src/HOL/Lattices.thy
author haftmann
Fri Mar 16 21:32:10 2007 +0100 (2007-03-16)
changeset 22454 c3654ba76a09
parent 22422 ee19cdb07528
child 22548 6ce4bddf3bcb
permissions -rw-r--r--
integrated with LOrder.thy
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(*  Title:      HOL/Lattices.thy
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    ID:         $Id$
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    Author:     Tobias Nipkow
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*)
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header {* Abstract lattices *}
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theory Lattices
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imports Orderings
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begin
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subsection{* Lattices *}
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text{*
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  This theory of lattices only defines binary sup and inf
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  operations. The extension to (finite) sets is done in theories @{text FixedPoint}
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  and @{text Finite_Set}.
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*}
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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" 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" and sup_ge2 [simp]: "y \<sqsubseteq> x \<squnion> y"
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  and sup_least: "y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<squnion> z \<sqsubseteq> x"
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class lattice = lower_semilattice + upper_semilattice
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subsubsection{* Intro and elim rules*}
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context lower_semilattice
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begin
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lemmas antisym_intro [intro!] = antisym
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lemmas (in -) [rule del] = antisym_intro
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lemma le_infI1[intro]: "a \<sqsubseteq> x \<Longrightarrow> a \<sqinter> b \<sqsubseteq> x"
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apply(subgoal_tac "a \<sqinter> b \<sqsubseteq> a")
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 apply(blast intro: order_trans)
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apply simp
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done
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lemmas (in -) [rule del] = le_infI1
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lemma le_infI2[intro]: "b \<sqsubseteq> x \<Longrightarrow> a \<sqinter> b \<sqsubseteq> x"
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apply(subgoal_tac "a \<sqinter> b \<sqsubseteq> b")
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 apply(blast intro: order_trans)
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apply simp
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done
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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 dest:eq_iff[THEN iffD1])
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end
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context upper_semilattice
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begin
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lemmas antisym_intro [intro!] = antisym
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lemmas (in -) [rule del] = antisym_intro
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lemma le_supI1[intro]: "x \<sqsubseteq> a \<Longrightarrow> x \<sqsubseteq> a \<squnion> b"
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apply(subgoal_tac "a \<sqsubseteq> a \<squnion> b")
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 apply(blast intro: order_trans)
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apply simp
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done
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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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apply(subgoal_tac "b \<sqsubseteq> a \<squnion> b")
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 apply(blast intro: order_trans)
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apply simp
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done
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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 dest:eq_iff[THEN iffD1])
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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
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lemma inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
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by blast
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lemma inf_idem[simp]: "x \<sqinter> x = x"
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by blast
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lemma inf_left_idem[simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
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by blast
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lemma inf_absorb1: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
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by blast
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lemma inf_absorb2: "y \<sqsubseteq> x \<Longrightarrow> x \<sqinter> y = y"
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by blast
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lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
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by blast
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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
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lemma sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
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by blast
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lemma sup_idem[simp]: "x \<squnion> x = x"
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by blast
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lemma sup_left_idem[simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
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by blast
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lemma sup_absorb1: "y \<sqsubseteq> x \<Longrightarrow> x \<squnion> y = x"
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by blast
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lemma sup_absorb2: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
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by blast
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lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
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by blast
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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 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 "inf x y = f x y"
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proof (rule antisym)
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  show "x \<triangle> y \<le> inf x y" by (rule le_infI) (rule le1 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 "inf x y \<le> x \<triangle> y" by (rule leI) simp_all
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qed
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lemma 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 "sup x y = f x y"
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proof (rule antisym)
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  show "sup x y \<le> x \<nabla> y" by (rule le_supI) (rule ge1 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> sup x y" by (rule leI) simp_all
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qed
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subsection {* min/max on linear orders as special case of inf/sup *}
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interpretation min_max:
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  distrib_lattice ["op \<le> \<Colon> 'a\<Colon>linorder \<Rightarrow> 'a \<Rightarrow> bool" "op <" min max]
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apply unfold_locales
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apply (simp add: min_def linorder_not_le order_less_imp_le)
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apply (simp add: min_def linorder_not_le order_less_imp_le)
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apply (simp add: min_def linorder_not_le order_less_imp_le)
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apply (simp add: max_def linorder_not_le order_less_imp_le)
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apply (simp add: max_def linorder_not_le order_less_imp_le)
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unfolding min_def max_def by auto
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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
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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
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lemmas le_maxI1 = min_max.sup_ge1
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lemmas le_maxI2 = min_max.sup_ge2
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lemmas max_ac = min_max.sup_assoc min_max.sup_commute
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  mk_left_commute [of max, OF min_max.sup_assoc min_max.sup_commute]
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lemmas min_ac = min_max.inf_assoc min_max.inf_commute
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  mk_left_commute [of min, OF min_max.inf_assoc min_max.inf_commute]
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text {*
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  Now we have inherited antisymmetry as an intro-rule on all
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  linear orders. This is a problem because it applies to bool, which is
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  undesirable.
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*}
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lemmas [rule del] = min_max.antisym_intro min_max.le_infI min_max.le_supI
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  min_max.le_supE min_max.le_infE min_max.le_supI1 min_max.le_supI2
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  min_max.le_infI1 min_max.le_infI2
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subsection {* Bool as lattice *}
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instance bool :: distrib_lattice
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  inf_bool_eq: "inf P Q \<equiv> P \<and> Q"
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  sup_bool_eq: "sup P Q \<equiv> P \<or> Q"
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  by intro_classes (auto simp add: inf_bool_eq sup_bool_eq le_bool_def)
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text {* duplicates *}
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lemmas inf_aci = inf_ACI
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lemmas sup_aci = sup_ACI
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text {* ML legacy bindings *}
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ML {*
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val Least_def = @{thm Least_def}
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val Least_equality = @{thm Least_equality}
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val min_def = @{thm min_def}
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val min_of_mono = @{thm min_of_mono}
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val max_def = @{thm max_def}
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val max_of_mono = @{thm max_of_mono}
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val min_leastL = @{thm min_leastL}
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val max_leastL = @{thm max_leastL}
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val min_leastR = @{thm min_leastR}
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val max_leastR = @{thm max_leastR}
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val le_max_iff_disj = @{thm le_max_iff_disj}
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val le_maxI1 = @{thm le_maxI1}
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val le_maxI2 = @{thm le_maxI2}
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val less_max_iff_disj = @{thm less_max_iff_disj}
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val max_less_iff_conj = @{thm max_less_iff_conj}
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val min_less_iff_conj = @{thm min_less_iff_conj}
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val min_le_iff_disj = @{thm min_le_iff_disj}
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val min_less_iff_disj = @{thm min_less_iff_disj}
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val split_min = @{thm split_min}
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val split_max = @{thm split_max}
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*}
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end