src/HOL/Lattices.thy
author haftmann
Fri Oct 10 06:45:53 2008 +0200 (2008-10-10)
changeset 28562 4e74209f113e
parent 27682 25aceefd4786
child 28685 275122631271
permissions -rw-r--r--
`code func` now just `code`
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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 Fun
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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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begin
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text {* Dual lattice *}
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lemma dual_lattice:
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  "lower_semilattice (op \<ge>) (op >) sup"
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by (rule lower_semilattice.intro, rule dual_order)
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  (unfold_locales, simp_all add: sup_least)
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end
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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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   312
haftmann@21249
   313
lemmas le_maxI1 = min_max.sup_ge1
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   314
lemmas le_maxI2 = min_max.sup_ge2
haftmann@21381
   315
 
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   316
lemmas max_ac = min_max.sup_assoc min_max.sup_commute
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   317
  mk_left_commute [of max, OF min_max.sup_assoc min_max.sup_commute]
haftmann@21249
   318
haftmann@21249
   319
lemmas min_ac = min_max.inf_assoc min_max.inf_commute
haftmann@22422
   320
  mk_left_commute [of min, OF min_max.inf_assoc min_max.inf_commute]
haftmann@21249
   321
haftmann@22454
   322
text {*
haftmann@22454
   323
  Now we have inherited antisymmetry as an intro-rule on all
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   324
  linear orders. This is a problem because it applies to bool, which is
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   325
  undesirable.
haftmann@22454
   326
*}
haftmann@22454
   327
haftmann@25102
   328
lemmas [rule del] = min_max.le_infI min_max.le_supI
haftmann@22454
   329
  min_max.le_supE min_max.le_infE min_max.le_supI1 min_max.le_supI2
haftmann@22454
   330
  min_max.le_infI1 min_max.le_infI2
haftmann@22454
   331
haftmann@22454
   332
haftmann@23878
   333
subsection {* Complete lattices *}
haftmann@23878
   334
haftmann@23878
   335
class complete_lattice = lattice +
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   336
  fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
haftmann@24345
   337
    and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
haftmann@23878
   338
  assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
haftmann@24345
   339
     and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
haftmann@24345
   340
  assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
haftmann@24345
   341
     and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
haftmann@23878
   342
begin
haftmann@23878
   343
haftmann@25062
   344
lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> a}"
haftmann@25102
   345
  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
haftmann@23878
   346
haftmann@25062
   347
lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<le> b}"
haftmann@25102
   348
  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
haftmann@23878
   349
haftmann@23878
   350
lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}"
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   351
  unfolding Sup_Inf by auto
haftmann@23878
   352
haftmann@23878
   353
lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}"
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   354
  unfolding Inf_Sup by auto
haftmann@23878
   355
haftmann@23878
   356
lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
haftmann@26233
   357
  by (auto intro: antisym Inf_greatest Inf_lower)
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   358
haftmann@24345
   359
lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
haftmann@26233
   360
  by (auto intro: antisym Sup_least Sup_upper)
haftmann@23878
   361
haftmann@23878
   362
lemma Inf_singleton [simp]:
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   363
  "\<Sqinter>{a} = a"
haftmann@23878
   364
  by (auto intro: antisym Inf_lower Inf_greatest)
haftmann@23878
   365
haftmann@24345
   366
lemma Sup_singleton [simp]:
haftmann@23878
   367
  "\<Squnion>{a} = a"
haftmann@23878
   368
  by (auto intro: antisym Sup_upper Sup_least)
haftmann@23878
   369
haftmann@23878
   370
lemma Inf_insert_simp:
haftmann@23878
   371
  "\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)"
haftmann@23878
   372
  by (cases "A = {}") (simp_all, simp add: Inf_insert)
haftmann@23878
   373
haftmann@23878
   374
lemma Sup_insert_simp:
haftmann@23878
   375
  "\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)"
haftmann@23878
   376
  by (cases "A = {}") (simp_all, simp add: Sup_insert)
haftmann@23878
   377
haftmann@23878
   378
lemma Inf_binary:
haftmann@23878
   379
  "\<Sqinter>{a, b} = a \<sqinter> b"
haftmann@23878
   380
  by (simp add: Inf_insert_simp)
haftmann@23878
   381
haftmann@23878
   382
lemma Sup_binary:
haftmann@23878
   383
  "\<Squnion>{a, b} = a \<squnion> b"
haftmann@23878
   384
  by (simp add: Sup_insert_simp)
haftmann@23878
   385
haftmann@23878
   386
definition
wenzelm@25382
   387
  top :: 'a where
haftmann@25206
   388
  "top = \<Sqinter>{}"
haftmann@23878
   389
haftmann@23878
   390
definition
wenzelm@25382
   391
  bot :: 'a where
haftmann@25206
   392
  "bot = \<Squnion>{}"
haftmann@23878
   393
haftmann@25062
   394
lemma top_greatest [simp]: "x \<le> top"
haftmann@23878
   395
  by (unfold top_def, rule Inf_greatest, simp)
haftmann@23878
   396
haftmann@25062
   397
lemma bot_least [simp]: "bot \<le> x"
haftmann@23878
   398
  by (unfold bot_def, rule Sup_least, simp)
haftmann@23878
   399
haftmann@23878
   400
definition
haftmann@24749
   401
  SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
haftmann@23878
   402
where
haftmann@25206
   403
  "SUPR A f == \<Squnion> (f ` A)"
haftmann@23878
   404
haftmann@23878
   405
definition
haftmann@24749
   406
  INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a"
haftmann@23878
   407
where
haftmann@25206
   408
  "INFI A f == \<Sqinter> (f ` A)"
haftmann@23878
   409
haftmann@24749
   410
end
haftmann@24749
   411
haftmann@23878
   412
syntax
haftmann@23878
   413
  "_SUP1"     :: "pttrns => 'b => 'b"           ("(3SUP _./ _)" [0, 10] 10)
haftmann@23878
   414
  "_SUP"      :: "pttrn => 'a set => 'b => 'b"  ("(3SUP _:_./ _)" [0, 10] 10)
haftmann@23878
   415
  "_INF1"     :: "pttrns => 'b => 'b"           ("(3INF _./ _)" [0, 10] 10)
haftmann@23878
   416
  "_INF"      :: "pttrn => 'a set => 'b => 'b"  ("(3INF _:_./ _)" [0, 10] 10)
haftmann@23878
   417
haftmann@23878
   418
translations
haftmann@23878
   419
  "SUP x y. B"   == "SUP x. SUP y. B"
haftmann@23878
   420
  "SUP x. B"     == "CONST SUPR UNIV (%x. B)"
haftmann@23878
   421
  "SUP x. B"     == "SUP x:UNIV. B"
haftmann@23878
   422
  "SUP x:A. B"   == "CONST SUPR A (%x. B)"
haftmann@23878
   423
  "INF x y. B"   == "INF x. INF y. B"
haftmann@23878
   424
  "INF x. B"     == "CONST INFI UNIV (%x. B)"
haftmann@23878
   425
  "INF x. B"     == "INF x:UNIV. B"
haftmann@23878
   426
  "INF x:A. B"   == "CONST INFI A (%x. B)"
haftmann@23878
   427
haftmann@23878
   428
(* To avoid eta-contraction of body: *)
haftmann@23878
   429
print_translation {*
haftmann@23878
   430
let
haftmann@23878
   431
  fun btr' syn (A :: Abs abs :: ts) =
haftmann@23878
   432
    let val (x,t) = atomic_abs_tr' abs
haftmann@23878
   433
    in list_comb (Syntax.const syn $ x $ A $ t, ts) end
haftmann@23878
   434
  val const_syntax_name = Sign.const_syntax_name @{theory} o fst o dest_Const
haftmann@23878
   435
in
haftmann@23878
   436
[(const_syntax_name @{term SUPR}, btr' "_SUP"),(const_syntax_name @{term "INFI"}, btr' "_INF")]
haftmann@23878
   437
end
haftmann@23878
   438
*}
haftmann@23878
   439
haftmann@25102
   440
context complete_lattice
haftmann@25102
   441
begin
haftmann@25102
   442
haftmann@23878
   443
lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)"
haftmann@23878
   444
  by (auto simp add: SUPR_def intro: Sup_upper)
haftmann@23878
   445
haftmann@23878
   446
lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u"
haftmann@23878
   447
  by (auto simp add: SUPR_def intro: Sup_least)
haftmann@23878
   448
haftmann@23878
   449
lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i"
haftmann@23878
   450
  by (auto simp add: INFI_def intro: Inf_lower)
haftmann@23878
   451
haftmann@23878
   452
lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)"
haftmann@23878
   453
  by (auto simp add: INFI_def intro: Inf_greatest)
haftmann@23878
   454
haftmann@23878
   455
lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"
haftmann@25102
   456
  by (auto intro: antisym SUP_leI le_SUPI)
haftmann@23878
   457
haftmann@23878
   458
lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"
haftmann@25102
   459
  by (auto intro: antisym INF_leI le_INFI)
haftmann@25102
   460
haftmann@25102
   461
end
haftmann@23878
   462
haftmann@23878
   463
haftmann@22454
   464
subsection {* Bool as lattice *}
haftmann@22454
   465
haftmann@25510
   466
instantiation bool :: distrib_lattice
haftmann@25510
   467
begin
haftmann@25510
   468
haftmann@25510
   469
definition
haftmann@25510
   470
  inf_bool_eq: "P \<sqinter> Q \<longleftrightarrow> P \<and> Q"
haftmann@25510
   471
haftmann@25510
   472
definition
haftmann@25510
   473
  sup_bool_eq: "P \<squnion> Q \<longleftrightarrow> P \<or> Q"
haftmann@25510
   474
haftmann@25510
   475
instance
haftmann@22454
   476
  by intro_classes (auto simp add: inf_bool_eq sup_bool_eq le_bool_def)
haftmann@22454
   477
haftmann@25510
   478
end
haftmann@25510
   479
haftmann@25510
   480
instantiation bool :: complete_lattice
haftmann@25510
   481
begin
haftmann@25510
   482
haftmann@25510
   483
definition
haftmann@25510
   484
  Inf_bool_def: "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
haftmann@25510
   485
haftmann@25510
   486
definition
haftmann@25510
   487
  Sup_bool_def: "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
haftmann@25510
   488
haftmann@25510
   489
instance
haftmann@24345
   490
  by intro_classes (auto simp add: Inf_bool_def Sup_bool_def le_bool_def)
haftmann@23878
   491
haftmann@25510
   492
end
haftmann@25510
   493
haftmann@23878
   494
lemma Inf_empty_bool [simp]:
haftmann@25206
   495
  "\<Sqinter>{}"
haftmann@23878
   496
  unfolding Inf_bool_def by auto
haftmann@23878
   497
haftmann@23878
   498
lemma not_Sup_empty_bool [simp]:
haftmann@23878
   499
  "\<not> Sup {}"
haftmann@24345
   500
  unfolding Sup_bool_def by auto
haftmann@23878
   501
haftmann@23878
   502
lemma top_bool_eq: "top = True"
haftmann@23878
   503
  by (iprover intro!: order_antisym le_boolI top_greatest)
haftmann@23878
   504
haftmann@23878
   505
lemma bot_bool_eq: "bot = False"
haftmann@23878
   506
  by (iprover intro!: order_antisym le_boolI bot_least)
haftmann@23878
   507
haftmann@23878
   508
haftmann@23878
   509
subsection {* Fun as lattice *}
haftmann@23878
   510
haftmann@25510
   511
instantiation "fun" :: (type, lattice) lattice
haftmann@25510
   512
begin
haftmann@25510
   513
haftmann@25510
   514
definition
haftmann@28562
   515
  inf_fun_eq [code del]: "f \<sqinter> g = (\<lambda>x. f x \<sqinter> g x)"
haftmann@25510
   516
haftmann@25510
   517
definition
haftmann@28562
   518
  sup_fun_eq [code del]: "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)"
haftmann@25510
   519
haftmann@25510
   520
instance
haftmann@23878
   521
apply intro_classes
haftmann@23878
   522
unfolding inf_fun_eq sup_fun_eq
haftmann@23878
   523
apply (auto intro: le_funI)
haftmann@23878
   524
apply (rule le_funI)
haftmann@23878
   525
apply (auto dest: le_funD)
haftmann@23878
   526
apply (rule le_funI)
haftmann@23878
   527
apply (auto dest: le_funD)
haftmann@23878
   528
done
haftmann@23878
   529
haftmann@25510
   530
end
haftmann@23878
   531
haftmann@23878
   532
instance "fun" :: (type, distrib_lattice) distrib_lattice
haftmann@23878
   533
  by default (auto simp add: inf_fun_eq sup_fun_eq sup_inf_distrib1)
haftmann@23878
   534
haftmann@25510
   535
instantiation "fun" :: (type, complete_lattice) complete_lattice
haftmann@25510
   536
begin
haftmann@25510
   537
haftmann@25510
   538
definition
haftmann@28562
   539
  Inf_fun_def [code del]: "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
haftmann@25510
   540
haftmann@25510
   541
definition
haftmann@28562
   542
  Sup_fun_def [code del]: "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
haftmann@25510
   543
haftmann@25510
   544
instance
haftmann@24345
   545
  by intro_classes
haftmann@24345
   546
    (auto simp add: Inf_fun_def Sup_fun_def le_fun_def
haftmann@24345
   547
      intro: Inf_lower Sup_upper Inf_greatest Sup_least)
haftmann@23878
   548
haftmann@25510
   549
end
haftmann@23878
   550
haftmann@23878
   551
lemma Inf_empty_fun:
haftmann@25206
   552
  "\<Sqinter>{} = (\<lambda>_. \<Sqinter>{})"
haftmann@23878
   553
  by rule (auto simp add: Inf_fun_def)
haftmann@23878
   554
haftmann@23878
   555
lemma Sup_empty_fun:
haftmann@25206
   556
  "\<Squnion>{} = (\<lambda>_. \<Squnion>{})"
haftmann@24345
   557
  by rule (auto simp add: Sup_fun_def)
haftmann@23878
   558
haftmann@23878
   559
lemma top_fun_eq: "top = (\<lambda>x. top)"
haftmann@23878
   560
  by (iprover intro!: order_antisym le_funI top_greatest)
haftmann@23878
   561
haftmann@23878
   562
lemma bot_fun_eq: "bot = (\<lambda>x. bot)"
haftmann@23878
   563
  by (iprover intro!: order_antisym le_funI bot_least)
haftmann@23878
   564
haftmann@23878
   565
berghofe@26794
   566
subsection {* Set as lattice *}
berghofe@26794
   567
berghofe@26794
   568
lemma inf_set_eq: "A \<sqinter> B = A \<inter> B"
berghofe@26794
   569
  apply (rule subset_antisym)
berghofe@26794
   570
  apply (rule Int_greatest)
berghofe@26794
   571
  apply (rule inf_le1)
berghofe@26794
   572
  apply (rule inf_le2)
berghofe@26794
   573
  apply (rule inf_greatest)
berghofe@26794
   574
  apply (rule Int_lower1)
berghofe@26794
   575
  apply (rule Int_lower2)
berghofe@26794
   576
  done
berghofe@26794
   577
berghofe@26794
   578
lemma sup_set_eq: "A \<squnion> B = A \<union> B"
berghofe@26794
   579
  apply (rule subset_antisym)
berghofe@26794
   580
  apply (rule sup_least)
berghofe@26794
   581
  apply (rule Un_upper1)
berghofe@26794
   582
  apply (rule Un_upper2)
berghofe@26794
   583
  apply (rule Un_least)
berghofe@26794
   584
  apply (rule sup_ge1)
berghofe@26794
   585
  apply (rule sup_ge2)
berghofe@26794
   586
  done
berghofe@26794
   587
berghofe@26794
   588
lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"
berghofe@26794
   589
  apply (fold inf_set_eq sup_set_eq)
berghofe@26794
   590
  apply (erule mono_inf)
berghofe@26794
   591
  done
berghofe@26794
   592
berghofe@26794
   593
lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)"
berghofe@26794
   594
  apply (fold inf_set_eq sup_set_eq)
berghofe@26794
   595
  apply (erule mono_sup)
berghofe@26794
   596
  done
berghofe@26794
   597
berghofe@26794
   598
lemma Inf_set_eq: "\<Sqinter>S = \<Inter>S"
berghofe@26794
   599
  apply (rule subset_antisym)
berghofe@26794
   600
  apply (rule Inter_greatest)
berghofe@26794
   601
  apply (erule Inf_lower)
berghofe@26794
   602
  apply (rule Inf_greatest)
berghofe@26794
   603
  apply (erule Inter_lower)
berghofe@26794
   604
  done
berghofe@26794
   605
berghofe@26794
   606
lemma Sup_set_eq: "\<Squnion>S = \<Union>S"
berghofe@26794
   607
  apply (rule subset_antisym)
berghofe@26794
   608
  apply (rule Sup_least)
berghofe@26794
   609
  apply (erule Union_upper)
berghofe@26794
   610
  apply (rule Union_least)
berghofe@26794
   611
  apply (erule Sup_upper)
berghofe@26794
   612
  done
berghofe@26794
   613
  
berghofe@26794
   614
lemma top_set_eq: "top = UNIV"
berghofe@26794
   615
  by (iprover intro!: subset_antisym subset_UNIV top_greatest)
berghofe@26794
   616
berghofe@26794
   617
lemma bot_set_eq: "bot = {}"
berghofe@26794
   618
  by (iprover intro!: subset_antisym empty_subsetI bot_least)
berghofe@26794
   619
berghofe@26794
   620
haftmann@23878
   621
text {* redundant bindings *}
haftmann@22454
   622
haftmann@22454
   623
lemmas inf_aci = inf_ACI
haftmann@22454
   624
lemmas sup_aci = sup_ACI
haftmann@22454
   625
haftmann@25062
   626
no_notation
wenzelm@25382
   627
  less_eq  (infix "\<sqsubseteq>" 50) and
wenzelm@25382
   628
  less (infix "\<sqsubset>" 50) and
wenzelm@25382
   629
  inf  (infixl "\<sqinter>" 70) and
wenzelm@25382
   630
  sup  (infixl "\<squnion>" 65) and
wenzelm@25382
   631
  Inf  ("\<Sqinter>_" [900] 900) and
wenzelm@25382
   632
  Sup  ("\<Squnion>_" [900] 900)
haftmann@25062
   633
haftmann@21249
   634
end