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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 *)
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15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/arity/lift1".
19 include "arity/props.ma".
21 include "drop1/defs.ma".
24 \forall (g: G).(\forall (a: A).(\forall (c2: C).(\forall (hds:
25 PList).(\forall (c1: C).(\forall (t: T).((drop1 hds c1 c2) \to ((arity g c2 t
26 a) \to (arity g c1 (lift1 hds t) a))))))))
28 \lambda (g: G).(\lambda (a: A).(\lambda (c2: C).(\lambda (hds:
29 PList).(PList_ind (\lambda (p: PList).(\forall (c1: C).(\forall (t:
30 T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1 (lift1 p t) a))))))
31 (\lambda (c1: C).(\lambda (t: T).(\lambda (H: (drop1 PNil c1 c2)).(\lambda
32 (H0: (arity g c2 t a)).(let H1 \def (match H in drop1 return (\lambda (p:
33 PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop1 p c c0)).((eq
34 PList p PNil) \to ((eq C c c1) \to ((eq C c0 c2) \to (arity g c1 t a))))))))
35 with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda
36 (H2: (eq C c c1)).(\lambda (H3: (eq C c c2)).(eq_ind C c1 (\lambda (c0:
37 C).((eq C c0 c2) \to (arity g c1 t a))) (\lambda (H4: (eq C c1 c2)).(eq_ind C
38 c2 (\lambda (c0: C).(arity g c0 t a)) H0 c1 (sym_eq C c1 c2 H4))) c (sym_eq C
39 c c1 H2) H3)))) | (drop1_cons c0 c3 h d H1 c4 hds0 H2) \Rightarrow (\lambda
40 (H3: (eq PList (PCons h d hds0) PNil)).(\lambda (H4: (eq C c0 c1)).(\lambda
41 (H5: (eq C c4 c2)).((let H6 \def (eq_ind PList (PCons h d hds0) (\lambda (e:
42 PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil
43 \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in
44 (False_ind ((eq C c0 c1) \to ((eq C c4 c2) \to ((drop h d c0 c3) \to ((drop1
45 hds0 c3 c4) \to (arity g c1 t a))))) H6)) H4 H5 H1 H2))))]) in (H1
46 (refl_equal PList PNil) (refl_equal C c1) (refl_equal C c2))))))) (\lambda
47 (n: nat).(\lambda (n0: nat).(\lambda (p: PList).(\lambda (H: ((\forall (c1:
48 C).(\forall (t: T).((drop1 p c1 c2) \to ((arity g c2 t a) \to (arity g c1
49 (lift1 p t) a))))))).(\lambda (c1: C).(\lambda (t: T).(\lambda (H0: (drop1
50 (PCons n n0 p) c1 c2)).(\lambda (H1: (arity g c2 t a)).(let H2 \def (match H0
51 in drop1 return (\lambda (p0: PList).(\lambda (c: C).(\lambda (c0:
52 C).(\lambda (_: (drop1 p0 c c0)).((eq PList p0 (PCons n n0 p)) \to ((eq C c
53 c1) \to ((eq C c0 c2) \to (arity g c1 (lift n n0 (lift1 p t)) a)))))))) with
54 [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil (PCons n n0
55 p))).(\lambda (H3: (eq C c c1)).(\lambda (H4: (eq C c c2)).((let H5 \def
56 (eq_ind PList PNil (\lambda (e: PList).(match e in PList return (\lambda (_:
57 PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow False]))
58 I (PCons n n0 p) H2) in (False_ind ((eq C c c1) \to ((eq C c c2) \to (arity g
59 c1 (lift n n0 (lift1 p t)) a))) H5)) H3 H4)))) | (drop1_cons c0 c3 h d H2 c4
60 hds0 H3) \Rightarrow (\lambda (H4: (eq PList (PCons h d hds0) (PCons n n0
61 p))).(\lambda (H5: (eq C c0 c1)).(\lambda (H6: (eq C c4 c2)).((let H7 \def
62 (f_equal PList PList (\lambda (e: PList).(match e in PList return (\lambda
63 (_: PList).PList) with [PNil \Rightarrow hds0 | (PCons _ _ p0) \Rightarrow
64 p0])) (PCons h d hds0) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat
65 (\lambda (e: PList).(match e in PList return (\lambda (_: PList).nat) with
66 [PNil \Rightarrow d | (PCons _ n1 _) \Rightarrow n1])) (PCons h d hds0)
67 (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e:
68 PList).(match e in PList return (\lambda (_: PList).nat) with [PNil
69 \Rightarrow h | (PCons n1 _ _) \Rightarrow n1])) (PCons h d hds0) (PCons n n0
70 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds0
71 p) \to ((eq C c0 c1) \to ((eq C c4 c2) \to ((drop n1 d c0 c3) \to ((drop1
72 hds0 c3 c4) \to (arity g c1 (lift n n0 (lift1 p t)) a)))))))) (\lambda (H10:
73 (eq nat d n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds0 p) \to ((eq
74 C c0 c1) \to ((eq C c4 c2) \to ((drop n n1 c0 c3) \to ((drop1 hds0 c3 c4) \to
75 (arity g c1 (lift n n0 (lift1 p t)) a))))))) (\lambda (H11: (eq PList hds0
76 p)).(eq_ind PList p (\lambda (p0: PList).((eq C c0 c1) \to ((eq C c4 c2) \to
77 ((drop n n0 c0 c3) \to ((drop1 p0 c3 c4) \to (arity g c1 (lift n n0 (lift1 p
78 t)) a)))))) (\lambda (H12: (eq C c0 c1)).(eq_ind C c1 (\lambda (c: C).((eq C
79 c4 c2) \to ((drop n n0 c c3) \to ((drop1 p c3 c4) \to (arity g c1 (lift n n0
80 (lift1 p t)) a))))) (\lambda (H13: (eq C c4 c2)).(eq_ind C c2 (\lambda (c:
81 C).((drop n n0 c1 c3) \to ((drop1 p c3 c) \to (arity g c1 (lift n n0 (lift1 p
82 t)) a)))) (\lambda (H14: (drop n n0 c1 c3)).(\lambda (H15: (drop1 p c3
83 c2)).(arity_lift g c3 (lift1 p t) a (H c3 t H15 H1) c1 n n0 H14))) c4 (sym_eq
84 C c4 c2 H13))) c0 (sym_eq C c0 c1 H12))) hds0 (sym_eq PList hds0 p H11))) d
85 (sym_eq nat d n0 H10))) h (sym_eq nat h n H9))) H8)) H7)) H5 H6 H2 H3))))])
86 in (H2 (refl_equal PList (PCons n n0 p)) (refl_equal C c1) (refl_equal C
87 c2))))))))))) hds)))).