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src/HOL/Library/Permutation.thy

author | haftmann |

Mon Oct 04 14:46:48 2010 +0200 (2010-10-04) | |

changeset 39916 | 8c83139a1433 |

parent 39078 | 39f8f6d1eb74 |

child 40122 | 1d8ad2ff3e01 |

permissions | -rw-r--r-- |

turned distinct and sorted into inductive predicates: yields nice induction principles for free

1 (* Title: HOL/Library/Permutation.thy

2 Author: Lawrence C Paulson and Thomas M Rasmussen and Norbert Voelker

3 *)

5 header {* Permutations *}

7 theory Permutation

8 imports Main Multiset

9 begin

11 inductive

12 perm :: "'a list => 'a list => bool" ("_ <~~> _" [50, 50] 50)

13 where

14 Nil [intro!]: "[] <~~> []"

15 | swap [intro!]: "y # x # l <~~> x # y # l"

16 | Cons [intro!]: "xs <~~> ys ==> z # xs <~~> z # ys"

17 | trans [intro]: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs"

19 lemma perm_refl [iff]: "l <~~> l"

20 by (induct l) auto

23 subsection {* Some examples of rule induction on permutations *}

25 lemma xperm_empty_imp: "[] <~~> ys ==> ys = []"

26 by (induct xs == "[]::'a list" ys pred: perm) simp_all

29 text {*

30 \medskip This more general theorem is easier to understand!

31 *}

33 lemma perm_length: "xs <~~> ys ==> length xs = length ys"

34 by (induct pred: perm) simp_all

36 lemma perm_empty_imp: "[] <~~> xs ==> xs = []"

37 by (drule perm_length) auto

39 lemma perm_sym: "xs <~~> ys ==> ys <~~> xs"

40 by (induct pred: perm) auto

43 subsection {* Ways of making new permutations *}

45 text {*

46 We can insert the head anywhere in the list.

47 *}

49 lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"

50 by (induct xs) auto

52 lemma perm_append_swap: "xs @ ys <~~> ys @ xs"

53 apply (induct xs)

54 apply simp_all

55 apply (blast intro: perm_append_Cons)

56 done

58 lemma perm_append_single: "a # xs <~~> xs @ [a]"

59 by (rule perm.trans [OF _ perm_append_swap]) simp

61 lemma perm_rev: "rev xs <~~> xs"

62 apply (induct xs)

63 apply simp_all

64 apply (blast intro!: perm_append_single intro: perm_sym)

65 done

67 lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys"

68 by (induct l) auto

70 lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l"

71 by (blast intro!: perm_append_swap perm_append1)

74 subsection {* Further results *}

76 lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])"

77 by (blast intro: perm_empty_imp)

79 lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])"

80 apply auto

81 apply (erule perm_sym [THEN perm_empty_imp])

82 done

84 lemma perm_sing_imp: "ys <~~> xs ==> xs = [y] ==> ys = [y]"

85 by (induct pred: perm) auto

87 lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])"

88 by (blast intro: perm_sing_imp)

90 lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])"

91 by (blast dest: perm_sym)

94 subsection {* Removing elements *}

96 lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove1 x ys"

97 by (induct ys) auto

100 text {* \medskip Congruence rule *}

102 lemma perm_remove_perm: "xs <~~> ys ==> remove1 z xs <~~> remove1 z ys"

103 by (induct pred: perm) auto

105 lemma remove_hd [simp]: "remove1 z (z # xs) = xs"

106 by auto

108 lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys"

109 by (drule_tac z = z in perm_remove_perm) auto

111 lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)"

112 by (blast intro: cons_perm_imp_perm)

114 lemma append_perm_imp_perm: "zs @ xs <~~> zs @ ys ==> xs <~~> ys"

115 apply (induct zs arbitrary: xs ys rule: rev_induct)

116 apply (simp_all (no_asm_use))

117 apply blast

118 done

120 lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)"

121 by (blast intro: append_perm_imp_perm perm_append1)

123 lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)"

124 apply (safe intro!: perm_append2)

125 apply (rule append_perm_imp_perm)

126 apply (rule perm_append_swap [THEN perm.trans])

127 -- {* the previous step helps this @{text blast} call succeed quickly *}

128 apply (blast intro: perm_append_swap)

129 done

131 lemma multiset_of_eq_perm: "(multiset_of xs = multiset_of ys) = (xs <~~> ys) "

132 apply (rule iffI)

133 apply (erule_tac [2] perm.induct, simp_all add: union_ac)

134 apply (erule rev_mp, rule_tac x=ys in spec)

135 apply (induct_tac xs, auto)

136 apply (erule_tac x = "remove1 a x" in allE, drule sym, simp)

137 apply (subgoal_tac "a \<in> set x")

138 apply (drule_tac z=a in perm.Cons)

139 apply (erule perm.trans, rule perm_sym, erule perm_remove)

140 apply (drule_tac f=set_of in arg_cong, simp)

141 done

143 lemma multiset_of_le_perm_append:

144 "multiset_of xs \<le> multiset_of ys \<longleftrightarrow> (\<exists>zs. xs @ zs <~~> ys)"

145 apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv)

146 apply (insert surj_multiset_of, drule surjD)

147 apply (blast intro: sym)+

148 done

150 lemma perm_set_eq: "xs <~~> ys ==> set xs = set ys"

151 by (metis multiset_of_eq_perm multiset_of_eq_setD)

153 lemma perm_distinct_iff: "xs <~~> ys ==> distinct xs = distinct ys"

154 apply (induct pred: perm)

155 apply simp_all

156 apply fastsimp

157 apply (metis perm_set_eq)

158 done

160 lemma eq_set_perm_remdups: "set xs = set ys ==> remdups xs <~~> remdups ys"

161 apply (induct xs arbitrary: ys rule: length_induct)

162 apply (case_tac "remdups xs", simp, simp)

163 apply (subgoal_tac "a : set (remdups ys)")

164 prefer 2 apply (metis set.simps(2) insert_iff set_remdups)

165 apply (drule split_list) apply(elim exE conjE)

166 apply (drule_tac x=list in spec) apply(erule impE) prefer 2

167 apply (drule_tac x="ysa@zs" in spec) apply(erule impE) prefer 2

168 apply simp

169 apply (subgoal_tac "a#list <~~> a#ysa@zs")

170 apply (metis Cons_eq_appendI perm_append_Cons trans)

171 apply (metis Cons Cons_eq_appendI distinct_simps(2)

172 distinct_remdups distinct_remdups_id perm_append_swap perm_distinct_iff)

173 apply (subgoal_tac "set (a#list) = set (ysa@a#zs) & distinct (a#list) & distinct (ysa@a#zs)")

174 apply (fastsimp simp add: insert_ident)

175 apply (metis distinct_remdups set_remdups)

176 apply (subgoal_tac "length (remdups xs) < Suc (length xs)")

177 apply simp

178 apply (subgoal_tac "length (remdups xs) \<le> length xs")

179 apply simp

180 apply (rule length_remdups_leq)

181 done

183 lemma perm_remdups_iff_eq_set: "remdups x <~~> remdups y = (set x = set y)"

184 by (metis List.set_remdups perm_set_eq eq_set_perm_remdups)

186 lemma permutation_Ex_bij:

187 assumes "xs <~~> ys"

188 shows "\<exists>f. bij_betw f {..<length xs} {..<length ys} \<and> (\<forall>i<length xs. xs ! i = ys ! (f i))"

189 using assms proof induct

190 case Nil then show ?case unfolding bij_betw_def by simp

191 next

192 case (swap y x l)

193 show ?case

194 proof (intro exI[of _ "Fun.swap 0 1 id"] conjI allI impI)

195 show "bij_betw (Fun.swap 0 1 id) {..<length (y # x # l)} {..<length (x # y # l)}"

196 by (auto simp: bij_betw_def bij_betw_swap_iff)

197 fix i assume "i < length(y#x#l)"

198 show "(y # x # l) ! i = (x # y # l) ! (Fun.swap 0 1 id) i"

199 by (cases i) (auto simp: Fun.swap_def gr0_conv_Suc)

200 qed

201 next

202 case (Cons xs ys z)

203 then obtain f where bij: "bij_betw f {..<length xs} {..<length ys}" and

204 perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" by blast

205 let "?f i" = "case i of Suc n \<Rightarrow> Suc (f n) | 0 \<Rightarrow> 0"

206 show ?case

207 proof (intro exI[of _ ?f] allI conjI impI)

208 have *: "{..<length (z#xs)} = {0} \<union> Suc ` {..<length xs}"

209 "{..<length (z#ys)} = {0} \<union> Suc ` {..<length ys}"

210 by (simp_all add: lessThan_Suc_eq_insert_0)

211 show "bij_betw ?f {..<length (z#xs)} {..<length (z#ys)}" unfolding *

212 proof (rule bij_betw_combine)

213 show "bij_betw ?f (Suc ` {..<length xs}) (Suc ` {..<length ys})"

214 using bij unfolding bij_betw_def

215 by (auto intro!: inj_onI imageI dest: inj_onD simp: image_compose[symmetric] comp_def)

216 qed (auto simp: bij_betw_def)

217 fix i assume "i < length (z#xs)"

218 then show "(z # xs) ! i = (z # ys) ! (?f i)"

219 using perm by (cases i) auto

220 qed

221 next

222 case (trans xs ys zs)

223 then obtain f g where

224 bij: "bij_betw f {..<length xs} {..<length ys}" "bij_betw g {..<length ys} {..<length zs}" and

225 perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" "\<forall>i<length ys. ys ! i = zs ! (g i)" by blast

226 show ?case

227 proof (intro exI[of _ "g\<circ>f"] conjI allI impI)

228 show "bij_betw (g \<circ> f) {..<length xs} {..<length zs}"

229 using bij by (rule bij_betw_trans)

230 fix i assume "i < length xs"

231 with bij have "f i < length ys" unfolding bij_betw_def by force

232 with `i < length xs` show "xs ! i = zs ! (g \<circ> f) i"

233 using trans(1,3)[THEN perm_length] perm by force

234 qed

235 qed

237 end