--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOLCF/Bifinite.thy Mon Jan 14 19:26:01 2008 +0100
@@ -0,0 +1,198 @@
+(* Title: HOLCF/Bifinite.thy
+ ID: $Id$
+ Author: Brian Huffman
+*)
+
+header {* Bifinite domains and approximation *}
+
+theory Bifinite
+imports Cfun
+begin
+
+subsection {* Bifinite domains *}
+
+axclass approx < pcpo
+
+consts approx :: "nat \<Rightarrow> 'a::approx \<rightarrow> 'a"
+
+axclass bifinite < approx
+ chain_approx_app: "chain (\<lambda>i. approx i\<cdot>x)"
+ lub_approx_app [simp]: "(\<Squnion>i. approx i\<cdot>x) = x"
+ approx_idem: "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x"
+ finite_fixes_approx: "finite {x. approx i\<cdot>x = x}"
+
+lemma finite_range_imp_finite_fixes:
+ "finite {x. \<exists>y. x = f y} \<Longrightarrow> finite {x. f x = x}"
+apply (subgoal_tac "{x. f x = x} \<subseteq> {x. \<exists>y. x = f y}")
+apply (erule (1) finite_subset)
+apply (clarify, erule subst, rule exI, rule refl)
+done
+
+lemma chain_approx [simp]:
+ "chain (approx :: nat \<Rightarrow> 'a::bifinite \<rightarrow> 'a)"
+apply (rule chainI)
+apply (rule less_cfun_ext)
+apply (rule chainE)
+apply (rule chain_approx_app)
+done
+
+lemma lub_approx [simp]: "(\<Squnion>i. approx i) = (\<Lambda>(x::'a::bifinite). x)"
+by (rule ext_cfun, simp add: contlub_cfun_fun)
+
+lemma approx_less: "approx i\<cdot>x \<sqsubseteq> (x::'a::bifinite)"
+apply (subgoal_tac "approx i\<cdot>x \<sqsubseteq> (\<Squnion>i. approx i\<cdot>x)", simp)
+apply (rule is_ub_thelub, simp)
+done
+
+lemma approx_strict [simp]: "approx i\<cdot>(\<bottom>::'a::bifinite) = \<bottom>"
+by (rule UU_I, rule approx_less)
+
+lemma approx_approx1:
+ "i \<le> j \<Longrightarrow> approx i\<cdot>(approx j\<cdot>x) = approx i\<cdot>(x::'a::bifinite)"
+apply (rule antisym_less)
+apply (rule monofun_cfun_arg [OF approx_less])
+apply (rule sq_ord_eq_less_trans [OF approx_idem [symmetric]])
+apply (rule monofun_cfun_arg)
+apply (rule monofun_cfun_fun)
+apply (erule chain_mono3 [OF chain_approx])
+done
+
+lemma approx_approx2:
+ "j \<le> i \<Longrightarrow> approx i\<cdot>(approx j\<cdot>x) = approx j\<cdot>(x::'a::bifinite)"
+apply (rule antisym_less)
+apply (rule approx_less)
+apply (rule sq_ord_eq_less_trans [OF approx_idem [symmetric]])
+apply (rule monofun_cfun_fun)
+apply (erule chain_mono3 [OF chain_approx])
+done
+
+lemma approx_approx [simp]:
+ "approx i\<cdot>(approx j\<cdot>x) = approx (min i j)\<cdot>(x::'a::bifinite)"
+apply (rule_tac x=i and y=j in linorder_le_cases)
+apply (simp add: approx_approx1 min_def)
+apply (simp add: approx_approx2 min_def)
+done
+
+lemma idem_fixes_eq_range:
+ "\<forall>x. f (f x) = f x \<Longrightarrow> {x. f x = x} = {y. \<exists>x. y = f x}"
+by (auto simp add: eq_sym_conv)
+
+lemma finite_approx: "finite {y::'a::bifinite. \<exists>x. y = approx n\<cdot>x}"
+using finite_fixes_approx by (simp add: idem_fixes_eq_range)
+
+lemma finite_range_approx:
+ "finite (range (\<lambda>x::'a::bifinite. approx n\<cdot>x))"
+by (simp add: image_def finite_approx)
+
+lemma compact_approx [simp]:
+ fixes x :: "'a::bifinite"
+ shows "compact (approx n\<cdot>x)"
+proof (rule compactI2)
+ fix Y::"nat \<Rightarrow> 'a"
+ assume Y: "chain Y"
+ have "finite_chain (\<lambda>i. approx n\<cdot>(Y i))"
+ proof (rule finite_range_imp_finch)
+ show "chain (\<lambda>i. approx n\<cdot>(Y i))"
+ using Y by simp
+ have "range (\<lambda>i. approx n\<cdot>(Y i)) \<subseteq> {x. approx n\<cdot>x = x}"
+ by clarsimp
+ thus "finite (range (\<lambda>i. approx n\<cdot>(Y i)))"
+ using finite_fixes_approx by (rule finite_subset)
+ qed
+ hence "\<exists>j. (\<Squnion>i. approx n\<cdot>(Y i)) = approx n\<cdot>(Y j)"
+ by (simp add: finite_chain_def maxinch_is_thelub Y)
+ then obtain j where j: "(\<Squnion>i. approx n\<cdot>(Y i)) = approx n\<cdot>(Y j)" ..
+
+ assume "approx n\<cdot>x \<sqsubseteq> (\<Squnion>i. Y i)"
+ hence "approx n\<cdot>(approx n\<cdot>x) \<sqsubseteq> approx n\<cdot>(\<Squnion>i. Y i)"
+ by (rule monofun_cfun_arg)
+ hence "approx n\<cdot>x \<sqsubseteq> (\<Squnion>i. approx n\<cdot>(Y i))"
+ by (simp add: contlub_cfun_arg Y)
+ hence "approx n\<cdot>x \<sqsubseteq> approx n\<cdot>(Y j)"
+ using j by simp
+ hence "approx n\<cdot>x \<sqsubseteq> Y j"
+ using approx_less by (rule trans_less)
+ thus "\<exists>j. approx n\<cdot>x \<sqsubseteq> Y j" ..
+qed
+
+lemma bifinite_compact_eq_approx:
+ fixes x :: "'a::bifinite"
+ assumes x: "compact x"
+ shows "\<exists>i. approx i\<cdot>x = x"
+proof -
+ have chain: "chain (\<lambda>i. approx i\<cdot>x)" by simp
+ have less: "x \<sqsubseteq> (\<Squnion>i. approx i\<cdot>x)" by simp
+ obtain i where i: "x \<sqsubseteq> approx i\<cdot>x"
+ using compactD2 [OF x chain less] ..
+ with approx_less have "approx i\<cdot>x = x"
+ by (rule antisym_less)
+ thus "\<exists>i. approx i\<cdot>x = x" ..
+qed
+
+lemma bifinite_compact_iff:
+ "compact (x::'a::bifinite) = (\<exists>n. approx n\<cdot>x = x)"
+ apply (rule iffI)
+ apply (erule bifinite_compact_eq_approx)
+ apply (erule exE)
+ apply (erule subst)
+ apply (rule compact_approx)
+done
+
+lemma approx_induct:
+ assumes adm: "adm P" and P: "\<And>n x. P (approx n\<cdot>x)"
+ shows "P (x::'a::bifinite)"
+proof -
+ have "P (\<Squnion>n. approx n\<cdot>x)"
+ by (rule admD [OF adm], simp, simp add: P)
+ thus "P x" by simp
+qed
+
+lemma bifinite_less_ext:
+ fixes x y :: "'a::bifinite"
+ shows "(\<And>i. approx i\<cdot>x \<sqsubseteq> approx i\<cdot>y) \<Longrightarrow> x \<sqsubseteq> y"
+apply (subgoal_tac "(\<Squnion>i. approx i\<cdot>x) \<sqsubseteq> (\<Squnion>i. approx i\<cdot>y)", simp)
+apply (rule lub_mono [rule_format], simp, simp, simp)
+done
+
+subsection {* Instance for continuous function space *}
+
+lemma finite_range_lemma:
+ fixes h :: "'a::cpo \<rightarrow> 'b::cpo"
+ fixes k :: "'c::cpo \<rightarrow> 'd::cpo"
+ shows "\<lbrakk>finite {y. \<exists>x. y = h\<cdot>x}; finite {y. \<exists>x. y = k\<cdot>x}\<rbrakk>
+ \<Longrightarrow> finite {g. \<exists>f. g = (\<Lambda> x. k\<cdot>(f\<cdot>(h\<cdot>x)))}"
+ apply (rule_tac f="\<lambda>g. {(h\<cdot>x, y) |x y. y = g\<cdot>x}" in finite_imageD)
+ apply (rule_tac B="Pow ({y. \<exists>x. y = h\<cdot>x} \<times> {y. \<exists>x. y = k\<cdot>x})"
+ in finite_subset)
+ apply (rule image_subsetI)
+ apply (clarsimp, fast)
+ apply simp
+ apply (rule inj_onI)
+ apply (clarsimp simp add: expand_set_eq)
+ apply (rule ext_cfun, simp)
+ apply (drule_tac x="h\<cdot>x" in spec)
+ apply (drule_tac x="k\<cdot>(f\<cdot>(h\<cdot>x))" in spec)
+ apply (drule iffD1, fast)
+ apply clarsimp
+done
+
+instance "->" :: (bifinite, bifinite) approx ..
+
+defs (overloaded)
+ approx_cfun_def:
+ "approx \<equiv> \<lambda>n. \<Lambda> f x. approx n\<cdot>(f\<cdot>(approx n\<cdot>x))"
+
+instance "->" :: (bifinite, bifinite) bifinite
+ apply (intro_classes, unfold approx_cfun_def)
+ apply simp
+ apply (simp add: lub_distribs eta_cfun)
+ apply simp
+ apply simp
+ apply (rule finite_range_imp_finite_fixes)
+ apply (intro finite_range_lemma finite_approx)
+done
+
+lemma approx_cfun: "approx n\<cdot>f\<cdot>x = approx n\<cdot>(f\<cdot>(approx n\<cdot>x))"
+by (simp add: approx_cfun_def)
+
+end