author | nipkow |
Sat, 08 Aug 1998 14:00:56 +0200 | |
changeset 5281 | f4d16517b360 |
parent 5278 | a903b66822e2 |
child 5283 | 0027ddfbc831 |
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 |
||
6 |
List lemmas |
|
7 |
*) |
|
8 |
||
4935 | 9 |
Goal "!x. xs ~= x#xs"; |
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10 |
by (induct_tac "xs" 1); |
5129 | 11 |
by (Auto_tac); |
2608 | 12 |
qed_spec_mp "not_Cons_self"; |
3574 | 13 |
bind_thm("not_Cons_self2",not_Cons_self RS not_sym); |
14 |
Addsimps [not_Cons_self,not_Cons_self2]; |
|
923 | 15 |
|
4935 | 16 |
Goal "(xs ~= []) = (? y ys. xs = y#ys)"; |
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17 |
by (induct_tac "xs" 1); |
5129 | 18 |
by (Auto_tac); |
923 | 19 |
qed "neq_Nil_conv"; |
20 |
||
4830 | 21 |
(* Induction over the length of a list: *) |
4935 | 22 |
val [prem] = Goal |
4911 | 23 |
"(!!xs. (!ys. length ys < length xs --> P ys) ==> P xs) ==> P(xs)"; |
5132 | 24 |
by (rtac measure_induct 1 THEN etac prem 1); |
4911 | 25 |
qed "length_induct"; |
26 |
||
923 | 27 |
|
3468 | 28 |
(** "lists": the list-forming operator over sets **) |
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29 |
|
5043 | 30 |
Goalw lists.defs "A<=B ==> lists A <= lists B"; |
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31 |
by (rtac lfp_mono 1); |
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32 |
by (REPEAT (ares_tac basic_monos 1)); |
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33 |
qed "lists_mono"; |
3196 | 34 |
|
3468 | 35 |
val listsE = lists.mk_cases list.simps "x#l : lists A"; |
36 |
AddSEs [listsE]; |
|
37 |
AddSIs lists.intrs; |
|
38 |
||
5043 | 39 |
Goal "l: lists A ==> l: lists B --> l: lists (A Int B)"; |
3468 | 40 |
by (etac lists.induct 1); |
41 |
by (ALLGOALS Blast_tac); |
|
42 |
qed_spec_mp "lists_IntI"; |
|
43 |
||
4935 | 44 |
Goal "lists (A Int B) = lists A Int lists B"; |
4423 | 45 |
by (rtac (mono_Int RS equalityI) 1); |
4089 | 46 |
by (simp_tac (simpset() addsimps [mono_def, lists_mono]) 1); |
47 |
by (blast_tac (claset() addSIs [lists_IntI]) 1); |
|
3468 | 48 |
qed "lists_Int_eq"; |
49 |
Addsimps [lists_Int_eq]; |
|
50 |
||
3196 | 51 |
|
4643 | 52 |
(** Case analysis **) |
53 |
section "Case analysis"; |
|
2608 | 54 |
|
4935 | 55 |
val prems = Goal "[| P([]); !!x xs. P(x#xs) |] ==> P(xs)"; |
3457 | 56 |
by (induct_tac "xs" 1); |
57 |
by (REPEAT(resolve_tac prems 1)); |
|
2608 | 58 |
qed "list_cases"; |
59 |
||
4935 | 60 |
Goal "(xs=[] --> P([])) & (!y ys. xs=y#ys --> P(y#ys)) --> P(xs)"; |
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by (induct_tac "xs" 1); |
2891 | 62 |
by (Blast_tac 1); |
63 |
by (Blast_tac 1); |
|
2608 | 64 |
bind_thm("list_eq_cases", |
65 |
impI RSN (2,allI RSN (2,allI RSN (2,impI RS (conjI RS (result() RS mp)))))); |
|
66 |
||
3860 | 67 |
(** length **) |
68 |
(* needs to come before "@" because of thm append_eq_append_conv *) |
|
69 |
||
70 |
section "length"; |
|
71 |
||
4935 | 72 |
Goal "length(xs@ys) = length(xs)+length(ys)"; |
3860 | 73 |
by (induct_tac "xs" 1); |
5129 | 74 |
by (Auto_tac); |
3860 | 75 |
qed"length_append"; |
76 |
Addsimps [length_append]; |
|
77 |
||
5129 | 78 |
Goal "length (map f xs) = length xs"; |
79 |
by (induct_tac "xs" 1); |
|
80 |
by (Auto_tac); |
|
3860 | 81 |
qed "length_map"; |
82 |
Addsimps [length_map]; |
|
83 |
||
4935 | 84 |
Goal "length(rev xs) = length(xs)"; |
3860 | 85 |
by (induct_tac "xs" 1); |
5129 | 86 |
by (Auto_tac); |
3860 | 87 |
qed "length_rev"; |
88 |
Addsimps [length_rev]; |
|
89 |
||
5043 | 90 |
Goal "xs ~= [] ==> length(tl xs) = (length xs) - 1"; |
4423 | 91 |
by (exhaust_tac "xs" 1); |
5129 | 92 |
by (Auto_tac); |
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93 |
qed "length_tl"; |
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94 |
Addsimps [length_tl]; |
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95 |
|
4935 | 96 |
Goal "(length xs = 0) = (xs = [])"; |
3860 | 97 |
by (induct_tac "xs" 1); |
5129 | 98 |
by (Auto_tac); |
3860 | 99 |
qed "length_0_conv"; |
100 |
AddIffs [length_0_conv]; |
|
101 |
||
4935 | 102 |
Goal "(0 = length xs) = (xs = [])"; |
3860 | 103 |
by (induct_tac "xs" 1); |
5129 | 104 |
by (Auto_tac); |
3860 | 105 |
qed "zero_length_conv"; |
106 |
AddIffs [zero_length_conv]; |
|
107 |
||
4935 | 108 |
Goal "(0 < length xs) = (xs ~= [])"; |
3860 | 109 |
by (induct_tac "xs" 1); |
5129 | 110 |
by (Auto_tac); |
3860 | 111 |
qed "length_greater_0_conv"; |
112 |
AddIffs [length_greater_0_conv]; |
|
113 |
||
923 | 114 |
(** @ - append **) |
115 |
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3467 | 116 |
section "@ - append"; |
117 |
||
4935 | 118 |
Goal "(xs@ys)@zs = xs@(ys@zs)"; |
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119 |
by (induct_tac "xs" 1); |
5129 | 120 |
by (Auto_tac); |
923 | 121 |
qed "append_assoc"; |
2512 | 122 |
Addsimps [append_assoc]; |
923 | 123 |
|
4935 | 124 |
Goal "xs @ [] = xs"; |
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125 |
by (induct_tac "xs" 1); |
5129 | 126 |
by (Auto_tac); |
923 | 127 |
qed "append_Nil2"; |
2512 | 128 |
Addsimps [append_Nil2]; |
923 | 129 |
|
4935 | 130 |
Goal "(xs@ys = []) = (xs=[] & ys=[])"; |
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131 |
by (induct_tac "xs" 1); |
5129 | 132 |
by (Auto_tac); |
2608 | 133 |
qed "append_is_Nil_conv"; |
134 |
AddIffs [append_is_Nil_conv]; |
|
135 |
||
4935 | 136 |
Goal "([] = xs@ys) = (xs=[] & ys=[])"; |
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137 |
by (induct_tac "xs" 1); |
5129 | 138 |
by (Auto_tac); |
2608 | 139 |
qed "Nil_is_append_conv"; |
140 |
AddIffs [Nil_is_append_conv]; |
|
923 | 141 |
|
4935 | 142 |
Goal "(xs @ ys = xs) = (ys=[])"; |
3574 | 143 |
by (induct_tac "xs" 1); |
5129 | 144 |
by (Auto_tac); |
3574 | 145 |
qed "append_self_conv"; |
146 |
||
4935 | 147 |
Goal "(xs = xs @ ys) = (ys=[])"; |
3574 | 148 |
by (induct_tac "xs" 1); |
5129 | 149 |
by (Auto_tac); |
3574 | 150 |
qed "self_append_conv"; |
151 |
AddIffs [append_self_conv,self_append_conv]; |
|
152 |
||
4935 | 153 |
Goal "!ys. length xs = length ys | length us = length vs \ |
3860 | 154 |
\ --> (xs@us = ys@vs) = (xs=ys & us=vs)"; |
4423 | 155 |
by (induct_tac "xs" 1); |
156 |
by (rtac allI 1); |
|
157 |
by (exhaust_tac "ys" 1); |
|
158 |
by (Asm_simp_tac 1); |
|
159 |
by (fast_tac (claset() addIs [less_add_Suc2] addss simpset() |
|
3860 | 160 |
addEs [less_not_refl2 RSN (2,rev_notE)]) 1); |
4423 | 161 |
by (rtac allI 1); |
162 |
by (exhaust_tac "ys" 1); |
|
163 |
by (fast_tac (claset() addIs [less_add_Suc2] addss simpset() |
|
3860 | 164 |
addEs [(less_not_refl2 RS not_sym) RSN (2,rev_notE)]) 1); |
4423 | 165 |
by (Asm_simp_tac 1); |
3860 | 166 |
qed_spec_mp "append_eq_append_conv"; |
167 |
Addsimps [append_eq_append_conv]; |
|
168 |
||
4935 | 169 |
Goal "(xs @ ys = xs @ zs) = (ys=zs)"; |
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170 |
by (Simp_tac 1); |
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171 |
qed "same_append_eq"; |
3860 | 172 |
|
4935 | 173 |
Goal "(xs @ [x] = ys @ [y]) = (xs = ys & x = y)"; |
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174 |
by (Simp_tac 1); |
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175 |
qed "append1_eq_conv"; |
2608 | 176 |
|
4935 | 177 |
Goal "(ys @ xs = zs @ xs) = (ys=zs)"; |
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178 |
by (Simp_tac 1); |
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179 |
qed "append_same_eq"; |
2608 | 180 |
|
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181 |
AddSIs |
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182 |
[same_append_eq RS iffD2, append1_eq_conv RS iffD2, append_same_eq RS iffD2]; |
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183 |
AddSDs |
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184 |
[same_append_eq RS iffD1, append1_eq_conv RS iffD1, append_same_eq RS iffD1]; |
3571 | 185 |
|
4935 | 186 |
Goal "(xs @ ys = ys) = (xs=[])"; |
5132 | 187 |
by (cut_inst_tac [("zs","[]")] append_same_eq 1); |
5129 | 188 |
by (Auto_tac); |
4647 | 189 |
qed "append_self_conv2"; |
190 |
||
4935 | 191 |
Goal "(ys = xs @ ys) = (xs=[])"; |
5132 | 192 |
by (simp_tac (simpset() addsimps |
4647 | 193 |
[simplify (simpset()) (read_instantiate[("ys","[]")]append_same_eq)]) 1); |
5132 | 194 |
by (Blast_tac 1); |
4647 | 195 |
qed "self_append_conv2"; |
196 |
AddIffs [append_self_conv2,self_append_conv2]; |
|
197 |
||
4935 | 198 |
Goal "xs ~= [] --> hd xs # tl xs = xs"; |
3457 | 199 |
by (induct_tac "xs" 1); |
5129 | 200 |
by (Auto_tac); |
2608 | 201 |
qed_spec_mp "hd_Cons_tl"; |
202 |
Addsimps [hd_Cons_tl]; |
|
923 | 203 |
|
4935 | 204 |
Goal "hd(xs@ys) = (if xs=[] then hd ys else hd xs)"; |
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205 |
by (induct_tac "xs" 1); |
5129 | 206 |
by (Auto_tac); |
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|
207 |
qed "hd_append"; |
923 | 208 |
|
5043 | 209 |
Goal "xs ~= [] ==> hd(xs @ ys) = hd xs"; |
4089 | 210 |
by (asm_simp_tac (simpset() addsimps [hd_append] |
5183 | 211 |
addsplits [list.split]) 1); |
3571 | 212 |
qed "hd_append2"; |
213 |
Addsimps [hd_append2]; |
|
214 |
||
4935 | 215 |
Goal "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)"; |
5183 | 216 |
by (simp_tac (simpset() addsplits [list.split]) 1); |
2608 | 217 |
qed "tl_append"; |
218 |
||
5043 | 219 |
Goal "xs ~= [] ==> tl(xs @ ys) = (tl xs) @ ys"; |
4089 | 220 |
by (asm_simp_tac (simpset() addsimps [tl_append] |
5183 | 221 |
addsplits [list.split]) 1); |
3571 | 222 |
qed "tl_append2"; |
223 |
Addsimps [tl_append2]; |
|
224 |
||
5272 | 225 |
(* trivial rules for solving @-equations automatically *) |
226 |
||
227 |
Goal "xs = ys ==> xs = [] @ ys"; |
|
228 |
by(Asm_simp_tac 1); |
|
229 |
qed "eq_Nil_appendI"; |
|
230 |
||
231 |
Goal "[| x#xs1 = ys; xs = xs1 @ zs |] ==> x#xs = ys@zs"; |
|
232 |
bd sym 1; |
|
233 |
by(Asm_simp_tac 1); |
|
234 |
qed "Cons_eq_appendI"; |
|
235 |
||
236 |
Goal "[| xs@xs1 = zs; ys = xs1 @ us |] ==> xs@ys = zs@us"; |
|
237 |
bd sym 1; |
|
238 |
by(Asm_simp_tac 1); |
|
239 |
qed "append_eq_appendI"; |
|
240 |
||
4830 | 241 |
|
2608 | 242 |
(** map **) |
243 |
||
3467 | 244 |
section "map"; |
245 |
||
5278 | 246 |
Goal "(!x. x : set xs --> f x = g x) --> map f xs = map g xs"; |
3457 | 247 |
by (induct_tac "xs" 1); |
5129 | 248 |
by (Auto_tac); |
2608 | 249 |
bind_thm("map_ext", impI RS (allI RS (result() RS mp))); |
250 |
||
4935 | 251 |
Goal "map (%x. x) = (%xs. xs)"; |
2608 | 252 |
by (rtac ext 1); |
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253 |
by (induct_tac "xs" 1); |
5129 | 254 |
by (Auto_tac); |
2608 | 255 |
qed "map_ident"; |
256 |
Addsimps[map_ident]; |
|
257 |
||
4935 | 258 |
Goal "map f (xs@ys) = map f xs @ map f ys"; |
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|
259 |
by (induct_tac "xs" 1); |
5129 | 260 |
by (Auto_tac); |
2608 | 261 |
qed "map_append"; |
262 |
Addsimps[map_append]; |
|
263 |
||
4935 | 264 |
Goalw [o_def] "map (f o g) xs = map f (map g xs)"; |
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|
265 |
by (induct_tac "xs" 1); |
5129 | 266 |
by (Auto_tac); |
2608 | 267 |
qed "map_compose"; |
268 |
Addsimps[map_compose]; |
|
269 |
||
4935 | 270 |
Goal "rev(map f xs) = map f (rev xs)"; |
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271 |
by (induct_tac "xs" 1); |
5129 | 272 |
by (Auto_tac); |
2608 | 273 |
qed "rev_map"; |
274 |
||
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|
275 |
(* a congruence rule for map: *) |
5278 | 276 |
Goal "(xs=ys) --> (!x. x : set ys --> f x = g x) --> map f xs = map g ys"; |
4423 | 277 |
by (rtac impI 1); |
278 |
by (hyp_subst_tac 1); |
|
279 |
by (induct_tac "ys" 1); |
|
5129 | 280 |
by (Auto_tac); |
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|
281 |
val lemma = result(); |
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Added function `replicate' and lemmas map_cong and set_replicate.
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|
282 |
bind_thm("map_cong",impI RSN (2,allI RSN (2,lemma RS mp RS mp))); |
244daa75f890
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|
283 |
|
4935 | 284 |
Goal "(map f xs = []) = (xs = [])"; |
4423 | 285 |
by (induct_tac "xs" 1); |
5129 | 286 |
by (Auto_tac); |
3860 | 287 |
qed "map_is_Nil_conv"; |
288 |
AddIffs [map_is_Nil_conv]; |
|
289 |
||
4935 | 290 |
Goal "([] = map f xs) = (xs = [])"; |
4423 | 291 |
by (induct_tac "xs" 1); |
5129 | 292 |
by (Auto_tac); |
3860 | 293 |
qed "Nil_is_map_conv"; |
294 |
AddIffs [Nil_is_map_conv]; |
|
295 |
||
296 |
||
1169 | 297 |
(** rev **) |
298 |
||
3467 | 299 |
section "rev"; |
300 |
||
4935 | 301 |
Goal "rev(xs@ys) = rev(ys) @ rev(xs)"; |
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|
302 |
by (induct_tac "xs" 1); |
5129 | 303 |
by (Auto_tac); |
1169 | 304 |
qed "rev_append"; |
2512 | 305 |
Addsimps[rev_append]; |
1169 | 306 |
|
4935 | 307 |
Goal "rev(rev l) = l"; |
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|
308 |
by (induct_tac "l" 1); |
5129 | 309 |
by (Auto_tac); |
1169 | 310 |
qed "rev_rev_ident"; |
2512 | 311 |
Addsimps[rev_rev_ident]; |
1169 | 312 |
|
4935 | 313 |
Goal "(rev xs = []) = (xs = [])"; |
4423 | 314 |
by (induct_tac "xs" 1); |
5129 | 315 |
by (Auto_tac); |
3860 | 316 |
qed "rev_is_Nil_conv"; |
317 |
AddIffs [rev_is_Nil_conv]; |
|
318 |
||
4935 | 319 |
Goal "([] = rev xs) = (xs = [])"; |
4423 | 320 |
by (induct_tac "xs" 1); |
5129 | 321 |
by (Auto_tac); |
3860 | 322 |
qed "Nil_is_rev_conv"; |
323 |
AddIffs [Nil_is_rev_conv]; |
|
324 |
||
4935 | 325 |
val prems = Goal "[| P []; !!x xs. P xs ==> P(xs@[x]) |] ==> P xs"; |
5132 | 326 |
by (stac (rev_rev_ident RS sym) 1); |
4935 | 327 |
br(read_instantiate [("P","%xs. ?P(rev xs)")]list.induct)1; |
5132 | 328 |
by (ALLGOALS Simp_tac); |
329 |
by (resolve_tac prems 1); |
|
330 |
by (eresolve_tac prems 1); |
|
4935 | 331 |
qed "rev_induct"; |
332 |
||
5272 | 333 |
fun rev_induct_tac xs = res_inst_tac [("xs",xs)] rev_induct; |
334 |
||
4935 | 335 |
Goal "(xs = [] --> P) --> (!ys y. xs = ys@[y] --> P) --> P"; |
5132 | 336 |
by (res_inst_tac [("xs","xs")] rev_induct 1); |
337 |
by (Auto_tac); |
|
4935 | 338 |
bind_thm ("rev_exhaust", |
339 |
impI RSN (2,allI RSN (2,allI RSN (2,impI RS (result() RS mp RS mp))))); |
|
340 |
||
2608 | 341 |
|
923 | 342 |
(** mem **) |
343 |
||
3467 | 344 |
section "mem"; |
345 |
||
4935 | 346 |
Goal "x mem (xs@ys) = (x mem xs | x mem ys)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
347 |
by (induct_tac "xs" 1); |
5129 | 348 |
by (Auto_tac); |
923 | 349 |
qed "mem_append"; |
2512 | 350 |
Addsimps[mem_append]; |
923 | 351 |
|
4935 | 352 |
Goal "x mem [x:xs. P(x)] = (x mem xs & P(x))"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
353 |
by (induct_tac "xs" 1); |
5129 | 354 |
by (Auto_tac); |
923 | 355 |
qed "mem_filter"; |
2512 | 356 |
Addsimps[mem_filter]; |
923 | 357 |
|
3465 | 358 |
(** set **) |
1812 | 359 |
|
3467 | 360 |
section "set"; |
361 |
||
4935 | 362 |
Goal "set (xs@ys) = (set xs Un set ys)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
363 |
by (induct_tac "xs" 1); |
5129 | 364 |
by (Auto_tac); |
3647
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
365 |
qed "set_append"; |
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
366 |
Addsimps[set_append]; |
1812 | 367 |
|
4935 | 368 |
Goal "(x mem xs) = (x: set xs)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
369 |
by (induct_tac "xs" 1); |
5129 | 370 |
by (Auto_tac); |
3647
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
371 |
qed "set_mem_eq"; |
1812 | 372 |
|
4935 | 373 |
Goal "set l <= set (x#l)"; |
5129 | 374 |
by (Auto_tac); |
3647
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
375 |
qed "set_subset_Cons"; |
1936 | 376 |
|
4935 | 377 |
Goal "(set xs = {}) = (xs = [])"; |
3457 | 378 |
by (induct_tac "xs" 1); |
5129 | 379 |
by (Auto_tac); |
3647
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
380 |
qed "set_empty"; |
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
381 |
Addsimps [set_empty]; |
2608 | 382 |
|
4935 | 383 |
Goal "set(rev xs) = set(xs)"; |
3457 | 384 |
by (induct_tac "xs" 1); |
5129 | 385 |
by (Auto_tac); |
3647
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
386 |
qed "set_rev"; |
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
387 |
Addsimps [set_rev]; |
2608 | 388 |
|
4935 | 389 |
Goal "set(map f xs) = f``(set xs)"; |
3457 | 390 |
by (induct_tac "xs" 1); |
5129 | 391 |
by (Auto_tac); |
3647
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
392 |
qed "set_map"; |
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
393 |
Addsimps [set_map]; |
2608 | 394 |
|
4935 | 395 |
Goal "(x : set(filter P xs)) = (x : set xs & P x)"; |
4605 | 396 |
by (induct_tac "xs" 1); |
5129 | 397 |
by (Auto_tac); |
4605 | 398 |
qed "in_set_filter"; |
399 |
Addsimps [in_set_filter]; |
|
400 |
||
5272 | 401 |
Goal "(x : set xs) = (? ys zs. xs = ys@x#zs)"; |
402 |
by(induct_tac "xs" 1); |
|
403 |
by(Simp_tac 1); |
|
404 |
by(Asm_simp_tac 1); |
|
405 |
br iffI 1; |
|
406 |
by(blast_tac (claset() addIs [eq_Nil_appendI,Cons_eq_appendI]) 1); |
|
407 |
by(REPEAT(etac exE 1)); |
|
408 |
by(exhaust_tac "ys" 1); |
|
409 |
by(Auto_tac); |
|
410 |
qed "in_set_conv_decomp"; |
|
411 |
||
412 |
(* eliminate `lists' in favour of `set' *) |
|
413 |
||
414 |
Goal "(xs : lists A) = (!x : set xs. x : A)"; |
|
415 |
by(induct_tac "xs" 1); |
|
416 |
by(Auto_tac); |
|
417 |
qed "in_lists_conv_set"; |
|
418 |
||
419 |
bind_thm("in_listsD",in_lists_conv_set RS iffD1); |
|
420 |
AddSDs [in_listsD]; |
|
421 |
bind_thm("in_listsI",in_lists_conv_set RS iffD2); |
|
422 |
AddSIs [in_listsI]; |
|
1812 | 423 |
|
923 | 424 |
(** list_all **) |
425 |
||
3467 | 426 |
section "list_all"; |
427 |
||
4935 | 428 |
Goal "list_all (%x. True) xs = True"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
429 |
by (induct_tac "xs" 1); |
5129 | 430 |
by (Auto_tac); |
923 | 431 |
qed "list_all_True"; |
2512 | 432 |
Addsimps [list_all_True]; |
923 | 433 |
|
4935 | 434 |
Goal "list_all p (xs@ys) = (list_all p xs & list_all p ys)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
435 |
by (induct_tac "xs" 1); |
5129 | 436 |
by (Auto_tac); |
2512 | 437 |
qed "list_all_append"; |
438 |
Addsimps [list_all_append]; |
|
923 | 439 |
|
4935 | 440 |
Goal "list_all P xs = (!x. x mem xs --> P(x))"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
441 |
by (induct_tac "xs" 1); |
5129 | 442 |
by (Auto_tac); |
923 | 443 |
qed "list_all_mem_conv"; |
444 |
||
445 |
||
2608 | 446 |
(** filter **) |
923 | 447 |
|
3467 | 448 |
section "filter"; |
449 |
||
4935 | 450 |
Goal "filter P (xs@ys) = filter P xs @ filter P ys"; |
3457 | 451 |
by (induct_tac "xs" 1); |
5129 | 452 |
by (Auto_tac); |
2608 | 453 |
qed "filter_append"; |
454 |
Addsimps [filter_append]; |
|
455 |
||
4935 | 456 |
Goal "filter (%x. True) xs = xs"; |
4605 | 457 |
by (induct_tac "xs" 1); |
5129 | 458 |
by (Auto_tac); |
4605 | 459 |
qed "filter_True"; |
460 |
Addsimps [filter_True]; |
|
461 |
||
4935 | 462 |
Goal "filter (%x. False) xs = []"; |
4605 | 463 |
by (induct_tac "xs" 1); |
5129 | 464 |
by (Auto_tac); |
4605 | 465 |
qed "filter_False"; |
466 |
Addsimps [filter_False]; |
|
467 |
||
4935 | 468 |
Goal "length (filter P xs) <= length xs"; |
3457 | 469 |
by (induct_tac "xs" 1); |
5129 | 470 |
by (Auto_tac); |
4605 | 471 |
qed "length_filter"; |
3383
7707cb7a5054
Corrected statement of filter_append; added filter_size
paulson
parents:
3342
diff
changeset
|
472 |
|
2608 | 473 |
|
474 |
(** concat **) |
|
475 |
||
3467 | 476 |
section "concat"; |
477 |
||
4935 | 478 |
Goal "concat(xs@ys) = concat(xs)@concat(ys)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
479 |
by (induct_tac "xs" 1); |
5129 | 480 |
by (Auto_tac); |
2608 | 481 |
qed"concat_append"; |
482 |
Addsimps [concat_append]; |
|
2512 | 483 |
|
4935 | 484 |
Goal "(concat xss = []) = (!xs:set xss. xs=[])"; |
4423 | 485 |
by (induct_tac "xss" 1); |
5129 | 486 |
by (Auto_tac); |
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
487 |
qed "concat_eq_Nil_conv"; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
488 |
AddIffs [concat_eq_Nil_conv]; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
489 |
|
4935 | 490 |
Goal "([] = concat xss) = (!xs:set xss. xs=[])"; |
4423 | 491 |
by (induct_tac "xss" 1); |
5129 | 492 |
by (Auto_tac); |
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
493 |
qed "Nil_eq_concat_conv"; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
494 |
AddIffs [Nil_eq_concat_conv]; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
495 |
|
4935 | 496 |
Goal "set(concat xs) = Union(set `` set xs)"; |
3467 | 497 |
by (induct_tac "xs" 1); |
5129 | 498 |
by (Auto_tac); |
3647
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
499 |
qed"set_concat"; |
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
500 |
Addsimps [set_concat]; |
3467 | 501 |
|
4935 | 502 |
Goal "map f (concat xs) = concat (map (map f) xs)"; |
3467 | 503 |
by (induct_tac "xs" 1); |
5129 | 504 |
by (Auto_tac); |
3467 | 505 |
qed "map_concat"; |
506 |
||
4935 | 507 |
Goal "filter p (concat xs) = concat (map (filter p) xs)"; |
3467 | 508 |
by (induct_tac "xs" 1); |
5129 | 509 |
by (Auto_tac); |
3467 | 510 |
qed"filter_concat"; |
511 |
||
4935 | 512 |
Goal "rev(concat xs) = concat (map rev (rev xs))"; |
3467 | 513 |
by (induct_tac "xs" 1); |
5129 | 514 |
by (Auto_tac); |
2608 | 515 |
qed "rev_concat"; |
923 | 516 |
|
517 |
(** nth **) |
|
518 |
||
3467 | 519 |
section "nth"; |
520 |
||
5278 | 521 |
Goal "!xs. (xs@ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"; |
5183 | 522 |
by (induct_tac "n" 1); |
3457 | 523 |
by (Asm_simp_tac 1); |
524 |
by (rtac allI 1); |
|
525 |
by (exhaust_tac "xs" 1); |
|
5129 | 526 |
by (Auto_tac); |
2608 | 527 |
qed_spec_mp "nth_append"; |
528 |
||
4935 | 529 |
Goal "!n. n < length xs --> (map f xs)!n = f(xs!n)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
530 |
by (induct_tac "xs" 1); |
1301 | 531 |
(* case [] *) |
532 |
by (Asm_full_simp_tac 1); |
|
533 |
(* case x#xl *) |
|
534 |
by (rtac allI 1); |
|
5183 | 535 |
by (induct_tac "n" 1); |
5129 | 536 |
by (Auto_tac); |
1485
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1465
diff
changeset
|
537 |
qed_spec_mp "nth_map"; |
1301 | 538 |
Addsimps [nth_map]; |
539 |
||
4935 | 540 |
Goal "!n. n < length xs --> list_all P xs --> P(xs!n)"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
541 |
by (induct_tac "xs" 1); |
1301 | 542 |
(* case [] *) |
543 |
by (Simp_tac 1); |
|
544 |
(* case x#xl *) |
|
545 |
by (rtac allI 1); |
|
5183 | 546 |
by (induct_tac "n" 1); |
5129 | 547 |
by (Auto_tac); |
1485
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1465
diff
changeset
|
548 |
qed_spec_mp "list_all_nth"; |
1301 | 549 |
|
4935 | 550 |
Goal "!n. n < length xs --> xs!n mem xs"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
551 |
by (induct_tac "xs" 1); |
1301 | 552 |
(* case [] *) |
553 |
by (Simp_tac 1); |
|
554 |
(* case x#xl *) |
|
555 |
by (rtac allI 1); |
|
5183 | 556 |
by (induct_tac "n" 1); |
1301 | 557 |
(* case 0 *) |
558 |
by (Asm_full_simp_tac 1); |
|
559 |
(* case Suc x *) |
|
4686 | 560 |
by (Asm_full_simp_tac 1); |
1485
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1465
diff
changeset
|
561 |
qed_spec_mp "nth_mem"; |
1301 | 562 |
Addsimps [nth_mem]; |
563 |
||
5077
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents:
5043
diff
changeset
|
564 |
(** list update **) |
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents:
5043
diff
changeset
|
565 |
|
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents:
5043
diff
changeset
|
566 |
section "list update"; |
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents:
5043
diff
changeset
|
567 |
|
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents:
5043
diff
changeset
|
568 |
Goal "!i. length(xs[i:=x]) = length xs"; |
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents:
5043
diff
changeset
|
569 |
by (induct_tac "xs" 1); |
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents:
5043
diff
changeset
|
570 |
by (Simp_tac 1); |
5183 | 571 |
by (asm_full_simp_tac (simpset() addsplits [nat.split]) 1); |
5077
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents:
5043
diff
changeset
|
572 |
qed_spec_mp "length_list_update"; |
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents:
5043
diff
changeset
|
573 |
Addsimps [length_list_update]; |
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents:
5043
diff
changeset
|
574 |
|
71043526295f
* HOL/List: new function list_update written xs[i:=v] that updates the i-th
nipkow
parents:
5043
diff
changeset
|
575 |
|
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
576 |
(** last & butlast **) |
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
577 |
|
4935 | 578 |
Goal "last(xs@[x]) = x"; |
4423 | 579 |
by (induct_tac "xs" 1); |
5129 | 580 |
by (Auto_tac); |
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
581 |
qed "last_snoc"; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
582 |
Addsimps [last_snoc]; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
583 |
|
4935 | 584 |
Goal "butlast(xs@[x]) = xs"; |
4423 | 585 |
by (induct_tac "xs" 1); |
5129 | 586 |
by (Auto_tac); |
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
587 |
qed "butlast_snoc"; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
588 |
Addsimps [butlast_snoc]; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
589 |
|
4935 | 590 |
Goal "length(butlast xs) = length xs - 1"; |
591 |
by (res_inst_tac [("xs","xs")] rev_induct 1); |
|
5129 | 592 |
by (Auto_tac); |
4643 | 593 |
qed "length_butlast"; |
594 |
Addsimps [length_butlast]; |
|
595 |
||
5278 | 596 |
Goal "!ys. butlast (xs@ys) = (if ys=[] then butlast xs else xs@butlast ys)"; |
4423 | 597 |
by (induct_tac "xs" 1); |
5129 | 598 |
by (Auto_tac); |
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
599 |
qed_spec_mp "butlast_append"; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
600 |
|
4935 | 601 |
Goal "x:set(butlast xs) --> x:set xs"; |
4423 | 602 |
by (induct_tac "xs" 1); |
5129 | 603 |
by (Auto_tac); |
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
604 |
qed_spec_mp "in_set_butlastD"; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
605 |
|
5043 | 606 |
Goal "x:set(butlast xs) ==> x:set(butlast(xs@ys))"; |
4686 | 607 |
by (asm_simp_tac (simpset() addsimps [butlast_append]) 1); |
4423 | 608 |
by (blast_tac (claset() addDs [in_set_butlastD]) 1); |
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
609 |
qed "in_set_butlast_appendI1"; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
610 |
|
5043 | 611 |
Goal "x:set(butlast ys) ==> x:set(butlast(xs@ys))"; |
4686 | 612 |
by (asm_simp_tac (simpset() addsimps [butlast_append]) 1); |
4423 | 613 |
by (Clarify_tac 1); |
614 |
by (Full_simp_tac 1); |
|
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
615 |
qed "in_set_butlast_appendI2"; |
3902 | 616 |
|
2608 | 617 |
(** take & drop **) |
618 |
section "take & drop"; |
|
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
619 |
|
4935 | 620 |
Goal "take 0 xs = []"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
621 |
by (induct_tac "xs" 1); |
5129 | 622 |
by (Auto_tac); |
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
623 |
qed "take_0"; |
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
624 |
|
4935 | 625 |
Goal "drop 0 xs = xs"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
626 |
by (induct_tac "xs" 1); |
5129 | 627 |
by (Auto_tac); |
2608 | 628 |
qed "drop_0"; |
629 |
||
4935 | 630 |
Goal "take (Suc n) (x#xs) = x # take n xs"; |
1552 | 631 |
by (Simp_tac 1); |
1419
a6a034a47a71
defined take/drop by induction over list rather than nat.
nipkow
parents:
1327
diff
changeset
|
632 |
qed "take_Suc_Cons"; |
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
633 |
|
4935 | 634 |
Goal "drop (Suc n) (x#xs) = drop n xs"; |
2608 | 635 |
by (Simp_tac 1); |
636 |
qed "drop_Suc_Cons"; |
|
637 |
||
638 |
Delsimps [take_Cons,drop_Cons]; |
|
639 |
Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons]; |
|
640 |
||
4935 | 641 |
Goal "!xs. length(take n xs) = min (length xs) n"; |
5183 | 642 |
by (induct_tac "n" 1); |
5129 | 643 |
by (Auto_tac); |
3457 | 644 |
by (exhaust_tac "xs" 1); |
5129 | 645 |
by (Auto_tac); |
2608 | 646 |
qed_spec_mp "length_take"; |
647 |
Addsimps [length_take]; |
|
923 | 648 |
|
4935 | 649 |
Goal "!xs. length(drop n xs) = (length xs - n)"; |
5183 | 650 |
by (induct_tac "n" 1); |
5129 | 651 |
by (Auto_tac); |
3457 | 652 |
by (exhaust_tac "xs" 1); |
5129 | 653 |
by (Auto_tac); |
2608 | 654 |
qed_spec_mp "length_drop"; |
655 |
Addsimps [length_drop]; |
|
656 |
||
4935 | 657 |
Goal "!xs. length xs <= n --> take n xs = xs"; |
5183 | 658 |
by (induct_tac "n" 1); |
5129 | 659 |
by (Auto_tac); |
3457 | 660 |
by (exhaust_tac "xs" 1); |
5129 | 661 |
by (Auto_tac); |
2608 | 662 |
qed_spec_mp "take_all"; |
923 | 663 |
|
4935 | 664 |
Goal "!xs. length xs <= n --> drop n xs = []"; |
5183 | 665 |
by (induct_tac "n" 1); |
5129 | 666 |
by (Auto_tac); |
3457 | 667 |
by (exhaust_tac "xs" 1); |
5129 | 668 |
by (Auto_tac); |
2608 | 669 |
qed_spec_mp "drop_all"; |
670 |
||
5278 | 671 |
Goal "!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"; |
5183 | 672 |
by (induct_tac "n" 1); |
5129 | 673 |
by (Auto_tac); |
3457 | 674 |
by (exhaust_tac "xs" 1); |
5129 | 675 |
by (Auto_tac); |
2608 | 676 |
qed_spec_mp "take_append"; |
677 |
Addsimps [take_append]; |
|
678 |
||
4935 | 679 |
Goal "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; |
5183 | 680 |
by (induct_tac "n" 1); |
5129 | 681 |
by (Auto_tac); |
3457 | 682 |
by (exhaust_tac "xs" 1); |
5129 | 683 |
by (Auto_tac); |
2608 | 684 |
qed_spec_mp "drop_append"; |
685 |
Addsimps [drop_append]; |
|
686 |
||
4935 | 687 |
Goal "!xs n. take n (take m xs) = take (min n m) xs"; |
5183 | 688 |
by (induct_tac "m" 1); |
5129 | 689 |
by (Auto_tac); |
3457 | 690 |
by (exhaust_tac "xs" 1); |
5129 | 691 |
by (Auto_tac); |
5183 | 692 |
by (exhaust_tac "na" 1); |
5129 | 693 |
by (Auto_tac); |
2608 | 694 |
qed_spec_mp "take_take"; |
695 |
||
4935 | 696 |
Goal "!xs. drop n (drop m xs) = drop (n + m) xs"; |
5183 | 697 |
by (induct_tac "m" 1); |
5129 | 698 |
by (Auto_tac); |
3457 | 699 |
by (exhaust_tac "xs" 1); |
5129 | 700 |
by (Auto_tac); |
2608 | 701 |
qed_spec_mp "drop_drop"; |
923 | 702 |
|
4935 | 703 |
Goal "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; |
5183 | 704 |
by (induct_tac "m" 1); |
5129 | 705 |
by (Auto_tac); |
3457 | 706 |
by (exhaust_tac "xs" 1); |
5129 | 707 |
by (Auto_tac); |
2608 | 708 |
qed_spec_mp "take_drop"; |
709 |
||
4935 | 710 |
Goal "!xs. take n (map f xs) = map f (take n xs)"; |
5183 | 711 |
by (induct_tac "n" 1); |
5129 | 712 |
by (Auto_tac); |
3457 | 713 |
by (exhaust_tac "xs" 1); |
5129 | 714 |
by (Auto_tac); |
2608 | 715 |
qed_spec_mp "take_map"; |
716 |
||
4935 | 717 |
Goal "!xs. drop n (map f xs) = map f (drop n xs)"; |
5183 | 718 |
by (induct_tac "n" 1); |
5129 | 719 |
by (Auto_tac); |
3457 | 720 |
by (exhaust_tac "xs" 1); |
5129 | 721 |
by (Auto_tac); |
2608 | 722 |
qed_spec_mp "drop_map"; |
723 |
||
4935 | 724 |
Goal "!n i. i < n --> (take n xs)!i = xs!i"; |
3457 | 725 |
by (induct_tac "xs" 1); |
5129 | 726 |
by (Auto_tac); |
3457 | 727 |
by (exhaust_tac "n" 1); |
728 |
by (Blast_tac 1); |
|
729 |
by (exhaust_tac "i" 1); |
|
5129 | 730 |
by (Auto_tac); |
2608 | 731 |
qed_spec_mp "nth_take"; |
732 |
Addsimps [nth_take]; |
|
923 | 733 |
|
4935 | 734 |
Goal "!xs i. n + i <= length xs --> (drop n xs)!i = xs!(n+i)"; |
5183 | 735 |
by (induct_tac "n" 1); |
5129 | 736 |
by (Auto_tac); |
3457 | 737 |
by (exhaust_tac "xs" 1); |
5129 | 738 |
by (Auto_tac); |
2608 | 739 |
qed_spec_mp "nth_drop"; |
740 |
Addsimps [nth_drop]; |
|
741 |
||
742 |
(** takeWhile & dropWhile **) |
|
743 |
||
3467 | 744 |
section "takeWhile & dropWhile"; |
745 |
||
4935 | 746 |
Goal "takeWhile P xs @ dropWhile P xs = xs"; |
3586 | 747 |
by (induct_tac "xs" 1); |
5129 | 748 |
by (Auto_tac); |
3586 | 749 |
qed "takeWhile_dropWhile_id"; |
750 |
Addsimps [takeWhile_dropWhile_id]; |
|
751 |
||
4935 | 752 |
Goal "x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs"; |
3457 | 753 |
by (induct_tac "xs" 1); |
5129 | 754 |
by (Auto_tac); |
2608 | 755 |
bind_thm("takeWhile_append1", conjI RS (result() RS mp)); |
756 |
Addsimps [takeWhile_append1]; |
|
923 | 757 |
|
4935 | 758 |
Goal "(!x:set xs. P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys"; |
3457 | 759 |
by (induct_tac "xs" 1); |
5129 | 760 |
by (Auto_tac); |
2608 | 761 |
bind_thm("takeWhile_append2", ballI RS (result() RS mp)); |
762 |
Addsimps [takeWhile_append2]; |
|
1169 | 763 |
|
4935 | 764 |
Goal "x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"; |
3457 | 765 |
by (induct_tac "xs" 1); |
5129 | 766 |
by (Auto_tac); |
2608 | 767 |
bind_thm("dropWhile_append1", conjI RS (result() RS mp)); |
768 |
Addsimps [dropWhile_append1]; |
|
769 |
||
4935 | 770 |
Goal "(!x:set xs. P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys"; |
3457 | 771 |
by (induct_tac "xs" 1); |
5129 | 772 |
by (Auto_tac); |
2608 | 773 |
bind_thm("dropWhile_append2", ballI RS (result() RS mp)); |
774 |
Addsimps [dropWhile_append2]; |
|
775 |
||
4935 | 776 |
Goal "x:set(takeWhile P xs) --> x:set xs & P x"; |
3457 | 777 |
by (induct_tac "xs" 1); |
5129 | 778 |
by (Auto_tac); |
3647
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
779 |
qed_spec_mp"set_take_whileD"; |
2608 | 780 |
|
4132 | 781 |
qed_goal "zip_Nil_Nil" thy "zip [] [] = []" (K [Simp_tac 1]); |
782 |
qed_goal "zip_Cons_Cons" thy "zip (x#xs) (y#ys) = (x,y)#zip xs ys" |
|
783 |
(K [Simp_tac 1]); |
|
4605 | 784 |
|
5272 | 785 |
|
786 |
(** foldl **) |
|
787 |
section "foldl"; |
|
788 |
||
789 |
Goal "!a. foldl f a (xs @ ys) = foldl f (foldl f a xs) ys"; |
|
790 |
by(induct_tac "xs" 1); |
|
791 |
by(Auto_tac); |
|
792 |
qed_spec_mp "foldl_append"; |
|
793 |
Addsimps [foldl_append]; |
|
794 |
||
795 |
(* Note: `n <= foldl op+ n ns' looks simpler, but is more difficult to use |
|
796 |
because it requires an additional transitivity step |
|
797 |
*) |
|
798 |
Goal "!n::nat. m <= n --> m <= foldl op+ n ns"; |
|
799 |
by(induct_tac "ns" 1); |
|
800 |
by(Simp_tac 1); |
|
801 |
by(Asm_full_simp_tac 1); |
|
802 |
by(blast_tac (claset() addIs [trans_le_add1]) 1); |
|
803 |
qed_spec_mp "start_le_sum"; |
|
804 |
||
805 |
Goal "n : set ns ==> n <= foldl op+ 0 ns"; |
|
806 |
by(auto_tac (claset() addIs [start_le_sum], |
|
807 |
simpset() addsimps [in_set_conv_decomp])); |
|
808 |
qed "elem_le_sum"; |
|
809 |
||
810 |
Goal "!m. (foldl op+ m ns = 0) = (m=0 & (!n : set ns. n=0))"; |
|
811 |
by(induct_tac "ns" 1); |
|
812 |
by(Auto_tac); |
|
813 |
qed_spec_mp "sum_eq_0_conv"; |
|
814 |
AddIffs [sum_eq_0_conv]; |
|
815 |
||
816 |
||
4605 | 817 |
(** nodups & remdups **) |
818 |
section "nodups & remdups"; |
|
819 |
||
4935 | 820 |
Goal "set(remdups xs) = set xs"; |
4605 | 821 |
by (induct_tac "xs" 1); |
822 |
by (Simp_tac 1); |
|
4686 | 823 |
by (asm_full_simp_tac (simpset() addsimps [insert_absorb]) 1); |
4605 | 824 |
qed "set_remdups"; |
825 |
Addsimps [set_remdups]; |
|
826 |
||
4935 | 827 |
Goal "nodups(remdups xs)"; |
4605 | 828 |
by (induct_tac "xs" 1); |
5129 | 829 |
by (Auto_tac); |
4605 | 830 |
qed "nodups_remdups"; |
831 |
||
4935 | 832 |
Goal "nodups xs --> nodups (filter P xs)"; |
4605 | 833 |
by (induct_tac "xs" 1); |
5129 | 834 |
by (Auto_tac); |
4605 | 835 |
qed_spec_mp "nodups_filter"; |
836 |
||
3589
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
837 |
(** replicate **) |
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
838 |
section "replicate"; |
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
839 |
|
4935 | 840 |
Goal "set(replicate (Suc n) x) = {x}"; |
4423 | 841 |
by (induct_tac "n" 1); |
5129 | 842 |
by (Auto_tac); |
3589
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
843 |
val lemma = result(); |
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
844 |
|
5043 | 845 |
Goal "n ~= 0 ==> set(replicate n x) = {x}"; |
4423 | 846 |
by (fast_tac (claset() addSDs [not0_implies_Suc] addSIs [lemma]) 1); |
3589
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
847 |
qed "set_replicate"; |
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
848 |
Addsimps [set_replicate]; |
5162 | 849 |
|
850 |
||
5281 | 851 |
(*** Lexcicographic orderings on lists ***) |
852 |
section"Lexcicographic orderings on lists"; |
|
853 |
||
854 |
Goal "wf r ==> wf(lexn r n)"; |
|
855 |
by(induct_tac "n" 1); |
|
856 |
by(Simp_tac 1); |
|
857 |
by(Simp_tac 1); |
|
858 |
br wf_subset 1; |
|
859 |
br Int_lower1 2; |
|
860 |
br wf_prod_fun_image 1; |
|
861 |
br injI 2; |
|
862 |
by(Auto_tac); |
|
863 |
qed "wf_lexn"; |
|
864 |
||
865 |
Goal "!xs ys. (xs,ys) : lexn r n --> length xs = n & length ys = n"; |
|
866 |
by(induct_tac "n" 1); |
|
867 |
by(Auto_tac); |
|
868 |
qed_spec_mp "lexn_length"; |
|
869 |
||
870 |
Goalw [lex_def] "wf r ==> wf(lex r)"; |
|
871 |
br wf_UN 1; |
|
872 |
by(blast_tac (claset() addIs [wf_lexn]) 1); |
|
873 |
by(Clarify_tac 1); |
|
874 |
by(rename_tac "m n" 1); |
|
875 |
by(subgoal_tac "m ~= n" 1); |
|
876 |
by(Blast_tac 2); |
|
877 |
by(blast_tac (claset() addDs [lexn_length,not_sym]) 1); |
|
878 |
qed "wf_lex"; |
|
879 |
AddSIs [wf_lex]; |
|
880 |
||
881 |
Goal |
|
882 |
"lexn r n = \ |
|
883 |
\ {(xs,ys). length xs = n & length ys = n & \ |
|
884 |
\ (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (x,y):r)}"; |
|
885 |
by(induct_tac "n" 1); |
|
886 |
by(Simp_tac 1); |
|
887 |
by(Blast_tac 1); |
|
888 |
by(asm_full_simp_tac (simpset() addsimps [lex_prod_def]) 1); |
|
889 |
by(Auto_tac); |
|
890 |
by(Blast_tac 1); |
|
891 |
by(rename_tac "a xys x xs' y ys'" 1); |
|
892 |
by(res_inst_tac [("x","a#xys")] exI 1); |
|
893 |
by(Simp_tac 1); |
|
894 |
by(exhaust_tac "xys" 1); |
|
895 |
by(ALLGOALS Asm_full_simp_tac); |
|
896 |
by(Blast_tac 1); |
|
897 |
qed "lexn_conv"; |
|
898 |
||
899 |
Goalw [lex_def] |
|
900 |
"lex r = \ |
|
901 |
\ {(xs,ys). length xs = length ys & \ |
|
902 |
\ (? xys x y xs' ys'. xs= xys @ x#xs' & ys= xys @ y#ys' & (x,y):r)}"; |
|
903 |
by(force_tac (claset(), simpset() addsimps [lexn_conv]) 1); |
|
904 |
qed "lex_conv"; |
|
905 |
||
906 |
Goalw [lexico_def] "wf r ==> wf(lexico r)"; |
|
907 |
by(Blast_tac 1); |
|
908 |
qed "wf_lexico"; |
|
909 |
AddSIs [wf_lexico]; |
|
910 |
||
911 |
Goalw |
|
912 |
[lexico_def,diag_def,lex_prod_def,measure_def,inv_image_def] |
|
913 |
"lexico r = {(xs,ys). length xs < length ys | \ |
|
914 |
\ length xs = length ys & (xs,ys) : lex r}"; |
|
915 |
by(Simp_tac 1); |
|
916 |
qed "lexico_conv"; |
|
917 |
||
5162 | 918 |
(*** |
919 |
Simplification procedure for all list equalities. |
|
920 |
Currently only tries to rearranges @ to see if |
|
921 |
- both lists end in a singleton list, |
|
922 |
- or both lists end in the same list. |
|
923 |
***) |
|
924 |
local |
|
925 |
||
926 |
val list_eq_pattern = |
|
927 |
read_cterm (sign_of List.thy) ("(xs::'a list) = ys",HOLogic.boolT); |
|
928 |
||
5183 | 929 |
fun last (cons as Const("List.list.op #",_) $ _ $ xs) = |
930 |
(case xs of Const("List.list.[]",_) => cons | _ => last xs) |
|
5200 | 931 |
| last (Const("List.op @",_) $ _ $ ys) = last ys |
5162 | 932 |
| last t = t; |
933 |
||
5183 | 934 |
fun list1 (Const("List.list.op #",_) $ _ $ Const("List.list.[]",_)) = true |
5162 | 935 |
| list1 _ = false; |
936 |
||
5183 | 937 |
fun butlast ((cons as Const("List.list.op #",_) $ x) $ xs) = |
938 |
(case xs of Const("List.list.[]",_) => xs | _ => cons $ butlast xs) |
|
5200 | 939 |
| butlast ((app as Const("List.op @",_) $ xs) $ ys) = app $ butlast ys |
5183 | 940 |
| butlast xs = Const("List.list.[]",fastype_of xs); |
5162 | 941 |
|
942 |
val rearr_tac = |
|
943 |
simp_tac (HOL_basic_ss addsimps [append_assoc,append_Nil,append_Cons]); |
|
944 |
||
945 |
fun list_eq sg _ (F as (eq as Const(_,eqT)) $ lhs $ rhs) = |
|
946 |
let |
|
947 |
val lastl = last lhs and lastr = last rhs |
|
948 |
fun rearr conv = |
|
949 |
let val lhs1 = butlast lhs and rhs1 = butlast rhs |
|
950 |
val Type(_,listT::_) = eqT |
|
951 |
val appT = [listT,listT] ---> listT |
|
5200 | 952 |
val app = Const("List.op @",appT) |
5162 | 953 |
val F2 = eq $ (app$lhs1$lastl) $ (app$rhs1$lastr) |
954 |
val ct = cterm_of sg (HOLogic.mk_Trueprop(HOLogic.mk_eq(F,F2))) |
|
955 |
val thm = prove_goalw_cterm [] ct (K [rearr_tac 1]) |
|
956 |
handle ERROR => |
|
957 |
error("The error(s) above occurred while trying to prove " ^ |
|
958 |
string_of_cterm ct) |
|
959 |
in Some((conv RS (thm RS trans)) RS eq_reflection) end |
|
960 |
||
961 |
in if list1 lastl andalso list1 lastr |
|
962 |
then rearr append1_eq_conv |
|
963 |
else |
|
964 |
if lastl aconv lastr |
|
965 |
then rearr append_same_eq |
|
966 |
else None |
|
967 |
end; |
|
968 |
in |
|
969 |
val list_eq_simproc = mk_simproc "list_eq" [list_eq_pattern] list_eq; |
|
970 |
end; |
|
971 |
||
972 |
Addsimprocs [list_eq_simproc]; |