# Theory Permutation

theory Permutation
imports Multiset
```(*  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 (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 (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 {..<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}"
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_comp 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 ∘ f"] conjI allI impI)
show "bij_betw (g ∘ 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 ∘ 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
```