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src/HOL/Library/Permutation.thy

author | nipkow |

Thu, 22 Oct 2020 08:39:08 +0200 | |

changeset 72501 | 70b420065a07 |

parent 70680 | b8cd7ea34e33 |

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

tuned names: t_ -> T_

(* Title: HOL/Library/Permutation.thy Author: Lawrence C Paulson and Thomas M Rasmussen and Norbert Voelker *) section \<open>Permutations\<close> theory Permutation imports Multiset begin inductive perm :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infixr \<open><~~>\<close> 50) where Nil [intro!]: "[] <~~> []" | swap [intro!]: "y # x # l <~~> x # y # l" | Cons [intro!]: "xs <~~> ys \<Longrightarrow> z # xs <~~> z # ys" | trans [intro]: "xs <~~> ys \<Longrightarrow> ys <~~> zs \<Longrightarrow> xs <~~> zs" proposition perm_refl [iff]: "l <~~> l" by (induct l) auto subsection \<open>Some examples of rule induction on permutations\<close> proposition perm_empty_imp: "[] <~~> ys \<Longrightarrow> ys = []" by (induction "[] :: 'a list" ys pred: perm) simp_all text \<open>\medskip This more general theorem is easier to understand!\<close> proposition perm_length: "xs <~~> ys \<Longrightarrow> length xs = length ys" by (induct pred: perm) simp_all proposition perm_sym: "xs <~~> ys \<Longrightarrow> ys <~~> xs" by (induct pred: perm) auto subsection \<open>Ways of making new permutations\<close> text \<open>We can insert the head anywhere in the list.\<close> proposition perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys" by (induct xs) auto proposition perm_append_swap: "xs @ ys <~~> ys @ xs" by (induct xs) (auto intro: perm_append_Cons) proposition perm_append_single: "a # xs <~~> xs @ [a]" by (rule perm.trans [OF _ perm_append_swap]) simp proposition perm_rev: "rev xs <~~> xs" by (induct xs) (auto intro!: perm_append_single intro: perm_sym) proposition perm_append1: "xs <~~> ys \<Longrightarrow> l @ xs <~~> l @ ys" by (induct l) auto proposition perm_append2: "xs <~~> ys \<Longrightarrow> xs @ l <~~> ys @ l" by (blast intro!: perm_append_swap perm_append1) subsection \<open>Further results\<close> proposition perm_empty [iff]: "[] <~~> xs \<longleftrightarrow> xs = []" by (blast intro: perm_empty_imp) proposition perm_empty2 [iff]: "xs <~~> [] \<longleftrightarrow> xs = []" using perm_sym by auto proposition perm_sing_imp: "ys <~~> xs \<Longrightarrow> xs = [y] \<Longrightarrow> ys = [y]" by (induct pred: perm) auto proposition perm_sing_eq [iff]: "ys <~~> [y] \<longleftrightarrow> ys = [y]" by (blast intro: perm_sing_imp) proposition perm_sing_eq2 [iff]: "[y] <~~> ys \<longleftrightarrow> ys = [y]" by (blast dest: perm_sym) subsection \<open>Removing elements\<close> proposition perm_remove: "x \<in> set ys \<Longrightarrow> ys <~~> x # remove1 x ys" by (induct ys) auto text \<open>\medskip Congruence rule\<close> proposition perm_remove_perm: "xs <~~> ys \<Longrightarrow> remove1 z xs <~~> remove1 z ys" by (induct pred: perm) auto proposition remove_hd [simp]: "remove1 z (z # xs) = xs" by auto proposition cons_perm_imp_perm: "z # xs <~~> z # ys \<Longrightarrow> xs <~~> ys" by (drule perm_remove_perm [where z = z]) auto proposition cons_perm_eq [iff]: "z#xs <~~> z#ys \<longleftrightarrow> xs <~~> ys" by (meson cons_perm_imp_perm perm.Cons) proposition append_perm_imp_perm: "zs @ xs <~~> zs @ ys \<Longrightarrow> xs <~~> ys" by (induct zs arbitrary: xs ys rule: rev_induct) auto proposition perm_append1_eq [iff]: "zs @ xs <~~> zs @ ys \<longleftrightarrow> xs <~~> ys" by (blast intro: append_perm_imp_perm perm_append1) proposition perm_append2_eq [iff]: "xs @ zs <~~> ys @ zs \<longleftrightarrow> xs <~~> ys" by (meson perm.trans perm_append1_eq perm_append_swap) theorem mset_eq_perm: "mset xs = mset ys \<longleftrightarrow> xs <~~> ys" proof assume "mset xs = mset ys" then show "xs <~~> ys" proof (induction xs arbitrary: ys) case (Cons x xs) then have "x \<in> set ys" using mset_eq_setD by fastforce then show ?case by (metis Cons.IH Cons.prems mset_remove1 perm.Cons perm.trans perm_remove perm_sym remove_hd) qed auto next assume "xs <~~> ys" then show "mset xs = mset ys" by induction (simp_all add: union_ac) qed proposition mset_le_perm_append: "mset xs \<subseteq># mset ys \<longleftrightarrow> (\<exists>zs. xs @ zs <~~> ys)" apply (rule iffI) apply (metis mset_append mset_eq_perm mset_subset_eq_exists_conv surjD surj_mset) by (metis mset_append mset_eq_perm mset_subset_eq_exists_conv) proposition perm_set_eq: "xs <~~> ys \<Longrightarrow> set xs = set ys" by (metis mset_eq_perm mset_eq_setD) proposition perm_distinct_iff: "xs <~~> ys \<Longrightarrow> distinct xs = distinct ys" by (metis card_distinct distinct_card perm_length perm_set_eq) theorem eq_set_perm_remdups: "set xs = set ys \<Longrightarrow> remdups xs <~~> remdups ys" proof (induction xs arbitrary: ys rule: length_induct) case (1 xs) show ?case proof (cases "remdups xs") case Nil with "1.prems" show ?thesis using "1.prems" by auto next case (Cons x us) then have "x \<in> set (remdups ys)" using "1.prems" set_remdups by fastforce then show ?thesis using "1.prems" mset_eq_perm set_eq_iff_mset_remdups_eq by blast qed qed proposition perm_remdups_iff_eq_set: "remdups x <~~> remdups y \<longleftrightarrow> set x = set y" by (metis List.set_remdups perm_set_eq eq_set_perm_remdups) theorem permutation_Ex_bij: assumes "xs <~~> ys" shows "\<exists>f. bij_betw f {..<length xs} {..<length ys} \<and> (\<forall>i<length xs. xs ! i = ys ! (f i))" using assms proof induct case Nil then show ?case unfolding bij_betw_def by simp next case (swap y x l) show ?case proof (intro exI[of _ "Fun.swap 0 1 id"] conjI allI impI) show "bij_betw (Fun.swap 0 1 id) {..<length (y # x # l)} {..<length (x # y # l)}" by (auto simp: bij_betw_def) fix i assume "i < length (y # x # l)" show "(y # x # l) ! i = (x # y # l) ! (Fun.swap 0 1 id) i" by (cases i) (auto simp: Fun.swap_def gr0_conv_Suc) qed next case (Cons xs ys z) then obtain f where bij: "bij_betw f {..<length xs} {..<length ys}" and perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" by blast let ?f = "\<lambda>i. case i of Suc n \<Rightarrow> Suc (f n) | 0 \<Rightarrow> 0" show ?case proof (intro exI[of _ ?f] allI conjI impI) have *: "{..<length (z#xs)} = {0} \<union> Suc ` {..<length xs}" "{..<length (z#ys)} = {0} \<union> Suc ` {..<length ys}" by (simp_all add: lessThan_Suc_eq_insert_0) show "bij_betw ?f {..<length (z#xs)} {..<length (z#ys)}" unfolding * proof (rule bij_betw_combine) show "bij_betw ?f (Suc ` {..<length xs}) (Suc ` {..<length ys})" using bij unfolding bij_betw_def by (auto intro!: inj_onI imageI dest: inj_onD simp: image_comp comp_def) qed (auto simp: bij_betw_def) fix i assume "i < length (z # xs)" then show "(z # xs) ! i = (z # ys) ! (?f i)" using perm by (cases i) auto qed next case (trans xs ys zs) then obtain f g where bij: "bij_betw f {..<length xs} {..<length ys}" "bij_betw g {..<length ys} {..<length zs}" and perm: "\<forall>i<length xs. xs ! i = ys ! (f i)" "\<forall>i<length ys. ys ! i = zs ! (g i)" by blast show ?case proof (intro exI[of _ "g \<circ> f"] conjI allI impI) show "bij_betw (g \<circ> f) {..<length xs} {..<length zs}" using bij by (rule bij_betw_trans) fix i assume "i < length xs" with bij have "f i < length ys" unfolding bij_betw_def by force with \<open>i < length xs\<close> show "xs ! i = zs ! (g \<circ> f) i" using trans(1,3)[THEN perm_length] perm by auto qed qed proposition perm_finite: "finite {B. B <~~> A}" proof (rule finite_subset[where B="{xs. set xs \<subseteq> set A \<and> length xs \<le> length A}"]) show "finite {xs. set xs \<subseteq> set A \<and> length xs \<le> length A}" using finite_lists_length_le by blast next show "{B. B <~~> A} \<subseteq> {xs. set xs \<subseteq> set A \<and> length xs \<le> length A}" by (clarsimp simp add: perm_length perm_set_eq) qed proposition perm_swap: assumes "i < length xs" "j < length xs" shows "xs[i := xs ! j, j := xs ! i] <~~> xs" using assms by (simp add: mset_eq_perm[symmetric] mset_swap) end