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author | haftmann |

Wed, 24 Feb 2021 13:31:28 +0000 | |

changeset 73300 | c52c5a5bf4e6 |

parent 73299 | 43ce3b8a25ee |

child 73301 | bfe92e4f6ea4 |

emphasize connection to multisets

--- a/src/HOL/Library/List_Permutation.thy Wed Feb 24 18:54:53 2021 +0100 +++ b/src/HOL/Library/List_Permutation.thy Wed Feb 24 13:31:28 2021 +0000 @@ -8,6 +8,8 @@ imports Multiset begin +subsection \<open>An inductive definition\<dots>\<close> + inductive perm :: "'a list \<Rightarrow> 'a list \<Rightarrow> bool" (infixr \<open><~~>\<close> 50) where Nil [intro!]: "[] <~~> []" @@ -18,139 +20,87 @@ 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 +text \<open>\<dots>that is equivalent to an already existing notion:\<close> -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) +lemma perm_iff_eq_mset: + \<open>xs <~~> ys \<longleftrightarrow> mset xs = mset ys\<close> +proof + assume \<open>mset xs = mset ys\<close> + then show \<open>xs <~~> ys\<close> + proof (induction xs arbitrary: ys) + case Nil + then show ?case + by simp + next + case (Cons x xs) + from Cons.prems [symmetric] have \<open>mset xs = mset (remove1 x ys)\<close> + by simp + then have \<open>xs <~~> remove1 x ys\<close> + by (rule Cons.IH) + then have \<open>x # xs <~~> x # remove1 x ys\<close> + by (rule perm.Cons) + moreover from Cons.prems have \<open>x \<in> set ys\<close> + by (auto dest: union_single_eq_member) + then have \<open>x # remove1 x ys <~~> ys\<close> + by (induction ys) auto + ultimately show \<open>x # xs <~~> ys\<close> + by (rule perm.trans) + qed +next + assume \<open>xs <~~> ys\<close> + then show \<open>mset xs = mset ys\<close> + by induction simp_all +qed -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 +lemma list_permuted_induct [consumes 1, case_names Nil swap Cons trans]: + \<open>P xs ys\<close> + if \<open>mset xs = mset ys\<close> + \<open>P [] []\<close> + \<open>\<And>y x zs. P (y # x # zs) (x # y # zs)\<close> + \<open>\<And>xs ys z. mset xs = mset ys \<Longrightarrow> P xs ys \<Longrightarrow> P (z # xs) (z # ys)\<close> + \<open>\<And>xs ys zs. mset xs = mset ys \<Longrightarrow> mset ys = mset zs \<Longrightarrow> P xs ys \<Longrightarrow> P ys zs \<Longrightarrow> P xs zs\<close> +proof - + from \<open>mset xs = mset ys\<close> have \<open>xs <~~> ys\<close> + by (simp add: perm_iff_eq_mset) + then show ?thesis + using that(2-3) apply (rule perm.induct) + apply (simp_all add: perm_iff_eq_mset) + apply (fact that(4)) + subgoal for xs ys zs + apply (rule that(5) [of xs ys zs]) + apply simp_all + done + done +qed -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 +subsection \<open>\<dots>that is equivalent to an already existing notion:\<close> -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) +theorem mset_eq_perm: \<open>mset xs = mset ys \<longleftrightarrow> xs <~~> ys\<close> + by (simp add: perm_iff_eq_mset) -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) +subsection \<open>Nontrivial conclusions\<close> -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 perm_swap: + \<open>xs[i := xs ! j, j := xs ! i] <~~> xs\<close> + if \<open>i < length xs\<close> \<open>j < length xs\<close> + using that by (cases \<open>i = j\<close>) (simp_all add: perm_iff_eq_mset mset_update) 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) + by (auto simp add: perm_iff_eq_mset mset_subset_eq_exists_conv ex_mset dest: sym) proposition perm_set_eq: "xs <~~> ys \<Longrightarrow> set xs = set ys" - by (metis mset_eq_perm mset_eq_setD) + by (rule mset_eq_setD) (simp add: perm_iff_eq_mset) proposition perm_distinct_iff: "xs <~~> ys \<Longrightarrow> distinct xs = distinct ys" - by (metis card_distinct distinct_card perm_length perm_set_eq) + by (auto simp add: perm_iff_eq_mset distinct_count_atmost_1 dest: mset_eq_setD) 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 + by (simp add: perm_iff_eq_mset set_eq_iff_mset_remdups_eq) 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) + by (simp add: perm_iff_eq_mset set_eq_iff_mset_remdups_eq) theorem permutation_Ex_bij: assumes "xs <~~> ys" @@ -209,22 +159,95 @@ 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 + using trans(1,3) 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 +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) + show "{B. B <~~> A} \<subseteq> {xs. set xs \<subseteq> set A \<and> length xs \<le> length A}" + by (auto simp add: perm_iff_eq_mset dest: mset_eq_setD mset_eq_length) 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) + +subsection \<open>Trivial conclusions:\<close> + +proposition perm_empty_imp: "[] <~~> ys \<Longrightarrow> ys = []" + by (simp add: perm_iff_eq_mset) + + +text \<open>\medskip This more general theorem is easier to understand!\<close> + +proposition perm_length: "xs <~~> ys \<Longrightarrow> length xs = length ys" + by (rule mset_eq_length) (simp add: perm_iff_eq_mset) + +proposition perm_sym: "xs <~~> ys \<Longrightarrow> ys <~~> xs" + by (simp add: perm_iff_eq_mset) + + +text \<open>We can insert the head anywhere in the list.\<close> + +proposition perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys" + by (simp add: perm_iff_eq_mset) + +proposition perm_append_swap: "xs @ ys <~~> ys @ xs" + by (simp add: perm_iff_eq_mset) + +proposition perm_append_single: "a # xs <~~> xs @ [a]" + by (simp add: perm_iff_eq_mset) + +proposition perm_rev: "rev xs <~~> xs" + by (simp add: perm_iff_eq_mset) + +proposition perm_append1: "xs <~~> ys \<Longrightarrow> l @ xs <~~> l @ ys" + by (simp add: perm_iff_eq_mset) + +proposition perm_append2: "xs <~~> ys \<Longrightarrow> xs @ l <~~> ys @ l" + by (simp add: perm_iff_eq_mset) + +proposition perm_empty [iff]: "[] <~~> xs \<longleftrightarrow> xs = []" + by (simp add: perm_iff_eq_mset) + +proposition perm_empty2 [iff]: "xs <~~> [] \<longleftrightarrow> xs = []" + by (simp add: perm_iff_eq_mset) + +proposition perm_sing_imp: "ys <~~> xs \<Longrightarrow> xs = [y] \<Longrightarrow> ys = [y]" + by (simp add: perm_iff_eq_mset) + +proposition perm_sing_eq [iff]: "ys <~~> [y] \<longleftrightarrow> ys = [y]" + by (simp add: perm_iff_eq_mset) + +proposition perm_sing_eq2 [iff]: "[y] <~~> ys \<longleftrightarrow> ys = [y]" + by (simp add: perm_iff_eq_mset) + +proposition perm_remove: "x \<in> set ys \<Longrightarrow> ys <~~> x # remove1 x ys" + by (simp add: perm_iff_eq_mset) + + +text \<open>\medskip Congruence rule\<close> + +proposition perm_remove_perm: "xs <~~> ys \<Longrightarrow> remove1 z xs <~~> remove1 z ys" + by (simp add: perm_iff_eq_mset) + +proposition remove_hd [simp]: "remove1 z (z # xs) = xs" + by (simp add: perm_iff_eq_mset) + +proposition cons_perm_imp_perm: "z # xs <~~> z # ys \<Longrightarrow> xs <~~> ys" + by (simp add: perm_iff_eq_mset) + +proposition cons_perm_eq [simp]: "z#xs <~~> z#ys \<longleftrightarrow> xs <~~> ys" + by (simp add: perm_iff_eq_mset) + +proposition append_perm_imp_perm: "zs @ xs <~~> zs @ ys \<Longrightarrow> xs <~~> ys" + by (simp add: perm_iff_eq_mset) + +proposition perm_append1_eq [iff]: "zs @ xs <~~> zs @ ys \<longleftrightarrow> xs <~~> ys" + by (simp add: perm_iff_eq_mset) + +proposition perm_append2_eq [iff]: "xs @ zs <~~> ys @ zs \<longleftrightarrow> xs <~~> ys" + by (simp add: perm_iff_eq_mset) end