(* 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) (* FIXME proper infix, without ambiguity!? *)
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 *}
lemma perm_remove: "x \ set ys \ ys <~~> x # remove1 x ys"
by (induct ys) auto
text {* \medskip Congruence rule *}
lemma perm_remove_perm: "xs <~~> ys \ remove1 z xs <~~> remove1 z ys"
by (induct pred: perm) auto
lemma remove_hd [simp]: "remove1 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"
by (induct zs arbitrary: xs ys rule: rev_induct) auto
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 = "remove1 a x" in allE, drule sym, simp)
apply (subgoal_tac "a \ 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 \ multiset_of ys \ (\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 fastforce
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")
apply simp_all
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 (fastforce simp add: insert_ident)
apply (metis distinct_remdups set_remdups)
apply (subgoal_tac "length (remdups xs) < Suc (length xs)")
apply simp
apply (subgoal_tac "length (remdups xs) \ length xs")
apply simp
apply (rule length_remdups_leq)
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)
lemma permutation_Ex_bij:
assumes "xs <~~> ys"
shows "\f. bij_betw f {.. (\iii. case i of Suc n \ Suc (f n) | 0 \ 0" show ?case proof (intro exi[of _ ?f] alli conji impi) have *: "{.. Suc ` {.. Suc ` {..ii f"] conjI allI impI)
show "bij_betw (g \ f) {.. f) i"
using trans(1,3)[THEN perm_length] perm by auto
qed
qed
end