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