src/HOL/Lifting_Set.thy
author wenzelm
Mon Jul 06 21:20:28 2015 +0200 (2015-07-06)
changeset 60676 92fd47ae2a67
parent 60229 4cd6462c1fda
child 60758 d8d85a8172b5
permissions -rw-r--r--
tuned proofs;
kuncar@53012
     1
(*  Title:      HOL/Lifting_Set.thy
kuncar@53012
     2
    Author:     Brian Huffman and Ondrej Kuncar
kuncar@53012
     3
*)
kuncar@53012
     4
wenzelm@58889
     5
section {* Setup for Lifting/Transfer for the set type *}
kuncar@53012
     6
kuncar@53012
     7
theory Lifting_Set
kuncar@53012
     8
imports Lifting
kuncar@53012
     9
begin
kuncar@53012
    10
kuncar@53012
    11
subsection {* Relator and predicator properties *}
kuncar@53012
    12
blanchet@55938
    13
lemma rel_setD1: "\<lbrakk> rel_set R A B; x \<in> A \<rbrakk> \<Longrightarrow> \<exists>y \<in> B. R x y"
blanchet@55938
    14
  and rel_setD2: "\<lbrakk> rel_set R A B; y \<in> B \<rbrakk> \<Longrightarrow> \<exists>x \<in> A. R x y"
wenzelm@60676
    15
  by (simp_all add: rel_set_def)
Andreas@53927
    16
blanchet@55938
    17
lemma rel_set_conversep [simp]: "rel_set A\<inverse>\<inverse> = (rel_set A)\<inverse>\<inverse>"
blanchet@55938
    18
  unfolding rel_set_def by auto
kuncar@53012
    19
blanchet@55938
    20
lemma rel_set_eq [relator_eq]: "rel_set (op =) = (op =)"
blanchet@55938
    21
  unfolding rel_set_def fun_eq_iff by auto
kuncar@53012
    22
blanchet@55938
    23
lemma rel_set_mono[relator_mono]:
kuncar@53012
    24
  assumes "A \<le> B"
blanchet@55938
    25
  shows "rel_set A \<le> rel_set B"
wenzelm@60676
    26
  using assms unfolding rel_set_def by blast
kuncar@53012
    27
blanchet@55938
    28
lemma rel_set_OO[relator_distr]: "rel_set R OO rel_set S = rel_set (R OO S)"
kuncar@53012
    29
  apply (rule sym)
wenzelm@60676
    30
  apply (intro ext)
wenzelm@60676
    31
  subgoal for X Z
wenzelm@60676
    32
    apply (rule iffI)
wenzelm@60676
    33
    apply (rule relcomppI [where b="{y. (\<exists>x\<in>X. R x y) \<and> (\<exists>z\<in>Z. S y z)}"])
wenzelm@60676
    34
    apply (simp add: rel_set_def, fast)+
wenzelm@60676
    35
    done
kuncar@53012
    36
  done
kuncar@53012
    37
kuncar@53012
    38
lemma Domainp_set[relator_domain]:
kuncar@56520
    39
  "Domainp (rel_set T) = (\<lambda>A. Ball A (Domainp T))"
wenzelm@60676
    40
  unfolding rel_set_def Domainp_iff[abs_def]
wenzelm@60676
    41
  apply (intro ext)
wenzelm@60676
    42
  apply (rule iffI) 
wenzelm@60676
    43
  apply blast
wenzelm@60676
    44
  subgoal for A by (rule exI [where x="{y. \<exists>x\<in>A. T x y}"]) fast
wenzelm@60676
    45
  done
kuncar@53012
    46
kuncar@56518
    47
lemma left_total_rel_set[transfer_rule]: 
blanchet@55938
    48
  "left_total A \<Longrightarrow> left_total (rel_set A)"
blanchet@55938
    49
  unfolding left_total_def rel_set_def
kuncar@53012
    50
  apply safe
wenzelm@60676
    51
  subgoal for X by (rule exI [where x="{y. \<exists>x\<in>X. A x y}"]) fast
wenzelm@60676
    52
  done
kuncar@53012
    53
kuncar@56518
    54
lemma left_unique_rel_set[transfer_rule]: 
blanchet@55938
    55
  "left_unique A \<Longrightarrow> left_unique (rel_set A)"
blanchet@55938
    56
  unfolding left_unique_def rel_set_def
kuncar@53012
    57
  by fast
kuncar@53012
    58
blanchet@55938
    59
lemma right_total_rel_set [transfer_rule]:
blanchet@55938
    60
  "right_total A \<Longrightarrow> right_total (rel_set A)"
wenzelm@60676
    61
  using left_total_rel_set[of "A\<inverse>\<inverse>"] by simp
kuncar@53012
    62
blanchet@55938
    63
lemma right_unique_rel_set [transfer_rule]:
blanchet@55938
    64
  "right_unique A \<Longrightarrow> right_unique (rel_set A)"
blanchet@55938
    65
  unfolding right_unique_def rel_set_def by fast
kuncar@53012
    66
blanchet@55938
    67
lemma bi_total_rel_set [transfer_rule]:
blanchet@55938
    68
  "bi_total A \<Longrightarrow> bi_total (rel_set A)"
wenzelm@60676
    69
  by(simp add: bi_total_alt_def left_total_rel_set right_total_rel_set)
kuncar@53012
    70
blanchet@55938
    71
lemma bi_unique_rel_set [transfer_rule]:
blanchet@55938
    72
  "bi_unique A \<Longrightarrow> bi_unique (rel_set A)"
blanchet@55938
    73
  unfolding bi_unique_def rel_set_def by fast
kuncar@53012
    74
kuncar@56519
    75
lemma set_relator_eq_onp [relator_eq_onp]:
kuncar@56519
    76
  "rel_set (eq_onp P) = eq_onp (\<lambda>A. Ball A P)"
kuncar@56519
    77
  unfolding fun_eq_iff rel_set_def eq_onp_def Ball_def by fast
kuncar@53012
    78
hoelzl@57129
    79
lemma bi_unique_rel_set_lemma:
hoelzl@57129
    80
  assumes "bi_unique R" and "rel_set R X Y"
hoelzl@57129
    81
  obtains f where "Y = image f X" and "inj_on f X" and "\<forall>x\<in>X. R x (f x)"
hoelzl@57129
    82
proof
hoelzl@57129
    83
  def f \<equiv> "\<lambda>x. THE y. R x y"
hoelzl@57129
    84
  { fix x assume "x \<in> X"
hoelzl@57129
    85
    with `rel_set R X Y` `bi_unique R` have "R x (f x)"
hoelzl@57129
    86
      by (simp add: bi_unique_def rel_set_def f_def) (metis theI)
hoelzl@57129
    87
    with assms `x \<in> X` 
hoelzl@57129
    88
    have  "R x (f x)" "\<forall>x'\<in>X. R x' (f x) \<longrightarrow> x = x'" "\<forall>y\<in>Y. R x y \<longrightarrow> y = f x" "f x \<in> Y"
hoelzl@57129
    89
      by (fastforce simp add: bi_unique_def rel_set_def)+ }
hoelzl@57129
    90
  note * = this
hoelzl@57129
    91
  moreover
hoelzl@57129
    92
  { fix y assume "y \<in> Y"
hoelzl@57129
    93
    with `rel_set R X Y` *(3) `y \<in> Y` have "\<exists>x\<in>X. y = f x"
hoelzl@57129
    94
      by (fastforce simp: rel_set_def) }
hoelzl@57129
    95
  ultimately show "\<forall>x\<in>X. R x (f x)" "Y = image f X" "inj_on f X"
hoelzl@57129
    96
    by (auto simp: inj_on_def image_iff)
hoelzl@57129
    97
qed
hoelzl@57129
    98
kuncar@53012
    99
subsection {* Quotient theorem for the Lifting package *}
kuncar@53012
   100
kuncar@53012
   101
lemma Quotient_set[quot_map]:
kuncar@53012
   102
  assumes "Quotient R Abs Rep T"
blanchet@55938
   103
  shows "Quotient (rel_set R) (image Abs) (image Rep) (rel_set T)"
kuncar@53012
   104
  using assms unfolding Quotient_alt_def4
blanchet@55938
   105
  apply (simp add: rel_set_OO[symmetric])
wenzelm@60676
   106
  apply (simp add: rel_set_def)
wenzelm@60676
   107
  apply fast
kuncar@53012
   108
  done
kuncar@53012
   109
wenzelm@60676
   110
kuncar@53012
   111
subsection {* Transfer rules for the Transfer package *}
kuncar@53012
   112
kuncar@53012
   113
subsubsection {* Unconditional transfer rules *}
kuncar@53012
   114
kuncar@53012
   115
context
kuncar@53012
   116
begin
wenzelm@60676
   117
kuncar@53012
   118
interpretation lifting_syntax .
kuncar@53012
   119
blanchet@55938
   120
lemma empty_transfer [transfer_rule]: "(rel_set A) {} {}"
blanchet@55938
   121
  unfolding rel_set_def by simp
kuncar@53012
   122
kuncar@53012
   123
lemma insert_transfer [transfer_rule]:
blanchet@55938
   124
  "(A ===> rel_set A ===> rel_set A) insert insert"
blanchet@55945
   125
  unfolding rel_fun_def rel_set_def by auto
kuncar@53012
   126
kuncar@53012
   127
lemma union_transfer [transfer_rule]:
blanchet@55938
   128
  "(rel_set A ===> rel_set A ===> rel_set A) union union"
blanchet@55945
   129
  unfolding rel_fun_def rel_set_def by auto
kuncar@53012
   130
kuncar@53012
   131
lemma Union_transfer [transfer_rule]:
blanchet@55938
   132
  "(rel_set (rel_set A) ===> rel_set A) Union Union"
blanchet@55945
   133
  unfolding rel_fun_def rel_set_def by simp fast
kuncar@53012
   134
kuncar@53012
   135
lemma image_transfer [transfer_rule]:
blanchet@55938
   136
  "((A ===> B) ===> rel_set A ===> rel_set B) image image"
blanchet@55945
   137
  unfolding rel_fun_def rel_set_def by simp fast
kuncar@53012
   138
kuncar@53012
   139
lemma UNION_transfer [transfer_rule]:
blanchet@55938
   140
  "(rel_set A ===> (A ===> rel_set B) ===> rel_set B) UNION UNION"
haftmann@56166
   141
  unfolding Union_image_eq [symmetric, abs_def] by transfer_prover
kuncar@53012
   142
kuncar@53012
   143
lemma Ball_transfer [transfer_rule]:
blanchet@55938
   144
  "(rel_set A ===> (A ===> op =) ===> op =) Ball Ball"
blanchet@55945
   145
  unfolding rel_set_def rel_fun_def by fast
kuncar@53012
   146
kuncar@53012
   147
lemma Bex_transfer [transfer_rule]:
blanchet@55938
   148
  "(rel_set A ===> (A ===> op =) ===> op =) Bex Bex"
blanchet@55945
   149
  unfolding rel_set_def rel_fun_def by fast
kuncar@53012
   150
kuncar@53012
   151
lemma Pow_transfer [transfer_rule]:
blanchet@55938
   152
  "(rel_set A ===> rel_set (rel_set A)) Pow Pow"
wenzelm@60676
   153
  apply (rule rel_funI)
wenzelm@60676
   154
  apply (rule rel_setI)
wenzelm@60676
   155
  subgoal for X Y X'
wenzelm@60676
   156
    apply (rule rev_bexI [where x="{y\<in>Y. \<exists>x\<in>X'. A x y}"])
wenzelm@60676
   157
    apply clarsimp
wenzelm@60676
   158
    apply (simp add: rel_set_def)
wenzelm@60676
   159
    apply fast
wenzelm@60676
   160
    done
wenzelm@60676
   161
  subgoal for X Y Y'
wenzelm@60676
   162
    apply (rule rev_bexI [where x="{x\<in>X. \<exists>y\<in>Y'. A x y}"])
wenzelm@60676
   163
    apply clarsimp
wenzelm@60676
   164
    apply (simp add: rel_set_def)
wenzelm@60676
   165
    apply fast
wenzelm@60676
   166
    done
kuncar@53012
   167
  done
kuncar@53012
   168
blanchet@55938
   169
lemma rel_set_transfer [transfer_rule]:
haftmann@56482
   170
  "((A ===> B ===> op =) ===> rel_set A ===> rel_set B ===> op =) rel_set rel_set"
blanchet@55945
   171
  unfolding rel_fun_def rel_set_def by fast
kuncar@53012
   172
kuncar@53952
   173
lemma bind_transfer [transfer_rule]:
blanchet@55938
   174
  "(rel_set A ===> (A ===> rel_set B) ===> rel_set B) Set.bind Set.bind"
haftmann@56482
   175
  unfolding bind_UNION [abs_def] by transfer_prover
haftmann@56482
   176
haftmann@56482
   177
lemma INF_parametric [transfer_rule]:
haftmann@56482
   178
  "(rel_set A ===> (A ===> HOL.eq) ===> HOL.eq) INFIMUM INFIMUM"
haftmann@56482
   179
  unfolding INF_def [abs_def] by transfer_prover
haftmann@56482
   180
haftmann@56482
   181
lemma SUP_parametric [transfer_rule]:
haftmann@56482
   182
  "(rel_set R ===> (R ===> HOL.eq) ===> HOL.eq) SUPREMUM SUPREMUM"
haftmann@56482
   183
  unfolding SUP_def [abs_def] by transfer_prover
haftmann@56482
   184
kuncar@53952
   185
kuncar@53012
   186
subsubsection {* Rules requiring bi-unique, bi-total or right-total relations *}
kuncar@53012
   187
kuncar@53012
   188
lemma member_transfer [transfer_rule]:
kuncar@53012
   189
  assumes "bi_unique A"
blanchet@55938
   190
  shows "(A ===> rel_set A ===> op =) (op \<in>) (op \<in>)"
blanchet@55945
   191
  using assms unfolding rel_fun_def rel_set_def bi_unique_def by fast
kuncar@53012
   192
kuncar@53012
   193
lemma right_total_Collect_transfer[transfer_rule]:
kuncar@53012
   194
  assumes "right_total A"
blanchet@55938
   195
  shows "((A ===> op =) ===> rel_set A) (\<lambda>P. Collect (\<lambda>x. P x \<and> Domainp A x)) Collect"
blanchet@55945
   196
  using assms unfolding right_total_def rel_set_def rel_fun_def Domainp_iff by fast
kuncar@53012
   197
kuncar@53012
   198
lemma Collect_transfer [transfer_rule]:
kuncar@53012
   199
  assumes "bi_total A"
blanchet@55938
   200
  shows "((A ===> op =) ===> rel_set A) Collect Collect"
blanchet@55945
   201
  using assms unfolding rel_fun_def rel_set_def bi_total_def by fast
kuncar@53012
   202
kuncar@53012
   203
lemma inter_transfer [transfer_rule]:
kuncar@53012
   204
  assumes "bi_unique A"
blanchet@55938
   205
  shows "(rel_set A ===> rel_set A ===> rel_set A) inter inter"
blanchet@55945
   206
  using assms unfolding rel_fun_def rel_set_def bi_unique_def by fast
kuncar@53012
   207
kuncar@53012
   208
lemma Diff_transfer [transfer_rule]:
kuncar@53012
   209
  assumes "bi_unique A"
blanchet@55938
   210
  shows "(rel_set A ===> rel_set A ===> rel_set A) (op -) (op -)"
blanchet@55945
   211
  using assms unfolding rel_fun_def rel_set_def bi_unique_def
kuncar@53012
   212
  unfolding Ball_def Bex_def Diff_eq
kuncar@53012
   213
  by (safe, simp, metis, simp, metis)
kuncar@53012
   214
kuncar@53012
   215
lemma subset_transfer [transfer_rule]:
kuncar@53012
   216
  assumes [transfer_rule]: "bi_unique A"
blanchet@55938
   217
  shows "(rel_set A ===> rel_set A ===> op =) (op \<subseteq>) (op \<subseteq>)"
kuncar@53012
   218
  unfolding subset_eq [abs_def] by transfer_prover
kuncar@53012
   219
kuncar@60229
   220
declare right_total_UNIV_transfer[transfer_rule]
kuncar@53012
   221
kuncar@53012
   222
lemma UNIV_transfer [transfer_rule]:
kuncar@53012
   223
  assumes "bi_total A"
blanchet@55938
   224
  shows "(rel_set A) UNIV UNIV"
blanchet@55938
   225
  using assms unfolding rel_set_def bi_total_def by simp
kuncar@53012
   226
kuncar@53012
   227
lemma right_total_Compl_transfer [transfer_rule]:
kuncar@53012
   228
  assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
blanchet@55938
   229
  shows "(rel_set A ===> rel_set A) (\<lambda>S. uminus S \<inter> Collect (Domainp A)) uminus"
kuncar@53012
   230
  unfolding Compl_eq [abs_def]
kuncar@53012
   231
  by (subst Collect_conj_eq[symmetric]) transfer_prover
kuncar@53012
   232
kuncar@53012
   233
lemma Compl_transfer [transfer_rule]:
kuncar@53012
   234
  assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
blanchet@55938
   235
  shows "(rel_set A ===> rel_set A) uminus uminus"
kuncar@53012
   236
  unfolding Compl_eq [abs_def] by transfer_prover
kuncar@53012
   237
kuncar@53012
   238
lemma right_total_Inter_transfer [transfer_rule]:
kuncar@53012
   239
  assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "right_total A"
blanchet@55938
   240
  shows "(rel_set (rel_set A) ===> rel_set A) (\<lambda>S. Inter S \<inter> Collect (Domainp A)) Inter"
kuncar@53012
   241
  unfolding Inter_eq[abs_def]
kuncar@53012
   242
  by (subst Collect_conj_eq[symmetric]) transfer_prover
kuncar@53012
   243
kuncar@53012
   244
lemma Inter_transfer [transfer_rule]:
kuncar@53012
   245
  assumes [transfer_rule]: "bi_unique A" and [transfer_rule]: "bi_total A"
blanchet@55938
   246
  shows "(rel_set (rel_set A) ===> rel_set A) Inter Inter"
kuncar@53012
   247
  unfolding Inter_eq [abs_def] by transfer_prover
kuncar@53012
   248
kuncar@53012
   249
lemma filter_transfer [transfer_rule]:
kuncar@53012
   250
  assumes [transfer_rule]: "bi_unique A"
blanchet@55938
   251
  shows "((A ===> op=) ===> rel_set A ===> rel_set A) Set.filter Set.filter"
blanchet@55945
   252
  unfolding Set.filter_def[abs_def] rel_fun_def rel_set_def by blast
kuncar@53012
   253
kuncar@53012
   254
lemma finite_transfer [transfer_rule]:
blanchet@55938
   255
  "bi_unique A \<Longrightarrow> (rel_set A ===> op =) finite finite"
hoelzl@57129
   256
  by (rule rel_funI, erule (1) bi_unique_rel_set_lemma)
hoelzl@57129
   257
     (auto dest: finite_imageD)
kuncar@53012
   258
kuncar@53012
   259
lemma card_transfer [transfer_rule]:
blanchet@55938
   260
  "bi_unique A \<Longrightarrow> (rel_set A ===> op =) card card"
hoelzl@57129
   261
  by (rule rel_funI, erule (1) bi_unique_rel_set_lemma)
hoelzl@57129
   262
     (simp add: card_image)
kuncar@53012
   263
Andreas@53927
   264
lemma vimage_parametric [transfer_rule]:
Andreas@53927
   265
  assumes [transfer_rule]: "bi_total A" "bi_unique B"
blanchet@55938
   266
  shows "((A ===> B) ===> rel_set B ===> rel_set A) vimage vimage"
hoelzl@57129
   267
  unfolding vimage_def[abs_def] by transfer_prover
Andreas@53927
   268
Andreas@57599
   269
lemma Image_parametric [transfer_rule]:
Andreas@57599
   270
  assumes "bi_unique A"
Andreas@57599
   271
  shows "(rel_set (rel_prod A B) ===> rel_set A ===> rel_set B) op `` op ``"
wenzelm@60676
   272
  by (intro rel_funI rel_setI)
wenzelm@60676
   273
    (force dest: rel_setD1 bi_uniqueDr[OF assms], force dest: rel_setD2 bi_uniqueDl[OF assms])
Andreas@57599
   274
kuncar@53012
   275
end
kuncar@53012
   276
hoelzl@57129
   277
lemma (in comm_monoid_set) F_parametric [transfer_rule]:
hoelzl@57129
   278
  fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"
hoelzl@57129
   279
  assumes "bi_unique A"
hoelzl@57129
   280
  shows "rel_fun (rel_fun A (op =)) (rel_fun (rel_set A) (op =)) F F"
wenzelm@60676
   281
proof (rule rel_funI)+
hoelzl@57129
   282
  fix f :: "'b \<Rightarrow> 'a" and g S T
hoelzl@57129
   283
  assume "rel_fun A (op =) f g" "rel_set A S T"
hoelzl@57129
   284
  with `bi_unique A` obtain i where "bij_betw i S T" "\<And>x. x \<in> S \<Longrightarrow> f x = g (i x)"
hoelzl@57129
   285
    by (auto elim: bi_unique_rel_set_lemma simp: rel_fun_def bij_betw_def)
hoelzl@57129
   286
  then show "F f S = F g T"
hoelzl@57129
   287
    by (simp add: reindex_bij_betw)
hoelzl@57129
   288
qed
hoelzl@57129
   289
hoelzl@57129
   290
lemmas setsum_parametric = setsum.F_parametric
hoelzl@57129
   291
lemmas setprod_parametric = setprod.F_parametric
hoelzl@57129
   292
Andreas@60057
   293
lemma rel_set_UNION:
Andreas@60057
   294
  assumes [transfer_rule]: "rel_set Q A B" "rel_fun Q (rel_set R) f g"
Andreas@60057
   295
  shows "rel_set R (UNION A f) (UNION B g)"
wenzelm@60676
   296
  by transfer_prover
Andreas@60057
   297
kuncar@53012
   298
end