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src/HOL/Library/Permutation.thy

changeset 11054 | a5404c70982f |

child 11153 | 950ede59c05a |

--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Library/Permutation.thy Sun Feb 04 19:42:54 2001 +0100 @@ -0,0 +1,199 @@ +(* Title: HOL/Library/Permutation.thy + ID: $Id$ + Author: Lawrence C Paulson, Cambridge University Computer Laboratory + Copyright 1995 University of Cambridge + +TODO: it would be nice to prove (for "multiset", defined on +HOL/ex/Sorting.thy) xs <~~> ys = (\<forall>x. multiset xs x = multiset ys x) +*) + +header {* + \title{Permutations} + \author{Lawrence C Paulson and Thomas M Rasmussen} +*} + +theory Permutation = Main: + +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 [intro] + Nil: "[] <~~> []" + swap: "y # x # l <~~> x # y # l" + Cons: "xs <~~> ys ==> z # xs <~~> z # ys" + trans: "xs <~~> ys ==> ys <~~> zs ==> xs <~~> zs" + +lemma perm_refl [iff]: "l <~~> l" + apply (induct l) + apply auto + done + + +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 = []" + apply (insert xperm_empty_imp_aux) + apply blast + done + + +text {* + \medskip This more general theorem is easier to understand! + *} + +lemma perm_length: "xs <~~> ys ==> length xs = length ys" + apply (erule perm.induct) + apply simp_all + done + +lemma perm_empty_imp: "[] <~~> xs ==> xs = []" + apply (drule perm_length) + apply auto + done + +lemma perm_sym: "xs <~~> ys ==> ys <~~> xs" + apply (erule perm.induct) + apply auto + done + +lemma perm_mem [rule_format]: "xs <~~> ys ==> x mem xs --> x mem ys" + apply (erule perm.induct) + apply auto + done + + +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" + apply (induct xs) + apply auto + done + +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]" + apply (rule perm.trans) + prefer 2 + apply (rule perm_append_swap) + apply simp + done + +lemma perm_rev: "rev xs <~~> xs" + apply (induct xs) + apply simp_all + apply (blast intro: perm_sym perm_append_single) + done + +lemma perm_append1: "xs <~~> ys ==> l @ xs <~~> l @ ys" + apply (induct l) + apply auto + done + +lemma perm_append2: "xs <~~> ys ==> xs @ l <~~> ys @ l" + apply (blast intro!: perm_append_swap perm_append1) + done + + +subsection {* Further results *} + +lemma perm_empty [iff]: "([] <~~> xs) = (xs = [])" + apply (blast intro: perm_empty_imp) + done + +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]" + apply (erule perm.induct) + apply auto + done + +lemma perm_sing_eq [iff]: "(ys <~~> [y]) = (ys = [y])" + apply (blast intro: perm_sing_imp) + done + +lemma perm_sing_eq2 [iff]: "([y] <~~> ys) = (ys = [y])" + apply (blast dest: perm_sym) + done + + +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" + apply (induct ys) + apply auto + done + +lemma remove_commute: "remove x (remove y l) = remove y (remove x l)" + apply (induct l) + apply auto + done + + +text {* \medskip Congruence rule *} + +lemma perm_remove_perm: "xs <~~> ys ==> remove z xs <~~> remove z ys" + apply (erule perm.induct) + apply auto + done + +lemma remove_hd [simp]: "remove z (z # xs) = xs" + apply auto + done + +lemma cons_perm_imp_perm: "z # xs <~~> z # ys ==> xs <~~> ys" + apply (drule_tac z = z in perm_remove_perm) + apply auto + done + +lemma cons_perm_eq [iff]: "(z#xs <~~> z#ys) = (xs <~~> ys)" + apply (blast intro: cons_perm_imp_perm) + done + +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)" + apply (blast intro: append_perm_imp_perm perm_append1) + done + +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 + +end