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src/HOL/Library/Permutation.thy

author | paulson |

Thu, 24 Jun 2004 17:54:53 +0200 | |

changeset 15005 | 546c8e7e28d4 |

parent 14706 | 71590b7733b7 |

child 15072 | 4861bf6af0b4 |

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

Norbert Voelker

(* Title: HOL/Library/Permutation.thy Author: Lawrence C Paulson and Thomas M Rasmussen and Norbert Voelker *) header {* Permutations *} theory Permutation = Multiset: consts perm :: "('a list * 'a list) set" syntax "_perm" :: "'a list => 'a list => bool" ("_ <~~> _" [50, 50] 50) translations "x <~~> y" == "(x, y) \<in> perm" inductive perm intros 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_aux: "xs <~~> ys ==> xs = [] --> ys = []" -- {* the form of the premise lets the induction bind @{term xs} and @{term ys} *} apply (erule perm.induct) apply (simp_all (no_asm_simp)) done lemma xperm_empty_imp: "[] <~~> ys ==> ys = []" by (insert xperm_empty_imp_aux, blast) text {* \medskip This more general theorem is easier to understand! *} lemma perm_length: "xs <~~> ys ==> length xs = length ys" by (erule perm.induct, simp_all) lemma perm_empty_imp: "[] <~~> xs ==> xs = []" by (drule perm_length, auto) lemma perm_sym: "xs <~~> ys ==> ys <~~> xs" by (erule perm.induct, auto) lemma perm_mem [rule_format]: "xs <~~> ys ==> x mem xs --> x mem ys" by (erule perm.induct, 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, simp_all) apply (blast intro: perm_append_Cons) done lemma perm_append_single: "a # xs <~~> xs @ [a]" apply (rule perm.trans) prefer 2 apply (rule perm_append_swap, simp) done lemma perm_rev: "rev xs <~~> xs" apply (induct xs, 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 [rule_format]: "ys <~~> xs ==> xs = [y] --> ys = [y]" by (erule perm.induct, 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) text {* \medskip Congruence rule *} lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys" by (erule perm.induct, auto) lemma remove_hd [simp]: "remove z (z # xs) = xs" apply auto done 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: "!!xs ys. zs @ xs <~~> zs @ ys ==> xs <~~> ys" apply (induct zs 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 (****************** Norbert Voelker 17 June 2004 **************) consts multiset_of :: "'a list \<Rightarrow> 'a multiset" primrec "multiset_of [] = {#}" "multiset_of (a # x) = multiset_of x + {# a #}" lemma multiset_of_zero_iff[simp]: "(multiset_of x = {#}) = (x = [])" by (induct_tac x, auto) lemma multiset_of_zero_iff_right[simp]: "({#} = multiset_of x) = (x = [])" by (induct_tac x, auto) lemma set_of_multiset_of[simp]: "set_of(multiset_of x) = set x" by (induct_tac x, auto) lemma multiset_of_remove[simp]: "multiset_of (remove a x) = multiset_of x - {#a#}" by (induct_tac x, auto simp: multiset_eq_conv_count_eq) 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, 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 set_count_multiset_of: "set x = {a. 0 < count (multiset_of x) a}" by (induct_tac x, auto) lemma distinct_count_multiset_of: "distinct x \<Longrightarrow> count (multiset_of x) a = (if a \<in> set x then 1 else 0)" by (erule rev_mp, induct_tac x, auto) lemma distinct_set_eq_iff_multiset_of_eq: "\<lbrakk>distinct x; distinct y\<rbrakk> \<Longrightarrow> (set x = set y) = (multiset_of x = multiset_of y)" by (auto simp: multiset_eq_conv_count_eq distinct_count_multiset_of) end