author | haftmann |
Fri, 09 Mar 2007 08:45:50 +0100 | |
changeset 22422 | ee19cdb07528 |
parent 22390 | 378f34b1e380 |
child 22430 | 6a56bf1b3a64 |
permissions | -rw-r--r-- |
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(* Title: HOL/FixedPoint.thy |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Author: Stefan Berghofer, TU Muenchen |
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Copyright 1992 University of Cambridge |
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*) |
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header{* Fixed Points and the Knaster-Tarski Theorem*} |
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theory FixedPoint |
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imports Product_Type LOrder |
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begin |
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subsection {* Complete lattices *} |
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consts |
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Inf :: "'a::order set \<Rightarrow> 'a" |
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definition |
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Sup :: "'a::order set \<Rightarrow> 'a" where |
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"Sup A = Inf {b. \<forall>a \<in> A. a \<le> b}" |
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definition |
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SUP :: "('a \<Rightarrow> 'b::order) \<Rightarrow> 'b" (binder "SUP " 10) where |
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"(SUP x. f x) = Sup (f ` UNIV)" |
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(* |
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abbreviation |
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bot :: "'a::order" where |
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"bot == Sup {}" |
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*) |
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class comp_lat = order + |
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assumes Inf_lower: "x \<in> A \<Longrightarrow> Inf A \<sqsubseteq> x" |
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assumes Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> Inf A" |
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theorem Sup_upper: "(x::'a::comp_lat) \<in> A \<Longrightarrow> x <= Sup A" |
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by (auto simp: Sup_def intro: Inf_greatest) |
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theorem Sup_least: "(\<And>x::'a::comp_lat. x \<in> A \<Longrightarrow> x <= z) \<Longrightarrow> Sup A <= z" |
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by (auto simp: Sup_def intro: Inf_lower) |
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text {* A complete lattice is a lattice *} |
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lemma is_meet_Inf: "is_meet (\<lambda>(x::'a::comp_lat) y. Inf {x, y})" |
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by (auto simp: is_meet_def intro: Inf_lower Inf_greatest) |
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lemma is_join_Sup: "is_join (\<lambda>(x::'a::comp_lat) y. Sup {x, y})" |
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by (auto simp: is_join_def intro: Sup_upper Sup_least) |
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instance comp_lat < lorder |
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proof |
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from is_meet_Inf show "\<exists>m::'a\<Rightarrow>'a\<Rightarrow>'a. is_meet m" by iprover |
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from is_join_Sup show "\<exists>j::'a\<Rightarrow>'a\<Rightarrow>'a. is_join j" by iprover |
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qed |
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subsubsection {* Properties *} |
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lemma mono_inf: "mono f \<Longrightarrow> f (inf A B) <= inf (f A) (f B)" |
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by (auto simp add: mono_def) |
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lemma mono_sup: "mono f \<Longrightarrow> sup (f A) (f B) <= f (sup A B)" |
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by (auto simp add: mono_def) |
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lemma Sup_insert[simp]: "Sup (insert (a::'a::comp_lat) A) = sup a (Sup A)" |
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apply(simp add:Sup_def) |
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apply(rule order_antisym) |
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apply(rule Inf_lower) |
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apply(clarsimp) |
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apply(rule le_supI2) |
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apply(rule Inf_greatest) |
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apply blast |
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apply simp |
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apply rule |
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apply(rule Inf_greatest)apply blast |
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apply(rule Inf_greatest) |
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apply(rule Inf_lower) |
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apply blast |
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done |
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lemma bot_least[simp]: "Sup{} \<le> (x::'a::comp_lat)" |
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apply(simp add: Sup_def) |
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apply(rule Inf_lower) |
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apply blast |
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done |
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(* |
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lemma Inf_singleton[simp]: "Inf{a} = (a::'a::comp_lat)" |
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apply(rule order_antisym) |
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apply(simp add: Inf_lower) |
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apply(rule Inf_greatest) |
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apply(simp) |
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done |
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*) |
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lemma le_SupI: "(l::'a::comp_lat) : M \<Longrightarrow> l \<le> Sup M" |
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apply(simp add:Sup_def) |
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apply(rule Inf_greatest) |
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apply(simp) |
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done |
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lemma le_SUPI: "(l::'a::comp_lat) = M i \<Longrightarrow> l \<le> (SUP i. M i)" |
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apply(simp add:SUP_def) |
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apply(blast intro:le_SupI) |
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done |
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lemma Sup_leI: "(!!x. x:M \<Longrightarrow> x \<le> u) \<Longrightarrow> Sup M \<le> (u::'a::comp_lat)" |
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apply(simp add:Sup_def) |
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apply(rule Inf_lower) |
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apply(blast) |
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done |
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lemma SUP_leI: "(!!i. M i \<le> u) \<Longrightarrow> (SUP i. M i) \<le> (u::'a::comp_lat)" |
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apply(simp add:SUP_def) |
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apply(blast intro!:Sup_leI) |
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done |
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lemma SUP_const[simp]: "(SUP i. M) = (M::'a::comp_lat)" |
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by(simp add:SUP_def image_constant_conv sup_absorb1) |
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subsection {* Some instances of the type class of complete lattices *} |
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subsubsection {* Booleans *} |
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defs |
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Inf_bool_def: "Inf A == ALL x:A. x" |
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instance bool :: comp_lat |
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apply intro_classes |
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apply (unfold Inf_bool_def) |
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apply (iprover intro!: le_boolI elim: ballE) |
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apply (iprover intro!: ballI le_boolI elim: ballE le_boolE) |
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done |
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theorem inf_bool_eq: "inf P Q \<longleftrightarrow> P \<and> Q" |
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apply (rule order_antisym) |
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apply (rule le_boolI) |
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apply (rule conjI) |
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apply (rule le_boolE) |
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apply (rule inf_le1) |
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apply assumption+ |
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apply (rule le_boolE) |
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apply (rule inf_le2) |
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apply assumption+ |
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apply (rule le_infI) |
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apply (rule le_boolI) |
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apply (erule conjunct1) |
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apply (rule le_boolI) |
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apply (erule conjunct2) |
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done |
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theorem sup_bool_eq: "sup P Q \<longleftrightarrow> P \<or> Q" |
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apply (rule order_antisym) |
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apply (rule le_supI) |
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apply (rule le_boolI) |
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apply (erule disjI1) |
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apply (rule le_boolI) |
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apply (erule disjI2) |
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apply (rule le_boolI) |
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apply (erule disjE) |
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apply (rule le_boolE) |
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apply (rule sup_ge1) |
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apply assumption+ |
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apply (rule le_boolE) |
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apply (rule sup_ge2) |
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apply assumption+ |
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done |
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theorem Sup_bool_eq: "Sup A \<longleftrightarrow> (\<exists>x \<in> A. x)" |
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apply (rule order_antisym) |
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apply (rule Sup_least) |
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apply (rule le_boolI) |
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apply (erule bexI, assumption) |
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apply (rule le_boolI) |
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apply (erule bexE) |
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apply (rule le_boolE) |
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apply (rule Sup_upper) |
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apply assumption+ |
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178 |
done |
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|
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|
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181 |
subsubsection {* Functions *} |
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|
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183 |
text {* |
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Handy introduction and elimination rules for @{text "\<le>"} |
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185 |
on unary and binary predicates |
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186 |
*} |
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187 |
|
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188 |
lemma predicate1I [Pure.intro!, intro!]: |
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189 |
assumes PQ: "\<And>x. P x \<Longrightarrow> Q x" |
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190 |
shows "P \<le> Q" |
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191 |
apply (rule le_funI) |
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192 |
apply (rule le_boolI) |
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193 |
apply (rule PQ) |
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|
194 |
apply assumption |
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195 |
done |
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196 |
|
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197 |
lemma predicate1D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x" |
21017
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198 |
apply (erule le_funE) |
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|
199 |
apply (erule le_boolE) |
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|
200 |
apply assumption+ |
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|
201 |
done |
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202 |
|
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203 |
lemma predicate2I [Pure.intro!, intro!]: |
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204 |
assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y" |
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205 |
shows "P \<le> Q" |
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|
206 |
apply (rule le_funI)+ |
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207 |
apply (rule le_boolI) |
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|
208 |
apply (rule PQ) |
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|
209 |
apply assumption |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
210 |
done |
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|
211 |
|
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212 |
lemma predicate2D [Pure.dest, dest]: "P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y" |
21017
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|
213 |
apply (erule le_funE)+ |
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|
214 |
apply (erule le_boolE) |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
215 |
apply assumption+ |
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|
216 |
done |
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|
217 |
|
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218 |
lemma rev_predicate1D: "P x ==> P <= Q ==> Q x" |
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|
219 |
by (rule predicate1D) |
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|
220 |
|
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|
221 |
lemma rev_predicate2D: "P x y ==> P <= Q ==> Q x y" |
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222 |
by (rule predicate2D) |
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|
223 |
|
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224 |
defs |
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225 |
Inf_fun_def: "Inf A == (\<lambda>x. Inf {y. EX f:A. y = f x})" |
21017
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226 |
|
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|
227 |
instance "fun" :: (type, comp_lat) comp_lat |
5693e4471c2b
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|
228 |
apply intro_classes |
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229 |
apply (unfold Inf_fun_def) |
21017
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230 |
apply (rule le_funI) |
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231 |
apply (rule Inf_lower) |
21017
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|
232 |
apply (rule CollectI) |
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|
233 |
apply (rule bexI) |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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|
234 |
apply (rule refl) |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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|
235 |
apply assumption |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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|
236 |
apply (rule le_funI) |
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237 |
apply (rule Inf_greatest) |
21017
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Generalized gfp and lfp to arbitrary complete lattices.
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|
238 |
apply (erule CollectE) |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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changeset
|
239 |
apply (erule bexE) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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|
240 |
apply (iprover elim: le_funE) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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|
241 |
done |
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|
242 |
|
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|
243 |
theorem inf_fun_eq: "inf f g = (\<lambda>x. inf (f x) (g x))" |
21017
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Generalized gfp and lfp to arbitrary complete lattices.
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|
244 |
apply (rule order_antisym) |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
245 |
apply (rule le_funI) |
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|
246 |
apply (rule le_infI) |
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|
247 |
apply (rule le_funD [OF inf_le1]) |
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|
248 |
apply (rule le_funD [OF inf_le2]) |
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|
249 |
apply (rule le_infI) |
21017
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Generalized gfp and lfp to arbitrary complete lattices.
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|
250 |
apply (rule le_funI) |
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|
251 |
apply (rule inf_le1) |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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|
252 |
apply (rule le_funI) |
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|
253 |
apply (rule inf_le2) |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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|
254 |
done |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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|
255 |
|
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|
256 |
theorem sup_fun_eq: "sup f g = (\<lambda>x. sup (f x) (g x))" |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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|
257 |
apply (rule order_antisym) |
22422
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|
258 |
apply (rule le_supI) |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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|
259 |
apply (rule le_funI) |
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|
260 |
apply (rule sup_ge1) |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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changeset
|
261 |
apply (rule le_funI) |
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changeset
|
262 |
apply (rule sup_ge2) |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
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|
263 |
apply (rule le_funI) |
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|
264 |
apply (rule le_supI) |
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changeset
|
265 |
apply (rule le_funD [OF sup_ge1]) |
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|
266 |
apply (rule le_funD [OF sup_ge2]) |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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changeset
|
267 |
done |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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|
268 |
|
21312 | 269 |
theorem Sup_fun_eq: "Sup A = (\<lambda>x. Sup {y::'a::comp_lat. EX f:A. y = f x})" |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
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|
270 |
apply (rule order_antisym) |
21312 | 271 |
apply (rule Sup_least) |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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|
272 |
apply (rule le_funI) |
21312 | 273 |
apply (rule Sup_upper) |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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changeset
|
274 |
apply fast |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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changeset
|
275 |
apply (rule le_funI) |
21312 | 276 |
apply (rule Sup_least) |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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|
277 |
apply (erule CollectE) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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changeset
|
278 |
apply (erule bexE) |
21312 | 279 |
apply (drule le_funD [OF Sup_upper]) |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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changeset
|
280 |
apply simp |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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changeset
|
281 |
done |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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changeset
|
282 |
|
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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changeset
|
283 |
subsubsection {* Sets *} |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
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changeset
|
284 |
|
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|
285 |
defs |
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|
286 |
Inf_set_def: "Inf S == \<Inter>S" |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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changeset
|
287 |
|
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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|
288 |
instance set :: (type) comp_lat |
22422
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|
289 |
by intro_classes (auto simp add: Inf_set_def) |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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changeset
|
290 |
|
22422
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|
291 |
theorem inf_set_eq: "inf A B = A \<inter> B" |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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changeset
|
292 |
apply (rule subset_antisym) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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changeset
|
293 |
apply (rule Int_greatest) |
22422
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haftmann
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changeset
|
294 |
apply (rule inf_le1) |
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haftmann
parents:
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changeset
|
295 |
apply (rule inf_le2) |
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haftmann
parents:
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changeset
|
296 |
apply (rule le_infI) |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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changeset
|
297 |
apply (rule Int_lower1) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
298 |
apply (rule Int_lower2) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
299 |
done |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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changeset
|
300 |
|
22422
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changeset
|
301 |
theorem sup_set_eq: "sup A B = A \<union> B" |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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changeset
|
302 |
apply (rule subset_antisym) |
22422
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haftmann
parents:
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changeset
|
303 |
apply (rule le_supI) |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
304 |
apply (rule Un_upper1) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
305 |
apply (rule Un_upper2) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
306 |
apply (rule Un_least) |
22422
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stepping towards uniform lattice theory development in HOL
haftmann
parents:
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diff
changeset
|
307 |
apply (rule sup_ge1) |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
308 |
apply (rule sup_ge2) |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
309 |
done |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
310 |
|
21312 | 311 |
theorem Sup_set_eq: "Sup S = \<Union>S" |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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changeset
|
312 |
apply (rule subset_antisym) |
21312 | 313 |
apply (rule Sup_least) |
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314 |
apply (erule Union_upper) |
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|
315 |
apply (rule Union_least) |
21312 | 316 |
apply (erule Sup_upper) |
21017
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|
317 |
done |
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|
318 |
|
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|
319 |
|
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|
320 |
subsection {* Least and greatest fixed points *} |
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|
321 |
|
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322 |
definition |
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323 |
lfp :: "('a\<Colon>comp_lat \<Rightarrow> 'a) \<Rightarrow> 'a" where |
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324 |
"lfp f = Inf {u. f u \<le> u}" --{*least fixed point*} |
17006 | 325 |
|
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326 |
definition |
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327 |
gfp :: "('a\<Colon>comp_lat \<Rightarrow> 'a) \<Rightarrow> 'a" where |
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328 |
"gfp f = Sup {u. u \<le> f u}" --{*greatest fixed point*} |
17006 | 329 |
|
330 |
||
331 |
subsection{*Proof of Knaster-Tarski Theorem using @{term lfp}*} |
|
332 |
||
333 |
text{*@{term "lfp f"} is the least upper bound of |
|
21017
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|
334 |
the set @{term "{u. f(u) \<le> u}"} *} |
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|
335 |
|
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|
336 |
lemma lfp_lowerbound: "f A \<le> A ==> lfp f \<le> A" |
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337 |
by (auto simp add: lfp_def intro: Inf_lower) |
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|
338 |
|
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|
339 |
lemma lfp_greatest: "(!!u. f u \<le> u ==> A \<le> u) ==> A \<le> lfp f" |
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340 |
by (auto simp add: lfp_def intro: Inf_greatest) |
17006 | 341 |
|
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|
342 |
lemma lfp_lemma2: "mono f ==> f (lfp f) \<le> lfp f" |
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|
343 |
by (iprover intro: lfp_greatest order_trans monoD lfp_lowerbound) |
17006 | 344 |
|
21017
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|
345 |
lemma lfp_lemma3: "mono f ==> lfp f \<le> f (lfp f)" |
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|
346 |
by (iprover intro: lfp_lemma2 monoD lfp_lowerbound) |
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|
347 |
|
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|
348 |
lemma lfp_unfold: "mono f ==> lfp f = f (lfp f)" |
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|
349 |
by (iprover intro: order_antisym lfp_lemma2 lfp_lemma3) |
17006 | 350 |
|
22356 | 351 |
lemma lfp_const: "lfp (\<lambda>x. t) = t" |
352 |
by (rule lfp_unfold) (simp add:mono_def) |
|
353 |
||
21017
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|
354 |
subsection{*General induction rules for least fixed points*} |
17006 | 355 |
|
21017
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|
356 |
theorem lfp_induct: |
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|
357 |
assumes mono: "mono f" and ind: "f (inf (lfp f) P) <= P" |
21017
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|
358 |
shows "lfp f <= P" |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
359 |
proof - |
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360 |
have "inf (lfp f) P <= lfp f" by (rule inf_le1) |
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|
361 |
with mono have "f (inf (lfp f) P) <= f (lfp f)" .. |
21017
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Generalized gfp and lfp to arbitrary complete lattices.
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|
362 |
also from mono have "f (lfp f) = lfp f" by (rule lfp_unfold [symmetric]) |
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|
363 |
finally have "f (inf (lfp f) P) <= lfp f" . |
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|
364 |
from this and ind have "f (inf (lfp f) P) <= inf (lfp f) P" by (rule le_infI) |
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|
365 |
hence "lfp f <= inf (lfp f) P" by (rule lfp_lowerbound) |
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|
366 |
also have "inf (lfp f) P <= P" by (rule inf_le2) |
21017
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|
367 |
finally show ?thesis . |
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Generalized gfp and lfp to arbitrary complete lattices.
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|
368 |
qed |
17006 | 369 |
|
21017
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|
370 |
lemma lfp_induct_set: |
17006 | 371 |
assumes lfp: "a: lfp(f)" |
372 |
and mono: "mono(f)" |
|
373 |
and indhyp: "!!x. [| x: f(lfp(f) Int {x. P(x)}) |] ==> P(x)" |
|
374 |
shows "P(a)" |
|
21017
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|
375 |
by (rule lfp_induct [THEN subsetD, THEN CollectD, OF mono _ lfp]) |
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376 |
(auto simp: inf_set_eq intro: indhyp) |
17006 | 377 |
|
378 |
||
379 |
text{*Version of induction for binary relations*} |
|
21017
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|
380 |
lemmas lfp_induct2 = lfp_induct_set [of "(a,b)", split_format (complete)] |
17006 | 381 |
|
382 |
||
383 |
lemma lfp_ordinal_induct: |
|
384 |
assumes mono: "mono f" |
|
385 |
shows "[| !!S. P S ==> P(f S); !!M. !S:M. P S ==> P(Union M) |] |
|
386 |
==> P(lfp f)" |
|
387 |
apply(subgoal_tac "lfp f = Union{S. S \<subseteq> lfp f & P S}") |
|
388 |
apply (erule ssubst, simp) |
|
389 |
apply(subgoal_tac "Union{S. S \<subseteq> lfp f & P S} \<subseteq> lfp f") |
|
390 |
prefer 2 apply blast |
|
391 |
apply(rule equalityI) |
|
392 |
prefer 2 apply assumption |
|
393 |
apply(drule mono [THEN monoD]) |
|
394 |
apply (cut_tac mono [THEN lfp_unfold], simp) |
|
395 |
apply (rule lfp_lowerbound, auto) |
|
396 |
done |
|
397 |
||
398 |
||
399 |
text{*Definition forms of @{text lfp_unfold} and @{text lfp_induct}, |
|
400 |
to control unfolding*} |
|
401 |
||
402 |
lemma def_lfp_unfold: "[| h==lfp(f); mono(f) |] ==> h = f(h)" |
|
403 |
by (auto intro!: lfp_unfold) |
|
404 |
||
405 |
lemma def_lfp_induct: |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
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|
406 |
"[| A == lfp(f); mono(f); |
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|
407 |
f (inf A P) \<le> P |
21017
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Generalized gfp and lfp to arbitrary complete lattices.
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|
408 |
|] ==> A \<le> P" |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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diff
changeset
|
409 |
by (blast intro: lfp_induct) |
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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changeset
|
410 |
|
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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changeset
|
411 |
lemma def_lfp_induct_set: |
17006 | 412 |
"[| A == lfp(f); mono(f); a:A; |
413 |
!!x. [| x: f(A Int {x. P(x)}) |] ==> P(x) |
|
414 |
|] ==> P(a)" |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
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changeset
|
415 |
by (blast intro: lfp_induct_set) |
17006 | 416 |
|
417 |
(*Monotonicity of lfp!*) |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
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|
418 |
lemma lfp_mono: "(!!Z. f Z \<le> g Z) ==> lfp f \<le> lfp g" |
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Generalized gfp and lfp to arbitrary complete lattices.
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changeset
|
419 |
by (rule lfp_lowerbound [THEN lfp_greatest], blast intro: order_trans) |
17006 | 420 |
|
421 |
||
422 |
subsection{*Proof of Knaster-Tarski Theorem using @{term gfp}*} |
|
423 |
||
424 |
||
425 |
text{*@{term "gfp f"} is the greatest lower bound of |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
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changeset
|
426 |
the set @{term "{u. u \<le> f(u)}"} *} |
17006 | 427 |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
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changeset
|
428 |
lemma gfp_upperbound: "X \<le> f X ==> X \<le> gfp f" |
21312 | 429 |
by (auto simp add: gfp_def intro: Sup_upper) |
17006 | 430 |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
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changeset
|
431 |
lemma gfp_least: "(!!u. u \<le> f u ==> u \<le> X) ==> gfp f \<le> X" |
21312 | 432 |
by (auto simp add: gfp_def intro: Sup_least) |
17006 | 433 |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
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changeset
|
434 |
lemma gfp_lemma2: "mono f ==> gfp f \<le> f (gfp f)" |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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diff
changeset
|
435 |
by (iprover intro: gfp_least order_trans monoD gfp_upperbound) |
17006 | 436 |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
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changeset
|
437 |
lemma gfp_lemma3: "mono f ==> f (gfp f) \<le> gfp f" |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
438 |
by (iprover intro: gfp_lemma2 monoD gfp_upperbound) |
17006 | 439 |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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changeset
|
440 |
lemma gfp_unfold: "mono f ==> gfp f = f (gfp f)" |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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diff
changeset
|
441 |
by (iprover intro: order_antisym gfp_lemma2 gfp_lemma3) |
17006 | 442 |
|
443 |
subsection{*Coinduction rules for greatest fixed points*} |
|
444 |
||
445 |
text{*weak version*} |
|
446 |
lemma weak_coinduct: "[| a: X; X \<subseteq> f(X) |] ==> a : gfp(f)" |
|
447 |
by (rule gfp_upperbound [THEN subsetD], auto) |
|
448 |
||
449 |
lemma weak_coinduct_image: "!!X. [| a : X; g`X \<subseteq> f (g`X) |] ==> g a : gfp f" |
|
450 |
apply (erule gfp_upperbound [THEN subsetD]) |
|
451 |
apply (erule imageI) |
|
452 |
done |
|
453 |
||
454 |
lemma coinduct_lemma: |
|
22422
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|
455 |
"[| X \<le> f (sup X (gfp f)); mono f |] ==> sup X (gfp f) \<le> f (sup X (gfp f))" |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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diff
changeset
|
456 |
apply (frule gfp_lemma2) |
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|
457 |
apply (drule mono_sup) |
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|
458 |
apply (rule le_supI) |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
459 |
apply assumption |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
460 |
apply (rule order_trans) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
461 |
apply (rule order_trans) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
462 |
apply assumption |
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|
463 |
apply (rule sup_ge2) |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
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parents:
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diff
changeset
|
464 |
apply assumption |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
465 |
done |
17006 | 466 |
|
467 |
text{*strong version, thanks to Coen and Frost*} |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
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changeset
|
468 |
lemma coinduct_set: "[| mono(f); a: X; X \<subseteq> f(X Un gfp(f)) |] ==> a : gfp(f)" |
22422
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changeset
|
469 |
by (blast intro: weak_coinduct [OF _ coinduct_lemma, simplified sup_set_eq]) |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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changeset
|
470 |
|
22422
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|
471 |
lemma coinduct: "[| mono(f); X \<le> f (sup X (gfp f)) |] ==> X \<le> gfp(f)" |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
472 |
apply (rule order_trans) |
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changeset
|
473 |
apply (rule sup_ge1) |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
474 |
apply (erule gfp_upperbound [OF coinduct_lemma]) |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
475 |
apply assumption |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
476 |
done |
17006 | 477 |
|
478 |
lemma gfp_fun_UnI2: "[| mono(f); a: gfp(f) |] ==> a: f(X Un gfp(f))" |
|
479 |
by (blast dest: gfp_lemma2 mono_Un) |
|
480 |
||
481 |
subsection{*Even Stronger Coinduction Rule, by Martin Coen*} |
|
482 |
||
483 |
text{* Weakens the condition @{term "X \<subseteq> f(X)"} to one expressed using both |
|
484 |
@{term lfp} and @{term gfp}*} |
|
485 |
||
486 |
lemma coinduct3_mono_lemma: "mono(f) ==> mono(%x. f(x) Un X Un B)" |
|
17589 | 487 |
by (iprover intro: subset_refl monoI Un_mono monoD) |
17006 | 488 |
|
489 |
lemma coinduct3_lemma: |
|
490 |
"[| X \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f))); mono(f) |] |
|
491 |
==> lfp(%x. f(x) Un X Un gfp(f)) \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f)))" |
|
492 |
apply (rule subset_trans) |
|
493 |
apply (erule coinduct3_mono_lemma [THEN lfp_lemma3]) |
|
494 |
apply (rule Un_least [THEN Un_least]) |
|
495 |
apply (rule subset_refl, assumption) |
|
496 |
apply (rule gfp_unfold [THEN equalityD1, THEN subset_trans], assumption) |
|
497 |
apply (rule monoD, assumption) |
|
498 |
apply (subst coinduct3_mono_lemma [THEN lfp_unfold], auto) |
|
499 |
done |
|
500 |
||
501 |
lemma coinduct3: |
|
502 |
"[| mono(f); a:X; X \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f))) |] ==> a : gfp(f)" |
|
503 |
apply (rule coinduct3_lemma [THEN [2] weak_coinduct]) |
|
504 |
apply (rule coinduct3_mono_lemma [THEN lfp_unfold, THEN ssubst], auto) |
|
505 |
done |
|
506 |
||
507 |
||
508 |
text{*Definition forms of @{text gfp_unfold} and @{text coinduct}, |
|
509 |
to control unfolding*} |
|
510 |
||
511 |
lemma def_gfp_unfold: "[| A==gfp(f); mono(f) |] ==> A = f(A)" |
|
512 |
by (auto intro!: gfp_unfold) |
|
513 |
||
514 |
lemma def_coinduct: |
|
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515 |
"[| A==gfp(f); mono(f); X \<le> f(sup X A) |] ==> X \<le> A" |
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Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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|
516 |
by (iprover intro!: coinduct) |
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Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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changeset
|
517 |
|
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Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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|
518 |
lemma def_coinduct_set: |
17006 | 519 |
"[| A==gfp(f); mono(f); a:X; X \<subseteq> f(X Un A) |] ==> a: A" |
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Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
520 |
by (auto intro!: coinduct_set) |
17006 | 521 |
|
522 |
(*The version used in the induction/coinduction package*) |
|
523 |
lemma def_Collect_coinduct: |
|
524 |
"[| A == gfp(%w. Collect(P(w))); mono(%w. Collect(P(w))); |
|
525 |
a: X; !!z. z: X ==> P (X Un A) z |] ==> |
|
526 |
a : A" |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
527 |
apply (erule def_coinduct_set, auto) |
17006 | 528 |
done |
529 |
||
530 |
lemma def_coinduct3: |
|
531 |
"[| A==gfp(f); mono(f); a:X; X \<subseteq> f(lfp(%x. f(x) Un X Un A)) |] ==> a: A" |
|
532 |
by (auto intro!: coinduct3) |
|
533 |
||
534 |
text{*Monotonicity of @{term gfp}!*} |
|
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Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
changeset
|
535 |
lemma gfp_mono: "(!!Z. f Z \<le> g Z) ==> gfp f \<le> gfp g" |
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Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
536 |
by (rule gfp_upperbound [THEN gfp_least], blast intro: order_trans) |
17006 | 537 |
|
538 |
||
21312 | 539 |
|
17006 | 540 |
ML |
541 |
{* |
|
542 |
val lfp_def = thm "lfp_def"; |
|
543 |
val lfp_lowerbound = thm "lfp_lowerbound"; |
|
544 |
val lfp_greatest = thm "lfp_greatest"; |
|
545 |
val lfp_unfold = thm "lfp_unfold"; |
|
546 |
val lfp_induct = thm "lfp_induct"; |
|
547 |
val lfp_induct2 = thm "lfp_induct2"; |
|
548 |
val lfp_ordinal_induct = thm "lfp_ordinal_induct"; |
|
549 |
val def_lfp_unfold = thm "def_lfp_unfold"; |
|
550 |
val def_lfp_induct = thm "def_lfp_induct"; |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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diff
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|
551 |
val def_lfp_induct_set = thm "def_lfp_induct_set"; |
17006 | 552 |
val lfp_mono = thm "lfp_mono"; |
553 |
val gfp_def = thm "gfp_def"; |
|
554 |
val gfp_upperbound = thm "gfp_upperbound"; |
|
555 |
val gfp_least = thm "gfp_least"; |
|
556 |
val gfp_unfold = thm "gfp_unfold"; |
|
557 |
val weak_coinduct = thm "weak_coinduct"; |
|
558 |
val weak_coinduct_image = thm "weak_coinduct_image"; |
|
559 |
val coinduct = thm "coinduct"; |
|
560 |
val gfp_fun_UnI2 = thm "gfp_fun_UnI2"; |
|
561 |
val coinduct3 = thm "coinduct3"; |
|
562 |
val def_gfp_unfold = thm "def_gfp_unfold"; |
|
563 |
val def_coinduct = thm "def_coinduct"; |
|
564 |
val def_Collect_coinduct = thm "def_Collect_coinduct"; |
|
565 |
val def_coinduct3 = thm "def_coinduct3"; |
|
566 |
val gfp_mono = thm "gfp_mono"; |
|
21017
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Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
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|
567 |
val le_funI = thm "le_funI"; |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
568 |
val le_boolI = thm "le_boolI"; |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
569 |
val le_boolI' = thm "le_boolI'"; |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
570 |
val inf_fun_eq = thm "inf_fun_eq"; |
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22390
diff
changeset
|
571 |
val inf_bool_eq = thm "inf_bool_eq"; |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
572 |
val le_funE = thm "le_funE"; |
22276
96a4db55a0b3
Introduction and elimination rules for <= on predicates
berghofe
parents:
21547
diff
changeset
|
573 |
val le_funD = thm "le_funD"; |
21017
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
574 |
val le_boolE = thm "le_boolE"; |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
575 |
val le_boolD = thm "le_boolD"; |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
576 |
val le_bool_def = thm "le_bool_def"; |
5693e4471c2b
Generalized gfp and lfp to arbitrary complete lattices.
berghofe
parents:
17589
diff
changeset
|
577 |
val le_fun_def = thm "le_fun_def"; |
17006 | 578 |
*} |
579 |
||
580 |
end |