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src/HOL/Library/Permutation.thy

author | wenzelm |

Sat, 10 Nov 2007 18:36:07 +0100 | |

changeset 25379 | 12bcf37252b1 |

parent 25287 | 094dab519ff5 |

child 26072 | f65a7fa2da6c |

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

tuned proofs;
tuned document;

(* 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) 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 *} consts remove :: "'a => 'a list => 'a list" primrec "remove x [] = []" "remove x (y # ys) = (if x = y then ys else y # remove x ys)" lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove x ys" by (induct ys) auto lemma remove_commute: "remove x (remove y l) = remove y (remove x l)" by (induct l) auto lemma multiset_of_remove [simp]: "multiset_of (remove a x) = multiset_of x - {#a#}" apply (induct x) apply (auto simp: multiset_eq_conv_count_eq) done text {* \medskip Congruence rule *} lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys" by (induct pred: perm) auto lemma remove_hd [simp]: "remove 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" apply (induct zs arbitrary: xs ys rule: rev_induct) apply (simp_all (no_asm_use)) apply blast done 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, simp_all add: union_ac) apply (erule rev_mp, rule_tac x=ys in spec) apply (induct_tac xs, auto) apply (erule_tac x = "remove a x" in allE, drule sym, simp) apply (subgoal_tac "a \<in> set x") apply (drule_tac z=a in perm.Cons) apply (erule perm.trans, rule perm_sym, erule perm_remove) apply (drule_tac f=set_of in arg_cong, simp) done lemma multiset_of_le_perm_append: "(multiset_of xs \<le># multiset_of ys) = (\<exists>zs. xs @ zs <~~> ys)"; apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv) apply (insert surj_multiset_of, 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 fastsimp 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", simp, simp) 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 (fastsimp simp add: insert_ident) apply (metis distinct_remdups set_remdups) apply (metis Nat.le_less_trans Suc_length_conv le_def length_remdups_leq less_Suc_eq) 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) end