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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 {* Lattices via Locales *}

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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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text{* This theory of lattice locales only defines binary sup and inf


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operations. The extension to finite sets is done in theory @{text


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Finite_Set}. In the longer term it may be better to define arbitrary


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sups and infs via @{text THE}. *}


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locale 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" 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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locale 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" 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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locale 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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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:trans)


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apply simp


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done

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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:trans)


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apply simp


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done


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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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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: trans)

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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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apply rule


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apply(simp add: antisym)


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apply(subgoal_tac "x \<sqinter> y \<sqsubseteq> y")


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apply(simp)


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apply(simp (no_asm))


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done

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end


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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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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:trans)


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apply simp


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done

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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:trans)


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apply simp


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done


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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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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: trans)

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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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apply rule


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apply(simp add: antisym)


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apply(subgoal_tac "x \<sqsubseteq> x \<squnion> y")


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apply(simp)


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apply(simp (no_asm))


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done


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end


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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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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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locale 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 {* min/max on linear orders as special case of inf/sup *}

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interpretation min_max:

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distrib_lattice ["op \<le>" "op <" "min \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> 'a" "max"]

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apply unfold_locales

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apply (simp add: min_def linorder_not_le order_less_imp_le)


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apply (simp add: min_def linorder_not_le order_less_imp_le)


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apply (simp add: min_def linorder_not_le order_less_imp_le)


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apply (simp add: max_def linorder_not_le order_less_imp_le)


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apply (simp add: max_def linorder_not_le order_less_imp_le)


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unfolding min_def max_def by auto

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text{* Now we have inherited antisymmetry as an introrule 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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declare


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min_max.antisym_intro[rule del]

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min_max.le_infI[rule del] min_max.le_supI[rule del]


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min_max.le_supE[rule del] min_max.le_infE[rule del]


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min_max.le_supI1[rule del] min_max.le_supI2[rule del]


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min_max.le_infI1[rule del] min_max.le_infI2[rule del]

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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 {* ML legacy bindings *}


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ML {*

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val Least_def = @{thm Least_def}


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val Least_equality = @{thm Least_equality}


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val min_def = @{thm min_def}


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val min_of_mono = @{thm min_of_mono}


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val max_def = @{thm max_def}


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val max_of_mono = @{thm max_of_mono}


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val min_leastL = @{thm min_leastL}


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val max_leastL = @{thm max_leastL}


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val min_leastR = @{thm min_leastR}


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val max_leastR = @{thm max_leastR}


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val le_max_iff_disj = @{thm le_max_iff_disj}


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val le_maxI1 = @{thm le_maxI1}


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val le_maxI2 = @{thm le_maxI2}


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val less_max_iff_disj = @{thm less_max_iff_disj}


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val max_less_iff_conj = @{thm max_less_iff_conj}


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val min_less_iff_conj = @{thm min_less_iff_conj}


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val min_le_iff_disj = @{thm min_le_iff_disj}


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val min_less_iff_disj = @{thm min_less_iff_disj}


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val split_min = @{thm split_min}


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val split_max = @{thm split_max}

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*}


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end
