src/HOL/Lattices.thy
author haftmann
Fri Nov 30 20:13:03 2007 +0100 (2007-11-30)
changeset 25510 38c15efe603b
parent 25482 4ed49eccb1eb
child 26014 00c2c3525bef
permissions -rw-r--r--
adjustions to due to instance target
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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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notation
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  less_eq  (infix "\<sqsubseteq>" 50) and
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  less  (infix "\<sqsubset>" 50)
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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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lemma le_infI1[intro]:
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  assumes "a \<sqsubseteq> x"
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  shows "a \<sqinter> b \<sqsubseteq> x"
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proof (rule order_trans)
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  from assms show "a \<sqsubseteq> x" .
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  show "a \<sqinter> b \<sqsubseteq> a" by simp 
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qed
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lemmas (in -) [rule del] = le_infI1
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lemma le_infI2[intro]:
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  assumes "b \<sqsubseteq> x"
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  shows "a \<sqinter> b \<sqsubseteq> x"
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proof (rule order_trans)
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  from assms show "b \<sqsubseteq> x" .
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  show "a \<sqinter> b \<sqsubseteq> b" by simp
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qed
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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 intro: antisym dest: eq_iff [THEN iffD1])
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lemma mono_inf:
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  fixes f :: "'a \<Rightarrow> 'b\<Colon>lower_semilattice"
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  shows "mono f \<Longrightarrow> f (A \<sqinter> B) \<le> f A \<sqinter> f B"
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  by (auto simp add: mono_def intro: Lattices.inf_greatest)
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end
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context upper_semilattice
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begin
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lemma le_supI1[intro]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
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  by (rule order_trans) auto
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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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  by (rule order_trans) auto 
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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 intro: antisym dest: eq_iff [THEN iffD1])
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lemma mono_sup:
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  fixes f :: "'a \<Rightarrow> 'b\<Colon>upper_semilattice"
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  shows "mono f \<Longrightarrow> f A \<squnion> f B \<le> f (A \<squnion> B)"
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  by (auto simp add: mono_def intro: Lattices.sup_least)
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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 intro: antisym)
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lemma inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
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  by (blast intro: antisym)
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lemma inf_idem[simp]: "x \<sqinter> x = x"
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  by (blast intro: antisym)
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lemma inf_left_idem[simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
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  by (blast intro: antisym)
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lemma inf_absorb1: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
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  by (blast intro: antisym)
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lemma inf_absorb2: "y \<sqsubseteq> x \<Longrightarrow> x \<sqinter> y = y"
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  by (blast intro: antisym)
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lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
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  by (blast intro: antisym)
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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 intro: antisym)
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lemma sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
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  by (blast intro: antisym)
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lemma sup_idem[simp]: "x \<squnion> x = x"
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  by (blast intro: antisym)
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lemma sup_left_idem[simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
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  by (blast intro: antisym)
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lemma sup_absorb1: "y \<sqsubseteq> x \<Longrightarrow> x \<squnion> y = x"
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  by (blast intro: antisym)
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lemma sup_absorb2: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
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  by (blast intro: antisym)
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lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
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  by (blast intro: antisym)
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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 \<le> x" and le2: "\<And>x y. x \<triangle> y \<le> y"
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  and greatest: "\<And>x y z. x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<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 \<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 \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> y \<triangle> z" by (blast intro: greatest)
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  show "x \<sqinter> y \<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 \<le> x \<nabla> y" and ge2: "\<And>x y. y \<le> x \<nabla> y"
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  and least: "\<And>x y z. y \<le> x \<Longrightarrow> z \<le> x \<Longrightarrow> y \<nabla> z \<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 \<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 \<le> z \<Longrightarrow> y \<le> z \<Longrightarrow> x \<nabla> y \<le> z" by (blast intro: least)
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  show "x \<nabla> y \<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 \<le>) (op <) min max"
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proof unfold_locales
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  have aux: "\<And>x y \<Colon> 'a. x < y \<Longrightarrow> y \<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 intro: antisym)
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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 intro: antisym)
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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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   310
  mk_left_commute [of min, OF min_max.inf_assoc min_max.inf_commute]
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   311
haftmann@22454
   312
text {*
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   313
  Now we have inherited antisymmetry as an intro-rule on all
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   314
  linear orders. This is a problem because it applies to bool, which is
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   315
  undesirable.
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   316
*}
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   317
haftmann@25102
   318
lemmas [rule del] = min_max.le_infI min_max.le_supI
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   319
  min_max.le_supE min_max.le_infE min_max.le_supI1 min_max.le_supI2
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   320
  min_max.le_infI1 min_max.le_infI2
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   321
haftmann@22454
   322
haftmann@23878
   323
subsection {* Complete lattices *}
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   324
haftmann@23878
   325
class complete_lattice = lattice +
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   326
  fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
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   327
    and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
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   328
  assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
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   329
     and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
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   330
  assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
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   331
     and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
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   332
begin
haftmann@23878
   333
haftmann@25062
   334
lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> a}"
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   335
  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
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   336
haftmann@25062
   337
lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<le> b}"
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   338
  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
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   339
haftmann@23878
   340
lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}"
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   341
  unfolding Sup_Inf by auto
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   342
haftmann@23878
   343
lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}"
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   344
  unfolding Inf_Sup by auto
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   345
haftmann@23878
   346
lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
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   347
  apply (rule antisym)
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   348
  apply (rule le_infI)
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   349
  apply (rule Inf_lower)
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   350
  apply simp
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   351
  apply (rule Inf_greatest)
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   352
  apply (rule Inf_lower)
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   353
  apply simp
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   354
  apply (rule Inf_greatest)
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   355
  apply (erule insertE)
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   356
  apply (rule le_infI1)
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   357
  apply simp
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   358
  apply (rule le_infI2)
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   359
  apply (erule Inf_lower)
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   360
  done
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   361
haftmann@24345
   362
lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
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   363
  apply (rule antisym)
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   364
  apply (rule Sup_least)
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   365
  apply (erule insertE)
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   366
  apply (rule le_supI1)
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   367
  apply simp
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   368
  apply (rule le_supI2)
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   369
  apply (erule Sup_upper)
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   370
  apply (rule le_supI)
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   371
  apply (rule Sup_upper)
haftmann@23878
   372
  apply simp
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   373
  apply (rule Sup_least)
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   374
  apply (rule Sup_upper)
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   375
  apply simp
haftmann@23878
   376
  done
haftmann@23878
   377
haftmann@23878
   378
lemma Inf_singleton [simp]:
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   379
  "\<Sqinter>{a} = a"
haftmann@23878
   380
  by (auto intro: antisym Inf_lower Inf_greatest)
haftmann@23878
   381
haftmann@24345
   382
lemma Sup_singleton [simp]:
haftmann@23878
   383
  "\<Squnion>{a} = a"
haftmann@23878
   384
  by (auto intro: antisym Sup_upper Sup_least)
haftmann@23878
   385
haftmann@23878
   386
lemma Inf_insert_simp:
haftmann@23878
   387
  "\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)"
haftmann@23878
   388
  by (cases "A = {}") (simp_all, simp add: Inf_insert)
haftmann@23878
   389
haftmann@23878
   390
lemma Sup_insert_simp:
haftmann@23878
   391
  "\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)"
haftmann@23878
   392
  by (cases "A = {}") (simp_all, simp add: Sup_insert)
haftmann@23878
   393
haftmann@23878
   394
lemma Inf_binary:
haftmann@23878
   395
  "\<Sqinter>{a, b} = a \<sqinter> b"
haftmann@23878
   396
  by (simp add: Inf_insert_simp)
haftmann@23878
   397
haftmann@23878
   398
lemma Sup_binary:
haftmann@23878
   399
  "\<Squnion>{a, b} = a \<squnion> b"
haftmann@23878
   400
  by (simp add: Sup_insert_simp)
haftmann@23878
   401
haftmann@23878
   402
definition
wenzelm@25382
   403
  top :: 'a where
haftmann@25206
   404
  "top = \<Sqinter>{}"
haftmann@23878
   405
haftmann@23878
   406
definition
wenzelm@25382
   407
  bot :: 'a where
haftmann@25206
   408
  "bot = \<Squnion>{}"
haftmann@23878
   409
haftmann@25062
   410
lemma top_greatest [simp]: "x \<le> top"
haftmann@23878
   411
  by (unfold top_def, rule Inf_greatest, simp)
haftmann@23878
   412
haftmann@25062
   413
lemma bot_least [simp]: "bot \<le> x"
haftmann@23878
   414
  by (unfold bot_def, rule Sup_least, simp)
haftmann@23878
   415
haftmann@23878
   416
definition
haftmann@24749
   417
  SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
haftmann@23878
   418
where
haftmann@25206
   419
  "SUPR A f == \<Squnion> (f ` A)"
haftmann@23878
   420
haftmann@23878
   421
definition
haftmann@24749
   422
  INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
haftmann@23878
   423
where
haftmann@25206
   424
  "INFI A f == \<Sqinter> (f ` A)"
haftmann@23878
   425
haftmann@24749
   426
end
haftmann@24749
   427
haftmann@23878
   428
syntax
haftmann@23878
   429
  "_SUP1"     :: "pttrns => 'b => 'b"           ("(3SUP _./ _)" [0, 10] 10)
haftmann@23878
   430
  "_SUP"      :: "pttrn => 'a set => 'b => 'b"  ("(3SUP _:_./ _)" [0, 10] 10)
haftmann@23878
   431
  "_INF1"     :: "pttrns => 'b => 'b"           ("(3INF _./ _)" [0, 10] 10)
haftmann@23878
   432
  "_INF"      :: "pttrn => 'a set => 'b => 'b"  ("(3INF _:_./ _)" [0, 10] 10)
haftmann@23878
   433
haftmann@23878
   434
translations
haftmann@23878
   435
  "SUP x y. B"   == "SUP x. SUP y. B"
haftmann@23878
   436
  "SUP x. B"     == "CONST SUPR UNIV (%x. B)"
haftmann@23878
   437
  "SUP x. B"     == "SUP x:UNIV. B"
haftmann@23878
   438
  "SUP x:A. B"   == "CONST SUPR A (%x. B)"
haftmann@23878
   439
  "INF x y. B"   == "INF x. INF y. B"
haftmann@23878
   440
  "INF x. B"     == "CONST INFI UNIV (%x. B)"
haftmann@23878
   441
  "INF x. B"     == "INF x:UNIV. B"
haftmann@23878
   442
  "INF x:A. B"   == "CONST INFI A (%x. B)"
haftmann@23878
   443
haftmann@23878
   444
(* To avoid eta-contraction of body: *)
haftmann@23878
   445
print_translation {*
haftmann@23878
   446
let
haftmann@23878
   447
  fun btr' syn (A :: Abs abs :: ts) =
haftmann@23878
   448
    let val (x,t) = atomic_abs_tr' abs
haftmann@23878
   449
    in list_comb (Syntax.const syn $ x $ A $ t, ts) end
haftmann@23878
   450
  val const_syntax_name = Sign.const_syntax_name @{theory} o fst o dest_Const
haftmann@23878
   451
in
haftmann@23878
   452
[(const_syntax_name @{term SUPR}, btr' "_SUP"),(const_syntax_name @{term "INFI"}, btr' "_INF")]
haftmann@23878
   453
end
haftmann@23878
   454
*}
haftmann@23878
   455
haftmann@25102
   456
context complete_lattice
haftmann@25102
   457
begin
haftmann@25102
   458
haftmann@23878
   459
lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)"
haftmann@23878
   460
  by (auto simp add: SUPR_def intro: Sup_upper)
haftmann@23878
   461
haftmann@23878
   462
lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u"
haftmann@23878
   463
  by (auto simp add: SUPR_def intro: Sup_least)
haftmann@23878
   464
haftmann@23878
   465
lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i"
haftmann@23878
   466
  by (auto simp add: INFI_def intro: Inf_lower)
haftmann@23878
   467
haftmann@23878
   468
lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)"
haftmann@23878
   469
  by (auto simp add: INFI_def intro: Inf_greatest)
haftmann@23878
   470
haftmann@23878
   471
lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"
haftmann@25102
   472
  by (auto intro: antisym SUP_leI le_SUPI)
haftmann@23878
   473
haftmann@23878
   474
lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"
haftmann@25102
   475
  by (auto intro: antisym INF_leI le_INFI)
haftmann@25102
   476
haftmann@25102
   477
end
haftmann@23878
   478
haftmann@23878
   479
haftmann@22454
   480
subsection {* Bool as lattice *}
haftmann@22454
   481
haftmann@25510
   482
instantiation bool :: distrib_lattice
haftmann@25510
   483
begin
haftmann@25510
   484
haftmann@25510
   485
definition
haftmann@25510
   486
  inf_bool_eq: "P \<sqinter> Q \<longleftrightarrow> P \<and> Q"
haftmann@25510
   487
haftmann@25510
   488
definition
haftmann@25510
   489
  sup_bool_eq: "P \<squnion> Q \<longleftrightarrow> P \<or> Q"
haftmann@25510
   490
haftmann@25510
   491
instance
haftmann@22454
   492
  by intro_classes (auto simp add: inf_bool_eq sup_bool_eq le_bool_def)
haftmann@22454
   493
haftmann@25510
   494
end
haftmann@25510
   495
haftmann@25510
   496
instantiation bool :: complete_lattice
haftmann@25510
   497
begin
haftmann@25510
   498
haftmann@25510
   499
definition
haftmann@25510
   500
  Inf_bool_def: "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
haftmann@25510
   501
haftmann@25510
   502
definition
haftmann@25510
   503
  Sup_bool_def: "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
haftmann@25510
   504
haftmann@25510
   505
instance
haftmann@24345
   506
  by intro_classes (auto simp add: Inf_bool_def Sup_bool_def le_bool_def)
haftmann@23878
   507
haftmann@25510
   508
end
haftmann@25510
   509
haftmann@23878
   510
lemma Inf_empty_bool [simp]:
haftmann@25206
   511
  "\<Sqinter>{}"
haftmann@23878
   512
  unfolding Inf_bool_def by auto
haftmann@23878
   513
haftmann@23878
   514
lemma not_Sup_empty_bool [simp]:
haftmann@23878
   515
  "\<not> Sup {}"
haftmann@24345
   516
  unfolding Sup_bool_def by auto
haftmann@23878
   517
haftmann@23878
   518
lemma top_bool_eq: "top = True"
haftmann@23878
   519
  by (iprover intro!: order_antisym le_boolI top_greatest)
haftmann@23878
   520
haftmann@23878
   521
lemma bot_bool_eq: "bot = False"
haftmann@23878
   522
  by (iprover intro!: order_antisym le_boolI bot_least)
haftmann@23878
   523
haftmann@23878
   524
haftmann@23878
   525
subsection {* Set as lattice *}
haftmann@23878
   526
haftmann@25510
   527
instantiation set :: (type) distrib_lattice
haftmann@25510
   528
begin
haftmann@25510
   529
haftmann@25510
   530
definition
haftmann@25510
   531
  inf_set_eq [code func del]: "A \<sqinter> B = A \<inter> B"
haftmann@25510
   532
haftmann@25510
   533
definition
haftmann@25510
   534
  sup_set_eq [code func del]: "A \<squnion> B = A \<union> B"
haftmann@25510
   535
haftmann@25510
   536
instance
haftmann@23878
   537
  by intro_classes (auto simp add: inf_set_eq sup_set_eq)
haftmann@23878
   538
haftmann@25510
   539
end
haftmann@23878
   540
wenzelm@24514
   541
lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"
wenzelm@24514
   542
  apply (fold inf_set_eq sup_set_eq)
wenzelm@24514
   543
  apply (erule mono_inf)
wenzelm@24514
   544
  done
haftmann@23878
   545
wenzelm@24514
   546
lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)"
wenzelm@24514
   547
  apply (fold inf_set_eq sup_set_eq)
wenzelm@24514
   548
  apply (erule mono_sup)
wenzelm@24514
   549
  done
haftmann@23878
   550
haftmann@25510
   551
instantiation set :: (type) complete_lattice
haftmann@25510
   552
begin
haftmann@25510
   553
haftmann@25510
   554
definition
haftmann@25510
   555
  Inf_set_def [code func del]: "\<Sqinter>S \<equiv> \<Inter>S"
haftmann@25510
   556
haftmann@25510
   557
definition
haftmann@25510
   558
  Sup_set_def [code func del]: "\<Squnion>S \<equiv> \<Union>S"
haftmann@25510
   559
haftmann@25510
   560
instance
haftmann@24345
   561
  by intro_classes (auto simp add: Inf_set_def Sup_set_def)
haftmann@23878
   562
haftmann@25510
   563
end
haftmann@23878
   564
haftmann@23878
   565
lemma top_set_eq: "top = UNIV"
haftmann@23878
   566
  by (iprover intro!: subset_antisym subset_UNIV top_greatest)
haftmann@23878
   567
haftmann@23878
   568
lemma bot_set_eq: "bot = {}"
haftmann@23878
   569
  by (iprover intro!: subset_antisym empty_subsetI bot_least)
haftmann@23878
   570
haftmann@23878
   571
haftmann@23878
   572
subsection {* Fun as lattice *}
haftmann@23878
   573
haftmann@25510
   574
instantiation "fun" :: (type, lattice) lattice
haftmann@25510
   575
begin
haftmann@25510
   576
haftmann@25510
   577
definition
haftmann@25510
   578
  inf_fun_eq [code func del]: "f \<sqinter> g = (\<lambda>x. f x \<sqinter> g x)"
haftmann@25510
   579
haftmann@25510
   580
definition
haftmann@25510
   581
  sup_fun_eq [code func del]: "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)"
haftmann@25510
   582
haftmann@25510
   583
instance
haftmann@23878
   584
apply intro_classes
haftmann@23878
   585
unfolding inf_fun_eq sup_fun_eq
haftmann@23878
   586
apply (auto intro: le_funI)
haftmann@23878
   587
apply (rule le_funI)
haftmann@23878
   588
apply (auto dest: le_funD)
haftmann@23878
   589
apply (rule le_funI)
haftmann@23878
   590
apply (auto dest: le_funD)
haftmann@23878
   591
done
haftmann@23878
   592
haftmann@25510
   593
end
haftmann@23878
   594
haftmann@23878
   595
instance "fun" :: (type, distrib_lattice) distrib_lattice
haftmann@23878
   596
  by default (auto simp add: inf_fun_eq sup_fun_eq sup_inf_distrib1)
haftmann@23878
   597
haftmann@25510
   598
instantiation "fun" :: (type, complete_lattice) complete_lattice
haftmann@25510
   599
begin
haftmann@25510
   600
haftmann@25510
   601
definition
haftmann@25510
   602
  Inf_fun_def [code func del]: "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
haftmann@25510
   603
haftmann@25510
   604
definition
haftmann@25510
   605
  Sup_fun_def [code func del]: "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
haftmann@25510
   606
haftmann@25510
   607
instance
haftmann@24345
   608
  by intro_classes
haftmann@24345
   609
    (auto simp add: Inf_fun_def Sup_fun_def le_fun_def
haftmann@24345
   610
      intro: Inf_lower Sup_upper Inf_greatest Sup_least)
haftmann@23878
   611
haftmann@25510
   612
end
haftmann@23878
   613
haftmann@23878
   614
lemma Inf_empty_fun:
haftmann@25206
   615
  "\<Sqinter>{} = (\<lambda>_. \<Sqinter>{})"
haftmann@23878
   616
  by rule (auto simp add: Inf_fun_def)
haftmann@23878
   617
haftmann@23878
   618
lemma Sup_empty_fun:
haftmann@25206
   619
  "\<Squnion>{} = (\<lambda>_. \<Squnion>{})"
haftmann@24345
   620
  by rule (auto simp add: Sup_fun_def)
haftmann@23878
   621
haftmann@23878
   622
lemma top_fun_eq: "top = (\<lambda>x. top)"
haftmann@23878
   623
  by (iprover intro!: order_antisym le_funI top_greatest)
haftmann@23878
   624
haftmann@23878
   625
lemma bot_fun_eq: "bot = (\<lambda>x. bot)"
haftmann@23878
   626
  by (iprover intro!: order_antisym le_funI bot_least)
haftmann@23878
   627
haftmann@23878
   628
haftmann@23878
   629
text {* redundant bindings *}
haftmann@22454
   630
haftmann@22454
   631
lemmas inf_aci = inf_ACI
haftmann@22454
   632
lemmas sup_aci = sup_ACI
haftmann@22454
   633
haftmann@25062
   634
no_notation
wenzelm@25382
   635
  less_eq  (infix "\<sqsubseteq>" 50) and
wenzelm@25382
   636
  less (infix "\<sqsubset>" 50) and
wenzelm@25382
   637
  inf  (infixl "\<sqinter>" 70) and
wenzelm@25382
   638
  sup  (infixl "\<squnion>" 65) and
wenzelm@25382
   639
  Inf  ("\<Sqinter>_" [900] 900) and
wenzelm@25382
   640
  Sup  ("\<Squnion>_" [900] 900)
haftmann@25062
   641
haftmann@21249
   642
end