(* 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 {..<length xs} {..<length ys} ∧ (∀i<length xs. xs ! i = ys ! (f i))"

using assms proof induct

case Nil

then show ?case unfolding bij_betw_def by simp

next

case (swap y x l)

show ?case

proof (intro exI[of _ "Fun.swap 0 1 id"] conjI allI impI)

show "bij_betw (Fun.swap 0 1 id) {..<length (y # x # l)} {..<length (x # y # l)}"

by (auto simp: bij_betw_def)

fix i

assume "i < length(y#x#l)"

show "(y # x # l) ! i = (x # y # l) ! (Fun.swap 0 1 id) i"

by (cases i) (auto simp: Fun.swap_def gr0_conv_Suc)

qed

next

case (Cons xs ys z)

then obtain f where bij: "bij_betw f {..<length xs} {..<length ys}" and

perm: "∀i<length xs. xs ! i = ys ! (f i)" by blast

let ?f = "λi. case i of Suc n => Suc (f n) | 0 => 0"

show ?case

proof (intro exI[of _ ?f] allI conjI impI)

have *: "{..<length (z#xs)} = {0} ∪ Suc ` {..<length xs}"

"{..<length (z#ys)} = {0} ∪ Suc ` {..<length ys}"

by (simp_all add: lessThan_Suc_eq_insert_0)

show "bij_betw ?f {..<length (z#xs)} {..<length (z#ys)}"

unfolding *

proof (rule bij_betw_combine)

show "bij_betw ?f (Suc ` {..<length xs}) (Suc ` {..<length ys})"

using bij unfolding bij_betw_def

by (auto intro!: inj_onI imageI dest: inj_onD simp: image_compose[symmetric] comp_def)

qed (auto simp: bij_betw_def)

fix i

assume "i < length (z#xs)"

then show "(z # xs) ! i = (z # ys) ! (?f i)"

using perm by (cases i) auto

qed

next

case (trans xs ys zs)

then obtain f g where

bij: "bij_betw f {..<length xs} {..<length ys}" "bij_betw g {..<length ys} {..<length zs}" and

perm: "∀i<length xs. xs ! i = ys ! (f i)" "∀i<length ys. ys ! i = zs ! (g i)" by blast

show ?case

proof (intro exI[of _ "g o f"] conjI allI impI)

show "bij_betw (g o f) {..<length xs} {..<length zs}"

using bij by (rule bij_betw_trans)

fix i assume "i < length xs"

with bij have "f i < length ys" unfolding bij_betw_def by force

with `i < length xs` show "xs ! i = zs ! (g o f) i"

using trans(1,3)[THEN perm_length] perm by auto

qed

qed

end