1.1 --- a/src/HOLCF/Bifinite.thy Mon Jun 30 21:52:17 2008 +0200
1.2 +++ b/src/HOLCF/Bifinite.thy Mon Jun 30 22:16:47 2008 +0200
1.3 @@ -6,7 +6,7 @@
1.4 header {* Bifinite domains and approximation *}
1.5
1.6 theory Bifinite
1.7 -imports Cfun
1.8 +imports Deflation
1.9 begin
1.10
1.11 subsection {* Omega-profinite and bifinite domains *}
1.12 @@ -20,40 +20,45 @@
1.13
1.14 class bifinite = profinite + pcpo
1.15
1.16 -lemma finite_range_imp_finite_fixes:
1.17 - "finite {x. \<exists>y. x = f y} \<Longrightarrow> finite {x. f x = x}"
1.18 -apply (subgoal_tac "{x. f x = x} \<subseteq> {x. \<exists>y. x = f y}")
1.19 -apply (erule (1) finite_subset)
1.20 -apply (clarify, erule subst, rule exI, rule refl)
1.21 -done
1.22 +lemma approx_less: "approx i\<cdot>x \<sqsubseteq> x"
1.23 +proof -
1.24 + have "chain (\<lambda>i. approx i\<cdot>x)" by simp
1.25 + hence "approx i\<cdot>x \<sqsubseteq> (\<Squnion>i. approx i\<cdot>x)" by (rule is_ub_thelub)
1.26 + thus "approx i\<cdot>x \<sqsubseteq> x" by simp
1.27 +qed
1.28 +
1.29 +lemma finite_deflation_approx: "finite_deflation (approx i)"
1.30 +proof
1.31 + fix x :: 'a
1.32 + show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
1.33 + by (rule approx_idem)
1.34 + show "approx i\<cdot>x \<sqsubseteq> x"
1.35 + by (rule approx_less)
1.36 + show "finite {x. approx i\<cdot>x = x}"
1.37 + by (rule finite_fixes_approx)
1.38 +qed
1.39 +
1.40 +interpretation approx: finite_deflation ["approx i"]
1.41 +by (rule finite_deflation_approx)
1.42 +
1.43 +lemma deflation_approx: "deflation (approx i)"
1.44 +by (rule approx.deflation_axioms)
1.45
1.46 lemma lub_approx [simp]: "(\<Squnion>i. approx i) = (\<Lambda> x. x)"
1.47 by (rule ext_cfun, simp add: contlub_cfun_fun)
1.48
1.49 -lemma approx_less: "approx i\<cdot>x \<sqsubseteq> x"
1.50 -apply (subgoal_tac "approx i\<cdot>x \<sqsubseteq> (\<Squnion>i. approx i\<cdot>x)", simp)
1.51 -apply (rule is_ub_thelub, simp)
1.52 -done
1.53 -
1.54 lemma approx_strict [simp]: "approx i\<cdot>\<bottom> = \<bottom>"
1.55 by (rule UU_I, rule approx_less)
1.56
1.57 lemma approx_approx1:
1.58 "i \<le> j \<Longrightarrow> approx i\<cdot>(approx j\<cdot>x) = approx i\<cdot>x"
1.59 -apply (rule antisym_less)
1.60 -apply (rule monofun_cfun_arg [OF approx_less])
1.61 -apply (rule sq_ord_eq_less_trans [OF approx_idem [symmetric]])
1.62 -apply (rule monofun_cfun_arg)
1.63 -apply (rule monofun_cfun_fun)
1.64 +apply (rule deflation_less_comp1 [OF deflation_approx deflation_approx])
1.65 apply (erule chain_mono [OF chain_approx])
1.66 done
1.67
1.68 lemma approx_approx2:
1.69 "j \<le> i \<Longrightarrow> approx i\<cdot>(approx j\<cdot>x) = approx j\<cdot>x"
1.70 -apply (rule antisym_less)
1.71 -apply (rule approx_less)
1.72 -apply (rule sq_ord_eq_less_trans [OF approx_idem [symmetric]])
1.73 -apply (rule monofun_cfun_fun)
1.74 +apply (rule deflation_less_comp2 [OF deflation_approx deflation_approx])
1.75 apply (erule chain_mono [OF chain_approx])
1.76 done
1.77
1.78 @@ -64,50 +69,17 @@
1.79 apply (simp add: approx_approx2 min_def)
1.80 done
1.81
1.82 -lemma idem_fixes_eq_range:
1.83 - "\<forall>x. f (f x) = f x \<Longrightarrow> {x. f x = x} = {y. \<exists>x. y = f x}"
1.84 -by (auto simp add: eq_sym_conv)
1.85 +lemma finite_image_approx: "finite ((\<lambda>x. approx n\<cdot>x) ` A)"
1.86 +by (rule approx.finite_image)
1.87
1.88 -lemma finite_approx: "finite {y. \<exists>x. y = approx n\<cdot>x}"
1.89 -using finite_fixes_approx by (simp add: idem_fixes_eq_range)
1.90 -
1.91 -lemma finite_image_approx: "finite ((\<lambda>x. approx n\<cdot>x) ` A)"
1.92 -by (rule finite_subset [OF _ finite_fixes_approx [where i=n]]) auto
1.93 -
1.94 -lemma finite_range_approx: "finite (range (\<lambda>x. approx n\<cdot>x))"
1.95 -by (rule finite_image_approx)
1.96 +lemma finite_range_approx: "finite (range (\<lambda>x. approx i\<cdot>x))"
1.97 +by (rule approx.finite_range)
1.98
1.99 lemma compact_approx [simp]: "compact (approx n\<cdot>x)"
1.100 -proof (rule compactI2)
1.101 - fix Y::"nat \<Rightarrow> 'a"
1.102 - assume Y: "chain Y"
1.103 - have "finite_chain (\<lambda>i. approx n\<cdot>(Y i))"
1.104 - proof (rule finite_range_imp_finch)
1.105 - show "chain (\<lambda>i. approx n\<cdot>(Y i))"
1.106 - using Y by simp
1.107 - have "range (\<lambda>i. approx n\<cdot>(Y i)) \<subseteq> {x. approx n\<cdot>x = x}"
1.108 - by clarsimp
1.109 - thus "finite (range (\<lambda>i. approx n\<cdot>(Y i)))"
1.110 - using finite_fixes_approx by (rule finite_subset)
1.111 - qed
1.112 - hence "\<exists>j. (\<Squnion>i. approx n\<cdot>(Y i)) = approx n\<cdot>(Y j)"
1.113 - by (simp add: finite_chain_def maxinch_is_thelub Y)
1.114 - then obtain j where j: "(\<Squnion>i. approx n\<cdot>(Y i)) = approx n\<cdot>(Y j)" ..
1.115 -
1.116 - assume "approx n\<cdot>x \<sqsubseteq> (\<Squnion>i. Y i)"
1.117 - hence "approx n\<cdot>(approx n\<cdot>x) \<sqsubseteq> approx n\<cdot>(\<Squnion>i. Y i)"
1.118 - by (rule monofun_cfun_arg)
1.119 - hence "approx n\<cdot>x \<sqsubseteq> (\<Squnion>i. approx n\<cdot>(Y i))"
1.120 - by (simp add: contlub_cfun_arg Y)
1.121 - hence "approx n\<cdot>x \<sqsubseteq> approx n\<cdot>(Y j)"
1.122 - using j by simp
1.123 - hence "approx n\<cdot>x \<sqsubseteq> Y j"
1.124 - using approx_less by (rule trans_less)
1.125 - thus "\<exists>j. approx n\<cdot>x \<sqsubseteq> Y j" ..
1.126 -qed
1.127 +by (rule approx.compact)
1.128
1.129 lemma profinite_compact_eq_approx: "compact x \<Longrightarrow> \<exists>i. approx i\<cdot>x = x"
1.130 -by (rule admD2) simp_all
1.131 +by (rule admD2, simp_all)
1.132
1.133 lemma profinite_compact_iff: "compact x \<longleftrightarrow> (\<exists>n. approx n\<cdot>x = x)"
1.134 apply (rule iffI)
1.135 @@ -133,25 +105,34 @@
1.136
1.137 subsection {* Instance for continuous function space *}
1.138
1.139 -lemma finite_range_lemma:
1.140 - fixes h :: "'a::cpo \<rightarrow> 'b::cpo"
1.141 - fixes k :: "'c::cpo \<rightarrow> 'd::cpo"
1.142 - shows "\<lbrakk>finite {y. \<exists>x. y = h\<cdot>x}; finite {y. \<exists>x. y = k\<cdot>x}\<rbrakk>
1.143 - \<Longrightarrow> finite {g. \<exists>f. g = (\<Lambda> x. k\<cdot>(f\<cdot>(h\<cdot>x)))}"
1.144 - apply (rule_tac f="\<lambda>g. {(h\<cdot>x, y) |x y. y = g\<cdot>x}" in finite_imageD)
1.145 - apply (rule_tac B="Pow ({y. \<exists>x. y = h\<cdot>x} \<times> {y. \<exists>x. y = k\<cdot>x})"
1.146 - in finite_subset)
1.147 - apply (rule image_subsetI)
1.148 - apply (clarsimp, fast)
1.149 - apply simp
1.150 - apply (rule inj_onI)
1.151 - apply (clarsimp simp add: expand_set_eq)
1.152 - apply (rule ext_cfun, simp)
1.153 - apply (drule_tac x="h\<cdot>x" in spec)
1.154 - apply (drule_tac x="k\<cdot>(f\<cdot>(h\<cdot>x))" in spec)
1.155 - apply (drule iffD1, fast)
1.156 - apply clarsimp
1.157 -done
1.158 +lemma finite_range_cfun_lemma:
1.159 + assumes a: "finite (range (\<lambda>x. a\<cdot>x))"
1.160 + assumes b: "finite (range (\<lambda>y. b\<cdot>y))"
1.161 + shows "finite (range (\<lambda>f. \<Lambda> x. b\<cdot>(f\<cdot>(a\<cdot>x))))" (is "finite (range ?h)")
1.162 +proof (rule finite_imageD)
1.163 + let ?f = "\<lambda>g. range (\<lambda>x. (a\<cdot>x, g\<cdot>x))"
1.164 + show "finite (?f ` range ?h)"
1.165 + proof (rule finite_subset)
1.166 + let ?B = "Pow (range (\<lambda>x. a\<cdot>x) \<times> range (\<lambda>y. b\<cdot>y))"
1.167 + show "?f ` range ?h \<subseteq> ?B"
1.168 + by clarsimp
1.169 + show "finite ?B"
1.170 + by (simp add: a b)
1.171 + qed
1.172 + show "inj_on ?f (range ?h)"
1.173 + proof (rule inj_onI, rule ext_cfun, clarsimp)
1.174 + fix x f g
1.175 + 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))))"
1.176 + 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))))"
1.177 + by (rule equalityD1)
1.178 + 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))))"
1.179 + by (simp add: subset_eq)
1.180 + then obtain y where "(a\<cdot>x, b\<cdot>(f\<cdot>(a\<cdot>x))) = (a\<cdot>y, b\<cdot>(g\<cdot>(a\<cdot>y)))"
1.181 + by (rule rangeE)
1.182 + thus "b\<cdot>(f\<cdot>(a\<cdot>x)) = b\<cdot>(g\<cdot>(a\<cdot>x))"
1.183 + by clarsimp
1.184 + qed
1.185 +qed
1.186
1.187 instantiation "->" :: (profinite, profinite) profinite
1.188 begin
1.189 @@ -160,15 +141,26 @@
1.190 approx_cfun_def:
1.191 "approx = (\<lambda>n. \<Lambda> f x. approx n\<cdot>(f\<cdot>(approx n\<cdot>x)))"
1.192
1.193 -instance
1.194 - apply (intro_classes, unfold approx_cfun_def)
1.195 - apply simp
1.196 - apply (simp add: lub_distribs eta_cfun)
1.197 - apply simp
1.198 - apply simp
1.199 - apply (rule finite_range_imp_finite_fixes)
1.200 - apply (intro finite_range_lemma finite_approx)
1.201 -done
1.202 +instance proof
1.203 + show "chain (approx :: nat \<Rightarrow> ('a \<rightarrow> 'b) \<rightarrow> ('a \<rightarrow> 'b))"
1.204 + unfolding approx_cfun_def by simp
1.205 +next
1.206 + fix x :: "'a \<rightarrow> 'b"
1.207 + show "(\<Squnion>i. approx i\<cdot>x) = x"
1.208 + unfolding approx_cfun_def
1.209 + by (simp add: lub_distribs eta_cfun)
1.210 +next
1.211 + fix i :: nat and x :: "'a \<rightarrow> 'b"
1.212 + show "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
1.213 + unfolding approx_cfun_def by simp
1.214 +next
1.215 + fix i :: nat
1.216 + show "finite {x::'a \<rightarrow> 'b. approx i\<cdot>x = x}"
1.217 + apply (rule finite_range_imp_finite_fixes)
1.218 + apply (simp add: approx_cfun_def)
1.219 + apply (intro finite_range_cfun_lemma finite_range_approx)
1.220 + done
1.221 +qed
1.222
1.223 end
1.224