(* Title: HOL/Library/Permutation.thy
Author: Lawrence C Paulson and Thomas M Rasmussen and Norbert Voelker
*)
section \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"
proposition perm_refl [iff]: "l <~~> l"
by (induct l) auto
subsection \Some examples of rule induction on permutations\
proposition 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!\
proposition perm_length: "xs <~~> ys \ length xs = length ys"
by (induct pred: perm) simp_all
proposition perm_empty_imp: "[] <~~> xs \ xs = []"
by (drule perm_length) auto
proposition 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.\
proposition perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
by (induct xs) auto
proposition perm_append_swap: "xs @ ys <~~> ys @ xs"
by (induct xs) (auto intro: perm_append_Cons)
proposition perm_append_single: "a # xs <~~> xs @ [a]"
by (rule perm.trans [OF _ perm_append_swap]) simp
proposition perm_rev: "rev xs <~~> xs"
by (induct xs) (auto intro!: perm_append_single intro: perm_sym)
proposition perm_append1: "xs <~~> ys \ l @ xs <~~> l @ ys"
by (induct l) auto
proposition perm_append2: "xs <~~> ys \ xs @ l <~~> ys @ l"
by (blast intro!: perm_append_swap perm_append1)
subsection \Further results\
proposition perm_empty [iff]: "[] <~~> xs \ xs = []"
by (blast intro: perm_empty_imp)
proposition perm_empty2 [iff]: "xs <~~> [] \ xs = []"
apply auto
apply (erule perm_sym [THEN perm_empty_imp])
done
proposition perm_sing_imp: "ys <~~> xs \ xs = [y] \ ys = [y]"
by (induct pred: perm) auto
proposition perm_sing_eq [iff]: "ys <~~> [y] \ ys = [y]"
by (blast intro: perm_sing_imp)
proposition perm_sing_eq2 [iff]: "[y] <~~> ys \ ys = [y]"
by (blast dest: perm_sym)
subsection \Removing elements\
proposition perm_remove: "x \ set ys \ ys <~~> x # remove1 x ys"
by (induct ys) auto
text \\medskip Congruence rule\
proposition perm_remove_perm: "xs <~~> ys \ remove1 z xs <~~> remove1 z ys"
by (induct pred: perm) auto
proposition remove_hd [simp]: "remove1 z (z # xs) = xs"
by auto
proposition cons_perm_imp_perm: "z # xs <~~> z # ys \ xs <~~> ys"
by (drule perm_remove_perm [where z = z]) auto
proposition cons_perm_eq [iff]: "z#xs <~~> z#ys \ xs <~~> ys"
by (blast intro: cons_perm_imp_perm)
proposition append_perm_imp_perm: "zs @ xs <~~> zs @ ys \ xs <~~> ys"
by (induct zs arbitrary: xs ys rule: rev_induct) auto
proposition perm_append1_eq [iff]: "zs @ xs <~~> zs @ ys \ xs <~~> ys"
by (blast intro: append_perm_imp_perm perm_append1)
proposition 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 \blast\ call succeed quickly\
apply (blast intro: perm_append_swap)
done
theorem mset_eq_perm: "mset xs = mset ys \ xs <~~> ys"
apply (rule iffI)
apply (erule_tac [2] perm.induct)
apply (simp_all add: union_ac)
apply (erule rev_mp)
apply (rule_tac x=ys in spec)
apply (induct_tac xs)
apply auto
apply (erule_tac x = "remove1 a x" in allE)
apply (drule sym)
apply simp
apply (subgoal_tac "a \ set x")
apply (drule_tac z = a in perm.Cons)
apply (erule perm.trans)
apply (rule perm_sym)
apply (erule perm_remove)
apply (drule_tac f=set_mset in arg_cong)
apply simp
done
proposition mset_le_perm_append: "mset xs \# mset ys \ (\zs. xs @ zs <~~> ys)"
apply (auto simp: mset_eq_perm[THEN sym] mset_subset_eq_exists_conv)
apply (insert surj_mset)
apply (drule surjD)
apply (blast intro: sym)+
done
proposition perm_set_eq: "xs <~~> ys \ set xs = set ys"
by (metis mset_eq_perm mset_eq_setD)
proposition perm_distinct_iff: "xs <~~> ys \ distinct xs = distinct ys"
apply (induct pred: perm)
apply simp_all
apply fastforce
apply (metis perm_set_eq)
done
theorem 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 list.set(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
proposition 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)
theorem 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) {..i < length xs\ show "xs ! i = zs ! (g \ f) i"
using trans(1,3)[THEN perm_length] perm by auto
qed
qed
proposition perm_finite: "finite {B. B <~~> A}"
proof (rule finite_subset[where B="{xs. set xs \ set A \ length xs \ length A}"])
show "finite {xs. set xs \ set A \ length xs \ length A}"
apply (cases A, simp)
apply (rule card_ge_0_finite)
apply (auto simp: card_lists_length_le)
done
next
show "{B. B <~~> A} \ {xs. set xs \ set A \ length xs \ length A}"
by (clarsimp simp add: perm_length perm_set_eq)
qed
proposition perm_swap:
assumes "i < length xs" "j < length xs"
shows "xs[i := xs ! j, j := xs ! i] <~~> xs"
using assms by (simp add: mset_eq_perm[symmetric] mset_swap)
end