author | nipkow |
Mon, 09 Feb 1998 14:40:59 +0100 | |
changeset 4612 | 26764de50c74 |
parent 4605 | 579e0ef2df6b |
child 4628 | 0c7e97836e3c |
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 |
*) |
|
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||
3708 | 9 |
open List; |
10 |
||
3011 | 11 |
goal thy "!x. xs ~= x#xs"; |
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by (induct_tac "xs" 1); |
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by (ALLGOALS Asm_simp_tac); |
2608 | 14 |
qed_spec_mp "not_Cons_self"; |
3574 | 15 |
bind_thm("not_Cons_self2",not_Cons_self RS not_sym); |
16 |
Addsimps [not_Cons_self,not_Cons_self2]; |
|
923 | 17 |
|
3011 | 18 |
goal thy "(xs ~= []) = (? y ys. xs = y#ys)"; |
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by (induct_tac "xs" 1); |
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by (Simp_tac 1); |
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by (Asm_simp_tac 1); |
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qed "neq_Nil_conv"; |
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||
24 |
||
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(** "lists": the list-forming operator over sets **) |
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goalw thy lists.defs "!!A B. A<=B ==> lists A <= lists B"; |
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28 |
by (rtac lfp_mono 1); |
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by (REPEAT (ares_tac basic_monos 1)); |
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qed "lists_mono"; |
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|
3468 | 32 |
val listsE = lists.mk_cases list.simps "x#l : lists A"; |
33 |
AddSEs [listsE]; |
|
34 |
AddSIs lists.intrs; |
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35 |
||
36 |
goal thy "!!l. l: lists A ==> l: lists B --> l: lists (A Int B)"; |
|
37 |
by (etac lists.induct 1); |
|
38 |
by (ALLGOALS Blast_tac); |
|
39 |
qed_spec_mp "lists_IntI"; |
|
40 |
||
41 |
goal thy "lists (A Int B) = lists A Int lists B"; |
|
4423 | 42 |
by (rtac (mono_Int RS equalityI) 1); |
4089 | 43 |
by (simp_tac (simpset() addsimps [mono_def, lists_mono]) 1); |
44 |
by (blast_tac (claset() addSIs [lists_IntI]) 1); |
|
3468 | 45 |
qed "lists_Int_eq"; |
46 |
Addsimps [lists_Int_eq]; |
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47 |
||
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(** list_case **) |
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||
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val prems = goal thy "[| P([]); !!x xs. P(x#xs) |] ==> P(xs)"; |
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by (induct_tac "xs" 1); |
53 |
by (REPEAT(resolve_tac prems 1)); |
|
2608 | 54 |
qed "list_cases"; |
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goal thy "(xs=[] --> P([])) & (!y ys. xs=y#ys --> P(y#ys)) --> P(xs)"; |
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by (induct_tac "xs" 1); |
2891 | 58 |
by (Blast_tac 1); |
59 |
by (Blast_tac 1); |
|
2608 | 60 |
bind_thm("list_eq_cases", |
61 |
impI RSN (2,allI RSN (2,allI RSN (2,impI RS (conjI RS (result() RS mp)))))); |
|
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||
63 |
||
3860 | 64 |
(** length **) |
65 |
(* needs to come before "@" because of thm append_eq_append_conv *) |
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||
67 |
section "length"; |
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||
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goal thy "length(xs@ys) = length(xs)+length(ys)"; |
|
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by (induct_tac "xs" 1); |
|
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by (ALLGOALS Asm_simp_tac); |
|
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qed"length_append"; |
|
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Addsimps [length_append]; |
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||
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goal thy "length (map f l) = length l"; |
|
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by (induct_tac "l" 1); |
|
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by (ALLGOALS Simp_tac); |
|
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qed "length_map"; |
|
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Addsimps [length_map]; |
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||
81 |
goal thy "length(rev xs) = length(xs)"; |
|
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by (induct_tac "xs" 1); |
|
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by (ALLGOALS Asm_simp_tac); |
|
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qed "length_rev"; |
|
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Addsimps [length_rev]; |
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||
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goal List.thy "!!xs. xs ~= [] ==> length(tl xs) = pred(length xs)"; |
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by (exhaust_tac "xs" 1); |
89 |
by (ALLGOALS Asm_full_simp_tac); |
|
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qed "length_tl"; |
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Addsimps [length_tl]; |
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goal thy "(length xs = 0) = (xs = [])"; |
94 |
by (induct_tac "xs" 1); |
|
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by (ALLGOALS Asm_simp_tac); |
|
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qed "length_0_conv"; |
|
97 |
AddIffs [length_0_conv]; |
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||
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goal thy "(0 = length xs) = (xs = [])"; |
|
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by (induct_tac "xs" 1); |
|
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by (ALLGOALS Asm_simp_tac); |
|
102 |
qed "zero_length_conv"; |
|
103 |
AddIffs [zero_length_conv]; |
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||
105 |
goal thy "(0 < length xs) = (xs ~= [])"; |
|
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by (induct_tac "xs" 1); |
|
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by (ALLGOALS Asm_simp_tac); |
|
108 |
qed "length_greater_0_conv"; |
|
109 |
AddIffs [length_greater_0_conv]; |
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110 |
||
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(** @ - append **) |
112 |
||
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section "@ - append"; |
114 |
||
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goal thy "(xs@ys)@zs = xs@(ys@zs)"; |
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by (induct_tac "xs" 1); |
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by (ALLGOALS Asm_simp_tac); |
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qed "append_assoc"; |
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Addsimps [append_assoc]; |
923 | 120 |
|
3011 | 121 |
goal thy "xs @ [] = xs"; |
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by (induct_tac "xs" 1); |
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by (ALLGOALS Asm_simp_tac); |
923 | 124 |
qed "append_Nil2"; |
2512 | 125 |
Addsimps [append_Nil2]; |
923 | 126 |
|
3011 | 127 |
goal thy "(xs@ys = []) = (xs=[] & ys=[])"; |
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by (induct_tac "xs" 1); |
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129 |
by (ALLGOALS Asm_simp_tac); |
2608 | 130 |
qed "append_is_Nil_conv"; |
131 |
AddIffs [append_is_Nil_conv]; |
|
132 |
||
3011 | 133 |
goal thy "([] = xs@ys) = (xs=[] & ys=[])"; |
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134 |
by (induct_tac "xs" 1); |
2608 | 135 |
by (ALLGOALS Asm_simp_tac); |
3457 | 136 |
by (Blast_tac 1); |
2608 | 137 |
qed "Nil_is_append_conv"; |
138 |
AddIffs [Nil_is_append_conv]; |
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923 | 139 |
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goal thy "(xs @ ys = xs) = (ys=[])"; |
141 |
by (induct_tac "xs" 1); |
|
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by (ALLGOALS Asm_simp_tac); |
|
143 |
qed "append_self_conv"; |
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||
145 |
goal thy "(xs = xs @ ys) = (ys=[])"; |
|
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by (induct_tac "xs" 1); |
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by (ALLGOALS Asm_simp_tac); |
|
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by (Blast_tac 1); |
|
149 |
qed "self_append_conv"; |
|
150 |
AddIffs [append_self_conv,self_append_conv]; |
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||
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goal thy "!ys. length xs = length ys | length us = length vs \ |
153 |
\ --> (xs@us = ys@vs) = (xs=ys & us=vs)"; |
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4423 | 154 |
by (induct_tac "xs" 1); |
155 |
by (rtac allI 1); |
|
156 |
by (exhaust_tac "ys" 1); |
|
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by (Asm_simp_tac 1); |
|
158 |
by (fast_tac (claset() addIs [less_add_Suc2] addss simpset() |
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3860 | 159 |
addEs [less_not_refl2 RSN (2,rev_notE)]) 1); |
4423 | 160 |
by (rtac allI 1); |
161 |
by (exhaust_tac "ys" 1); |
|
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by (fast_tac (claset() addIs [less_add_Suc2] addss simpset() |
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3860 | 163 |
addEs [(less_not_refl2 RS not_sym) RSN (2,rev_notE)]) 1); |
4423 | 164 |
by (Asm_simp_tac 1); |
3860 | 165 |
qed_spec_mp "append_eq_append_conv"; |
166 |
Addsimps [append_eq_append_conv]; |
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167 |
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goal thy "(xs @ ys = xs @ zs) = (ys=zs)"; |
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by (Simp_tac 1); |
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170 |
qed "same_append_eq"; |
3860 | 171 |
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goal thy "(xs @ [x] = ys @ [y]) = (xs = ys & x = y)"; |
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173 |
by (Simp_tac 1); |
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qed "append1_eq_conv"; |
2608 | 175 |
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goal thy "(ys @ xs = zs @ xs) = (ys=zs)"; |
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by (Simp_tac 1); |
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qed "append_same_eq"; |
2608 | 179 |
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AddSIs |
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[same_append_eq RS iffD2, append1_eq_conv RS iffD2, append_same_eq RS iffD2]; |
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AddSDs |
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183 |
[same_append_eq RS iffD1, append1_eq_conv RS iffD1, append_same_eq RS iffD1]; |
3571 | 184 |
|
3011 | 185 |
goal thy "xs ~= [] --> hd xs # tl xs = xs"; |
3457 | 186 |
by (induct_tac "xs" 1); |
187 |
by (ALLGOALS Asm_simp_tac); |
|
2608 | 188 |
qed_spec_mp "hd_Cons_tl"; |
189 |
Addsimps [hd_Cons_tl]; |
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923 | 190 |
|
3011 | 191 |
goal thy "hd(xs@ys) = (if xs=[] then hd ys else hd xs)"; |
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by (induct_tac "xs" 1); |
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by (ALLGOALS Asm_simp_tac); |
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194 |
qed "hd_append"; |
923 | 195 |
|
3571 | 196 |
goal thy "!!xs. xs ~= [] ==> hd(xs @ ys) = hd xs"; |
4089 | 197 |
by (asm_simp_tac (simpset() addsimps [hd_append] |
4069 | 198 |
addsplits [split_list_case]) 1); |
3571 | 199 |
qed "hd_append2"; |
200 |
Addsimps [hd_append2]; |
|
201 |
||
3011 | 202 |
goal thy "tl(xs@ys) = (case xs of [] => tl(ys) | z#zs => zs@ys)"; |
4089 | 203 |
by (simp_tac (simpset() addsplits [split_list_case]) 1); |
2608 | 204 |
qed "tl_append"; |
205 |
||
3571 | 206 |
goal thy "!!xs. xs ~= [] ==> tl(xs @ ys) = (tl xs) @ ys"; |
4089 | 207 |
by (asm_simp_tac (simpset() addsimps [tl_append] |
4069 | 208 |
addsplits [split_list_case]) 1); |
3571 | 209 |
qed "tl_append2"; |
210 |
Addsimps [tl_append2]; |
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211 |
||
2608 | 212 |
(** map **) |
213 |
||
3467 | 214 |
section "map"; |
215 |
||
3011 | 216 |
goal thy |
3465 | 217 |
"(!x. x : set xs --> f x = g x) --> map f xs = map g xs"; |
3457 | 218 |
by (induct_tac "xs" 1); |
219 |
by (ALLGOALS Asm_simp_tac); |
|
2608 | 220 |
bind_thm("map_ext", impI RS (allI RS (result() RS mp))); |
221 |
||
3842 | 222 |
goal thy "map (%x. x) = (%xs. xs)"; |
2608 | 223 |
by (rtac ext 1); |
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224 |
by (induct_tac "xs" 1); |
2608 | 225 |
by (ALLGOALS Asm_simp_tac); |
226 |
qed "map_ident"; |
|
227 |
Addsimps[map_ident]; |
|
228 |
||
3011 | 229 |
goal thy "map f (xs@ys) = map f xs @ map f ys"; |
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230 |
by (induct_tac "xs" 1); |
2608 | 231 |
by (ALLGOALS Asm_simp_tac); |
232 |
qed "map_append"; |
|
233 |
Addsimps[map_append]; |
|
234 |
||
3011 | 235 |
goalw thy [o_def] "map (f o g) xs = map f (map g xs)"; |
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236 |
by (induct_tac "xs" 1); |
2608 | 237 |
by (ALLGOALS Asm_simp_tac); |
238 |
qed "map_compose"; |
|
239 |
Addsimps[map_compose]; |
|
240 |
||
3011 | 241 |
goal thy "rev(map f xs) = map f (rev xs)"; |
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242 |
by (induct_tac "xs" 1); |
2608 | 243 |
by (ALLGOALS Asm_simp_tac); |
244 |
qed "rev_map"; |
|
245 |
||
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246 |
(* a congruence rule for map: *) |
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247 |
goal thy |
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248 |
"(xs=ys) --> (!x. x : set ys --> f x = g x) --> map f xs = map g ys"; |
4423 | 249 |
by (rtac impI 1); |
250 |
by (hyp_subst_tac 1); |
|
251 |
by (induct_tac "ys" 1); |
|
252 |
by (ALLGOALS Asm_simp_tac); |
|
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253 |
val lemma = result(); |
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254 |
bind_thm("map_cong",impI RSN (2,allI RSN (2,lemma RS mp RS mp))); |
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255 |
|
3860 | 256 |
goal List.thy "(map f xs = []) = (xs = [])"; |
4423 | 257 |
by (induct_tac "xs" 1); |
258 |
by (ALLGOALS Asm_simp_tac); |
|
3860 | 259 |
qed "map_is_Nil_conv"; |
260 |
AddIffs [map_is_Nil_conv]; |
|
261 |
||
262 |
goal List.thy "([] = map f xs) = (xs = [])"; |
|
4423 | 263 |
by (induct_tac "xs" 1); |
264 |
by (ALLGOALS Asm_simp_tac); |
|
3860 | 265 |
qed "Nil_is_map_conv"; |
266 |
AddIffs [Nil_is_map_conv]; |
|
267 |
||
268 |
||
1169 | 269 |
(** rev **) |
270 |
||
3467 | 271 |
section "rev"; |
272 |
||
3011 | 273 |
goal thy "rev(xs@ys) = rev(ys) @ rev(xs)"; |
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274 |
by (induct_tac "xs" 1); |
2512 | 275 |
by (ALLGOALS Asm_simp_tac); |
1169 | 276 |
qed "rev_append"; |
2512 | 277 |
Addsimps[rev_append]; |
1169 | 278 |
|
3011 | 279 |
goal thy "rev(rev l) = l"; |
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280 |
by (induct_tac "l" 1); |
2512 | 281 |
by (ALLGOALS Asm_simp_tac); |
1169 | 282 |
qed "rev_rev_ident"; |
2512 | 283 |
Addsimps[rev_rev_ident]; |
1169 | 284 |
|
3860 | 285 |
goal thy "(rev xs = []) = (xs = [])"; |
4423 | 286 |
by (induct_tac "xs" 1); |
287 |
by (ALLGOALS Asm_simp_tac); |
|
3860 | 288 |
qed "rev_is_Nil_conv"; |
289 |
AddIffs [rev_is_Nil_conv]; |
|
290 |
||
291 |
goal thy "([] = rev xs) = (xs = [])"; |
|
4423 | 292 |
by (induct_tac "xs" 1); |
293 |
by (ALLGOALS Asm_simp_tac); |
|
3860 | 294 |
qed "Nil_is_rev_conv"; |
295 |
AddIffs [Nil_is_rev_conv]; |
|
296 |
||
2608 | 297 |
|
923 | 298 |
(** mem **) |
299 |
||
3467 | 300 |
section "mem"; |
301 |
||
3011 | 302 |
goal thy "x mem (xs@ys) = (x mem xs | x mem ys)"; |
3040
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Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
303 |
by (induct_tac "xs" 1); |
4089 | 304 |
by (ALLGOALS (asm_simp_tac (simpset() addsplits [expand_if]))); |
923 | 305 |
qed "mem_append"; |
2512 | 306 |
Addsimps[mem_append]; |
923 | 307 |
|
3842 | 308 |
goal thy "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
|
309 |
by (induct_tac "xs" 1); |
4089 | 310 |
by (ALLGOALS (asm_simp_tac (simpset() addsplits [expand_if]))); |
923 | 311 |
qed "mem_filter"; |
2512 | 312 |
Addsimps[mem_filter]; |
923 | 313 |
|
3465 | 314 |
(** set **) |
1812 | 315 |
|
3467 | 316 |
section "set"; |
317 |
||
3465 | 318 |
goal thy "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
|
319 |
by (induct_tac "xs" 1); |
1812 | 320 |
by (ALLGOALS Asm_simp_tac); |
3647
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
321 |
qed "set_append"; |
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
322 |
Addsimps[set_append]; |
1812 | 323 |
|
3465 | 324 |
goal thy "(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
|
325 |
by (induct_tac "xs" 1); |
4089 | 326 |
by (ALLGOALS (asm_simp_tac (simpset() addsplits [expand_if]))); |
2891 | 327 |
by (Blast_tac 1); |
3647
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
328 |
qed "set_mem_eq"; |
1812 | 329 |
|
3465 | 330 |
goal thy "set l <= set (x#l)"; |
1936 | 331 |
by (Simp_tac 1); |
2891 | 332 |
by (Blast_tac 1); |
3647
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
333 |
qed "set_subset_Cons"; |
1936 | 334 |
|
3465 | 335 |
goal thy "(set xs = {}) = (xs = [])"; |
3457 | 336 |
by (induct_tac "xs" 1); |
337 |
by (ALLGOALS Asm_simp_tac); |
|
3647
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
338 |
qed "set_empty"; |
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
339 |
Addsimps [set_empty]; |
2608 | 340 |
|
3465 | 341 |
goal thy "set(rev xs) = set(xs)"; |
3457 | 342 |
by (induct_tac "xs" 1); |
343 |
by (ALLGOALS Asm_simp_tac); |
|
3647
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
344 |
qed "set_rev"; |
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
345 |
Addsimps [set_rev]; |
2608 | 346 |
|
3465 | 347 |
goal thy "set(map f xs) = f``(set xs)"; |
3457 | 348 |
by (induct_tac "xs" 1); |
349 |
by (ALLGOALS Asm_simp_tac); |
|
3647
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
350 |
qed "set_map"; |
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
351 |
Addsimps [set_map]; |
2608 | 352 |
|
4605 | 353 |
goal thy "set(map f xs) = f``(set xs)"; |
354 |
by (induct_tac "xs" 1); |
|
355 |
by (ALLGOALS Asm_simp_tac); |
|
356 |
qed "set_map"; |
|
357 |
Addsimps [set_map]; |
|
358 |
||
359 |
goal thy "(x : set(filter P xs)) = (x : set xs & P x)"; |
|
360 |
by (induct_tac "xs" 1); |
|
361 |
by (ALLGOALS (asm_simp_tac (simpset() addsplits [expand_if]))); |
|
362 |
by(Blast_tac 1); |
|
363 |
qed "in_set_filter"; |
|
364 |
Addsimps [in_set_filter]; |
|
365 |
||
1812 | 366 |
|
923 | 367 |
(** list_all **) |
368 |
||
3467 | 369 |
section "list_all"; |
370 |
||
3842 | 371 |
goal thy "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
|
372 |
by (induct_tac "xs" 1); |
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1202
diff
changeset
|
373 |
by (ALLGOALS Asm_simp_tac); |
923 | 374 |
qed "list_all_True"; |
2512 | 375 |
Addsimps [list_all_True]; |
923 | 376 |
|
3011 | 377 |
goal thy "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
|
378 |
by (induct_tac "xs" 1); |
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1202
diff
changeset
|
379 |
by (ALLGOALS Asm_simp_tac); |
2512 | 380 |
qed "list_all_append"; |
381 |
Addsimps [list_all_append]; |
|
923 | 382 |
|
3011 | 383 |
goal thy "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
|
384 |
by (induct_tac "xs" 1); |
4089 | 385 |
by (ALLGOALS (asm_simp_tac (simpset() addsplits [expand_if]))); |
2891 | 386 |
by (Blast_tac 1); |
923 | 387 |
qed "list_all_mem_conv"; |
388 |
||
389 |
||
2608 | 390 |
(** filter **) |
923 | 391 |
|
3467 | 392 |
section "filter"; |
393 |
||
3383
7707cb7a5054
Corrected statement of filter_append; added filter_size
paulson
parents:
3342
diff
changeset
|
394 |
goal thy "filter P (xs@ys) = filter P xs @ filter P ys"; |
3457 | 395 |
by (induct_tac "xs" 1); |
4089 | 396 |
by (ALLGOALS (asm_simp_tac (simpset() addsplits [expand_if]))); |
2608 | 397 |
qed "filter_append"; |
398 |
Addsimps [filter_append]; |
|
399 |
||
4605 | 400 |
goal thy "filter (%x. True) xs = xs"; |
401 |
by (induct_tac "xs" 1); |
|
402 |
by (ALLGOALS Asm_simp_tac); |
|
403 |
qed "filter_True"; |
|
404 |
Addsimps [filter_True]; |
|
405 |
||
406 |
goal thy "filter (%x. False) xs = []"; |
|
407 |
by (induct_tac "xs" 1); |
|
408 |
by (ALLGOALS Asm_simp_tac); |
|
409 |
qed "filter_False"; |
|
410 |
Addsimps [filter_False]; |
|
411 |
||
412 |
goal thy "length (filter P xs) <= length xs"; |
|
3457 | 413 |
by (induct_tac "xs" 1); |
4089 | 414 |
by (ALLGOALS (asm_simp_tac (simpset() addsplits [expand_if]))); |
4605 | 415 |
qed "length_filter"; |
3383
7707cb7a5054
Corrected statement of filter_append; added filter_size
paulson
parents:
3342
diff
changeset
|
416 |
|
2608 | 417 |
|
418 |
(** concat **) |
|
419 |
||
3467 | 420 |
section "concat"; |
421 |
||
3011 | 422 |
goal thy "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
|
423 |
by (induct_tac "xs" 1); |
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1202
diff
changeset
|
424 |
by (ALLGOALS Asm_simp_tac); |
2608 | 425 |
qed"concat_append"; |
426 |
Addsimps [concat_append]; |
|
2512 | 427 |
|
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
428 |
goal thy "(concat xss = []) = (!xs:set xss. xs=[])"; |
4423 | 429 |
by (induct_tac "xss" 1); |
430 |
by (ALLGOALS Asm_simp_tac); |
|
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
431 |
qed "concat_eq_Nil_conv"; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
432 |
AddIffs [concat_eq_Nil_conv]; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
433 |
|
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
434 |
goal thy "([] = concat xss) = (!xs:set xss. xs=[])"; |
4423 | 435 |
by (induct_tac "xss" 1); |
436 |
by (ALLGOALS Asm_simp_tac); |
|
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
437 |
qed "Nil_eq_concat_conv"; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
438 |
AddIffs [Nil_eq_concat_conv]; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
439 |
|
3467 | 440 |
goal thy "set(concat xs) = Union(set `` set xs)"; |
441 |
by (induct_tac "xs" 1); |
|
442 |
by (ALLGOALS Asm_simp_tac); |
|
3647
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
443 |
qed"set_concat"; |
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
444 |
Addsimps [set_concat]; |
3467 | 445 |
|
446 |
goal thy "map f (concat xs) = concat (map (map f) xs)"; |
|
447 |
by (induct_tac "xs" 1); |
|
448 |
by (ALLGOALS Asm_simp_tac); |
|
449 |
qed "map_concat"; |
|
450 |
||
451 |
goal thy "filter p (concat xs) = concat (map (filter p) xs)"; |
|
452 |
by (induct_tac "xs" 1); |
|
453 |
by (ALLGOALS Asm_simp_tac); |
|
454 |
qed"filter_concat"; |
|
455 |
||
456 |
goal thy "rev(concat xs) = concat (map rev (rev xs))"; |
|
457 |
by (induct_tac "xs" 1); |
|
2512 | 458 |
by (ALLGOALS Asm_simp_tac); |
2608 | 459 |
qed "rev_concat"; |
923 | 460 |
|
461 |
(** nth **) |
|
462 |
||
3467 | 463 |
section "nth"; |
464 |
||
3011 | 465 |
goal thy |
4502 | 466 |
"!xs. (xs@ys)!n = (if n < length xs then xs!n else ys!(n - length xs))"; |
3457 | 467 |
by (nat_ind_tac "n" 1); |
468 |
by (Asm_simp_tac 1); |
|
469 |
by (rtac allI 1); |
|
470 |
by (exhaust_tac "xs" 1); |
|
471 |
by (ALLGOALS Asm_simp_tac); |
|
472 |
by (rtac allI 1); |
|
473 |
by (exhaust_tac "xs" 1); |
|
474 |
by (ALLGOALS Asm_simp_tac); |
|
2608 | 475 |
qed_spec_mp "nth_append"; |
476 |
||
4502 | 477 |
goal thy "!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
|
478 |
by (induct_tac "xs" 1); |
1301 | 479 |
(* case [] *) |
480 |
by (Asm_full_simp_tac 1); |
|
481 |
(* case x#xl *) |
|
482 |
by (rtac allI 1); |
|
483 |
by (nat_ind_tac "n" 1); |
|
484 |
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
|
485 |
qed_spec_mp "nth_map"; |
1301 | 486 |
Addsimps [nth_map]; |
487 |
||
4502 | 488 |
goal thy "!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
|
489 |
by (induct_tac "xs" 1); |
1301 | 490 |
(* case [] *) |
491 |
by (Simp_tac 1); |
|
492 |
(* case x#xl *) |
|
493 |
by (rtac allI 1); |
|
494 |
by (nat_ind_tac "n" 1); |
|
495 |
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
|
496 |
qed_spec_mp "list_all_nth"; |
1301 | 497 |
|
4502 | 498 |
goal thy "!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
|
499 |
by (induct_tac "xs" 1); |
1301 | 500 |
(* case [] *) |
501 |
by (Simp_tac 1); |
|
502 |
(* case x#xl *) |
|
503 |
by (rtac allI 1); |
|
504 |
by (nat_ind_tac "n" 1); |
|
505 |
(* case 0 *) |
|
506 |
by (Asm_full_simp_tac 1); |
|
507 |
(* case Suc x *) |
|
4089 | 508 |
by (asm_full_simp_tac (simpset() addsplits [expand_if]) 1); |
1485
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1465
diff
changeset
|
509 |
qed_spec_mp "nth_mem"; |
1301 | 510 |
Addsimps [nth_mem]; |
511 |
||
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
512 |
(** last & butlast **) |
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
513 |
|
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
514 |
goal thy "last(xs@[x]) = x"; |
4423 | 515 |
by (induct_tac "xs" 1); |
516 |
by (ALLGOALS (asm_simp_tac (simpset() addsplits [expand_if]))); |
|
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
517 |
qed "last_snoc"; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
518 |
Addsimps [last_snoc]; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
519 |
|
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
520 |
goal thy "butlast(xs@[x]) = xs"; |
4423 | 521 |
by (induct_tac "xs" 1); |
522 |
by (ALLGOALS (asm_simp_tac (simpset() addsplits [expand_if]))); |
|
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
523 |
qed "butlast_snoc"; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
524 |
Addsimps [butlast_snoc]; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
525 |
|
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
526 |
goal thy |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
527 |
"!ys. butlast (xs@ys) = (if ys=[] then butlast xs else xs@butlast ys)"; |
4423 | 528 |
by (induct_tac "xs" 1); |
529 |
by (ALLGOALS(asm_simp_tac (simpset() addsplits [expand_if]))); |
|
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
530 |
qed_spec_mp "butlast_append"; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
531 |
|
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
532 |
goal thy "x:set(butlast xs) --> x:set xs"; |
4423 | 533 |
by (induct_tac "xs" 1); |
534 |
by (ALLGOALS (asm_simp_tac (simpset() addsplits [expand_if]))); |
|
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
535 |
qed_spec_mp "in_set_butlastD"; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
536 |
|
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
537 |
goal thy "!!xs. x:set(butlast xs) ==> x:set(butlast(xs@ys))"; |
4423 | 538 |
by (asm_simp_tac (simpset() addsimps [butlast_append] |
3919 | 539 |
addsplits [expand_if]) 1); |
4423 | 540 |
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
|
541 |
qed "in_set_butlast_appendI1"; |
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
542 |
|
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
543 |
goal thy "!!xs. x:set(butlast ys) ==> x:set(butlast(xs@ys))"; |
4423 | 544 |
by (asm_simp_tac (simpset() addsimps [butlast_append] |
3919 | 545 |
addsplits [expand_if]) 1); |
4423 | 546 |
by (Clarify_tac 1); |
547 |
by (Full_simp_tac 1); |
|
3896
ee8ebb74ec00
Various new lemmas. Improved conversion of equations to rewrite rules:
nipkow
parents:
3860
diff
changeset
|
548 |
qed "in_set_butlast_appendI2"; |
3902 | 549 |
|
2608 | 550 |
(** take & drop **) |
551 |
section "take & drop"; |
|
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
552 |
|
1419
a6a034a47a71
defined take/drop by induction over list rather than nat.
nipkow
parents:
1327
diff
changeset
|
553 |
goal thy "take 0 xs = []"; |
3040
7d48671753da
Introduced a generic "induct_tac" which picks up the right induction scheme
nipkow
parents:
3011
diff
changeset
|
554 |
by (induct_tac "xs" 1); |
1419
a6a034a47a71
defined take/drop by induction over list rather than nat.
nipkow
parents:
1327
diff
changeset
|
555 |
by (ALLGOALS Asm_simp_tac); |
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
556 |
qed "take_0"; |
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
557 |
|
2608 | 558 |
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
|
559 |
by (induct_tac "xs" 1); |
2608 | 560 |
by (ALLGOALS Asm_simp_tac); |
561 |
qed "drop_0"; |
|
562 |
||
1419
a6a034a47a71
defined take/drop by induction over list rather than nat.
nipkow
parents:
1327
diff
changeset
|
563 |
goal thy "take (Suc n) (x#xs) = x # take n xs"; |
1552 | 564 |
by (Simp_tac 1); |
1419
a6a034a47a71
defined take/drop by induction over list rather than nat.
nipkow
parents:
1327
diff
changeset
|
565 |
qed "take_Suc_Cons"; |
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
566 |
|
2608 | 567 |
goal thy "drop (Suc n) (x#xs) = drop n xs"; |
568 |
by (Simp_tac 1); |
|
569 |
qed "drop_Suc_Cons"; |
|
570 |
||
571 |
Delsimps [take_Cons,drop_Cons]; |
|
572 |
Addsimps [take_0,take_Suc_Cons,drop_0,drop_Suc_Cons]; |
|
573 |
||
3011 | 574 |
goal thy "!xs. length(take n xs) = min (length xs) n"; |
3457 | 575 |
by (nat_ind_tac "n" 1); |
576 |
by (ALLGOALS Asm_simp_tac); |
|
577 |
by (rtac allI 1); |
|
578 |
by (exhaust_tac "xs" 1); |
|
579 |
by (ALLGOALS Asm_simp_tac); |
|
2608 | 580 |
qed_spec_mp "length_take"; |
581 |
Addsimps [length_take]; |
|
923 | 582 |
|
3011 | 583 |
goal thy "!xs. length(drop n xs) = (length xs - n)"; |
3457 | 584 |
by (nat_ind_tac "n" 1); |
585 |
by (ALLGOALS Asm_simp_tac); |
|
586 |
by (rtac allI 1); |
|
587 |
by (exhaust_tac "xs" 1); |
|
588 |
by (ALLGOALS Asm_simp_tac); |
|
2608 | 589 |
qed_spec_mp "length_drop"; |
590 |
Addsimps [length_drop]; |
|
591 |
||
3011 | 592 |
goal thy "!xs. length xs <= n --> take n xs = xs"; |
3457 | 593 |
by (nat_ind_tac "n" 1); |
594 |
by (ALLGOALS Asm_simp_tac); |
|
595 |
by (rtac allI 1); |
|
596 |
by (exhaust_tac "xs" 1); |
|
597 |
by (ALLGOALS Asm_simp_tac); |
|
2608 | 598 |
qed_spec_mp "take_all"; |
923 | 599 |
|
3011 | 600 |
goal thy "!xs. length xs <= n --> drop n xs = []"; |
3457 | 601 |
by (nat_ind_tac "n" 1); |
602 |
by (ALLGOALS Asm_simp_tac); |
|
603 |
by (rtac allI 1); |
|
604 |
by (exhaust_tac "xs" 1); |
|
605 |
by (ALLGOALS Asm_simp_tac); |
|
2608 | 606 |
qed_spec_mp "drop_all"; |
607 |
||
3011 | 608 |
goal thy |
2608 | 609 |
"!xs. take n (xs @ ys) = (take n xs @ take (n - length xs) ys)"; |
3457 | 610 |
by (nat_ind_tac "n" 1); |
611 |
by (ALLGOALS Asm_simp_tac); |
|
612 |
by (rtac allI 1); |
|
613 |
by (exhaust_tac "xs" 1); |
|
614 |
by (ALLGOALS Asm_simp_tac); |
|
2608 | 615 |
qed_spec_mp "take_append"; |
616 |
Addsimps [take_append]; |
|
617 |
||
3011 | 618 |
goal thy "!xs. drop n (xs@ys) = drop n xs @ drop (n - length xs) ys"; |
3457 | 619 |
by (nat_ind_tac "n" 1); |
620 |
by (ALLGOALS Asm_simp_tac); |
|
621 |
by (rtac allI 1); |
|
622 |
by (exhaust_tac "xs" 1); |
|
623 |
by (ALLGOALS Asm_simp_tac); |
|
2608 | 624 |
qed_spec_mp "drop_append"; |
625 |
Addsimps [drop_append]; |
|
626 |
||
3011 | 627 |
goal thy "!xs n. take n (take m xs) = take (min n m) xs"; |
3457 | 628 |
by (nat_ind_tac "m" 1); |
629 |
by (ALLGOALS Asm_simp_tac); |
|
630 |
by (rtac allI 1); |
|
631 |
by (exhaust_tac "xs" 1); |
|
632 |
by (ALLGOALS Asm_simp_tac); |
|
633 |
by (rtac allI 1); |
|
634 |
by (exhaust_tac "n" 1); |
|
635 |
by (ALLGOALS Asm_simp_tac); |
|
2608 | 636 |
qed_spec_mp "take_take"; |
637 |
||
3011 | 638 |
goal thy "!xs. drop n (drop m xs) = drop (n + m) xs"; |
3457 | 639 |
by (nat_ind_tac "m" 1); |
640 |
by (ALLGOALS Asm_simp_tac); |
|
641 |
by (rtac allI 1); |
|
642 |
by (exhaust_tac "xs" 1); |
|
643 |
by (ALLGOALS Asm_simp_tac); |
|
2608 | 644 |
qed_spec_mp "drop_drop"; |
923 | 645 |
|
3011 | 646 |
goal thy "!xs n. take n (drop m xs) = drop m (take (n + m) xs)"; |
3457 | 647 |
by (nat_ind_tac "m" 1); |
648 |
by (ALLGOALS Asm_simp_tac); |
|
649 |
by (rtac allI 1); |
|
650 |
by (exhaust_tac "xs" 1); |
|
651 |
by (ALLGOALS Asm_simp_tac); |
|
2608 | 652 |
qed_spec_mp "take_drop"; |
653 |
||
3011 | 654 |
goal thy "!xs. take n (map f xs) = map f (take n xs)"; |
3457 | 655 |
by (nat_ind_tac "n" 1); |
656 |
by (ALLGOALS Asm_simp_tac); |
|
657 |
by (rtac allI 1); |
|
658 |
by (exhaust_tac "xs" 1); |
|
659 |
by (ALLGOALS Asm_simp_tac); |
|
2608 | 660 |
qed_spec_mp "take_map"; |
661 |
||
3011 | 662 |
goal thy "!xs. drop n (map f xs) = map f (drop n xs)"; |
3457 | 663 |
by (nat_ind_tac "n" 1); |
664 |
by (ALLGOALS Asm_simp_tac); |
|
665 |
by (rtac allI 1); |
|
666 |
by (exhaust_tac "xs" 1); |
|
667 |
by (ALLGOALS Asm_simp_tac); |
|
2608 | 668 |
qed_spec_mp "drop_map"; |
669 |
||
4502 | 670 |
goal thy "!n i. i < n --> (take n xs)!i = xs!i"; |
3457 | 671 |
by (induct_tac "xs" 1); |
672 |
by (ALLGOALS Asm_simp_tac); |
|
3708 | 673 |
by (Clarify_tac 1); |
3457 | 674 |
by (exhaust_tac "n" 1); |
675 |
by (Blast_tac 1); |
|
676 |
by (exhaust_tac "i" 1); |
|
677 |
by (ALLGOALS Asm_full_simp_tac); |
|
2608 | 678 |
qed_spec_mp "nth_take"; |
679 |
Addsimps [nth_take]; |
|
923 | 680 |
|
4502 | 681 |
goal thy "!xs i. n + i <= length xs --> (drop n xs)!i = xs!(n+i)"; |
3457 | 682 |
by (nat_ind_tac "n" 1); |
683 |
by (ALLGOALS Asm_simp_tac); |
|
684 |
by (rtac allI 1); |
|
685 |
by (exhaust_tac "xs" 1); |
|
686 |
by (ALLGOALS Asm_simp_tac); |
|
2608 | 687 |
qed_spec_mp "nth_drop"; |
688 |
Addsimps [nth_drop]; |
|
689 |
||
690 |
(** takeWhile & dropWhile **) |
|
691 |
||
3467 | 692 |
section "takeWhile & dropWhile"; |
693 |
||
3586 | 694 |
goal thy "takeWhile P xs @ dropWhile P xs = xs"; |
695 |
by (induct_tac "xs" 1); |
|
696 |
by (Simp_tac 1); |
|
4089 | 697 |
by (asm_full_simp_tac (simpset() addsplits [expand_if]) 1); |
3586 | 698 |
qed "takeWhile_dropWhile_id"; |
699 |
Addsimps [takeWhile_dropWhile_id]; |
|
700 |
||
701 |
goal thy "x:set xs & ~P(x) --> takeWhile P (xs @ ys) = takeWhile P xs"; |
|
3457 | 702 |
by (induct_tac "xs" 1); |
703 |
by (Simp_tac 1); |
|
4089 | 704 |
by (asm_full_simp_tac (simpset() addsplits [expand_if]) 1); |
3457 | 705 |
by (Blast_tac 1); |
2608 | 706 |
bind_thm("takeWhile_append1", conjI RS (result() RS mp)); |
707 |
Addsimps [takeWhile_append1]; |
|
923 | 708 |
|
3011 | 709 |
goal thy |
3842 | 710 |
"(!x:set xs. P(x)) --> takeWhile P (xs @ ys) = xs @ takeWhile P ys"; |
3457 | 711 |
by (induct_tac "xs" 1); |
712 |
by (Simp_tac 1); |
|
4089 | 713 |
by (asm_full_simp_tac (simpset() addsplits [expand_if]) 1); |
2608 | 714 |
bind_thm("takeWhile_append2", ballI RS (result() RS mp)); |
715 |
Addsimps [takeWhile_append2]; |
|
1169 | 716 |
|
3011 | 717 |
goal thy |
3465 | 718 |
"x:set xs & ~P(x) --> dropWhile P (xs @ ys) = (dropWhile P xs)@ys"; |
3457 | 719 |
by (induct_tac "xs" 1); |
720 |
by (Simp_tac 1); |
|
4089 | 721 |
by (asm_full_simp_tac (simpset() addsplits [expand_if]) 1); |
3457 | 722 |
by (Blast_tac 1); |
2608 | 723 |
bind_thm("dropWhile_append1", conjI RS (result() RS mp)); |
724 |
Addsimps [dropWhile_append1]; |
|
725 |
||
3011 | 726 |
goal thy |
3842 | 727 |
"(!x:set xs. P(x)) --> dropWhile P (xs @ ys) = dropWhile P ys"; |
3457 | 728 |
by (induct_tac "xs" 1); |
729 |
by (Simp_tac 1); |
|
4089 | 730 |
by (asm_full_simp_tac (simpset() addsplits [expand_if]) 1); |
2608 | 731 |
bind_thm("dropWhile_append2", ballI RS (result() RS mp)); |
732 |
Addsimps [dropWhile_append2]; |
|
733 |
||
3465 | 734 |
goal thy "x:set(takeWhile P xs) --> x:set xs & P x"; |
3457 | 735 |
by (induct_tac "xs" 1); |
736 |
by (Simp_tac 1); |
|
4089 | 737 |
by (asm_full_simp_tac (simpset() addsplits [expand_if]) 1); |
3647
a64c8fbcd98f
Renamed theorems of the form set_of_list_XXX to set_XXX
paulson
parents:
3589
diff
changeset
|
738 |
qed_spec_mp"set_take_whileD"; |
2608 | 739 |
|
4132 | 740 |
qed_goal "zip_Nil_Nil" thy "zip [] [] = []" (K [Simp_tac 1]); |
741 |
qed_goal "zip_Cons_Cons" thy "zip (x#xs) (y#ys) = (x,y)#zip xs ys" |
|
742 |
(K [Simp_tac 1]); |
|
4605 | 743 |
|
744 |
(** nodups & remdups **) |
|
745 |
section "nodups & remdups"; |
|
746 |
||
747 |
goal thy "set(remdups xs) = set xs"; |
|
748 |
by (induct_tac "xs" 1); |
|
749 |
by (Simp_tac 1); |
|
750 |
by (asm_full_simp_tac (simpset() addsimps [insert_absorb] |
|
751 |
addsplits [expand_if]) 1); |
|
752 |
qed "set_remdups"; |
|
753 |
Addsimps [set_remdups]; |
|
754 |
||
755 |
goal thy "nodups(remdups xs)"; |
|
756 |
by (induct_tac "xs" 1); |
|
757 |
by (Simp_tac 1); |
|
758 |
by (asm_full_simp_tac (simpset() addsplits [expand_if]) 1); |
|
759 |
qed "nodups_remdups"; |
|
760 |
||
761 |
goal thy "nodups xs --> nodups (filter P xs)"; |
|
762 |
by (induct_tac "xs" 1); |
|
763 |
by (Simp_tac 1); |
|
764 |
by (asm_full_simp_tac (simpset() addsplits [expand_if]) 1); |
|
765 |
qed_spec_mp "nodups_filter"; |
|
766 |
||
3589
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
767 |
(** replicate **) |
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
768 |
section "replicate"; |
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
769 |
|
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
770 |
goal thy "set(replicate (Suc n) x) = {x}"; |
4423 | 771 |
by (induct_tac "n" 1); |
772 |
by (ALLGOALS Asm_full_simp_tac); |
|
3589
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
773 |
val lemma = result(); |
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
774 |
|
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
775 |
goal thy "!!n. n ~= 0 ==> set(replicate n x) = {x}"; |
4423 | 776 |
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
|
777 |
qed "set_replicate"; |
244daa75f890
Added function `replicate' and lemmas map_cong and set_replicate.
nipkow
parents:
3586
diff
changeset
|
778 |
Addsimps [set_replicate]; |