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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 *)
13 (**************************************************************************)
15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/LambdaDelta/drop1/getl".
19 include "drop1/defs.ma".
21 include "getl/drop.ma".
23 theorem drop1_getl_trans:
24 \forall (hds: PList).(\forall (c1: C).(\forall (c2: C).((drop1 hds c2 c1)
25 \to (\forall (b: B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl
26 i c1 (CHead e1 (Bind b) v)) \to (ex2 C (\lambda (e2: C).(drop1 (ptrans hds i)
27 e2 e1)) (\lambda (e2: C).(getl (trans hds i) c2 (CHead e2 (Bind b) (lift1
28 (ptrans hds i) v)))))))))))))
30 \lambda (hds: PList).(PList_ind (\lambda (p: PList).(\forall (c1:
31 C).(\forall (c2: C).((drop1 p c2 c1) \to (\forall (b: B).(\forall (e1:
32 C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1 (Bind b) v)) \to
33 (ex2 C (\lambda (e2: C).(drop1 (ptrans p i) e2 e1)) (\lambda (e2: C).(getl
34 (trans p i) c2 (CHead e2 (Bind b) (lift1 (ptrans p i) v))))))))))))))
35 (\lambda (c1: C).(\lambda (c2: C).(\lambda (H: (drop1 PNil c2 c1)).(\lambda
36 (b: B).(\lambda (e1: C).(\lambda (v: T).(\lambda (i: nat).(\lambda (H0: (getl
37 i c1 (CHead e1 (Bind b) v))).(let H1 \def (match H in drop1 return (\lambda
38 (p: PList).(\lambda (c: C).(\lambda (c0: C).(\lambda (_: (drop1 p c c0)).((eq
39 PList p PNil) \to ((eq C c c2) \to ((eq C c0 c1) \to (ex2 C (\lambda (e2:
40 C).(drop1 PNil e2 e1)) (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b)
41 v))))))))))) with [(drop1_nil c) \Rightarrow (\lambda (_: (eq PList PNil
42 PNil)).(\lambda (H2: (eq C c c2)).(\lambda (H3: (eq C c c1)).(eq_ind C c2
43 (\lambda (c0: C).((eq C c0 c1) \to (ex2 C (\lambda (e2: C).(drop1 PNil e2
44 e1)) (\lambda (e2: C).(getl i c2 (CHead e2 (Bind b) v)))))) (\lambda (H4: (eq
45 C c2 c1)).(eq_ind C c1 (\lambda (c0: C).(ex2 C (\lambda (e2: C).(drop1 PNil
46 e2 e1)) (\lambda (e2: C).(getl i c0 (CHead e2 (Bind b) v))))) (ex_intro2 C
47 (\lambda (e2: C).(drop1 PNil e2 e1)) (\lambda (e2: C).(getl i c1 (CHead e2
48 (Bind b) v))) e1 (drop1_nil e1) H0) c2 (sym_eq C c2 c1 H4))) c (sym_eq C c c2
49 H2) H3)))) | (drop1_cons c0 c3 h d H1 c4 hds0 H2) \Rightarrow (\lambda (H3:
50 (eq PList (PCons h d hds0) PNil)).(\lambda (H4: (eq C c0 c2)).(\lambda (H5:
51 (eq C c4 c1)).((let H6 \def (eq_ind PList (PCons h d hds0) (\lambda (e:
52 PList).(match e in PList return (\lambda (_: PList).Prop) with [PNil
53 \Rightarrow False | (PCons _ _ _) \Rightarrow True])) I PNil H3) in
54 (False_ind ((eq C c0 c2) \to ((eq C c4 c1) \to ((drop h d c0 c3) \to ((drop1
55 hds0 c3 c4) \to (ex2 C (\lambda (e2: C).(drop1 PNil e2 e1)) (\lambda (e2:
56 C).(getl i c2 (CHead e2 (Bind b) v)))))))) H6)) H4 H5 H1 H2))))]) in (H1
57 (refl_equal PList PNil) (refl_equal C c2) (refl_equal C c1)))))))))))
58 (\lambda (h: nat).(\lambda (d: nat).(\lambda (hds0: PList).(\lambda (H:
59 ((\forall (c1: C).(\forall (c2: C).((drop1 hds0 c2 c1) \to (\forall (b:
60 B).(\forall (e1: C).(\forall (v: T).(\forall (i: nat).((getl i c1 (CHead e1
61 (Bind b) v)) \to (ex2 C (\lambda (e2: C).(drop1 (ptrans hds0 i) e2 e1))
62 (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift1 (ptrans
63 hds0 i) v))))))))))))))).(\lambda (c1: C).(\lambda (c2: C).(\lambda (H0:
64 (drop1 (PCons h d hds0) c2 c1)).(\lambda (b: B).(\lambda (e1: C).(\lambda (v:
65 T).(\lambda (i: nat).(\lambda (H1: (getl i c1 (CHead e1 (Bind b) v))).(let H2
66 \def (match H0 in drop1 return (\lambda (p: PList).(\lambda (c: C).(\lambda
67 (c0: C).(\lambda (_: (drop1 p c c0)).((eq PList p (PCons h d hds0)) \to ((eq
68 C c c2) \to ((eq C c0 c1) \to (ex2 C (\lambda (e2: C).(drop1 (match (blt
69 (trans hds0 i) d) with [true \Rightarrow (PCons h (minus d (S (trans hds0
70 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) e2 e1)) (\lambda
71 (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans
72 hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b)
73 (lift1 (match (blt (trans hds0 i) d) with [true \Rightarrow (PCons h (minus d
74 (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)])
75 v)))))))))))) with [(drop1_nil c) \Rightarrow (\lambda (H2: (eq PList PNil
76 (PCons h d hds0))).(\lambda (H3: (eq C c c2)).(\lambda (H4: (eq C c
77 c1)).((let H5 \def (eq_ind PList PNil (\lambda (e: PList).(match e in PList
78 return (\lambda (_: PList).Prop) with [PNil \Rightarrow True | (PCons _ _ _)
79 \Rightarrow False])) I (PCons h d hds0) H2) in (False_ind ((eq C c c2) \to
80 ((eq C c c1) \to (ex2 C (\lambda (e2: C).(drop1 (match (blt (trans hds0 i) d)
81 with [true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i))
82 | false \Rightarrow (ptrans hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match
83 (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i) | false
84 \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (lift1 (match
85 (blt (trans hds0 i) d) with [true \Rightarrow (PCons h (minus d (S (trans
86 hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) v)))))))
87 H5)) H3 H4)))) | (drop1_cons c0 c3 h0 d0 H2 c4 hds1 H3) \Rightarrow (\lambda
88 (H4: (eq PList (PCons h0 d0 hds1) (PCons h d hds0))).(\lambda (H5: (eq C c0
89 c2)).(\lambda (H6: (eq C c4 c1)).((let H7 \def (f_equal PList PList (\lambda
90 (e: PList).(match e in PList return (\lambda (_: PList).PList) with [PNil
91 \Rightarrow hds1 | (PCons _ _ p) \Rightarrow p])) (PCons h0 d0 hds1) (PCons h
92 d hds0) H4) in ((let H8 \def (f_equal PList nat (\lambda (e: PList).(match e
93 in PList return (\lambda (_: PList).nat) with [PNil \Rightarrow d0 | (PCons _
94 n _) \Rightarrow n])) (PCons h0 d0 hds1) (PCons h d hds0) H4) in ((let H9
95 \def (f_equal PList nat (\lambda (e: PList).(match e in PList return (\lambda
96 (_: PList).nat) with [PNil \Rightarrow h0 | (PCons n _ _) \Rightarrow n]))
97 (PCons h0 d0 hds1) (PCons h d hds0) H4) in (eq_ind nat h (\lambda (n:
98 nat).((eq nat d0 d) \to ((eq PList hds1 hds0) \to ((eq C c0 c2) \to ((eq C c4
99 c1) \to ((drop n d0 c0 c3) \to ((drop1 hds1 c3 c4) \to (ex2 C (\lambda (e2:
100 C).(drop1 (match (blt (trans hds0 i) d) with [true \Rightarrow (PCons h
101 (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans
102 hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with
103 [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i)
104 h)]) c2 (CHead e2 (Bind b) (lift1 (match (blt (trans hds0 i) d) with [true
105 \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false
106 \Rightarrow (ptrans hds0 i)]) v)))))))))))) (\lambda (H10: (eq nat d0
107 d)).(eq_ind nat d (\lambda (n: nat).((eq PList hds1 hds0) \to ((eq C c0 c2)
108 \to ((eq C c4 c1) \to ((drop h n c0 c3) \to ((drop1 hds1 c3 c4) \to (ex2 C
109 (\lambda (e2: C).(drop1 (match (blt (trans hds0 i) d) with [true \Rightarrow
110 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow
111 (ptrans hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match (blt (trans hds0 i)
112 d) with [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans
113 hds0 i) h)]) c2 (CHead e2 (Bind b) (lift1 (match (blt (trans hds0 i) d) with
114 [true \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) |
115 false \Rightarrow (ptrans hds0 i)]) v))))))))))) (\lambda (H11: (eq PList
116 hds1 hds0)).(eq_ind PList hds0 (\lambda (p: PList).((eq C c0 c2) \to ((eq C
117 c4 c1) \to ((drop h d c0 c3) \to ((drop1 p c3 c4) \to (ex2 C (\lambda (e2:
118 C).(drop1 (match (blt (trans hds0 i) d) with [true \Rightarrow (PCons h
119 (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans
120 hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with
121 [true \Rightarrow (trans hds0 i) | false \Rightarrow (plus (trans hds0 i)
122 h)]) c2 (CHead e2 (Bind b) (lift1 (match (blt (trans hds0 i) d) with [true
123 \Rightarrow (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false
124 \Rightarrow (ptrans hds0 i)]) v)))))))))) (\lambda (H12: (eq C c0
125 c2)).(eq_ind C c2 (\lambda (c: C).((eq C c4 c1) \to ((drop h d c c3) \to
126 ((drop1 hds0 c3 c4) \to (ex2 C (\lambda (e2: C).(drop1 (match (blt (trans
127 hds0 i) d) with [true \Rightarrow (PCons h (minus d (S (trans hds0 i)))
128 (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) e2 e1)) (\lambda (e2:
129 C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow (trans hds0 i)
130 | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (lift1
131 (match (blt (trans hds0 i) d) with [true \Rightarrow (PCons h (minus d (S
132 (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)])
133 v))))))))) (\lambda (H13: (eq C c4 c1)).(eq_ind C c1 (\lambda (c: C).((drop h
134 d c2 c3) \to ((drop1 hds0 c3 c) \to (ex2 C (\lambda (e2: C).(drop1 (match
135 (blt (trans hds0 i) d) with [true \Rightarrow (PCons h (minus d (S (trans
136 hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) e2 e1))
137 (\lambda (e2: C).(getl (match (blt (trans hds0 i) d) with [true \Rightarrow
138 (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2
139 (Bind b) (lift1 (match (blt (trans hds0 i) d) with [true \Rightarrow (PCons h
140 (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans
141 hds0 i)]) v)))))))) (\lambda (H14: (drop h d c2 c3)).(\lambda (H15: (drop1
142 hds0 c3 c1)).(xinduction bool (blt (trans hds0 i) d) (\lambda (b0: bool).(ex2
143 C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow (PCons h (minus d
144 (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) e2
145 e1)) (\lambda (e2: C).(getl (match b0 with [true \Rightarrow (trans hds0 i) |
146 false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2 (Bind b) (lift1
147 (match b0 with [true \Rightarrow (PCons h (minus d (S (trans hds0 i)))
148 (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) v)))))) (\lambda (x_x:
149 bool).(bool_ind (\lambda (b0: bool).((eq bool (blt (trans hds0 i) d) b0) \to
150 (ex2 C (\lambda (e2: C).(drop1 (match b0 with [true \Rightarrow (PCons h
151 (minus d (S (trans hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans
152 hds0 i)]) e2 e1)) (\lambda (e2: C).(getl (match b0 with [true \Rightarrow
153 (trans hds0 i) | false \Rightarrow (plus (trans hds0 i) h)]) c2 (CHead e2
154 (Bind b) (lift1 (match b0 with [true \Rightarrow (PCons h (minus d (S (trans
155 hds0 i))) (ptrans hds0 i)) | false \Rightarrow (ptrans hds0 i)]) v)))))))
156 (\lambda (H16: (eq bool (blt (trans hds0 i) d) true)).(let H_x \def (H c1 c3
157 H15 b e1 v i H1) in (let H17 \def H_x in (ex2_ind C (\lambda (e2: C).(drop1
158 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) c3 (CHead e2
159 (Bind b) (lift1 (ptrans hds0 i) v)))) (ex2 C (\lambda (e2: C).(drop1 (PCons h
160 (minus d (S (trans hds0 i))) (ptrans hds0 i)) e2 e1)) (\lambda (e2: C).(getl
161 (trans hds0 i) c2 (CHead e2 (Bind b) (lift1 (PCons h (minus d (S (trans hds0
162 i))) (ptrans hds0 i)) v))))) (\lambda (x: C).(\lambda (H18: (drop1 (ptrans
163 hds0 i) x e1)).(\lambda (H19: (getl (trans hds0 i) c3 (CHead x (Bind b)
164 (lift1 (ptrans hds0 i) v)))).(let H_x0 \def (drop_getl_trans_lt (trans hds0
165 i) d (le_S_n (S (trans hds0 i)) d (lt_le_S (S (trans hds0 i)) (S d) (blt_lt
166 (S d) (S (trans hds0 i)) H16))) c2 c3 h H14 b x (lift1 (ptrans hds0 i) v)
167 H19) in (let H20 \def H_x0 in (ex2_ind C (\lambda (e2: C).(getl (trans hds0
168 i) c2 (CHead e2 (Bind b) (lift h (minus d (S (trans hds0 i))) (lift1 (ptrans
169 hds0 i) v))))) (\lambda (e2: C).(drop h (minus d (S (trans hds0 i))) e2 x))
170 (ex2 C (\lambda (e2: C).(drop1 (PCons h (minus d (S (trans hds0 i))) (ptrans
171 hds0 i)) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b)
172 (lift1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) v))))) (\lambda
173 (x0: C).(\lambda (H21: (getl (trans hds0 i) c2 (CHead x0 (Bind b) (lift h
174 (minus d (S (trans hds0 i))) (lift1 (ptrans hds0 i) v))))).(\lambda (H22:
175 (drop h (minus d (S (trans hds0 i))) x0 x)).(ex_intro2 C (\lambda (e2:
176 C).(drop1 (PCons h (minus d (S (trans hds0 i))) (ptrans hds0 i)) e2 e1))
177 (\lambda (e2: C).(getl (trans hds0 i) c2 (CHead e2 (Bind b) (lift1 (PCons h
178 (minus d (S (trans hds0 i))) (ptrans hds0 i)) v)))) x0 (drop1_cons x0 x h
179 (minus d (S (trans hds0 i))) H22 e1 (ptrans hds0 i) H18) H21)))) H20))))))
180 H17)))) (\lambda (H16: (eq bool (blt (trans hds0 i) d) false)).(let H_x \def
181 (H c1 c3 H15 b e1 v i H1) in (let H17 \def H_x in (ex2_ind C (\lambda (e2:
182 C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (trans hds0 i) c3
183 (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v)))) (ex2 C (\lambda (e2:
184 C).(drop1 (ptrans hds0 i) e2 e1)) (\lambda (e2: C).(getl (plus (trans hds0 i)
185 h) c2 (CHead e2 (Bind b) (lift1 (ptrans hds0 i) v))))) (\lambda (x:
186 C).(\lambda (H18: (drop1 (ptrans hds0 i) x e1)).(\lambda (H19: (getl (trans
187 hds0 i) c3 (CHead x (Bind b) (lift1 (ptrans hds0 i) v)))).(let H20 \def
188 (drop_getl_trans_ge (trans hds0 i) c2 c3 d h H14 (CHead x (Bind b) (lift1
189 (ptrans hds0 i) v)) H19) in (ex_intro2 C (\lambda (e2: C).(drop1 (ptrans hds0
190 i) e2 e1)) (\lambda (e2: C).(getl (plus (trans hds0 i) h) c2 (CHead e2 (Bind
191 b) (lift1 (ptrans hds0 i) v)))) x H18 (H20 (bge_le d (trans hds0 i)
192 H16))))))) H17)))) x_x))))) c4 (sym_eq C c4 c1 H13))) c0 (sym_eq C c0 c2
193 H12))) hds1 (sym_eq PList hds1 hds0 H11))) d0 (sym_eq nat d0 d H10))) h0
194 (sym_eq nat h0 h H9))) H8)) H7)) H5 H6 H2 H3))))]) in (H2 (refl_equal PList
195 (PCons h d hds0)) (refl_equal C c2) (refl_equal C c1))))))))))))))) hds).