(* This file was automatically generated: do not edit *********************)
-
+set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/drop/fwd".
include "drop/defs.ma".
\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:
+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 (CSort n)) (eq nat n0 O) (eq nat n1 O))))))) (\lambda (c:
+\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 (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)
+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 (CHead e k u) (CSort n)) (eq nat h0
-O) (eq nat (S d0) O)) H4))))))))))) h d y x H0))) H))))).
+(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)) (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
+(\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 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))).
+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:
\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 (H6: (eq
-nat O (S h))).(let H7 \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) H6) in (False_ind (drop (r k h) O c c0) H7)))]) 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
+(\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 (_: (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 (H6: (eq nat O (S
+h))).(let H7 \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) H6) in (False_ind (drop (r k h) O c c0) H7)))]) 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
(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:
+((eq C c0 (CHead c k u)) \to (drop (r k h) (r k0 d) 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
+O) \to (drop (r k h) (S d) c (CHead e k0 u0))))) with [refl_equal \Rightarrow
(\lambda (H8: (eq nat (S d) O)).(let H9 \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 H8) in (False_ind (drop (r k h) O c
+False | (S _) \Rightarrow True])) I O H8) in (False_ind (drop (r k h) (S d) c
(CHead e k0 u0)) H9)))]) in (H8 (refl_equal nat O)))))))))))))) y1 y0 y x
H2))) H1))) H0))) H)))))).
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