author | berghofe |
Wed, 07 May 2008 10:56:36 +0200 | |
changeset 26794 | 354c3844dfde |
parent 26233 | 3751b3dbb67c |
child 27682 | 25aceefd4786 |
permissions | -rw-r--r-- |
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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 unfold_locales |
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(auto simp 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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|
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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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||
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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)" |
207 |
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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||
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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)" |
219 |
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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||
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(* seems unused *) |
230 |
lemma modular_le: "x \<sqsubseteq> z \<Longrightarrow> x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> z" |
|
231 |
by blast |
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232 |
||
21733 | 233 |
end |
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||
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subsection {* Distributive lattices *} |
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|
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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)" |
240 |
||
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context distrib_lattice |
242 |
begin |
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243 |
||
244 |
lemma sup_inf_distrib2: |
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"(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)" |
246 |
by(simp add:ACI sup_inf_distrib1) |
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247 |
||
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lemma inf_sup_distrib1: |
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"x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" |
250 |
by(rule distrib_imp2[OF sup_inf_distrib1]) |
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251 |
||
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lemma inf_sup_distrib2: |
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"(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)" |
254 |
by(simp add:ACI inf_sup_distrib1) |
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255 |
||
21733 | 256 |
lemmas distrib = |
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sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2 |
258 |
||
21733 | 259 |
end |
260 |
||
21249 | 261 |
|
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subsection {* Uniqueness of inf and sup *} |
263 |
||
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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" |
267 |
and greatest: "\<And>x y z. x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> y \<triangle> z" |
|
22737 | 268 |
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) |
22454 | 271 |
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) |
273 |
show "x \<sqinter> y \<le> x \<triangle> y" by (rule leI) simp_all |
|
22454 | 274 |
qed |
275 |
||
22737 | 276 |
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" |
279 |
and least: "\<And>x y z. y \<le> x \<Longrightarrow> z \<le> x \<Longrightarrow> y \<nabla> z \<le> x" |
|
22737 | 280 |
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 |
25062 | 284 |
have leI: "\<And>x y z. x \<le> z \<Longrightarrow> y \<le> z \<Longrightarrow> x \<nabla> y \<le> z" by (blast intro: least) |
285 |
show "x \<nabla> y \<le> x \<squnion> y" by (rule leI) simp_all |
|
22454 | 286 |
qed |
287 |
||
288 |
||
22916 | 289 |
subsection {* @{const min}/@{const max} on linear orders as |
290 |
special case of @{const inf}/@{const sup} *} |
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291 |
||
292 |
lemma (in linorder) distrib_lattice_min_max: |
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25062 | 293 |
"distrib_lattice (op \<le>) (op <) min max" |
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proof unfold_locales |
25062 | 295 |
have aux: "\<And>x y \<Colon> 'a. x < y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y" |
22916 | 296 |
by (auto simp add: less_le antisym) |
297 |
fix x y z |
|
298 |
show "max x (min y z) = min (max x y) (max x z)" |
|
299 |
unfolding min_def max_def |
|
24640
85a6c200ecd3
Simplified proofs due to transitivity reasoner setup.
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parents:
24514
diff
changeset
|
300 |
by auto |
22916 | 301 |
qed (auto simp add: min_def max_def not_le less_imp_le) |
21249 | 302 |
|
303 |
interpretation min_max: |
|
22454 | 304 |
distrib_lattice ["op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool" "op <" min max] |
23948 | 305 |
by (rule distrib_lattice_min_max) |
21249 | 306 |
|
22454 | 307 |
lemma inf_min: "inf = (min \<Colon> 'a\<Colon>{lower_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)" |
25102
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changeset
|
308 |
by (rule ext)+ (auto intro: antisym) |
21733 | 309 |
|
22454 | 310 |
lemma sup_max: "sup = (max \<Colon> 'a\<Colon>{upper_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)" |
25102
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parents:
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changeset
|
311 |
by (rule ext)+ (auto intro: antisym) |
21733 | 312 |
|
21249 | 313 |
lemmas le_maxI1 = min_max.sup_ge1 |
314 |
lemmas le_maxI2 = min_max.sup_ge2 |
|
21381 | 315 |
|
21249 | 316 |
lemmas max_ac = min_max.sup_assoc min_max.sup_commute |
22422
ee19cdb07528
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haftmann
parents:
22384
diff
changeset
|
317 |
mk_left_commute [of max, OF min_max.sup_assoc min_max.sup_commute] |
21249 | 318 |
|
319 |
lemmas min_ac = min_max.inf_assoc min_max.inf_commute |
|
22422
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changeset
|
320 |
mk_left_commute [of min, OF min_max.inf_assoc min_max.inf_commute] |
21249 | 321 |
|
22454 | 322 |
text {* |
323 |
Now we have inherited antisymmetry as an intro-rule on all |
|
324 |
linear orders. This is a problem because it applies to bool, which is |
|
325 |
undesirable. |
|
326 |
*} |
|
327 |
||
25102
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|
328 |
lemmas [rule del] = min_max.le_infI min_max.le_supI |
22454 | 329 |
min_max.le_supE min_max.le_infE min_max.le_supI1 min_max.le_supI2 |
330 |
min_max.le_infI1 min_max.le_infI2 |
|
331 |
||
332 |
||
23878 | 333 |
subsection {* Complete lattices *} |
334 |
||
335 |
class complete_lattice = lattice + |
|
336 |
fixes Inf :: "'a set \<Rightarrow> 'a" ("\<Sqinter>_" [900] 900) |
|
24345 | 337 |
and Sup :: "'a set \<Rightarrow> 'a" ("\<Squnion>_" [900] 900) |
23878 | 338 |
assumes Inf_lower: "x \<in> A \<Longrightarrow> \<Sqinter>A \<sqsubseteq> x" |
24345 | 339 |
and Inf_greatest: "(\<And>x. x \<in> A \<Longrightarrow> z \<sqsubseteq> x) \<Longrightarrow> z \<sqsubseteq> \<Sqinter>A" |
340 |
assumes Sup_upper: "x \<in> A \<Longrightarrow> x \<sqsubseteq> \<Squnion>A" |
|
341 |
and Sup_least: "(\<And>x. x \<in> A \<Longrightarrow> x \<sqsubseteq> z) \<Longrightarrow> \<Squnion>A \<sqsubseteq> z" |
|
23878 | 342 |
begin |
343 |
||
25062 | 344 |
lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<le> a}" |
25102
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changeset
|
345 |
by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) |
23878 | 346 |
|
25062 | 347 |
lemma Sup_Inf: "\<Squnion>A = \<Sqinter>{b. \<forall>a \<in> A. a \<le> b}" |
25102
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|
348 |
by (auto intro: antisym Inf_lower Inf_greatest Sup_upper Sup_least) |
23878 | 349 |
|
350 |
lemma Inf_Univ: "\<Sqinter>UNIV = \<Squnion>{}" |
|
24345 | 351 |
unfolding Sup_Inf by auto |
23878 | 352 |
|
353 |
lemma Sup_Univ: "\<Squnion>UNIV = \<Sqinter>{}" |
|
354 |
unfolding Inf_Sup by auto |
|
355 |
||
356 |
lemma Inf_insert: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A" |
|
26233 | 357 |
by (auto intro: antisym Inf_greatest Inf_lower) |
23878 | 358 |
|
24345 | 359 |
lemma Sup_insert: "\<Squnion>insert a A = a \<squnion> \<Squnion>A" |
26233 | 360 |
by (auto intro: antisym Sup_least Sup_upper) |
23878 | 361 |
|
362 |
lemma Inf_singleton [simp]: |
|
363 |
"\<Sqinter>{a} = a" |
|
364 |
by (auto intro: antisym Inf_lower Inf_greatest) |
|
365 |
||
24345 | 366 |
lemma Sup_singleton [simp]: |
23878 | 367 |
"\<Squnion>{a} = a" |
368 |
by (auto intro: antisym Sup_upper Sup_least) |
|
369 |
||
370 |
lemma Inf_insert_simp: |
|
371 |
"\<Sqinter>insert a A = (if A = {} then a else a \<sqinter> \<Sqinter>A)" |
|
372 |
by (cases "A = {}") (simp_all, simp add: Inf_insert) |
|
373 |
||
374 |
lemma Sup_insert_simp: |
|
375 |
"\<Squnion>insert a A = (if A = {} then a else a \<squnion> \<Squnion>A)" |
|
376 |
by (cases "A = {}") (simp_all, simp add: Sup_insert) |
|
377 |
||
378 |
lemma Inf_binary: |
|
379 |
"\<Sqinter>{a, b} = a \<sqinter> b" |
|
380 |
by (simp add: Inf_insert_simp) |
|
381 |
||
382 |
lemma Sup_binary: |
|
383 |
"\<Squnion>{a, b} = a \<squnion> b" |
|
384 |
by (simp add: Sup_insert_simp) |
|
385 |
||
386 |
definition |
|
25382 | 387 |
top :: 'a where |
25206 | 388 |
"top = \<Sqinter>{}" |
23878 | 389 |
|
390 |
definition |
|
25382 | 391 |
bot :: 'a where |
25206 | 392 |
"bot = \<Squnion>{}" |
23878 | 393 |
|
25062 | 394 |
lemma top_greatest [simp]: "x \<le> top" |
23878 | 395 |
by (unfold top_def, rule Inf_greatest, simp) |
396 |
||
25062 | 397 |
lemma bot_least [simp]: "bot \<le> x" |
23878 | 398 |
by (unfold bot_def, rule Sup_least, simp) |
399 |
||
400 |
definition |
|
24749 | 401 |
SUPR :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" |
23878 | 402 |
where |
25206 | 403 |
"SUPR A f == \<Squnion> (f ` A)" |
23878 | 404 |
|
405 |
definition |
|
24749 | 406 |
INFI :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'a" |
23878 | 407 |
where |
25206 | 408 |
"INFI A f == \<Sqinter> (f ` A)" |
23878 | 409 |
|
24749 | 410 |
end |
411 |
||
23878 | 412 |
syntax |
413 |
"_SUP1" :: "pttrns => 'b => 'b" ("(3SUP _./ _)" [0, 10] 10) |
|
414 |
"_SUP" :: "pttrn => 'a set => 'b => 'b" ("(3SUP _:_./ _)" [0, 10] 10) |
|
415 |
"_INF1" :: "pttrns => 'b => 'b" ("(3INF _./ _)" [0, 10] 10) |
|
416 |
"_INF" :: "pttrn => 'a set => 'b => 'b" ("(3INF _:_./ _)" [0, 10] 10) |
|
417 |
||
418 |
translations |
|
419 |
"SUP x y. B" == "SUP x. SUP y. B" |
|
420 |
"SUP x. B" == "CONST SUPR UNIV (%x. B)" |
|
421 |
"SUP x. B" == "SUP x:UNIV. B" |
|
422 |
"SUP x:A. B" == "CONST SUPR A (%x. B)" |
|
423 |
"INF x y. B" == "INF x. INF y. B" |
|
424 |
"INF x. B" == "CONST INFI UNIV (%x. B)" |
|
425 |
"INF x. B" == "INF x:UNIV. B" |
|
426 |
"INF x:A. B" == "CONST INFI A (%x. B)" |
|
427 |
||
428 |
(* To avoid eta-contraction of body: *) |
|
429 |
print_translation {* |
|
430 |
let |
|
431 |
fun btr' syn (A :: Abs abs :: ts) = |
|
432 |
let val (x,t) = atomic_abs_tr' abs |
|
433 |
in list_comb (Syntax.const syn $ x $ A $ t, ts) end |
|
434 |
val const_syntax_name = Sign.const_syntax_name @{theory} o fst o dest_Const |
|
435 |
in |
|
436 |
[(const_syntax_name @{term SUPR}, btr' "_SUP"),(const_syntax_name @{term "INFI"}, btr' "_INF")] |
|
437 |
end |
|
438 |
*} |
|
439 |
||
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|
440 |
context complete_lattice |
db3e412c4cb1
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|
441 |
begin |
db3e412c4cb1
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changeset
|
442 |
|
23878 | 443 |
lemma le_SUPI: "i : A \<Longrightarrow> M i \<le> (SUP i:A. M i)" |
444 |
by (auto simp add: SUPR_def intro: Sup_upper) |
|
445 |
||
446 |
lemma SUP_leI: "(\<And>i. i : A \<Longrightarrow> M i \<le> u) \<Longrightarrow> (SUP i:A. M i) \<le> u" |
|
447 |
by (auto simp add: SUPR_def intro: Sup_least) |
|
448 |
||
449 |
lemma INF_leI: "i : A \<Longrightarrow> (INF i:A. M i) \<le> M i" |
|
450 |
by (auto simp add: INFI_def intro: Inf_lower) |
|
451 |
||
452 |
lemma le_INFI: "(\<And>i. i : A \<Longrightarrow> u \<le> M i) \<Longrightarrow> u \<le> (INF i:A. M i)" |
|
453 |
by (auto simp add: INFI_def intro: Inf_greatest) |
|
454 |
||
455 |
lemma SUP_const[simp]: "A \<noteq> {} \<Longrightarrow> (SUP i:A. M) = M" |
|
25102
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parents:
25062
diff
changeset
|
456 |
by (auto intro: antisym SUP_leI le_SUPI) |
23878 | 457 |
|
458 |
lemma INF_const[simp]: "A \<noteq> {} \<Longrightarrow> (INF i:A. M) = M" |
|
25102
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antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
459 |
by (auto intro: antisym INF_leI le_INFI) |
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
460 |
|
db3e412c4cb1
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parents:
25062
diff
changeset
|
461 |
end |
23878 | 462 |
|
463 |
||
22454 | 464 |
subsection {* Bool as lattice *} |
465 |
||
25510 | 466 |
instantiation bool :: distrib_lattice |
467 |
begin |
|
468 |
||
469 |
definition |
|
470 |
inf_bool_eq: "P \<sqinter> Q \<longleftrightarrow> P \<and> Q" |
|
471 |
||
472 |
definition |
|
473 |
sup_bool_eq: "P \<squnion> Q \<longleftrightarrow> P \<or> Q" |
|
474 |
||
475 |
instance |
|
22454 | 476 |
by intro_classes (auto simp add: inf_bool_eq sup_bool_eq le_bool_def) |
477 |
||
25510 | 478 |
end |
479 |
||
480 |
instantiation bool :: complete_lattice |
|
481 |
begin |
|
482 |
||
483 |
definition |
|
484 |
Inf_bool_def: "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)" |
|
485 |
||
486 |
definition |
|
487 |
Sup_bool_def: "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)" |
|
488 |
||
489 |
instance |
|
24345 | 490 |
by intro_classes (auto simp add: Inf_bool_def Sup_bool_def le_bool_def) |
23878 | 491 |
|
25510 | 492 |
end |
493 |
||
23878 | 494 |
lemma Inf_empty_bool [simp]: |
25206 | 495 |
"\<Sqinter>{}" |
23878 | 496 |
unfolding Inf_bool_def by auto |
497 |
||
498 |
lemma not_Sup_empty_bool [simp]: |
|
499 |
"\<not> Sup {}" |
|
24345 | 500 |
unfolding Sup_bool_def by auto |
23878 | 501 |
|
502 |
lemma top_bool_eq: "top = True" |
|
503 |
by (iprover intro!: order_antisym le_boolI top_greatest) |
|
504 |
||
505 |
lemma bot_bool_eq: "bot = False" |
|
506 |
by (iprover intro!: order_antisym le_boolI bot_least) |
|
507 |
||
508 |
||
509 |
subsection {* Fun as lattice *} |
|
510 |
||
25510 | 511 |
instantiation "fun" :: (type, lattice) lattice |
512 |
begin |
|
513 |
||
514 |
definition |
|
515 |
inf_fun_eq [code func del]: "f \<sqinter> g = (\<lambda>x. f x \<sqinter> g x)" |
|
516 |
||
517 |
definition |
|
518 |
sup_fun_eq [code func del]: "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)" |
|
519 |
||
520 |
instance |
|
23878 | 521 |
apply intro_classes |
522 |
unfolding inf_fun_eq sup_fun_eq |
|
523 |
apply (auto intro: le_funI) |
|
524 |
apply (rule le_funI) |
|
525 |
apply (auto dest: le_funD) |
|
526 |
apply (rule le_funI) |
|
527 |
apply (auto dest: le_funD) |
|
528 |
done |
|
529 |
||
25510 | 530 |
end |
23878 | 531 |
|
532 |
instance "fun" :: (type, distrib_lattice) distrib_lattice |
|
533 |
by default (auto simp add: inf_fun_eq sup_fun_eq sup_inf_distrib1) |
|
534 |
||
25510 | 535 |
instantiation "fun" :: (type, complete_lattice) complete_lattice |
536 |
begin |
|
537 |
||
538 |
definition |
|
539 |
Inf_fun_def [code func del]: "\<Sqinter>A = (\<lambda>x. \<Sqinter>{y. \<exists>f\<in>A. y = f x})" |
|
540 |
||
541 |
definition |
|
542 |
Sup_fun_def [code func del]: "\<Squnion>A = (\<lambda>x. \<Squnion>{y. \<exists>f\<in>A. y = f x})" |
|
543 |
||
544 |
instance |
|
24345 | 545 |
by intro_classes |
546 |
(auto simp add: Inf_fun_def Sup_fun_def le_fun_def |
|
547 |
intro: Inf_lower Sup_upper Inf_greatest Sup_least) |
|
23878 | 548 |
|
25510 | 549 |
end |
23878 | 550 |
|
551 |
lemma Inf_empty_fun: |
|
25206 | 552 |
"\<Sqinter>{} = (\<lambda>_. \<Sqinter>{})" |
23878 | 553 |
by rule (auto simp add: Inf_fun_def) |
554 |
||
555 |
lemma Sup_empty_fun: |
|
25206 | 556 |
"\<Squnion>{} = (\<lambda>_. \<Squnion>{})" |
24345 | 557 |
by rule (auto simp add: Sup_fun_def) |
23878 | 558 |
|
559 |
lemma top_fun_eq: "top = (\<lambda>x. top)" |
|
560 |
by (iprover intro!: order_antisym le_funI top_greatest) |
|
561 |
||
562 |
lemma bot_fun_eq: "bot = (\<lambda>x. bot)" |
|
563 |
by (iprover intro!: order_antisym le_funI bot_least) |
|
564 |
||
565 |
||
26794 | 566 |
subsection {* Set as lattice *} |
567 |
||
568 |
lemma inf_set_eq: "A \<sqinter> B = A \<inter> B" |
|
569 |
apply (rule subset_antisym) |
|
570 |
apply (rule Int_greatest) |
|
571 |
apply (rule inf_le1) |
|
572 |
apply (rule inf_le2) |
|
573 |
apply (rule inf_greatest) |
|
574 |
apply (rule Int_lower1) |
|
575 |
apply (rule Int_lower2) |
|
576 |
done |
|
577 |
||
578 |
lemma sup_set_eq: "A \<squnion> B = A \<union> B" |
|
579 |
apply (rule subset_antisym) |
|
580 |
apply (rule sup_least) |
|
581 |
apply (rule Un_upper1) |
|
582 |
apply (rule Un_upper2) |
|
583 |
apply (rule Un_least) |
|
584 |
apply (rule sup_ge1) |
|
585 |
apply (rule sup_ge2) |
|
586 |
done |
|
587 |
||
588 |
lemma mono_Int: "mono f \<Longrightarrow> f (A \<inter> B) \<subseteq> f A \<inter> f B" |
|
589 |
apply (fold inf_set_eq sup_set_eq) |
|
590 |
apply (erule mono_inf) |
|
591 |
done |
|
592 |
||
593 |
lemma mono_Un: "mono f \<Longrightarrow> f A \<union> f B \<subseteq> f (A \<union> B)" |
|
594 |
apply (fold inf_set_eq sup_set_eq) |
|
595 |
apply (erule mono_sup) |
|
596 |
done |
|
597 |
||
598 |
lemma Inf_set_eq: "\<Sqinter>S = \<Inter>S" |
|
599 |
apply (rule subset_antisym) |
|
600 |
apply (rule Inter_greatest) |
|
601 |
apply (erule Inf_lower) |
|
602 |
apply (rule Inf_greatest) |
|
603 |
apply (erule Inter_lower) |
|
604 |
done |
|
605 |
||
606 |
lemma Sup_set_eq: "\<Squnion>S = \<Union>S" |
|
607 |
apply (rule subset_antisym) |
|
608 |
apply (rule Sup_least) |
|
609 |
apply (erule Union_upper) |
|
610 |
apply (rule Union_least) |
|
611 |
apply (erule Sup_upper) |
|
612 |
done |
|
613 |
||
614 |
lemma top_set_eq: "top = UNIV" |
|
615 |
by (iprover intro!: subset_antisym subset_UNIV top_greatest) |
|
616 |
||
617 |
lemma bot_set_eq: "bot = {}" |
|
618 |
by (iprover intro!: subset_antisym empty_subsetI bot_least) |
|
619 |
||
620 |
||
23878 | 621 |
text {* redundant bindings *} |
22454 | 622 |
|
623 |
lemmas inf_aci = inf_ACI |
|
624 |
lemmas sup_aci = sup_ACI |
|
625 |
||
25062 | 626 |
no_notation |
25382 | 627 |
less_eq (infix "\<sqsubseteq>" 50) and |
628 |
less (infix "\<sqsubset>" 50) and |
|
629 |
inf (infixl "\<sqinter>" 70) and |
|
630 |
sup (infixl "\<squnion>" 65) and |
|
631 |
Inf ("\<Sqinter>_" [900] 900) and |
|
632 |
Sup ("\<Squnion>_" [900] 900) |
|
25062 | 633 |
|
21249 | 634 |
end |