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 set "baseuri" "cic:/matita/LAMBDA-TYPES/LambdaDelta-1/sn3/lift1".
19 include "sn3/props.ma".
21 include "drop1/defs.ma".
23 include "lift1/fwd.ma".
26 \forall (e: C).(\forall (hds: PList).(\forall (c: C).((drop1 hds c e) \to
27 (\forall (ts: TList).((sns3 e ts) \to (sns3 c (lifts1 hds ts)))))))
29 \lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall
30 (c: C).((drop1 p c e) \to (\forall (ts: TList).((sns3 e ts) \to (sns3 c
31 (lifts1 p ts))))))) (\lambda (c: C).(\lambda (H: (drop1 PNil c e)).(\lambda
32 (ts: TList).(\lambda (H0: (sns3 e ts)).(let H1 \def (match H in drop1 return
33 (\lambda (p: PList).(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (drop1 p
34 c0 c1)).((eq PList p PNil) \to ((eq C c0 c) \to ((eq C c1 e) \to (sns3 c
35 (lifts1 PNil ts))))))))) with [(drop1_nil c0) \Rightarrow (\lambda (_: (eq
36 PList PNil PNil)).(\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq C c0
37 e)).(eq_ind C c (\lambda (c1: C).((eq C c1 e) \to (sns3 c (lifts1 PNil ts))))
38 (\lambda (H4: (eq C c e)).(eq_ind C e (\lambda (c1: C).(sns3 c1 (lifts1 PNil
39 ts))) (eq_ind_r TList ts (\lambda (t: TList).(sns3 e t)) H0 (lifts1 PNil ts)
40 (lifts1_nil ts)) c (sym_eq C c e H4))) c0 (sym_eq C c0 c H2) H3)))) |
41 (drop1_cons c1 c2 h d H1 c3 hds0 H2) \Rightarrow (\lambda (H3: (eq PList
42 (PCons h d hds0) PNil)).(\lambda (H4: (eq C c1 c)).(\lambda (H5: (eq C c3
43 e)).((let H6 \def (eq_ind PList (PCons h d hds0) (\lambda (e0: PList).(match
44 e0 in PList return (\lambda (_: PList).Prop) with [PNil \Rightarrow False |
45 (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c1 c) \to
46 ((eq C c3 e) \to ((drop h d c1 c2) \to ((drop1 hds0 c2 c3) \to (sns3 c
47 (lifts1 PNil ts)))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList PNil)
48 (refl_equal C c) (refl_equal C e))))))) (\lambda (n: nat).(\lambda (n0:
49 nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).((drop1 p c e) \to
50 (\forall (ts: TList).((sns3 e ts) \to (sns3 c (lifts1 p ts)))))))).(\lambda
51 (c: C).(\lambda (H0: (drop1 (PCons n n0 p) c e)).(\lambda (ts:
52 TList).(\lambda (H1: (sns3 e ts)).(let H2 \def (match H0 in drop1 return
53 (\lambda (p0: PList).(\lambda (c0: C).(\lambda (c1: C).(\lambda (_: (drop1 p0
54 c0 c1)).((eq PList p0 (PCons n n0 p)) \to ((eq C c0 c) \to ((eq C c1 e) \to
55 (sns3 c (lifts1 (PCons n n0 p) ts))))))))) with [(drop1_nil c0) \Rightarrow
56 (\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c0
57 c)).(\lambda (H4: (eq C c0 e)).((let H5 \def (eq_ind PList PNil (\lambda (e0:
58 PList).(match e0 in PList return (\lambda (_: PList).Prop) with [PNil
59 \Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in
60 (False_ind ((eq C c0 c) \to ((eq C c0 e) \to (sns3 c (lifts1 (PCons n n0 p)
61 ts)))) H5)) H3 H4)))) | (drop1_cons c1 c2 h d H2 c3 hds0 H3) \Rightarrow
62 (\lambda (H4: (eq PList (PCons h d hds0) (PCons n n0 p))).(\lambda (H5: (eq C
63 c1 c)).(\lambda (H6: (eq C c3 e)).((let H7 \def (f_equal PList PList (\lambda
64 (e0: PList).(match e0 in PList return (\lambda (_: PList).PList) with [PNil
65 \Rightarrow hds0 | (PCons _ _ p0) \Rightarrow p0])) (PCons h d hds0) (PCons n
66 n0 p) H4) in ((let H8 \def (f_equal PList nat (\lambda (e0: PList).(match e0
67 in PList return (\lambda (_: PList).nat) with [PNil \Rightarrow d | (PCons _
68 n1 _) \Rightarrow n1])) (PCons h d hds0) (PCons n n0 p) H4) in ((let H9 \def
69 (f_equal PList nat (\lambda (e0: PList).(match e0 in PList return (\lambda
70 (_: PList).nat) with [PNil \Rightarrow h | (PCons n1 _ _) \Rightarrow n1]))
71 (PCons h d hds0) (PCons n n0 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq
72 nat d n0) \to ((eq PList hds0 p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop
73 n1 d c1 c2) \to ((drop1 hds0 c2 c3) \to (sns3 c (lifts1 (PCons n n0 p)
74 ts))))))))) (\lambda (H10: (eq nat d n0)).(eq_ind nat n0 (\lambda (n1:
75 nat).((eq PList hds0 p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n n1 c1
76 c2) \to ((drop1 hds0 c2 c3) \to (sns3 c (lifts1 (PCons n n0 p) ts))))))))
77 (\lambda (H11: (eq PList hds0 p)).(eq_ind PList p (\lambda (p0: PList).((eq C
78 c1 c) \to ((eq C c3 e) \to ((drop n n0 c1 c2) \to ((drop1 p0 c2 c3) \to (sns3
79 c (lifts1 (PCons n n0 p) ts))))))) (\lambda (H12: (eq C c1 c)).(eq_ind C c
80 (\lambda (c0: C).((eq C c3 e) \to ((drop n n0 c0 c2) \to ((drop1 p c2 c3) \to
81 (sns3 c (lifts1 (PCons n n0 p) ts)))))) (\lambda (H13: (eq C c3 e)).(eq_ind C
82 e (\lambda (c0: C).((drop n n0 c c2) \to ((drop1 p c2 c0) \to (sns3 c (lifts1
83 (PCons n n0 p) ts))))) (\lambda (H14: (drop n n0 c c2)).(\lambda (H15: (drop1
84 p c2 e)).(eq_ind_r TList (lifts n n0 (lifts1 p ts)) (\lambda (t: TList).(sns3
85 c t)) (sns3_lifts c c2 n n0 H14 (lifts1 p ts) (H c2 H15 ts H1)) (lifts1
86 (PCons n n0 p) ts) (lifts1_cons n n0 p ts)))) c3 (sym_eq C c3 e H13))) c1
87 (sym_eq C c1 c H12))) hds0 (sym_eq PList hds0 p H11))) d (sym_eq nat d n0
88 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal
89 PList (PCons n n0 p)) (refl_equal C c) (refl_equal C e))))))))))) hds)).