author | haftmann |
Wed, 28 Jan 2009 11:04:10 +0100 | |
changeset 29650 | cc3958d31b1d |
parent 27368 | 9f90ac19e32b |
child 30738 | 0842e906300c |
permissions | -rw-r--r-- |
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(* Title: HOL/Library/Permutation.thy |
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Author: Lawrence C Paulson and Thomas M Rasmussen and Norbert Voelker |
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*) |
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header {* Permutations *} |
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theory Permutation |
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imports Plain Multiset |
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begin |
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inductive |
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perm :: "'a list => 'a list => bool" ("_ <~~> _" [50, 50] 50) |
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where |
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Nil [intro!]: "[] <~~> []" |
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| swap [intro!]: "y # x # l <~~> x # y # l" |
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| Cons [intro!]: "xs <~~> ys ==> z # xs <~~> z # ys" |
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| trans [intro]: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs" |
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lemma perm_refl [iff]: "l <~~> l" |
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by (induct l) auto |
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subsection {* Some examples of rule induction on permutations *} |
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lemma xperm_empty_imp: "[] <~~> ys ==> ys = []" |
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by (induct xs == "[]::'a list" ys pred: perm) simp_all |
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text {* |
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\medskip This more general theorem is easier to understand! |
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*} |
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lemma perm_length: "xs <~~> ys ==> length xs = length ys" |
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by (induct pred: perm) simp_all |
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lemma perm_empty_imp: "[] <~~> xs ==> xs = []" |
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by (drule perm_length) auto |
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lemma perm_sym: "xs <~~> ys ==> ys <~~> xs" |
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by (induct pred: perm) auto |
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subsection {* Ways of making new permutations *} |
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text {* |
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We can insert the head anywhere in the list. |
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*} |
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lemma perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys" |
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by (induct xs) auto |
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lemma perm_append_swap: "xs @ ys <~~> ys @ xs" |
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apply (induct xs) |
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apply simp_all |
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apply (blast intro: perm_append_Cons) |
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done |
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lemma perm_append_single: "a # xs <~~> xs @ [a]" |
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by (rule perm.trans [OF _ perm_append_swap]) simp |
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lemma perm_rev: "rev xs <~~> xs" |
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apply (induct xs) |
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apply simp_all |
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apply (blast intro!: perm_append_single intro: perm_sym) |
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done |
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lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys" |
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by (induct l) auto |
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lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l" |
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by (blast intro!: perm_append_swap perm_append1) |
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subsection {* Further results *} |
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lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])" |
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by (blast intro: perm_empty_imp) |
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lemma perm_empty2 [iff]: "(xs <~~> []) = (xs = [])" |
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apply auto |
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apply (erule perm_sym [THEN perm_empty_imp]) |
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done |
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lemma perm_sing_imp: "ys <~~> xs ==> xs = [y] ==> ys = [y]" |
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by (induct pred: perm) auto |
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lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])" |
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by (blast intro: perm_sing_imp) |
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lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])" |
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by (blast dest: perm_sym) |
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subsection {* Removing elements *} |
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consts |
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remove :: "'a => 'a list => 'a list" |
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primrec |
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"remove x [] = []" |
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"remove x (y # ys) = (if x = y then ys else y # remove x ys)" |
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lemma perm_remove: "x \<in> set ys ==> ys <~~> x # remove x ys" |
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by (induct ys) auto |
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lemma remove_commute: "remove x (remove y l) = remove y (remove x l)" |
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by (induct l) auto |
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lemma multiset_of_remove [simp]: |
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"multiset_of (remove a x) = multiset_of x - {#a#}" |
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apply (induct x) |
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apply (auto simp: multiset_eq_conv_count_eq) |
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done |
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text {* \medskip Congruence rule *} |
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lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys" |
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by (induct pred: perm) auto |
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lemma remove_hd [simp]: "remove z (z # xs) = xs" |
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by auto |
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lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys" |
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by (drule_tac z = z in perm_remove_perm) auto |
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lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)" |
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by (blast intro: cons_perm_imp_perm) |
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lemma append_perm_imp_perm: "zs @ xs <~~> zs @ ys ==> xs <~~> ys" |
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apply (induct zs arbitrary: xs ys rule: rev_induct) |
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apply (simp_all (no_asm_use)) |
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apply blast |
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done |
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lemma perm_append1_eq [iff]: "(zs @ xs <~~> zs @ ys) = (xs <~~> ys)" |
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by (blast intro: append_perm_imp_perm perm_append1) |
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lemma perm_append2_eq [iff]: "(xs @ zs <~~> ys @ zs) = (xs <~~> ys)" |
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apply (safe intro!: perm_append2) |
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apply (rule append_perm_imp_perm) |
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apply (rule perm_append_swap [THEN perm.trans]) |
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-- {* the previous step helps this @{text blast} call succeed quickly *} |
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apply (blast intro: perm_append_swap) |
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done |
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lemma multiset_of_eq_perm: "(multiset_of xs = multiset_of ys) = (xs <~~> ys) " |
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apply (rule iffI) |
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apply (erule_tac [2] perm.induct, simp_all add: union_ac) |
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apply (erule rev_mp, rule_tac x=ys in spec) |
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apply (induct_tac xs, auto) |
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apply (erule_tac x = "remove a x" in allE, drule sym, simp) |
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apply (subgoal_tac "a \<in> set x") |
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apply (drule_tac z=a in perm.Cons) |
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apply (erule perm.trans, rule perm_sym, erule perm_remove) |
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apply (drule_tac f=set_of in arg_cong, simp) |
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done |
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lemma multiset_of_le_perm_append: |
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"(multiset_of xs \<le># multiset_of ys) = (\<exists>zs. xs @ zs <~~> ys)"; |
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apply (auto simp: multiset_of_eq_perm[THEN sym] mset_le_exists_conv) |
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apply (insert surj_multiset_of, drule surjD) |
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apply (blast intro: sym)+ |
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done |
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lemma perm_set_eq: "xs <~~> ys ==> set xs = set ys" |
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by (metis multiset_of_eq_perm multiset_of_eq_setD) |
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lemma perm_distinct_iff: "xs <~~> ys ==> distinct xs = distinct ys" |
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apply (induct pred: perm) |
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apply simp_all |
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apply fastsimp |
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apply (metis perm_set_eq) |
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done |
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lemma eq_set_perm_remdups: "set xs = set ys ==> remdups xs <~~> remdups ys" |
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apply (induct xs arbitrary: ys rule: length_induct) |
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apply (case_tac "remdups xs", simp, simp) |
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apply (subgoal_tac "a : set (remdups ys)") |
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prefer 2 apply (metis set.simps(2) insert_iff set_remdups) |
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apply (drule split_list) apply(elim exE conjE) |
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apply (drule_tac x=list in spec) apply(erule impE) prefer 2 |
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apply (drule_tac x="ysa@zs" in spec) apply(erule impE) prefer 2 |
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apply simp |
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apply (subgoal_tac "a#list <~~> a#ysa@zs") |
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apply (metis Cons_eq_appendI perm_append_Cons trans) |
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apply (metis Cons Cons_eq_appendI distinct.simps(2) |
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distinct_remdups distinct_remdups_id perm_append_swap perm_distinct_iff) |
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apply (subgoal_tac "set (a#list) = set (ysa@a#zs) & distinct (a#list) & distinct (ysa@a#zs)") |
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apply (fastsimp simp add: insert_ident) |
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apply (metis distinct_remdups set_remdups) |
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9e9e67e33557
removed redundant less_trans, less_linear, le_imp_less_or_eq, le_less_trans, less_le_trans (cf. Orderings.thy);
wenzelm
parents:
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changeset
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apply (metis le_less_trans Suc_length_conv length_remdups_leq less_Suc_eq nat_less_le) |
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done |
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lemma perm_remdups_iff_eq_set: "remdups x <~~> remdups y = (set x = set y)" |
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by (metis List.set_remdups perm_set_eq eq_set_perm_remdups) |
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end |