(* Title: HOL/Library/Permutation.thy Author: Lawrence C Paulson and Thomas M Rasmussen and Norbert Voelker *) section ‹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" proposition perm_refl [iff]: "l <~~> l" by (induct l) auto subsection ‹Some examples of rule induction on permutations› proposition 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!› proposition perm_length: "xs <~~> ys ⟹ length xs = length ys" by (induct pred: perm) simp_all proposition perm_empty_imp: "[] <~~> xs ⟹ xs = []" by (drule perm_length) auto proposition 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.› 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 ⟹ l @ xs <~~> l @ ys" by (induct l) auto proposition perm_append2: "xs <~~> ys ⟹ xs @ l <~~> ys @ l" by (blast intro!: perm_append_swap perm_append1) subsection ‹Further results› proposition perm_empty [iff]: "[] <~~> xs ⟷ xs = []" by (blast intro: perm_empty_imp) proposition perm_empty2 [iff]: "xs <~~> [] ⟷ xs = []" apply auto apply (erule perm_sym [THEN perm_empty_imp]) done proposition perm_sing_imp: "ys <~~> xs ⟹ xs = [y] ⟹ ys = [y]" by (induct pred: perm) auto proposition perm_sing_eq [iff]: "ys <~~> [y] ⟷ ys = [y]" by (blast intro: perm_sing_imp) proposition perm_sing_eq2 [iff]: "[y] <~~> ys ⟷ ys = [y]" by (blast dest: perm_sym) subsection ‹Removing elements› proposition perm_remove: "x ∈ set ys ⟹ ys <~~> x # remove1 x ys" by (induct ys) auto text ‹\medskip Congruence rule› proposition perm_remove_perm: "xs <~~> ys ⟹ 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 ⟹ xs <~~> ys" by (drule perm_remove_perm [where z = z]) auto proposition cons_perm_eq [iff]: "z#xs <~~> z#ys ⟷ xs <~~> ys" by (blast intro: cons_perm_imp_perm) proposition append_perm_imp_perm: "zs @ xs <~~> zs @ ys ⟹ xs <~~> ys" by (induct zs arbitrary: xs ys rule: rev_induct) auto proposition perm_append1_eq [iff]: "zs @ xs <~~> zs @ ys ⟷ xs <~~> ys" by (blast intro: append_perm_imp_perm perm_append1) proposition 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 ‹blast› call succeed quickly› apply (blast intro: perm_append_swap) done theorem mset_eq_perm: "mset xs = mset 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_mset in arg_cong) apply simp done proposition mset_le_perm_append: "mset xs ≤# mset ys ⟷ (∃zs. xs @ zs <~~> ys)" apply (auto simp: mset_eq_perm[THEN sym] mset_subset_eq_exists_conv) apply (insert surj_mset) apply (drule surjD) apply (blast intro: sym)+ done proposition perm_set_eq: "xs <~~> ys ⟹ set xs = set ys" by (metis mset_eq_perm mset_eq_setD) proposition perm_distinct_iff: "xs <~~> ys ⟹ distinct xs = distinct ys" apply (induct pred: perm) apply simp_all apply fastforce apply (metis perm_set_eq) done theorem 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 list.set(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 proposition 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) theorem 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 ∘ f"] conjI allI impI) show "bij_betw (g ∘ 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 ∘ 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 ⊆ set A ∧ length xs ≤ length A}"]) show "finite {xs. set xs ⊆ set A ∧ length xs ≤ length A}" apply (cases A, simp) apply (rule card_ge_0_finite) apply (auto simp: card_lists_length_le) done next show "{B. B <~~> A} ⊆ {xs. set xs ⊆ set A ∧ length xs ≤ 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