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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 include "LambdaDelta-1/drop1/defs.ma".
19 theorem drop1_gen_pnil:
20 \forall (c1: C).(\forall (c2: C).((drop1 PNil c1 c2) \to (eq C c1 c2)))
22 \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c1 c2)).(insert_eq
23 PList PNil (\lambda (p: PList).(drop1 p c1 c2)) (\lambda (_: PList).(eq C c1
24 c2)) (\lambda (y: PList).(\lambda (H0: (drop1 y c1 c2)).(drop1_ind (\lambda
25 (p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p PNil) \to (eq C c
26 c0))))) (\lambda (c: C).(\lambda (_: (eq PList PNil PNil)).(refl_equal C c)))
27 (\lambda (c3: C).(\lambda (c4: C).(\lambda (h: nat).(\lambda (d:
28 nat).(\lambda (_: (drop h d c3 c4)).(\lambda (c5: C).(\lambda (hds:
29 PList).(\lambda (_: (drop1 hds c4 c5)).(\lambda (_: (((eq PList hds PNil) \to
30 (eq C c4 c5)))).(\lambda (H4: (eq PList (PCons h d hds) PNil)).(let H5 \def
31 (eq_ind PList (PCons h d hds) (\lambda (ee: PList).(match ee in PList return
32 (\lambda (_: PList).Prop) with [PNil \Rightarrow False | (PCons _ _ _)
33 \Rightarrow True])) I PNil H4) in (False_ind (eq C c3 c5) H5)))))))))))) y c1
36 theorem drop1_gen_pcons:
37 \forall (c1: C).(\forall (c3: C).(\forall (hds: PList).(\forall (h:
38 nat).(\forall (d: nat).((drop1 (PCons h d hds) c1 c3) \to (ex2 C (\lambda
39 (c2: C).(drop h d c1 c2)) (\lambda (c2: C).(drop1 hds c2 c3))))))))
41 \lambda (c1: C).(\lambda (c3: C).(\lambda (hds: PList).(\lambda (h:
42 nat).(\lambda (d: nat).(\lambda (H: (drop1 (PCons h d hds) c1 c3)).(insert_eq
43 PList (PCons h d hds) (\lambda (p: PList).(drop1 p c1 c3)) (\lambda (_:
44 PList).(ex2 C (\lambda (c2: C).(drop h d c1 c2)) (\lambda (c2: C).(drop1 hds
45 c2 c3)))) (\lambda (y: PList).(\lambda (H0: (drop1 y c1 c3)).(drop1_ind
46 (\lambda (p: PList).(\lambda (c: C).(\lambda (c0: C).((eq PList p (PCons h d
47 hds)) \to (ex2 C (\lambda (c2: C).(drop h d c c2)) (\lambda (c2: C).(drop1
48 hds c2 c0))))))) (\lambda (c: C).(\lambda (H1: (eq PList PNil (PCons h d
49 hds))).(let H2 \def (eq_ind PList PNil (\lambda (ee: PList).(match ee in
50 PList return (\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _
51 _ _) \Rightarrow False])) I (PCons h d hds) H1) in (False_ind (ex2 C (\lambda
52 (c2: C).(drop h d c c2)) (\lambda (c2: C).(drop1 hds c2 c))) H2)))) (\lambda
53 (c2: C).(\lambda (c4: C).(\lambda (h0: nat).(\lambda (d0: nat).(\lambda (H1:
54 (drop h0 d0 c2 c4)).(\lambda (c5: C).(\lambda (hds0: PList).(\lambda (H2:
55 (drop1 hds0 c4 c5)).(\lambda (H3: (((eq PList hds0 (PCons h d hds)) \to (ex2
56 C (\lambda (c6: C).(drop h d c4 c6)) (\lambda (c6: C).(drop1 hds c6
57 c5)))))).(\lambda (H4: (eq PList (PCons h0 d0 hds0) (PCons h d hds))).(let H5
58 \def (f_equal PList nat (\lambda (e: PList).(match e in PList return (\lambda
59 (_: PList).nat) with [PNil \Rightarrow h0 | (PCons n _ _) \Rightarrow n]))
60 (PCons h0 d0 hds0) (PCons h d hds) H4) in ((let H6 \def (f_equal PList nat
61 (\lambda (e: PList).(match e in PList return (\lambda (_: PList).nat) with
62 [PNil \Rightarrow d0 | (PCons _ n _) \Rightarrow n])) (PCons h0 d0 hds0)
63 (PCons h d hds) H4) in ((let H7 \def (f_equal PList PList (\lambda (e:
64 PList).(match e in PList return (\lambda (_: PList).PList) with [PNil
65 \Rightarrow hds0 | (PCons _ _ p) \Rightarrow p])) (PCons h0 d0 hds0) (PCons h
66 d hds) H4) in (\lambda (H8: (eq nat d0 d)).(\lambda (H9: (eq nat h0 h)).(let
67 H10 \def (eq_ind PList hds0 (\lambda (p: PList).((eq PList p (PCons h d hds))
68 \to (ex2 C (\lambda (c6: C).(drop h d c4 c6)) (\lambda (c6: C).(drop1 hds c6
69 c5))))) H3 hds H7) in (let H11 \def (eq_ind PList hds0 (\lambda (p:
70 PList).(drop1 p c4 c5)) H2 hds H7) in (let H12 \def (eq_ind nat d0 (\lambda
71 (n: nat).(drop h0 n c2 c4)) H1 d H8) in (let H13 \def (eq_ind nat h0 (\lambda
72 (n: nat).(drop n d c2 c4)) H12 h H9) in (ex_intro2 C (\lambda (c6: C).(drop h
73 d c2 c6)) (\lambda (c6: C).(drop1 hds c6 c5)) c4 H13 H11)))))))) H6))
74 H5)))))))))))) y c1 c3 H0))) H)))))).