src/HOL/Lattices.thy
author wenzelm
Thu Oct 04 20:29:42 2007 +0200 (2007-10-04)
changeset 24850 0cfd722ab579
parent 24749 151b3758f576
child 25062 af5ef0d4d655
permissions -rw-r--r--
Name.uu, Name.aT;
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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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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)"
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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 .
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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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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
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  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
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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)
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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 intro-rule on all
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  linear orders. This is a problem because it applies to bool, which is
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  undesirable.
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*}
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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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    and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
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  assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
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     and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
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  assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
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     and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
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begin
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lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<^loc>\<le> a}"
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  by (auto intro: Inf_lower Inf_greatest Sup_upper Sup_least)
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lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<^loc>\<le> b}"
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  by (auto intro: Inf_lower Inf_greatest Sup_upper Sup_least)
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lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}"
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  unfolding Sup_Inf 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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lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
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  apply (rule antisym)
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  apply (rule le_infI)
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  apply (rule Inf_lower)
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  apply simp
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  apply (rule Inf_greatest)
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  apply (rule Inf_lower)
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  apply simp
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  apply (rule Inf_greatest)
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  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)
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  done
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   363
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lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
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  apply (rule antisym)
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  apply (rule Sup_least)
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  apply (erule insertE)
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  apply (rule le_supI1)
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  apply simp
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  apply (rule le_supI2)
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  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
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  apply (rule Sup_least)
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  apply (rule Sup_upper)
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  apply simp
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  done
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   380
lemma Inf_singleton [simp]:
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  "\<Sqinter>{a} = a"
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  by (auto intro: antisym Inf_lower Inf_greatest)
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lemma Sup_singleton [simp]:
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  "\<Squnion>{a} = a"
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  by (auto intro: antisym Sup_upper Sup_least)
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   387
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lemma Inf_insert_simp:
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  "\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)"
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  by (cases "A = {}") (simp_all, simp add: Inf_insert)
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   391
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   392
lemma Sup_insert_simp:
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  "\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)"
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  by (cases "A = {}") (simp_all, simp add: Sup_insert)
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   395
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   396
lemma Inf_binary:
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  "\<Sqinter>{a, b} = a \<sqinter> b"
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   398
  by (simp add: Inf_insert_simp)
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   399
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   400
lemma Sup_binary:
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   401
  "\<Squnion>{a, b} = a \<squnion> b"
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  by (simp add: Sup_insert_simp)
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   403
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   404
definition
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  top :: 'a
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   406
where
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  "top = Inf {}"
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   408
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   409
definition
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  bot :: 'a
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   411
where
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   412
  "bot = Sup {}"
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   413
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   414
lemma top_greatest [simp]: "x \<^loc>\<le> top"
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   415
  by (unfold top_def, rule Inf_greatest, simp)
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   416
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   417
lemma bot_least [simp]: "bot \<^loc>\<le> x"
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   418
  by (unfold bot_def, rule Sup_least, simp)
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   419
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   420
definition
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   421
  SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
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   422
where
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   423
  "SUPR A f == Sup (f ` A)"
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   424
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   425
definition
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   426
  INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
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   427
where
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   428
  "INFI A f == Inf (f ` A)"
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   429
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   430
end
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   431
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   432
syntax
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   433
  "_SUP1"     :: "pttrns => 'b => 'b"           ("(3SUP _./ _)" [0, 10] 10)
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   434
  "_SUP"      :: "pttrn => 'a set => 'b => 'b"  ("(3SUP _:_./ _)" [0, 10] 10)
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   435
  "_INF1"     :: "pttrns => 'b => 'b"           ("(3INF _./ _)" [0, 10] 10)
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   436
  "_INF"      :: "pttrn => 'a set => 'b => 'b"  ("(3INF _:_./ _)" [0, 10] 10)
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   437
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   438
translations
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   439
  "SUP x y. B"   == "SUP x. SUP y. B"
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   440
  "SUP x. B"     == "CONST SUPR UNIV (%x. B)"
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   441
  "SUP x. B"     == "SUP x:UNIV. B"
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   442
  "SUP x:A. B"   == "CONST SUPR A (%x. B)"
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   443
  "INF x y. B"   == "INF x. INF y. B"
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   444
  "INF x. B"     == "CONST INFI UNIV (%x. B)"
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   445
  "INF x. B"     == "INF x:UNIV. B"
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   446
  "INF x:A. B"   == "CONST INFI A (%x. B)"
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   447
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   448
(* To avoid eta-contraction of body: *)
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   449
print_translation {*
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   450
let
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   451
  fun btr' syn (A :: Abs abs :: ts) =
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   452
    let val (x,t) = atomic_abs_tr' abs
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   453
    in list_comb (Syntax.const syn $ x $ A $ t, ts) end
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   454
  val const_syntax_name = Sign.const_syntax_name @{theory} o fst o dest_Const
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   455
in
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   456
[(const_syntax_name @{term SUPR}, btr' "_SUP"),(const_syntax_name @{term "INFI"}, btr' "_INF")]
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   457
end
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   458
*}
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   459
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   460
lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)"
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   461
  by (auto simp add: SUPR_def intro: Sup_upper)
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   462
haftmann@23878
   463
lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u"
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   464
  by (auto simp add: SUPR_def intro: Sup_least)
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   465
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   466
lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i"
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   467
  by (auto simp add: INFI_def intro: Inf_lower)
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   468
haftmann@23878
   469
lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)"
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   470
  by (auto simp add: INFI_def intro: Inf_greatest)
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   471
haftmann@23878
   472
lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"
haftmann@23878
   473
  by (auto intro: order_antisym SUP_leI le_SUPI)
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   474
haftmann@23878
   475
lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"
haftmann@23878
   476
  by (auto intro: order_antisym INF_leI le_INFI)
haftmann@23878
   477
haftmann@23878
   478
haftmann@22454
   479
subsection {* Bool as lattice *}
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   480
haftmann@22454
   481
instance bool :: distrib_lattice
haftmann@22454
   482
  inf_bool_eq: "inf P Q \<equiv> P \<and> Q"
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   483
  sup_bool_eq: "sup P Q \<equiv> P \<or> Q"
haftmann@22454
   484
  by intro_classes (auto simp add: inf_bool_eq sup_bool_eq le_bool_def)
haftmann@22454
   485
haftmann@23878
   486
instance bool :: complete_lattice
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   487
  Inf_bool_def: "Inf A \<equiv> \<forall>x\<in>A. x"
haftmann@24345
   488
  Sup_bool_def: "Sup A \<equiv> \<exists>x\<in>A. x"
haftmann@24345
   489
  by intro_classes (auto simp add: Inf_bool_def Sup_bool_def le_bool_def)
haftmann@23878
   490
haftmann@23878
   491
lemma Inf_empty_bool [simp]:
haftmann@23878
   492
  "Inf {}"
haftmann@23878
   493
  unfolding Inf_bool_def by auto
haftmann@23878
   494
haftmann@23878
   495
lemma not_Sup_empty_bool [simp]:
haftmann@23878
   496
  "\<not> Sup {}"
haftmann@24345
   497
  unfolding Sup_bool_def by auto
haftmann@23878
   498
haftmann@23878
   499
lemma top_bool_eq: "top = True"
haftmann@23878
   500
  by (iprover intro!: order_antisym le_boolI top_greatest)
haftmann@23878
   501
haftmann@23878
   502
lemma bot_bool_eq: "bot = False"
haftmann@23878
   503
  by (iprover intro!: order_antisym le_boolI bot_least)
haftmann@23878
   504
haftmann@23878
   505
haftmann@23878
   506
subsection {* Set as lattice *}
haftmann@23878
   507
haftmann@23878
   508
instance set :: (type) distrib_lattice
haftmann@23878
   509
  inf_set_eq: "inf A B \<equiv> A \<inter> B"
haftmann@23878
   510
  sup_set_eq: "sup A B \<equiv> A \<union> B"
haftmann@23878
   511
  by intro_classes (auto simp add: inf_set_eq sup_set_eq)
haftmann@23878
   512
haftmann@23878
   513
lemmas [code func del] = inf_set_eq sup_set_eq
haftmann@23878
   514
wenzelm@24514
   515
lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"
wenzelm@24514
   516
  apply (fold inf_set_eq sup_set_eq)
wenzelm@24514
   517
  apply (erule mono_inf)
wenzelm@24514
   518
  done
haftmann@23878
   519
wenzelm@24514
   520
lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)"
wenzelm@24514
   521
  apply (fold inf_set_eq sup_set_eq)
wenzelm@24514
   522
  apply (erule mono_sup)
wenzelm@24514
   523
  done
haftmann@23878
   524
haftmann@23878
   525
instance set :: (type) complete_lattice
haftmann@23878
   526
  Inf_set_def: "Inf S \<equiv> \<Inter>S"
haftmann@24345
   527
  Sup_set_def: "Sup S \<equiv> \<Union>S"
haftmann@24345
   528
  by intro_classes (auto simp add: Inf_set_def Sup_set_def)
haftmann@23878
   529
haftmann@24345
   530
lemmas [code func del] = Inf_set_def Sup_set_def
haftmann@23878
   531
haftmann@23878
   532
lemma top_set_eq: "top = UNIV"
haftmann@23878
   533
  by (iprover intro!: subset_antisym subset_UNIV top_greatest)
haftmann@23878
   534
haftmann@23878
   535
lemma bot_set_eq: "bot = {}"
haftmann@23878
   536
  by (iprover intro!: subset_antisym empty_subsetI bot_least)
haftmann@23878
   537
haftmann@23878
   538
haftmann@23878
   539
subsection {* Fun as lattice *}
haftmann@23878
   540
haftmann@23878
   541
instance "fun" :: (type, lattice) lattice
haftmann@23878
   542
  inf_fun_eq: "inf f g \<equiv> (\<lambda>x. inf (f x) (g x))"
haftmann@23878
   543
  sup_fun_eq: "sup f g \<equiv> (\<lambda>x. sup (f x) (g x))"
haftmann@23878
   544
apply intro_classes
haftmann@23878
   545
unfolding inf_fun_eq sup_fun_eq
haftmann@23878
   546
apply (auto intro: le_funI)
haftmann@23878
   547
apply (rule le_funI)
haftmann@23878
   548
apply (auto dest: le_funD)
haftmann@23878
   549
apply (rule le_funI)
haftmann@23878
   550
apply (auto dest: le_funD)
haftmann@23878
   551
done
haftmann@23878
   552
haftmann@23878
   553
lemmas [code func del] = inf_fun_eq sup_fun_eq
haftmann@23878
   554
haftmann@23878
   555
instance "fun" :: (type, distrib_lattice) distrib_lattice
haftmann@23878
   556
  by default (auto simp add: inf_fun_eq sup_fun_eq sup_inf_distrib1)
haftmann@23878
   557
haftmann@23878
   558
instance "fun" :: (type, complete_lattice) complete_lattice
haftmann@23878
   559
  Inf_fun_def: "Inf A \<equiv> (\<lambda>x. Inf {y. \<exists>f\<in>A. y = f x})"
haftmann@24345
   560
  Sup_fun_def: "Sup A \<equiv> (\<lambda>x. Sup {y. \<exists>f\<in>A. y = f x})"
haftmann@24345
   561
  by intro_classes
haftmann@24345
   562
    (auto simp add: Inf_fun_def Sup_fun_def le_fun_def
haftmann@24345
   563
      intro: Inf_lower Sup_upper Inf_greatest Sup_least)
haftmann@23878
   564
haftmann@24345
   565
lemmas [code func del] = Inf_fun_def Sup_fun_def
haftmann@23878
   566
haftmann@23878
   567
lemma Inf_empty_fun:
haftmann@23878
   568
  "Inf {} = (\<lambda>_. Inf {})"
haftmann@23878
   569
  by rule (auto simp add: Inf_fun_def)
haftmann@23878
   570
haftmann@23878
   571
lemma Sup_empty_fun:
haftmann@23878
   572
  "Sup {} = (\<lambda>_. Sup {})"
haftmann@24345
   573
  by rule (auto simp add: Sup_fun_def)
haftmann@23878
   574
haftmann@23878
   575
lemma top_fun_eq: "top = (\<lambda>x. top)"
haftmann@23878
   576
  by (iprover intro!: order_antisym le_funI top_greatest)
haftmann@23878
   577
haftmann@23878
   578
lemma bot_fun_eq: "bot = (\<lambda>x. bot)"
haftmann@23878
   579
  by (iprover intro!: order_antisym le_funI bot_least)
haftmann@23878
   580
haftmann@23878
   581
haftmann@23878
   582
text {* redundant bindings *}
haftmann@22454
   583
haftmann@22454
   584
lemmas inf_aci = inf_ACI
haftmann@22454
   585
lemmas sup_aci = sup_ACI
haftmann@22454
   586
haftmann@21249
   587
end