1 (**************************************************************************)
4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 include "LambdaDelta-1/nf2/props.ma".
19 include "LambdaDelta-1/drop1/defs.ma".
22 \forall (e: C).(\forall (hds: PList).(\forall (c: C).(\forall (t: T).((drop1
23 hds c e) \to ((nf2 e t) \to (nf2 c (lift1 hds t)))))))
25 \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
26 (c: C).(\forall (t: T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p
27 t))))))) (\lambda (c: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c
28 e)).(\lambda (H0: (nf2 e t)).(let H1 \def (match H in drop1 return (\lambda
29 (p: PList).(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (drop1 p c0
30 c1)).((eq PList p PNil) \to ((eq C c0 c) \to ((eq C c1 e) \to (nf2 c
31 t)))))))) with [(drop1_nil c0) \Rightarrow (\lambda (_: (eq PList PNil
32 PNil)).(\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq C c0 e)).(eq_ind C c
33 (\lambda (c1: C).((eq C c1 e) \to (nf2 c t))) (\lambda (H4: (eq C c
34 e)).(eq_ind C e (\lambda (c1: C).(nf2 c1 t)) H0 c (sym_eq C c e H4))) c0
35 (sym_eq C c0 c H2) H3)))) | (drop1_cons c1 c2 h d H1 c3 hds0 H2) \Rightarrow
36 (\lambda (H3: (eq PList (PCons h d hds0) PNil)).(\lambda (H4: (eq C c1
37 c)).(\lambda (H5: (eq C c3 e)).((let H6 \def (eq_ind PList (PCons h d hds0)
38 (\lambda (e0: PList).(match e0 in PList return (\lambda (_: PList).Prop) with
39 [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in
40 (False_ind ((eq C c1 c) \to ((eq C c3 e) \to ((drop h d c1 c2) \to ((drop1
41 hds0 c2 c3) \to (nf2 c t))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList
42 PNil) (refl_equal C c) (refl_equal C e))))))) (\lambda (n: nat).(\lambda (n0:
43 nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).(\forall (t:
44 T).((drop1 p c e) \to ((nf2 e t) \to (nf2 c (lift1 p t)))))))).(\lambda (c:
45 C).(\lambda (t: T).(\lambda (H0: (drop1 (PCons n n0 p) c e)).(\lambda (H1:
46 (nf2 e t)).(let H2 \def (match H0 in drop1 return (\lambda (p0:
47 PList).(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (drop1 p0 c0 c1)).((eq
48 PList p0 (PCons n n0 p)) \to ((eq C c0 c) \to ((eq C c1 e) \to (nf2 c (lift n
49 n0 (lift1 p t)))))))))) with [(drop1_nil c0) \Rightarrow (\lambda (H2: (eq
50 PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c0 c)).(\lambda (H4: (eq C c0
51 e)).((let H5 \def (eq_ind PList PNil (\lambda (e0: PList).(match e0 in PList
52 return (\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _)
53 \Rightarrow False])) I (PCons n n0 p) H2) in (False_ind ((eq C c0 c) \to ((eq
54 C c0 e) \to (nf2 c (lift n n0 (lift1 p t))))) H5)) H3 H4)))) | (drop1_cons c1
55 c2 h d H2 c3 hds0 H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds0)
56 (PCons n n0 p))).(\lambda (H5: (eq C c1 c)).(\lambda (H6: (eq C c3 e)).((let
57 H7 \def (f_equal PList PList (\lambda (e0: PList).(match e0 in PList return
58 (\lambda (_: PList).PList) with [PNil \Rightarrow hds0 | (PCons _ _ p0)
59 \Rightarrow p0])) (PCons h d hds0) (PCons n n0 p) H4) in ((let H8 \def
60 (f_equal PList nat (\lambda (e0: PList).(match e0 in PList return (\lambda
61 (_: PList).nat) with [PNil \Rightarrow d | (PCons _ n1 _) \Rightarrow n1]))
62 (PCons h d hds0) (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat
63 (\lambda (e0: PList).(match e0 in PList return (\lambda (_: PList).nat) with
64 [PNil \Rightarrow h | (PCons n1 _ _) \Rightarrow n1])) (PCons h d hds0)
65 (PCons n n0 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to
66 ((eq PList hds0 p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n1 d c1 c2)
67 \to ((drop1 hds0 c2 c3) \to (nf2 c (lift n n0 (lift1 p t)))))))))) (\lambda
68 (H10: (eq nat d n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds0 p) \to
69 ((eq C c1 c) \to ((eq C c3 e) \to ((drop n n1 c1 c2) \to ((drop1 hds0 c2 c3)
70 \to (nf2 c (lift n n0 (lift1 p t))))))))) (\lambda (H11: (eq PList hds0
71 p)).(eq_ind PList p (\lambda (p0: PList).((eq C c1 c) \to ((eq C c3 e) \to
72 ((drop n n0 c1 c2) \to ((drop1 p0 c2 c3) \to (nf2 c (lift n n0 (lift1 p
73 t)))))))) (\lambda (H12: (eq C c1 c)).(eq_ind C c (\lambda (c0: C).((eq C c3
74 e) \to ((drop n n0 c0 c2) \to ((drop1 p c2 c3) \to (nf2 c (lift n n0 (lift1 p
75 t))))))) (\lambda (H13: (eq C c3 e)).(eq_ind C e (\lambda (c0: C).((drop n n0
76 c c2) \to ((drop1 p c2 c0) \to (nf2 c (lift n n0 (lift1 p t)))))) (\lambda
77 (H14: (drop n n0 c c2)).(\lambda (H15: (drop1 p c2 e)).(nf2_lift c2 (lift1 p
78 t) (H c2 t H15 H1) c n n0 H14))) c3 (sym_eq C c3 e H13))) c1 (sym_eq C c1 c
79 H12))) hds0 (sym_eq PList hds0 p H11))) d (sym_eq nat d n0 H10))) h (sym_eq
80 nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n
81 n0 p)) (refl_equal C c) (refl_equal C e))))))))))) hds)).