(* Title: HOL/Library/Permutation.thy ID: $Id: Permutation.thy,v 1.2 2001/02/16 12:37:21 paulson Exp $ 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 = (∀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) ∈ 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" 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_append_single intro: perm_sym) 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 ∈ 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

lemma perm_refl:

l <~~> l

lemma xperm_empty_imp_aux:

xs <~~> ys ==> xs = [] --> ys = []

lemma xperm_empty_imp:

[] <~~> ys ==> ys = []

lemma perm_length:

xs <~~> ys ==> length xs = length ys

lemma perm_empty_imp:

[] <~~> xs ==> xs = []

lemma perm_sym:

xs <~~> ys ==> ys <~~> xs

lemma perm_mem:

[| xs <~~> ys; x mem xs |] ==> x mem ys

lemma perm_append_Cons:

a # xs @ ys <~~> xs @ a # ys

lemma perm_append_swap:

xs @ ys <~~> ys @ xs

lemma perm_append_single:

a # xs <~~> xs @ [a]

lemma perm_rev:

rev xs <~~> xs

lemma perm_append1:

xs <~~> ys ==> l @ xs <~~> l @ ys

lemma perm_append2:

xs <~~> ys ==> xs @ l <~~> ys @ l

lemma perm_empty:

([] <~~> xs) = (xs = [])

lemma perm_empty2:

(xs <~~> []) = (xs = [])

lemma perm_sing_imp:

[| ys <~~> xs; xs = [y] |] ==> ys = [y]

lemma perm_sing_eq:

(ys <~~> [y]) = (ys = [y])

lemma perm_sing_eq2:

([y] <~~> ys) = (ys = [y])

lemma perm_remove:

x ∈ set ys ==> ys <~~> x # remove x ys

lemma remove_commute:

remove x (remove y l) = remove y (remove x l)

lemma perm_remove_perm:

xs <~~> ys ==> remove z xs <~~> remove z ys

lemma remove_hd:

remove z (z # xs) = xs

lemma cons_perm_imp_perm:

z # xs <~~> z # ys ==> xs <~~> ys

lemma cons_perm_eq:

(z # xs <~~> z # ys) = (xs <~~> ys)

lemma append_perm_imp_perm:

zs @ xs <~~> zs @ ys ==> xs <~~> ys

lemma perm_append1_eq:

(zs @ xs <~~> zs @ ys) = (xs <~~> ys)

lemma perm_append2_eq:

(xs @ zs <~~> ys @ zs) = (xs <~~> ys)