author | paulson |
Mon, 26 May 1997 12:38:29 +0200 | |
changeset 3342 | ec3b55fcb165 |
parent 3292 | 8b143c196d42 |
child 3383 | 7707cb7a5054 |
permissions | -rw-r--r-- |
1465 | 1 |
(* Title: HOL/List |
923 | 2 |
ID: $Id$ |
1465 | 3 |
Author: Tobias Nipkow |
923 | 4 |
Copyright 1994 TU Muenchen |
5 |
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6 |
List lemmas |
|
7 |
*) |
|
8 |
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3011 | 9 |
goal thy "!x. xs ~= x#xs"; |
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10 |
by (induct_tac "xs" 1); |
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11 |
by (ALLGOALS Asm_simp_tac); |
2608 | 12 |
qed_spec_mp "not_Cons_self"; |
2512 | 13 |
Addsimps [not_Cons_self]; |
923 | 14 |
|
3011 | 15 |
goal thy "(xs ~= []) = (? y ys. xs = y#ys)"; |
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16 |
by (induct_tac "xs" 1); |
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17 |
by (Simp_tac 1); |
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18 |
by (Asm_simp_tac 1); |
923 | 19 |
qed "neq_Nil_conv"; |
20 |
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21 |
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22 |
(** List operator over sets **) |
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23 |
|
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24 |
goalw thy lists.defs "!!A B. A<=B ==> lists A <= lists B"; |
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25 |
by (rtac lfp_mono 1); |
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26 |
by (REPEAT (ares_tac basic_monos 1)); |
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27 |
qed "lists_mono"; |
3196 | 28 |
|
29 |
||
2608 | 30 |
(** list_case **) |
31 |
||
3011 | 32 |
goal thy |
2608 | 33 |
"P(list_case a f xs) = ((xs=[] --> P(a)) & \ |
2891 | 34 |
\ (!y ys. xs=y#ys --> P(f y ys)))"; |
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35 |
by (induct_tac "xs" 1); |
2608 | 36 |
by (ALLGOALS Asm_simp_tac); |
2891 | 37 |
by (Blast_tac 1); |
2608 | 38 |
qed "expand_list_case"; |
39 |
||
3011 | 40 |
val prems = goal thy "[| P([]); !!x xs. P(x#xs) |] ==> P(xs)"; |
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41 |
by(induct_tac "xs" 1); |
2608 | 42 |
by(REPEAT(resolve_tac prems 1)); |
43 |
qed "list_cases"; |
|
44 |
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3011 | 45 |
goal thy "(xs=[] --> P([])) & (!y ys. xs=y#ys --> P(y#ys)) --> P(xs)"; |
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46 |
by (induct_tac "xs" 1); |
2891 | 47 |
by (Blast_tac 1); |
48 |
by (Blast_tac 1); |
|
2608 | 49 |
bind_thm("list_eq_cases", |
50 |
impI RSN (2,allI RSN (2,allI RSN (2,impI RS (conjI RS (result() RS mp)))))); |
|
51 |
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52 |
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923 | 53 |
(** @ - append **) |
54 |
||
3011 | 55 |
goal thy "(xs@ys)@zs = xs@(ys@zs)"; |
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by (induct_tac "xs" 1); |
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57 |
by (ALLGOALS Asm_simp_tac); |
923 | 58 |
qed "append_assoc"; |
2512 | 59 |
Addsimps [append_assoc]; |
923 | 60 |
|
3011 | 61 |
goal thy "xs @ [] = xs"; |
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62 |
by (induct_tac "xs" 1); |
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63 |
by (ALLGOALS Asm_simp_tac); |
923 | 64 |
qed "append_Nil2"; |
2512 | 65 |
Addsimps [append_Nil2]; |
923 | 66 |
|
3011 | 67 |
goal thy "(xs@ys = []) = (xs=[] & ys=[])"; |
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by (induct_tac "xs" 1); |
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69 |
by (ALLGOALS Asm_simp_tac); |
2608 | 70 |
qed "append_is_Nil_conv"; |
71 |
AddIffs [append_is_Nil_conv]; |
|
72 |
||
3011 | 73 |
goal thy "([] = xs@ys) = (xs=[] & ys=[])"; |
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74 |
by (induct_tac "xs" 1); |
2608 | 75 |
by (ALLGOALS Asm_simp_tac); |
2891 | 76 |
by(Blast_tac 1); |
2608 | 77 |
qed "Nil_is_append_conv"; |
78 |
AddIffs [Nil_is_append_conv]; |
|
923 | 79 |
|
3011 | 80 |
goal thy "(xs @ ys = xs @ zs) = (ys=zs)"; |
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81 |
by (induct_tac "xs" 1); |
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82 |
by (ALLGOALS Asm_simp_tac); |
923 | 83 |
qed "same_append_eq"; |
2608 | 84 |
AddIffs [same_append_eq]; |
85 |
||
3011 | 86 |
goal thy "!ys. (xs @ [x] = ys @ [y]) = (xs = ys & x = y)"; |
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87 |
by(induct_tac "xs" 1); |
2608 | 88 |
br allI 1; |
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89 |
by(induct_tac "ys" 1); |
2608 | 90 |
by(ALLGOALS Asm_simp_tac); |
91 |
br allI 1; |
|
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92 |
by(induct_tac "ys" 1); |
2608 | 93 |
by(ALLGOALS Asm_simp_tac); |
94 |
qed_spec_mp "append1_eq_conv"; |
|
95 |
AddIffs [append1_eq_conv]; |
|
96 |
||
3011 | 97 |
goal thy "xs ~= [] --> hd xs # tl xs = xs"; |
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98 |
by(induct_tac "xs" 1); |
2608 | 99 |
by(ALLGOALS Asm_simp_tac); |
100 |
qed_spec_mp "hd_Cons_tl"; |
|
101 |
Addsimps [hd_Cons_tl]; |
|
923 | 102 |
|
3011 | 103 |
goal thy "hd(xs@ys) = (if xs=[] then hd ys else hd xs)"; |
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104 |
by (induct_tac "xs" 1); |
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105 |
by (ALLGOALS Asm_simp_tac); |
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106 |
qed "hd_append"; |
923 | 107 |
|
3011 | 108 |
goal thy "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)"; |
2608 | 109 |
by(simp_tac (!simpset setloop(split_tac[expand_list_case])) 1); |
110 |
qed "tl_append"; |
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111 |
||
112 |
(** map **) |
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113 |
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3011 | 114 |
goal thy |
2608 | 115 |
"(!x. x : set_of_list xs --> f x = g x) --> map f xs = map g xs"; |
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116 |
by(induct_tac "xs" 1); |
2608 | 117 |
by(ALLGOALS Asm_simp_tac); |
118 |
bind_thm("map_ext", impI RS (allI RS (result() RS mp))); |
|
119 |
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3011 | 120 |
goal thy "map (%x.x) = (%xs.xs)"; |
2608 | 121 |
by (rtac ext 1); |
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122 |
by (induct_tac "xs" 1); |
2608 | 123 |
by (ALLGOALS Asm_simp_tac); |
124 |
qed "map_ident"; |
|
125 |
Addsimps[map_ident]; |
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126 |
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3011 | 127 |
goal thy "map f (xs@ys) = map f xs @ map f ys"; |
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128 |
by (induct_tac "xs" 1); |
2608 | 129 |
by (ALLGOALS Asm_simp_tac); |
130 |
qed "map_append"; |
|
131 |
Addsimps[map_append]; |
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132 |
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3011 | 133 |
goalw thy [o_def] "map (f o g) xs = map f (map g xs)"; |
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134 |
by (induct_tac "xs" 1); |
2608 | 135 |
by (ALLGOALS Asm_simp_tac); |
136 |
qed "map_compose"; |
|
137 |
Addsimps[map_compose]; |
|
138 |
||
3011 | 139 |
goal thy "rev(map f xs) = map f (rev xs)"; |
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140 |
by (induct_tac "xs" 1); |
2608 | 141 |
by (ALLGOALS Asm_simp_tac); |
142 |
qed "rev_map"; |
|
143 |
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1169 | 144 |
(** rev **) |
145 |
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3011 | 146 |
goal thy "rev(xs@ys) = rev(ys) @ rev(xs)"; |
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147 |
by (induct_tac "xs" 1); |
2512 | 148 |
by (ALLGOALS Asm_simp_tac); |
1169 | 149 |
qed "rev_append"; |
2512 | 150 |
Addsimps[rev_append]; |
1169 | 151 |
|
3011 | 152 |
goal thy "rev(rev l) = l"; |
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153 |
by (induct_tac "l" 1); |
2512 | 154 |
by (ALLGOALS Asm_simp_tac); |
1169 | 155 |
qed "rev_rev_ident"; |
2512 | 156 |
Addsimps[rev_rev_ident]; |
1169 | 157 |
|
2608 | 158 |
|
923 | 159 |
(** mem **) |
160 |
||
3011 | 161 |
goal thy "x mem (xs@ys) = (x mem xs | x mem ys)"; |
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by (induct_tac "xs" 1); |
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163 |
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if])))); |
923 | 164 |
qed "mem_append"; |
2512 | 165 |
Addsimps[mem_append]; |
923 | 166 |
|
3011 | 167 |
goal thy "x mem [x:xs.P(x)] = (x mem xs & P(x))"; |
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168 |
by (induct_tac "xs" 1); |
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169 |
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if])))); |
923 | 170 |
qed "mem_filter"; |
2512 | 171 |
Addsimps[mem_filter]; |
923 | 172 |
|
1908 | 173 |
(** set_of_list **) |
1812 | 174 |
|
1908 | 175 |
goal thy "set_of_list (xs@ys) = (set_of_list xs Un set_of_list ys)"; |
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176 |
by (induct_tac "xs" 1); |
1812 | 177 |
by (ALLGOALS Asm_simp_tac); |
2891 | 178 |
by (Blast_tac 1); |
1908 | 179 |
qed "set_of_list_append"; |
2512 | 180 |
Addsimps[set_of_list_append]; |
1812 | 181 |
|
1908 | 182 |
goal thy "(x mem xs) = (x: set_of_list xs)"; |
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183 |
by (induct_tac "xs" 1); |
1812 | 184 |
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if])))); |
2891 | 185 |
by (Blast_tac 1); |
1908 | 186 |
qed "set_of_list_mem_eq"; |
1812 | 187 |
|
3011 | 188 |
goal thy "set_of_list l <= set_of_list (x#l)"; |
1936 | 189 |
by (Simp_tac 1); |
2891 | 190 |
by (Blast_tac 1); |
1936 | 191 |
qed "set_of_list_subset_Cons"; |
192 |
||
3011 | 193 |
goal thy "(set_of_list xs = {}) = (xs = [])"; |
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194 |
by(induct_tac "xs" 1); |
2608 | 195 |
by(ALLGOALS Asm_simp_tac); |
196 |
qed "set_of_list_empty"; |
|
197 |
Addsimps [set_of_list_empty]; |
|
198 |
||
3011 | 199 |
goal thy "set_of_list(rev xs) = set_of_list(xs)"; |
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200 |
by(induct_tac "xs" 1); |
2608 | 201 |
by(ALLGOALS Asm_simp_tac); |
2891 | 202 |
by(Blast_tac 1); |
2608 | 203 |
qed "set_of_list_rev"; |
204 |
Addsimps [set_of_list_rev]; |
|
205 |
||
3011 | 206 |
goal thy "set_of_list(map f xs) = f``(set_of_list xs)"; |
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207 |
by(induct_tac "xs" 1); |
2608 | 208 |
by(ALLGOALS Asm_simp_tac); |
209 |
qed "set_of_list_map"; |
|
210 |
Addsimps [set_of_list_map]; |
|
211 |
||
1812 | 212 |
|
923 | 213 |
(** list_all **) |
214 |
||
3011 | 215 |
goal thy "list_all (%x.True) xs = True"; |
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216 |
by (induct_tac "xs" 1); |
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217 |
by (ALLGOALS Asm_simp_tac); |
923 | 218 |
qed "list_all_True"; |
2512 | 219 |
Addsimps [list_all_True]; |
923 | 220 |
|
3011 | 221 |
goal thy "list_all p (xs@ys) = (list_all p xs & list_all p ys)"; |
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222 |
by (induct_tac "xs" 1); |
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223 |
by (ALLGOALS Asm_simp_tac); |
2512 | 224 |
qed "list_all_append"; |
225 |
Addsimps [list_all_append]; |
|
923 | 226 |
|
3011 | 227 |
goal thy "list_all P xs = (!x. x mem xs --> P(x))"; |
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228 |
by (induct_tac "xs" 1); |
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229 |
by (ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if])))); |
2891 | 230 |
by (Blast_tac 1); |
923 | 231 |
qed "list_all_mem_conv"; |
232 |
||
233 |
||
2608 | 234 |
(** filter **) |
923 | 235 |
|
3011 | 236 |
goal thy "[x:xs@ys . P] = [x:xs . P] @ [y:ys . P]"; |
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237 |
by(induct_tac "xs" 1); |
2608 | 238 |
by(ALLGOALS (asm_simp_tac (!simpset setloop (split_tac [expand_if])))); |
239 |
qed "filter_append"; |
|
240 |
Addsimps [filter_append]; |
|
241 |
||
242 |
||
243 |
(** concat **) |
|
244 |
||
3011 | 245 |
goal thy "concat(xs@ys) = concat(xs)@concat(ys)"; |
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246 |
by (induct_tac "xs" 1); |
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247 |
by (ALLGOALS Asm_simp_tac); |
2608 | 248 |
qed"concat_append"; |
249 |
Addsimps [concat_append]; |
|
2512 | 250 |
|
3011 | 251 |
goal thy "rev(concat ls) = concat (map rev (rev ls))"; |
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252 |
by (induct_tac "ls" 1); |
2512 | 253 |
by (ALLGOALS Asm_simp_tac); |
2608 | 254 |
qed "rev_concat"; |
923 | 255 |
|
962 | 256 |
(** length **) |
257 |
||
3011 | 258 |
goal thy "length(xs@ys) = length(xs)+length(ys)"; |
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259 |
by (induct_tac "xs" 1); |
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diff
changeset
|
260 |
by (ALLGOALS Asm_simp_tac); |
962 | 261 |
qed"length_append"; |
1301 | 262 |
Addsimps [length_append]; |
263 |
||
3011 | 264 |
goal thy "length (map f l) = length l"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
265 |
by (induct_tac "l" 1); |
1301 | 266 |
by (ALLGOALS Simp_tac); |
267 |
qed "length_map"; |
|
268 |
Addsimps [length_map]; |
|
962 | 269 |
|
3011 | 270 |
goal thy "length(rev xs) = length(xs)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
271 |
by (induct_tac "xs" 1); |
1301 | 272 |
by (ALLGOALS Asm_simp_tac); |
1169 | 273 |
qed "length_rev"; |
1301 | 274 |
Addsimps [length_rev]; |
1169 | 275 |
|
3011 | 276 |
goal thy "(length xs = 0) = (xs = [])"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
277 |
by(induct_tac "xs" 1); |
2608 | 278 |
by(ALLGOALS Asm_simp_tac); |
279 |
qed "length_0_conv"; |
|
280 |
AddIffs [length_0_conv]; |
|
281 |
||
3011 | 282 |
goal thy "(0 < length xs) = (xs ~= [])"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
283 |
by(induct_tac "xs" 1); |
2608 | 284 |
by(ALLGOALS Asm_simp_tac); |
285 |
qed "length_greater_0_conv"; |
|
286 |
AddIffs [length_greater_0_conv]; |
|
287 |
||
288 |
||
923 | 289 |
(** nth **) |
290 |
||
3011 | 291 |
goal thy |
2608 | 292 |
"!xs. nth n (xs@ys) = \ |
293 |
\ (if n < length xs then nth n xs else nth (n - length xs) ys)"; |
|
294 |
by(nat_ind_tac "n" 1); |
|
295 |
by(Asm_simp_tac 1); |
|
296 |
br allI 1; |
|
3283
0db086394024
Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents:
3196
diff
changeset
|
297 |
by(exhaust_tac "xs" 1); |
2608 | 298 |
by(ALLGOALS Asm_simp_tac); |
299 |
br allI 1; |
|
3283
0db086394024
Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents:
3196
diff
changeset
|
300 |
by(exhaust_tac "xs" 1); |
2608 | 301 |
by(ALLGOALS Asm_simp_tac); |
302 |
qed_spec_mp "nth_append"; |
|
303 |
||
3011 | 304 |
goal thy "!n. n < length xs --> nth n (map f xs) = f (nth n xs)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
305 |
by (induct_tac "xs" 1); |
1301 | 306 |
(* case [] *) |
307 |
by (Asm_full_simp_tac 1); |
|
308 |
(* case x#xl *) |
|
309 |
by (rtac allI 1); |
|
310 |
by (nat_ind_tac "n" 1); |
|
311 |
by (ALLGOALS Asm_full_simp_tac); |
|
1485
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1465
diff
changeset
|
312 |
qed_spec_mp "nth_map"; |
1301 | 313 |
Addsimps [nth_map]; |
314 |
||
3011 | 315 |
goal thy "!n. n < length xs --> list_all P xs --> P(nth n xs)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
316 |
by (induct_tac "xs" 1); |
1301 | 317 |
(* case [] *) |
318 |
by (Simp_tac 1); |
|
319 |
(* case x#xl *) |
|
320 |
by (rtac allI 1); |
|
321 |
by (nat_ind_tac "n" 1); |
|
322 |
by (ALLGOALS Asm_full_simp_tac); |
|
1485
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1465
diff
changeset
|
323 |
qed_spec_mp "list_all_nth"; |
1301 | 324 |
|
3011 | 325 |
goal thy "!n. n < length xs --> (nth n xs) mem xs"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
326 |
by (induct_tac "xs" 1); |
1301 | 327 |
(* case [] *) |
328 |
by (Simp_tac 1); |
|
329 |
(* case x#xl *) |
|
330 |
by (rtac allI 1); |
|
331 |
by (nat_ind_tac "n" 1); |
|
332 |
(* case 0 *) |
|
333 |
by (Asm_full_simp_tac 1); |
|
334 |
(* case Suc x *) |
|
335 |
by (asm_full_simp_tac (!simpset setloop (split_tac [expand_if])) 1); |
|
1485
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1465
diff
changeset
|
336 |
qed_spec_mp "nth_mem"; |
1301 | 337 |
Addsimps [nth_mem]; |
338 |
||
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
339 |
|
2608 | 340 |
(** take & drop **) |
341 |
section "take & drop"; |
|
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
342 |
|
1419
a6a034a47a71
defined take/drop by induction over list rather than nat.
nipkow
parents:
1327
diff
changeset
|
343 |
goal thy "take 0 xs = []"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
344 |
by (induct_tac "xs" 1); |
1419
a6a034a47a71
defined take/drop by induction over list rather than nat.
nipkow
parents:
1327
diff
changeset
|
345 |
by (ALLGOALS Asm_simp_tac); |
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
346 |
qed "take_0"; |
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
347 |
|
2608 | 348 |
goal thy "drop 0 xs = xs"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
349 |
by (induct_tac "xs" 1); |
2608 | 350 |
by (ALLGOALS Asm_simp_tac); |
351 |
qed "drop_0"; |
|
352 |
||
1419
a6a034a47a71
defined take/drop by induction over list rather than nat.
nipkow
parents:
1327
diff
changeset
|
353 |
goal thy "take (Suc n) (x#xs) = x # take n xs"; |
1552 | 354 |
by (Simp_tac 1); |
1419
a6a034a47a71
defined take/drop by induction over list rather than nat.
nipkow
parents:
1327
diff
changeset
|
355 |
qed "take_Suc_Cons"; |
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
356 |
|
2608 | 357 |
goal thy "drop (Suc n) (x#xs) = drop n xs"; |
358 |
by (Simp_tac 1); |
|
359 |
qed "drop_Suc_Cons"; |
|
360 |
||
361 |
Delsimps [take_Cons,drop_Cons]; |
|
362 |
Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons]; |
|
363 |
||
3011 | 364 |
goal thy "!xs. length(take n xs) = min (length xs) n"; |
2608 | 365 |
by(nat_ind_tac "n" 1); |
366 |
by(ALLGOALS Asm_simp_tac); |
|
367 |
br allI 1; |
|
3283
0db086394024
Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents:
3196
diff
changeset
|
368 |
by(exhaust_tac "xs" 1); |
2608 | 369 |
by(ALLGOALS Asm_simp_tac); |
370 |
qed_spec_mp "length_take"; |
|
371 |
Addsimps [length_take]; |
|
923 | 372 |
|
3011 | 373 |
goal thy "!xs. length(drop n xs) = (length xs - n)"; |
2608 | 374 |
by(nat_ind_tac "n" 1); |
375 |
by(ALLGOALS Asm_simp_tac); |
|
376 |
br allI 1; |
|
3283
0db086394024
Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents:
3196
diff
changeset
|
377 |
by(exhaust_tac "xs" 1); |
2608 | 378 |
by(ALLGOALS Asm_simp_tac); |
379 |
qed_spec_mp "length_drop"; |
|
380 |
Addsimps [length_drop]; |
|
381 |
||
3011 | 382 |
goal thy "!xs. length xs <= n --> take n xs = xs"; |
2608 | 383 |
by(nat_ind_tac "n" 1); |
384 |
by(ALLGOALS Asm_simp_tac); |
|
385 |
br allI 1; |
|
3283
0db086394024
Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents:
3196
diff
changeset
|
386 |
by(exhaust_tac "xs" 1); |
2608 | 387 |
by(ALLGOALS Asm_simp_tac); |
388 |
qed_spec_mp "take_all"; |
|
923 | 389 |
|
3011 | 390 |
goal thy "!xs. length xs <= n --> drop n xs = []"; |
2608 | 391 |
by(nat_ind_tac "n" 1); |
392 |
by(ALLGOALS Asm_simp_tac); |
|
393 |
br allI 1; |
|
3283
0db086394024
Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents:
3196
diff
changeset
|
394 |
by(exhaust_tac "xs" 1); |
2608 | 395 |
by(ALLGOALS Asm_simp_tac); |
396 |
qed_spec_mp "drop_all"; |
|
397 |
||
3011 | 398 |
goal thy |
2608 | 399 |
"!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"; |
400 |
by(nat_ind_tac "n" 1); |
|
401 |
by(ALLGOALS Asm_simp_tac); |
|
402 |
br allI 1; |
|
3283
0db086394024
Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents:
3196
diff
changeset
|
403 |
by(exhaust_tac "xs" 1); |
2608 | 404 |
by(ALLGOALS Asm_simp_tac); |
405 |
qed_spec_mp "take_append"; |
|
406 |
Addsimps [take_append]; |
|
407 |
||
3011 | 408 |
goal thy "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; |
2608 | 409 |
by(nat_ind_tac "n" 1); |
410 |
by(ALLGOALS Asm_simp_tac); |
|
411 |
br allI 1; |
|
3283
0db086394024
Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents:
3196
diff
changeset
|
412 |
by(exhaust_tac "xs" 1); |
2608 | 413 |
by(ALLGOALS Asm_simp_tac); |
414 |
qed_spec_mp "drop_append"; |
|
415 |
Addsimps [drop_append]; |
|
416 |
||
3011 | 417 |
goal thy "!xs n. take n (take m xs) = take (min n m) xs"; |
2608 | 418 |
by(nat_ind_tac "m" 1); |
419 |
by(ALLGOALS Asm_simp_tac); |
|
420 |
br allI 1; |
|
3283
0db086394024
Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents:
3196
diff
changeset
|
421 |
by(exhaust_tac "xs" 1); |
2608 | 422 |
by(ALLGOALS Asm_simp_tac); |
423 |
br allI 1; |
|
3283
0db086394024
Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents:
3196
diff
changeset
|
424 |
by(exhaust_tac "n" 1); |
2608 | 425 |
by(ALLGOALS Asm_simp_tac); |
426 |
qed_spec_mp "take_take"; |
|
427 |
||
3011 | 428 |
goal thy "!xs. drop n (drop m xs) = drop (n + m) xs"; |
2608 | 429 |
by(nat_ind_tac "m" 1); |
430 |
by(ALLGOALS Asm_simp_tac); |
|
431 |
br allI 1; |
|
3283
0db086394024
Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents:
3196
diff
changeset
|
432 |
by(exhaust_tac "xs" 1); |
2608 | 433 |
by(ALLGOALS Asm_simp_tac); |
434 |
qed_spec_mp "drop_drop"; |
|
923 | 435 |
|
3011 | 436 |
goal thy "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; |
2608 | 437 |
by(nat_ind_tac "m" 1); |
438 |
by(ALLGOALS Asm_simp_tac); |
|
439 |
br allI 1; |
|
3283
0db086394024
Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents:
3196
diff
changeset
|
440 |
by(exhaust_tac "xs" 1); |
2608 | 441 |
by(ALLGOALS Asm_simp_tac); |
442 |
qed_spec_mp "take_drop"; |
|
443 |
||
3011 | 444 |
goal thy "!xs. take n (map f xs) = map f (take n xs)"; |
2608 | 445 |
by(nat_ind_tac "n" 1); |
446 |
by(ALLGOALS Asm_simp_tac); |
|
447 |
br allI 1; |
|
3283
0db086394024
Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents:
3196
diff
changeset
|
448 |
by(exhaust_tac "xs" 1); |
2608 | 449 |
by(ALLGOALS Asm_simp_tac); |
450 |
qed_spec_mp "take_map"; |
|
451 |
||
3011 | 452 |
goal thy "!xs. drop n (map f xs) = map f (drop n xs)"; |
2608 | 453 |
by(nat_ind_tac "n" 1); |
454 |
by(ALLGOALS Asm_simp_tac); |
|
455 |
br allI 1; |
|
3283
0db086394024
Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents:
3196
diff
changeset
|
456 |
by(exhaust_tac "xs" 1); |
2608 | 457 |
by(ALLGOALS Asm_simp_tac); |
458 |
qed_spec_mp "drop_map"; |
|
459 |
||
3283
0db086394024
Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents:
3196
diff
changeset
|
460 |
goal thy "!n i. i < n --> nth i (take n xs) = nth i xs"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
461 |
by(induct_tac "xs" 1); |
2608 | 462 |
by(ALLGOALS Asm_simp_tac); |
463 |
by(strip_tac 1); |
|
3283
0db086394024
Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents:
3196
diff
changeset
|
464 |
by(exhaust_tac "n" 1); |
2891 | 465 |
by(Blast_tac 1); |
3283
0db086394024
Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents:
3196
diff
changeset
|
466 |
by(exhaust_tac "i" 1); |
3292 | 467 |
by(ALLGOALS Asm_full_simp_tac); |
2608 | 468 |
qed_spec_mp "nth_take"; |
469 |
Addsimps [nth_take]; |
|
923 | 470 |
|
3283
0db086394024
Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents:
3196
diff
changeset
|
471 |
goal thy "!xs i. n + i < length xs --> nth i (drop n xs) = nth (n + i) xs"; |
2608 | 472 |
by(nat_ind_tac "n" 1); |
473 |
by(ALLGOALS Asm_simp_tac); |
|
474 |
br allI 1; |
|
3283
0db086394024
Replaced res_inst-list_cases by generic exhaust_tac.
nipkow
parents:
3196
diff
changeset
|
475 |
by(exhaust_tac "xs" 1); |
2608 | 476 |
by(ALLGOALS Asm_simp_tac); |
477 |
qed_spec_mp "nth_drop"; |
|
478 |
Addsimps [nth_drop]; |
|
479 |
||
480 |
(** takeWhile & dropWhile **) |
|
481 |
||
3011 | 482 |
goal thy |
2608 | 483 |
"x:set_of_list xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
484 |
by(induct_tac "xs" 1); |
2608 | 485 |
by(Simp_tac 1); |
486 |
by(asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1); |
|
2891 | 487 |
by(Blast_tac 1); |
2608 | 488 |
bind_thm("takeWhile_append1", conjI RS (result() RS mp)); |
489 |
Addsimps [takeWhile_append1]; |
|
923 | 490 |
|
3011 | 491 |
goal thy |
2608 | 492 |
"(!x:set_of_list xs.P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
493 |
by(induct_tac "xs" 1); |
2608 | 494 |
by(Simp_tac 1); |
495 |
by(asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1); |
|
496 |
bind_thm("takeWhile_append2", ballI RS (result() RS mp)); |
|
497 |
Addsimps [takeWhile_append2]; |
|
1169 | 498 |
|
3011 | 499 |
goal thy |
2608 | 500 |
"x:set_of_list xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
501 |
by(induct_tac "xs" 1); |
2608 | 502 |
by(Simp_tac 1); |
503 |
by(asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1); |
|
2891 | 504 |
by(Blast_tac 1); |
2608 | 505 |
bind_thm("dropWhile_append1", conjI RS (result() RS mp)); |
506 |
Addsimps [dropWhile_append1]; |
|
507 |
||
3011 | 508 |
goal thy |
2608 | 509 |
"(!x:set_of_list xs.P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
510 |
by(induct_tac "xs" 1); |
2608 | 511 |
by(Simp_tac 1); |
512 |
by(asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1); |
|
513 |
bind_thm("dropWhile_append2", ballI RS (result() RS mp)); |
|
514 |
Addsimps [dropWhile_append2]; |
|
515 |
||
3011 | 516 |
goal thy "x:set_of_list(takeWhile P xs) --> x:set_of_list xs & P x"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
517 |
by(induct_tac "xs" 1); |
2608 | 518 |
by(Simp_tac 1); |
519 |
by(asm_full_simp_tac (!simpset setloop (split_tac[expand_if])) 1); |
|
520 |
qed_spec_mp"set_of_list_take_whileD"; |
|
521 |