author | huffman |
Mon, 19 May 2008 23:49:20 +0200 | |
changeset 26962 | c8b20f615d6c |
parent 26407 | 562a1d615336 |
child 27186 | 416d66c36d8f |
permissions | -rw-r--r-- |
25903 | 1 |
(* Title: HOLCF/Bifinite.thy |
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ID: $Id$ |
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Author: Brian Huffman |
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*) |
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header {* Bifinite domains and approximation *} |
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theory Bifinite |
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imports Cfun |
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begin |
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subsection {* Omega-profinite and bifinite domains *} |
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class profinite = cpo + |
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fixes approx :: "nat \<Rightarrow> 'a \<rightarrow> 'a" |
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assumes chain_approx_app: "chain (\<lambda>i. approx i\<cdot>x)" |
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assumes lub_approx_app [simp]: "(\<Squnion>i. approx i\<cdot>x) = x" |
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assumes approx_idem: "approx i\<cdot>(approx i\<cdot>x) = approx i\<cdot>x" |
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assumes finite_fixes_approx: "finite {x. approx i\<cdot>x = x}" |
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class bifinite = profinite + pcpo |
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lemma finite_range_imp_finite_fixes: |
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"finite {x. \<exists>y. x = f y} \<Longrightarrow> finite {x. f x = x}" |
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apply (subgoal_tac "{x. f x = x} \<subseteq> {x. \<exists>y. x = f y}") |
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apply (erule (1) finite_subset) |
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apply (clarify, erule subst, rule exI, rule refl) |
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done |
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lemma chain_approx [simp]: |
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"chain (approx :: nat \<Rightarrow> 'a::profinite \<rightarrow> 'a)" |
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apply (rule chainI) |
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apply (rule less_cfun_ext) |
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apply (rule chainE) |
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apply (rule chain_approx_app) |
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done |
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||
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lemma lub_approx [simp]: "(\<Squnion>i. approx i) = (\<Lambda>(x::'a::profinite). x)" |
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by (rule ext_cfun, simp add: contlub_cfun_fun) |
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lemma approx_less: "approx i\<cdot>x \<sqsubseteq> (x::'a::profinite)" |
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apply (subgoal_tac "approx i\<cdot>x \<sqsubseteq> (\<Squnion>i. approx i\<cdot>x)", simp) |
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apply (rule is_ub_thelub, simp) |
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done |
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lemma approx_strict [simp]: "approx i\<cdot>(\<bottom>::'a::bifinite) = \<bottom>" |
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by (rule UU_I, rule approx_less) |
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lemma approx_approx1: |
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"i \<le> j \<Longrightarrow> approx i\<cdot>(approx j\<cdot>x) = approx i\<cdot>(x::'a::profinite)" |
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apply (rule antisym_less) |
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apply (rule monofun_cfun_arg [OF approx_less]) |
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apply (rule sq_ord_eq_less_trans [OF approx_idem [symmetric]]) |
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apply (rule monofun_cfun_arg) |
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apply (rule monofun_cfun_fun) |
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apply (erule chain_mono [OF chain_approx]) |
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done |
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lemma approx_approx2: |
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"j \<le> i \<Longrightarrow> approx i\<cdot>(approx j\<cdot>x) = approx j\<cdot>(x::'a::profinite)" |
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apply (rule antisym_less) |
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apply (rule approx_less) |
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apply (rule sq_ord_eq_less_trans [OF approx_idem [symmetric]]) |
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apply (rule monofun_cfun_fun) |
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apply (erule chain_mono [OF chain_approx]) |
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done |
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lemma approx_approx [simp]: |
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"approx i\<cdot>(approx j\<cdot>x) = approx (min i j)\<cdot>(x::'a::profinite)" |
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apply (rule_tac x=i and y=j in linorder_le_cases) |
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apply (simp add: approx_approx1 min_def) |
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apply (simp add: approx_approx2 min_def) |
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done |
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lemma idem_fixes_eq_range: |
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"\<forall>x. f (f x) = f x \<Longrightarrow> {x. f x = x} = {y. \<exists>x. y = f x}" |
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by (auto simp add: eq_sym_conv) |
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lemma finite_approx: "finite {y::'a::profinite. \<exists>x. y = approx n\<cdot>x}" |
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using finite_fixes_approx by (simp add: idem_fixes_eq_range) |
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lemma finite_range_approx: |
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"finite (range (\<lambda>x::'a::profinite. approx n\<cdot>x))" |
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by (simp add: image_def finite_approx) |
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lemma compact_approx [simp]: |
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fixes x :: "'a::profinite" |
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shows "compact (approx n\<cdot>x)" |
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proof (rule compactI2) |
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fix Y::"nat \<Rightarrow> 'a" |
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assume Y: "chain Y" |
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have "finite_chain (\<lambda>i. approx n\<cdot>(Y i))" |
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proof (rule finite_range_imp_finch) |
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show "chain (\<lambda>i. approx n\<cdot>(Y i))" |
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using Y by simp |
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have "range (\<lambda>i. approx n\<cdot>(Y i)) \<subseteq> {x. approx n\<cdot>x = x}" |
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by clarsimp |
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thus "finite (range (\<lambda>i. approx n\<cdot>(Y i)))" |
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using finite_fixes_approx by (rule finite_subset) |
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qed |
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hence "\<exists>j. (\<Squnion>i. approx n\<cdot>(Y i)) = approx n\<cdot>(Y j)" |
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by (simp add: finite_chain_def maxinch_is_thelub Y) |
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then obtain j where j: "(\<Squnion>i. approx n\<cdot>(Y i)) = approx n\<cdot>(Y j)" .. |
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assume "approx n\<cdot>x \<sqsubseteq> (\<Squnion>i. Y i)" |
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hence "approx n\<cdot>(approx n\<cdot>x) \<sqsubseteq> approx n\<cdot>(\<Squnion>i. Y i)" |
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by (rule monofun_cfun_arg) |
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hence "approx n\<cdot>x \<sqsubseteq> (\<Squnion>i. approx n\<cdot>(Y i))" |
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by (simp add: contlub_cfun_arg Y) |
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hence "approx n\<cdot>x \<sqsubseteq> approx n\<cdot>(Y j)" |
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using j by simp |
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hence "approx n\<cdot>x \<sqsubseteq> Y j" |
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using approx_less by (rule trans_less) |
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thus "\<exists>j. approx n\<cdot>x \<sqsubseteq> Y j" .. |
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qed |
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lemma bifinite_compact_eq_approx: |
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fixes x :: "'a::profinite" |
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assumes x: "compact x" |
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shows "\<exists>i. approx i\<cdot>x = x" |
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proof - |
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have chain: "chain (\<lambda>i. approx i\<cdot>x)" by simp |
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have less: "x \<sqsubseteq> (\<Squnion>i. approx i\<cdot>x)" by simp |
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obtain i where i: "x \<sqsubseteq> approx i\<cdot>x" |
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using compactD2 [OF x chain less] .. |
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with approx_less have "approx i\<cdot>x = x" |
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by (rule antisym_less) |
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thus "\<exists>i. approx i\<cdot>x = x" .. |
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qed |
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lemma bifinite_compact_iff: |
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"compact (x::'a::profinite) = (\<exists>n. approx n\<cdot>x = x)" |
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apply (rule iffI) |
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apply (erule bifinite_compact_eq_approx) |
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apply (erule exE) |
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apply (erule subst) |
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apply (rule compact_approx) |
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done |
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lemma approx_induct: |
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assumes adm: "adm P" and P: "\<And>n x. P (approx n\<cdot>x)" |
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shows "P (x::'a::profinite)" |
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proof - |
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have "P (\<Squnion>n. approx n\<cdot>x)" |
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by (rule admD [OF adm], simp, simp add: P) |
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thus "P x" by simp |
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qed |
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lemma bifinite_less_ext: |
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fixes x y :: "'a::profinite" |
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shows "(\<And>i. approx i\<cdot>x \<sqsubseteq> approx i\<cdot>y) \<Longrightarrow> x \<sqsubseteq> y" |
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apply (subgoal_tac "(\<Squnion>i. approx i\<cdot>x) \<sqsubseteq> (\<Squnion>i. approx i\<cdot>y)", simp) |
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apply (rule lub_mono, simp, simp, simp) |
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done |
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subsection {* Instance for continuous function space *} |
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lemma finite_range_lemma: |
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fixes h :: "'a::cpo \<rightarrow> 'b::cpo" |
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fixes k :: "'c::cpo \<rightarrow> 'd::cpo" |
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shows "\<lbrakk>finite {y. \<exists>x. y = h\<cdot>x}; finite {y. \<exists>x. y = k\<cdot>x}\<rbrakk> |
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\<Longrightarrow> finite {g. \<exists>f. g = (\<Lambda> x. k\<cdot>(f\<cdot>(h\<cdot>x)))}" |
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apply (rule_tac f="\<lambda>g. {(h\<cdot>x, y) |x y. y = g\<cdot>x}" in finite_imageD) |
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apply (rule_tac B="Pow ({y. \<exists>x. y = h\<cdot>x} \<times> {y. \<exists>x. y = k\<cdot>x})" |
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in finite_subset) |
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apply (rule image_subsetI) |
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apply (clarsimp, fast) |
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apply simp |
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apply (rule inj_onI) |
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apply (clarsimp simp add: expand_set_eq) |
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apply (rule ext_cfun, simp) |
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apply (drule_tac x="h\<cdot>x" in spec) |
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apply (drule_tac x="k\<cdot>(f\<cdot>(h\<cdot>x))" in spec) |
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apply (drule iffD1, fast) |
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apply clarsimp |
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done |
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instantiation "->" :: (profinite, profinite) profinite |
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begin |
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definition |
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approx_cfun_def: |
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"approx = (\<lambda>n. \<Lambda> f x. approx n\<cdot>(f\<cdot>(approx n\<cdot>x)))" |
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instance |
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apply (intro_classes, unfold approx_cfun_def) |
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apply simp |
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apply (simp add: lub_distribs eta_cfun) |
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apply simp |
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apply simp |
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apply (rule finite_range_imp_finite_fixes) |
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apply (intro finite_range_lemma finite_approx) |
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done |
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parents:
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end |
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huffman
parents:
26407
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changeset
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instance "->" :: (profinite, bifinite) bifinite .. |
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lemma approx_cfun: "approx n\<cdot>f\<cdot>x = approx n\<cdot>(f\<cdot>(approx n\<cdot>x))" |
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by (simp add: approx_cfun_def) |
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end |