src/HOL/Library/Permutations.thy
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(*  Title:      HOL/Library/Permutations.thy
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    Author:     Amine Chaieb, University of Cambridge
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*)
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section \<open>Permutations, both general and specifically on finite sets.\<close>
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theory Permutations
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imports Binomial Multiset Disjoint_Sets
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begin
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subsection \<open>Transpositions\<close>
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lemma swap_id_idempotent [simp]:
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  "Fun.swap a b id \<circ> Fun.swap a b id = id"
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  by (rule ext, auto simp add: Fun.swap_def)
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lemma inv_swap_id:
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  "inv (Fun.swap a b id) = Fun.swap a b id"
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  by (rule inv_unique_comp) simp_all
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lemma swap_id_eq:
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  "Fun.swap a b id x = (if x = a then b else if x = b then a else x)"
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  by (simp add: Fun.swap_def)
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lemma bij_inv_eq_iff: "bij p \<Longrightarrow> x = inv p y \<longleftrightarrow> p x = y"
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  using surj_f_inv_f[of p] by (auto simp add: bij_def)
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lemma bij_swap_comp:
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  assumes bp: "bij p"
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  shows "Fun.swap a b id \<circ> p = Fun.swap (inv p a) (inv p b) p"
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  using surj_f_inv_f[OF bij_is_surj[OF bp]]
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  by (simp add: fun_eq_iff Fun.swap_def bij_inv_eq_iff[OF bp])
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lemma bij_swap_compose_bij: "bij p \<Longrightarrow> bij (Fun.swap a b id \<circ> p)"
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proof -
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  assume H: "bij p"
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  show ?thesis
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    unfolding bij_swap_comp[OF H] bij_swap_iff
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    using H .
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qed
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subsection \<open>Basic consequences of the definition\<close>
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definition permutes  (infixr "permutes" 41)
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  where "(p permutes S) \<longleftrightarrow> (\<forall>x. x \<notin> S \<longrightarrow> p x = x) \<and> (\<forall>y. \<exists>!x. p x = y)"
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lemma permutes_in_image: "p permutes S \<Longrightarrow> p x \<in> S \<longleftrightarrow> x \<in> S"
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  unfolding permutes_def by metis
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lemma permutes_not_in:
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  assumes "f permutes S" "x \<notin> S" shows "f x = x"
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  using assms by (auto simp: permutes_def)
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lemma permutes_image: "p permutes S \<Longrightarrow> p ` S = S"
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  unfolding permutes_def
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  apply (rule set_eqI)
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  apply (simp add: image_iff)
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  apply metis
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  done
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lemma permutes_inj: "p permutes S \<Longrightarrow> inj p"
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  unfolding permutes_def inj_def by blast
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lemma permutes_inj_on: "f permutes S \<Longrightarrow> inj_on f A"
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  unfolding permutes_def inj_on_def by auto
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lemma permutes_surj: "p permutes s \<Longrightarrow> surj p"
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  unfolding permutes_def surj_def by metis
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lemma permutes_bij: "p permutes s \<Longrightarrow> bij p"
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unfolding bij_def by (metis permutes_inj permutes_surj)
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lemma permutes_imp_bij: "p permutes S \<Longrightarrow> bij_betw p S S"
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by (metis UNIV_I bij_betw_subset permutes_bij permutes_image subsetI)
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lemma bij_imp_permutes: "bij_betw p S S \<Longrightarrow> (\<And>x. x \<notin> S \<Longrightarrow> p x = x) \<Longrightarrow> p permutes S"
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  unfolding permutes_def bij_betw_def inj_on_def
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  by auto (metis image_iff)+
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lemma permutes_inv_o:
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  assumes pS: "p permutes S"
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  shows "p \<circ> inv p = id"
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    and "inv p \<circ> p = id"
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  using permutes_inj[OF pS] permutes_surj[OF pS]
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  unfolding inj_iff[symmetric] surj_iff[symmetric] by blast+
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lemma permutes_inverses:
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  fixes p :: "'a \<Rightarrow> 'a"
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  assumes pS: "p permutes S"
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  shows "p (inv p x) = x"
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    and "inv p (p x) = x"
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  using permutes_inv_o[OF pS, unfolded fun_eq_iff o_def] by auto
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lemma permutes_subset: "p permutes S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> p permutes T"
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  unfolding permutes_def by blast
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lemma permutes_empty[simp]: "p permutes {} \<longleftrightarrow> p = id"
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  unfolding fun_eq_iff permutes_def by simp metis
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lemma permutes_sing[simp]: "p permutes {a} \<longleftrightarrow> p = id"
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  unfolding fun_eq_iff permutes_def by simp metis
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lemma permutes_univ: "p permutes UNIV \<longleftrightarrow> (\<forall>y. \<exists>!x. p x = y)"
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  unfolding permutes_def by simp
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lemma permutes_inv_eq: "p permutes S \<Longrightarrow> inv p y = x \<longleftrightarrow> p x = y"
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  unfolding permutes_def inv_def
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  apply auto
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  apply (erule allE[where x=y])
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  apply (erule allE[where x=y])
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  apply (rule someI_ex)
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  apply blast
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  apply (rule some1_equality)
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  apply blast
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  apply blast
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  done
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lemma permutes_swap_id: "a \<in> S \<Longrightarrow> b \<in> S \<Longrightarrow> Fun.swap a b id permutes S"
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  unfolding permutes_def Fun.swap_def fun_upd_def by auto metis
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lemma permutes_superset: "p permutes S \<Longrightarrow> (\<forall>x \<in> S - T. p x = x) \<Longrightarrow> p permutes T"
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  by (simp add: Ball_def permutes_def) metis
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(* Next three lemmas contributed by Lukas Bulwahn *)
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lemma permutes_bij_inv_into:
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  fixes A :: "'a set" and B :: "'b set"
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  assumes "p permutes A"
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  assumes "bij_betw f A B"
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  shows "(\<lambda>x. if x \<in> B then f (p (inv_into A f x)) else x) permutes B"
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proof (rule bij_imp_permutes)
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  have "bij_betw p A A" "bij_betw f A B" "bij_betw (inv_into A f) B A"
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    using assms by (auto simp add: permutes_imp_bij bij_betw_inv_into)
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  from this have "bij_betw (f o p o inv_into A f) B B" by (simp add: bij_betw_trans)
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  from this show "bij_betw (\<lambda>x. if x \<in> B then f (p (inv_into A f x)) else x) B B"
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    by (subst bij_betw_cong[where g="f o p o inv_into A f"]) auto
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next
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  fix x
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   139
  assume "x \<notin> B"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   140
  from this show "(if x \<in> B then f (p (inv_into A f x)) else x) = x" by auto
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   141
qed
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   142
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   143
lemma permutes_image_mset:
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   144
  assumes "p permutes A"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   145
  shows "image_mset p (mset_set A) = mset_set A"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   146
using assms by (metis image_mset_mset_set bij_betw_imp_inj_on permutes_imp_bij permutes_image)
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   147
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   148
lemma permutes_implies_image_mset_eq:
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   149
  assumes "p permutes A" "\<And>x. x \<in> A \<Longrightarrow> f x = f' (p x)"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   150
  shows "image_mset f' (mset_set A) = image_mset f (mset_set A)"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   151
proof -
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   152
  have "f x = f' (p x)" if x: "x \<in># mset_set A" for x
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   153
    using assms(2)[of x] x by (cases "finite A") auto
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   154
  from this have "image_mset f (mset_set A) = image_mset (f' o p) (mset_set A)"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   155
    using assms by (auto intro!: image_mset_cong)
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   156
  also have "\<dots> = image_mset f' (image_mset p (mset_set A))"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   157
    by (simp add: image_mset.compositionality)
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   158
  also have "\<dots> = image_mset f' (mset_set A)"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   159
  proof -
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   160
    from assms have "image_mset p (mset_set A) = mset_set A"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   161
      using permutes_image_mset by blast
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   162
    from this show ?thesis by simp
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   163
  qed
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   164
  finally show ?thesis ..
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   165
qed
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   166
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   167
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   168
subsection \<open>Group properties\<close>
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   169
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   170
lemma permutes_id: "id permutes S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   171
  unfolding permutes_def by simp
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   172
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   173
lemma permutes_compose: "p permutes S \<Longrightarrow> q permutes S \<Longrightarrow> q \<circ> p permutes S"
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   174
  unfolding permutes_def o_def by metis
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   175
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   176
lemma permutes_inv:
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   177
  assumes pS: "p permutes S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   178
  shows "inv p permutes S"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30267
diff changeset
   179
  using pS unfolding permutes_def permutes_inv_eq[OF pS] by metis
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   180
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   181
lemma permutes_inv_inv:
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   182
  assumes pS: "p permutes S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   183
  shows "inv (inv p) = p"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   184
  unfolding fun_eq_iff permutes_inv_eq[OF pS] permutes_inv_eq[OF permutes_inv[OF pS]]
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   185
  by blast
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   186
64284
f3b905b2eee2 HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents: 64267
diff changeset
   187
lemma permutes_invI:
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   188
  assumes perm: "p permutes S"
64284
f3b905b2eee2 HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents: 64267
diff changeset
   189
      and inv:  "\<And>x. x \<in> S \<Longrightarrow> p' (p x) = x"
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   190
      and outside: "\<And>x. x \<notin> S \<Longrightarrow> p' x = x"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   191
  shows   "inv p = p'"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   192
proof
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   193
  fix x show "inv p x = p' x"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   194
  proof (cases "x \<in> S")
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   195
    assume [simp]: "x \<in> S"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   196
    from assms have "p' x = p' (p (inv p x))" by (simp add: permutes_inverses)
64284
f3b905b2eee2 HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents: 64267
diff changeset
   197
    also from permutes_inv[OF perm]
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   198
      have "\<dots> = inv p x" by (subst inv) (simp_all add: permutes_in_image)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   199
    finally show "inv p x = p' x" ..
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   200
  qed (insert permutes_inv[OF perm], simp_all add: outside permutes_not_in)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   201
qed
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   202
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   203
lemma permutes_vimage: "f permutes A \<Longrightarrow> f -` A = A"
64284
f3b905b2eee2 HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents: 64267
diff changeset
   204
  by (simp add: bij_vimage_eq_inv_image permutes_bij permutes_image[OF permutes_inv])
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   205
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   206
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   207
subsection \<open>The number of permutations on a finite set\<close>
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   208
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30267
diff changeset
   209
lemma permutes_insert_lemma:
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   210
  assumes pS: "p permutes (insert a S)"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   211
  shows "Fun.swap a (p a) id \<circ> p permutes S"
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   212
  apply (rule permutes_superset[where S = "insert a S"])
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   213
  apply (rule permutes_compose[OF pS])
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   214
  apply (rule permutes_swap_id, simp)
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   215
  using permutes_in_image[OF pS, of a]
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   216
  apply simp
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 54681
diff changeset
   217
  apply (auto simp add: Ball_def Fun.swap_def)
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   218
  done
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   219
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   220
lemma permutes_insert: "{p. p permutes (insert a S)} =
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   221
  (\<lambda>(b,p). Fun.swap a b id \<circ> p) ` {(b,p). b \<in> insert a S \<and> p \<in> {p. p permutes S}}"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   222
proof -
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   223
  {
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   224
    fix p
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   225
    {
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   226
      assume pS: "p permutes insert a S"
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   227
      let ?b = "p a"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   228
      let ?q = "Fun.swap a (p a) id \<circ> p"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   229
      have th0: "p = Fun.swap a ?b id \<circ> ?q"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   230
        unfolding fun_eq_iff o_assoc by simp
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   231
      have th1: "?b \<in> insert a S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   232
        unfolding permutes_in_image[OF pS] by simp
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   233
      from permutes_insert_lemma[OF pS] th0 th1
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   234
      have "\<exists>b q. p = Fun.swap a b id \<circ> q \<and> b \<in> insert a S \<and> q permutes S" by blast
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   235
    }
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   236
    moreover
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   237
    {
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   238
      fix b q
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   239
      assume bq: "p = Fun.swap a b id \<circ> q" "b \<in> insert a S" "q permutes S"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30267
diff changeset
   240
      from permutes_subset[OF bq(3), of "insert a S"]
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   241
      have qS: "q permutes insert a S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   242
        by auto
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   243
      have aS: "a \<in> insert a S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   244
        by simp
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   245
      from bq(1) permutes_compose[OF qS permutes_swap_id[OF aS bq(2)]]
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   246
      have "p permutes insert a S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   247
        by simp
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   248
    }
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   249
    ultimately have "p permutes insert a S \<longleftrightarrow>
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   250
        (\<exists>b q. p = Fun.swap a b id \<circ> q \<and> b \<in> insert a S \<and> q permutes S)"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   251
      by blast
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   252
  }
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   253
  then show ?thesis
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   254
    by auto
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   255
qed
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   256
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   257
lemma card_permutations:
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   258
  assumes Sn: "card S = n"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   259
    and fS: "finite S"
33715
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33057
diff changeset
   260
  shows "card {p. p permutes S} = fact n"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   261
  using fS Sn
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   262
proof (induct arbitrary: n)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   263
  case empty
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   264
  then show ?case by simp
33715
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33057
diff changeset
   265
next
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33057
diff changeset
   266
  case (insert x F)
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   267
  {
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   268
    fix n
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   269
    assume H0: "card (insert x F) = n"
33715
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33057
diff changeset
   270
    let ?xF = "{p. p permutes insert x F}"
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33057
diff changeset
   271
    let ?pF = "{p. p permutes F}"
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33057
diff changeset
   272
    let ?pF' = "{(b, p). b \<in> insert x F \<and> p \<in> ?pF}"
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33057
diff changeset
   273
    let ?g = "(\<lambda>(b, p). Fun.swap x b id \<circ> p)"
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33057
diff changeset
   274
    from permutes_insert[of x F]
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33057
diff changeset
   275
    have xfgpF': "?xF = ?g ` ?pF'" .
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   276
    have Fs: "card F = n - 1"
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   277
      using \<open>x \<notin> F\<close> H0 \<open>finite F\<close> by auto
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   278
    from insert.hyps Fs have pFs: "card ?pF = fact (n - 1)"
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   279
      using \<open>finite F\<close> by auto
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   280
    then have "finite ?pF"
59730
b7c394c7a619 The factorial function, "fact", now has type "nat => 'a"
paulson <lp15@cam.ac.uk>
parents: 59669
diff changeset
   281
      by (auto intro: card_ge_0_finite)
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   282
    then have pF'f: "finite ?pF'"
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   283
      using H0 \<open>finite F\<close>
61424
c3658c18b7bc prod_case as canonical name for product type eliminator
haftmann
parents: 60601
diff changeset
   284
      apply (simp only: Collect_case_prod Collect_mem_eq)
33715
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33057
diff changeset
   285
      apply (rule finite_cartesian_product)
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33057
diff changeset
   286
      apply simp_all
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33057
diff changeset
   287
      done
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   288
33715
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33057
diff changeset
   289
    have ginj: "inj_on ?g ?pF'"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   290
    proof -
33715
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33057
diff changeset
   291
      {
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   292
        fix b p c q
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   293
        assume bp: "(b,p) \<in> ?pF'"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   294
        assume cq: "(c,q) \<in> ?pF'"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   295
        assume eq: "?g (b,p) = ?g (c,q)"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   296
        from bp cq have ths: "b \<in> insert x F" "c \<in> insert x F" "x \<in> insert x F"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   297
          "p permutes F" "q permutes F"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   298
          by auto
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   299
        from ths(4) \<open>x \<notin> F\<close> eq have "b = ?g (b,p) x"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   300
          unfolding permutes_def
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 54681
diff changeset
   301
          by (auto simp add: Fun.swap_def fun_upd_def fun_eq_iff)
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   302
        also have "\<dots> = ?g (c,q) x"
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   303
          using ths(5) \<open>x \<notin> F\<close> eq
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   304
          by (auto simp add: swap_def fun_upd_def fun_eq_iff)
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   305
        also have "\<dots> = c"
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   306
          using ths(5) \<open>x \<notin> F\<close>
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   307
          unfolding permutes_def
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 54681
diff changeset
   308
          by (auto simp add: Fun.swap_def fun_upd_def fun_eq_iff)
33715
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33057
diff changeset
   309
        finally have bc: "b = c" .
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   310
        then have "Fun.swap x b id = Fun.swap x c id"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   311
          by simp
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   312
        with eq have "Fun.swap x b id \<circ> p = Fun.swap x b id \<circ> q"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   313
          by simp
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   314
        then have "Fun.swap x b id \<circ> (Fun.swap x b id \<circ> p) =
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   315
          Fun.swap x b id \<circ> (Fun.swap x b id \<circ> q)"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   316
          by simp
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   317
        then have "p = q"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   318
          by (simp add: o_assoc)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   319
        with bc have "(b, p) = (c, q)"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   320
          by simp
33715
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33057
diff changeset
   321
      }
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   322
      then show ?thesis
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   323
        unfolding inj_on_def by blast
33715
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33057
diff changeset
   324
    qed
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   325
    from \<open>x \<notin> F\<close> H0 have n0: "n \<noteq> 0"
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   326
      using \<open>finite F\<close> by auto
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   327
    then have "\<exists>m. n = Suc m"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   328
      by presburger
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   329
    then obtain m where n[simp]: "n = Suc m"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   330
      by blast
33715
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33057
diff changeset
   331
    from pFs H0 have xFc: "card ?xF = fact n"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   332
      unfolding xfgpF' card_image[OF ginj]
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   333
      using \<open>finite F\<close> \<open>finite ?pF\<close>
61424
c3658c18b7bc prod_case as canonical name for product type eliminator
haftmann
parents: 60601
diff changeset
   334
      apply (simp only: Collect_case_prod Collect_mem_eq card_cartesian_product)
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   335
      apply simp
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   336
      done
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   337
    from finite_imageI[OF pF'f, of ?g] have xFf: "finite ?xF"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   338
      unfolding xfgpF' by simp
33715
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33057
diff changeset
   339
    have "card ?xF = fact n"
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33057
diff changeset
   340
      using xFf xFc unfolding xFf by blast
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33057
diff changeset
   341
  }
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   342
  then show ?case
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   343
    using insert by simp
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   344
qed
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   345
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   346
lemma finite_permutations:
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   347
  assumes fS: "finite S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   348
  shows "finite {p. p permutes S}"
64284
f3b905b2eee2 HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents: 64267
diff changeset
   349
  using card_permutations[OF refl fS]
33715
8cce3a34c122 removed hassize predicate
hoelzl
parents: 33057
diff changeset
   350
  by (auto intro: card_ge_0_finite)
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   351
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   352
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   353
subsection \<open>Permutations of index set for iterated operations\<close>
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   354
51489
f738e6dbd844 fundamental revision of big operators on sets
haftmann
parents: 49739
diff changeset
   355
lemma (in comm_monoid_set) permute:
f738e6dbd844 fundamental revision of big operators on sets
haftmann
parents: 49739
diff changeset
   356
  assumes "p permutes S"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   357
  shows "F g S = F (g \<circ> p) S"
51489
f738e6dbd844 fundamental revision of big operators on sets
haftmann
parents: 49739
diff changeset
   358
proof -
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   359
  from \<open>p permutes S\<close> have "inj p"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   360
    by (rule permutes_inj)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   361
  then have "inj_on p S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   362
    by (auto intro: subset_inj_on)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   363
  then have "F g (p ` S) = F (g \<circ> p) S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   364
    by (rule reindex)
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   365
  moreover from \<open>p permutes S\<close> have "p ` S = S"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   366
    by (rule permutes_image)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   367
  ultimately show ?thesis
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   368
    by simp
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   369
qed
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   370
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   371
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   372
subsection \<open>Various combinations of transpositions with 2, 1 and 0 common elements\<close>
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   373
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   374
lemma swap_id_common:" a \<noteq> c \<Longrightarrow> b \<noteq> c \<Longrightarrow>
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   375
  Fun.swap a b id \<circ> Fun.swap a c id = Fun.swap b c id \<circ> Fun.swap a b id"
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 54681
diff changeset
   376
  by (simp add: fun_eq_iff Fun.swap_def)
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   377
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   378
lemma swap_id_common': "a \<noteq> b \<Longrightarrow> a \<noteq> c \<Longrightarrow>
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   379
  Fun.swap a c id \<circ> Fun.swap b c id = Fun.swap b c id \<circ> Fun.swap a b id"
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 54681
diff changeset
   380
  by (simp add: fun_eq_iff Fun.swap_def)
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   381
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   382
lemma swap_id_independent: "a \<noteq> c \<Longrightarrow> a \<noteq> d \<Longrightarrow> b \<noteq> c \<Longrightarrow> b \<noteq> d \<Longrightarrow>
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   383
  Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap c d id \<circ> Fun.swap a b id"
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 54681
diff changeset
   384
  by (simp add: fun_eq_iff Fun.swap_def)
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   385
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   386
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   387
subsection \<open>Permutations as transposition sequences\<close>
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   388
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   389
inductive swapidseq :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> bool"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   390
where
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   391
  id[simp]: "swapidseq 0 id"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   392
| comp_Suc: "swapidseq n p \<Longrightarrow> a \<noteq> b \<Longrightarrow> swapidseq (Suc n) (Fun.swap a b id \<circ> p)"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   393
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   394
declare id[unfolded id_def, simp]
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   395
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   396
definition "permutation p \<longleftrightarrow> (\<exists>n. swapidseq n p)"
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   397
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   398
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   399
subsection \<open>Some closure properties of the set of permutations, with lengths\<close>
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   400
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   401
lemma permutation_id[simp]: "permutation id"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   402
  unfolding permutation_def by (rule exI[where x=0]) simp
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   403
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   404
declare permutation_id[unfolded id_def, simp]
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   405
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   406
lemma swapidseq_swap: "swapidseq (if a = b then 0 else 1) (Fun.swap a b id)"
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   407
  apply clarsimp
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   408
  using comp_Suc[of 0 id a b]
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   409
  apply simp
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   410
  done
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   411
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   412
lemma permutation_swap_id: "permutation (Fun.swap a b id)"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   413
  apply (cases "a = b")
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   414
  apply simp_all
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   415
  unfolding permutation_def
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   416
  using swapidseq_swap[of a b]
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   417
  apply blast
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   418
  done
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   419
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   420
lemma swapidseq_comp_add: "swapidseq n p \<Longrightarrow> swapidseq m q \<Longrightarrow> swapidseq (n + m) (p \<circ> q)"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   421
proof (induct n p arbitrary: m q rule: swapidseq.induct)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   422
  case (id m q)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   423
  then show ?case by simp
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   424
next
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   425
  case (comp_Suc n p a b m q)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   426
  have th: "Suc n + m = Suc (n + m)"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   427
    by arith
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   428
  show ?case
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   429
    unfolding th comp_assoc
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   430
    apply (rule swapidseq.comp_Suc)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   431
    using comp_Suc.hyps(2)[OF comp_Suc.prems] comp_Suc.hyps(3)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   432
    apply blast+
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   433
    done
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   434
qed
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   435
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   436
lemma permutation_compose: "permutation p \<Longrightarrow> permutation q \<Longrightarrow> permutation (p \<circ> q)"
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   437
  unfolding permutation_def using swapidseq_comp_add[of _ p _ q] by metis
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   438
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   439
lemma swapidseq_endswap: "swapidseq n p \<Longrightarrow> a \<noteq> b \<Longrightarrow> swapidseq (Suc n) (p \<circ> Fun.swap a b id)"
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   440
  apply (induct n p rule: swapidseq.induct)
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   441
  using swapidseq_swap[of a b]
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   442
  apply (auto simp add: comp_assoc intro: swapidseq.comp_Suc)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   443
  done
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   444
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   445
lemma swapidseq_inverse_exists: "swapidseq n p \<Longrightarrow> \<exists>q. swapidseq n q \<and> p \<circ> q = id \<and> q \<circ> p = id"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   446
proof (induct n p rule: swapidseq.induct)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   447
  case id
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   448
  then show ?case
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   449
    by (rule exI[where x=id]) simp
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30267
diff changeset
   450
next
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   451
  case (comp_Suc n p a b)
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   452
  from comp_Suc.hyps obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   453
    by blast
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   454
  let ?q = "q \<circ> Fun.swap a b id"
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   455
  note H = comp_Suc.hyps
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   456
  from swapidseq_swap[of a b] H(3) have th0: "swapidseq 1 (Fun.swap a b id)"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   457
    by simp
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   458
  from swapidseq_comp_add[OF q(1) th0] have th1: "swapidseq (Suc n) ?q"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   459
    by simp
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   460
  have "Fun.swap a b id \<circ> p \<circ> ?q = Fun.swap a b id \<circ> (p \<circ> q) \<circ> Fun.swap a b id"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   461
    by (simp add: o_assoc)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   462
  also have "\<dots> = id"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   463
    by (simp add: q(2))
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   464
  finally have th2: "Fun.swap a b id \<circ> p \<circ> ?q = id" .
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   465
  have "?q \<circ> (Fun.swap a b id \<circ> p) = q \<circ> (Fun.swap a b id \<circ> Fun.swap a b id) \<circ> p"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   466
    by (simp only: o_assoc)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   467
  then have "?q \<circ> (Fun.swap a b id \<circ> p) = id"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   468
    by (simp add: q(3))
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   469
  with th1 th2 show ?case
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   470
    by blast
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   471
qed
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   472
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   473
lemma swapidseq_inverse:
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   474
  assumes H: "swapidseq n p"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   475
  shows "swapidseq n (inv p)"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   476
  using swapidseq_inverse_exists[OF H] inv_unique_comp[of p] by auto
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   477
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   478
lemma permutation_inverse: "permutation p \<Longrightarrow> permutation (inv p)"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   479
  using permutation_def swapidseq_inverse by blast
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   480
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   481
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   482
subsection \<open>The identity map only has even transposition sequences\<close>
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   483
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   484
lemma symmetry_lemma:
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   485
  assumes "\<And>a b c d. P a b c d \<Longrightarrow> P a b d c"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   486
    and "\<And>a b c d. a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow>
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   487
      a = c \<and> b = d \<or> a = c \<and> b \<noteq> d \<or> a \<noteq> c \<and> b = d \<or> a \<noteq> c \<and> a \<noteq> d \<and> b \<noteq> c \<and> b \<noteq> d \<Longrightarrow>
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   488
      P a b c d"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   489
  shows "\<And>a b c d. a \<noteq> b \<longrightarrow> c \<noteq> d \<longrightarrow>  P a b c d"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   490
  using assms by metis
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   491
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   492
lemma swap_general: "a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow>
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   493
  Fun.swap a b id \<circ> Fun.swap c d id = id \<or>
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   494
  (\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and>
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   495
    Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap x y id \<circ> Fun.swap a z id)"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   496
proof -
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   497
  assume H: "a \<noteq> b" "c \<noteq> d"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   498
  have "a \<noteq> b \<longrightarrow> c \<noteq> d \<longrightarrow>
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   499
    (Fun.swap a b id \<circ> Fun.swap c d id = id \<or>
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   500
      (\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and>
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   501
        Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap x y id \<circ> Fun.swap a z id))"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   502
    apply (rule symmetry_lemma[where a=a and b=b and c=c and d=d])
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 54681
diff changeset
   503
    apply (simp_all only: swap_commute)
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   504
    apply (case_tac "a = c \<and> b = d")
56608
8e3c848008fa more simp rules for Fun.swap
haftmann
parents: 56545
diff changeset
   505
    apply (clarsimp simp only: swap_commute swap_id_idempotent)
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   506
    apply (case_tac "a = c \<and> b \<noteq> d")
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   507
    apply (rule disjI2)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   508
    apply (rule_tac x="b" in exI)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   509
    apply (rule_tac x="d" in exI)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   510
    apply (rule_tac x="b" in exI)
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 54681
diff changeset
   511
    apply (clarsimp simp add: fun_eq_iff Fun.swap_def)
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   512
    apply (case_tac "a \<noteq> c \<and> b = d")
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   513
    apply (rule disjI2)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   514
    apply (rule_tac x="c" in exI)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   515
    apply (rule_tac x="d" in exI)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   516
    apply (rule_tac x="c" in exI)
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 54681
diff changeset
   517
    apply (clarsimp simp add: fun_eq_iff Fun.swap_def)
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   518
    apply (rule disjI2)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   519
    apply (rule_tac x="c" in exI)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   520
    apply (rule_tac x="d" in exI)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   521
    apply (rule_tac x="b" in exI)
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 54681
diff changeset
   522
    apply (clarsimp simp add: fun_eq_iff Fun.swap_def)
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   523
    done
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   524
  with H show ?thesis by metis
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   525
qed
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   526
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   527
lemma swapidseq_id_iff[simp]: "swapidseq 0 p \<longleftrightarrow> p = id"
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   528
  using swapidseq.cases[of 0 p "p = id"]
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   529
  by auto
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   530
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   531
lemma swapidseq_cases: "swapidseq n p \<longleftrightarrow>
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   532
  n = 0 \<and> p = id \<or> (\<exists>a b q m. n = Suc m \<and> p = Fun.swap a b id \<circ> q \<and> swapidseq m q \<and> a \<noteq> b)"
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   533
  apply (rule iffI)
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   534
  apply (erule swapidseq.cases[of n p])
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   535
  apply simp
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   536
  apply (rule disjI2)
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   537
  apply (rule_tac x= "a" in exI)
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   538
  apply (rule_tac x= "b" in exI)
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   539
  apply (rule_tac x= "pa" in exI)
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   540
  apply (rule_tac x= "na" in exI)
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   541
  apply simp
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   542
  apply auto
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   543
  apply (rule comp_Suc, simp_all)
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   544
  done
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   545
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   546
lemma fixing_swapidseq_decrease:
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   547
  assumes spn: "swapidseq n p"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   548
    and ab: "a \<noteq> b"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   549
    and pa: "(Fun.swap a b id \<circ> p) a = a"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   550
  shows "n \<noteq> 0 \<and> swapidseq (n - 1) (Fun.swap a b id \<circ> p)"
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   551
  using spn ab pa
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   552
proof (induct n arbitrary: p a b)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   553
  case 0
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   554
  then show ?case
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 54681
diff changeset
   555
    by (auto simp add: Fun.swap_def fun_upd_def)
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   556
next
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   557
  case (Suc n p a b)
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   558
  from Suc.prems(1) swapidseq_cases[of "Suc n" p]
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   559
  obtain c d q m where
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   560
    cdqm: "Suc n = Suc m" "p = Fun.swap c d id \<circ> q" "swapidseq m q" "c \<noteq> d" "n = m"
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   561
    by auto
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   562
  {
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   563
    assume H: "Fun.swap a b id \<circ> Fun.swap c d id = id"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   564
    have ?case by (simp only: cdqm o_assoc H) (simp add: cdqm)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   565
  }
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   566
  moreover
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   567
  {
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   568
    fix x y z
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   569
    assume H: "x \<noteq> a" "y \<noteq> a" "z \<noteq> a" "x \<noteq> y"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   570
      "Fun.swap a b id \<circ> Fun.swap c d id = Fun.swap x y id \<circ> Fun.swap a z id"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   571
    from H have az: "a \<noteq> z"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   572
      by simp
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   573
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   574
    {
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   575
      fix h
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   576
      have "(Fun.swap x y id \<circ> h) a = a \<longleftrightarrow> h a = a"
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 54681
diff changeset
   577
        using H by (simp add: Fun.swap_def)
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   578
    }
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   579
    note th3 = this
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   580
    from cdqm(2) have "Fun.swap a b id \<circ> p = Fun.swap a b id \<circ> (Fun.swap c d id \<circ> q)"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   581
      by simp
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   582
    then have "Fun.swap a b id \<circ> p = Fun.swap x y id \<circ> (Fun.swap a z id \<circ> q)"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   583
      by (simp add: o_assoc H)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   584
    then have "(Fun.swap a b id \<circ> p) a = (Fun.swap x y id \<circ> (Fun.swap a z id \<circ> q)) a"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   585
      by simp
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   586
    then have "(Fun.swap x y id \<circ> (Fun.swap a z id \<circ> q)) a = a"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   587
      unfolding Suc by metis
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   588
    then have th1: "(Fun.swap a z id \<circ> q) a = a"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   589
      unfolding th3 .
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   590
    from Suc.hyps[OF cdqm(3)[ unfolded cdqm(5)[symmetric]] az th1]
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   591
    have th2: "swapidseq (n - 1) (Fun.swap a z id \<circ> q)" "n \<noteq> 0"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   592
      by blast+
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   593
    have th: "Suc n - 1 = Suc (n - 1)"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   594
      using th2(2) by auto
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   595
    have ?case
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   596
      unfolding cdqm(2) H o_assoc th
49739
13aa6d8268ec consolidated names of theorems on composition;
haftmann
parents: 45922
diff changeset
   597
      apply (simp only: Suc_not_Zero simp_thms comp_assoc)
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   598
      apply (rule comp_Suc)
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   599
      using th2 H
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   600
      apply blast+
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   601
      done
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   602
  }
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   603
  ultimately show ?case
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   604
    using swap_general[OF Suc.prems(2) cdqm(4)] by metis
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   605
qed
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   606
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30267
diff changeset
   607
lemma swapidseq_identity_even:
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   608
  assumes "swapidseq n (id :: 'a \<Rightarrow> 'a)"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   609
  shows "even n"
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   610
  using \<open>swapidseq n id\<close>
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   611
proof (induct n rule: nat_less_induct)
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   612
  fix n
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   613
  assume H: "\<forall>m<n. swapidseq m (id::'a \<Rightarrow> 'a) \<longrightarrow> even m" "swapidseq n (id :: 'a \<Rightarrow> 'a)"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   614
  {
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   615
    assume "n = 0"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   616
    then have "even n" by presburger
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   617
  }
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30267
diff changeset
   618
  moreover
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   619
  {
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   620
    fix a b :: 'a and q m
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   621
    assume h: "n = Suc m" "(id :: 'a \<Rightarrow> 'a) = Fun.swap a b id \<circ> q" "swapidseq m q" "a \<noteq> b"
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   622
    from fixing_swapidseq_decrease[OF h(3,4), unfolded h(2)[symmetric]]
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   623
    have m: "m \<noteq> 0" "swapidseq (m - 1) (id :: 'a \<Rightarrow> 'a)"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   624
      by auto
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   625
    from h m have mn: "m - 1 < n"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   626
      by arith
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   627
    from H(1)[rule_format, OF mn m(2)] h(1) m(1) have "even n"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   628
      by presburger
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   629
  }
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   630
  ultimately show "even n"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   631
    using H(2)[unfolded swapidseq_cases[of n id]] by auto
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   632
qed
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   633
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   634
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   635
subsection \<open>Therefore we have a welldefined notion of parity\<close>
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   636
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   637
definition "evenperm p = even (SOME n. swapidseq n p)"
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   638
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   639
lemma swapidseq_even_even:
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   640
  assumes m: "swapidseq m p"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   641
    and n: "swapidseq n p"
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   642
  shows "even m \<longleftrightarrow> even n"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   643
proof -
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   644
  from swapidseq_inverse_exists[OF n]
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   645
  obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   646
    by blast
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   647
  from swapidseq_identity_even[OF swapidseq_comp_add[OF m q(1), unfolded q]]
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   648
  show ?thesis
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   649
    by arith
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   650
qed
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   651
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   652
lemma evenperm_unique:
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   653
  assumes p: "swapidseq n p"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   654
    and n:"even n = b"
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   655
  shows "evenperm p = b"
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   656
  unfolding n[symmetric] evenperm_def
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   657
  apply (rule swapidseq_even_even[where p = p])
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   658
  apply (rule someI[where x = n])
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   659
  using p
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   660
  apply blast+
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   661
  done
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   662
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   663
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   664
subsection \<open>And it has the expected composition properties\<close>
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   665
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   666
lemma evenperm_id[simp]: "evenperm id = True"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   667
  by (rule evenperm_unique[where n = 0]) simp_all
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   668
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   669
lemma evenperm_swap: "evenperm (Fun.swap a b id) = (a = b)"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   670
  by (rule evenperm_unique[where n="if a = b then 0 else 1"]) (simp_all add: swapidseq_swap)
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   671
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30267
diff changeset
   672
lemma evenperm_comp:
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   673
  assumes p: "permutation p"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   674
    and q:"permutation q"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   675
  shows "evenperm (p \<circ> q) = (evenperm p = evenperm q)"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   676
proof -
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   677
  from p q obtain n m where n: "swapidseq n p" and m: "swapidseq m q"
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   678
    unfolding permutation_def by blast
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   679
  note nm =  swapidseq_comp_add[OF n m]
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   680
  have th: "even (n + m) = (even n \<longleftrightarrow> even m)"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   681
    by arith
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   682
  from evenperm_unique[OF n refl] evenperm_unique[OF m refl]
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   683
    evenperm_unique[OF nm th]
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   684
  show ?thesis
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   685
    by blast
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   686
qed
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   687
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   688
lemma evenperm_inv:
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   689
  assumes p: "permutation p"
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   690
  shows "evenperm (inv p) = evenperm p"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   691
proof -
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   692
  from p obtain n where n: "swapidseq n p"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   693
    unfolding permutation_def by blast
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   694
  from evenperm_unique[OF swapidseq_inverse[OF n] evenperm_unique[OF n refl, symmetric]]
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   695
  show ?thesis .
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   696
qed
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   697
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   698
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   699
subsection \<open>A more abstract characterization of permutations\<close>
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   700
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   701
lemma bij_iff: "bij f \<longleftrightarrow> (\<forall>x. \<exists>!y. f y = x)"
64966
d53d7ca3303e added inj_def (redundant, analogous to surj_def, bij_def);
wenzelm
parents: 64543
diff changeset
   702
  unfolding bij_def inj_def surj_def
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   703
  apply auto
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   704
  apply metis
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   705
  apply metis
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   706
  done
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   707
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30267
diff changeset
   708
lemma permutation_bijective:
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30267
diff changeset
   709
  assumes p: "permutation p"
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   710
  shows "bij p"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   711
proof -
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   712
  from p obtain n where n: "swapidseq n p"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   713
    unfolding permutation_def by blast
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   714
  from swapidseq_inverse_exists[OF n]
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   715
  obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   716
    by blast
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   717
  then show ?thesis unfolding bij_iff
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   718
    apply (auto simp add: fun_eq_iff)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   719
    apply metis
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   720
    done
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30267
diff changeset
   721
qed
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   722
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   723
lemma permutation_finite_support:
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   724
  assumes p: "permutation p"
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   725
  shows "finite {x. p x \<noteq> x}"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   726
proof -
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   727
  from p obtain n where n: "swapidseq n p"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   728
    unfolding permutation_def by blast
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   729
  from n show ?thesis
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   730
  proof (induct n p rule: swapidseq.induct)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   731
    case id
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   732
    then show ?case by simp
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   733
  next
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   734
    case (comp_Suc n p a b)
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   735
    let ?S = "insert a (insert b {x. p x \<noteq> x})"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   736
    from comp_Suc.hyps(2) have fS: "finite ?S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   737
      by simp
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   738
    from \<open>a \<noteq> b\<close> have th: "{x. (Fun.swap a b id \<circ> p) x \<noteq> x} \<subseteq> ?S"
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 54681
diff changeset
   739
      by (auto simp add: Fun.swap_def)
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   740
    from finite_subset[OF th fS] show ?case  .
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   741
  qed
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   742
qed
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   743
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30267
diff changeset
   744
lemma permutation_lemma:
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   745
  assumes fS: "finite S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   746
    and p: "bij p"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   747
    and pS: "\<forall>x. x\<notin> S \<longrightarrow> p x = x"
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   748
  shows "permutation p"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   749
  using fS p pS
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   750
proof (induct S arbitrary: p rule: finite_induct)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   751
  case (empty p)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   752
  then show ?case by simp
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   753
next
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   754
  case (insert a F p)
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   755
  let ?r = "Fun.swap a (p a) id \<circ> p"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   756
  let ?q = "Fun.swap a (p a) id \<circ> ?r"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   757
  have raa: "?r a = a"
56545
8f1e7596deb7 more operations and lemmas
haftmann
parents: 54681
diff changeset
   758
    by (simp add: Fun.swap_def)
64543
6b13586ef1a2 remove typo in bij_swap_compose_bij theorem name; tune proof
bulwahn
parents: 64284
diff changeset
   759
  from bij_swap_compose_bij[OF insert(4)] have br: "bij ?r"  .
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30267
diff changeset
   760
  from insert raa have th: "\<forall>x. x \<notin> F \<longrightarrow> ?r x = x"
64966
d53d7ca3303e added inj_def (redundant, analogous to surj_def, bij_def);
wenzelm
parents: 64543
diff changeset
   761
    by (metis bij_pointE comp_apply id_apply insert_iff swap_apply(3))
64543
6b13586ef1a2 remove typo in bij_swap_compose_bij theorem name; tune proof
bulwahn
parents: 64284
diff changeset
   762
  from insert(3)[OF br th] have rp: "permutation ?r" .
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   763
  have "permutation ?q"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   764
    by (simp add: permutation_compose permutation_swap_id rp)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   765
  then show ?case
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   766
    by (simp add: o_assoc)
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   767
qed
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   768
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30267
diff changeset
   769
lemma permutation: "permutation p \<longleftrightarrow> bij p \<and> finite {x. p x \<noteq> x}"
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   770
  (is "?lhs \<longleftrightarrow> ?b \<and> ?f")
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   771
proof
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   772
  assume p: ?lhs
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   773
  from p permutation_bijective permutation_finite_support show "?b \<and> ?f"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   774
    by auto
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   775
next
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   776
  assume "?b \<and> ?f"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   777
  then have "?f" "?b" by blast+
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   778
  from permutation_lemma[OF this] show ?lhs
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   779
    by blast
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   780
qed
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   781
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   782
lemma permutation_inverse_works:
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   783
  assumes p: "permutation p"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   784
  shows "inv p \<circ> p = id"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   785
    and "p \<circ> inv p = id"
44227
78e033e8ba05 get Library/Permutations.thy compiled and working again
huffman
parents: 41959
diff changeset
   786
  using permutation_bijective [OF p]
78e033e8ba05 get Library/Permutations.thy compiled and working again
huffman
parents: 41959
diff changeset
   787
  unfolding bij_def inj_iff surj_iff by auto
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   788
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   789
lemma permutation_inverse_compose:
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   790
  assumes p: "permutation p"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   791
    and q: "permutation q"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   792
  shows "inv (p \<circ> q) = inv q \<circ> inv p"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   793
proof -
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   794
  note ps = permutation_inverse_works[OF p]
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   795
  note qs = permutation_inverse_works[OF q]
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   796
  have "p \<circ> q \<circ> (inv q \<circ> inv p) = p \<circ> (q \<circ> inv q) \<circ> inv p"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   797
    by (simp add: o_assoc)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   798
  also have "\<dots> = id"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   799
    by (simp add: ps qs)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   800
  finally have th0: "p \<circ> q \<circ> (inv q \<circ> inv p) = id" .
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   801
  have "inv q \<circ> inv p \<circ> (p \<circ> q) = inv q \<circ> (inv p \<circ> p) \<circ> q"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   802
    by (simp add: o_assoc)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   803
  also have "\<dots> = id"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   804
    by (simp add: ps qs)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   805
  finally have th1: "inv q \<circ> inv p \<circ> (p \<circ> q) = id" .
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   806
  from inv_unique_comp[OF th0 th1] show ?thesis .
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   807
qed
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   808
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   809
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   810
subsection \<open>Relation to "permutes"\<close>
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   811
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   812
lemma permutation_permutes: "permutation p \<longleftrightarrow> (\<exists>S. finite S \<and> p permutes S)"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   813
  unfolding permutation permutes_def bij_iff[symmetric]
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   814
  apply (rule iffI, clarify)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   815
  apply (rule exI[where x="{x. p x \<noteq> x}"])
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   816
  apply simp
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   817
  apply clarsimp
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   818
  apply (rule_tac B="S" in finite_subset)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   819
  apply auto
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   820
  done
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   821
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   822
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   823
subsection \<open>Hence a sort of induction principle composing by swaps\<close>
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   824
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   825
lemma permutes_induct: "finite S \<Longrightarrow> P id \<Longrightarrow>
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   826
  (\<And> a b p. a \<in> S \<Longrightarrow> b \<in> S \<Longrightarrow> P p \<Longrightarrow> P p \<Longrightarrow> permutation p \<Longrightarrow> P (Fun.swap a b id \<circ> p)) \<Longrightarrow>
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   827
  (\<And>p. p permutes S \<Longrightarrow> P p)"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   828
proof (induct S rule: finite_induct)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   829
  case empty
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   830
  then show ?case by auto
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30267
diff changeset
   831
next
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   832
  case (insert x F p)
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   833
  let ?r = "Fun.swap x (p x) id \<circ> p"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   834
  let ?q = "Fun.swap x (p x) id \<circ> ?r"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   835
  have qp: "?q = p"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   836
    by (simp add: o_assoc)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   837
  from permutes_insert_lemma[OF insert.prems(3)] insert have Pr: "P ?r"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   838
    by blast
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30267
diff changeset
   839
  from permutes_in_image[OF insert.prems(3), of x]
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   840
  have pxF: "p x \<in> insert x F"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   841
    by simp
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   842
  have xF: "x \<in> insert x F"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   843
    by simp
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   844
  have rp: "permutation ?r"
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30267
diff changeset
   845
    unfolding permutation_permutes using insert.hyps(1)
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   846
      permutes_insert_lemma[OF insert.prems(3)]
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   847
    by blast
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30267
diff changeset
   848
  from insert.prems(2)[OF xF pxF Pr Pr rp]
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   849
  show ?case
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   850
    unfolding qp .
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   851
qed
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   852
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   853
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   854
subsection \<open>Sign of a permutation as a real number\<close>
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   855
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   856
definition "sign p = (if evenperm p then (1::int) else -1)"
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   857
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   858
lemma sign_nz: "sign p \<noteq> 0"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   859
  by (simp add: sign_def)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   860
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   861
lemma sign_id: "sign id = 1"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   862
  by (simp add: sign_def)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   863
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   864
lemma sign_inverse: "permutation p \<Longrightarrow> sign (inv p) = sign p"
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   865
  by (simp add: sign_def evenperm_inv)
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   866
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   867
lemma sign_compose: "permutation p \<Longrightarrow> permutation q \<Longrightarrow> sign (p \<circ> q) = sign p * sign q"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   868
  by (simp add: sign_def evenperm_comp)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   869
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   870
lemma sign_swap_id: "sign (Fun.swap a b id) = (if a = b then 1 else -1)"
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   871
  by (simp add: sign_def evenperm_swap)
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   872
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   873
lemma sign_idempotent: "sign p * sign p = 1"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   874
  by (simp add: sign_def)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   875
64284
f3b905b2eee2 HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents: 64267
diff changeset
   876
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   877
subsection \<open>Permuting a list\<close>
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   878
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   879
text \<open>This function permutes a list by applying a permutation to the indices.\<close>
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   880
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   881
definition permute_list :: "(nat \<Rightarrow> nat) \<Rightarrow> 'a list \<Rightarrow> 'a list" where
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   882
  "permute_list f xs = map (\<lambda>i. xs ! (f i)) [0..<length xs]"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   883
64284
f3b905b2eee2 HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents: 64267
diff changeset
   884
lemma permute_list_map:
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   885
  assumes "f permutes {..<length xs}"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   886
  shows   "permute_list f (map g xs) = map g (permute_list f xs)"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   887
  using permutes_in_image[OF assms] by (auto simp: permute_list_def)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   888
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   889
lemma permute_list_nth:
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   890
  assumes "f permutes {..<length xs}" "i < length xs"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   891
  shows   "permute_list f xs ! i = xs ! f i"
64284
f3b905b2eee2 HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents: 64267
diff changeset
   892
  using permutes_in_image[OF assms(1)] assms(2)
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   893
  by (simp add: permute_list_def)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   894
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   895
lemma permute_list_Nil [simp]: "permute_list f [] = []"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   896
  by (simp add: permute_list_def)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   897
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   898
lemma length_permute_list [simp]: "length (permute_list f xs) = length xs"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   899
  by (simp add: permute_list_def)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   900
64284
f3b905b2eee2 HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents: 64267
diff changeset
   901
lemma permute_list_compose:
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   902
  assumes "g permutes {..<length xs}"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   903
  shows   "permute_list (f \<circ> g) xs = permute_list g (permute_list f xs)"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   904
  using assms[THEN permutes_in_image] by (auto simp add: permute_list_def)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   905
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   906
lemma permute_list_ident [simp]: "permute_list (\<lambda>x. x) xs = xs"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   907
  by (simp add: permute_list_def map_nth)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   908
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   909
lemma permute_list_id [simp]: "permute_list id xs = xs"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   910
  by (simp add: id_def)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   911
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   912
lemma mset_permute_list [simp]:
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   913
  assumes "f permutes {..<length (xs :: 'a list)}"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   914
  shows   "mset (permute_list f xs) = mset xs"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   915
proof (rule multiset_eqI)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   916
  fix y :: 'a
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   917
  from assms have [simp]: "f x < length xs \<longleftrightarrow> x < length xs" for x
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   918
    using permutes_in_image[OF assms] by auto
64284
f3b905b2eee2 HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents: 64267
diff changeset
   919
  have "count (mset (permute_list f xs)) y =
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   920
          card ((\<lambda>i. xs ! f i) -` {y} \<inter> {..<length xs})"
64543
6b13586ef1a2 remove typo in bij_swap_compose_bij theorem name; tune proof
bulwahn
parents: 64284
diff changeset
   921
    by (simp add: permute_list_def count_image_mset atLeast0LessThan)
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   922
  also have "(\<lambda>i. xs ! f i) -` {y} \<inter> {..<length xs} = f -` {i. i < length xs \<and> y = xs ! i}"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   923
    by auto
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   924
  also from assms have "card \<dots> = card {i. i < length xs \<and> y = xs ! i}"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   925
    by (intro card_vimage_inj) (auto simp: permutes_inj permutes_surj)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   926
  also have "\<dots> = count (mset xs) y" by (simp add: count_mset length_filter_conv_card)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   927
  finally show "count (mset (permute_list f xs)) y = count (mset xs) y" by simp
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   928
qed
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   929
64284
f3b905b2eee2 HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents: 64267
diff changeset
   930
lemma set_permute_list [simp]:
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   931
  assumes "f permutes {..<length xs}"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   932
  shows   "set (permute_list f xs) = set xs"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   933
  by (rule mset_eq_setD[OF mset_permute_list]) fact
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   934
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   935
lemma distinct_permute_list [simp]:
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   936
  assumes "f permutes {..<length xs}"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   937
  shows   "distinct (permute_list f xs) = distinct xs"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   938
  by (simp add: distinct_count_atmost_1 assms)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   939
64284
f3b905b2eee2 HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents: 64267
diff changeset
   940
lemma permute_list_zip:
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   941
  assumes "f permutes A" "A = {..<length xs}"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   942
  assumes [simp]: "length xs = length ys"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   943
  shows   "permute_list f (zip xs ys) = zip (permute_list f xs) (permute_list f ys)"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   944
proof -
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   945
  from permutes_in_image[OF assms(1)] assms(2)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   946
    have [simp]: "f i < length ys \<longleftrightarrow> i < length ys" for i by simp
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   947
  have "permute_list f (zip xs ys) = map (\<lambda>i. zip xs ys ! f i) [0..<length ys]"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   948
    by (simp_all add: permute_list_def zip_map_map)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   949
  also have "\<dots> = map (\<lambda>(x, y). (xs ! f x, ys ! f y)) (zip [0..<length ys] [0..<length ys])"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   950
    by (intro nth_equalityI) simp_all
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   951
  also have "\<dots> = zip (permute_list f xs) (permute_list f ys)"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   952
    by (simp_all add: permute_list_def zip_map_map)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   953
  finally show ?thesis .
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   954
qed
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   955
64284
f3b905b2eee2 HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents: 64267
diff changeset
   956
lemma map_of_permute:
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   957
  assumes "\<sigma> permutes fst ` set xs"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   958
  shows   "map_of xs \<circ> \<sigma> = map_of (map (\<lambda>(x,y). (inv \<sigma> x, y)) xs)" (is "_ = map_of (map ?f _)")
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   959
proof
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   960
  fix x
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   961
  from assms have "inj \<sigma>" "surj \<sigma>" by (simp_all add: permutes_inj permutes_surj)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   962
  thus "(map_of xs \<circ> \<sigma>) x = map_of (map ?f xs) x"
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   963
    by (induction xs) (auto simp: inv_f_f surj_f_inv_f)
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   964
qed
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   965
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
   966
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
   967
subsection \<open>More lemmas about permutations\<close>
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
   968
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   969
text \<open>
63921
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   970
  The following few lemmas were contributed by Lukas Bulwahn.
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
   971
\<close>
63921
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   972
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   973
lemma count_image_mset_eq_card_vimage:
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   974
  assumes "finite A"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   975
  shows "count (image_mset f (mset_set A)) b = card {a \<in> A. f a = b}"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   976
  using assms
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   977
proof (induct A)
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   978
  case empty
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   979
  show ?case by simp
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   980
next
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   981
  case (insert x F)
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   982
  show ?case
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   983
  proof cases
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   984
    assume "f x = b"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   985
    from this have "count (image_mset f (mset_set (insert x F))) b = Suc (card {a \<in> F. f a = f x})"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   986
      using insert.hyps by auto
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   987
    also have "\<dots> = card (insert x {a \<in> F. f a = f x})"
64284
f3b905b2eee2 HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents: 64267
diff changeset
   988
      using insert.hyps(1,2) by simp
63921
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   989
    also have "card (insert x {a \<in> F. f a = f x}) = card {a \<in> insert x F. f a = b}"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   990
      using \<open>f x = b\<close> by (auto intro: arg_cong[where f="card"])
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   991
    finally show ?thesis using insert by auto
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   992
  next
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   993
    assume A: "f x \<noteq> b"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   994
    hence "{a \<in> F. f a = b} = {a \<in> insert x F. f a = b}" by auto
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   995
    with insert A show ?thesis by simp
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   996
  qed
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   997
qed
64284
f3b905b2eee2 HOL-Analysis: more theorems from Sébastien Gouëzel's Ergodic_Theory
hoelzl
parents: 64267
diff changeset
   998
63921
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
   999
(* Prove image_mset_eq_implies_permutes *)
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1000
lemma image_mset_eq_implies_permutes:
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1001
  fixes f :: "'a \<Rightarrow> 'b"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1002
  assumes "finite A"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1003
  assumes mset_eq: "image_mset f (mset_set A) = image_mset f' (mset_set A)"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1004
  obtains p where "p permutes A" and "\<forall>x\<in>A. f x = f' (p x)"
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
  1005
proof -
63921
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1006
  from \<open>finite A\<close> have [simp]: "finite {a \<in> A. f a = (b::'b)}" for f b by auto
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1007
  have "f ` A = f' ` A"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1008
  proof -
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1009
    have "f ` A = f ` (set_mset (mset_set A))" using \<open>finite A\<close> by simp
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1010
    also have "\<dots> = f' ` (set_mset (mset_set A))"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1011
      by (metis mset_eq multiset.set_map)
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1012
    also have "\<dots> = f' ` A" using \<open>finite A\<close> by simp
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1013
    finally show ?thesis .
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1014
  qed
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1015
  have "\<forall>b\<in>(f ` A). \<exists>p. bij_betw p {a \<in> A. f a = b} {a \<in> A. f' a = b}"
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
  1016
  proof
63921
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1017
    fix b
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1018
    from mset_eq have
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1019
      "count (image_mset f (mset_set A)) b = count (image_mset f' (mset_set A)) b" by simp
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1020
    from this  have "card {a \<in> A. f a = b} = card {a \<in> A. f' a = b}"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1021
      using \<open>finite A\<close>
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1022
      by (simp add: count_image_mset_eq_card_vimage)
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1023
    from this show "\<exists>p. bij_betw p {a\<in>A. f a = b} {a \<in> A. f' a = b}"
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
  1024
      by (intro finite_same_card_bij) simp_all
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
  1025
  qed
63921
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1026
  hence "\<exists>p. \<forall>b\<in>f ` A. bij_betw (p b) {a \<in> A. f a = b} {a \<in> A. f' a = b}"
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
  1027
    by (rule bchoice)
63921
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1028
  then guess p .. note p = this
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1029
  define p' where "p' = (\<lambda>a. if a \<in> A then p (f a) a else a)"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1030
  have "p' permutes A"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1031
  proof (rule bij_imp_permutes)
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1032
    have "disjoint_family_on (\<lambda>i. {a \<in> A. f' a = i}) (f ` A)"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1033
      unfolding disjoint_family_on_def by auto
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1034
    moreover have "bij_betw (\<lambda>a. p (f a) a) {a \<in> A. f a = b} {a \<in> A. f' a = b}" if b: "b \<in> f ` A" for b
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1035
      using p b by (subst bij_betw_cong[where g="p b"]) auto
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1036
    ultimately have "bij_betw (\<lambda>a. p (f a) a) (\<Union>b\<in>f ` A. {a \<in> A. f a = b}) (\<Union>b\<in>f ` A. {a \<in> A. f' a = b})"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1037
      by (rule bij_betw_UNION_disjoint)
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1038
    moreover have "(\<Union>b\<in>f ` A. {a \<in> A. f a = b}) = A" by auto
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1039
    moreover have "(\<Union>b\<in>f ` A. {a \<in> A. f' a = b}) = A" using \<open>f ` A = f' ` A\<close> by auto
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1040
    ultimately show "bij_betw p' A A"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1041
      unfolding p'_def by (subst bij_betw_cong[where g="(\<lambda>a. p (f a) a)"]) auto
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1042
  next
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1043
    fix x
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1044
    assume "x \<notin> A"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1045
    from this show "p' x = x"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1046
      unfolding p'_def by simp
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
  1047
  qed
63921
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1048
  moreover from p have "\<forall>x\<in>A. f x = f' (p' x)"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1049
    unfolding p'_def using bij_betwE by fastforce
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1050
  ultimately show ?thesis by (rule that)
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1051
qed
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
  1052
63921
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1053
lemma mset_set_upto_eq_mset_upto:
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1054
  "mset_set {..<n} = mset [0..<n]"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1055
  by (induct n) (auto simp add: add.commute lessThan_Suc)
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
  1056
63921
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1057
(* and derive the existing property: *)
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1058
lemma mset_eq_permutation:
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1059
  assumes mset_eq: "mset (xs::'a list) = mset ys"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1060
  obtains p where "p permutes {..<length ys}" "permute_list p ys = xs"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1061
proof -
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1062
  from mset_eq have length_eq: "length xs = length ys"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1063
    using mset_eq_length by blast
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1064
  have "mset_set {..<length ys} = mset [0..<length ys]"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1065
    using mset_set_upto_eq_mset_upto by blast
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1066
  from mset_eq length_eq this have
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1067
    "image_mset (\<lambda>i. xs ! i) (mset_set {..<length ys}) = image_mset (\<lambda>i. ys ! i) (mset_set {..<length ys})"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1068
    by (metis map_nth mset_map)
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1069
  from image_mset_eq_implies_permutes[OF _ this]
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1070
    obtain p where "p permutes {..<length ys}"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1071
    and "\<forall>i\<in>{..<length ys}. xs ! i = ys ! (p i)" by auto
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1072
  moreover from this length_eq have "permute_list p ys = xs"
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1073
    by (auto intro!: nth_equalityI simp add: permute_list_nth)
0a5184877cb7 Additions to permutations (contributed by Lukas Bulwahn)
eberlm <eberlm@in.tum.de>
parents: 63539
diff changeset
  1074
  ultimately show thesis using that by blast
63099
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
  1075
qed
af0e964aad7b Moved material from AFP/Randomised_Social_Choice to distribution
eberlm
parents: 62390
diff changeset
  1076
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1077
lemma permutes_natset_le:
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1078
  fixes S :: "'a::wellorder set"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1079
  assumes p: "p permutes S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1080
    and le: "\<forall>i \<in> S. p i \<le> i"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1081
  shows "p = id"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1082
proof -
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1083
  {
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1084
    fix n
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30267
diff changeset
  1085
    have "p n = n"
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1086
      using p le
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1087
    proof (induct n arbitrary: S rule: less_induct)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1088
      fix n S
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1089
      assume H:
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1090
        "\<And>m S. m < n \<Longrightarrow> p permutes S \<Longrightarrow> \<forall>i\<in>S. p i \<le> i \<Longrightarrow> p m = m"
32960
69916a850301 eliminated hard tabulators, guessing at each author's individual tab-width;
wenzelm
parents: 32456
diff changeset
  1091
        "p permutes S" "\<forall>i \<in>S. p i \<le> i"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1092
      {
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1093
        assume "n \<notin> S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1094
        with H(2) have "p n = n"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1095
          unfolding permutes_def by metis
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1096
      }
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1097
      moreover
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1098
      {
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1099
        assume ns: "n \<in> S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1100
        from H(3)  ns have "p n < n \<or> p n = n"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1101
          by auto
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1102
        moreover {
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1103
          assume h: "p n < n"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1104
          from H h have "p (p n) = p n"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1105
            by metis
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1106
          with permutes_inj[OF H(2)] have "p n = n"
64966
d53d7ca3303e added inj_def (redundant, analogous to surj_def, bij_def);
wenzelm
parents: 64543
diff changeset
  1107
            unfolding inj_def by blast
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1108
          with h have False
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1109
            by simp
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1110
        }
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1111
        ultimately have "p n = n"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1112
          by blast
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1113
      }
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1114
      ultimately show "p n = n"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1115
        by blast
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1116
    qed
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1117
  }
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1118
  then show ?thesis
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1119
    by (auto simp add: fun_eq_iff)
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1120
qed
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1121
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1122
lemma permutes_natset_ge:
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1123
  fixes S :: "'a::wellorder set"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1124
  assumes p: "p permutes S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1125
    and le: "\<forall>i \<in> S. p i \<ge> i"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1126
  shows "p = id"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1127
proof -
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1128
  {
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1129
    fix i
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1130
    assume i: "i \<in> S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1131
    from i permutes_in_image[OF permutes_inv[OF p]] have "inv p i \<in> S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1132
      by simp
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1133
    with le have "p (inv p i) \<ge> inv p i"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1134
      by blast
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1135
    with permutes_inverses[OF p] have "i \<ge> inv p i"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1136
      by simp
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1137
  }
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1138
  then have th: "\<forall>i\<in>S. inv p i \<le> i"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1139
    by blast
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30267
diff changeset
  1140
  from permutes_natset_le[OF permutes_inv[OF p] th]
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1141
  have "inv p = inv id"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1142
    by simp
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30267
diff changeset
  1143
  then show ?thesis
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1144
    apply (subst permutes_inv_inv[OF p, symmetric])
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1145
    apply (rule inv_unique_comp)
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1146
    apply simp_all
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1147
    done
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1148
qed
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1149
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1150
lemma image_inverse_permutations: "{inv p |p. p permutes S} = {p. p permutes S}"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1151
  apply (rule set_eqI)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1152
  apply auto
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1153
  using permutes_inv_inv permutes_inv
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1154
  apply auto
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1155
  apply (rule_tac x="inv x" in exI)
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1156
  apply auto
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1157
  done
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1158
30488
5c4c3a9e9102 remove trailing spaces
huffman
parents: 30267
diff changeset
  1159
lemma image_compose_permutations_left:
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1160
  assumes q: "q permutes S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1161
  shows "{q \<circ> p | p. p permutes S} = {p . p permutes S}"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1162
  apply (rule set_eqI)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1163
  apply auto
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1164
  apply (rule permutes_compose)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1165
  using q
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1166
  apply auto
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1167
  apply (rule_tac x = "inv q \<circ> x" in exI)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1168
  apply (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1169
  done
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1170
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1171
lemma image_compose_permutations_right:
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1172
  assumes q: "q permutes S"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1173
  shows "{p \<circ> q | p. p permutes S} = {p . p permutes S}"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1174
  apply (rule set_eqI)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1175
  apply auto
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1176
  apply (rule permutes_compose)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1177
  using q
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1178
  apply auto
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1179
  apply (rule_tac x = "x \<circ> inv q" in exI)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1180
  apply (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o comp_assoc)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1181
  done
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1182
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1183
lemma permutes_in_seg: "p permutes {1 ..n} \<Longrightarrow> i \<in> {1..n} \<Longrightarrow> 1 \<le> p i \<and> p i \<le> n"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1184
  by (simp add: permutes_def) metis
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1185
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63921
diff changeset
  1186
lemma sum_permutations_inverse:
b9a1486e79be setsum -> sum
nipkow
parents: 63921
diff changeset
  1187
  "sum f {p. p permutes S} = sum (\<lambda>p. f(inv p)) {p. p permutes S}"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1188
  (is "?lhs = ?rhs")
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1189
proof -
30036
3a074e3a9a18 generalize some lemmas
huffman
parents: 29840
diff changeset
  1190
  let ?S = "{p . p permutes S}"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1191
  have th0: "inj_on inv ?S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1192
  proof (auto simp add: inj_on_def)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1193
    fix q r
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1194
    assume q: "q permutes S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1195
      and r: "r permutes S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1196
      and qr: "inv q = inv r"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1197
    then have "inv (inv q) = inv (inv r)"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1198
      by simp
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1199
    with permutes_inv_inv[OF q] permutes_inv_inv[OF r] show "q = r"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1200
      by metis
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1201
  qed
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1202
  have th1: "inv ` ?S = ?S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1203
    using image_inverse_permutations by blast
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63921
diff changeset
  1204
  have th2: "?rhs = sum (f \<circ> inv) ?S"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1205
    by (simp add: o_def)
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63921
diff changeset
  1206
  from sum.reindex[OF th0, of f] show ?thesis unfolding th1 th2 .
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1207
qed
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1208
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1209
lemma setum_permutations_compose_left:
30036
3a074e3a9a18 generalize some lemmas
huffman
parents: 29840
diff changeset
  1210
  assumes q: "q permutes S"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63921
diff changeset
  1211
  shows "sum f {p. p permutes S} = sum (\<lambda>p. f(q \<circ> p)) {p. p permutes S}"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1212
  (is "?lhs = ?rhs")
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1213
proof -
30036
3a074e3a9a18 generalize some lemmas
huffman
parents: 29840
diff changeset
  1214
  let ?S = "{p. p permutes S}"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63921
diff changeset
  1215
  have th0: "?rhs = sum (f \<circ> (op \<circ> q)) ?S"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1216
    by (simp add: o_def)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1217
  have th1: "inj_on (op \<circ> q) ?S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1218
  proof (auto simp add: inj_on_def)
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1219
    fix p r
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1220
    assume "p permutes S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1221
      and r: "r permutes S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1222
      and rp: "q \<circ> p = q \<circ> r"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1223
    then have "inv q \<circ> q \<circ> p = inv q \<circ> q \<circ> r"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1224
      by (simp add: comp_assoc)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1225
    with permutes_inj[OF q, unfolded inj_iff] show "p = r"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1226
      by simp
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1227
  qed
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1228
  have th3: "(op \<circ> q) ` ?S = ?S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1229
    using image_compose_permutations_left[OF q] by auto
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63921
diff changeset
  1230
  from sum.reindex[OF th1, of f] show ?thesis unfolding th0 th1 th3 .
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1231
qed
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1232
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1233
lemma sum_permutations_compose_right:
30036
3a074e3a9a18 generalize some lemmas
huffman
parents: 29840
diff changeset
  1234
  assumes q: "q permutes S"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63921
diff changeset
  1235
  shows "sum f {p. p permutes S} = sum (\<lambda>p. f(p \<circ> q)) {p. p permutes S}"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1236
  (is "?lhs = ?rhs")
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1237
proof -
30036
3a074e3a9a18 generalize some lemmas
huffman
parents: 29840
diff changeset
  1238
  let ?S = "{p. p permutes S}"
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63921
diff changeset
  1239
  have th0: "?rhs = sum (f \<circ> (\<lambda>p. p \<circ> q)) ?S"
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1240
    by (simp add: o_def)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1241
  have th1: "inj_on (\<lambda>p. p \<circ> q) ?S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1242
  proof (auto simp add: inj_on_def)
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1243
    fix p r
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1244
    assume "p permutes S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1245
      and r: "r permutes S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1246
      and rp: "p \<circ> q = r \<circ> q"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1247
    then have "p \<circ> (q \<circ> inv q) = r \<circ> (q \<circ> inv q)"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1248
      by (simp add: o_assoc)
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1249
    with permutes_surj[OF q, unfolded surj_iff] show "p = r"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1250
      by simp
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1251
  qed
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1252
  have th3: "(\<lambda>p. p \<circ> q) ` ?S = ?S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1253
    using image_compose_permutations_right[OF q] by auto
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63921
diff changeset
  1254
  from sum.reindex[OF th1, of f]
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1255
  show ?thesis unfolding th0 th1 th3 .
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1256
qed
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1257
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1258
60500
903bb1495239 isabelle update_cartouches;
wenzelm
parents: 59730
diff changeset
  1259
subsection \<open>Sum over a set of permutations (could generalize to iteration)\<close>
29840
cfab6a76aa13 Permutations, both general and specifically on finite sets.
chaieb
parents:
diff changeset
  1260
64267
b9a1486e79be setsum -> sum
nipkow
parents: 63921
diff changeset
  1261
lemma sum_over_permutations_insert:
54681
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset
  1262
  assumes fS: "finite S"
8a8e6db7f391 tuned proofs;
wenzelm
parents: 53374
diff changeset