lambda
+(_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift
+h (S d) z)))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T
+(THead (Bind b) u t) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T
+(lift h d (THead k t0 t1)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0))
+H1 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in
+(let H3 \def (match H2 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
+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)))).
+
+theorem lift_gen_flat:
+ \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h:
+nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d x)) \to (ex3_2 T
+T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda
+(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h d z)))))))))))
+\def
+ \lambda (f: F).(\lambda (u: T).(\lambda (t: T).(\lambda (x: T).(T_ind
+(\lambda (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead (Flat f) u
+t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0
+(THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))))))) (\lambda
+(n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T (THead (Flat
+f) u t) (lift h d (TSort n)))).(let H0 \def (match H in eq return (\lambda
+(t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (lift h d (TSort n))) \to (ex3_2
+T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead (Flat f) y z))))
+(\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T t (lift h d z)))))))) with [refl_equal \Rightarrow
+(\lambda (H0: (eq T (THead (Flat f) u t) (lift h d (TSort n)))).(let H1 \def
+(eq_ind T (THead (Flat f) u t) (\lambda (e: T).(match e in T return (\lambda
+(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False
+| (THead _ _ _) \Rightarrow True])) I (lift h d (TSort n)) H0) in (False_ind
+(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TSort n) (THead (Flat f) y
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T t (lift h d z))))) H1)))]) in (H0 (refl_equal T
+(lift h d (TSort n))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H: (eq T (THead (Flat f) u t) (lift h d (TLRef n)))).(lt_le_e
+n d (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat
+f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
+(_: T).(\lambda (z: T).(eq T t (lift h d z))))) (\lambda (H0: (lt n d)).(let
+H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead (Flat f)
+u t) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def (match H1 in
+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
+(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
+(\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y z))))
+(\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T t (lift h d z))))) H2)))]) in (H2 (refl_equal T
+(TLRef n)))))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d
+(TLRef n)) (\lambda (t0: T).(eq T (THead (Flat f) u t) t0)) H (TLRef (plus n
+h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 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 (Flat f) y z))))
+(\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T t (lift h d z)))))))) with [refl_equal \Rightarrow
+(\lambda (H1: (eq T (THead (Flat f) u t) (TLRef (plus n h)))).(let H2 \def
+(eq_ind T (THead (Flat f) u t) (\lambda (e: T).(match e 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 (Flat f) y
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T t (lift h d z))))) H2)))]) in (H2 (refl_equal T
+(TLRef (plus n h)))))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_:
+((\forall (h: nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d
+t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Flat f)
+y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_:
+T).(\lambda (z: T).(eq T t (lift h d z)))))))))).(\lambda (t1: T).(\lambda
+(_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h
+d t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Flat
+f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda
+(_: T).(\lambda (z: T).(eq T t (lift h d z)))))))))).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead (Flat f) u t) (lift h d
+(THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda
+(t0: T).(eq T (THead (Flat f) u t) t0)) H1 (THead k (lift h d t0) (lift h (s
+k d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def (match H2 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:
+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)
+with [(TSort _) \Rightarrow (Flat f) | (TLRef _) \Rightarrow (Flat f) |
+(THead k _ _) \Rightarrow k])) (THead (Flat f) u t) (THead k (lift h d t0)
+(lift h (s k d) t1)) H2) in (eq_ind K (Flat f) (\lambda (k: K).((eq T u (lift
+h d t0)) \to ((eq T t (lift h (s k d) t1)) \to (ex3_2 T T (\lambda (y:
+T).(\lambda (z: T).(eq T (THead k t0 t1) (THead (Flat f) y z)))) (\lambda (y:
+T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z:
+T).(eq T t (lift h d z)))))))) (\lambda (H6: (eq T u (lift h d t0))).(eq_ind
+T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s (Flat f) d) t1)) \to
+(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) t0 t1)
+(THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d
+y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))))) (\lambda
+(H7: (eq T t (lift h (s (Flat f) d) t1))).(eq_ind T (lift h (s (Flat f) d)
+t1) (\lambda (t: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead
+(Flat f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T
+(lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h
+d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat
+f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift
+h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s
+(Flat f) d) t1) (lift h d z)))) t0 t1 (refl_equal T (THead (Flat f) t0 t1))
+(refl_equal T (lift h d t0)) (refl_equal T (lift h d t1))) t (sym_eq T t
+(lift h (s (Flat f) d) t1) H7))) u (sym_eq T u (lift h d t0) H6))) k H5))
+H4)) H3)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h (s k d)
+t1)))))))))))))) x)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift/props".
+
+include "lift/fwd.ma".
+
+theorem thead_x_lift_y_y:
+ \forall (k: K).(\forall (t: T).(\forall (v: T).(\forall (h: nat).(\forall
+(d: nat).((eq T (THead k v (lift h d t)) t) \to (\forall (P: Prop).P))))))
+\def
+ \lambda (k: K).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (v:
+T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift h d t0)) t0)
+\to (\forall (P: Prop).P)))))) (\lambda (n: nat).(\lambda (v: T).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H: (eq T (THead k v (lift h d (TSort n)))
+(TSort n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead k v (lift h d
+(TSort n))) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with
+[(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _)
+\Rightarrow True])) I (TSort n) H) in (False_ind P H0)))))))) (\lambda (n:
+nat).(\lambda (v: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (eq T
+(THead k v (lift h d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def
+(eq_ind T (THead k v (lift h d (TLRef n))) (\lambda (ee: T).(match ee in T
+return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _)
+\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in
+(False_ind P H0)))))))) (\lambda (k0: K).(\lambda (t0: T).(\lambda (_:
+((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift
+h d t0)) t0) \to (\forall (P: Prop).P))))))).(\lambda (t1: T).(\lambda (H0:
+((\forall (v: T).(\forall (h: nat).(\forall (d: nat).((eq T (THead k v (lift
+h d t1)) t1) \to (\forall (P: Prop).P))))))).(\lambda (v: T).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k v (lift h d (THead k0 t0
+t1))) (THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \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 v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1) in ((let H3 \def
+(f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with
+[(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t _)
+\Rightarrow t])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1)
+in ((let H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda
+(_: T).T) with [(TSort _) \Rightarrow (THead k0 ((let rec lref_map (f: ((nat
+\to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow
+(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true
+\Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow
+(THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda
+(x: nat).(plus x h)) d t0) ((let rec lref_map (f: ((nat \to nat))) (d: nat)
+(t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef
+i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false
+\Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u)
+(lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x h)) (s k0
+d) t1)) | (TLRef _) \Rightarrow (THead k0 ((let rec lref_map (f: ((nat \to
+nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow
+(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true
+\Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow
+(THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda
+(x: nat).(plus x h)) d t0) ((let rec lref_map (f: ((nat \to nat))) (d: nat)
+(t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef
+i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false
+\Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u)
+(lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x h)) (s k0
+d) t1)) | (THead _ _ t) \Rightarrow t])) (THead k v (lift h d (THead k0 t0
+t1))) (THead k0 t0 t1) H1) in (\lambda (_: (eq T v t0)).(\lambda (H6: (eq K k
+k0)).(let H7 \def (eq_ind K k (\lambda (k: K).(\forall (v: T).(\forall (h:
+nat).(\forall (d: nat).((eq T (THead k v (lift h d t1)) t1) \to (\forall (P:
+Prop).P)))))) H0 k0 H6) in (let H8 \def (eq_ind T (lift h d (THead k0 t0 t1))
+(\lambda (t: T).(eq T t t1)) H4 (THead k0 (lift h d t0) (lift h (s k0 d) t1))
+(lift_head k0 t0 t1 h d)) in (H7 (lift h d t0) h (s k0 d) H8 P)))))) H3))
+H2)))))))))))) t)).
+
+theorem lift_r:
+ \forall (t: T).(\forall (d: nat).(eq T (lift O d t) t))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (d: nat).(eq T (lift O d t0)
+t0))) (\lambda (n: nat).(\lambda (_: nat).(refl_equal T (TSort n)))) (\lambda
+(n: nat).(\lambda (d: nat).(lt_le_e n d (eq T (lift O d (TLRef n)) (TLRef n))
+(\lambda (H: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (TLRef
+n))) (refl_equal T (TLRef n)) (lift O d (TLRef n)) (lift_lref_lt n O d H)))
+(\lambda (H: (le d n)).(eq_ind_r T (TLRef (plus n O)) (\lambda (t0: T).(eq T
+t0 (TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O)
+(plus_n_O n))) (lift O d (TLRef n)) (lift_lref_ge n O d H)))))) (\lambda (k:
+K).(\lambda (t0: T).(\lambda (H: ((\forall (d: nat).(eq T (lift O d t0)
+t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (lift O d t1)
+t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift O d t0) (lift O (s k d)
+t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (sym_equal T (THead k t0 t1)
+(THead k (lift O d t0) (lift O (s k d) t1)) (sym_equal T (THead k (lift O d
+t0) (lift O (s k d) t1)) (THead k t0 t1) (sym_equal T (THead k t0 t1) (THead
+k (lift O d t0) (lift O (s k d) t1)) (f_equal3 K T T T THead k k t0 (lift O d
+t0) t1 (lift O (s k d) t1) (refl_equal K k) (sym_eq T (lift O d t0) t0 (H d))
+(sym_eq T (lift O (s k d) t1) t1 (H0 (s k d))))))) (lift O d (THead k t0 t1))
+(lift_head k t0 t1 O d)))))))) t).
+
+theorem lift_lref_gt:
+ \forall (d: nat).(\forall (n: nat).((lt d n) \to (eq T (lift (S O) d (TLRef
+(pred n))) (TLRef n))))
+\def
+ \lambda (d: nat).(\lambda (n: nat).(\lambda (H: (lt d n)).(eq_ind_r T (TLRef
+(plus (pred n) (S O))) (\lambda (t: T).(eq T t (TLRef n))) (eq_ind nat (plus
+(S O) (pred n)) (\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (eq_ind nat n
+(\lambda (n0: nat).(eq T (TLRef n0) (TLRef n))) (refl_equal T (TLRef n)) (S
+(pred n)) (S_pred n d H)) (plus (pred n) (S O)) (plus_comm (S O) (pred n)))
+(lift (S O) d (TLRef (pred n))) (lift_lref_ge (pred n) (S O) d (le_S_n d
+(pred n) (eq_ind nat n (\lambda (n0: nat).(le (S d) n0)) H (S (pred n))
+(S_pred n d H))))))).
+
+theorem lift_inj:
+ \forall (x: T).(\forall (t: T).(\forall (h: nat).(\forall (d: nat).((eq T
+(lift h d x) (lift h d t)) \to (eq T x t)))))
+\def
+ \lambda (x: T).(T_ind (\lambda (t: T).(\forall (t0: T).(\forall (h:
+nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to (eq T t
+t0)))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H: (eq T (lift h d (TSort n)) (lift h d t))).(let H0 \def
+(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H
+(TSort n) (lift_sort n h d)) in (sym_eq T t (TSort n) (lift_gen_sort h d n t
+H0)))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H: (eq T (lift h d (TLRef n)) (lift h d t))).(lt_le_e n d (eq
+T (TLRef n) t) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind T (lift h d
+(TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef n) (lift_lref_lt
+n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_lt h d n (lt_le_trans n d
+d H0 (le_n d)) t H1)))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift
+h d (TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef (plus n h))
+(lift_lref_ge n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_ge h d n H0
+t H1)))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t:
+T).(((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t)
+(lift h d t0)) \to (eq T t t0)))))) \to (\forall (t0: T).(((\forall (t:
+T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t)) \to
+(eq T t0 t)))))) \to (\forall (t1: T).(\forall (h: nat).(\forall (d:
+nat).((eq T (lift h d (THead k0 t t0)) (lift h d t1)) \to (eq T (THead k0 t
+t0) t1)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H: ((\forall (t0:
+T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to
+(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t: T).(\forall
+(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t)) \to (eq T t0
+t))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1:
+(eq T (lift h d (THead (Bind b) t t0)) (lift h d t1))).(let H2 \def (eq_ind T
+(lift h d (THead (Bind b) t t0)) (\lambda (t: T).(eq T t (lift h d t1))) H1
+(THead (Bind b) (lift h d t) (lift h (S d) t0)) (lift_bind b t t0 h d)) in
+(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Bind b) y
+z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y))))
+(\lambda (_: T).(\lambda (z: T).(eq T (lift h (S d) t0) (lift h (S d) z))))
+(eq T (THead (Bind b) t t0) t1) (\lambda (x0: T).(\lambda (x1: T).(\lambda
+(H3: (eq T t1 (THead (Bind b) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift
+h d x0))).(\lambda (H5: (eq T (lift h (S d) t0) (lift h (S d) x1))).(eq_ind_r
+T (THead (Bind b) x0 x1) (\lambda (t2: T).(eq T (THead (Bind b) t t0) t2))
+(sym_equal T (THead (Bind b) x0 x1) (THead (Bind b) t t0) (sym_equal T (THead
+(Bind b) t t0) (THead (Bind b) x0 x1) (sym_equal T (THead (Bind b) x0 x1)
+(THead (Bind b) t t0) (f_equal3 K T T T THead (Bind b) (Bind b) x0 t x1 t0
+(refl_equal K (Bind b)) (sym_eq T t x0 (H x0 h d H4)) (sym_eq T t0 x1 (H0 x1
+h (S d) H5)))))) t1 H3)))))) (lift_gen_bind b (lift h d t) (lift h (S d) t0)
+t1 h d H2)))))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (H: ((\forall
+(t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d
+t0)) \to (eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t:
+T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t)) \to
+(eq T t0 t))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d:
+nat).(\lambda (H1: (eq T (lift h d (THead (Flat f) t t0)) (lift h d
+t1))).(let H2 \def (eq_ind T (lift h d (THead (Flat f) t t0)) (\lambda (t:
+T).(eq T t (lift h d t1))) H1 (THead (Flat f) (lift h d t) (lift h d t0))
+(lift_flat f t t0 h d)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq
+T t1 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d
+t) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h d t0) (lift
+h d z)))) (eq T (THead (Flat f) t t0) t1) (\lambda (x0: T).(\lambda (x1:
+T).(\lambda (H3: (eq T t1 (THead (Flat f) x0 x1))).(\lambda (H4: (eq T (lift
+h d t) (lift h d x0))).(\lambda (H5: (eq T (lift h d t0) (lift h d
+x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t2: T).(eq T (THead (Flat
+f) t t0) t2)) (sym_equal T (THead (Flat f) x0 x1) (THead (Flat f) t t0)
+(sym_equal T (THead (Flat f) t t0) (THead (Flat f) x0 x1) (sym_equal T (THead
+(Flat f) x0 x1) (THead (Flat f) t t0) (f_equal3 K T T T THead (Flat f) (Flat
+f) x0 t x1 t0 (refl_equal K (Flat f)) (sym_eq T t x0 (H x0 h d H4)) (sym_eq T
+t0 x1 (H0 x1 h d H5)))))) t1 H3)))))) (lift_gen_flat f (lift h d t) (lift h d
+t0) t1 h d H2)))))))))))) k)) x).
+
+theorem lift_gen_lift:
+ \forall (t1: T).(\forall (x: T).(\forall (h1: nat).(\forall (h2:
+nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1
+t1) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1
+t2))) (\lambda (t2: T).(eq T t1 (lift h2 d2 t2)))))))))))
+\def
+ \lambda (t1: T).(T_ind (\lambda (t: T).(\forall (x: T).(\forall (h1:
+nat).(\forall (h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to
+((eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2:
+T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2
+t2)))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda (h1: nat).(\lambda
+(h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda (_: (le d1
+d2)).(\lambda (H0: (eq T (lift h1 d1 (TSort n)) (lift h2 (plus d2 h1)
+x))).(let H1 \def (eq_ind T (lift h1 d1 (TSort n)) (\lambda (t: T).(eq T t
+(lift h2 (plus d2 h1) x))) H0 (TSort n) (lift_sort n h1 d1)) in (eq_ind_r T
+(TSort n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h1 d1 t2)))
+(\lambda (t2: T).(eq T (TSort n) (lift h2 d2 t2))))) (ex_intro2 T (\lambda
+(t2: T).(eq T (TSort n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TSort n)
+(lift h2 d2 t2))) (TSort n) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T
+(TSort n) t)) (refl_equal T (TSort n)) (lift h1 d1 (TSort n)) (lift_sort n h1
+d1)) (eq_ind_r T (TSort n) (\lambda (t: T).(eq T (TSort n) t)) (refl_equal T
+(TSort n)) (lift h2 d2 (TSort n)) (lift_sort n h2 d2))) x (lift_gen_sort h2
+(plus d2 h1) n x H1))))))))))) (\lambda (n: nat).(\lambda (x: T).(\lambda
+(h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2: nat).(\lambda
+(H: (le d1 d2)).(\lambda (H0: (eq T (lift h1 d1 (TLRef n)) (lift h2 (plus d2
+h1) x))).(lt_le_e n d1 (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2)))
+(\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))) (\lambda (H1: (lt n
+d1)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n)) (\lambda (t: T).(eq T t
+(lift h2 (plus d2 h1) x))) H0 (TLRef n) (lift_lref_lt n h1 d1 H1)) in
+(eq_ind_r T (TLRef n) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift
+h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) (ex_intro2 T
+(\lambda (t2: T).(eq T (TLRef n) (lift h1 d1 t2))) (\lambda (t2: T).(eq T
+(TLRef n) (lift h2 d2 t2))) (TLRef n) (eq_ind_r T (TLRef n) (\lambda (t:
+T).(eq T (TLRef n) t)) (refl_equal T (TLRef n)) (lift h1 d1 (TLRef n))
+(lift_lref_lt n h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef
+n) t)) (refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2
+(lt_le_trans n d1 d2 H1 H)))) x (lift_gen_lref_lt h2 (plus d2 h1) n
+(lt_le_trans n d1 (plus d2 h1) H1 (le_plus_trans d1 d2 h1 H)) x H2))))
+(\lambda (H1: (le d1 n)).(let H2 \def (eq_ind T (lift h1 d1 (TLRef n))
+(\lambda (t: T).(eq T t (lift h2 (plus d2 h1) x))) H0 (TLRef (plus n h1))
+(lift_lref_ge n h1 d1 H1)) in (lt_le_e n d2 (ex2 T (\lambda (t2: T).(eq T x
+(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))))
+(\lambda (H3: (lt n d2)).(eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(ex2
+T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n)
+(lift h2 d2 t2))))) (ex_intro2 T (\lambda (t2: T).(eq T (TLRef (plus n h1))
+(lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2))) (TLRef
+n) (eq_ind_r T (TLRef (plus n h1)) (\lambda (t: T).(eq T (TLRef (plus n h1))
+t)) (refl_equal T (TLRef (plus n h1))) (lift h1 d1 (TLRef n)) (lift_lref_ge n
+h1 d1 H1)) (eq_ind_r T (TLRef n) (\lambda (t: T).(eq T (TLRef n) t))
+(refl_equal T (TLRef n)) (lift h2 d2 (TLRef n)) (lift_lref_lt n h2 d2 H3))) x
+(lift_gen_lref_lt h2 (plus d2 h1) (plus n h1) (plus_lt_compat_r n d2 h1 H3) x
+H2))) (\lambda (H3: (le d2 n)).(lt_le_e n (plus d2 h2) (ex2 T (\lambda (t2:
+T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2
+t2)))) (\lambda (H4: (lt n (plus d2 h2))).(lift_gen_lref_false h2 (plus d2
+h1) (plus n h1) (le_S_n (plus d2 h1) (plus n h1) (lt_le_S (plus d2 h1) (S
+(plus n h1)) (le_lt_n_Sm (plus d2 h1) (plus n h1) (plus_le_compat d2 n h1 h1
+H3 (le_n h1))))) (eq_ind_r nat (plus (plus d2 h2) h1) (\lambda (n0: nat).(lt
+(plus n h1) n0)) (lt_le_S (plus n h1) (plus (plus d2 h2) h1)
+(plus_lt_compat_r n (plus d2 h2) h1 H4)) (plus (plus d2 h1) h2)
+(plus_permute_2_in_3 d2 h1 h2)) x H2 (ex2 T (\lambda (t2: T).(eq T x (lift h1
+d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))))) (\lambda (H4:
+(le (plus d2 h2) n)).(let H5 \def (eq_ind nat (plus n h1) (\lambda (n:
+nat).(eq T (TLRef n) (lift h2 (plus d2 h1) x))) H2 (plus (minus (plus n h1)
+h2) h2) (le_plus_minus_sym h2 (plus n h1) (le_plus_trans h2 n h1
+(le_trans_plus_r d2 h2 n H4)))) in (eq_ind_r T (TLRef (minus (plus n h1) h2))
+(\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda
+(t2: T).(eq T (TLRef n) (lift h2 d2 t2))))) (ex_intro2 T (\lambda (t2: T).(eq
+T (TLRef (minus (plus n h1) h2)) (lift h1 d1 t2))) (\lambda (t2: T).(eq T
+(TLRef n) (lift h2 d2 t2))) (TLRef (minus n h2)) (eq_ind_r nat (plus (minus n
+h2) h1) (\lambda (n0: nat).(eq T (TLRef n0) (lift h1 d1 (TLRef (minus n
+h2))))) (eq_ind_r T (TLRef (plus (minus n h2) h1)) (\lambda (t: T).(eq T
+(TLRef (plus (minus n h2) h1)) t)) (refl_equal T (TLRef (plus (minus n h2)
+h1))) (lift h1 d1 (TLRef (minus n h2))) (lift_lref_ge (minus n h2) h1 d1
+(le_trans d1 d2 (minus n h2) H (le_minus d2 n h2 H4)))) (minus (plus n h1)
+h2) (le_minus_plus h2 n (le_trans_plus_r d2 h2 n H4) h1)) (eq_ind_r nat (plus
+(minus n h2) h2) (\lambda (n0: nat).(eq T (TLRef n0) (lift h2 d2 (TLRef
+(minus n0 h2))))) (eq_ind_r T (TLRef (plus (minus (plus (minus n h2) h2) h2)
+h2)) (\lambda (t: T).(eq T (TLRef (plus (minus n h2) h2)) t)) (f_equal nat T
+TLRef (plus (minus n h2) h2) (plus (minus (plus (minus n h2) h2) h2) h2)
+(f_equal2 nat nat nat plus (minus n h2) (minus (plus (minus n h2) h2) h2) h2
+h2 (sym_eq nat (minus (plus (minus n h2) h2) h2) (minus n h2) (minus_plus_r
+(minus n h2) h2)) (refl_equal nat h2))) (lift h2 d2 (TLRef (minus (plus
+(minus n h2) h2) h2))) (lift_lref_ge (minus (plus (minus n h2) h2) h2) h2 d2
+(le_minus d2 (plus (minus n h2) h2) h2 (plus_le_compat d2 (minus n h2) h2 h2
+(le_minus d2 n h2 H4) (le_n h2))))) n (le_plus_minus_sym h2 n
+(le_trans_plus_r d2 h2 n H4)))) x (lift_gen_lref_ge h2 (plus d2 h1) (minus
+(plus n h1) h2) (arith0 h2 d2 n H4 h1) x H5)))))))))))))))))) (\lambda (k:
+K).(\lambda (t: T).(\lambda (H: ((\forall (x: T).(\forall (h1: nat).(\forall
+(h2: nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift
+h1 d1 t) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift
+h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2 d2 t2))))))))))))).(\lambda
+(t0: T).(\lambda (H0: ((\forall (x: T).(\forall (h1: nat).(\forall (h2:
+nat).(\forall (d1: nat).(\forall (d2: nat).((le d1 d2) \to ((eq T (lift h1 d1
+t0) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1
+t2))) (\lambda (t2: T).(eq T t0 (lift h2 d2 t2))))))))))))).(\lambda (x:
+T).(\lambda (h1: nat).(\lambda (h2: nat).(\lambda (d1: nat).(\lambda (d2:
+nat).(\lambda (H1: (le d1 d2)).(\lambda (H2: (eq T (lift h1 d1 (THead k t
+t0)) (lift h2 (plus d2 h1) x))).(K_ind (\lambda (k0: K).((eq T (lift h1 d1
+(THead k0 t t0)) (lift h2 (plus d2 h1) x)) \to (ex2 T (\lambda (t2: T).(eq T
+x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead k0 t t0) (lift h2 d2
+t2)))))) (\lambda (b: B).(\lambda (H3: (eq T (lift h1 d1 (THead (Bind b) t
+t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind T (lift h1 d1 (THead
+(Bind b) t t0)) (\lambda (t: T).(eq T t (lift h2 (plus d2 h1) x))) H3 (THead
+(Bind b) (lift h1 d1 t) (lift h1 (S d1) t0)) (lift_bind b t t0 h1 d1)) 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 (lift h1 d1 t) (lift h2 (plus d2
+h1) y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h1 (S d1) t0) (lift h2
+(S (plus d2 h1)) z)))) (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2)))
+(\lambda (t2: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t2)))) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H5: (eq T x (THead (Bind b) x0 x1))).(\lambda
+(H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x0))).(\lambda (H7: (eq T
+(lift h1 (S d1) t0) (lift h2 (S (plus d2 h1)) x1))).(eq_ind_r T (THead (Bind
+b) x0 x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h1 d1 t3)))
+(\lambda (t3: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t3))))) (ex2_ind T
+(\lambda (t2: T).(eq T x0 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2
+d2 t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Bind b) x0 x1) (lift h1 d1
+t2))) (\lambda (t2: T).(eq T (THead (Bind b) t t0) (lift h2 d2 t2))))
+(\lambda (x2: T).(\lambda (H8: (eq T x0 (lift h1 d1 x2))).(\lambda (H9: (eq T
+t (lift h2 d2 x2))).(eq_ind_r T (lift h1 d1 x2) (\lambda (t2: T).(ex2 T
+(\lambda (t3: T).(eq T (THead (Bind b) t2 x1) (lift h1 d1 t3))) (\lambda (t3:
+T).(eq T (THead (Bind b) t t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 d2
+x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift h1
+d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) t2 t0)
+(lift h2 d2 t3))))) (let H10 \def (refl_equal nat (plus (S d2) h1)) in (let
+H11 \def (eq_ind nat (S (plus d2 h1)) (\lambda (n: nat).(eq T (lift h1 (S d1)
+t0) (lift h2 n x1))) H7 (plus (S d2) h1) H10) in (ex2_ind T (\lambda (t2:
+T).(eq T x1 (lift h1 (S d1) t2))) (\lambda (t2: T).(eq T t0 (lift h2 (S d2)
+t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2) x1) (lift
+h1 d1 t2))) (\lambda (t2: T).(eq T (THead (Bind b) (lift h2 d2 x2) t0) (lift
+h2 d2 t2)))) (\lambda (x3: T).(\lambda (H12: (eq T x1 (lift h1 (S d1)
+x3))).(\lambda (H13: (eq T t0 (lift h2 (S d2) x3))).(eq_ind_r T (lift h1 (S
+d1) x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift
+h1 d1 x2) t2) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift
+h2 d2 x2) t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 (S d2) x3) (\lambda
+(t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Bind b) (lift h1 d1 x2) (lift
+h1 (S d1) x3)) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Bind b) (lift
+h2 d2 x2) t2) (lift h2 d2 t3))))) (ex_intro2 T (\lambda (t2: T).(eq T (THead
+(Bind b) (lift h1 d1 x2) (lift h1 (S d1) x3)) (lift h1 d1 t2))) (\lambda (t2:
+T).(eq T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)) (lift h2 d2
+t2))) (THead (Bind b) x2 x3) (eq_ind_r T (THead (Bind b) (lift h1 d1 x2)
+(lift h1 (S d1) x3)) (\lambda (t2: T).(eq T (THead (Bind b) (lift h1 d1 x2)
+(lift h1 (S d1) x3)) t2)) (refl_equal T (THead (Bind b) (lift h1 d1 x2) (lift
+h1 (S d1) x3))) (lift h1 d1 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h1
+d1)) (eq_ind_r T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3))
+(\lambda (t2: T).(eq T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3))
+t2)) (refl_equal T (THead (Bind b) (lift h2 d2 x2) (lift h2 (S d2) x3)))
+(lift h2 d2 (THead (Bind b) x2 x3)) (lift_bind b x2 x3 h2 d2))) t0 H13) x1
+H12)))) (H0 x1 h1 h2 (S d1) (S d2) (le_S_n (S d1) (S d2) (lt_le_S (S d1) (S
+(S d2)) (lt_n_S d1 (S d2) (le_lt_n_Sm d1 d2 H1)))) H11)))) t H9) x0 H8)))) (H
+x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_bind b (lift h1 d1 t) (lift h1 (S
+d1) t0) x h2 (plus d2 h1) H4))))) (\lambda (f: F).(\lambda (H3: (eq T (lift
+h1 d1 (THead (Flat f) t t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind
+T (lift h1 d1 (THead (Flat f) t t0)) (\lambda (t: T).(eq T t (lift h2 (plus
+d2 h1) x))) H3 (THead (Flat f) (lift h1 d1 t) (lift h1 d1 t0)) (lift_flat f t
+t0 h1 d1)) 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 (lift h1 d1 t) (lift
+h2 (plus d2 h1) y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h1 d1 t0)
+(lift h2 (plus d2 h1) z)))) (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2)))
+(\lambda (t2: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t2)))) (\lambda (x0:
+T).(\lambda (x1: T).(\lambda (H5: (eq T x (THead (Flat f) x0 x1))).(\lambda
+(H6: (eq T (lift h1 d1 t) (lift h2 (plus d2 h1) x0))).(\lambda (H7: (eq T
+(lift h1 d1 t0) (lift h2 (plus d2 h1) x1))).(eq_ind_r T (THead (Flat f) x0
+x1) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T t2 (lift h1 d1 t3)))
+(\lambda (t3: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t3))))) (ex2_ind T
+(\lambda (t2: T).(eq T x0 (lift h1 d1 t2))) (\lambda (t2: T).(eq T t (lift h2
+d2 t2))) (ex2 T (\lambda (t2: T).(eq T (THead (Flat f) x0 x1) (lift h1 d1
+t2))) (\lambda (t2: T).(eq T (THead (Flat f) t t0) (lift h2 d2 t2))))
+(\lambda (x2: T).(\lambda (H8: (eq T x0 (lift h1 d1 x2))).(\lambda (H9: (eq T
+t (lift h2 d2 x2))).(eq_ind_r T (lift h1 d1 x2) (\lambda (t2: T).(ex2 T
+(\lambda (t3: T).(eq T (THead (Flat f) t2 x1) (lift h1 d1 t3))) (\lambda (t3:
+T).(eq T (THead (Flat f) t t0) (lift h2 d2 t3))))) (eq_ind_r T (lift h2 d2
+x2) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat f) (lift h1
+d1 x2) x1) (lift h1 d1 t3))) (\lambda (t3: T).(eq T (THead (Flat f) t2 t0)
+(lift h2 d2 t3))))) (ex2_ind T (\lambda (t2: T).(eq T x1 (lift h1 d1 t2)))
+(\lambda (t2: T).(eq T t0 (lift h2 d2 t2))) (ex2 T (\lambda (t2: T).(eq T
+(THead (Flat f) (lift h1 d1 x2) x1) (lift h1 d1 t2))) (\lambda (t2: T).(eq T
+(THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2 t2)))) (\lambda (x3:
+T).(\lambda (H10: (eq T x1 (lift h1 d1 x3))).(\lambda (H11: (eq T t0 (lift h2
+d2 x3))).(eq_ind_r T (lift h1 d1 x3) (\lambda (t2: T).(ex2 T (\lambda (t3:
+T).(eq T (THead (Flat f) (lift h1 d1 x2) t2) (lift h1 d1 t3))) (\lambda (t3:
+T).(eq T (THead (Flat f) (lift h2 d2 x2) t0) (lift h2 d2 t3))))) (eq_ind_r T
+(lift h2 d2 x3) (\lambda (t2: T).(ex2 T (\lambda (t3: T).(eq T (THead (Flat
+f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1 t3))) (\lambda (t3: T).(eq T
+(THead (Flat f) (lift h2 d2 x2) t2) (lift h2 d2 t3))))) (ex_intro2 T (\lambda
+(t2: T).(eq T (THead (Flat f) (lift h1 d1 x2) (lift h1 d1 x3)) (lift h1 d1
+t2))) (\lambda (t2: T).(eq T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3))
+(lift h2 d2 t2))) (THead (Flat f) x2 x3) (eq_ind_r T (THead (Flat f) (lift h1
+d1 x2) (lift h1 d1 x3)) (\lambda (t2: T).(eq T (THead (Flat f) (lift h1 d1
+x2) (lift h1 d1 x3)) t2)) (refl_equal T (THead (Flat f) (lift h1 d1 x2) (lift
+h1 d1 x3))) (lift h1 d1 (THead (Flat f) x2 x3)) (lift_flat f x2 x3 h1 d1))
+(eq_ind_r T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3)) (\lambda (t2:
+T).(eq T (THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3)) t2)) (refl_equal T
+(THead (Flat f) (lift h2 d2 x2) (lift h2 d2 x3))) (lift h2 d2 (THead (Flat f)
+x2 x3)) (lift_flat f x2 x3 h2 d2))) t0 H11) x1 H10)))) (H0 x1 h1 h2 d1 d2 H1
+H7)) t H9) x0 H8)))) (H x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_flat f
+(lift h1 d1 t) (lift h1 d1 t0) x h2 (plus d2 h1) H4))))) k H2)))))))))))))
+t1).
+
+theorem lift_free:
+ \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d:
+nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e
+(lift h d t)) (lift (plus k h) d t))))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k:
+nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to
+(eq T (lift k e (lift h d t0)) (lift (plus k h) d t0))))))))) (\lambda (n:
+nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e:
+nat).(\lambda (_: (le e (plus d h))).(\lambda (_: (le d e)).(eq_ind_r T
+(TSort n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TSort
+n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d
+(TSort n)))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (TSort n) t0))
+(refl_equal T (TSort n)) (lift (plus k h) d (TSort n)) (lift_sort n (plus k
+h) d)) (lift k e (TSort n)) (lift_sort n k e)) (lift h d (TSort n))
+(lift_sort n h d))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (k:
+nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H: (le e (plus d
+h))).(\lambda (H0: (le d e)).(lt_le_e n d (eq T (lift k e (lift h d (TLRef
+n))) (lift (plus k h) d (TLRef n))) (\lambda (H1: (lt n d)).(eq_ind_r T
+(TLRef n) (\lambda (t0: T).(eq T (lift k e t0) (lift (plus k h) d (TLRef
+n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift (plus k h) d
+(TLRef n)))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0))
+(refl_equal T (TLRef n)) (lift (plus k h) d (TLRef n)) (lift_lref_lt n (plus
+k h) d H1)) (lift k e (TLRef n)) (lift_lref_lt n k e (lt_le_trans n d e H1
+H0))) (lift h d (TLRef n)) (lift_lref_lt n h d H1))) (\lambda (H1: (le d
+n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (lift k e t0) (lift
+(plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus (plus n h) k)) (\lambda
+(t0: T).(eq T t0 (lift (plus k h) d (TLRef n)))) (eq_ind_r T (TLRef (plus n
+(plus k h))) (\lambda (t0: T).(eq T (TLRef (plus (plus n h) k)) t0)) (f_equal
+nat T TLRef (plus (plus n h) k) (plus n (plus k h))
+(plus_permute_2_in_3_assoc n h k)) (lift (plus k h) d (TLRef n))
+(lift_lref_ge n (plus k h) d H1)) (lift k e (TLRef (plus n h))) (lift_lref_ge
+(plus n h) k e (le_trans e (plus d h) (plus n h) H (plus_le_compat d n h h H1
+(le_n h))))) (lift h d (TLRef n)) (lift_lref_ge n h d H1))))))))))) (\lambda
+(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k:
+nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to
+(eq T (lift k e (lift h d t0)) (lift (plus k h) d t0)))))))))).(\lambda (t1:
+T).(\lambda (H0: ((\forall (h: nat).(\forall (k: nat).(\forall (d:
+nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e
+(lift h d t1)) (lift (plus k h) d t1)))))))))).(\lambda (h: nat).(\lambda
+(k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e (plus d
+h))).(\lambda (H2: (le d e)).(eq_ind_r T (THead k (lift h d t0) (lift h (s k
+d) t1)) (\lambda (t2: T).(eq T (lift k0 e t2) (lift (plus k0 h) d (THead k t0
+t1)))) (eq_ind_r T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift
+h (s k d) t1))) (\lambda (t2: T).(eq T t2 (lift (plus k0 h) d (THead k t0
+t1)))) (eq_ind_r T (THead k (lift (plus k0 h) d t0) (lift (plus k0 h) (s k d)
+t1)) (\lambda (t2: T).(eq T (THead k (lift k0 e (lift h d t0)) (lift k0 (s k
+e) (lift h (s k d) t1))) t2)) (f_equal3 K T T T THead k k (lift k0 e (lift h
+d t0)) (lift (plus k0 h) d t0) (lift k0 (s k e) (lift h (s k d) t1)) (lift
+(plus k0 h) (s k d) t1) (refl_equal K k) (H h k0 d e H1 H2) (H0 h k0 (s k d)
+(s k e) (eq_ind nat (s k (plus d h)) (\lambda (n: nat).(le (s k e) n)) (s_le
+k e (plus d h) H1) (plus (s k d) h) (s_plus k d h)) (s_le k d e H2))) (lift
+(plus k0 h) d (THead k t0 t1)) (lift_head k t0 t1 (plus k0 h) d)) (lift k0 e
+(THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k (lift h d t0) (lift
+h (s k d) t1) k0 e)) (lift h d (THead k t0 t1)) (lift_head k t0 t1 h
+d))))))))))))) t).
+
+theorem lift_d:
+ \forall (t: T).(\forall (h: nat).(\forall (k: nat).(\forall (d:
+nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k d) (lift k e t))
+(lift k e (lift h d t))))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (k:
+nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k
+d) (lift k e t0)) (lift k e (lift h d t0))))))))) (\lambda (n: nat).(\lambda
+(h: nat).(\lambda (k: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (_:
+(le e d)).(eq_ind_r T (TSort n) (\lambda (t0: T).(eq T (lift h (plus k d) t0)
+(lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq
+T t0 (lift k e (lift h d (TSort n))))) (eq_ind_r T (TSort n) (\lambda (t0:
+T).(eq T (TSort n) (lift k e t0))) (eq_ind_r T (TSort n) (\lambda (t0: T).(eq
+T (TSort n) t0)) (refl_equal T (TSort n)) (lift k e (TSort n)) (lift_sort n k
+e)) (lift h d (TSort n)) (lift_sort n h d)) (lift h (plus k d) (TSort n))
+(lift_sort n h (plus k d))) (lift k e (TSort n)) (lift_sort n k e))))))))
+(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(\lambda (d:
+nat).(\lambda (e: nat).(\lambda (H: (le e d)).(lt_le_e n e (eq T (lift h
+(plus k d) (lift k e (TLRef n))) (lift k e (lift h d (TLRef n)))) (\lambda
+(H0: (lt n e)).(let H1 \def (lt_le_trans n e d H0 H) in (eq_ind_r T (TLRef n)
+(\lambda (t0: T).(eq T (lift h (plus k d) t0) (lift k e (lift h d (TLRef
+n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T t0 (lift k e (lift h d
+(TLRef n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) (lift k
+e t0))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq T (TLRef n) t0))
+(refl_equal T (TLRef n)) (lift k e (TLRef n)) (lift_lref_lt n k e H0)) (lift
+h d (TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus k d) (TLRef n))
+(lift_lref_lt n h (plus k d) (lt_le_trans n d (plus k d) H1 (le_plus_r k
+d)))) (lift k e (TLRef n)) (lift_lref_lt n k e H0)))) (\lambda (H0: (le e
+n)).(eq_ind_r T (TLRef (plus n k)) (\lambda (t0: T).(eq T (lift h (plus k d)
+t0) (lift k e (lift h d (TLRef n))))) (eq_ind_r nat (plus d k) (\lambda (n0:
+nat).(eq T (lift h n0 (TLRef (plus n k))) (lift k e (lift h d (TLRef n)))))
+(lt_le_e n d (eq T (lift h (plus d k) (TLRef (plus n k))) (lift k e (lift h d
+(TLRef n)))) (\lambda (H1: (lt n d)).(eq_ind_r T (TLRef (plus n k)) (\lambda
+(t0: T).(eq T t0 (lift k e (lift h d (TLRef n))))) (eq_ind_r T (TLRef n)
+(\lambda (t0: T).(eq T (TLRef (plus n k)) (lift k e t0))) (eq_ind_r T (TLRef
+(plus n k)) (\lambda (t0: T).(eq T (TLRef (plus n k)) t0)) (refl_equal T
+(TLRef (plus n k))) (lift k e (TLRef n)) (lift_lref_ge n k e H0)) (lift h d
+(TLRef n)) (lift_lref_lt n h d H1)) (lift h (plus d k) (TLRef (plus n k)))
+(lift_lref_lt (plus n k) h (plus d k) (lt_le_S (plus n k) (plus d k)
+(plus_lt_compat_r n d k H1))))) (\lambda (H1: (le d n)).(eq_ind_r T (TLRef
+(plus (plus n k) h)) (\lambda (t0: T).(eq T t0 (lift k e (lift h d (TLRef
+n))))) (eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (TLRef (plus
+(plus n k) h)) (lift k e t0))) (eq_ind_r T (TLRef (plus (plus n h) k))
+(\lambda (t0: T).(eq T (TLRef (plus (plus n k) h)) t0)) (f_equal nat T TLRef
+(plus (plus n k) h) (plus (plus n h) k) (sym_eq nat (plus (plus n h) k) (plus
+(plus n k) h) (plus_permute_2_in_3 n h k))) (lift k e (TLRef (plus n h)))
+(lift_lref_ge (plus n h) k e (le_S_n e (plus n h) (lt_le_S e (S (plus n h))
+(le_lt_n_Sm e (plus n h) (le_plus_trans e n h H0)))))) (lift h d (TLRef n))
+(lift_lref_ge n h d H1)) (lift h (plus d k) (TLRef (plus n k))) (lift_lref_ge
+(plus n k) h (plus d k) (le_S_n (plus d k) (plus n k) (lt_le_S (plus d k) (S
+(plus n k)) (le_lt_n_Sm (plus d k) (plus n k) (plus_le_compat d n k k H1
+(le_n k))))))))) (plus k d) (plus_comm k d)) (lift k e (TLRef n))
+(lift_lref_ge n k e H0)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda
+(H: ((\forall (h: nat).(\forall (k: nat).(\forall (d: nat).(\forall (e:
+nat).((le e d) \to (eq T (lift h (plus k d) (lift k e t0)) (lift k e (lift h
+d t0)))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k:
+nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k
+d) (lift k e t1)) (lift k e (lift h d t1)))))))))).(\lambda (h: nat).(\lambda
+(k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e
+d)).(eq_ind_r T (THead k (lift k0 e t0) (lift k0 (s k e) t1)) (\lambda (t2:
+T).(eq T (lift h (plus k0 d) t2) (lift k0 e (lift h d (THead k t0 t1)))))
+(eq_ind_r T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus
+k0 d)) (lift k0 (s k e) t1))) (\lambda (t2: T).(eq T t2 (lift k0 e (lift h d
+(THead k t0 t1))))) (eq_ind_r T (THead k (lift h d t0) (lift h (s k d) t1))
+(\lambda (t2: T).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h
+(s k (plus k0 d)) (lift k0 (s k e) t1))) (lift k0 e t2))) (eq_ind_r T (THead
+k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h (s k d) t1))) (\lambda
+(t2: T).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus
+k0 d)) (lift k0 (s k e) t1))) t2)) (eq_ind_r nat (plus k0 (s k d)) (\lambda
+(n: nat).(eq T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h n (lift
+k0 (s k e) t1))) (THead k (lift k0 e (lift h d t0)) (lift k0 (s k e) (lift h
+(s k d) t1))))) (f_equal3 K T T T THead k k (lift h (plus k0 d) (lift k0 e
+t0)) (lift k0 e (lift h d t0)) (lift h (plus k0 (s k d)) (lift k0 (s k e)
+t1)) (lift k0 (s k e) (lift h (s k d) t1)) (refl_equal K k) (H h k0 d e H1)
+(H0 h k0 (s k d) (s k e) (s_le k e d H1))) (s k (plus k0 d)) (s_plus_sym k k0
+d)) (lift k0 e (THead k (lift h d t0) (lift h (s k d) t1))) (lift_head k
+(lift h d t0) (lift h (s k d) t1) k0 e)) (lift h d (THead k t0 t1))
+(lift_head k t0 t1 h d)) (lift h (plus k0 d) (THead k (lift k0 e t0) (lift k0
+(s k e) t1))) (lift_head k (lift k0 e t0) (lift k0 (s k e) t1) h (plus k0
+d))) (lift k0 e (THead k t0 t1)) (lift_head k t0 t1 k0 e)))))))))))) t).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift/tlt".
+
+include "lift/defs.ma".
+
+include "tlt/defs.ma".
+
+theorem lift_weight_map:
+ \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to
+nat))).(((\forall (m: nat).((le d m) \to (eq nat (f m) O)))) \to (eq nat
+(weight_map f (lift h d t)) (weight_map f t))))))
+\def
+ \lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h: nat).(\forall (d:
+nat).(\forall (f: ((nat \to nat))).(((\forall (m: nat).((le d m) \to (eq nat
+(f m) O)))) \to (eq nat (weight_map f (lift h d t0)) (weight_map f t0)))))))
+(\lambda (n: nat).(\lambda (_: nat).(\lambda (d: nat).(\lambda (f: ((nat \to
+nat))).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (f m)
+O))))).(refl_equal nat (weight_map f (TSort n)))))))) (\lambda (n:
+nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to
+nat))).(\lambda (H: ((\forall (m: nat).((le d m) \to (eq nat (f m)
+O))))).(lt_le_e n d (eq nat (weight_map f (lift h d (TLRef n))) (weight_map f
+(TLRef n))) (\lambda (H0: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0:
+T).(eq nat (weight_map f t0) (weight_map f (TLRef n)))) (refl_equal nat
+(weight_map f (TLRef n))) (lift h d (TLRef n)) (lift_lref_lt n h d H0)))
+(\lambda (H0: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq
+nat (weight_map f t0) (weight_map f (TLRef n)))) (eq_ind_r nat O (\lambda
+(n0: nat).(eq nat (f (plus n h)) n0)) (H (plus n h) (le_S_n d (plus n h)
+(le_n_S d (plus n h) (le_plus_trans d n h H0)))) (f n) (H n H0)) (lift h d
+(TLRef n)) (lift_lref_ge n h d H0))))))))) (\lambda (k: K).(\lambda (t0:
+T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to
+nat))).(((\forall (m: nat).((le d m) \to (eq nat (f m) O)))) \to (eq nat
+(weight_map f (lift h d t0)) (weight_map f t0)))))))).(\lambda (t1:
+T).(\lambda (H0: ((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to
+nat))).(((\forall (m: nat).((le d m) \to (eq nat (f m) O)))) \to (eq nat
+(weight_map f (lift h d t1)) (weight_map f t1)))))))).(\lambda (h:
+nat).(\lambda (d: nat).(\lambda (f: ((nat \to nat))).(\lambda (H1: ((\forall
+(m: nat).((le d m) \to (eq nat (f m) O))))).(K_ind (\lambda (k0: K).(eq nat
+(weight_map f (lift h d (THead k0 t0 t1))) (weight_map f (THead k0 t0 t1))))
+(\lambda (b: B).(eq_ind_r T (THead (Bind b) (lift h d t0) (lift h (s (Bind b)
+d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) (weight_map f (THead (Bind
+b) t0 t1)))) (B_ind (\lambda (b0: B).(eq nat (match b0 with [Abbr \Rightarrow
+(S (plus (weight_map f (lift h d t0)) (weight_map (wadd f (S (weight_map f
+(lift h d t0)))) (lift h (S d) t1)))) | Abst \Rightarrow (S (plus (weight_map
+f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1)))) | Void
+\Rightarrow (S (plus (weight_map f (lift h d t0)) (weight_map (wadd f O)
+(lift h (S d) t1))))]) (match b0 with [Abbr \Rightarrow (S (plus (weight_map
+f t0) (weight_map (wadd f (S (weight_map f t0))) t1))) | Abst \Rightarrow (S
+(plus (weight_map f t0) (weight_map (wadd f O) t1))) | Void \Rightarrow (S
+(plus (weight_map f t0) (weight_map (wadd f O) t1)))]))) (eq_ind_r nat
+(weight_map f t0) (\lambda (n: nat).(eq nat (S (plus n (weight_map (wadd f (S
+n)) (lift h (S d) t1)))) (S (plus (weight_map f t0) (weight_map (wadd f (S
+(weight_map f t0))) t1))))) (eq_ind_r nat (weight_map (wadd f (S (weight_map
+f t0))) t1) (\lambda (n: nat).(eq nat (S (plus (weight_map f t0) n)) (S (plus
+(weight_map f t0) (weight_map (wadd f (S (weight_map f t0))) t1)))))
+(refl_equal nat (S (plus (weight_map f t0) (weight_map (wadd f (S (weight_map
+f t0))) t1)))) (weight_map (wadd f (S (weight_map f t0))) (lift h (S d) t1))
+(H0 h (S d) (wadd f (S (weight_map f t0))) (\lambda (m: nat).(\lambda (H2:
+(le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n:
+nat).(le d n)) (eq nat (wadd f (S (weight_map f t0)) m) O) (\lambda (x:
+nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S
+x) (\lambda (n: nat).(eq nat (wadd f (S (weight_map f t0)) n) O)) (H1 x H4) m
+H3)))) (le_gen_S d m H2)))))) (weight_map f (lift h d t0)) (H h d f H1))
+(eq_ind_r nat (weight_map (wadd f O) t1) (\lambda (n: nat).(eq nat (S (plus
+(weight_map f (lift h d t0)) n)) (S (plus (weight_map f t0) (weight_map (wadd
+f O) t1))))) (f_equal nat nat S (plus (weight_map f (lift h d t0))
+(weight_map (wadd f O) t1)) (plus (weight_map f t0) (weight_map (wadd f O)
+t1)) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map f
+t0) (weight_map (wadd f O) t1) (weight_map (wadd f O) t1) (H h d f H1)
+(refl_equal nat (weight_map (wadd f O) t1)))) (weight_map (wadd f O) (lift h
+(S d) t1)) (H0 h (S d) (wadd f O) (\lambda (m: nat).(\lambda (H2: (le (S d)
+m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d
+n)) (eq nat (wadd f O m) O) (\lambda (x: nat).(\lambda (H3: (eq nat m (S
+x))).(\lambda (H4: (le d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat
+(wadd f O n) O)) (H1 x H4) m H3)))) (le_gen_S d m H2)))))) (eq_ind_r nat
+(weight_map (wadd f O) t1) (\lambda (n: nat).(eq nat (S (plus (weight_map f
+(lift h d t0)) n)) (S (plus (weight_map f t0) (weight_map (wadd f O) t1)))))
+(f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map (wadd f O)
+t1)) (plus (weight_map f t0) (weight_map (wadd f O) t1)) (f_equal2 nat nat
+nat plus (weight_map f (lift h d t0)) (weight_map f t0) (weight_map (wadd f
+O) t1) (weight_map (wadd f O) t1) (H h d f H1) (refl_equal nat (weight_map
+(wadd f O) t1)))) (weight_map (wadd f O) (lift h (S d) t1)) (H0 h (S d) (wadd
+f O) (\lambda (m: nat).(\lambda (H2: (le (S d) m)).(ex2_ind nat (\lambda (n:
+nat).(eq nat m (S n))) (\lambda (n: nat).(le d n)) (eq nat (wadd f O m) O)
+(\lambda (x: nat).(\lambda (H3: (eq nat m (S x))).(\lambda (H4: (le d
+x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd f O n) O)) (H1 x H4)
+m H3)))) (le_gen_S d m H2)))))) b) (lift h d (THead (Bind b) t0 t1))
+(lift_head (Bind b) t0 t1 h d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat
+f0) (lift h d t0) (lift h (s (Flat f0) d) t1)) (\lambda (t2: T).(eq nat
+(weight_map f t2) (weight_map f (THead (Flat f0) t0 t1)))) (f_equal nat nat S
+(plus (weight_map f (lift h d t0)) (weight_map f (lift h d t1))) (plus
+(weight_map f t0) (weight_map f t1)) (f_equal2 nat nat nat plus (weight_map f
+(lift h d t0)) (weight_map f t0) (weight_map f (lift h d t1)) (weight_map f
+t1) (H h d f H1) (H0 h d f H1))) (lift h d (THead (Flat f0) t0 t1))
+(lift_head (Flat f0) t0 t1 h d))) k)))))))))) t).
+
+theorem lift_weight:
+ \forall (t: T).(\forall (h: nat).(\forall (d: nat).(eq nat (weight (lift h d
+t)) (weight t))))
+\def
+ \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(lift_weight_map t h d
+(\lambda (_: nat).O) (\lambda (m: nat).(\lambda (_: (le d m)).(refl_equal nat
+O)))))).
+
+theorem lift_weight_add:
+ \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (d:
+nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to nat))).(((\forall
+(m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to
+(((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m))))) \to (eq nat
+(weight_map f (lift h d t)) (weight_map g (lift (S h) d t)))))))))))
+\def
+ \lambda (w: nat).(\lambda (t: T).(T_ind (\lambda (t0: T).(\forall (h:
+nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall (g: ((nat \to
+nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f m))))) \to ((eq nat
+(g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m)))))
+\to (eq nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d
+t0))))))))))) (\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda
+(f: ((nat \to nat))).(\lambda (g: ((nat \to nat))).(\lambda (_: ((\forall (m:
+nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d)
+w)).(\lambda (_: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f
+m)))))).(refl_equal nat (weight_map g (lift (S h) d (TSort n))))))))))))
+(\lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat \to
+nat))).(\lambda (g: ((nat \to nat))).(\lambda (H: ((\forall (m: nat).((lt m
+d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d) w)).(\lambda (H1:
+((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f m)))))).(lt_le_e n d
+(eq nat (weight_map f (lift h d (TLRef n))) (weight_map g (lift (S h) d
+(TLRef n)))) (\lambda (H2: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0:
+T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n)))))
+(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq nat (weight_map f (TLRef n))
+(weight_map g t0))) (sym_equal nat (g n) (f n) (H n H2)) (lift (S h) d (TLRef
+n)) (lift_lref_lt n (S h) d H2)) (lift h d (TLRef n)) (lift_lref_lt n h d
+H2))) (\lambda (H2: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0:
+T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n)))))
+(eq_ind_r T (TLRef (plus n (S h))) (\lambda (t0: T).(eq nat (weight_map f
+(TLRef (plus n h))) (weight_map g t0))) (eq_ind nat (S (plus n h)) (\lambda
+(n0: nat).(eq nat (f (plus n h)) (g n0))) (sym_equal nat (g (S (plus n h)))
+(f (plus n h)) (H1 (plus n h) (le_plus_trans d n h H2))) (plus n (S h))
+(plus_n_Sm n h)) (lift (S h) d (TLRef n)) (lift_lref_ge n (S h) d H2)) (lift
+h d (TLRef n)) (lift_lref_ge n h d H2)))))))))))) (\lambda (k: K).(\lambda
+(t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat
+\to nat))).(\forall (g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to
+(eq nat (g m) (f m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d
+m) \to (eq nat (g (S m)) (f m))))) \to (eq nat (weight_map f (lift h d t0))
+(weight_map g (lift (S h) d t0)))))))))))).(\lambda (t1: T).(\lambda (H0:
+((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat \to nat))).(\forall
+(g: ((nat \to nat))).(((\forall (m: nat).((lt m d) \to (eq nat (g m) (f
+m))))) \to ((eq nat (g d) w) \to (((\forall (m: nat).((le d m) \to (eq nat (g
+(S m)) (f m))))) \to (eq nat (weight_map f (lift h d t1)) (weight_map g (lift
+(S h) d t1)))))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (f: ((nat
+\to nat))).(\lambda (g: ((nat \to nat))).(\lambda (H1: ((\forall (m:
+nat).((lt m d) \to (eq nat (g m) (f m)))))).(\lambda (H2: (eq nat (g d)
+w)).(\lambda (H3: ((\forall (m: nat).((le d m) \to (eq nat (g (S m)) (f
+m)))))).(K_ind (\lambda (k0: K).(eq nat (weight_map f (lift h d (THead k0 t0
+t1))) (weight_map g (lift (S h) d (THead k0 t0 t1))))) (\lambda (b:
+B).(eq_ind_r T (THead (Bind b) (lift h d t0) (lift h (s (Bind b) d) t1))
+(\lambda (t2: T).(eq nat (weight_map f t2) (weight_map g (lift (S h) d (THead
+(Bind b) t0 t1))))) (eq_ind_r T (THead (Bind b) (lift (S h) d t0) (lift (S h)
+(s (Bind b) d) t1)) (\lambda (t2: T).(eq nat (weight_map f (THead (Bind b)
+(lift h d t0) (lift h (s (Bind b) d) t1))) (weight_map g t2))) (B_ind
+(\lambda (b0: B).(eq nat (match b0 with [Abbr \Rightarrow (S (plus
+(weight_map f (lift h d t0)) (weight_map (wadd f (S (weight_map f (lift h d
+t0)))) (lift h (S d) t1)))) | Abst \Rightarrow (S (plus (weight_map f (lift h
+d t0)) (weight_map (wadd f O) (lift h (S d) t1)))) | Void \Rightarrow (S
+(plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d)
+t1))))]) (match b0 with [Abbr \Rightarrow (S (plus (weight_map g (lift (S h)
+d t0)) (weight_map (wadd g (S (weight_map g (lift (S h) d t0)))) (lift (S h)
+(S d) t1)))) | Abst \Rightarrow (S (plus (weight_map g (lift (S h) d t0))
+(weight_map (wadd g O) (lift (S h) (S d) t1)))) | Void \Rightarrow (S (plus
+(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d)
+t1))))]))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map
+(wadd f (S (weight_map f (lift h d t0)))) (lift h (S d) t1))) (plus
+(weight_map g (lift (S h) d t0)) (weight_map (wadd g (S (weight_map g (lift
+(S h) d t0)))) (lift (S h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map
+f (lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map (wadd f (S
+(weight_map f (lift h d t0)))) (lift h (S d) t1)) (weight_map (wadd g (S
+(weight_map g (lift (S h) d t0)))) (lift (S h) (S d) t1)) (H h d f g H1 H2
+H3) (H0 h (S d) (wadd f (S (weight_map f (lift h d t0)))) (wadd g (S
+(weight_map g (lift (S h) d t0)))) (\lambda (m: nat).(\lambda (H4: (lt m (S
+d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0)))
+(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g (S (weight_map g (lift (S h) d
+t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (H5: (eq nat m
+O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift
+(S h) d t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (f_equal nat
+nat S (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t0))
+(sym_equal nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d t0))
+(H h d f g H1 H2 H3))) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq
+nat m (S m0))) (\lambda (m: nat).(lt m d)))).(ex2_ind nat (\lambda (m0:
+nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g (S
+(weight_map g (lift (S h) d t0))) m) (wadd f (S (weight_map f (lift h d t0)))
+m)) (\lambda (x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x
+d)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd g (S (weight_map g
+(lift (S h) d t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (H1 x
+H7) m H6)))) H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4:
+(le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n:
+nat).(le d n)) (eq nat (g m) (wadd f (S (weight_map f (lift h d t0))) m))
+(\lambda (x: nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d
+x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (g n) (wadd f (S
+(weight_map f (lift h d t0))) n))) (H3 x H6) m H5)))) (le_gen_S d m H4)))))))
+(f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map (wadd f O)
+(lift h (S d) t1))) (plus (weight_map g (lift (S h) d t0)) (weight_map (wadd
+g O) (lift (S h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map f (lift h
+d t0)) (weight_map g (lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d)
+t1)) (weight_map (wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h
+(S d) (wadd f O) (wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S
+d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0)))
+(\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g O m) (wadd f O m)) (\lambda
+(H5: (eq nat m O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g O n)
+(wadd f O n))) (refl_equal nat O) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0:
+nat).(eq nat m (S m0))) (\lambda (m: nat).(lt m d)))).(ex2_ind nat (\lambda
+(m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O
+m) (wadd f O m)) (\lambda (x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda
+(H7: (lt x d)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd g O n)
+(wadd f O n))) (H1 x H7) m H6)))) H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m:
+nat).(\lambda (H4: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S
+n))) (\lambda (n: nat).(le d n)) (eq nat (g m) (wadd f O m)) (\lambda (x:
+nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S
+x) (\lambda (n: nat).(eq nat (g n) (wadd f O n))) (H3 x H6) m H5))))
+(le_gen_S d m H4))))))) (f_equal nat nat S (plus (weight_map f (lift h d t0))
+(weight_map (wadd f O) (lift h (S d) t1))) (plus (weight_map g (lift (S h) d
+t0)) (weight_map (wadd g O) (lift (S h) (S d) t1))) (f_equal2 nat nat nat
+plus (weight_map f (lift h d t0)) (weight_map g (lift (S h) d t0))
+(weight_map (wadd f O) (lift h (S d) t1)) (weight_map (wadd g O) (lift (S h)
+(S d) t1)) (H h d f g H1 H2 H3) (H0 h (S d) (wadd f O) (wadd g O) (\lambda
+(m: nat).(\lambda (H4: (lt m (S d))).(or_ind (eq nat m O) (ex2 nat (\lambda
+(m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g O
+m) (wadd f O m)) (\lambda (H5: (eq nat m O)).(eq_ind_r nat O (\lambda (n:
+nat).(eq nat (wadd g O n) (wadd f O n))) (refl_equal nat O) m H5)) (\lambda
+(H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m: nat).(lt m
+d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m0:
+nat).(lt m0 d)) (eq nat (wadd g O m) (wadd f O m)) (\lambda (x: nat).(\lambda
+(H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r nat (S x) (\lambda
+(n: nat).(eq nat (wadd g O n) (wadd f O n))) (H1 x H7) m H6)))) H5))
+(lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d)
+m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d
+n)) (eq nat (g m) (wadd f O m)) (\lambda (x: nat).(\lambda (H5: (eq nat m (S
+x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (g
+n) (wadd f O n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) b) (lift (S h) d
+(THead (Bind b) t0 t1)) (lift_head (Bind b) t0 t1 (S h) d)) (lift h d (THead
+(Bind b) t0 t1)) (lift_head (Bind b) t0 t1 h d))) (\lambda (f0: F).(eq_ind_r
+T (THead (Flat f0) (lift h d t0) (lift h (s (Flat f0) d) t1)) (\lambda (t2:
+T).(eq nat (weight_map f t2) (weight_map g (lift (S h) d (THead (Flat f0) t0
+t1))))) (eq_ind_r T (THead (Flat f0) (lift (S h) d t0) (lift (S h) (s (Flat
+f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f (THead (Flat f0) (lift h d
+t0) (lift h (s (Flat f0) d) t1))) (weight_map g t2))) (f_equal nat nat S
+(plus (weight_map f (lift h d t0)) (weight_map f (lift h d t1))) (plus
+(weight_map g (lift (S h) d t0)) (weight_map g (lift (S h) d t1))) (f_equal2
+nat nat nat plus (weight_map f (lift h d t0)) (weight_map g (lift (S h) d
+t0)) (weight_map f (lift h d t1)) (weight_map g (lift (S h) d t1)) (H h d f g
+H1 H2 H3) (H0 h d f g H1 H2 H3))) (lift (S h) d (THead (Flat f0) t0 t1))
+(lift_head (Flat f0) t0 t1 (S h) d)) (lift h d (THead (Flat f0) t0 t1))
+(lift_head (Flat f0) t0 t1 h d))) k))))))))))))) t)).
+
+theorem lift_weight_add_O:
+ \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (f: ((nat \to
+nat))).(eq nat (weight_map f (lift h O t)) (weight_map (wadd f w) (lift (S h)
+O t))))))
+\def
+ \lambda (w: nat).(\lambda (t: T).(\lambda (h: nat).(\lambda (f: ((nat \to
+nat))).(lift_weight_add (plus (wadd f w O) O) t h O f (wadd f w) (\lambda (m:
+nat).(\lambda (H: (lt m O)).(let H0 \def (match H in le return (\lambda (n:
+nat).(\lambda (_: (le ? n)).((eq nat n O) \to (eq nat (wadd f w m) (f m)))))
+with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) O)).(let H1 \def (eq_ind
+nat (S m) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop)
+with [O \Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind
+(eq nat (wadd f w m) (f m)) H1))) | (le_S m0 H0) \Rightarrow (\lambda (H1:
+(eq nat (S m0) O)).((let H2 \def (eq_ind nat (S m0) (\lambda (e: nat).(match
+e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _)
+\Rightarrow True])) I O H1) in (False_ind ((le (S m) m0) \to (eq nat (wadd f
+w m) (f m))) H2)) H0))]) in (H0 (refl_equal nat O))))) (plus_n_O (wadd f w
+O)) (\lambda (m: nat).(\lambda (_: (le O m)).(refl_equal nat (f m)))))))).
+
+theorem lift_tlt_dx:
+ \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (h: nat).(\forall
+(d: nat).(tlt t (THead k u (lift h d t)))))))
+\def
+ \lambda (k: K).(\lambda (u: T).(\lambda (t: T).(\lambda (h: nat).(\lambda
+(d: nat).(eq_ind nat (weight (lift h d t)) (\lambda (n: nat).(lt n (weight
+(THead k u (lift h d t))))) (tlt_head_dx k u (lift h d t)) (weight t)
+(lift_weight t h d)))))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift1/defs".
+
+include "lift/defs.ma".
+
+definition trans:
+ PList \to (nat \to nat)
+\def
+ let rec trans (hds: PList) on hds: (nat \to nat) \def (\lambda (i:
+nat).(match hds with [PNil \Rightarrow i | (PCons h d hds0) \Rightarrow (let
+j \def (trans hds0 i) in (match (blt j d) with [true \Rightarrow j | false
+\Rightarrow (plus j h)]))])) in trans.
+
+definition lift1:
+ PList \to (T \to T)
+\def
+ let rec lift1 (hds: PList) on hds: (T \to T) \def (\lambda (t: T).(match hds
+with [PNil \Rightarrow t | (PCons h d hds0) \Rightarrow (lift h d (lift1 hds0
+t))])) in lift1.
+
+definition lifts1:
+ PList \to (TList \to TList)
+\def
+ let rec lifts1 (hds: PList) (ts: TList) on ts: TList \def (match ts with
+[TNil \Rightarrow TNil | (TCons t ts0) \Rightarrow (TCons (lift1 hds t)
+(lifts1 hds ts0))]) in lifts1.
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift1/fwd".
+
+include "lift1/defs.ma".
+
+theorem lift1_lref:
+ \forall (hds: PList).(\forall (i: nat).(eq T (lift1 hds (TLRef i)) (TLRef
+(trans hds i))))
+\def
+ \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (i: nat).(eq T
+(lift1 p (TLRef i)) (TLRef (trans p i))))) (\lambda (i: nat).(refl_equal T
+(TLRef (trans PNil i)))) (\lambda (h: nat).(\lambda (d: nat).(\lambda (p:
+PList).(\lambda (H: ((\forall (i: nat).(eq T (lift1 p (TLRef i)) (TLRef
+(trans p i)))))).(\lambda (i: nat).(eq_ind_r T (TLRef (trans p i)) (\lambda
+(t: T).(eq T (lift h d t) (TLRef (match (blt (trans p i) d) with [true
+\Rightarrow (trans p i) | false \Rightarrow (plus (trans p i) h)]))))
+(refl_equal T (TLRef (match (blt (trans p i) d) with [true \Rightarrow (trans
+p i) | false \Rightarrow (plus (trans p i) h)]))) (lift1 p (TLRef i)) (H
+i))))))) hds).
+
+theorem lift1_bind:
+ \forall (b: B).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T
+(lift1 hds (THead (Bind b) u t)) (THead (Bind b) (lift1 hds u) (lift1 (Ss
+hds) t))))))
+\def
+ \lambda (b: B).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
+(u: T).(\forall (t: T).(eq T (lift1 p (THead (Bind b) u t)) (THead (Bind b)
+(lift1 p u) (lift1 (Ss p) t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal
+T (THead (Bind b) (lift1 PNil u) (lift1 (Ss PNil) t))))) (\lambda (h:
+nat).(\lambda (d: nat).(\lambda (p: PList).(\lambda (H: ((\forall (u:
+T).(\forall (t: T).(eq T (lift1 p (THead (Bind b) u t)) (THead (Bind b)
+(lift1 p u) (lift1 (Ss p) t))))))).(\lambda (u: T).(\lambda (t: T).(eq_ind_r
+T (THead (Bind b) (lift1 p u) (lift1 (Ss p) t)) (\lambda (t0: T).(eq T (lift
+h d t0) (THead (Bind b) (lift h d (lift1 p u)) (lift h (S d) (lift1 (Ss p)
+t))))) (eq_ind_r T (THead (Bind b) (lift h d (lift1 p u)) (lift h (S d)
+(lift1 (Ss p) t))) (\lambda (t0: T).(eq T t0 (THead (Bind b) (lift h d (lift1
+p u)) (lift h (S d) (lift1 (Ss p) t))))) (refl_equal T (THead (Bind b) (lift
+h d (lift1 p u)) (lift h (S d) (lift1 (Ss p) t)))) (lift h d (THead (Bind b)
+(lift1 p u) (lift1 (Ss p) t))) (lift_bind b (lift1 p u) (lift1 (Ss p) t) h
+d)) (lift1 p (THead (Bind b) u t)) (H u t)))))))) hds)).
+
+theorem lift1_flat:
+ \forall (f: F).(\forall (hds: PList).(\forall (u: T).(\forall (t: T).(eq T
+(lift1 hds (THead (Flat f) u t)) (THead (Flat f) (lift1 hds u) (lift1 hds
+t))))))
+\def
+ \lambda (f: F).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
+(u: T).(\forall (t: T).(eq T (lift1 p (THead (Flat f) u t)) (THead (Flat f)
+(lift1 p u) (lift1 p t)))))) (\lambda (u: T).(\lambda (t: T).(refl_equal T
+(THead (Flat f) (lift1 PNil u) (lift1 PNil t))))) (\lambda (h: nat).(\lambda
+(d: nat).(\lambda (p: PList).(\lambda (H: ((\forall (u: T).(\forall (t:
+T).(eq T (lift1 p (THead (Flat f) u t)) (THead (Flat f) (lift1 p u) (lift1 p
+t))))))).(\lambda (u: T).(\lambda (t: T).(eq_ind_r T (THead (Flat f) (lift1 p
+u) (lift1 p t)) (\lambda (t0: T).(eq T (lift h d t0) (THead (Flat f) (lift h
+d (lift1 p u)) (lift h d (lift1 p t))))) (eq_ind_r T (THead (Flat f) (lift h
+d (lift1 p u)) (lift h d (lift1 p t))) (\lambda (t0: T).(eq T t0 (THead (Flat
+f) (lift h d (lift1 p u)) (lift h d (lift1 p t))))) (refl_equal T (THead
+(Flat f) (lift h d (lift1 p u)) (lift h d (lift1 p t)))) (lift h d (THead
+(Flat f) (lift1 p u) (lift1 p t))) (lift_flat f (lift1 p u) (lift1 p t) h d))
+(lift1 p (THead (Flat f) u t)) (H u t)))))))) hds)).
+
+theorem lift1_cons_tail:
+ \forall (t: T).(\forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(eq
+T (lift1 (PConsTail hds h d) t) (lift1 hds (lift h d t))))))
+\def
+ \lambda (t: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (hds:
+PList).(PList_ind (\lambda (p: PList).(eq T (lift1 (PConsTail p h d) t)
+(lift1 p (lift h d t)))) (refl_equal T (lift h d t)) (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: (eq T (lift1
+(PConsTail p h d) t) (lift1 p (lift h d t)))).(eq_ind_r T (lift1 p (lift h d
+t)) (\lambda (t0: T).(eq T (lift n n0 t0) (lift n n0 (lift1 p (lift h d
+t))))) (refl_equal T (lift n n0 (lift1 p (lift h d t)))) (lift1 (PConsTail p
+h d) t) H))))) hds)))).
+
+theorem lifts1_flat:
+ \forall (f: F).(\forall (hds: PList).(\forall (t: T).(\forall (ts:
+TList).(eq T (lift1 hds (THeads (Flat f) ts t)) (THeads (Flat f) (lifts1 hds
+ts) (lift1 hds t))))))
+\def
+ \lambda (f: F).(\lambda (hds: PList).(\lambda (t: T).(\lambda (ts:
+TList).(TList_ind (\lambda (t0: TList).(eq T (lift1 hds (THeads (Flat f) t0
+t)) (THeads (Flat f) (lifts1 hds t0) (lift1 hds t)))) (refl_equal T (lift1
+hds t)) (\lambda (t0: T).(\lambda (t1: TList).(\lambda (H: (eq T (lift1 hds
+(THeads (Flat f) t1 t)) (THeads (Flat f) (lifts1 hds t1) (lift1 hds
+t)))).(eq_ind_r T (THead (Flat f) (lift1 hds t0) (lift1 hds (THeads (Flat f)
+t1 t))) (\lambda (t2: T).(eq T t2 (THead (Flat f) (lift1 hds t0) (THeads
+(Flat f) (lifts1 hds t1) (lift1 hds t))))) (eq_ind_r T (THeads (Flat f)
+(lifts1 hds t1) (lift1 hds t)) (\lambda (t2: T).(eq T (THead (Flat f) (lift1
+hds t0) t2) (THead (Flat f) (lift1 hds t0) (THeads (Flat f) (lifts1 hds t1)
+(lift1 hds t))))) (refl_equal T (THead (Flat f) (lift1 hds t0) (THeads (Flat
+f) (lifts1 hds t1) (lift1 hds t)))) (lift1 hds (THeads (Flat f) t1 t)) H)
+(lift1 hds (THead (Flat f) t0 (THeads (Flat f) t1 t))) (lift1_flat f hds t0
+(THeads (Flat f) t1 t)))))) ts)))).
+
+theorem lifts1_nil:
+ \forall (ts: TList).(eq TList (lifts1 PNil ts) ts)
+\def
+ \lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 PNil t)
+t)) (refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H:
+(eq TList (lifts1 PNil t0) t0)).(eq_ind_r TList t0 (\lambda (t1: TList).(eq
+TList (TCons t t1) (TCons t t0))) (refl_equal TList (TCons t t0)) (lifts1
+PNil t0) H)))) ts).
+
+theorem lifts1_cons:
+ \forall (h: nat).(\forall (d: nat).(\forall (hds: PList).(\forall (ts:
+TList).(eq TList (lifts1 (PCons h d hds) ts) (lifts h d (lifts1 hds ts))))))
+\def
+ \lambda (h: nat).(\lambda (d: nat).(\lambda (hds: PList).(\lambda (ts:
+TList).(TList_ind (\lambda (t: TList).(eq TList (lifts1 (PCons h d hds) t)
+(lifts h d (lifts1 hds t)))) (refl_equal TList TNil) (\lambda (t: T).(\lambda
+(t0: TList).(\lambda (H: (eq TList (lifts1 (PCons h d hds) t0) (lifts h d
+(lifts1 hds t0)))).(eq_ind_r TList (lifts h d (lifts1 hds t0)) (\lambda (t1:
+TList).(eq TList (TCons (lift h d (lift1 hds t)) t1) (TCons (lift h d (lift1
+hds t)) (lifts h d (lifts1 hds t0))))) (refl_equal TList (TCons (lift h d
+(lift1 hds t)) (lifts h d (lifts1 hds t0)))) (lifts1 (PCons h d hds) t0)
+H)))) ts)))).
+
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 *********************)
+
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift1/props".
+
+include "lift1/defs.ma".
+
+theorem lift1_xhg:
+ \forall (hds: PList).(\forall (t: T).(eq T (lift1 (Ss hds) (lift (S O) O t))
+(lift (S O) O (lift1 hds t))))
+\def
+ \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (t: T).(eq T
+(lift1 (Ss p) (lift (S O) O t)) (lift (S O) O (lift1 p t))))) (\lambda (t:
+T).(refl_equal T (lift (S O) O t))) (\lambda (h: nat).(\lambda (d:
+nat).(\lambda (p: PList).(\lambda (H: ((\forall (t: T).(eq T (lift1 (Ss p)
+(lift (S O) O t)) (lift (S O) O (lift1 p t)))))).(\lambda (t: T).(eq_ind_r T
+(lift (S O) O (lift1 p t)) (\lambda (t0: T).(eq T (lift h (S d) t0) (lift (S
+O) O (lift h d (lift1 p t))))) (eq_ind nat (plus (S O) d) (\lambda (n:
+nat).(eq T (lift h n (lift (S O) O (lift1 p t))) (lift (S O) O (lift h d
+(lift1 p t))))) (eq_ind_r T (lift (S O) O (lift h d (lift1 p t))) (\lambda
+(t0: T).(eq T t0 (lift (S O) O (lift h d (lift1 p t))))) (refl_equal T (lift
+(S O) O (lift h d (lift1 p t)))) (lift h (plus (S O) d) (lift (S O) O (lift1
+p t))) (lift_d (lift1 p t) h (S O) d O (le_O_n d))) (S d) (refl_equal nat (S
+d))) (lift1 (Ss p) (lift (S O) O t)) (H t))))))) hds).
+
+theorem lifts1_xhg:
+ \forall (hds: PList).(\forall (ts: TList).(eq TList (lifts1 (Ss hds) (lifts
+(S O) O ts)) (lifts (S O) O (lifts1 hds ts))))
+\def
+ \lambda (hds: PList).(\lambda (ts: TList).(TList_ind (\lambda (t: TList).(eq
+TList (lifts1 (Ss hds) (lifts (S O) O t)) (lifts (S O) O (lifts1 hds t))))
+(refl_equal TList TNil) (\lambda (t: T).(\lambda (t0: TList).(\lambda (H: (eq
+TList (lifts1 (Ss hds) (lifts (S O) O t0)) (lifts (S O) O (lifts1 hds
+t0)))).(eq_ind_r T (lift (S O) O (lift1 hds t)) (\lambda (t1: T).(eq TList
+(TCons t1 (lifts1 (Ss hds) (lifts (S O) O t0))) (TCons (lift (S O) O (lift1
+hds t)) (lifts (S O) O (lifts1 hds t0))))) (eq_ind_r TList (lifts (S O) O
+(lifts1 hds t0)) (\lambda (t1: TList).(eq TList (TCons (lift (S O) O (lift1
+hds t)) t1) (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O (lifts1 hds
+t0))))) (refl_equal TList (TCons (lift (S O) O (lift1 hds t)) (lifts (S O) O
+(lifts1 hds t0)))) (lifts1 (Ss hds) (lifts (S O) O t0)) H) (lift1 (Ss hds)
+(lift (S O) O t)) (lift1_xhg hds t))))) ts)).
+
--- /dev/null
+set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/preamble".
+
+include "../Base/theory.ma".
+
+alias id "s" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/s/defs/s.con".
+alias id "lift_sort" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd/lift_sort.con".
+alias id "lift_head" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd/lift_head.con".
+alias id "r" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/r/defs/r.con".
+alias id "r_S" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/r/props/r_S.con".
+alias id "drop_gen_sort" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/drop/fwd/drop_gen_sort.con".
+alias id "drop_gen_refl" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/drop/fwd/drop_gen_refl.con".
+alias id "drop_gen_drop" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/drop/fwd/drop_gen_drop.con".
+alias id "drop_gen_skip_l" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/drop/fwd/drop_gen_skip_l.con".
+alias id "r_plus_sym" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/r/props/r_plus_sym.con".
+alias id "lift_bind" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd/lift_bind.con".
+alias id "lift_flat" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd/lift_flat.con".
+alias id "lift_d" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift/props/lift_d.con".
+alias id "s" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/s/defs/s.con".
+alias id "s_plus" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/s/props/s_plus.con".
+alias id "s_le" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/s/props/s_le.con".
+alias id "s_plus_sym" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/s/props/s_plus_sym.con".
+alias id "lift_lref_lt" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd/lift_lref_lt.con".
+alias id "lift_lref_ge" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd/lift_lref_ge.con".
+alias id "lift_head" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd/lift_head.con".
+alias id "tlt_head_dx" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/tlt/props/tlt_head_dx.con".
--- /dev/null
+alias id "s" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/s/defs/s.con".
+alias id "lift_sort" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd/lift_sort.con".
+alias id "lift_head" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd/lift_head.con".
+alias id "r" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/r/defs/r.con".
+alias id "r_S" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/r/props/r_S.con".
+alias id "drop_gen_sort" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/drop/fwd/drop_gen_sort.con".
+alias id "drop_gen_refl" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/drop/fwd/drop_gen_refl.con".
+alias id "drop_gen_drop" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/drop/fwd/drop_gen_drop.con".
+alias id "drop_gen_skip_l" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/drop/fwd/drop_gen_skip_l.con".
+alias id "r_plus_sym" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/r/props/r_plus_sym.con".
+alias id "lift_bind" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd/lift_bind.con".
+alias id "lift_flat" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd/lift_flat.con".
+alias id "lift_d" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift/props/lift_d.con".
+alias id "s" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/s/defs/s.con".
+alias id "s_plus" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/s/props/s_plus.con".
+alias id "s_le" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/s/props/s_le.con".
+alias id "s_plus_sym" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/s/props/s_plus_sym.con".
+alias id "lift_lref_lt" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd/lift_lref_lt.con".
+alias id "lift_lref_ge" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd/lift_lref_ge.con".
+alias id "lift_head" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd/lift_head.con".
+alias id "tlt_head_dx" = "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/tlt/props/tlt_head_dx.con".
include "s/props.ma".
+include "tlt/defs.ma".
+
+include "tlt/props.ma".
+
+include "iso/defs.ma".
+
+include "iso/fwd.ma".
+
+include "iso/props.ma".
+
+include "C/defs.ma".
+
+include "C/props.ma".
+
+include "r/defs.ma".
+
+include "r/props.ma".
+
+include "clen/defs.ma".
+
+include "flt/defs.ma".
+
+include "flt/props.ma".
+
+include "lift/defs.ma".
+
+include "lift/fwd.ma".
+
+include "lift/props.ma".
+
+include "lift/tlt.ma".
+
+include "lift1/defs.ma".
+
+include "lift1/fwd.ma".
+
+include "lift1/props.ma".
+
+include "cnt/defs.ma".
+
+include "cnt/props.ma".
+
+include "drop/defs.ma".
+
+include "drop/fwd.ma".
+
+include "drop/props.ma".
+
include "G/defs.ma".
include "next_plus/defs.ma".
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