src/HOLCF/Bifinite.thy
author huffman
Mon May 11 12:26:33 2009 -0700 (2009-05-11)
changeset 31113 15cf300a742f
parent 31076 99fe356cbbc2
child 33504 b4210cc3ac97
permissions -rw-r--r--
move bifinite instance for product type from Cprod.thy to Bifinite.thy
huffman@25903
     1
(*  Title:      HOLCF/Bifinite.thy
huffman@25903
     2
    Author:     Brian Huffman
huffman@25903
     3
*)
huffman@25903
     4
huffman@25903
     5
header {* Bifinite domains and approximation *}
huffman@25903
     6
huffman@25903
     7
theory Bifinite
huffman@27402
     8
imports Deflation
huffman@25903
     9
begin
huffman@25903
    10
huffman@26407
    11
subsection {* Omega-profinite and bifinite domains *}
huffman@25903
    12
haftmann@29614
    13
class profinite =
huffman@26962
    14
  fixes approx :: "nat \<Rightarrow> 'a \<rightarrow> 'a"
huffman@27310
    15
  assumes chain_approx [simp]: "chain approx"
huffman@26962
    16
  assumes lub_approx_app [simp]: "(\<Squnion>i. approx i\<cdot>x) = x"
huffman@26962
    17
  assumes approx_idem: "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
huffman@26962
    18
  assumes finite_fixes_approx: "finite {x. approx i\<cdot>x = x}"
huffman@25903
    19
huffman@26962
    20
class bifinite = profinite + pcpo
huffman@25909
    21
huffman@31076
    22
lemma approx_below: "approx i\<cdot>x \<sqsubseteq> x"
huffman@27402
    23
proof -
huffman@27402
    24
  have "chain (\<lambda>i. approx i\<cdot>x)" by simp
huffman@27402
    25
  hence "approx i\<cdot>x \<sqsubseteq> (\<Squnion>i. approx i\<cdot>x)" by (rule is_ub_thelub)
huffman@27402
    26
  thus "approx i\<cdot>x \<sqsubseteq> x" by simp
huffman@27402
    27
qed
huffman@27402
    28
huffman@27402
    29
lemma finite_deflation_approx: "finite_deflation (approx i)"
huffman@27402
    30
proof
huffman@27402
    31
  fix x :: 'a
huffman@27402
    32
  show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
huffman@27402
    33
    by (rule approx_idem)
huffman@27402
    34
  show "approx i\<cdot>x \<sqsubseteq> x"
huffman@31076
    35
    by (rule approx_below)
huffman@27402
    36
  show "finite {x. approx i\<cdot>x = x}"
huffman@27402
    37
    by (rule finite_fixes_approx)
huffman@27402
    38
qed
huffman@27402
    39
wenzelm@30729
    40
interpretation approx: finite_deflation "approx i"
huffman@27402
    41
by (rule finite_deflation_approx)
huffman@27402
    42
ballarin@28234
    43
lemma (in deflation) deflation: "deflation d" ..
ballarin@28234
    44
huffman@27402
    45
lemma deflation_approx: "deflation (approx i)"
ballarin@28234
    46
by (rule approx.deflation)
huffman@25903
    47
huffman@27186
    48
lemma lub_approx [simp]: "(\<Squnion>i. approx i) = (\<Lambda> x. x)"
huffman@25903
    49
by (rule ext_cfun, simp add: contlub_cfun_fun)
huffman@25903
    50
huffman@27309
    51
lemma approx_strict [simp]: "approx i\<cdot>\<bottom> = \<bottom>"
huffman@31076
    52
by (rule UU_I, rule approx_below)
huffman@25903
    53
huffman@25903
    54
lemma approx_approx1:
huffman@27186
    55
  "i \<le> j \<Longrightarrow> approx i\<cdot>(approx j\<cdot>x) = approx i\<cdot>x"
huffman@31076
    56
apply (rule deflation_below_comp1 [OF deflation_approx deflation_approx])
huffman@25922
    57
apply (erule chain_mono [OF chain_approx])
huffman@25903
    58
done
huffman@25903
    59
huffman@25903
    60
lemma approx_approx2:
huffman@27186
    61
  "j \<le> i \<Longrightarrow> approx i\<cdot>(approx j\<cdot>x) = approx j\<cdot>x"
huffman@31076
    62
apply (rule deflation_below_comp2 [OF deflation_approx deflation_approx])
huffman@25922
    63
apply (erule chain_mono [OF chain_approx])
huffman@25903
    64
done
huffman@25903
    65
huffman@25903
    66
lemma approx_approx [simp]:
huffman@27186
    67
  "approx i\<cdot>(approx j\<cdot>x) = approx (min i j)\<cdot>x"
huffman@25903
    68
apply (rule_tac x=i and y=j in linorder_le_cases)
huffman@25903
    69
apply (simp add: approx_approx1 min_def)
huffman@25903
    70
apply (simp add: approx_approx2 min_def)
huffman@25903
    71
done
huffman@25903
    72
huffman@27402
    73
lemma finite_image_approx: "finite ((\<lambda>x. approx n\<cdot>x) ` A)"
huffman@27402
    74
by (rule approx.finite_image)
huffman@25903
    75
huffman@27402
    76
lemma finite_range_approx: "finite (range (\<lambda>x. approx i\<cdot>x))"
huffman@27402
    77
by (rule approx.finite_range)
huffman@27186
    78
huffman@27186
    79
lemma compact_approx [simp]: "compact (approx n\<cdot>x)"
huffman@27402
    80
by (rule approx.compact)
huffman@25903
    81
huffman@27309
    82
lemma profinite_compact_eq_approx: "compact x \<Longrightarrow> \<exists>i. approx i\<cdot>x = x"
huffman@27402
    83
by (rule admD2, simp_all)
huffman@25903
    84
huffman@27309
    85
lemma profinite_compact_iff: "compact x \<longleftrightarrow> (\<exists>n. approx n\<cdot>x = x)"
huffman@25903
    86
 apply (rule iffI)
huffman@27309
    87
  apply (erule profinite_compact_eq_approx)
huffman@25903
    88
 apply (erule exE)
huffman@25903
    89
 apply (erule subst)
huffman@25903
    90
 apply (rule compact_approx)
huffman@25903
    91
done
huffman@25903
    92
huffman@25903
    93
lemma approx_induct:
huffman@25903
    94
  assumes adm: "adm P" and P: "\<And>n x. P (approx n\<cdot>x)"
huffman@27186
    95
  shows "P x"
huffman@25903
    96
proof -
huffman@25903
    97
  have "P (\<Squnion>n. approx n\<cdot>x)"
huffman@25903
    98
    by (rule admD [OF adm], simp, simp add: P)
huffman@25903
    99
  thus "P x" by simp
huffman@25903
   100
qed
huffman@25903
   101
huffman@31076
   102
lemma profinite_below_ext: "(\<And>i. approx i\<cdot>x \<sqsubseteq> approx i\<cdot>y) \<Longrightarrow> x \<sqsubseteq> y"
huffman@25903
   103
apply (subgoal_tac "(\<Squnion>i. approx i\<cdot>x) \<sqsubseteq> (\<Squnion>i. approx i\<cdot>y)", simp)
huffman@25923
   104
apply (rule lub_mono, simp, simp, simp)
huffman@25903
   105
done
huffman@25903
   106
huffman@31113
   107
subsection {* Instance for product type *}
huffman@31113
   108
huffman@31113
   109
instantiation "*" :: (profinite, profinite) profinite
huffman@31113
   110
begin
huffman@31113
   111
huffman@31113
   112
definition approx_prod_def:
huffman@31113
   113
  "approx = (\<lambda>n. \<Lambda> x. (approx n\<cdot>(fst x), approx n\<cdot>(snd x)))"
huffman@31113
   114
huffman@31113
   115
instance proof
huffman@31113
   116
  fix i :: nat and x :: "'a \<times> 'b"
huffman@31113
   117
  show "chain (approx :: nat \<Rightarrow> 'a \<times> 'b \<rightarrow> 'a \<times> 'b)"
huffman@31113
   118
    unfolding approx_prod_def by simp
huffman@31113
   119
  show "(\<Squnion>i. approx i\<cdot>x) = x"
huffman@31113
   120
    unfolding approx_prod_def
huffman@31113
   121
    by (simp add: lub_distribs thelub_Pair)
huffman@31113
   122
  show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
huffman@31113
   123
    unfolding approx_prod_def by simp
huffman@31113
   124
  have "{x::'a \<times> 'b. approx i\<cdot>x = x} \<subseteq>
huffman@31113
   125
        {x::'a. approx i\<cdot>x = x} \<times> {x::'b. approx i\<cdot>x = x}"
huffman@31113
   126
    unfolding approx_prod_def by clarsimp
huffman@31113
   127
  thus "finite {x::'a \<times> 'b. approx i\<cdot>x = x}"
huffman@31113
   128
    by (rule finite_subset,
huffman@31113
   129
        intro finite_cartesian_product finite_fixes_approx)
huffman@31113
   130
qed
huffman@31113
   131
huffman@31113
   132
end
huffman@31113
   133
huffman@31113
   134
instance "*" :: (bifinite, bifinite) bifinite ..
huffman@31113
   135
huffman@31113
   136
lemma approx_Pair [simp]:
huffman@31113
   137
  "approx i\<cdot>(x, y) = (approx i\<cdot>x, approx i\<cdot>y)"
huffman@31113
   138
unfolding approx_prod_def by simp
huffman@31113
   139
huffman@31113
   140
lemma fst_approx: "fst (approx i\<cdot>p) = approx i\<cdot>(fst p)"
huffman@31113
   141
by (induct p, simp)
huffman@31113
   142
huffman@31113
   143
lemma snd_approx: "snd (approx i\<cdot>p) = approx i\<cdot>(snd p)"
huffman@31113
   144
by (induct p, simp)
huffman@31113
   145
huffman@31113
   146
huffman@25903
   147
subsection {* Instance for continuous function space *}
huffman@25903
   148
huffman@27402
   149
lemma finite_range_cfun_lemma:
huffman@27402
   150
  assumes a: "finite (range (\<lambda>x. a\<cdot>x))"
huffman@27402
   151
  assumes b: "finite (range (\<lambda>y. b\<cdot>y))"
huffman@27402
   152
  shows "finite (range (\<lambda>f. \<Lambda> x. b\<cdot>(f\<cdot>(a\<cdot>x))))"  (is "finite (range ?h)")
huffman@27402
   153
proof (rule finite_imageD)
huffman@27402
   154
  let ?f = "\<lambda>g. range (\<lambda>x. (a\<cdot>x, g\<cdot>x))"
huffman@27402
   155
  show "finite (?f ` range ?h)"
huffman@27402
   156
  proof (rule finite_subset)
huffman@27402
   157
    let ?B = "Pow (range (\<lambda>x. a\<cdot>x) \<times> range (\<lambda>y. b\<cdot>y))"
huffman@27402
   158
    show "?f ` range ?h \<subseteq> ?B"
huffman@27402
   159
      by clarsimp
huffman@27402
   160
    show "finite ?B"
huffman@27402
   161
      by (simp add: a b)
huffman@27402
   162
  qed
huffman@27402
   163
  show "inj_on ?f (range ?h)"
huffman@27402
   164
  proof (rule inj_onI, rule ext_cfun, clarsimp)
huffman@27402
   165
    fix x f g
huffman@27402
   166
    assume "range (\<lambda>x. (a\<cdot>x, b\<cdot>(f\<cdot>(a\<cdot>x)))) = range (\<lambda>x. (a\<cdot>x, b\<cdot>(g\<cdot>(a\<cdot>x))))"
huffman@27402
   167
    hence "range (\<lambda>x. (a\<cdot>x, b\<cdot>(f\<cdot>(a\<cdot>x)))) \<subseteq> range (\<lambda>x. (a\<cdot>x, b\<cdot>(g\<cdot>(a\<cdot>x))))"
huffman@27402
   168
      by (rule equalityD1)
huffman@27402
   169
    hence "(a\<cdot>x, b\<cdot>(f\<cdot>(a\<cdot>x))) \<in> range (\<lambda>x. (a\<cdot>x, b\<cdot>(g\<cdot>(a\<cdot>x))))"
huffman@27402
   170
      by (simp add: subset_eq)
huffman@27402
   171
    then obtain y where "(a\<cdot>x, b\<cdot>(f\<cdot>(a\<cdot>x))) = (a\<cdot>y, b\<cdot>(g\<cdot>(a\<cdot>y)))"
huffman@27402
   172
      by (rule rangeE)
huffman@27402
   173
    thus "b\<cdot>(f\<cdot>(a\<cdot>x)) = b\<cdot>(g\<cdot>(a\<cdot>x))"
huffman@27402
   174
      by clarsimp
huffman@27402
   175
  qed
huffman@27402
   176
qed
huffman@25903
   177
huffman@26962
   178
instantiation "->" :: (profinite, profinite) profinite
huffman@26962
   179
begin
huffman@25903
   180
huffman@26962
   181
definition
huffman@25903
   182
  approx_cfun_def:
huffman@26962
   183
    "approx = (\<lambda>n. \<Lambda> f x. approx n\<cdot>(f\<cdot>(approx n\<cdot>x)))"
huffman@25903
   184
huffman@27402
   185
instance proof
huffman@27402
   186
  show "chain (approx :: nat \<Rightarrow> ('a \<rightarrow> 'b) \<rightarrow> ('a \<rightarrow> 'b))"
huffman@27402
   187
    unfolding approx_cfun_def by simp
huffman@27402
   188
next
huffman@27402
   189
  fix x :: "'a \<rightarrow> 'b"
huffman@27402
   190
  show "(\<Squnion>i. approx i\<cdot>x) = x"
huffman@27402
   191
    unfolding approx_cfun_def
huffman@27402
   192
    by (simp add: lub_distribs eta_cfun)
huffman@27402
   193
next
huffman@27402
   194
  fix i :: nat and x :: "'a \<rightarrow> 'b"
huffman@27402
   195
  show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
huffman@27402
   196
    unfolding approx_cfun_def by simp
huffman@27402
   197
next
huffman@27402
   198
  fix i :: nat
huffman@27402
   199
  show "finite {x::'a \<rightarrow> 'b. approx i\<cdot>x = x}"
huffman@27402
   200
    apply (rule finite_range_imp_finite_fixes)
huffman@27402
   201
    apply (simp add: approx_cfun_def)
huffman@27402
   202
    apply (intro finite_range_cfun_lemma finite_range_approx)
huffman@27402
   203
    done
huffman@27402
   204
qed
huffman@25903
   205
huffman@26962
   206
end
huffman@26962
   207
huffman@26407
   208
instance "->" :: (profinite, bifinite) bifinite ..
huffman@25909
   209
huffman@25903
   210
lemma approx_cfun: "approx n\<cdot>f\<cdot>x = approx n\<cdot>(f\<cdot>(approx n\<cdot>x))"
huffman@25903
   211
by (simp add: approx_cfun_def)
huffman@25903
   212
huffman@25903
   213
end