src/HOL/Library/Permutations.thy
author huffman
Sun Apr 25 23:22:29 2010 -0700 (2010-04-25)
changeset 36361 1debc8e29f6a
parent 36335 ceea7e15c04b
child 39198 f967a16dfcdd
permissions -rw-r--r--
fix duplicate simp rule warnings
chaieb@29840
     1
(* Title:      Library/Permutations
chaieb@29840
     2
   Author:     Amine Chaieb, University of Cambridge
chaieb@29840
     3
*)
chaieb@29840
     4
chaieb@29840
     5
header {* Permutations, both general and specifically on finite sets.*}
chaieb@29840
     6
chaieb@29840
     7
theory Permutations
huffman@36335
     8
imports Parity Fact
chaieb@29840
     9
begin
chaieb@29840
    10
chaieb@29840
    11
definition permutes (infixr "permutes" 41) where
chaieb@29840
    12
  "(p permutes S) \<longleftrightarrow> (\<forall>x. x \<notin> S \<longrightarrow> p x = x) \<and> (\<forall>y. \<exists>!x. p x = y)"
chaieb@29840
    13
chaieb@29840
    14
(* ------------------------------------------------------------------------- *)
chaieb@29840
    15
(* Transpositions.                                                           *)
chaieb@29840
    16
(* ------------------------------------------------------------------------- *)
chaieb@29840
    17
huffman@30488
    18
lemma swapid_sym: "Fun.swap a b id = Fun.swap b a id"
chaieb@29840
    19
  by (auto simp add: expand_fun_eq swap_def fun_upd_def)
chaieb@29840
    20
lemma swap_id_refl: "Fun.swap a a id = id" by simp
chaieb@29840
    21
lemma swap_id_sym: "Fun.swap a b id = Fun.swap b a id"
chaieb@29840
    22
  by (rule ext, simp add: swap_def)
chaieb@29840
    23
lemma swap_id_idempotent[simp]: "Fun.swap a b id o Fun.swap a b id = id"
chaieb@29840
    24
  by (rule ext, auto simp add: swap_def)
chaieb@29840
    25
chaieb@29840
    26
lemma inv_unique_comp: assumes fg: "f o g = id" and gf: "g o f = id"
chaieb@29840
    27
  shows "inv f = g"
chaieb@29840
    28
  using fg gf inv_equality[of g f] by (auto simp add: expand_fun_eq)
chaieb@29840
    29
chaieb@29840
    30
lemma inverse_swap_id: "inv (Fun.swap a b id) = Fun.swap a b id"
chaieb@29840
    31
  by (rule inv_unique_comp, simp_all)
chaieb@29840
    32
chaieb@29840
    33
lemma swap_id_eq: "Fun.swap a b id x = (if x = a then b else if x = b then a else x)"
chaieb@29840
    34
  by (simp add: swap_def)
chaieb@29840
    35
chaieb@29840
    36
(* ------------------------------------------------------------------------- *)
chaieb@29840
    37
(* Basic consequences of the definition.                                     *)
chaieb@29840
    38
(* ------------------------------------------------------------------------- *)
chaieb@29840
    39
chaieb@29840
    40
lemma permutes_in_image: "p permutes S \<Longrightarrow> p x \<in> S \<longleftrightarrow> x \<in> S"
chaieb@29840
    41
  unfolding permutes_def by metis
chaieb@29840
    42
chaieb@29840
    43
lemma permutes_image: assumes pS: "p permutes S" shows "p ` S = S"
chaieb@29840
    44
  using pS
huffman@30488
    45
  unfolding permutes_def
huffman@30488
    46
  apply -
huffman@30488
    47
  apply (rule set_ext)
chaieb@29840
    48
  apply (simp add: image_iff)
chaieb@29840
    49
  apply metis
chaieb@29840
    50
  done
chaieb@29840
    51
huffman@30488
    52
lemma permutes_inj: "p permutes S ==> inj p "
huffman@30488
    53
  unfolding permutes_def inj_on_def by blast
chaieb@29840
    54
huffman@30488
    55
lemma permutes_surj: "p permutes s ==> surj p"
huffman@30488
    56
  unfolding permutes_def surj_def by metis
chaieb@29840
    57
chaieb@29840
    58
lemma permutes_inv_o: assumes pS: "p permutes S"
chaieb@29840
    59
  shows " p o inv p = id"
chaieb@29840
    60
  and "inv p o p = id"
chaieb@29840
    61
  using permutes_inj[OF pS] permutes_surj[OF pS]
chaieb@29840
    62
  unfolding inj_iff[symmetric] surj_iff[symmetric] by blast+
chaieb@29840
    63
chaieb@29840
    64
huffman@30488
    65
lemma permutes_inverses:
chaieb@29840
    66
  fixes p :: "'a \<Rightarrow> 'a"
chaieb@29840
    67
  assumes pS: "p permutes S"
chaieb@29840
    68
  shows "p (inv p x) = x"
chaieb@29840
    69
  and "inv p (p x) = x"
chaieb@29840
    70
  using permutes_inv_o[OF pS, unfolded expand_fun_eq o_def] by auto
chaieb@29840
    71
chaieb@29840
    72
lemma permutes_subset: "p permutes S \<Longrightarrow> S \<subseteq> T ==> p permutes T"
chaieb@29840
    73
  unfolding permutes_def by blast
chaieb@29840
    74
chaieb@29840
    75
lemma permutes_empty[simp]: "p permutes {} \<longleftrightarrow> p = id"
huffman@30488
    76
  unfolding expand_fun_eq permutes_def apply simp by metis
chaieb@29840
    77
chaieb@29840
    78
lemma permutes_sing[simp]: "p permutes {a} \<longleftrightarrow> p = id"
chaieb@29840
    79
  unfolding expand_fun_eq permutes_def apply simp by metis
huffman@30488
    80
chaieb@29840
    81
lemma permutes_univ: "p permutes UNIV \<longleftrightarrow> (\<forall>y. \<exists>!x. p x = y)"
chaieb@29840
    82
  unfolding permutes_def by simp
chaieb@29840
    83
chaieb@29840
    84
lemma permutes_inv_eq: "p permutes S ==> inv p y = x \<longleftrightarrow> p x = y"
nipkow@33057
    85
  unfolding permutes_def inv_def apply auto
chaieb@29840
    86
  apply (erule allE[where x=y])
chaieb@29840
    87
  apply (erule allE[where x=y])
chaieb@29840
    88
  apply (rule someI_ex) apply blast
chaieb@29840
    89
  apply (rule some1_equality)
chaieb@29840
    90
  apply blast
chaieb@29840
    91
  apply blast
chaieb@29840
    92
  done
chaieb@29840
    93
chaieb@29840
    94
lemma permutes_swap_id: "a \<in> S \<Longrightarrow> b \<in> S ==> Fun.swap a b id permutes S"
nipkow@32988
    95
  unfolding permutes_def swap_def fun_upd_def  by auto metis
chaieb@29840
    96
nipkow@32988
    97
lemma permutes_superset:
nipkow@32988
    98
  "p permutes S \<Longrightarrow> (\<forall>x \<in> S - T. p x = x) \<Longrightarrow> p permutes T"
huffman@36361
    99
by (simp add: Ball_def permutes_def) metis
chaieb@29840
   100
chaieb@29840
   101
(* ------------------------------------------------------------------------- *)
chaieb@29840
   102
(* Group properties.                                                         *)
chaieb@29840
   103
(* ------------------------------------------------------------------------- *)
chaieb@29840
   104
huffman@30488
   105
lemma permutes_id: "id permutes S" unfolding permutes_def by simp
chaieb@29840
   106
chaieb@29840
   107
lemma permutes_compose: "p permutes S \<Longrightarrow> q permutes S ==> q o p permutes S"
chaieb@29840
   108
  unfolding permutes_def o_def by metis
chaieb@29840
   109
chaieb@29840
   110
lemma permutes_inv: assumes pS: "p permutes S" shows "inv p permutes S"
huffman@30488
   111
  using pS unfolding permutes_def permutes_inv_eq[OF pS] by metis
chaieb@29840
   112
chaieb@29840
   113
lemma permutes_inv_inv: assumes pS: "p permutes S" shows "inv (inv p) = p"
chaieb@29840
   114
  unfolding expand_fun_eq permutes_inv_eq[OF pS] permutes_inv_eq[OF permutes_inv[OF pS]]
chaieb@29840
   115
  by blast
chaieb@29840
   116
chaieb@29840
   117
(* ------------------------------------------------------------------------- *)
chaieb@29840
   118
(* The number of permutations on a finite set.                               *)
chaieb@29840
   119
(* ------------------------------------------------------------------------- *)
chaieb@29840
   120
huffman@30488
   121
lemma permutes_insert_lemma:
chaieb@29840
   122
  assumes pS: "p permutes (insert a S)"
chaieb@29840
   123
  shows "Fun.swap a (p a) id o p permutes S"
chaieb@29840
   124
  apply (rule permutes_superset[where S = "insert a S"])
chaieb@29840
   125
  apply (rule permutes_compose[OF pS])
chaieb@29840
   126
  apply (rule permutes_swap_id, simp)
chaieb@29840
   127
  using permutes_in_image[OF pS, of a] apply simp
huffman@36361
   128
  apply (auto simp add: Ball_def swap_def)
chaieb@29840
   129
  done
chaieb@29840
   130
chaieb@29840
   131
lemma permutes_insert: "{p. p permutes (insert a S)} =
chaieb@29840
   132
        (\<lambda>(b,p). Fun.swap a b id o p) ` {(b,p). b \<in> insert a S \<and> p \<in> {p. p permutes S}}"
chaieb@29840
   133
proof-
chaieb@29840
   134
huffman@30488
   135
  {fix p
chaieb@29840
   136
    {assume pS: "p permutes insert a S"
chaieb@29840
   137
      let ?b = "p a"
chaieb@29840
   138
      let ?q = "Fun.swap a (p a) id o p"
huffman@30488
   139
      have th0: "p = Fun.swap a ?b id o ?q" unfolding expand_fun_eq o_assoc by simp
huffman@30488
   140
      have th1: "?b \<in> insert a S " unfolding permutes_in_image[OF pS] by simp
chaieb@29840
   141
      from permutes_insert_lemma[OF pS] th0 th1
chaieb@29840
   142
      have "\<exists> b q. p = Fun.swap a b id o q \<and> b \<in> insert a S \<and> q permutes S" by blast}
chaieb@29840
   143
    moreover
chaieb@29840
   144
    {fix b q assume bq: "p = Fun.swap a b id o q" "b \<in> insert a S" "q permutes S"
huffman@30488
   145
      from permutes_subset[OF bq(3), of "insert a S"]
chaieb@29840
   146
      have qS: "q permutes insert a S" by auto
chaieb@29840
   147
      have aS: "a \<in> insert a S" by simp
chaieb@29840
   148
      from bq(1) permutes_compose[OF qS permutes_swap_id[OF aS bq(2)]]
chaieb@29840
   149
      have "p permutes insert a S"  by simp }
chaieb@29840
   150
    ultimately have "p permutes insert a S \<longleftrightarrow> (\<exists> b q. p = Fun.swap a b id o q \<and> b \<in> insert a S \<and> q permutes S)" by blast}
chaieb@29840
   151
  thus ?thesis by auto
chaieb@29840
   152
qed
chaieb@29840
   153
hoelzl@33715
   154
lemma card_permutations: assumes Sn: "card S = n" and fS: "finite S"
hoelzl@33715
   155
  shows "card {p. p permutes S} = fact n"
hoelzl@33715
   156
using fS Sn proof (induct arbitrary: n)
huffman@36361
   157
  case empty thus ?case by simp
hoelzl@33715
   158
next
hoelzl@33715
   159
  case (insert x F)
hoelzl@33715
   160
  { fix n assume H0: "card (insert x F) = n"
hoelzl@33715
   161
    let ?xF = "{p. p permutes insert x F}"
hoelzl@33715
   162
    let ?pF = "{p. p permutes F}"
hoelzl@33715
   163
    let ?pF' = "{(b, p). b \<in> insert x F \<and> p \<in> ?pF}"
hoelzl@33715
   164
    let ?g = "(\<lambda>(b, p). Fun.swap x b id \<circ> p)"
hoelzl@33715
   165
    from permutes_insert[of x F]
hoelzl@33715
   166
    have xfgpF': "?xF = ?g ` ?pF'" .
hoelzl@33715
   167
    have Fs: "card F = n - 1" using `x \<notin> F` H0 `finite F` by auto
hoelzl@33715
   168
    from insert.hyps Fs have pFs: "card ?pF = fact (n - 1)" using `finite F` by auto
hoelzl@33715
   169
    moreover hence "finite ?pF" using fact_gt_zero_nat by (auto intro: card_ge_0_finite)
hoelzl@33715
   170
    ultimately have pF'f: "finite ?pF'" using H0 `finite F`
hoelzl@33715
   171
      apply (simp only: Collect_split Collect_mem_eq)
hoelzl@33715
   172
      apply (rule finite_cartesian_product)
hoelzl@33715
   173
      apply simp_all
hoelzl@33715
   174
      done
chaieb@29840
   175
hoelzl@33715
   176
    have ginj: "inj_on ?g ?pF'"
hoelzl@33715
   177
    proof-
hoelzl@33715
   178
      {
hoelzl@33715
   179
        fix b p c q assume bp: "(b,p) \<in> ?pF'" and cq: "(c,q) \<in> ?pF'"
hoelzl@33715
   180
          and eq: "?g (b,p) = ?g (c,q)"
hoelzl@33715
   181
        from bp cq have ths: "b \<in> insert x F" "c \<in> insert x F" "x \<in> insert x F" "p permutes F" "q permutes F" by auto
hoelzl@33715
   182
        from ths(4) `x \<notin> F` eq have "b = ?g (b,p) x" unfolding permutes_def
hoelzl@33715
   183
          by (auto simp add: swap_def fun_upd_def expand_fun_eq)
hoelzl@33715
   184
        also have "\<dots> = ?g (c,q) x" using ths(5) `x \<notin> F` eq
hoelzl@33715
   185
          by (auto simp add: swap_def fun_upd_def expand_fun_eq)
hoelzl@33715
   186
        also have "\<dots> = c"using ths(5) `x \<notin> F` unfolding permutes_def
hoelzl@33715
   187
          by (auto simp add: swap_def fun_upd_def expand_fun_eq)
hoelzl@33715
   188
        finally have bc: "b = c" .
hoelzl@33715
   189
        hence "Fun.swap x b id = Fun.swap x c id" by simp
hoelzl@33715
   190
        with eq have "Fun.swap x b id o p = Fun.swap x b id o q" by simp
hoelzl@33715
   191
        hence "Fun.swap x b id o (Fun.swap x b id o p) = Fun.swap x b id o (Fun.swap x b id o q)" by simp
hoelzl@33715
   192
        hence "p = q" by (simp add: o_assoc)
hoelzl@33715
   193
        with bc have "(b,p) = (c,q)" by simp
hoelzl@33715
   194
      }
hoelzl@33715
   195
      thus ?thesis  unfolding inj_on_def by blast
hoelzl@33715
   196
    qed
hoelzl@33715
   197
    from `x \<notin> F` H0 have n0: "n \<noteq> 0 " using `finite F` by auto
hoelzl@33715
   198
    hence "\<exists>m. n = Suc m" by arith
hoelzl@33715
   199
    then obtain m where n[simp]: "n = Suc m" by blast
hoelzl@33715
   200
    from pFs H0 have xFc: "card ?xF = fact n"
hoelzl@33715
   201
      unfolding xfgpF' card_image[OF ginj] using `finite F` `finite ?pF`
hoelzl@33715
   202
      apply (simp only: Collect_split Collect_mem_eq card_cartesian_product)
hoelzl@33715
   203
      by simp
hoelzl@33715
   204
    from finite_imageI[OF pF'f, of ?g] have xFf: "finite ?xF" unfolding xfgpF' by simp
hoelzl@33715
   205
    have "card ?xF = fact n"
hoelzl@33715
   206
      using xFf xFc unfolding xFf by blast
hoelzl@33715
   207
  }
hoelzl@33715
   208
  thus ?case using insert by simp
chaieb@29840
   209
qed
chaieb@29840
   210
hoelzl@33715
   211
lemma finite_permutations: assumes fS: "finite S" shows "finite {p. p permutes S}"
hoelzl@33715
   212
  using card_permutations[OF refl fS] fact_gt_zero_nat
hoelzl@33715
   213
  by (auto intro: card_ge_0_finite)
chaieb@29840
   214
chaieb@29840
   215
(* ------------------------------------------------------------------------- *)
chaieb@29840
   216
(* Permutations of index set for iterated operations.                        *)
chaieb@29840
   217
(* ------------------------------------------------------------------------- *)
chaieb@29840
   218
huffman@30488
   219
lemma (in ab_semigroup_mult) fold_image_permute: assumes fS: "finite S" and pS: "p permutes S"
chaieb@29840
   220
  shows "fold_image times f z S = fold_image times (f o p) z S"
chaieb@29840
   221
  using fold_image_reindex[OF fS subset_inj_on[OF permutes_inj[OF pS], of S, simplified], of f z]
chaieb@29840
   222
  unfolding permutes_image[OF pS] .
huffman@30488
   223
lemma (in ab_semigroup_add) fold_image_permute: assumes fS: "finite S" and pS: "p permutes S"
chaieb@29840
   224
  shows "fold_image plus f z S = fold_image plus (f o p) z S"
chaieb@29840
   225
proof-
chaieb@29840
   226
  interpret ab_semigroup_mult plus apply unfold_locales apply (simp add: add_assoc)
chaieb@29840
   227
    apply (simp add: add_commute) done
chaieb@29840
   228
  from fold_image_reindex[OF fS subset_inj_on[OF permutes_inj[OF pS], of S, simplified], of f z]
chaieb@29840
   229
  show ?thesis
chaieb@29840
   230
  unfolding permutes_image[OF pS] .
chaieb@29840
   231
qed
chaieb@29840
   232
huffman@30488
   233
lemma setsum_permute: assumes pS: "p permutes S"
chaieb@29840
   234
  shows "setsum f S = setsum (f o p) S"
chaieb@29840
   235
  unfolding setsum_def using fold_image_permute[of S p f 0] pS by clarsimp
chaieb@29840
   236
huffman@30488
   237
lemma setsum_permute_natseg:assumes pS: "p permutes {m .. n}"
chaieb@29840
   238
  shows "setsum f {m .. n} = setsum (f o p) {m .. n}"
huffman@30488
   239
  using setsum_permute[OF pS, of f ] pS by blast
chaieb@29840
   240
huffman@30488
   241
lemma setprod_permute: assumes pS: "p permutes S"
chaieb@29840
   242
  shows "setprod f S = setprod (f o p) S"
huffman@30488
   243
  unfolding setprod_def
chaieb@29840
   244
  using ab_semigroup_mult_class.fold_image_permute[of S p f 1] pS by clarsimp
chaieb@29840
   245
huffman@30488
   246
lemma setprod_permute_natseg:assumes pS: "p permutes {m .. n}"
chaieb@29840
   247
  shows "setprod f {m .. n} = setprod (f o p) {m .. n}"
huffman@30488
   248
  using setprod_permute[OF pS, of f ] pS by blast
chaieb@29840
   249
chaieb@29840
   250
(* ------------------------------------------------------------------------- *)
chaieb@29840
   251
(* Various combinations of transpositions with 2, 1 and 0 common elements.   *)
chaieb@29840
   252
(* ------------------------------------------------------------------------- *)
chaieb@29840
   253
chaieb@29840
   254
lemma swap_id_common:" a \<noteq> c \<Longrightarrow> b \<noteq> c \<Longrightarrow>  Fun.swap a b id o Fun.swap a c id = Fun.swap b c id o Fun.swap a b id" by (simp add: expand_fun_eq swap_def)
chaieb@29840
   255
chaieb@29840
   256
lemma swap_id_common': "~(a = b) \<Longrightarrow> ~(a = c) \<Longrightarrow> Fun.swap a c id o Fun.swap b c id = Fun.swap b c id o Fun.swap a b id" by (simp add: expand_fun_eq swap_def)
chaieb@29840
   257
chaieb@29840
   258
lemma swap_id_independent: "~(a = c) \<Longrightarrow> ~(a = d) \<Longrightarrow> ~(b = c) \<Longrightarrow> ~(b = d) ==> Fun.swap a b id o Fun.swap c d id = Fun.swap c d id o Fun.swap a b id"
chaieb@29840
   259
  by (simp add: swap_def expand_fun_eq)
chaieb@29840
   260
chaieb@29840
   261
(* ------------------------------------------------------------------------- *)
chaieb@29840
   262
(* Permutations as transposition sequences.                                  *)
chaieb@29840
   263
(* ------------------------------------------------------------------------- *)
chaieb@29840
   264
chaieb@29840
   265
chaieb@29840
   266
inductive swapidseq :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> bool" where
chaieb@29840
   267
  id[simp]: "swapidseq 0 id"
chaieb@29840
   268
| comp_Suc: "swapidseq n p \<Longrightarrow> a \<noteq> b \<Longrightarrow> swapidseq (Suc n) (Fun.swap a b id o p)"
chaieb@29840
   269
chaieb@29840
   270
declare id[unfolded id_def, simp]
chaieb@29840
   271
definition "permutation p \<longleftrightarrow> (\<exists>n. swapidseq n p)"
chaieb@29840
   272
chaieb@29840
   273
(* ------------------------------------------------------------------------- *)
chaieb@29840
   274
(* Some closure properties of the set of permutations, with lengths.         *)
chaieb@29840
   275
(* ------------------------------------------------------------------------- *)
chaieb@29840
   276
chaieb@29840
   277
lemma permutation_id[simp]: "permutation id"unfolding permutation_def
chaieb@29840
   278
  by (rule exI[where x=0], simp)
chaieb@29840
   279
declare permutation_id[unfolded id_def, simp]
chaieb@29840
   280
chaieb@29840
   281
lemma swapidseq_swap: "swapidseq (if a = b then 0 else 1) (Fun.swap a b id)"
chaieb@29840
   282
  apply clarsimp
chaieb@29840
   283
  using comp_Suc[of 0 id a b] by simp
chaieb@29840
   284
chaieb@29840
   285
lemma permutation_swap_id: "permutation (Fun.swap a b id)"
chaieb@29840
   286
  apply (cases "a=b", simp_all)
huffman@30488
   287
  unfolding permutation_def using swapidseq_swap[of a b] by blast
chaieb@29840
   288
chaieb@29840
   289
lemma swapidseq_comp_add: "swapidseq n p \<Longrightarrow> swapidseq m q ==> swapidseq (n + m) (p o q)"
chaieb@29840
   290
  proof (induct n p arbitrary: m q rule: swapidseq.induct)
chaieb@29840
   291
    case (id m q) thus ?case by simp
chaieb@29840
   292
  next
huffman@30488
   293
    case (comp_Suc n p a b m q)
chaieb@29840
   294
    have th: "Suc n + m = Suc (n + m)" by arith
huffman@30488
   295
    show ?case unfolding th o_assoc[symmetric]
huffman@30488
   296
      apply (rule swapidseq.comp_Suc) using comp_Suc.hyps(2)[OF comp_Suc.prems]  comp_Suc.hyps(3) by blast+
chaieb@29840
   297
qed
chaieb@29840
   298
chaieb@29840
   299
lemma permutation_compose: "permutation p \<Longrightarrow> permutation q ==> permutation(p o q)"
chaieb@29840
   300
  unfolding permutation_def using swapidseq_comp_add[of _ p _ q] by metis
chaieb@29840
   301
chaieb@29840
   302
lemma swapidseq_endswap: "swapidseq n p \<Longrightarrow> a \<noteq> b ==> swapidseq (Suc n) (p o Fun.swap a b id)"
chaieb@29840
   303
  apply (induct n p rule: swapidseq.induct)
chaieb@29840
   304
  using swapidseq_swap[of a b]
chaieb@29840
   305
  by (auto simp add: o_assoc[symmetric] intro: swapidseq.comp_Suc)
chaieb@29840
   306
chaieb@29840
   307
lemma swapidseq_inverse_exists: "swapidseq n p ==> \<exists>q. swapidseq n q \<and> p o q = id \<and> q o p = id"
chaieb@29840
   308
proof(induct n p rule: swapidseq.induct)
chaieb@29840
   309
  case id  thus ?case by (rule exI[where x=id], simp)
huffman@30488
   310
next
chaieb@29840
   311
  case (comp_Suc n p a b)
chaieb@29840
   312
  from comp_Suc.hyps obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id" by blast
chaieb@29840
   313
  let ?q = "q o Fun.swap a b id"
chaieb@29840
   314
  note H = comp_Suc.hyps
chaieb@29840
   315
  from swapidseq_swap[of a b] H(3)  have th0: "swapidseq 1 (Fun.swap a b id)" by simp
huffman@30488
   316
  from swapidseq_comp_add[OF q(1) th0] have th1:"swapidseq (Suc n) ?q" by simp
chaieb@29840
   317
  have "Fun.swap a b id o p o ?q = Fun.swap a b id o (p o q) o Fun.swap a b id" by (simp add: o_assoc)
chaieb@29840
   318
  also have "\<dots> = id" by (simp add: q(2))
chaieb@29840
   319
  finally have th2: "Fun.swap a b id o p o ?q = id" .
huffman@30488
   320
  have "?q \<circ> (Fun.swap a b id \<circ> p) = q \<circ> (Fun.swap a b id o Fun.swap a b id) \<circ> p" by (simp only: o_assoc)
chaieb@29840
   321
  hence "?q \<circ> (Fun.swap a b id \<circ> p) = id" by (simp add: q(3))
chaieb@29840
   322
  with th1 th2 show ?case by blast
chaieb@29840
   323
qed
chaieb@29840
   324
chaieb@29840
   325
chaieb@29840
   326
lemma swapidseq_inverse: assumes H: "swapidseq n p" shows "swapidseq n (inv p)"
chaieb@29840
   327
  using swapidseq_inverse_exists[OF H] inv_unique_comp[of p] by auto
chaieb@29840
   328
chaieb@29840
   329
lemma permutation_inverse: "permutation p ==> permutation (inv p)"
chaieb@29840
   330
  using permutation_def swapidseq_inverse by blast
chaieb@29840
   331
chaieb@29840
   332
(* ------------------------------------------------------------------------- *)
chaieb@29840
   333
(* The identity map only has even transposition sequences.                   *)
chaieb@29840
   334
(* ------------------------------------------------------------------------- *)
chaieb@29840
   335
chaieb@29840
   336
lemma symmetry_lemma:"(\<And>a b c d. P a b c d ==> P a b d c) \<Longrightarrow>
chaieb@29840
   337
   (\<And>a b c d. a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow> (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) ==> P a b c d)
chaieb@29840
   338
   ==> (\<And>a b c d. a \<noteq> b --> c \<noteq> d \<longrightarrow>  P a b c d)" by metis
chaieb@29840
   339
huffman@30488
   340
lemma swap_general: "a \<noteq> b \<Longrightarrow> c \<noteq> d \<Longrightarrow> Fun.swap a b id o Fun.swap c d id = id \<or>
huffman@30488
   341
  (\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and> Fun.swap a b id o Fun.swap c d id = Fun.swap x y id o Fun.swap a z id)"
chaieb@29840
   342
proof-
chaieb@29840
   343
  assume H: "a\<noteq>b" "c\<noteq>d"
huffman@30488
   344
have "a \<noteq> b \<longrightarrow> c \<noteq> d \<longrightarrow>
huffman@30488
   345
(  Fun.swap a b id o Fun.swap c d id = id \<or>
huffman@30488
   346
  (\<exists>x y z. x \<noteq> a \<and> y \<noteq> a \<and> z \<noteq> a \<and> x \<noteq> y \<and> Fun.swap a b id o Fun.swap c d id = Fun.swap x y id o Fun.swap a z id))"
chaieb@29840
   347
  apply (rule symmetry_lemma[where a=a and b=b and c=c and d=d])
huffman@30488
   348
  apply (simp_all only: swapid_sym)
chaieb@29840
   349
  apply (case_tac "a = c \<and> b = d", clarsimp simp only: swapid_sym swap_id_idempotent)
chaieb@29840
   350
  apply (case_tac "a = c \<and> b \<noteq> d")
chaieb@29840
   351
  apply (rule disjI2)
chaieb@29840
   352
  apply (rule_tac x="b" in exI)
chaieb@29840
   353
  apply (rule_tac x="d" in exI)
chaieb@29840
   354
  apply (rule_tac x="b" in exI)
chaieb@29840
   355
  apply (clarsimp simp add: expand_fun_eq swap_def)
chaieb@29840
   356
  apply (case_tac "a \<noteq> c \<and> b = d")
chaieb@29840
   357
  apply (rule disjI2)
chaieb@29840
   358
  apply (rule_tac x="c" in exI)
chaieb@29840
   359
  apply (rule_tac x="d" in exI)
chaieb@29840
   360
  apply (rule_tac x="c" in exI)
chaieb@29840
   361
  apply (clarsimp simp add: expand_fun_eq swap_def)
chaieb@29840
   362
  apply (rule disjI2)
chaieb@29840
   363
  apply (rule_tac x="c" in exI)
chaieb@29840
   364
  apply (rule_tac x="d" in exI)
chaieb@29840
   365
  apply (rule_tac x="b" in exI)
chaieb@29840
   366
  apply (clarsimp simp add: expand_fun_eq swap_def)
chaieb@29840
   367
  done
huffman@30488
   368
with H show ?thesis by metis
chaieb@29840
   369
qed
chaieb@29840
   370
chaieb@29840
   371
lemma swapidseq_id_iff[simp]: "swapidseq 0 p \<longleftrightarrow> p = id"
chaieb@29840
   372
  using swapidseq.cases[of 0 p "p = id"]
chaieb@29840
   373
  by auto
chaieb@29840
   374
chaieb@29840
   375
lemma swapidseq_cases: "swapidseq n p \<longleftrightarrow> (n=0 \<and> p = id \<or> (\<exists>a b q m. n = Suc m \<and> p = Fun.swap a b id o q \<and> swapidseq m q \<and> a\<noteq> b))"
chaieb@29840
   376
  apply (rule iffI)
chaieb@29840
   377
  apply (erule swapidseq.cases[of n p])
chaieb@29840
   378
  apply simp
chaieb@29840
   379
  apply (rule disjI2)
chaieb@29840
   380
  apply (rule_tac x= "a" in exI)
chaieb@29840
   381
  apply (rule_tac x= "b" in exI)
chaieb@29840
   382
  apply (rule_tac x= "pa" in exI)
chaieb@29840
   383
  apply (rule_tac x= "na" in exI)
chaieb@29840
   384
  apply simp
chaieb@29840
   385
  apply auto
chaieb@29840
   386
  apply (rule comp_Suc, simp_all)
chaieb@29840
   387
  done
chaieb@29840
   388
lemma fixing_swapidseq_decrease:
chaieb@29840
   389
  assumes spn: "swapidseq n p" and ab: "a\<noteq>b" and pa: "(Fun.swap a b id o p) a = a"
chaieb@29840
   390
  shows "n \<noteq> 0 \<and> swapidseq (n - 1) (Fun.swap a b id o p)"
chaieb@29840
   391
  using spn ab pa
chaieb@29840
   392
proof(induct n arbitrary: p a b)
chaieb@29840
   393
  case 0 thus ?case by (auto simp add: swap_def fun_upd_def)
chaieb@29840
   394
next
chaieb@29840
   395
  case (Suc n p a b)
chaieb@29840
   396
  from Suc.prems(1) swapidseq_cases[of "Suc n" p] obtain
chaieb@29840
   397
    c d q m where cdqm: "Suc n = Suc m" "p = Fun.swap c d id o q" "swapidseq m q" "c \<noteq> d" "n = m"
chaieb@29840
   398
    by auto
chaieb@29840
   399
  {assume H: "Fun.swap a b id o Fun.swap c d id = id"
huffman@30488
   400
huffman@30488
   401
    have ?case apply (simp only: cdqm o_assoc H)
chaieb@29840
   402
      by (simp add: cdqm)}
chaieb@29840
   403
  moreover
chaieb@29840
   404
  { fix x y z
huffman@30488
   405
    assume H: "x\<noteq>a" "y\<noteq>a" "z \<noteq>a" "x \<noteq>y"
chaieb@29840
   406
      "Fun.swap a b id o Fun.swap c d id = Fun.swap x y id o Fun.swap a z id"
chaieb@29840
   407
    from H have az: "a \<noteq> z" by simp
chaieb@29840
   408
chaieb@29840
   409
    {fix h have "(Fun.swap x y id o h) a = a \<longleftrightarrow> h a = a"
chaieb@29840
   410
      using H by (simp add: swap_def)}
chaieb@29840
   411
    note th3 = this
chaieb@29840
   412
    from cdqm(2) have "Fun.swap a b id o p = Fun.swap a b id o (Fun.swap c d id o q)" by simp
chaieb@29840
   413
    hence "Fun.swap a b id o p = Fun.swap x y id o (Fun.swap a z id o q)" by (simp add: o_assoc H)
chaieb@29840
   414
    hence "(Fun.swap a b id o p) a = (Fun.swap x y id o (Fun.swap a z id o q)) a" by simp
chaieb@29840
   415
    hence "(Fun.swap x y id o (Fun.swap a z id o q)) a  = a" unfolding Suc by metis
chaieb@29840
   416
    hence th1: "(Fun.swap a z id o q) a = a" unfolding th3 .
chaieb@29840
   417
    from Suc.hyps[OF cdqm(3)[ unfolded cdqm(5)[symmetric]] az th1]
chaieb@29840
   418
    have th2: "swapidseq (n - 1) (Fun.swap a z id o q)" "n \<noteq> 0" by blast+
huffman@30488
   419
    have th: "Suc n - 1 = Suc (n - 1)" using th2(2) by auto
chaieb@29840
   420
    have ?case unfolding cdqm(2) H o_assoc th
chaieb@29840
   421
      apply (simp only: Suc_not_Zero simp_thms o_assoc[symmetric])
chaieb@29840
   422
      apply (rule comp_Suc)
chaieb@29840
   423
      using th2 H apply blast+
chaieb@29840
   424
      done}
huffman@30488
   425
  ultimately show ?case using swap_general[OF Suc.prems(2) cdqm(4)] by metis
chaieb@29840
   426
qed
chaieb@29840
   427
huffman@30488
   428
lemma swapidseq_identity_even:
chaieb@29840
   429
  assumes "swapidseq n (id :: 'a \<Rightarrow> 'a)" shows "even n"
chaieb@29840
   430
  using `swapidseq n id`
chaieb@29840
   431
proof(induct n rule: nat_less_induct)
chaieb@29840
   432
  fix n
chaieb@29840
   433
  assume H: "\<forall>m<n. swapidseq m (id::'a \<Rightarrow> 'a) \<longrightarrow> even m" "swapidseq n (id :: 'a \<Rightarrow> 'a)"
huffman@30488
   434
  {assume "n = 0" hence "even n" by arith}
huffman@30488
   435
  moreover
chaieb@29840
   436
  {fix a b :: 'a and q m
chaieb@29840
   437
    assume h: "n = Suc m" "(id :: 'a \<Rightarrow> 'a) = Fun.swap a b id \<circ> q" "swapidseq m q" "a \<noteq> b"
chaieb@29840
   438
    from fixing_swapidseq_decrease[OF h(3,4), unfolded h(2)[symmetric]]
chaieb@29840
   439
    have m: "m \<noteq> 0" "swapidseq (m - 1) (id :: 'a \<Rightarrow> 'a)" by auto
chaieb@29840
   440
    from h m have mn: "m - 1 < n" by arith
chaieb@29840
   441
    from H(1)[rule_format, OF mn m(2)] h(1) m(1) have "even n" apply arith done}
chaieb@29840
   442
  ultimately show "even n" using H(2)[unfolded swapidseq_cases[of n id]] by auto
chaieb@29840
   443
qed
chaieb@29840
   444
chaieb@29840
   445
(* ------------------------------------------------------------------------- *)
chaieb@29840
   446
(* Therefore we have a welldefined notion of parity.                         *)
chaieb@29840
   447
(* ------------------------------------------------------------------------- *)
chaieb@29840
   448
chaieb@29840
   449
definition "evenperm p = even (SOME n. swapidseq n p)"
chaieb@29840
   450
huffman@30488
   451
lemma swapidseq_even_even: assumes
chaieb@29840
   452
  m: "swapidseq m p" and n: "swapidseq n p"
chaieb@29840
   453
  shows "even m \<longleftrightarrow> even n"
chaieb@29840
   454
proof-
chaieb@29840
   455
  from swapidseq_inverse_exists[OF n]
chaieb@29840
   456
  obtain q where q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id" by blast
huffman@30488
   457
chaieb@29840
   458
  from swapidseq_identity_even[OF swapidseq_comp_add[OF m q(1), unfolded q]]
chaieb@29840
   459
  show ?thesis by arith
chaieb@29840
   460
qed
chaieb@29840
   461
chaieb@29840
   462
lemma evenperm_unique: assumes p: "swapidseq n p" and n:"even n = b"
chaieb@29840
   463
  shows "evenperm p = b"
chaieb@29840
   464
  unfolding n[symmetric] evenperm_def
chaieb@29840
   465
  apply (rule swapidseq_even_even[where p = p])
chaieb@29840
   466
  apply (rule someI[where x = n])
chaieb@29840
   467
  using p by blast+
chaieb@29840
   468
chaieb@29840
   469
(* ------------------------------------------------------------------------- *)
chaieb@29840
   470
(* And it has the expected composition properties.                           *)
chaieb@29840
   471
(* ------------------------------------------------------------------------- *)
chaieb@29840
   472
chaieb@29840
   473
lemma evenperm_id[simp]: "evenperm id = True"
chaieb@29840
   474
  apply (rule evenperm_unique[where n = 0]) by simp_all
chaieb@29840
   475
chaieb@29840
   476
lemma evenperm_swap: "evenperm (Fun.swap a b id) = (a = b)"
chaieb@29840
   477
apply (rule evenperm_unique[where n="if a = b then 0 else 1"])
chaieb@29840
   478
by (simp_all add: swapidseq_swap)
chaieb@29840
   479
huffman@30488
   480
lemma evenperm_comp:
chaieb@29840
   481
  assumes p: "permutation p" and q:"permutation q"
chaieb@29840
   482
  shows "evenperm (p o q) = (evenperm p = evenperm q)"
chaieb@29840
   483
proof-
huffman@30488
   484
  from p q obtain
huffman@30488
   485
    n m where n: "swapidseq n p" and m: "swapidseq m q"
chaieb@29840
   486
    unfolding permutation_def by blast
chaieb@29840
   487
  note nm =  swapidseq_comp_add[OF n m]
chaieb@29840
   488
  have th: "even (n + m) = (even n \<longleftrightarrow> even m)" by arith
chaieb@29840
   489
  from evenperm_unique[OF n refl] evenperm_unique[OF m refl]
chaieb@29840
   490
    evenperm_unique[OF nm th]
chaieb@29840
   491
  show ?thesis by blast
chaieb@29840
   492
qed
chaieb@29840
   493
chaieb@29840
   494
lemma evenperm_inv: assumes p: "permutation p"
chaieb@29840
   495
  shows "evenperm (inv p) = evenperm p"
chaieb@29840
   496
proof-
chaieb@29840
   497
  from p obtain n where n: "swapidseq n p" unfolding permutation_def by blast
chaieb@29840
   498
  from evenperm_unique[OF swapidseq_inverse[OF n] evenperm_unique[OF n refl, symmetric]]
chaieb@29840
   499
  show ?thesis .
chaieb@29840
   500
qed
chaieb@29840
   501
chaieb@29840
   502
(* ------------------------------------------------------------------------- *)
chaieb@29840
   503
(* A more abstract characterization of permutations.                         *)
chaieb@29840
   504
(* ------------------------------------------------------------------------- *)
chaieb@29840
   505
chaieb@29840
   506
chaieb@29840
   507
lemma bij_iff: "bij f \<longleftrightarrow> (\<forall>x. \<exists>!y. f y = x)"
chaieb@29840
   508
  unfolding bij_def inj_on_def surj_def
chaieb@29840
   509
  apply auto
chaieb@29840
   510
  apply metis
chaieb@29840
   511
  apply metis
chaieb@29840
   512
  done
chaieb@29840
   513
huffman@30488
   514
lemma permutation_bijective:
huffman@30488
   515
  assumes p: "permutation p"
chaieb@29840
   516
  shows "bij p"
chaieb@29840
   517
proof-
chaieb@29840
   518
  from p obtain n where n: "swapidseq n p" unfolding permutation_def by blast
huffman@30488
   519
  from swapidseq_inverse_exists[OF n] obtain q where
chaieb@29840
   520
    q: "swapidseq n q" "p \<circ> q = id" "q \<circ> p = id" by blast
chaieb@29840
   521
  thus ?thesis unfolding bij_iff  apply (auto simp add: expand_fun_eq) apply metis done
huffman@30488
   522
qed
chaieb@29840
   523
chaieb@29840
   524
lemma permutation_finite_support: assumes p: "permutation p"
chaieb@29840
   525
  shows "finite {x. p x \<noteq> x}"
chaieb@29840
   526
proof-
chaieb@29840
   527
  from p obtain n where n: "swapidseq n p" unfolding permutation_def by blast
chaieb@29840
   528
  from n show ?thesis
chaieb@29840
   529
  proof(induct n p rule: swapidseq.induct)
chaieb@29840
   530
    case id thus ?case by simp
chaieb@29840
   531
  next
chaieb@29840
   532
    case (comp_Suc n p a b)
chaieb@29840
   533
    let ?S = "insert a (insert b {x. p x \<noteq> x})"
chaieb@29840
   534
    from comp_Suc.hyps(2) have fS: "finite ?S" by simp
chaieb@29840
   535
    from `a \<noteq> b` have th: "{x. (Fun.swap a b id o p) x \<noteq> x} \<subseteq> ?S"
chaieb@29840
   536
      by (auto simp add: swap_def)
chaieb@29840
   537
    from finite_subset[OF th fS] show ?case  .
chaieb@29840
   538
qed
chaieb@29840
   539
qed
chaieb@29840
   540
chaieb@29840
   541
lemma bij_inv_eq_iff: "bij p ==> x = inv p y \<longleftrightarrow> p x = y"
chaieb@29840
   542
  using surj_f_inv_f[of p] inv_f_f[of f] by (auto simp add: bij_def)
chaieb@29840
   543
huffman@30488
   544
lemma bij_swap_comp:
chaieb@29840
   545
  assumes bp: "bij p" shows "Fun.swap a b id o p = Fun.swap (inv p a) (inv p b) p"
chaieb@29840
   546
  using surj_f_inv_f[OF bij_is_surj[OF bp]]
chaieb@29840
   547
  by (simp add: expand_fun_eq swap_def bij_inv_eq_iff[OF bp])
chaieb@29840
   548
chaieb@29840
   549
lemma bij_swap_ompose_bij: "bij p \<Longrightarrow> bij (Fun.swap a b id o p)"
chaieb@29840
   550
proof-
chaieb@29840
   551
  assume H: "bij p"
huffman@30488
   552
  show ?thesis
chaieb@29840
   553
    unfolding bij_swap_comp[OF H] bij_swap_iff
chaieb@29840
   554
    using H .
chaieb@29840
   555
qed
chaieb@29840
   556
huffman@30488
   557
lemma permutation_lemma:
chaieb@29840
   558
  assumes fS: "finite S" and p: "bij p" and pS: "\<forall>x. x\<notin> S \<longrightarrow> p x = x"
chaieb@29840
   559
  shows "permutation p"
chaieb@29840
   560
using fS p pS
chaieb@29840
   561
proof(induct S arbitrary: p rule: finite_induct)
chaieb@29840
   562
  case (empty p) thus ?case by simp
chaieb@29840
   563
next
chaieb@29840
   564
  case (insert a F p)
chaieb@29840
   565
  let ?r = "Fun.swap a (p a) id o p"
chaieb@29840
   566
  let ?q = "Fun.swap a (p a) id o ?r "
chaieb@29840
   567
  have raa: "?r a = a" by (simp add: swap_def)
chaieb@29840
   568
  from bij_swap_ompose_bij[OF insert(4)]
huffman@30488
   569
  have br: "bij ?r"  .
huffman@30488
   570
huffman@30488
   571
  from insert raa have th: "\<forall>x. x \<notin> F \<longrightarrow> ?r x = x"
chaieb@29840
   572
    apply (clarsimp simp add: swap_def)
chaieb@29840
   573
    apply (erule_tac x="x" in allE)
chaieb@29840
   574
    apply auto
chaieb@29840
   575
    unfolding bij_iff apply metis
chaieb@29840
   576
    done
chaieb@29840
   577
  from insert(3)[OF br th]
chaieb@29840
   578
  have rp: "permutation ?r" .
chaieb@29840
   579
  have "permutation ?q" by (simp add: permutation_compose permutation_swap_id rp)
chaieb@29840
   580
  thus ?case by (simp add: o_assoc)
chaieb@29840
   581
qed
chaieb@29840
   582
huffman@30488
   583
lemma permutation: "permutation p \<longleftrightarrow> bij p \<and> finite {x. p x \<noteq> x}"
chaieb@29840
   584
  (is "?lhs \<longleftrightarrow> ?b \<and> ?f")
chaieb@29840
   585
proof
chaieb@29840
   586
  assume p: ?lhs
chaieb@29840
   587
  from p permutation_bijective permutation_finite_support show "?b \<and> ?f" by auto
chaieb@29840
   588
next
chaieb@29840
   589
  assume bf: "?b \<and> ?f"
chaieb@29840
   590
  hence bf: "?f" "?b" by blast+
chaieb@29840
   591
  from permutation_lemma[OF bf] show ?lhs by blast
chaieb@29840
   592
qed
chaieb@29840
   593
chaieb@29840
   594
lemma permutation_inverse_works: assumes p: "permutation p"
chaieb@29840
   595
  shows "inv p o p = id" "p o inv p = id"
chaieb@29840
   596
using permutation_bijective[OF p] surj_iff bij_def inj_iff by auto
chaieb@29840
   597
chaieb@29840
   598
lemma permutation_inverse_compose:
chaieb@29840
   599
  assumes p: "permutation p" and q: "permutation q"
chaieb@29840
   600
  shows "inv (p o q) = inv q o inv p"
chaieb@29840
   601
proof-
chaieb@29840
   602
  note ps = permutation_inverse_works[OF p]
chaieb@29840
   603
  note qs = permutation_inverse_works[OF q]
chaieb@29840
   604
  have "p o q o (inv q o inv p) = p o (q o inv q) o inv p" by (simp add: o_assoc)
chaieb@29840
   605
  also have "\<dots> = id" by (simp add: ps qs)
chaieb@29840
   606
  finally have th0: "p o q o (inv q o inv p) = id" .
chaieb@29840
   607
  have "inv q o inv p o (p o q) = inv q o (inv p o p) o q" by (simp add: o_assoc)
chaieb@29840
   608
  also have "\<dots> = id" by (simp add: ps qs)
huffman@30488
   609
  finally have th1: "inv q o inv p o (p o q) = id" .
chaieb@29840
   610
  from inv_unique_comp[OF th0 th1] show ?thesis .
chaieb@29840
   611
qed
chaieb@29840
   612
chaieb@29840
   613
(* ------------------------------------------------------------------------- *)
chaieb@29840
   614
(* Relation to "permutes".                                                   *)
chaieb@29840
   615
(* ------------------------------------------------------------------------- *)
chaieb@29840
   616
chaieb@29840
   617
lemma permutation_permutes: "permutation p \<longleftrightarrow> (\<exists>S. finite S \<and> p permutes S)"
chaieb@29840
   618
unfolding permutation permutes_def bij_iff[symmetric]
chaieb@29840
   619
apply (rule iffI, clarify)
chaieb@29840
   620
apply (rule exI[where x="{x. p x \<noteq> x}"])
chaieb@29840
   621
apply simp
chaieb@29840
   622
apply clarsimp
chaieb@29840
   623
apply (rule_tac B="S" in finite_subset)
chaieb@29840
   624
apply auto
chaieb@29840
   625
done
chaieb@29840
   626
chaieb@29840
   627
(* ------------------------------------------------------------------------- *)
chaieb@29840
   628
(* Hence a sort of induction principle composing by swaps.                   *)
chaieb@29840
   629
(* ------------------------------------------------------------------------- *)
chaieb@29840
   630
chaieb@29840
   631
lemma permutes_induct: "finite S \<Longrightarrow>  P id  \<Longrightarrow> (\<And> a b p. a \<in> S \<Longrightarrow> b \<in> S \<Longrightarrow> P p \<Longrightarrow> P p \<Longrightarrow> permutation p ==> P (Fun.swap a b id o p))
chaieb@29840
   632
         ==> (\<And>p. p permutes S ==> P p)"
chaieb@29840
   633
proof(induct S rule: finite_induct)
chaieb@29840
   634
  case empty thus ?case by auto
huffman@30488
   635
next
chaieb@29840
   636
  case (insert x F p)
chaieb@29840
   637
  let ?r = "Fun.swap x (p x) id o p"
chaieb@29840
   638
  let ?q = "Fun.swap x (p x) id o ?r"
chaieb@29840
   639
  have qp: "?q = p" by (simp add: o_assoc)
chaieb@29840
   640
  from permutes_insert_lemma[OF insert.prems(3)] insert have Pr: "P ?r" by blast
huffman@30488
   641
  from permutes_in_image[OF insert.prems(3), of x]
chaieb@29840
   642
  have pxF: "p x \<in> insert x F" by simp
chaieb@29840
   643
  have xF: "x \<in> insert x F" by simp
chaieb@29840
   644
  have rp: "permutation ?r"
huffman@30488
   645
    unfolding permutation_permutes using insert.hyps(1)
chaieb@29840
   646
      permutes_insert_lemma[OF insert.prems(3)] by blast
huffman@30488
   647
  from insert.prems(2)[OF xF pxF Pr Pr rp]
huffman@30488
   648
  show ?case  unfolding qp .
chaieb@29840
   649
qed
chaieb@29840
   650
chaieb@29840
   651
(* ------------------------------------------------------------------------- *)
chaieb@29840
   652
(* Sign of a permutation as a real number.                                   *)
chaieb@29840
   653
(* ------------------------------------------------------------------------- *)
chaieb@29840
   654
chaieb@29840
   655
definition "sign p = (if evenperm p then (1::int) else -1)"
chaieb@29840
   656
huffman@30488
   657
lemma sign_nz: "sign p \<noteq> 0" by (simp add: sign_def)
chaieb@29840
   658
lemma sign_id: "sign id = 1" by (simp add: sign_def)
chaieb@29840
   659
lemma sign_inverse: "permutation p ==> sign (inv p) = sign p"
chaieb@29840
   660
  by (simp add: sign_def evenperm_inv)
chaieb@29840
   661
lemma sign_compose: "permutation p \<Longrightarrow> permutation q ==> sign (p o q) = sign(p) * sign(q)" by (simp add: sign_def evenperm_comp)
chaieb@29840
   662
lemma sign_swap_id: "sign (Fun.swap a b id) = (if a = b then 1 else -1)"
chaieb@29840
   663
  by (simp add: sign_def evenperm_swap)
chaieb@29840
   664
lemma sign_idempotent: "sign p * sign p = 1" by (simp add: sign_def)
chaieb@29840
   665
chaieb@29840
   666
(* ------------------------------------------------------------------------- *)
chaieb@29840
   667
(* More lemmas about permutations.                                           *)
chaieb@29840
   668
(* ------------------------------------------------------------------------- *)
chaieb@29840
   669
chaieb@29840
   670
lemma permutes_natset_le:
huffman@30037
   671
  assumes p: "p permutes (S::'a::wellorder set)" and le: "\<forall>i \<in> S.  p i <= i" shows "p = id"
chaieb@29840
   672
proof-
chaieb@29840
   673
  {fix n
huffman@30488
   674
    have "p n = n"
chaieb@29840
   675
      using p le
huffman@30037
   676
    proof(induct n arbitrary: S rule: less_induct)
huffman@30488
   677
      fix n S assume H: "\<And>m S. \<lbrakk>m < n; p permutes S; \<forall>i\<in>S. p i \<le> i\<rbrakk> \<Longrightarrow> p m = m"
wenzelm@32960
   678
        "p permutes S" "\<forall>i \<in>S. p i \<le> i"
chaieb@29840
   679
      {assume "n \<notin> S"
wenzelm@32960
   680
        with H(2) have "p n = n" unfolding permutes_def by metis}
chaieb@29840
   681
      moreover
chaieb@29840
   682
      {assume ns: "n \<in> S"
wenzelm@32960
   683
        from H(3)  ns have "p n < n \<or> p n = n" by auto
wenzelm@32960
   684
        moreover{assume h: "p n < n"
wenzelm@32960
   685
          from H h have "p (p n) = p n" by metis
wenzelm@32960
   686
          with permutes_inj[OF H(2)] have "p n = n" unfolding inj_on_def by blast
wenzelm@32960
   687
          with h have False by simp}
wenzelm@32960
   688
        ultimately have "p n = n" by blast }
chaieb@29840
   689
      ultimately show "p n = n"  by blast
chaieb@29840
   690
    qed}
chaieb@29840
   691
  thus ?thesis by (auto simp add: expand_fun_eq)
chaieb@29840
   692
qed
chaieb@29840
   693
chaieb@29840
   694
lemma permutes_natset_ge:
huffman@30037
   695
  assumes p: "p permutes (S::'a::wellorder set)" and le: "\<forall>i \<in> S.  p i \<ge> i" shows "p = id"
chaieb@29840
   696
proof-
chaieb@29840
   697
  {fix i assume i: "i \<in> S"
chaieb@29840
   698
    from i permutes_in_image[OF permutes_inv[OF p]] have "inv p i \<in> S" by simp
chaieb@29840
   699
    with le have "p (inv p i) \<ge> inv p i" by blast
chaieb@29840
   700
    with permutes_inverses[OF p] have "i \<ge> inv p i" by simp}
chaieb@29840
   701
  then have th: "\<forall>i\<in>S. inv p i \<le> i"  by blast
huffman@30488
   702
  from permutes_natset_le[OF permutes_inv[OF p] th]
chaieb@29840
   703
  have "inv p = inv id" by simp
huffman@30488
   704
  then show ?thesis
chaieb@29840
   705
    apply (subst permutes_inv_inv[OF p, symmetric])
chaieb@29840
   706
    apply (rule inv_unique_comp)
chaieb@29840
   707
    apply simp_all
chaieb@29840
   708
    done
chaieb@29840
   709
qed
chaieb@29840
   710
chaieb@29840
   711
lemma image_inverse_permutations: "{inv p |p. p permutes S} = {p. p permutes S}"
chaieb@29840
   712
apply (rule set_ext)
chaieb@29840
   713
apply auto
chaieb@29840
   714
  using permutes_inv_inv permutes_inv apply auto
chaieb@29840
   715
  apply (rule_tac x="inv x" in exI)
chaieb@29840
   716
  apply auto
chaieb@29840
   717
  done
chaieb@29840
   718
huffman@30488
   719
lemma image_compose_permutations_left:
chaieb@29840
   720
  assumes q: "q permutes S" shows "{q o p | p. p permutes S} = {p . p permutes S}"
chaieb@29840
   721
apply (rule set_ext)
chaieb@29840
   722
apply auto
chaieb@29840
   723
apply (rule permutes_compose)
chaieb@29840
   724
using q apply auto
chaieb@29840
   725
apply (rule_tac x = "inv q o x" in exI)
chaieb@29840
   726
by (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o)
chaieb@29840
   727
chaieb@29840
   728
lemma image_compose_permutations_right:
chaieb@29840
   729
  assumes q: "q permutes S"
chaieb@29840
   730
  shows "{p o q | p. p permutes S} = {p . p permutes S}"
chaieb@29840
   731
apply (rule set_ext)
chaieb@29840
   732
apply auto
chaieb@29840
   733
apply (rule permutes_compose)
chaieb@29840
   734
using q apply auto
chaieb@29840
   735
apply (rule_tac x = "x o inv q" in exI)
chaieb@29840
   736
by (simp add: o_assoc permutes_inv permutes_compose permutes_inv_o o_assoc[symmetric])
chaieb@29840
   737
chaieb@29840
   738
lemma permutes_in_seg: "p permutes {1 ..n} \<Longrightarrow> i \<in> {1..n} ==> 1 <= p i \<and> p i <= n"
chaieb@29840
   739
chaieb@29840
   740
apply (simp add: permutes_def)
chaieb@29840
   741
apply metis
chaieb@29840
   742
done
chaieb@29840
   743
chaieb@29840
   744
term setsum
huffman@30036
   745
lemma setsum_permutations_inverse: "setsum f {p. p permutes S} = setsum (\<lambda>p. f(inv p)) {p. p permutes S}" (is "?lhs = ?rhs")
chaieb@29840
   746
proof-
huffman@30036
   747
  let ?S = "{p . p permutes S}"
huffman@30488
   748
have th0: "inj_on inv ?S"
chaieb@29840
   749
proof(auto simp add: inj_on_def)
chaieb@29840
   750
  fix q r
huffman@30036
   751
  assume q: "q permutes S" and r: "r permutes S" and qr: "inv q = inv r"
chaieb@29840
   752
  hence "inv (inv q) = inv (inv r)" by simp
chaieb@29840
   753
  with permutes_inv_inv[OF q] permutes_inv_inv[OF r]
chaieb@29840
   754
  show "q = r" by metis
chaieb@29840
   755
qed
chaieb@29840
   756
  have th1: "inv ` ?S = ?S" using image_inverse_permutations by blast
chaieb@29840
   757
  have th2: "?rhs = setsum (f o inv) ?S" by (simp add: o_def)
chaieb@29840
   758
  from setsum_reindex[OF th0, of f]  show ?thesis unfolding th1 th2 .
chaieb@29840
   759
qed
chaieb@29840
   760
chaieb@29840
   761
lemma setum_permutations_compose_left:
huffman@30036
   762
  assumes q: "q permutes S"
huffman@30036
   763
  shows "setsum f {p. p permutes S} =
huffman@30036
   764
            setsum (\<lambda>p. f(q o p)) {p. p permutes S}" (is "?lhs = ?rhs")
chaieb@29840
   765
proof-
huffman@30036
   766
  let ?S = "{p. p permutes S}"
chaieb@29840
   767
  have th0: "?rhs = setsum (f o (op o q)) ?S" by (simp add: o_def)
chaieb@29840
   768
  have th1: "inj_on (op o q) ?S"
chaieb@29840
   769
    apply (auto simp add: inj_on_def)
chaieb@29840
   770
  proof-
chaieb@29840
   771
    fix p r
huffman@30036
   772
    assume "p permutes S" and r:"r permutes S" and rp: "q \<circ> p = q \<circ> r"
chaieb@29840
   773
    hence "inv q o q o p = inv q o q o r" by (simp add: o_assoc[symmetric])
chaieb@29840
   774
    with permutes_inj[OF q, unfolded inj_iff]
chaieb@29840
   775
chaieb@29840
   776
    show "p = r" by simp
chaieb@29840
   777
  qed
chaieb@29840
   778
  have th3: "(op o q) ` ?S = ?S" using image_compose_permutations_left[OF q] by auto
chaieb@29840
   779
  from setsum_reindex[OF th1, of f]
chaieb@29840
   780
  show ?thesis unfolding th0 th1 th3 .
chaieb@29840
   781
qed
chaieb@29840
   782
chaieb@29840
   783
lemma sum_permutations_compose_right:
huffman@30036
   784
  assumes q: "q permutes S"
huffman@30036
   785
  shows "setsum f {p. p permutes S} =
huffman@30036
   786
            setsum (\<lambda>p. f(p o q)) {p. p permutes S}" (is "?lhs = ?rhs")
chaieb@29840
   787
proof-
huffman@30036
   788
  let ?S = "{p. p permutes S}"
chaieb@29840
   789
  have th0: "?rhs = setsum (f o (\<lambda>p. p o q)) ?S" by (simp add: o_def)
chaieb@29840
   790
  have th1: "inj_on (\<lambda>p. p o q) ?S"
chaieb@29840
   791
    apply (auto simp add: inj_on_def)
chaieb@29840
   792
  proof-
chaieb@29840
   793
    fix p r
huffman@30036
   794
    assume "p permutes S" and r:"r permutes S" and rp: "p o q = r o q"
chaieb@29840
   795
    hence "p o (q o inv q)  = r o (q o inv q)" by (simp add: o_assoc)
chaieb@29840
   796
    with permutes_surj[OF q, unfolded surj_iff]
chaieb@29840
   797
chaieb@29840
   798
    show "p = r" by simp
chaieb@29840
   799
  qed
chaieb@29840
   800
  have th3: "(\<lambda>p. p o q) ` ?S = ?S" using image_compose_permutations_right[OF q] by auto
chaieb@29840
   801
  from setsum_reindex[OF th1, of f]
chaieb@29840
   802
  show ?thesis unfolding th0 th1 th3 .
chaieb@29840
   803
qed
chaieb@29840
   804
chaieb@29840
   805
(* ------------------------------------------------------------------------- *)
chaieb@29840
   806
(* Sum over a set of permutations (could generalize to iteration).           *)
chaieb@29840
   807
(* ------------------------------------------------------------------------- *)
chaieb@29840
   808
chaieb@29840
   809
lemma setsum_over_permutations_insert:
chaieb@29840
   810
  assumes fS: "finite S" and aS: "a \<notin> S"
chaieb@29840
   811
  shows "setsum f {p. p permutes (insert a S)} = setsum (\<lambda>b. setsum (\<lambda>q. f (Fun.swap a b id o q)) {p. p permutes S}) (insert a S)"
chaieb@29840
   812
proof-
chaieb@29840
   813
  have th0: "\<And>f a b. (\<lambda>(b,p). f (Fun.swap a b id o p)) = f o (\<lambda>(b,p). Fun.swap a b id o p)"
chaieb@29840
   814
    by (simp add: expand_fun_eq)
chaieb@29840
   815
  have th1: "\<And>P Q. P \<times> Q = {(a,b). a \<in> P \<and> b \<in> Q}" by blast
chaieb@29840
   816
  have th2: "\<And>P Q. P \<Longrightarrow> (P \<Longrightarrow> Q) \<Longrightarrow> P \<and> Q" by blast
huffman@30488
   817
  show ?thesis
huffman@30488
   818
    unfolding permutes_insert
chaieb@29840
   819
    unfolding setsum_cartesian_product
chaieb@29840
   820
    unfolding  th1[symmetric]
chaieb@29840
   821
    unfolding th0
chaieb@29840
   822
  proof(rule setsum_reindex)
chaieb@29840
   823
    let ?f = "(\<lambda>(b, y). Fun.swap a b id \<circ> y)"
chaieb@29840
   824
    let ?P = "{p. p permutes S}"
huffman@30488
   825
    {fix b c p q assume b: "b \<in> insert a S" and c: "c \<in> insert a S"
huffman@30488
   826
      and p: "p permutes S" and q: "q permutes S"
chaieb@29840
   827
      and eq: "Fun.swap a b id o p = Fun.swap a c id o q"
chaieb@29840
   828
      from p q aS have pa: "p a = a" and qa: "q a = a"
wenzelm@32960
   829
        unfolding permutes_def by metis+
chaieb@29840
   830
      from eq have "(Fun.swap a b id o p) a  = (Fun.swap a c id o q) a" by simp
chaieb@29840
   831
      hence bc: "b = c"
wenzelm@32960
   832
        by (simp add: permutes_def pa qa o_def fun_upd_def swap_def id_def cong del: if_weak_cong split: split_if_asm)
chaieb@29840
   833
      from eq[unfolded bc] have "(\<lambda>p. Fun.swap a c id o p) (Fun.swap a c id o p) = (\<lambda>p. Fun.swap a c id o p) (Fun.swap a c id o q)" by simp
chaieb@29840
   834
      hence "p = q" unfolding o_assoc swap_id_idempotent
wenzelm@32960
   835
        by (simp add: o_def)
chaieb@29840
   836
      with bc have "b = c \<and> p = q" by blast
chaieb@29840
   837
    }
huffman@30488
   838
huffman@30488
   839
    then show "inj_on ?f (insert a S \<times> ?P)"
chaieb@29840
   840
      unfolding inj_on_def
chaieb@29840
   841
      apply clarify by metis
chaieb@29840
   842
  qed
chaieb@29840
   843
qed
chaieb@29840
   844
chaieb@29840
   845
end