--- /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 *********************)
+
+include "LambdaDelta-1/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)) (\lambda (c: C).(and3 (eq C x c) (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 c) (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 c0) (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 c) (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 (CHead c k u)) (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 c) (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) (CHead c k (lift h0 (r k d0) u))) (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)) (\lambda (_: nat).(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)) (\lambda (n: nat).((eq nat y n) \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 n0) \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 (r k d)) \to (eq C c
+e0))))).(\lambda (u: T).(\lambda (H4: (eq nat (S d) O)).(\lambda (H5: (eq nat
+h (S d))).(let H6 \def (f_equal nat nat (\lambda (e1: nat).e1) h (S d) H5) in
+(let H7 \def (eq_ind nat h (\lambda (n: nat).((eq nat (r k d) O) \to ((eq nat
+n (r k d)) \to (eq C c e0)))) H3 (S d) H6) in (let H8 \def (eq_ind nat h
+(\lambda (n: nat).(drop n (r k d) c e0)) H2 (S d) H6) in (eq_ind_r nat (S d)
+(\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 (S d) (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)) (\lambda (_: C).(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)) (\lambda (n: nat).((eq C y (CHead c k u)) \to
+(drop (r k h) n 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)) (\lambda (_:
+nat).((eq nat y0 O) \to ((eq C y (CHead c k u)) \to (drop (r k h) y0 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) n0 c
+c1)))))))) (\lambda (c0: C).(\lambda (H3: (eq nat O (S h))).(\lambda (_: (eq
+nat O O)).(\lambda (H5: (eq C c0 (CHead c k u))).(eq_ind_r C (CHead c k u)
+(\lambda (c1: C).(drop (r k h) O c c1)) (let H6 \def (eq_ind nat O (\lambda
+(ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow
+True | (S _) \Rightarrow False])) I (S h) H3) in (False_ind (drop (r k h) O c
+(CHead c k u)) H6)) c0 H5))))) (\lambda (k0: K).(\lambda (h0: nat).(\lambda
+(c0: C).(\lambda (e: C).(\lambda (H3: (drop (r k0 h0) O c0 e)).(\lambda (H4:
+(((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 (f_equal C C (\lambda (e0: C).(match e0 in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _)
+\Rightarrow c1])) (CHead c0 k0 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 _ k1 _) \Rightarrow k1])) (CHead c0 k0 u0)
+(CHead c k u) H7) in ((let H10 \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) H7) in (\lambda (H11: (eq K
+k0 k)).(\lambda (H12: (eq C c0 c)).(let H13 \def (eq_ind C c0 (\lambda (c1:
+C).((eq nat (r k0 h0) (S h)) \to ((eq nat O O) \to ((eq C c1 (CHead c k u))
+\to (drop (r k h) O c e))))) H4 c H12) in (let H14 \def (eq_ind C c0 (\lambda
+(c1: C).(drop (r k0 h0) O c1 e)) H3 c H12) in (let H15 \def (eq_ind K k0
+(\lambda (k1: K).((eq nat (r k1 h0) (S h)) \to ((eq nat O O) \to ((eq C c
+(CHead c k u)) \to (drop (r k h) O c e))))) H13 k H11) in (let H16 \def
+(eq_ind K k0 (\lambda (k1: K).(drop (r k1 h0) O c e)) H14 k H11) in (let H17
+\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) H5) in
+(let H18 \def (eq_ind nat h0 (\lambda (n: nat).((eq nat (r k n) (S h)) \to
+((eq nat O O) \to ((eq C c (CHead c k u)) \to (drop (r k h) O c e))))) H15 h
+H17) in (let H19 \def (eq_ind nat h0 (\lambda (n: nat).(drop (r k n) O c e))
+H16 h H17) in H19)))))))))) H9)) H8)))))))))))) (\lambda (k0: K).(\lambda
+(h0: nat).(\lambda (d: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H3:
+(drop h0 (r k0 d) c0 e)).(\lambda (H4: (((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) (r k0 d) c
+e)))))).(\lambda (u0: T).(\lambda (H5: (eq nat h0 (S h))).(\lambda (H6: (eq
+nat (S d) O)).(\lambda (H7: (eq C (CHead c0 k0 (lift h0 (r k0 d) u0)) (CHead
+c k u))).(let H8 \def (eq_ind nat h0 (\lambda (n: nat).(eq C (CHead c0 k0
+(lift n (r k0 d) u0)) (CHead c k u))) H7 (S h) H5) in (let H9 \def (eq_ind
+nat h0 (\lambda (n: nat).((eq nat n (S h)) \to ((eq nat (r k0 d) O) \to ((eq
+C c0 (CHead c k u)) \to (drop (r k h) (r k0 d) c e))))) H4 (S h) H5) in (let
+H10 \def (eq_ind nat h0 (\lambda (n: nat).(drop n (r k0 d) c0 e)) H3 (S h)
+H5) in (let H11 \def (f_equal C C (\lambda (e0: C).(match e0 in C return
+(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _)
+\Rightarrow c1])) (CHead c0 k0 (lift (S h) (r k0 d) u0)) (CHead c k u) H8) in
+((let H12 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda
+(_: C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1]))
+(CHead c0 k0 (lift (S h) (r k0 d) u0)) (CHead c k u) H8) in ((let H13 \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))) (d0: nat) (t:
+T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef i)
+\Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i | false
+\Rightarrow (f i)])) | (THead k1 u1 t0) \Rightarrow (THead k1 (lref_map f d0
+u1) (lref_map f (s k1 d0) t0))]) in lref_map) (\lambda (x0: nat).(plus x0 (S
+h))) (r k0 d) u0) | (CHead _ _ t) \Rightarrow t])) (CHead c0 k0 (lift (S h)
+(r k0 d) u0)) (CHead c k u) H8) in (\lambda (H14: (eq K k0 k)).(\lambda (H15:
+(eq C c0 c)).(let H16 \def (eq_ind C c0 (\lambda (c1: C).((eq nat (S h) (S
+h)) \to ((eq nat (r k0 d) O) \to ((eq C c1 (CHead c k u)) \to (drop (r k h)
+(r k0 d) c e))))) H9 c H15) in (let H17 \def (eq_ind C c0 (\lambda (c1:
+C).(drop (S h) (r k0 d) c1 e)) H10 c H15) in (let H18 \def (eq_ind K k0
+(\lambda (k1: K).(eq T (lift (S h) (r k1 d) u0) u)) H13 k H14) in (let H19
+\def (eq_ind K k0 (\lambda (k1: K).((eq nat (S h) (S h)) \to ((eq nat (r k1
+d) O) \to ((eq C c (CHead c k u)) \to (drop (r k h) (r k1 d) c e))))) H16 k
+H14) in (let H20 \def (eq_ind K k0 (\lambda (k1: K).(drop (S h) (r k1 d) c
+e)) H17 k H14) in (eq_ind_r K k (\lambda (k1: K).(drop (r k h) (S d) c (CHead
+e k1 u0))) (let H21 \def (eq_ind_r T u (\lambda (t: T).((eq nat (S h) (S h))
+\to ((eq nat (r k d) O) \to ((eq C c (CHead c k t)) \to (drop (r k h) (r k d)
+c e))))) H19 (lift (S h) (r k d) u0) H18) in (let H22 \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 H6) in (False_ind (drop (r
+k h) (S d) c (CHead e k u0)) H22))) k0 H14))))))))) H12)) H11))))))))))))))))
+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))).(insert_eq C (CHead c k u) (\lambda (c0: C).(drop h (S d) x c0))
+(\lambda (_: C).(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)))) (\lambda (y: C).(\lambda (H0:
+(drop h (S d) x y)).(insert_eq nat (S d) (\lambda (n: nat).(drop h n x y))
+(\lambda (_: nat).((eq C y (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)))))
+(\lambda (y0: nat).(\lambda (H1: (drop h y0 x y)).(drop_ind (\lambda (n:
+nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq nat n0 (S d))
+\to ((eq C c1 (CHead c k u)) \to (ex2 C (\lambda (e: C).(eq C c0 (CHead e k
+(lift n (r k d) u)))) (\lambda (e: C).(drop n (r k d) e c))))))))) (\lambda
+(c0: C).(\lambda (H2: (eq nat O (S d))).(\lambda (H3: (eq C c0 (CHead c k
+u))).(eq_ind_r C (CHead c k u) (\lambda (c1: C).(ex2 C (\lambda (e: C).(eq C
+c1 (CHead e k (lift O (r k d) u)))) (\lambda (e: C).(drop O (r k d) e c))))
+(let H4 \def (eq_ind nat O (\lambda (ee: nat).(match ee in nat return
+(\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) \Rightarrow False]))
+I (S d) H2) in (False_ind (ex2 C (\lambda (e: C).(eq C (CHead c k u) (CHead e
+k (lift O (r k d) u)))) (\lambda (e: C).(drop O (r k d) e c))) H4)) c0 H3))))
+(\lambda (k0: K).(\lambda (h0: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda
+(H2: (drop (r k0 h0) O c0 e)).(\lambda (H3: (((eq nat O (S d)) \to ((eq C e
+(CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift (r k0
+h0) (r k d) u)))) (\lambda (e0: C).(drop (r k0 h0) (r k d) e0
+c))))))).(\lambda (u0: T).(\lambda (H4: (eq nat O (S d))).(\lambda (H5: (eq C
+e (CHead c k u))).(let H6 \def (eq_ind C e (\lambda (c1: C).((eq nat O (S d))
+\to ((eq C c1 (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k
+(lift (r k0 h0) (r k d) u)))) (\lambda (e0: C).(drop (r k0 h0) (r k d) e0
+c)))))) H3 (CHead c k u) H5) in (let H7 \def (eq_ind C e (\lambda (c1:
+C).(drop (r k0 h0) O c0 c1)) H2 (CHead c k u) H5) in (let H8 \def (eq_ind nat
+O (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow True | (S _) \Rightarrow False])) I (S d) H4) in (False_ind (ex2
+C (\lambda (e0: C).(eq C (CHead c0 k0 u0) (CHead e0 k (lift (S h0) (r k d)
+u)))) (\lambda (e0: C).(drop (S h0) (r k d) e0 c))) H8))))))))))))) (\lambda
+(k0: K).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (c0: C).(\lambda (e:
+C).(\lambda (H2: (drop h0 (r k0 d0) c0 e)).(\lambda (H3: (((eq nat (r k0 d0)
+(S d)) \to ((eq C e (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0
+(CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0
+c))))))).(\lambda (u0: T).(\lambda (H4: (eq nat (S d0) (S d))).(\lambda (H5:
+(eq C (CHead e k0 u0) (CHead c k u))).(let H6 \def (f_equal C C (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow e |
+(CHead c1 _ _) \Rightarrow c1])) (CHead e k0 u0) (CHead c k u) H5) in ((let
+H7 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_:
+C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1]))
+(CHead e k0 u0) (CHead c k u) H5) in ((let H8 \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) H5) in (\lambda
+(H9: (eq K k0 k)).(\lambda (H10: (eq C e c)).(eq_ind_r T u (\lambda (t:
+T).(ex2 C (\lambda (e0: C).(eq C (CHead c0 k0 (lift h0 (r k0 d0) t)) (CHead
+e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))) (let
+H11 \def (eq_ind C e (\lambda (c1: C).((eq nat (r k0 d0) (S d)) \to ((eq C c1
+(CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift h0 (r k
+d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))))) H3 c H10) in (let H12
+\def (eq_ind C e (\lambda (c1: C).(drop h0 (r k0 d0) c0 c1)) H2 c H10) in
+(let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq nat (r k1 d0) (S d)) \to
+((eq C c (CHead c k u)) \to (ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k
+(lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c)))))) H11 k H9)
+in (let H14 \def (eq_ind K k0 (\lambda (k1: K).(drop h0 (r k1 d0) c0 c)) H12
+k H9) in (eq_ind_r K k (\lambda (k1: K).(ex2 C (\lambda (e0: C).(eq C (CHead
+c0 k1 (lift h0 (r k1 d0) u)) (CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0:
+C).(drop h0 (r k d) e0 c)))) (let H15 \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) H4) in (let H16 \def (eq_ind nat d0
+(\lambda (n: nat).((eq nat (r k n) (S d)) \to ((eq C c (CHead c k u)) \to
+(ex2 C (\lambda (e0: C).(eq C c0 (CHead e0 k (lift h0 (r k d) u)))) (\lambda
+(e0: C).(drop h0 (r k d) e0 c)))))) H13 d H15) in (let H17 \def (eq_ind nat
+d0 (\lambda (n: nat).(drop h0 (r k n) c0 c)) H14 d H15) in (eq_ind_r nat d
+(\lambda (n: nat).(ex2 C (\lambda (e0: C).(eq C (CHead c0 k (lift h0 (r k n)
+u)) (CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0
+c)))) (ex_intro2 C (\lambda (e0: C).(eq C (CHead c0 k (lift h0 (r k d) u))
+(CHead e0 k (lift h0 (r k d) u)))) (\lambda (e0: C).(drop h0 (r k d) e0 c))
+c0 (refl_equal C (CHead c0 k (lift h0 (r k d) u))) H17) d0 H15)))) k0 H9)))))
+u0 H8)))) H7)) H6)))))))))))) h y0 x y H1))) H0))) H))))))).
+
+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)).(insert_eq C (CHead c k u) (\lambda (c0: C).(drop h (S d) c0 x)) (\lambda
+(_: C).(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))))) (\lambda (y: C).(\lambda (H0:
+(drop h (S d) y x)).(insert_eq nat (S d) (\lambda (n: nat).(drop h n y x))
+(\lambda (_: nat).((eq C y (CHead c k u)) \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)))))) (\lambda (y0: nat).(\lambda (H1: (drop h y0 y x)).(drop_ind
+(\lambda (n: nat).(\lambda (n0: nat).(\lambda (c0: C).(\lambda (c1: C).((eq
+nat n0 (S d)) \to ((eq C c0 (CHead c k u)) \to (ex3_2 C T (\lambda (e:
+C).(\lambda (v: T).(eq C c1 (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 (c0: C).(\lambda (H2: (eq nat O (S d))).(\lambda
+(H3: (eq C c0 (CHead c k u))).(eq_ind_r C (CHead c k u) (\lambda (c1:
+C).(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C c1 (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))))) (let H4 \def (eq_ind nat O
+(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O
+\Rightarrow True | (S _) \Rightarrow False])) I (S d) H2) in (False_ind
+(ex3_2 C T (\lambda (e: C).(\lambda (v: T).(eq C (CHead c k u) (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)))) H4)) c0 H3)))) (\lambda (k0:
+K).(\lambda (h0: nat).(\lambda (c0: C).(\lambda (e: C).(\lambda (H2: (drop (r
+k0 h0) O c0 e)).(\lambda (H3: (((eq nat O (S d)) \to ((eq C c0 (CHead c k u))
+\to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v))))
+(\lambda (_: C).(\lambda (v: T).(eq T u (lift (r k0 h0) (r k d) v))))
+(\lambda (e0: C).(\lambda (_: T).(drop (r k0 h0) (r k d) c
+e0)))))))).(\lambda (u0: T).(\lambda (H4: (eq nat O (S d))).(\lambda (H5: (eq
+C (CHead c0 k0 u0) (CHead c k u))).(let H6 \def (f_equal C C (\lambda (e0:
+C).(match e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow c0 |
+(CHead c1 _ _) \Rightarrow c1])) (CHead c0 k0 u0) (CHead c k u) H5) in ((let
+H7 \def (f_equal C K (\lambda (e0: C).(match e0 in C return (\lambda (_:
+C).K) with [(CSort _) \Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1]))
+(CHead c0 k0 u0) (CHead c k u) H5) in ((let H8 \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) H5) in
+(\lambda (H9: (eq K k0 k)).(\lambda (H10: (eq C c0 c)).(let H11 \def (eq_ind
+C c0 (\lambda (c1: C).((eq nat O (S d)) \to ((eq C c1 (CHead c k u)) \to
+(ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v))))
+(\lambda (_: C).(\lambda (v: T).(eq T u (lift (r k0 h0) (r k d) v))))
+(\lambda (e0: C).(\lambda (_: T).(drop (r k0 h0) (r k d) c e0))))))) H3 c
+H10) in (let H12 \def (eq_ind C c0 (\lambda (c1: C).(drop (r k0 h0) O c1 e))
+H2 c H10) in (let H13 \def (eq_ind K k0 (\lambda (k1: K).((eq nat O (S d))
+\to ((eq C c (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v:
+T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift (r
+k1 h0) (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop (r k1 h0) (r k d)
+c e0))))))) H11 k H9) in (let H14 \def (eq_ind K k0 (\lambda (k1: K).(drop (r
+k1 h0) O c e)) H12 k H9) in (let H15 \def (eq_ind nat O (\lambda (ee:
+nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True
+| (S _) \Rightarrow False])) I (S d) H4) in (False_ind (ex3_2 C T (\lambda
+(e0: C).(\lambda (v: T).(eq C e (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)))) H15))))))))) H7)) H6))))))))))) (\lambda
+(k0: K).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (c0: C).(\lambda (e:
+C).(\lambda (H2: (drop h0 (r k0 d0) c0 e)).(\lambda (H3: (((eq nat (r k0 d0)
+(S d)) \to ((eq C c0 (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda
+(v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u
+(lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c
+e0)))))))).(\lambda (u0: T).(\lambda (H4: (eq nat (S d0) (S d))).(\lambda
+(H5: (eq C (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u))).(let H6 \def
+(f_equal C C (\lambda (e0: C).(match e0 in C return (\lambda (_: C).C) with
+[(CSort _) \Rightarrow c0 | (CHead c1 _ _) \Rightarrow c1])) (CHead c0 k0
+(lift h0 (r k0 d0) u0)) (CHead c k u) H5) in ((let H7 \def (f_equal C K
+(\lambda (e0: C).(match e0 in C return (\lambda (_: C).K) with [(CSort _)
+\Rightarrow k0 | (CHead _ k1 _) \Rightarrow k1])) (CHead c0 k0 (lift h0 (r k0
+d0) u0)) (CHead c k u) H5) in ((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))) (d1: nat) (t: T) on t: T \def (match t with
+[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i
+d1) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u1 t0)
+\Rightarrow (THead k1 (lref_map f d1 u1) (lref_map f (s k1 d1) t0))]) in
+lref_map) (\lambda (x0: nat).(plus x0 h0)) (r k0 d0) u0) | (CHead _ _ t)
+\Rightarrow t])) (CHead c0 k0 (lift h0 (r k0 d0) u0)) (CHead c k u) H5) in
+(\lambda (H9: (eq K k0 k)).(\lambda (H10: (eq C c0 c)).(let H11 \def (eq_ind
+C c0 (\lambda (c1: C).((eq nat (r k0 d0) (S d)) \to ((eq C c1 (CHead c k u))
+\to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v))))
+(\lambda (_: C).(\lambda (v: T).(eq T u (lift h0 (r k d) v)))) (\lambda (e0:
+C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H3 c H10) in (let H12 \def
+(eq_ind C c0 (\lambda (c1: C).(drop h0 (r k0 d0) c1 e)) H2 c H10) in (let H13
+\def (eq_ind K k0 (\lambda (k1: K).(eq T (lift h0 (r k1 d0) u0) u)) H8 k H9)
+in (let H14 \def (eq_ind K k0 (\lambda (k1: K).((eq nat (r k1 d0) (S d)) \to
+((eq C c (CHead c k u)) \to (ex3_2 C T (\lambda (e0: C).(\lambda (v: T).(eq C
+e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h0 (r k d)
+v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H11 k H9)
+in (let H15 \def (eq_ind K k0 (\lambda (k1: K).(drop h0 (r k1 d0) c e)) H12 k
+H9) in (eq_ind_r K k (\lambda (k1: K).(ex3_2 C T (\lambda (e0: C).(\lambda
+(v: T).(eq C (CHead e k1 u0) (CHead e0 k v)))) (\lambda (_: C).(\lambda (v:
+T).(eq T u (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0
+(r k d) c e0))))) (let H16 \def (eq_ind_r T u (\lambda (t: T).((eq nat (r k
+d0) (S d)) \to ((eq C c (CHead c k t)) \to (ex3_2 C T (\lambda (e0:
+C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v:
+T).(eq T t (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0
+(r k d) c e0))))))) H14 (lift h0 (r k d0) u0) H13) in (eq_ind T (lift h0 (r k
+d0) u0) (\lambda (t: T).(ex3_2 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 t
+(lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h0 (r k d) c
+e0))))) (let H17 \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) H4) in (let H18 \def (eq_ind nat d0 (\lambda (n: nat).((eq nat
+(r k n) (S d)) \to ((eq C c (CHead c k (lift h0 (r k n) u0))) \to (ex3_2 C T
+(\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_:
+C).(\lambda (v: T).(eq T (lift h0 (r k n) u0) (lift h0 (r k d) v)))) (\lambda
+(e0: C).(\lambda (_: T).(drop h0 (r k d) c e0))))))) H16 d H17) in (let H19
+\def (eq_ind nat d0 (\lambda (n: nat).(drop h0 (r k n) c e)) H15 d H17) in
+(eq_ind_r nat d (\lambda (n: nat).(ex3_2 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 h0 (r k n) u0) (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_:
+T).(drop h0 (r k d) c e0))))) (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 h0 (r k d) u0) (lift h0 (r k d) v)))) (\lambda (e0: C).(\lambda (_:
+T).(drop h0 (r k d) c e0))) e u0 (refl_equal C (CHead e k u0)) (refl_equal T
+(lift h0 (r k d) u0)) H19) d0 H17)))) u H13)) k0 H9))))))))) H7))
+H6)))))))))))) h y0 y x H1))) H0))) H))))))).
+
projects / helm.git / blobdiff