# Theory HOL-Combinatorics.List_Permutation

```(*  Author:     Lawrence C Paulson and Thomas M Rasmussen and Norbert Voelker
*)

section ‹Permuted Lists›

theory List_Permutation
imports Permutations
begin

text ‹
Note that multisets already provide the notion of permutated list and hence
this theory mostly echoes material already logically present in theory
\<^text>‹Permutations›; it should be seldom needed.
›

subsection ‹An existing notion›

abbreviation (input) perm :: ‹'a list ⇒ 'a list ⇒ bool›  (infixr ‹<~~>› 50)
where ‹xs <~~> ys ≡ mset xs = mset ys›

subsection ‹Nontrivial conclusions›

proposition perm_swap:
‹xs[i := xs ! j, j := xs ! i] <~~> xs›
if ‹i < length xs› ‹j < length xs›
using that by (simp add: mset_swap)

proposition mset_le_perm_append: "mset xs ⊆# mset ys ⟷ (∃zs. xs @ zs <~~> ys)"
by (auto simp add: mset_subset_eq_exists_conv ex_mset dest: sym)

proposition perm_set_eq: "xs <~~> ys ⟹ set xs = set ys"
by (rule mset_eq_setD) simp

proposition perm_distinct_iff: "xs <~~> ys ⟹ distinct xs ⟷ distinct ys"
by (rule mset_eq_imp_distinct_iff) simp

theorem eq_set_perm_remdups: "set xs = set ys ⟹ remdups xs <~~> remdups ys"
by (simp add: set_eq_iff_mset_remdups_eq)

proposition perm_remdups_iff_eq_set: "remdups x <~~> remdups y ⟷ set x = set y"
by (simp add: set_eq_iff_mset_remdups_eq)

theorem permutation_Ex_bij:
assumes "xs <~~> ys"
shows "∃f. bij_betw f {..<length xs} {..<length ys} ∧ (∀i<length xs. xs ! i = ys ! (f i))"
proof -
from assms have ‹mset xs = mset ys› ‹length xs = length ys›
by (auto simp add: dest: mset_eq_length)
from ‹mset xs = mset ys› obtain p where ‹p permutes {..<length ys}› ‹permute_list p ys = xs›
by (rule mset_eq_permutation)
then have ‹bij_betw p {..<length xs} {..<length ys}›
by (simp add: ‹length xs = length ys› permutes_imp_bij)
moreover have ‹∀i<length xs. xs ! i = ys ! (p i)›
using ‹permute_list p ys = xs› ‹length xs = length ys› ‹p permutes {..<length ys}› permute_list_nth
by auto
ultimately show ?thesis
by blast
qed

proposition perm_finite: "finite {B. B <~~> A}"
using mset_eq_finite by auto

subsection ‹Trivial conclusions:›

proposition perm_empty_imp: "[] <~~> ys ⟹ ys = []"
by simp

text ‹\medskip This more general theorem is easier to understand!›

proposition perm_length: "xs <~~> ys ⟹ length xs = length ys"
by (rule mset_eq_length) simp

proposition perm_sym: "xs <~~> ys ⟹ ys <~~> xs"
by simp

text ‹We can insert the head anywhere in the list.›

proposition perm_append_Cons: "a # xs @ ys <~~> xs @ a # ys"
by simp

proposition perm_append_swap: "xs @ ys <~~> ys @ xs"
by simp

proposition perm_append_single: "a # xs <~~> xs @ [a]"
by simp

proposition perm_rev: "rev xs <~~> xs"
by simp

proposition perm_append1: "xs <~~> ys ⟹ l @ xs <~~> l @ ys"
by simp

proposition perm_append2: "xs <~~> ys ⟹ xs @ l <~~> ys @ l"
by simp

proposition perm_empty [iff]: "[] <~~> xs ⟷ xs = []"
by simp

proposition perm_empty2 [iff]: "xs <~~> [] ⟷ xs = []"
by simp

proposition perm_sing_imp: "ys <~~> xs ⟹ xs = [y] ⟹ ys = [y]"
by simp

proposition perm_sing_eq [iff]: "ys <~~> [y] ⟷ ys = [y]"
by simp

proposition perm_sing_eq2 [iff]: "[y] <~~> ys ⟷ ys = [y]"
by simp

proposition perm_remove: "x ∈ set ys ⟹ ys <~~> x # remove1 x ys"
by simp

text ‹\medskip Congruence rule›

proposition perm_remove_perm: "xs <~~> ys ⟹ remove1 z xs <~~> remove1 z ys"
by simp

proposition remove_hd [simp]: "remove1 z (z # xs) = xs"
by simp

proposition cons_perm_imp_perm: "z # xs <~~> z # ys ⟹ xs <~~> ys"
by simp

proposition cons_perm_eq [simp]: "z#xs <~~> z#ys ⟷ xs <~~> ys"
by simp

proposition append_perm_imp_perm: "zs @ xs <~~> zs @ ys ⟹ xs <~~> ys"
by simp

proposition perm_append1_eq [iff]: "zs @ xs <~~> zs @ ys ⟷ xs <~~> ys"
by simp

proposition perm_append2_eq [iff]: "xs @ zs <~~> ys @ zs ⟷ xs <~~> ys"
by simp

end
```