author  haftmann 
Fri, 20 Jul 2007 14:27:56 +0200  
changeset 23878  bd651ecd4b8a 
parent 23389  aaca6a8e5414 
child 23948  261bd4678076 
permissions  rwrr 
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(* Title: HOL/Lattices.thy 
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ID: $Id$ 

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Author: Tobias Nipkow 

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*) 

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header {* Abstract lattices *} 
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theory Lattices 

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imports Orderings 

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begin 

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subsection{* Lattices *} 

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class lower_semilattice = order + 
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fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70) 
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assumes inf_le1 [simp]: "x \<sqinter> y \<sqsubseteq> x" 
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and inf_le2 [simp]: "x \<sqinter> y \<sqsubseteq> y" 

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and inf_greatest: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<sqinter> z" 
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class upper_semilattice = order + 
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fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65) 
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assumes sup_ge1 [simp]: "x \<sqsubseteq> x \<squnion> y" 
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and sup_ge2 [simp]: "y \<sqsubseteq> x \<squnion> y" 

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and sup_least: "y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<squnion> z \<sqsubseteq> x" 
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class lattice = lower_semilattice + upper_semilattice 
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subsubsection{* Intro and elim rules*} 
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context lower_semilattice 

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begin 

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lemmas antisym_intro [intro!] = antisym 
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lemmas (in ) [rule del] = antisym_intro 
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lemma le_infI1[intro]: "a \<sqsubseteq> x \<Longrightarrow> a \<sqinter> b \<sqsubseteq> x" 
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apply(subgoal_tac "a \<sqinter> b \<sqsubseteq> a") 
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apply(blast intro: order_trans) 
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apply simp 
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done 

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lemmas (in ) [rule del] = le_infI1 
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lemma le_infI2[intro]: "b \<sqsubseteq> x \<Longrightarrow> a \<sqinter> b \<sqsubseteq> x" 
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apply(subgoal_tac "a \<sqinter> b \<sqsubseteq> b") 
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apply(blast intro: order_trans) 
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apply simp 
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done 

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lemmas (in ) [rule del] = le_infI2 
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lemma le_infI[intro!]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<sqinter> b" 
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by(blast intro: inf_greatest) 
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lemmas (in ) [rule del] = le_infI 
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lemma le_infE [elim!]: "x \<sqsubseteq> a \<sqinter> b \<Longrightarrow> (x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> b \<Longrightarrow> P) \<Longrightarrow> P" 
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by (blast intro: order_trans) 
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lemmas (in ) [rule del] = le_infE 
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lemma le_inf_iff [simp]: 
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"x \<sqsubseteq> y \<sqinter> z = (x \<sqsubseteq> y \<and> x \<sqsubseteq> z)" 
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by blast 

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lemma le_iff_inf: "(x \<sqsubseteq> y) = (x \<sqinter> y = x)" 
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by(blast dest:eq_iff[THEN iffD1]) 
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end 
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lemma mono_inf: "mono f \<Longrightarrow> f (inf A B) \<le> inf (f A) (f B)" 
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by (auto simp add: mono_def) 

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context upper_semilattice 

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begin 

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lemmas antisym_intro [intro!] = antisym 
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lemmas (in ) [rule del] = antisym_intro 
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lemma le_supI1[intro]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> a \<squnion> b" 
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apply(subgoal_tac "a \<sqsubseteq> a \<squnion> b") 
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apply(blast intro: order_trans) 
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apply simp 
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done 

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lemmas (in ) [rule del] = le_supI1 
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lemma le_supI2[intro]: "x \<sqsubseteq> b \<Longrightarrow> x \<sqsubseteq> a \<squnion> b" 
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apply(subgoal_tac "b \<sqsubseteq> a \<squnion> b") 
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apply(blast intro: order_trans) 
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apply simp 
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done 

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lemmas (in ) [rule del] = le_supI2 
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lemma le_supI[intro!]: "a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> a \<squnion> b \<sqsubseteq> x" 
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by(blast intro: sup_least) 
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lemmas (in ) [rule del] = le_supI 
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lemma le_supE[elim!]: "a \<squnion> b \<sqsubseteq> x \<Longrightarrow> (a \<sqsubseteq> x \<Longrightarrow> b \<sqsubseteq> x \<Longrightarrow> P) \<Longrightarrow> P" 
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by (blast intro: order_trans) 
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lemmas (in ) [rule del] = le_supE 
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lemma ge_sup_conv[simp]: 
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"x \<squnion> y \<sqsubseteq> z = (x \<sqsubseteq> z \<and> y \<sqsubseteq> z)" 
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by blast 

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lemma le_iff_sup: "(x \<sqsubseteq> y) = (x \<squnion> y = y)" 
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by(blast dest:eq_iff[THEN iffD1]) 
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end 
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lemma mono_sup: "mono f \<Longrightarrow> sup (f A) (f B) \<le> f (sup A B)" 
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by (auto simp add: mono_def) 

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subsubsection{* Equational laws *} 

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context lower_semilattice 
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begin 

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lemma inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)" 

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by blast 

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lemma inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)" 

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by blast 

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lemma inf_idem[simp]: "x \<sqinter> x = x" 

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by blast 

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lemma inf_left_idem[simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y" 

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by blast 

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lemma inf_absorb1: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x" 

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by blast 

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lemma inf_absorb2: "y \<sqsubseteq> x \<Longrightarrow> x \<sqinter> y = y" 

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by blast 

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lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)" 

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by blast 

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lemmas inf_ACI = inf_commute inf_assoc inf_left_commute inf_left_idem 

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142 
end 

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context upper_semilattice 

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begin 

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lemma sup_commute: "(x \<squnion> y) = (y \<squnion> x)" 
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by blast 

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lemma sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)" 

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by blast 

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lemma sup_idem[simp]: "x \<squnion> x = x" 

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by blast 

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lemma sup_left_idem[simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y" 

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by blast 

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lemma sup_absorb1: "y \<sqsubseteq> x \<Longrightarrow> x \<squnion> y = x" 

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by blast 

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lemma sup_absorb2: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y" 

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by blast 

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lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)" 
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by blast 

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lemmas sup_ACI = sup_commute sup_assoc sup_left_commute sup_left_idem 

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end 

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context lattice 
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begin 

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lemma inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x" 

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by(blast intro: antisym inf_le1 inf_greatest sup_ge1) 

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lemma sup_inf_absorb: "x \<squnion> (x \<sqinter> y) = x" 

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by(blast intro: antisym sup_ge1 sup_least inf_le1) 

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lemmas ACI = inf_ACI sup_ACI 
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lemmas inf_sup_ord = inf_le1 inf_le2 sup_ge1 sup_ge2 
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text{* Towards distributivity *} 
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lemma distrib_sup_le: "x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> (x \<squnion> z)" 
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by blast 

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lemma distrib_inf_le: "(x \<sqinter> y) \<squnion> (x \<sqinter> z) \<sqsubseteq> x \<sqinter> (y \<squnion> z)" 

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by blast 

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text{* If you have one of them, you have them all. *} 

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lemma distrib_imp1: 
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assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" 
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shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)" 

200 
proof 

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have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by(simp add:sup_inf_absorb) 

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also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))" by(simp add:D inf_commute sup_assoc) 

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also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)" 

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by(simp add:inf_sup_absorb inf_commute) 

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also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D) 

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finally show ?thesis . 

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qed 

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lemma distrib_imp2: 
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assumes D: "!!x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)" 
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shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" 

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proof 

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have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by(simp add:inf_sup_absorb) 

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also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))" by(simp add:D sup_commute inf_assoc) 

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also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)" 

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by(simp add:sup_inf_absorb sup_commute) 

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also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D) 

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finally show ?thesis . 

219 
qed 

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(* seems unused *) 
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lemma modular_le: "x \<sqsubseteq> z \<Longrightarrow> x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> z" 

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by blast 

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end 
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subsection{* Distributive lattices *} 

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class distrib_lattice = lattice + 
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assumes sup_inf_distrib1: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)" 
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context distrib_lattice 
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begin 

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236 
lemma sup_inf_distrib2: 

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"(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)" 
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by(simp add:ACI sup_inf_distrib1) 

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lemma inf_sup_distrib1: 
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"x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" 
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by(rule distrib_imp2[OF sup_inf_distrib1]) 

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lemma inf_sup_distrib2: 
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"(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)" 
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by(simp add:ACI inf_sup_distrib1) 

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lemmas distrib = 
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sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2 
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end 
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subsection {* Uniqueness of inf and sup *} 
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lemma (in lower_semilattice) inf_unique: 
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fixes f (infixl "\<triangle>" 70) 
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assumes le1: "\<And>x y. x \<triangle> y \<^loc>\<le> x" and le2: "\<And>x y. x \<triangle> y \<^loc>\<le> y" 
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and greatest: "\<And>x y z. x \<^loc>\<le> y \<Longrightarrow> x \<^loc>\<le> z \<Longrightarrow> x \<^loc>\<le> y \<triangle> z" 

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shows "x \<sqinter> y = x \<triangle> y" 

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proof (rule antisym) 
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show "x \<triangle> y \<^loc>\<le> x \<sqinter> y" by (rule le_infI) (rule le1, rule le2) 
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next 
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have leI: "\<And>x y z. x \<^loc>\<le> y \<Longrightarrow> x \<^loc>\<le> z \<Longrightarrow> x \<^loc>\<le> y \<triangle> z" by (blast intro: greatest) 
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show "x \<sqinter> y \<^loc>\<le> x \<triangle> y" by (rule leI) simp_all 

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qed 
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lemma (in upper_semilattice) sup_unique: 
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fixes f (infixl "\<nabla>" 70) 
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assumes ge1 [simp]: "\<And>x y. x \<^loc>\<le> x \<nabla> y" and ge2: "\<And>x y. y \<^loc>\<le> x \<nabla> y" 
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and least: "\<And>x y z. y \<^loc>\<le> x \<Longrightarrow> z \<^loc>\<le> x \<Longrightarrow> y \<nabla> z \<^loc>\<le> x" 

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shows "x \<squnion> y = x \<nabla> y" 

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proof (rule antisym) 
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show "x \<squnion> y \<^loc>\<le> x \<nabla> y" by (rule le_supI) (rule ge1, rule ge2) 
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next 
22737  276 
have leI: "\<And>x y z. x \<^loc>\<le> z \<Longrightarrow> y \<^loc>\<le> z \<Longrightarrow> x \<nabla> y \<^loc>\<le> z" by (blast intro: least) 
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show "x \<nabla> y \<^loc>\<le> x \<squnion> y" by (rule leI) simp_all 

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qed 
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subsection {* @{const min}/@{const max} on linear orders as 
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special case of @{const inf}/@{const sup} *} 

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lemma (in linorder) distrib_lattice_min_max: 

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"distrib_lattice (op \<^loc>\<le>) (op \<^loc><) min max" 
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proof unfold_locales 
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have aux: "\<And>x y \<Colon> 'a. x \<^loc>< y \<Longrightarrow> y \<^loc>\<le> x \<Longrightarrow> x = y" 

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by (auto simp add: less_le antisym) 

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fix x y z 

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show "max x (min y z) = min (max x y) (max x z)" 

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unfolding min_def max_def 

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by (auto simp add: intro: antisym, unfold not_le, 

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auto intro: less_trans le_less_trans aux) 

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qed (auto simp add: min_def max_def not_le less_imp_le) 

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interpretation min_max: 

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distrib_lattice ["op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool" "op <" min max] 
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by (rule distrib_lattice_min_max [folded ord_class.min ord_class.max]) 
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lemma inf_min: "inf = (min \<Colon> 'a\<Colon>{lower_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)" 
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by (rule ext)+ auto 

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lemma sup_max: "sup = (max \<Colon> 'a\<Colon>{upper_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)" 
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by (rule ext)+ auto 

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lemmas le_maxI1 = min_max.sup_ge1 
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lemmas le_maxI2 = min_max.sup_ge2 

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lemmas max_ac = min_max.sup_assoc min_max.sup_commute 
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mk_left_commute [of max, OF min_max.sup_assoc min_max.sup_commute] 
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lemmas min_ac = min_max.inf_assoc min_max.inf_commute 

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mk_left_commute [of min, OF min_max.inf_assoc min_max.inf_commute] 
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text {* 
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Now we have inherited antisymmetry as an introrule on all 

317 
linear orders. This is a problem because it applies to bool, which is 

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undesirable. 

319 
*} 

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lemmas [rule del] = min_max.antisym_intro min_max.le_infI min_max.le_supI 

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min_max.le_supE min_max.le_infE min_max.le_supI1 min_max.le_supI2 

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min_max.le_infI1 min_max.le_infI2 

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subsection {* Complete lattices *} 
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class complete_lattice = lattice + 

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fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900) 

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assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x" 

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assumes Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A" 

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begin 

333 

334 
definition 

335 
Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900) 

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where 

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"\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<^loc>\<le> b}" 

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lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<^loc>\<le> a}" 

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unfolding Sup_def by (auto intro: Inf_greatest Inf_lower) 

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lemma Sup_upper: "x \<in> A \<Longrightarrow> x \<^loc>\<le> \<Squnion>A" 

343 
by (auto simp: Sup_def intro: Inf_greatest) 

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345 
lemma Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<^loc>\<le> z) \<Longrightarrow> \<Squnion>A \<^loc>\<le> z" 

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by (auto simp: Sup_def intro: Inf_lower) 

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lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}" 

349 
unfolding Sup_def by auto 

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lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}" 

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unfolding Inf_Sup by auto 

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354 
lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A" 

355 
apply (rule antisym) 

356 
apply (rule le_infI) 

357 
apply (rule Inf_lower) 

358 
apply simp 

359 
apply (rule Inf_greatest) 

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apply (rule Inf_lower) 

361 
apply simp 

362 
apply (rule Inf_greatest) 

363 
apply (erule insertE) 

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apply (rule le_infI1) 

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apply simp 

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apply (rule le_infI2) 

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apply (erule Inf_lower) 

368 
done 

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370 
lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A" 

371 
apply (rule antisym) 

372 
apply (rule Sup_least) 

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apply (erule insertE) 

374 
apply (rule le_supI1) 

375 
apply simp 

376 
apply (rule le_supI2) 

377 
apply (erule Sup_upper) 

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apply (rule le_supI) 

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apply (rule Sup_upper) 

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apply simp 

381 
apply (rule Sup_least) 

382 
apply (rule Sup_upper) 

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apply simp 

384 
done 

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386 
lemma Inf_singleton [simp]: 

387 
"\<Sqinter>{a} = a" 

388 
by (auto intro: antisym Inf_lower Inf_greatest) 

389 

390 
lemma Sup_singleton [simp]: 

391 
"\<Squnion>{a} = a" 

392 
by (auto intro: antisym Sup_upper Sup_least) 

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394 
lemma Inf_insert_simp: 

395 
"\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)" 

396 
by (cases "A = {}") (simp_all, simp add: Inf_insert) 

397 

398 
lemma Sup_insert_simp: 

399 
"\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)" 

400 
by (cases "A = {}") (simp_all, simp add: Sup_insert) 

401 

402 
lemma Inf_binary: 

403 
"\<Sqinter>{a, b} = a \<sqinter> b" 

404 
by (simp add: Inf_insert_simp) 

405 

406 
lemma Sup_binary: 

407 
"\<Squnion>{a, b} = a \<squnion> b" 

408 
by (simp add: Sup_insert_simp) 

409 

410 
end 

411 

412 
lemmas Sup_def = Sup_def [folded complete_lattice_class.Sup] 

413 
lemmas Sup_upper = Sup_upper [folded complete_lattice_class.Sup] 

414 
lemmas Sup_least = Sup_least [folded complete_lattice_class.Sup] 

415 

416 
lemmas Sup_insert [code func] = Sup_insert [folded complete_lattice_class.Sup] 

417 
lemmas Sup_singleton [simp, code func] = Sup_singleton [folded complete_lattice_class.Sup] 

418 
lemmas Sup_insert_simp = Sup_insert_simp [folded complete_lattice_class.Sup] 

419 
lemmas Sup_binary = Sup_binary [folded complete_lattice_class.Sup] 

420 

421 
definition 

422 
top :: "'a::complete_lattice" 

423 
where 

424 
"top = Inf {}" 

425 

426 
definition 

427 
bot :: "'a::complete_lattice" 

428 
where 

429 
"bot = Sup {}" 

430 

431 
lemma top_greatest [simp]: "x \<le> top" 

432 
by (unfold top_def, rule Inf_greatest, simp) 

433 

434 
lemma bot_least [simp]: "bot \<le> x" 

435 
by (unfold bot_def, rule Sup_least, simp) 

436 

437 
definition 

438 
SUPR :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b::complete_lattice) \<Rightarrow> 'b" 

439 
where 

440 
"SUPR A f == Sup (f ` A)" 

441 

442 
definition 

443 
INFI :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b::complete_lattice) \<Rightarrow> 'b" 

444 
where 

445 
"INFI A f == Inf (f ` A)" 

446 

447 
syntax 

448 
"_SUP1" :: "pttrns => 'b => 'b" ("(3SUP _./ _)" [0, 10] 10) 

449 
"_SUP" :: "pttrn => 'a set => 'b => 'b" ("(3SUP _:_./ _)" [0, 10] 10) 

450 
"_INF1" :: "pttrns => 'b => 'b" ("(3INF _./ _)" [0, 10] 10) 

451 
"_INF" :: "pttrn => 'a set => 'b => 'b" ("(3INF _:_./ _)" [0, 10] 10) 

452 

453 
translations 

454 
"SUP x y. B" == "SUP x. SUP y. B" 

455 
"SUP x. B" == "CONST SUPR UNIV (%x. B)" 

456 
"SUP x. B" == "SUP x:UNIV. B" 

457 
"SUP x:A. B" == "CONST SUPR A (%x. B)" 

458 
"INF x y. B" == "INF x. INF y. B" 

459 
"INF x. B" == "CONST INFI UNIV (%x. B)" 

460 
"INF x. B" == "INF x:UNIV. B" 

461 
"INF x:A. B" == "CONST INFI A (%x. B)" 

462 

463 
(* To avoid etacontraction of body: *) 

464 
print_translation {* 

465 
let 

466 
fun btr' syn (A :: Abs abs :: ts) = 

467 
let val (x,t) = atomic_abs_tr' abs 

468 
in list_comb (Syntax.const syn $ x $ A $ t, ts) end 

469 
val const_syntax_name = Sign.const_syntax_name @{theory} o fst o dest_Const 

470 
in 

471 
[(const_syntax_name @{term SUPR}, btr' "_SUP"),(const_syntax_name @{term "INFI"}, btr' "_INF")] 

472 
end 

473 
*} 

474 

475 
lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)" 

476 
by (auto simp add: SUPR_def intro: Sup_upper) 

477 

478 
lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u" 

479 
by (auto simp add: SUPR_def intro: Sup_least) 

480 

481 
lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i" 

482 
by (auto simp add: INFI_def intro: Inf_lower) 

483 

484 
lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)" 

485 
by (auto simp add: INFI_def intro: Inf_greatest) 

486 

487 
lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M" 

488 
by (auto intro: order_antisym SUP_leI le_SUPI) 

489 

490 
lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M" 

491 
by (auto intro: order_antisym INF_leI le_INFI) 

492 

493 

22454  494 
subsection {* Bool as lattice *} 
495 

496 
instance bool :: distrib_lattice 

497 
inf_bool_eq: "inf P Q \<equiv> P \<and> Q" 

498 
sup_bool_eq: "sup P Q \<equiv> P \<or> Q" 

499 
by intro_classes (auto simp add: inf_bool_eq sup_bool_eq le_bool_def) 

500 

23878  501 
instance bool :: complete_lattice 
502 
Inf_bool_def: "Inf A \<equiv> \<forall>x\<in>A. x" 

503 
apply intro_classes 

504 
apply (unfold Inf_bool_def) 

505 
apply (iprover intro!: le_boolI elim: ballE) 

506 
apply (iprover intro!: ballI le_boolI elim: ballE le_boolE) 

507 
done 

22454  508 

23878  509 
theorem Sup_bool_eq: "Sup A \<longleftrightarrow> (\<exists>x\<in>A. x)" 
510 
apply (rule order_antisym) 

511 
apply (rule Sup_least) 

512 
apply (rule le_boolI) 

513 
apply (erule bexI, assumption) 

514 
apply (rule le_boolI) 

515 
apply (erule bexE) 

516 
apply (rule le_boolE) 

517 
apply (rule Sup_upper) 

518 
apply assumption+ 

519 
done 

520 

521 
lemma Inf_empty_bool [simp]: 

522 
"Inf {}" 

523 
unfolding Inf_bool_def by auto 

524 

525 
lemma not_Sup_empty_bool [simp]: 

526 
"\<not> Sup {}" 

527 
unfolding Sup_def Inf_bool_def by auto 

528 

529 
lemma top_bool_eq: "top = True" 

530 
by (iprover intro!: order_antisym le_boolI top_greatest) 

531 

532 
lemma bot_bool_eq: "bot = False" 

533 
by (iprover intro!: order_antisym le_boolI bot_least) 

534 

535 

536 
subsection {* Set as lattice *} 

537 

538 
instance set :: (type) distrib_lattice 

539 
inf_set_eq: "inf A B \<equiv> A \<inter> B" 

540 
sup_set_eq: "sup A B \<equiv> A \<union> B" 

541 
by intro_classes (auto simp add: inf_set_eq sup_set_eq) 

542 

543 
lemmas [code func del] = inf_set_eq sup_set_eq 

544 

545 
lemmas mono_Int = mono_inf 

546 
[where ?'a="?'a set", where ?'b="?'b set", unfolded inf_set_eq sup_set_eq] 

547 

548 
lemmas mono_Un = mono_sup 

549 
[where ?'a="?'a set", where ?'b="?'b set", unfolded inf_set_eq sup_set_eq] 

550 

551 
instance set :: (type) complete_lattice 

552 
Inf_set_def: "Inf S \<equiv> \<Inter>S" 

553 
by intro_classes (auto simp add: Inf_set_def) 

554 

555 
lemmas [code func del] = Inf_set_def 

556 

557 
theorem Sup_set_eq: "Sup S = \<Union>S" 

558 
apply (rule subset_antisym) 

559 
apply (rule Sup_least) 

560 
apply (erule Union_upper) 

561 
apply (rule Union_least) 

562 
apply (erule Sup_upper) 

563 
done 

564 

565 
lemma top_set_eq: "top = UNIV" 

566 
by (iprover intro!: subset_antisym subset_UNIV top_greatest) 

567 

568 
lemma bot_set_eq: "bot = {}" 

569 
by (iprover intro!: subset_antisym empty_subsetI bot_least) 

570 

571 

572 
subsection {* Fun as lattice *} 

573 

574 
instance "fun" :: (type, lattice) lattice 

575 
inf_fun_eq: "inf f g \<equiv> (\<lambda>x. inf (f x) (g x))" 

576 
sup_fun_eq: "sup f g \<equiv> (\<lambda>x. sup (f x) (g x))" 

577 
apply intro_classes 

578 
unfolding inf_fun_eq sup_fun_eq 

579 
apply (auto intro: le_funI) 

580 
apply (rule le_funI) 

581 
apply (auto dest: le_funD) 

582 
apply (rule le_funI) 

583 
apply (auto dest: le_funD) 

584 
done 

585 

586 
lemmas [code func del] = inf_fun_eq sup_fun_eq 

587 

588 
instance "fun" :: (type, distrib_lattice) distrib_lattice 

589 
by default (auto simp add: inf_fun_eq sup_fun_eq sup_inf_distrib1) 

590 

591 
instance "fun" :: (type, complete_lattice) complete_lattice 

592 
Inf_fun_def: "Inf A \<equiv> (\<lambda>x. Inf {y. \<exists>f\<in>A. y = f x})" 

593 
apply intro_classes 

594 
apply (unfold Inf_fun_def) 

595 
apply (rule le_funI) 

596 
apply (rule Inf_lower) 

597 
apply (rule CollectI) 

598 
apply (rule bexI) 

599 
apply (rule refl) 

600 
apply assumption 

601 
apply (rule le_funI) 

602 
apply (rule Inf_greatest) 

603 
apply (erule CollectE) 

604 
apply (erule bexE) 

605 
apply (iprover elim: le_funE) 

606 
done 

607 

608 
lemmas [code func del] = Inf_fun_def 

609 

610 
theorem Sup_fun_eq: "Sup A = (\<lambda>x. Sup {y. \<exists>f\<in>A. y = f x})" 

611 
apply (rule order_antisym) 

612 
apply (rule Sup_least) 

613 
apply (rule le_funI) 

614 
apply (rule Sup_upper) 

615 
apply fast 

616 
apply (rule le_funI) 

617 
apply (rule Sup_least) 

618 
apply (erule CollectE) 

619 
apply (erule bexE) 

620 
apply (drule le_funD [OF Sup_upper]) 

621 
apply simp 

622 
done 

623 

624 
lemma Inf_empty_fun: 

625 
"Inf {} = (\<lambda>_. Inf {})" 

626 
by rule (auto simp add: Inf_fun_def) 

627 

628 
lemma Sup_empty_fun: 

629 
"Sup {} = (\<lambda>_. Sup {})" 

630 
proof  

631 
have aux: "\<And>x. {y. \<exists>f. y = f x} = UNIV" by auto 

632 
show ?thesis 

633 
by (auto simp add: Sup_def Inf_fun_def Inf_binary inf_bool_eq aux) 

634 
qed 

635 

636 
lemma top_fun_eq: "top = (\<lambda>x. top)" 

637 
by (iprover intro!: order_antisym le_funI top_greatest) 

638 

639 
lemma bot_fun_eq: "bot = (\<lambda>x. bot)" 

640 
by (iprover intro!: order_antisym le_funI bot_least) 

641 

642 

643 
text {* redundant bindings *} 

22454  644 

645 
lemmas inf_aci = inf_ACI 

646 
lemmas sup_aci = sup_ACI 

647 

23878  648 
ML {* 
649 
val sup_fun_eq = @{thm sup_fun_eq} 

650 
val sup_bool_eq = @{thm sup_bool_eq} 

651 
*} 

652 

21249  653 
end 