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 "basic_1/drop1/fwd.ma".
19 include "basic_1/drop/props.ma".
21 include "basic_1/getl/defs.ma".
23 theorem drop1_skip_bind:
24 \forall (b: B).(\forall (e: C).(\forall (hds: PList).(\forall (c:
25 C).(\forall (u: T).((drop1 hds c e) \to (drop1 (Ss hds) (CHead c (Bind b)
26 (lift1 hds u)) (CHead e (Bind b) u)))))))
28 \lambda (b: B).(\lambda (e: C).(\lambda (hds: PList).(let TMP_7 \def
29 (\lambda (p: PList).(\forall (c: C).(\forall (u: T).((drop1 p c e) \to (let
30 TMP_1 \def (Ss p) in (let TMP_2 \def (Bind b) in (let TMP_3 \def (lift1 p u)
31 in (let TMP_4 \def (CHead c TMP_2 TMP_3) in (let TMP_5 \def (Bind b) in (let
32 TMP_6 \def (CHead e TMP_5 u) in (drop1 TMP_1 TMP_4 TMP_6))))))))))) in (let
33 TMP_16 \def (\lambda (c: C).(\lambda (u: T).(\lambda (H: (drop1 PNil c
34 e)).(let H_y \def (drop1_gen_pnil c e H) in (let TMP_12 \def (\lambda (c0:
35 C).(let TMP_8 \def (Bind b) in (let TMP_9 \def (CHead c0 TMP_8 u) in (let
36 TMP_10 \def (Bind b) in (let TMP_11 \def (CHead e TMP_10 u) in (drop1 PNil
37 TMP_9 TMP_11)))))) in (let TMP_13 \def (Bind b) in (let TMP_14 \def (CHead e
38 TMP_13 u) in (let TMP_15 \def (drop1_nil TMP_14) in (eq_ind_r C e TMP_12
39 TMP_15 c H_y))))))))) in (let TMP_44 \def (\lambda (n: nat).(\lambda (n0:
40 nat).(\lambda (p: PList).(\lambda (H: ((\forall (c: C).(\forall (u:
41 T).((drop1 p c e) \to (drop1 (Ss p) (CHead c (Bind b) (lift1 p u)) (CHead e
42 (Bind b) u))))))).(\lambda (c: C).(\lambda (u: T).(\lambda (H0: (drop1 (PCons
43 n n0 p) c e)).(let H_x \def (drop1_gen_pcons c e p n n0 H0) in (let H1 \def
44 H_x in (let TMP_17 \def (\lambda (c2: C).(drop n n0 c c2)) in (let TMP_18
45 \def (\lambda (c2: C).(drop1 p c2 e)) in (let TMP_19 \def (S n0) in (let
46 TMP_20 \def (Ss p) in (let TMP_21 \def (PCons n TMP_19 TMP_20) in (let TMP_22
47 \def (Bind b) in (let TMP_23 \def (lift1 p u) in (let TMP_24 \def (lift n n0
48 TMP_23) in (let TMP_25 \def (CHead c TMP_22 TMP_24) in (let TMP_26 \def (Bind
49 b) in (let TMP_27 \def (CHead e TMP_26 u) in (let TMP_28 \def (drop1 TMP_21
50 TMP_25 TMP_27) in (let TMP_43 \def (\lambda (x: C).(\lambda (H2: (drop n n0 c
51 x)).(\lambda (H3: (drop1 p x e)).(let TMP_29 \def (Bind b) in (let TMP_30
52 \def (lift1 p u) in (let TMP_31 \def (lift n n0 TMP_30) in (let TMP_32 \def
53 (CHead c TMP_29 TMP_31) in (let TMP_33 \def (Bind b) in (let TMP_34 \def
54 (lift1 p u) in (let TMP_35 \def (CHead x TMP_33 TMP_34) in (let TMP_36 \def
55 (S n0) in (let TMP_37 \def (lift1 p u) in (let TMP_38 \def (drop_skip_bind n
56 n0 c x H2 b TMP_37) in (let TMP_39 \def (Bind b) in (let TMP_40 \def (CHead e
57 TMP_39 u) in (let TMP_41 \def (Ss p) in (let TMP_42 \def (H x u H3) in
58 (drop1_cons TMP_32 TMP_35 n TMP_36 TMP_38 TMP_40 TMP_41
59 TMP_42)))))))))))))))))) in (ex2_ind C TMP_17 TMP_18 TMP_28 TMP_43
60 H1))))))))))))))))))))))) in (PList_ind TMP_7 TMP_16 TMP_44 hds)))))).
62 theorem drop1_cons_tail:
63 \forall (c2: C).(\forall (c3: C).(\forall (h: nat).(\forall (d: nat).((drop
64 h d c2 c3) \to (\forall (hds: PList).(\forall (c1: C).((drop1 hds c1 c2) \to
65 (drop1 (PConsTail hds h d) c1 c3))))))))
67 \lambda (c2: C).(\lambda (c3: C).(\lambda (h: nat).(\lambda (d:
68 nat).(\lambda (H: (drop h d c2 c3)).(\lambda (hds: PList).(let TMP_2 \def
69 (\lambda (p: PList).(\forall (c1: C).((drop1 p c1 c2) \to (let TMP_1 \def
70 (PConsTail p h d) in (drop1 TMP_1 c1 c3))))) in (let TMP_7 \def (\lambda (c1:
71 C).(\lambda (H0: (drop1 PNil c1 c2)).(let H_y \def (drop1_gen_pnil c1 c2 H0)
72 in (let TMP_4 \def (\lambda (c: C).(let TMP_3 \def (PCons h d PNil) in (drop1
73 TMP_3 c c3))) in (let TMP_5 \def (drop1_nil c3) in (let TMP_6 \def
74 (drop1_cons c2 c3 h d H c3 PNil TMP_5) in (eq_ind_r C c2 TMP_4 TMP_6 c1
75 H_y))))))) in (let TMP_16 \def (\lambda (n: nat).(\lambda (n0: nat).(\lambda
76 (p: PList).(\lambda (H0: ((\forall (c1: C).((drop1 p c1 c2) \to (drop1
77 (PConsTail p h d) c1 c3))))).(\lambda (c1: C).(\lambda (H1: (drop1 (PCons n
78 n0 p) c1 c2)).(let H_x \def (drop1_gen_pcons c1 c2 p n n0 H1) in (let H2 \def
79 H_x in (let TMP_8 \def (\lambda (c4: C).(drop n n0 c1 c4)) in (let TMP_9 \def
80 (\lambda (c4: C).(drop1 p c4 c2)) in (let TMP_10 \def (PConsTail p h d) in
81 (let TMP_11 \def (PCons n n0 TMP_10) in (let TMP_12 \def (drop1 TMP_11 c1 c3)
82 in (let TMP_15 \def (\lambda (x: C).(\lambda (H3: (drop n n0 c1 x)).(\lambda
83 (H4: (drop1 p x c2)).(let TMP_13 \def (PConsTail p h d) in (let TMP_14 \def
84 (H0 x H4) in (drop1_cons c1 x n n0 H3 c3 TMP_13 TMP_14)))))) in (ex2_ind C
85 TMP_8 TMP_9 TMP_12 TMP_15 H2))))))))))))))) in (PList_ind TMP_2 TMP_7 TMP_16
89 \forall (is1: PList).(\forall (c1: C).(\forall (c0: C).((drop1 is1 c1 c0)
90 \to (\forall (is2: PList).(\forall (c2: C).((drop1 is2 c0 c2) \to (drop1
91 (papp is1 is2) c1 c2)))))))
93 \lambda (is1: PList).(let TMP_2 \def (\lambda (p: PList).(\forall (c1:
94 C).(\forall (c0: C).((drop1 p c1 c0) \to (\forall (is2: PList).(\forall (c2:
95 C).((drop1 is2 c0 c2) \to (let TMP_1 \def (papp p is2) in (drop1 TMP_1 c1
96 c2))))))))) in (let TMP_4 \def (\lambda (c1: C).(\lambda (c0: C).(\lambda (H:
97 (drop1 PNil c1 c0)).(\lambda (is2: PList).(\lambda (c2: C).(\lambda (H0:
98 (drop1 is2 c0 c2)).(let H_y \def (drop1_gen_pnil c1 c0 H) in (let TMP_3 \def
99 (\lambda (c: C).(drop1 is2 c c2)) in (let H1 \def (eq_ind_r C c0 TMP_3 H0 c1
100 H_y) in H1))))))))) in (let TMP_13 \def (\lambda (n: nat).(\lambda (n0:
101 nat).(\lambda (p: PList).(\lambda (H: ((\forall (c1: C).(\forall (c0:
102 C).((drop1 p c1 c0) \to (\forall (is2: PList).(\forall (c2: C).((drop1 is2 c0
103 c2) \to (drop1 (papp p is2) c1 c2))))))))).(\lambda (c1: C).(\lambda (c0:
104 C).(\lambda (H0: (drop1 (PCons n n0 p) c1 c0)).(\lambda (is2: PList).(\lambda
105 (c2: C).(\lambda (H1: (drop1 is2 c0 c2)).(let H_x \def (drop1_gen_pcons c1 c0
106 p n n0 H0) in (let H2 \def H_x in (let TMP_5 \def (\lambda (c3: C).(drop n n0
107 c1 c3)) in (let TMP_6 \def (\lambda (c3: C).(drop1 p c3 c0)) in (let TMP_7
108 \def (papp p is2) in (let TMP_8 \def (PCons n n0 TMP_7) in (let TMP_9 \def
109 (drop1 TMP_8 c1 c2) in (let TMP_12 \def (\lambda (x: C).(\lambda (H3: (drop n
110 n0 c1 x)).(\lambda (H4: (drop1 p x c0)).(let TMP_10 \def (papp p is2) in (let
111 TMP_11 \def (H x c0 H4 is2 c2 H1) in (drop1_cons c1 x n n0 H3 c2 TMP_10
112 TMP_11)))))) in (ex2_ind C TMP_5 TMP_6 TMP_9 TMP_12 H2))))))))))))))))))) in
113 (PList_ind TMP_2 TMP_4 TMP_13 is1)))).