author | haftmann |
Mon, 13 Jul 2009 08:25:43 +0200 | |
changeset 32063 | 2aab4f2af536 |
parent 31991 | 37390299214a |
child 32064 | 53ca12ff305d |
permissions | -rw-r--r-- |
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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 Orderings |
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begin |
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subsection {* Lattices *} |
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notation |
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less_eq (infix "\<sqsubseteq>" 50) and |
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less (infix "\<sqsubset>" 50) |
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class lower_semilattice = order + |
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fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70) |
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assumes inf_le1 [simp]: "x \<sqinter> y \<sqsubseteq> x" |
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and inf_le2 [simp]: "x \<sqinter> y \<sqsubseteq> y" |
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and inf_greatest: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<sqinter> z" |
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class upper_semilattice = order + |
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fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65) |
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assumes sup_ge1 [simp]: "x \<sqsubseteq> x \<squnion> y" |
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and sup_ge2 [simp]: "y \<sqsubseteq> x \<squnion> y" |
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and sup_least: "y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<squnion> z \<sqsubseteq> x" |
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begin |
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text {* Dual lattice *} |
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lemma dual_semilattice: |
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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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|
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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 dual_lattice: |
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"lattice (op \<ge>) (op >) sup inf" |
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by (rule lattice.intro, rule dual_semilattice, rule upper_semilattice.intro, rule dual_order) |
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(unfold_locales, auto) |
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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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||
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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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219 |
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)" |
223 |
shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" |
|
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proof- |
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225 |
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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231 |
qed |
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||
21734 | 233 |
(* seems unused *) |
234 |
lemma modular_le: "x \<sqsubseteq> z \<Longrightarrow> x \<squnion> (y \<sqinter> z) \<sqsubseteq> (x \<squnion> y) \<sqinter> z" |
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235 |
by blast |
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||
21733 | 237 |
end |
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239 |
||
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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)" |
244 |
||
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context distrib_lattice |
246 |
begin |
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247 |
||
248 |
lemma sup_inf_distrib2: |
|
21249 | 249 |
"(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)" |
250 |
by(simp add:ACI sup_inf_distrib1) |
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251 |
||
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lemma inf_sup_distrib1: |
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"x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" |
254 |
by(rule distrib_imp2[OF sup_inf_distrib1]) |
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255 |
||
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lemma inf_sup_distrib2: |
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"(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)" |
258 |
by(simp add:ACI inf_sup_distrib1) |
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lemma dual_distrib_lattice: |
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"distrib_lattice (op \<ge>) (op >) sup inf" |
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by (rule distrib_lattice.intro, rule dual_lattice) |
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(unfold_locales, fact inf_sup_distrib1) |
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264 |
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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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21733 | 268 |
end |
269 |
||
21249 | 270 |
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subsection {* Boolean algebras *} |
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class boolean_algebra = distrib_lattice + top + bot + minus + uminus + |
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added boolean_algebra type class; tuned lattice duals
haftmann
parents:
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diff
changeset
|
274 |
assumes inf_compl_bot: "x \<sqinter> - x = bot" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
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diff
changeset
|
275 |
and sup_compl_top: "x \<squnion> - x = top" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
276 |
assumes diff_eq: "x - y = x \<sqinter> - y" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
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diff
changeset
|
277 |
begin |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
278 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
279 |
lemma dual_boolean_algebra: |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
280 |
"boolean_algebra (\<lambda>x y. x \<squnion> - y) uminus (op \<ge>) (op >) (op \<squnion>) (op \<sqinter>) top bot" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
281 |
by (rule boolean_algebra.intro, rule dual_distrib_lattice) |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
282 |
(unfold_locales, |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
283 |
auto simp add: inf_compl_bot sup_compl_top diff_eq less_le_not_le) |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
284 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
285 |
lemma compl_inf_bot: |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
286 |
"- x \<sqinter> x = bot" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
287 |
by (simp add: inf_commute inf_compl_bot) |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
288 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
289 |
lemma compl_sup_top: |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
290 |
"- x \<squnion> x = top" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
291 |
by (simp add: sup_commute sup_compl_top) |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
292 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
293 |
lemma inf_bot_left [simp]: |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
294 |
"bot \<sqinter> x = bot" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
295 |
by (rule inf_absorb1) simp |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
296 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
297 |
lemma inf_bot_right [simp]: |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
298 |
"x \<sqinter> bot = bot" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
299 |
by (rule inf_absorb2) simp |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
300 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
301 |
lemma sup_top_left [simp]: |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
302 |
"top \<squnion> x = top" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
303 |
by (rule sup_absorb1) simp |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
304 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
305 |
lemma sup_top_right [simp]: |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
306 |
"x \<squnion> top = top" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
307 |
by (rule sup_absorb2) simp |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
308 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
309 |
lemma inf_top_left [simp]: |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
310 |
"top \<sqinter> x = x" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
311 |
by (rule inf_absorb2) simp |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
312 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
313 |
lemma inf_top_right [simp]: |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
314 |
"x \<sqinter> top = x" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
315 |
by (rule inf_absorb1) simp |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
316 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
317 |
lemma sup_bot_left [simp]: |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
318 |
"bot \<squnion> x = x" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
319 |
by (rule sup_absorb2) simp |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
320 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
321 |
lemma sup_bot_right [simp]: |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
322 |
"x \<squnion> bot = x" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
323 |
by (rule sup_absorb1) simp |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
324 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
325 |
lemma compl_unique: |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
326 |
assumes "x \<sqinter> y = bot" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
327 |
and "x \<squnion> y = top" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
328 |
shows "- x = y" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
329 |
proof - |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
330 |
have "(x \<sqinter> - x) \<squnion> (- x \<sqinter> y) = (x \<sqinter> y) \<squnion> (- x \<sqinter> y)" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
331 |
using inf_compl_bot assms(1) by simp |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
332 |
then have "(- x \<sqinter> x) \<squnion> (- x \<sqinter> y) = (y \<sqinter> x) \<squnion> (y \<sqinter> - x)" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
333 |
by (simp add: inf_commute) |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
334 |
then have "- x \<sqinter> (x \<squnion> y) = y \<sqinter> (x \<squnion> - x)" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
335 |
by (simp add: inf_sup_distrib1) |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
336 |
then have "- x \<sqinter> top = y \<sqinter> top" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
337 |
using sup_compl_top assms(2) by simp |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
338 |
then show "- x = y" by (simp add: inf_top_right) |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
339 |
qed |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
340 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
341 |
lemma double_compl [simp]: |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
342 |
"- (- x) = x" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
343 |
using compl_inf_bot compl_sup_top by (rule compl_unique) |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
344 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
345 |
lemma compl_eq_compl_iff [simp]: |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
346 |
"- x = - y \<longleftrightarrow> x = y" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
347 |
proof |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
348 |
assume "- x = - y" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
349 |
then have "- x \<sqinter> y = bot" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
350 |
and "- x \<squnion> y = top" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
351 |
by (simp_all add: compl_inf_bot compl_sup_top) |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
352 |
then have "- (- x) = y" by (rule compl_unique) |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
353 |
then show "x = y" by simp |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
354 |
next |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
355 |
assume "x = y" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
356 |
then show "- x = - y" by simp |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
357 |
qed |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
358 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
359 |
lemma compl_bot_eq [simp]: |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
360 |
"- bot = top" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
361 |
proof - |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
362 |
from sup_compl_top have "bot \<squnion> - bot = top" . |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
363 |
then show ?thesis by simp |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
364 |
qed |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
365 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
366 |
lemma compl_top_eq [simp]: |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
367 |
"- top = bot" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
368 |
proof - |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
369 |
from inf_compl_bot have "top \<sqinter> - top = bot" . |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
370 |
then show ?thesis by simp |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
371 |
qed |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
372 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
373 |
lemma compl_inf [simp]: |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
374 |
"- (x \<sqinter> y) = - x \<squnion> - y" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
375 |
proof (rule compl_unique) |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
376 |
have "(x \<sqinter> y) \<sqinter> (- x \<squnion> - y) = ((x \<sqinter> y) \<sqinter> - x) \<squnion> ((x \<sqinter> y) \<sqinter> - y)" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
377 |
by (rule inf_sup_distrib1) |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
378 |
also have "... = (y \<sqinter> (x \<sqinter> - x)) \<squnion> (x \<sqinter> (y \<sqinter> - y))" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
379 |
by (simp only: inf_commute inf_assoc inf_left_commute) |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
380 |
finally show "(x \<sqinter> y) \<sqinter> (- x \<squnion> - y) = bot" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
381 |
by (simp add: inf_compl_bot) |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
382 |
next |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
383 |
have "(x \<sqinter> y) \<squnion> (- x \<squnion> - y) = (x \<squnion> (- x \<squnion> - y)) \<sqinter> (y \<squnion> (- x \<squnion> - y))" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
384 |
by (rule sup_inf_distrib2) |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
385 |
also have "... = (- y \<squnion> (x \<squnion> - x)) \<sqinter> (- x \<squnion> (y \<squnion> - y))" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
386 |
by (simp only: sup_commute sup_assoc sup_left_commute) |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
387 |
finally show "(x \<sqinter> y) \<squnion> (- x \<squnion> - y) = top" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
388 |
by (simp add: sup_compl_top) |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
389 |
qed |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
390 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
391 |
lemma compl_sup [simp]: |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
392 |
"- (x \<squnion> y) = - x \<sqinter> - y" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
393 |
proof - |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
394 |
interpret boolean_algebra "\<lambda>x y. x \<squnion> - y" uminus "op \<ge>" "op >" "op \<squnion>" "op \<sqinter>" top bot |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
395 |
by (rule dual_boolean_algebra) |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
396 |
then show ?thesis by simp |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
397 |
qed |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
398 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
399 |
end |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
400 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
401 |
|
22454 | 402 |
subsection {* Uniqueness of inf and sup *} |
403 |
||
22737 | 404 |
lemma (in lower_semilattice) inf_unique: |
22454 | 405 |
fixes f (infixl "\<triangle>" 70) |
25062 | 406 |
assumes le1: "\<And>x y. x \<triangle> y \<le> x" and le2: "\<And>x y. x \<triangle> y \<le> y" |
407 |
and greatest: "\<And>x y z. x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> y \<triangle> z" |
|
22737 | 408 |
shows "x \<sqinter> y = x \<triangle> y" |
22454 | 409 |
proof (rule antisym) |
25062 | 410 |
show "x \<triangle> y \<le> x \<sqinter> y" by (rule le_infI) (rule le1, rule le2) |
22454 | 411 |
next |
25062 | 412 |
have leI: "\<And>x y z. x \<le> y \<Longrightarrow> x \<le> z \<Longrightarrow> x \<le> y \<triangle> z" by (blast intro: greatest) |
413 |
show "x \<sqinter> y \<le> x \<triangle> y" by (rule leI) simp_all |
|
22454 | 414 |
qed |
415 |
||
22737 | 416 |
lemma (in upper_semilattice) sup_unique: |
22454 | 417 |
fixes f (infixl "\<nabla>" 70) |
25062 | 418 |
assumes ge1 [simp]: "\<And>x y. x \<le> x \<nabla> y" and ge2: "\<And>x y. y \<le> x \<nabla> y" |
419 |
and least: "\<And>x y z. y \<le> x \<Longrightarrow> z \<le> x \<Longrightarrow> y \<nabla> z \<le> x" |
|
22737 | 420 |
shows "x \<squnion> y = x \<nabla> y" |
22454 | 421 |
proof (rule antisym) |
25062 | 422 |
show "x \<squnion> y \<le> x \<nabla> y" by (rule le_supI) (rule ge1, rule ge2) |
22454 | 423 |
next |
25062 | 424 |
have leI: "\<And>x y z. x \<le> z \<Longrightarrow> y \<le> z \<Longrightarrow> x \<nabla> y \<le> z" by (blast intro: least) |
425 |
show "x \<nabla> y \<le> x \<squnion> y" by (rule leI) simp_all |
|
22454 | 426 |
qed |
427 |
||
428 |
||
22916 | 429 |
subsection {* @{const min}/@{const max} on linear orders as |
430 |
special case of @{const inf}/@{const sup} *} |
|
431 |
||
432 |
lemma (in linorder) distrib_lattice_min_max: |
|
25062 | 433 |
"distrib_lattice (op \<le>) (op <) min max" |
28823 | 434 |
proof |
25062 | 435 |
have aux: "\<And>x y \<Colon> 'a. x < y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y" |
22916 | 436 |
by (auto simp add: less_le antisym) |
437 |
fix x y z |
|
438 |
show "max x (min y z) = min (max x y) (max x z)" |
|
439 |
unfolding min_def max_def |
|
24640
85a6c200ecd3
Simplified proofs due to transitivity reasoner setup.
ballarin
parents:
24514
diff
changeset
|
440 |
by auto |
22916 | 441 |
qed (auto simp add: min_def max_def not_le less_imp_le) |
21249 | 442 |
|
30729
461ee3e49ad3
interpretation/interpret: prefixes are mandatory by default;
wenzelm
parents:
30302
diff
changeset
|
443 |
interpretation min_max: distrib_lattice "op \<le> :: 'a::linorder \<Rightarrow> 'a \<Rightarrow> bool" "op <" min max |
23948 | 444 |
by (rule distrib_lattice_min_max) |
21249 | 445 |
|
22454 | 446 |
lemma inf_min: "inf = (min \<Colon> 'a\<Colon>{lower_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)" |
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
447 |
by (rule ext)+ (auto intro: antisym) |
21733 | 448 |
|
22454 | 449 |
lemma sup_max: "sup = (max \<Colon> 'a\<Colon>{upper_semilattice, linorder} \<Rightarrow> 'a \<Rightarrow> 'a)" |
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
450 |
by (rule ext)+ (auto intro: antisym) |
21733 | 451 |
|
21249 | 452 |
lemmas le_maxI1 = min_max.sup_ge1 |
453 |
lemmas le_maxI2 = min_max.sup_ge2 |
|
21381 | 454 |
|
21249 | 455 |
lemmas max_ac = min_max.sup_assoc min_max.sup_commute |
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22384
diff
changeset
|
456 |
mk_left_commute [of max, OF min_max.sup_assoc min_max.sup_commute] |
21249 | 457 |
|
458 |
lemmas min_ac = min_max.inf_assoc min_max.inf_commute |
|
22422
ee19cdb07528
stepping towards uniform lattice theory development in HOL
haftmann
parents:
22384
diff
changeset
|
459 |
mk_left_commute [of min, OF min_max.inf_assoc min_max.inf_commute] |
21249 | 460 |
|
25102
db3e412c4cb1
antisymmetry not a default intro rule any longer
haftmann
parents:
25062
diff
changeset
|
461 |
lemmas [rule del] = min_max.le_infI min_max.le_supI |
22454 | 462 |
min_max.le_supE min_max.le_infE min_max.le_supI1 min_max.le_supI2 |
463 |
min_max.le_infI1 min_max.le_infI2 |
|
464 |
||
465 |
||
466 |
subsection {* Bool as lattice *} |
|
467 |
||
31991
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
468 |
instantiation bool :: boolean_algebra |
25510 | 469 |
begin |
470 |
||
471 |
definition |
|
31991
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
472 |
bool_Compl_def: "uminus = Not" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
473 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
474 |
definition |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
475 |
bool_diff_def: "A - B \<longleftrightarrow> A \<and> \<not> B" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
476 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
477 |
definition |
25510 | 478 |
inf_bool_eq: "P \<sqinter> Q \<longleftrightarrow> P \<and> Q" |
479 |
||
480 |
definition |
|
481 |
sup_bool_eq: "P \<squnion> Q \<longleftrightarrow> P \<or> Q" |
|
482 |
||
31991
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
483 |
instance proof |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
484 |
qed (simp_all add: inf_bool_eq sup_bool_eq le_bool_def |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
485 |
bot_bool_eq top_bool_eq bool_Compl_def bool_diff_def, auto) |
22454 | 486 |
|
25510 | 487 |
end |
488 |
||
23878 | 489 |
|
490 |
subsection {* Fun as lattice *} |
|
491 |
||
25510 | 492 |
instantiation "fun" :: (type, lattice) lattice |
493 |
begin |
|
494 |
||
495 |
definition |
|
28562 | 496 |
inf_fun_eq [code del]: "f \<sqinter> g = (\<lambda>x. f x \<sqinter> g x)" |
25510 | 497 |
|
498 |
definition |
|
28562 | 499 |
sup_fun_eq [code del]: "f \<squnion> g = (\<lambda>x. f x \<squnion> g x)" |
25510 | 500 |
|
501 |
instance |
|
23878 | 502 |
apply intro_classes |
503 |
unfolding inf_fun_eq sup_fun_eq |
|
504 |
apply (auto intro: le_funI) |
|
505 |
apply (rule le_funI) |
|
506 |
apply (auto dest: le_funD) |
|
507 |
apply (rule le_funI) |
|
508 |
apply (auto dest: le_funD) |
|
509 |
done |
|
510 |
||
25510 | 511 |
end |
23878 | 512 |
|
513 |
instance "fun" :: (type, distrib_lattice) distrib_lattice |
|
31991
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
514 |
proof |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
515 |
qed (auto simp add: inf_fun_eq sup_fun_eq sup_inf_distrib1) |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
516 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
517 |
instantiation "fun" :: (type, uminus) uminus |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
518 |
begin |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
519 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
520 |
definition |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
521 |
fun_Compl_def: "- A = (\<lambda>x. - A x)" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
522 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
523 |
instance .. |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
524 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
525 |
end |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
526 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
527 |
instantiation "fun" :: (type, minus) minus |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
528 |
begin |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
529 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
530 |
definition |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
531 |
fun_diff_def: "A - B = (\<lambda>x. A x - B x)" |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
532 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
533 |
instance .. |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
534 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
535 |
end |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
536 |
|
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
537 |
instance "fun" :: (type, boolean_algebra) boolean_algebra |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
538 |
proof |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
539 |
qed (simp_all add: inf_fun_eq sup_fun_eq bot_fun_eq top_fun_eq fun_Compl_def fun_diff_def |
37390299214a
added boolean_algebra type class; tuned lattice duals
haftmann
parents:
30729
diff
changeset
|
540 |
inf_compl_bot sup_compl_top diff_eq) |
23878 | 541 |
|
26794 | 542 |
|
23878 | 543 |
text {* redundant bindings *} |
22454 | 544 |
|
545 |
lemmas inf_aci = inf_ACI |
|
546 |
lemmas sup_aci = sup_ACI |
|
547 |
||
25062 | 548 |
no_notation |
25382 | 549 |
less_eq (infix "\<sqsubseteq>" 50) and |
550 |
less (infix "\<sqsubset>" 50) and |
|
551 |
inf (infixl "\<sqinter>" 70) and |
|
30302 | 552 |
sup (infixl "\<squnion>" 65) |
25062 | 553 |
|
21249 | 554 |
end |