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7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
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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/drop1/props".
19 include "drop1/defs.ma".
21 include "drop/props.ma".
23 include "getl/defs.ma".
25 theorem drop1_skip_bind:
26 \forall (b: B).(\forall (e: C).(\forall (hds: PList).(\forall (c:
27 C).(\forall (u: T).((drop1 hds c e) \to (drop1 (Ss hds) (CHead c (Bind b)
28 (lift1 hds u)) (CHead e (Bind b) u)))))))
30 \lambda (b: B).(\lambda (e: C).(\lambda (hds: PList).(PList_ind (\lambda (p:
31 PList).(\forall (c: C).(\forall (u: T).((drop1 p c e) \to (drop1 (Ss p)
32 (CHead c (Bind b) (lift1 p u)) (CHead e (Bind b) u)))))) (\lambda (c:
33 C).(\lambda (u: T).(\lambda (H: (drop1 PNil c e)).(let H0 \def (match H in
34 drop1 return (\lambda (p: PList).(\lambda (c0: C).(\lambda (c1: C).(\lambda
35 (_: (drop1 p c0 c1)).((eq PList p PNil) \to ((eq C c0 c) \to ((eq C c1 e) \to
36 (drop1 PNil (CHead c (Bind b) u) (CHead e (Bind b) u))))))))) with
37 [(drop1_nil c0) \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H1:
38 (eq C c0 c)).(\lambda (H2: (eq C c0 e)).(eq_ind C c (\lambda (c1: C).((eq C
39 c1 e) \to (drop1 PNil (CHead c (Bind b) u) (CHead e (Bind b) u)))) (\lambda
40 (H3: (eq C c e)).(eq_ind C e (\lambda (c: C).(drop1 PNil (CHead c (Bind b) u)
41 (CHead e (Bind b) u))) (drop1_nil (CHead e (Bind b) u)) c (sym_eq C c e H3)))
42 c0 (sym_eq C c0 c H1) H2)))) | (drop1_cons c1 c2 h d H0 c3 hds H1)
43 \Rightarrow (\lambda (H2: (eq PList (PCons h d hds) PNil)).(\lambda (H3: (eq
44 C c1 c)).(\lambda (H4: (eq C c3 e)).((let H5 \def (eq_ind PList (PCons h d
45 hds) (\lambda (e0: PList).(match e0 in PList return (\lambda (_: PList).Prop)
46 with [PNil \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H2)
47 in (False_ind ((eq C c1 c) \to ((eq C c3 e) \to ((drop h d c1 c2) \to ((drop1
48 hds c2 c3) \to (drop1 PNil (CHead c (Bind b) u) (CHead e (Bind b) u))))))
49 H5)) H3 H4 H0 H1))))]) in (H0 (refl_equal PList PNil) (refl_equal C c)
50 (refl_equal C e)))))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda (p:
51 PList).(\lambda (H: ((\forall (c: C).(\forall (u: T).((drop1 p c e) \to
52 (drop1 (Ss p) (CHead c (Bind b) (lift1 p u)) (CHead e (Bind b)
53 u))))))).(\lambda (c: C).(\lambda (u: T).(\lambda (H0: (drop1 (PCons n n0 p)
54 c e)).(let H1 \def (match H0 in drop1 return (\lambda (p0: PList).(\lambda
55 (c0: C).(\lambda (c1: C).(\lambda (_: (drop1 p0 c0 c1)).((eq PList p0 (PCons
56 n n0 p)) \to ((eq C c0 c) \to ((eq C c1 e) \to (drop1 (PCons n (S n0) (Ss p))
57 (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u))))))))) with
58 [(drop1_nil c0) \Rightarrow (\lambda (H1: (eq PList PNil (PCons n n0
59 p))).(\lambda (H2: (eq C c0 c)).(\lambda (H3: (eq C c0 e)).((let H4 \def
60 (eq_ind PList PNil (\lambda (e0: PList).(match e0 in PList return (\lambda
61 (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _) \Rightarrow
62 False])) I (PCons n n0 p) H1) in (False_ind ((eq C c0 c) \to ((eq C c0 e) \to
63 (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u)))
64 (CHead e (Bind b) u)))) H4)) H2 H3)))) | (drop1_cons c1 c2 h d H1 c3 hds H2)
65 \Rightarrow (\lambda (H3: (eq PList (PCons h d hds) (PCons n n0 p))).(\lambda
66 (H4: (eq C c1 c)).(\lambda (H5: (eq C c3 e)).((let H6 \def (f_equal PList
67 PList (\lambda (e0: PList).(match e0 in PList return (\lambda (_:
68 PList).PList) with [PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p]))
69 (PCons h d hds) (PCons n n0 p) H3) in ((let H7 \def (f_equal PList nat
70 (\lambda (e0: PList).(match e0 in PList return (\lambda (_: PList).nat) with
71 [PNil \Rightarrow d | (PCons _ n _) \Rightarrow n])) (PCons h d hds) (PCons n
72 n0 p) H3) in ((let H8 \def (f_equal PList nat (\lambda (e0: PList).(match e0
73 in PList return (\lambda (_: PList).nat) with [PNil \Rightarrow h | (PCons n
74 _ _) \Rightarrow n])) (PCons h d hds) (PCons n n0 p) H3) in (eq_ind nat n
75 (\lambda (n1: nat).((eq nat d n0) \to ((eq PList hds p) \to ((eq C c1 c) \to
76 ((eq C c3 e) \to ((drop n1 d c1 c2) \to ((drop1 hds c2 c3) \to (drop1 (PCons
77 n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b)
78 u))))))))) (\lambda (H9: (eq nat d n0)).(eq_ind nat n0 (\lambda (n1:
79 nat).((eq PList hds p) \to ((eq C c1 c) \to ((eq C c3 e) \to ((drop n n1 c1
80 c2) \to ((drop1 hds c2 c3) \to (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind
81 b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u)))))))) (\lambda (H10: (eq
82 PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c1 c) \to ((eq C c3
83 e) \to ((drop n n0 c1 c2) \to ((drop1 p0 c2 c3) \to (drop1 (PCons n (S n0)
84 (Ss p)) (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u)))))))
85 (\lambda (H11: (eq C c1 c)).(eq_ind C c (\lambda (c0: C).((eq C c3 e) \to
86 ((drop n n0 c0 c2) \to ((drop1 p c2 c3) \to (drop1 (PCons n (S n0) (Ss p))
87 (CHead c (Bind b) (lift n n0 (lift1 p u))) (CHead e (Bind b) u)))))) (\lambda
88 (H12: (eq C c3 e)).(eq_ind C e (\lambda (c0: C).((drop n n0 c c2) \to ((drop1
89 p c2 c0) \to (drop1 (PCons n (S n0) (Ss p)) (CHead c (Bind b) (lift n n0
90 (lift1 p u))) (CHead e (Bind b) u))))) (\lambda (H13: (drop n n0 c
91 c2)).(\lambda (H14: (drop1 p c2 e)).(drop1_cons (CHead c (Bind b) (lift n n0
92 (lift1 p u))) (CHead c2 (Bind b) (lift1 p u)) n (S n0) (drop_skip_bind n n0 c
93 c2 H13 b (lift1 p u)) (CHead e (Bind b) u) (Ss p) (H c2 u H14)))) c3 (sym_eq
94 C c3 e H12))) c1 (sym_eq C c1 c H11))) hds (sym_eq PList hds p H10))) d
95 (sym_eq nat d n0 H9))) h (sym_eq nat h n H8))) H7)) H6)) H4 H5 H1 H2))))]) in
96 (H1 (refl_equal PList (PCons n n0 p)) (refl_equal C c) (refl_equal C
99 theorem drop1_cons_tail:
100 \forall (c2: C).(\forall (c3: C).(\forall (h: nat).(\forall (d: nat).((drop
101 h d c2 c3) \to (\forall (hds: PList).(\forall (c1: C).((drop1 hds c1 c2) \to
102 (drop1 (PConsTail hds h d) c1 c3))))))))
104 \lambda (c2: C).(\lambda (c3: C).(\lambda (h: nat).(\lambda (d:
105 nat).(\lambda (H: (drop h d c2 c3)).(\lambda (hds: PList).(PList_ind (\lambda
106 (p: PList).(\forall (c1: C).((drop1 p c1 c2) \to (drop1 (PConsTail p h d) c1
107 c3)))) (\lambda (c1: C).(\lambda (H0: (drop1 PNil c1 c2)).(let H1 \def (match
108 H0 in drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda (c0:
109 C).(\lambda (_: (drop1 p c c0)).((eq PList p PNil) \to ((eq C c c1) \to ((eq
110 C c0 c2) \to (drop1 (PCons h d PNil) c1 c3)))))))) with [(drop1_nil c)
111 \Rightarrow (\lambda (_: (eq PList PNil PNil)).(\lambda (H2: (eq C c
112 c1)).(\lambda (H3: (eq C c c2)).(eq_ind C c1 (\lambda (c0: C).((eq C c0 c2)
113 \to (drop1 (PCons h d PNil) c1 c3))) (\lambda (H4: (eq C c1 c2)).(eq_ind C c2
114 (\lambda (c0: C).(drop1 (PCons h d PNil) c0 c3)) (drop1_cons c2 c3 h d H c3
115 PNil (drop1_nil c3)) c1 (sym_eq C c1 c2 H4))) c (sym_eq C c c1 H2) H3)))) |
116 (drop1_cons c0 c4 h0 d0 H1 c5 hds H2) \Rightarrow (\lambda (H3: (eq PList
117 (PCons h0 d0 hds) PNil)).(\lambda (H4: (eq C c0 c1)).(\lambda (H5: (eq C c5
118 c2)).((let H6 \def (eq_ind PList (PCons h0 d0 hds) (\lambda (e: PList).(match
119 e in PList return (\lambda (_: PList).Prop) with [PNil \Rightarrow False |
120 (PCons _ _ _) \Rightarrow True])) I PNil H3) in (False_ind ((eq C c0 c1) \to
121 ((eq C c5 c2) \to ((drop h0 d0 c0 c4) \to ((drop1 hds c4 c5) \to (drop1
122 (PCons h d PNil) c1 c3))))) H6)) H4 H5 H1 H2))))]) in (H1 (refl_equal PList
123 PNil) (refl_equal C c1) (refl_equal C c2))))) (\lambda (n: nat).(\lambda (n0:
124 nat).(\lambda (p: PList).(\lambda (H0: ((\forall (c1: C).((drop1 p c1 c2) \to
125 (drop1 (PConsTail p h d) c1 c3))))).(\lambda (c1: C).(\lambda (H1: (drop1
126 (PCons n n0 p) c1 c2)).(let H2 \def (match H1 in drop1 return (\lambda (p0:
127 PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop1 p0 c c0)).((eq
128 PList p0 (PCons n n0 p)) \to ((eq C c c1) \to ((eq C c0 c2) \to (drop1 (PCons
129 n n0 (PConsTail p h d)) c1 c3)))))))) with [(drop1_nil c) \Rightarrow
130 (\lambda (H2: (eq PList PNil (PCons n n0 p))).(\lambda (H3: (eq C c
131 c1)).(\lambda (H4: (eq C c c2)).((let H5 \def (eq_ind PList PNil (\lambda (e:
132 PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil
133 \Rightarrow True | (PCons _ _ _) \Rightarrow False])) I (PCons n n0 p) H2) in
134 (False_ind ((eq C c c1) \to ((eq C c c2) \to (drop1 (PCons n n0 (PConsTail p
135 h d)) c1 c3))) H5)) H3 H4)))) | (drop1_cons c0 c4 h0 d0 H2 c5 hds H3)
136 \Rightarrow (\lambda (H4: (eq PList (PCons h0 d0 hds) (PCons n n0
137 p))).(\lambda (H5: (eq C c0 c1)).(\lambda (H6: (eq C c5 c2)).((let H7 \def
138 (f_equal PList PList (\lambda (e: PList).(match e in PList return (\lambda
139 (_: PList).PList) with [PNil \Rightarrow hds | (PCons _ _ p) \Rightarrow p]))
140 (PCons h0 d0 hds) (PCons n n0 p) H4) in ((let H8 \def (f_equal PList nat
141 (\lambda (e: PList).(match e in PList return (\lambda (_: PList).nat) with
142 [PNil \Rightarrow d0 | (PCons _ n _) \Rightarrow n])) (PCons h0 d0 hds)
143 (PCons n n0 p) H4) in ((let H9 \def (f_equal PList nat (\lambda (e:
144 PList).(match e in PList return (\lambda (_: PList).nat) with [PNil
145 \Rightarrow h0 | (PCons n _ _) \Rightarrow n])) (PCons h0 d0 hds) (PCons n n0
146 p) H4) in (eq_ind nat n (\lambda (n1: nat).((eq nat d0 n0) \to ((eq PList hds
147 p) \to ((eq C c0 c1) \to ((eq C c5 c2) \to ((drop n1 d0 c0 c4) \to ((drop1
148 hds c4 c5) \to (drop1 (PCons n n0 (PConsTail p h d)) c1 c3)))))))) (\lambda
149 (H10: (eq nat d0 n0)).(eq_ind nat n0 (\lambda (n1: nat).((eq PList hds p) \to
150 ((eq C c0 c1) \to ((eq C c5 c2) \to ((drop n n1 c0 c4) \to ((drop1 hds c4 c5)
151 \to (drop1 (PCons n n0 (PConsTail p h d)) c1 c3))))))) (\lambda (H11: (eq
152 PList hds p)).(eq_ind PList p (\lambda (p0: PList).((eq C c0 c1) \to ((eq C
153 c5 c2) \to ((drop n n0 c0 c4) \to ((drop1 p0 c4 c5) \to (drop1 (PCons n n0
154 (PConsTail p h d)) c1 c3)))))) (\lambda (H12: (eq C c0 c1)).(eq_ind C c1
155 (\lambda (c: C).((eq C c5 c2) \to ((drop n n0 c c4) \to ((drop1 p c4 c5) \to
156 (drop1 (PCons n n0 (PConsTail p h d)) c1 c3))))) (\lambda (H13: (eq C c5
157 c2)).(eq_ind C c2 (\lambda (c: C).((drop n n0 c1 c4) \to ((drop1 p c4 c) \to
158 (drop1 (PCons n n0 (PConsTail p h d)) c1 c3)))) (\lambda (H14: (drop n n0 c1
159 c4)).(\lambda (H15: (drop1 p c4 c2)).(drop1_cons c1 c4 n n0 H14 c3 (PConsTail
160 p h d) (H0 c4 H15)))) c5 (sym_eq C c5 c2 H13))) c0 (sym_eq C c0 c1 H12))) hds
161 (sym_eq PList hds p H11))) d0 (sym_eq nat d0 n0 H10))) h0 (sym_eq nat h0 n
162 H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList (PCons n n0 p))
163 (refl_equal C c1) (refl_equal C c2))))))))) hds)))))).