(* Title: HOL/Library/Permutation.thy Author: Lawrence C Paulson and Thomas M Rasmussen and Norbert Voelker *) header {* Permutations *} theory Permutation imports Multiset begin inductive perm :: "'a list => 'a list => bool" ("_ <~~> _" [50, 50] 50) (* FIXME proper infix, without ambiguity!? *) where Nil [intro!]: "[] <~~> []" | swap [intro!]: "y # x # l <~~> x # y # l" | Cons [intro!]: "xs <~~> ys ==> z # xs <~~> z # ys" | trans [intro]: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs" lemma perm_refl [iff]: "l <~~> l" by (induct l) auto subsection {* Some examples of rule induction on permutations *} lemma xperm_empty_imp: "[] <~~> ys ==> ys = []" by (induct xs == "[] :: 'a list" ys pred: perm) simp_all text {* \medskip This more general theorem is easier to understand! *} lemma perm_length: "xs <~~> ys ==> length xs = length ys" by (induct pred: perm) simp_all lemma perm_empty_imp: "[] <~~> xs ==> xs = []" by (drule perm_length) auto lemma perm_sym: "xs <~~> ys ==> ys <~~> xs" by (induct pred: perm) auto subsection {* Ways of making new permutations *} text {* We can insert the head anywhere in the list. *} lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys" by (induct xs) auto lemma perm_append_swap: "xs @ ys <~~> ys @ xs" apply (induct xs) apply simp_all apply (blast intro: perm_append_Cons) done lemma perm_append_single: "a # xs <~~> xs @ [a]" by (rule perm.trans [OF _ perm_append_swap]) simp lemma perm_rev: "rev xs <~~> xs" apply (induct xs) apply simp_all apply (blast intro!: perm_append_single intro: perm_sym) done lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys" by (induct l) auto lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l" by (blast intro!: perm_append_swap perm_append1) subsection {* Further results *} lemma perm_empty [iff]: "[] <~~> xs <-> xs = []" by (blast intro: perm_empty_imp) lemma perm_empty2 [iff]: "xs <~~> [] <-> xs = []" apply auto apply (erule perm_sym [THEN perm_empty_imp]) done lemma perm_sing_imp: "ys <~~> xs ==> xs = [y] ==> ys = [y]" by (induct pred: perm) auto lemma perm_sing_eq [iff]: "ys <~~> [y] <-> ys = [y]" by (blast intro: perm_sing_imp) lemma perm_sing_eq2 [iff]: "[y] <~~> ys <-> ys = [y]" by (blast dest: perm_sym) subsection {* Removing elements *} lemma perm_remove: "x ∈ set ys ==> ys <~~> x # remove1 x ys" by (induct ys) auto text {* \medskip Congruence rule *} lemma perm_remove_perm: "xs <~~> ys ==> remove1 z xs <~~> remove1 z ys" by (induct pred: perm) auto lemma remove_hd [simp]: "remove1 z (z # xs) = xs" by auto lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys" by (drule_tac z = z in perm_remove_perm) auto lemma cons_perm_eq [iff]: "z#xs <~~> z#ys <-> xs <~~> ys" by (blast intro: cons_perm_imp_perm) lemma append_perm_imp_perm: "zs @ xs <~~> zs @ ys ==> xs <~~> ys" by (induct zs arbitrary: xs ys rule: rev_induct) auto lemma perm_append1_eq [iff]: "zs @ xs <~~> zs @ ys <-> xs <~~> ys" by (blast intro: append_perm_imp_perm perm_append1) lemma perm_append2_eq [iff]: "xs @ zs <~~> ys @ zs <-> xs <~~> ys" apply (safe intro!: perm_append2) apply (rule append_perm_imp_perm) apply (rule perm_append_swap [THEN perm.trans]) -- {* the previous step helps this @{text blast} call succeed quickly *} apply (blast intro: perm_append_swap) done lemma multiset_of_eq_perm: "multiset_of xs = multiset_of ys <-> xs <~~> ys" apply (rule iffI) apply (erule_tac [2] perm.induct) apply (simp_all add: union_ac) apply (erule rev_mp) apply (rule_tac x=ys in spec) apply (induct_tac xs) apply auto apply (erule_tac x = "remove1 a x" in allE) apply (drule sym) apply simp apply (subgoal_tac "a ∈ set x") apply (drule_tac z = a in perm.Cons) apply (erule perm.trans) apply (rule perm_sym) apply (erule perm_remove) apply (drule_tac f=set_of in arg_cong) apply simp done lemma multiset_of_le_perm_append: "multiset_of xs ≤ multiset_of ys <-> (∃zs. xs @ zs <~~> ys)" apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv) apply (insert surj_multiset_of) apply (drule surjD) apply (blast intro: sym)+ done lemma perm_set_eq: "xs <~~> ys ==> set xs = set ys" by (metis multiset_of_eq_perm multiset_of_eq_setD) lemma perm_distinct_iff: "xs <~~> ys ==> distinct xs = distinct ys" apply (induct pred: perm) apply simp_all apply fastforce apply (metis perm_set_eq) done lemma eq_set_perm_remdups: "set xs = set ys ==> remdups xs <~~> remdups ys" apply (induct xs arbitrary: ys rule: length_induct) apply (case_tac "remdups xs") apply simp_all apply (subgoal_tac "a ∈ set (remdups ys)") prefer 2 apply (metis set_simps(2) insert_iff set_remdups) apply (drule split_list) apply (elim exE conjE) apply (drule_tac x = list in spec) apply (erule impE) prefer 2 apply (drule_tac x = "ysa @ zs" in spec) apply (erule impE) prefer 2 apply simp apply (subgoal_tac "a # list <~~> a # ysa @ zs") apply (metis Cons_eq_appendI perm_append_Cons trans) apply (metis Cons Cons_eq_appendI distinct.simps(2) distinct_remdups distinct_remdups_id perm_append_swap perm_distinct_iff) apply (subgoal_tac "set (a # list) = set (ysa @ a # zs) ∧ distinct (a # list) ∧ distinct (ysa @ a # zs)") apply (fastforce simp add: insert_ident) apply (metis distinct_remdups set_remdups) apply (subgoal_tac "length (remdups xs) < Suc (length xs)") apply simp apply (subgoal_tac "length (remdups xs) ≤ length xs") apply simp apply (rule length_remdups_leq) done lemma perm_remdups_iff_eq_set: "remdups x <~~> remdups y <-> set x = set y" by (metis List.set_remdups perm_set_eq eq_set_perm_remdups) lemma permutation_Ex_bij: assumes "xs <~~> ys" shows "∃f. bij_betw f {..<length xs} {..<length ys} ∧ (∀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: "∀i<length xs. xs ! i = ys ! (f i)" by blast let ?f = "λi. case i of Suc n => Suc (f n) | 0 => 0" show ?case proof (intro exI[of _ ?f] allI conjI impI) have *: "{..<length (z#xs)} = {0} ∪ Suc ` {..<length xs}" "{..<length (z#ys)} = {0} ∪ 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: "∀i<length xs. xs ! i = ys ! (f i)" "∀i<length ys. ys ! i = zs ! (g i)" by blast show ?case proof (intro exI[of _ "g o f"] conjI allI impI) show "bij_betw (g o 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 `i < length xs` show "xs ! i = zs ! (g o f) i" using trans(1,3)[THEN perm_length] perm by auto qed qed end