src/HOL/Lattices.thy
author haftmann
Wed Jan 21 16:47:31 2009 +0100 (2009-01-21)
changeset 29580 117b88da143c
parent 29509 1ff0f3f08a7b
child 30302 5ffa9d4dbea7
permissions -rw-r--r--
dropped ID
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(*  Title:      HOL/Lattices.thy
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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
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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!: distrib_lattice "op \<le> :: 'a::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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   313
 
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   314
lemmas max_ac = min_max.sup_assoc min_max.sup_commute
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   315
  mk_left_commute [of max, OF min_max.sup_assoc min_max.sup_commute]
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   316
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   317
lemmas min_ac = min_max.inf_assoc min_max.inf_commute
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   318
  mk_left_commute [of min, OF min_max.inf_assoc min_max.inf_commute]
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   319
haftmann@22454
   320
text {*
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   321
  Now we have inherited antisymmetry as an intro-rule on all
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   322
  linear orders. This is a problem because it applies to bool, which is
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   323
  undesirable.
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   324
*}
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   325
haftmann@25102
   326
lemmas [rule del] = min_max.le_infI min_max.le_supI
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   327
  min_max.le_supE min_max.le_infE min_max.le_supI1 min_max.le_supI2
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   328
  min_max.le_infI1 min_max.le_infI2
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   329
haftmann@22454
   330
haftmann@23878
   331
subsection {* Complete lattices *}
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   332
haftmann@28692
   333
class complete_lattice = lattice + bot + top +
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   334
  fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900)
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   335
    and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900)
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   336
  assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x"
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   337
     and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A"
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   338
  assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A"
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   339
     and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z"
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   340
begin
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   341
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   342
lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> a}"
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   343
  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
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   344
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   345
lemma Sup_Inf:  "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<le> b}"
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   346
  by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least)
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   347
haftmann@23878
   348
lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}"
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   349
  unfolding Sup_Inf by auto
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   350
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   351
lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}"
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   352
  unfolding Inf_Sup by auto
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   353
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   354
lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
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   355
  by (auto intro: antisym Inf_greatest Inf_lower)
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   356
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   357
lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
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   358
  by (auto intro: antisym Sup_least Sup_upper)
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   359
haftmann@23878
   360
lemma Inf_singleton [simp]:
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   361
  "\<Sqinter>{a} = a"
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   362
  by (auto intro: antisym Inf_lower Inf_greatest)
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   363
haftmann@24345
   364
lemma Sup_singleton [simp]:
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   365
  "\<Squnion>{a} = a"
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   366
  by (auto intro: antisym Sup_upper Sup_least)
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   367
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   368
lemma Inf_insert_simp:
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   369
  "\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)"
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   370
  by (cases "A = {}") (simp_all, simp add: Inf_insert)
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   371
haftmann@23878
   372
lemma Sup_insert_simp:
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   373
  "\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)"
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   374
  by (cases "A = {}") (simp_all, simp add: Sup_insert)
haftmann@23878
   375
haftmann@23878
   376
lemma Inf_binary:
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   377
  "\<Sqinter>{a, b} = a \<sqinter> b"
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   378
  by (simp add: Inf_insert_simp)
haftmann@23878
   379
haftmann@23878
   380
lemma Sup_binary:
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   381
  "\<Squnion>{a, b} = a \<squnion> b"
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   382
  by (simp add: Sup_insert_simp)
haftmann@23878
   383
haftmann@28685
   384
lemma bot_def:
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   385
  "bot = \<Squnion>{}"
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   386
  by (auto intro: antisym Sup_least)
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   387
haftmann@28692
   388
lemma top_def:
haftmann@28692
   389
  "top = \<Sqinter>{}"
haftmann@28692
   390
  by (auto intro: antisym Inf_greatest)
haftmann@28692
   391
haftmann@28692
   392
lemma sup_bot [simp]:
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   393
  "x \<squnion> bot = x"
haftmann@28692
   394
  using bot_least [of x] by (simp add: le_iff_sup sup_commute)
haftmann@28692
   395
haftmann@28692
   396
lemma inf_top [simp]:
haftmann@28692
   397
  "x \<sqinter> top = x"
haftmann@28692
   398
  using top_greatest [of x] by (simp add: le_iff_inf inf_commute)
haftmann@28692
   399
haftmann@28692
   400
definition SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
haftmann@25206
   401
  "SUPR A f == \<Squnion> (f ` A)"
haftmann@23878
   402
haftmann@28692
   403
definition INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" where
haftmann@25206
   404
  "INFI A f == \<Sqinter> (f ` A)"
haftmann@23878
   405
haftmann@24749
   406
end
haftmann@24749
   407
haftmann@23878
   408
syntax
haftmann@23878
   409
  "_SUP1"     :: "pttrns => 'b => 'b"           ("(3SUP _./ _)" [0, 10] 10)
haftmann@23878
   410
  "_SUP"      :: "pttrn => 'a set => 'b => 'b"  ("(3SUP _:_./ _)" [0, 10] 10)
haftmann@23878
   411
  "_INF1"     :: "pttrns => 'b => 'b"           ("(3INF _./ _)" [0, 10] 10)
haftmann@23878
   412
  "_INF"      :: "pttrn => 'a set => 'b => 'b"  ("(3INF _:_./ _)" [0, 10] 10)
haftmann@23878
   413
haftmann@23878
   414
translations
haftmann@23878
   415
  "SUP x y. B"   == "SUP x. SUP y. B"
haftmann@23878
   416
  "SUP x. B"     == "CONST SUPR UNIV (%x. B)"
haftmann@23878
   417
  "SUP x. B"     == "SUP x:UNIV. B"
haftmann@23878
   418
  "SUP x:A. B"   == "CONST SUPR A (%x. B)"
haftmann@23878
   419
  "INF x y. B"   == "INF x. INF y. B"
haftmann@23878
   420
  "INF x. B"     == "CONST INFI UNIV (%x. B)"
haftmann@23878
   421
  "INF x. B"     == "INF x:UNIV. B"
haftmann@23878
   422
  "INF x:A. B"   == "CONST INFI A (%x. B)"
haftmann@23878
   423
haftmann@23878
   424
(* To avoid eta-contraction of body: *)
haftmann@23878
   425
print_translation {*
haftmann@23878
   426
let
haftmann@23878
   427
  fun btr' syn (A :: Abs abs :: ts) =
haftmann@23878
   428
    let val (x,t) = atomic_abs_tr' abs
haftmann@23878
   429
    in list_comb (Syntax.const syn $ x $ A $ t, ts) end
haftmann@23878
   430
  val const_syntax_name = Sign.const_syntax_name @{theory} o fst o dest_Const
haftmann@23878
   431
in
haftmann@23878
   432
[(const_syntax_name @{term SUPR}, btr' "_SUP"),(const_syntax_name @{term "INFI"}, btr' "_INF")]
haftmann@23878
   433
end
haftmann@23878
   434
*}
haftmann@23878
   435
haftmann@25102
   436
context complete_lattice
haftmann@25102
   437
begin
haftmann@25102
   438
haftmann@23878
   439
lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)"
haftmann@23878
   440
  by (auto simp add: SUPR_def intro: Sup_upper)
haftmann@23878
   441
haftmann@23878
   442
lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u"
haftmann@23878
   443
  by (auto simp add: SUPR_def intro: Sup_least)
haftmann@23878
   444
haftmann@23878
   445
lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i"
haftmann@23878
   446
  by (auto simp add: INFI_def intro: Inf_lower)
haftmann@23878
   447
haftmann@23878
   448
lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)"
haftmann@23878
   449
  by (auto simp add: INFI_def intro: Inf_greatest)
haftmann@23878
   450
haftmann@23878
   451
lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M"
haftmann@25102
   452
  by (auto intro: antisym SUP_leI le_SUPI)
haftmann@23878
   453
haftmann@23878
   454
lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M"
haftmann@25102
   455
  by (auto intro: antisym INF_leI le_INFI)
haftmann@25102
   456
haftmann@25102
   457
end
haftmann@23878
   458
haftmann@23878
   459
haftmann@22454
   460
subsection {* Bool as lattice *}
haftmann@22454
   461
haftmann@25510
   462
instantiation bool :: distrib_lattice
haftmann@25510
   463
begin
haftmann@25510
   464
haftmann@25510
   465
definition
haftmann@25510
   466
  inf_bool_eq: "P \<sqinter> Q \<longleftrightarrow> P \<and> Q"
haftmann@25510
   467
haftmann@25510
   468
definition
haftmann@25510
   469
  sup_bool_eq: "P \<squnion> Q \<longleftrightarrow> P \<or> Q"
haftmann@25510
   470
haftmann@25510
   471
instance
haftmann@22454
   472
  by intro_classes (auto simp add: inf_bool_eq sup_bool_eq le_bool_def)
haftmann@22454
   473
haftmann@25510
   474
end
haftmann@25510
   475
haftmann@25510
   476
instantiation bool :: complete_lattice
haftmann@25510
   477
begin
haftmann@25510
   478
haftmann@25510
   479
definition
haftmann@25510
   480
  Inf_bool_def: "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
haftmann@25510
   481
haftmann@25510
   482
definition
haftmann@25510
   483
  Sup_bool_def: "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
haftmann@25510
   484
haftmann@25510
   485
instance
haftmann@24345
   486
  by intro_classes (auto simp add: Inf_bool_def Sup_bool_def le_bool_def)
haftmann@23878
   487
haftmann@25510
   488
end
haftmann@25510
   489
haftmann@23878
   490
lemma Inf_empty_bool [simp]:
haftmann@25206
   491
  "\<Sqinter>{}"
haftmann@23878
   492
  unfolding Inf_bool_def by auto
haftmann@23878
   493
haftmann@23878
   494
lemma not_Sup_empty_bool [simp]:
haftmann@23878
   495
  "\<not> Sup {}"
haftmann@24345
   496
  unfolding Sup_bool_def by auto
haftmann@23878
   497
haftmann@23878
   498
haftmann@23878
   499
subsection {* Fun as lattice *}
haftmann@23878
   500
haftmann@25510
   501
instantiation "fun" :: (type, lattice) lattice
haftmann@25510
   502
begin
haftmann@25510
   503
haftmann@25510
   504
definition
haftmann@28562
   505
  inf_fun_eq [code del]: "f \<sqinter> g = (\<lambda>x. f x \<sqinter> g x)"
haftmann@25510
   506
haftmann@25510
   507
definition
haftmann@28562
   508
  sup_fun_eq [code del]: "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)"
haftmann@25510
   509
haftmann@25510
   510
instance
haftmann@23878
   511
apply intro_classes
haftmann@23878
   512
unfolding inf_fun_eq sup_fun_eq
haftmann@23878
   513
apply (auto intro: le_funI)
haftmann@23878
   514
apply (rule le_funI)
haftmann@23878
   515
apply (auto dest: le_funD)
haftmann@23878
   516
apply (rule le_funI)
haftmann@23878
   517
apply (auto dest: le_funD)
haftmann@23878
   518
done
haftmann@23878
   519
haftmann@25510
   520
end
haftmann@23878
   521
haftmann@23878
   522
instance "fun" :: (type, distrib_lattice) distrib_lattice
haftmann@23878
   523
  by default (auto simp add: inf_fun_eq sup_fun_eq sup_inf_distrib1)
haftmann@23878
   524
haftmann@25510
   525
instantiation "fun" :: (type, complete_lattice) complete_lattice
haftmann@25510
   526
begin
haftmann@25510
   527
haftmann@25510
   528
definition
haftmann@28562
   529
  Inf_fun_def [code del]: "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})"
haftmann@25510
   530
haftmann@25510
   531
definition
haftmann@28562
   532
  Sup_fun_def [code del]: "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})"
haftmann@25510
   533
haftmann@25510
   534
instance
haftmann@24345
   535
  by intro_classes
haftmann@24345
   536
    (auto simp add: Inf_fun_def Sup_fun_def le_fun_def
haftmann@24345
   537
      intro: Inf_lower Sup_upper Inf_greatest Sup_least)
haftmann@23878
   538
haftmann@25510
   539
end
haftmann@23878
   540
haftmann@23878
   541
lemma Inf_empty_fun:
haftmann@25206
   542
  "\<Sqinter>{} = (\<lambda>_. \<Sqinter>{})"
haftmann@23878
   543
  by rule (auto simp add: Inf_fun_def)
haftmann@23878
   544
haftmann@23878
   545
lemma Sup_empty_fun:
haftmann@25206
   546
  "\<Squnion>{} = (\<lambda>_. \<Squnion>{})"
haftmann@24345
   547
  by rule (auto simp add: Sup_fun_def)
haftmann@23878
   548
haftmann@23878
   549
berghofe@26794
   550
subsection {* Set as lattice *}
berghofe@26794
   551
berghofe@26794
   552
lemma inf_set_eq: "A \<sqinter> B = A \<inter> B"
berghofe@26794
   553
  apply (rule subset_antisym)
berghofe@26794
   554
  apply (rule Int_greatest)
berghofe@26794
   555
  apply (rule inf_le1)
berghofe@26794
   556
  apply (rule inf_le2)
berghofe@26794
   557
  apply (rule inf_greatest)
berghofe@26794
   558
  apply (rule Int_lower1)
berghofe@26794
   559
  apply (rule Int_lower2)
berghofe@26794
   560
  done
berghofe@26794
   561
berghofe@26794
   562
lemma sup_set_eq: "A \<squnion> B = A \<union> B"
berghofe@26794
   563
  apply (rule subset_antisym)
berghofe@26794
   564
  apply (rule sup_least)
berghofe@26794
   565
  apply (rule Un_upper1)
berghofe@26794
   566
  apply (rule Un_upper2)
berghofe@26794
   567
  apply (rule Un_least)
berghofe@26794
   568
  apply (rule sup_ge1)
berghofe@26794
   569
  apply (rule sup_ge2)
berghofe@26794
   570
  done
berghofe@26794
   571
berghofe@26794
   572
lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B"
berghofe@26794
   573
  apply (fold inf_set_eq sup_set_eq)
berghofe@26794
   574
  apply (erule mono_inf)
berghofe@26794
   575
  done
berghofe@26794
   576
berghofe@26794
   577
lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)"
berghofe@26794
   578
  apply (fold inf_set_eq sup_set_eq)
berghofe@26794
   579
  apply (erule mono_sup)
berghofe@26794
   580
  done
berghofe@26794
   581
berghofe@26794
   582
lemma Inf_set_eq: "\<Sqinter>S = \<Inter>S"
berghofe@26794
   583
  apply (rule subset_antisym)
berghofe@26794
   584
  apply (rule Inter_greatest)
berghofe@26794
   585
  apply (erule Inf_lower)
berghofe@26794
   586
  apply (rule Inf_greatest)
berghofe@26794
   587
  apply (erule Inter_lower)
berghofe@26794
   588
  done
berghofe@26794
   589
berghofe@26794
   590
lemma Sup_set_eq: "\<Squnion>S = \<Union>S"
berghofe@26794
   591
  apply (rule subset_antisym)
berghofe@26794
   592
  apply (rule Sup_least)
berghofe@26794
   593
  apply (erule Union_upper)
berghofe@26794
   594
  apply (rule Union_least)
berghofe@26794
   595
  apply (erule Sup_upper)
berghofe@26794
   596
  done
berghofe@26794
   597
  
berghofe@26794
   598
lemma top_set_eq: "top = UNIV"
berghofe@26794
   599
  by (iprover intro!: subset_antisym subset_UNIV top_greatest)
berghofe@26794
   600
berghofe@26794
   601
lemma bot_set_eq: "bot = {}"
berghofe@26794
   602
  by (iprover intro!: subset_antisym empty_subsetI bot_least)
berghofe@26794
   603
berghofe@26794
   604
haftmann@23878
   605
text {* redundant bindings *}
haftmann@22454
   606
haftmann@22454
   607
lemmas inf_aci = inf_ACI
haftmann@22454
   608
lemmas sup_aci = sup_ACI
haftmann@22454
   609
haftmann@25062
   610
no_notation
wenzelm@25382
   611
  less_eq  (infix "\<sqsubseteq>" 50) and
wenzelm@25382
   612
  less (infix "\<sqsubset>" 50) and
wenzelm@25382
   613
  inf  (infixl "\<sqinter>" 70) and
wenzelm@25382
   614
  sup  (infixl "\<squnion>" 65) and
wenzelm@25382
   615
  Inf  ("\<Sqinter>_" [900] 900) and
wenzelm@25382
   616
  Sup  ("\<Squnion>_" [900] 900)
haftmann@25062
   617
haftmann@21249
   618
end