Theory HOL-Combinatorics.List_Permutation

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

section ‹Permuted Lists›

theory List_Permutation
imports Permutations

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

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