(* Title: HOL/Lattices.thy
ID: $Id$
Author: Tobias Nipkow
*)
header {* Lattices via Locales *}
theory Lattice_Locales
imports Set
begin
subsection{* Lattices *}
text{* This theory of lattice locales only defines binary sup and inf
operations. The extension to finite sets is done in theory @{text
Finite_Set}. In the longer term it may be better to define arbitrary
sups and infs via @{text THE}. *}
locale partial_order =
fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infixl "\<sqsubseteq>" 50)
assumes refl[iff]: "x \<sqsubseteq> x"
and trans: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> z"
and antisym: "x \<sqsubseteq> y \<Longrightarrow> y \<sqsubseteq> x \<Longrightarrow> x = y"
locale lower_semilattice = partial_order +
fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<sqinter>" 70)
assumes inf_le1: "x \<sqinter> y \<sqsubseteq> x" and inf_le2: "x \<sqinter> y \<sqsubseteq> y"
and inf_least: "x \<sqsubseteq> y \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> x \<sqsubseteq> y \<sqinter> z"
locale upper_semilattice = partial_order +
fixes sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "\<squnion>" 65)
assumes sup_ge1: "x \<sqsubseteq> x \<squnion> y" and sup_ge2: "y \<sqsubseteq> x \<squnion> y"
and sup_greatest: "y \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> y \<squnion> z \<sqsubseteq> x"
locale lattice = lower_semilattice + upper_semilattice
lemma (in lower_semilattice) inf_commute: "(x \<sqinter> y) = (y \<sqinter> x)"
by(blast intro: antisym inf_le1 inf_le2 inf_least)
lemma (in upper_semilattice) sup_commute: "(x \<squnion> y) = (y \<squnion> x)"
by(blast intro: antisym sup_ge1 sup_ge2 sup_greatest)
lemma (in lower_semilattice) inf_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
by(blast intro: antisym inf_le1 inf_le2 inf_least trans del:refl)
lemma (in upper_semilattice) sup_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
by(blast intro!: antisym sup_ge1 sup_ge2 intro: sup_greatest trans del:refl)
lemma (in lower_semilattice) inf_idem[simp]: "x \<sqinter> x = x"
by(blast intro: antisym inf_le1 inf_le2 inf_least refl)
lemma (in upper_semilattice) sup_idem[simp]: "x \<squnion> x = x"
by(blast intro: antisym sup_ge1 sup_ge2 sup_greatest refl)
lemma (in lattice) inf_sup_absorb: "x \<sqinter> (x \<squnion> y) = x"
by(blast intro: antisym inf_le1 inf_least sup_ge1)
lemma (in lattice) sup_inf_absorb: "x \<squnion> (x \<sqinter> y) = x"
by(blast intro: antisym sup_ge1 sup_greatest inf_le1)
lemma (in lower_semilattice) inf_absorb: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
by(blast intro: antisym inf_le1 inf_least refl)
lemma (in upper_semilattice) sup_absorb: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
by(blast intro: antisym sup_ge2 sup_greatest refl)
text{* Towards distributivity: if you have one of them, you have them all. *}
lemma (in lattice) distrib_imp1:
assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
proof-
have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by(simp add:sup_inf_absorb)
also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))" by(simp add:D inf_commute sup_assoc)
also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"
by(simp add:inf_sup_absorb inf_commute)
also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D)
finally show ?thesis .
qed
lemma (in lattice) distrib_imp2:
assumes D: "!!x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
proof-
have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by(simp add:inf_sup_absorb)
also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))" by(simp add:D sup_commute inf_assoc)
also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"
by(simp add:sup_inf_absorb sup_commute)
also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D)
finally show ?thesis .
qed
text{* A package of rewrite rules for deciding equivalence wrt ACI: *}
lemma (in lower_semilattice) inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
proof -
have "x \<sqinter> (y \<sqinter> z) = (y \<sqinter> z) \<sqinter> x" by (simp only: inf_commute)
also have "... = y \<sqinter> (z \<sqinter> x)" by (simp only: inf_assoc)
also have "z \<sqinter> x = x \<sqinter> z" by (simp only: inf_commute)
finally show ?thesis .
qed
lemma (in upper_semilattice) sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
proof -
have "x \<squnion> (y \<squnion> z) = (y \<squnion> z) \<squnion> x" by (simp only: sup_commute)
also have "... = y \<squnion> (z \<squnion> x)" by (simp only: sup_assoc)
also have "z \<squnion> x = x \<squnion> z" by (simp only: sup_commute)
finally show ?thesis .
qed
lemma (in lower_semilattice) inf_left_idem: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
proof -
have "x \<sqinter> (x \<sqinter> y) = (x \<sqinter> x) \<sqinter> y" by(simp only:inf_assoc)
also have "\<dots> = x \<sqinter> y" by(simp)
finally show ?thesis .
qed
lemma (in upper_semilattice) sup_left_idem: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
proof -
have "x \<squnion> (x \<squnion> y) = (x \<squnion> x) \<squnion> y" by(simp only:sup_assoc)
also have "\<dots> = x \<squnion> y" by(simp)
finally show ?thesis .
qed
lemmas (in lower_semilattice) inf_ACI =
inf_commute inf_assoc inf_left_commute inf_left_idem
lemmas (in upper_semilattice) sup_ACI =
sup_commute sup_assoc sup_left_commute sup_left_idem
lemmas (in lattice) ACI = inf_ACI sup_ACI
subsection{* Distributive lattices *}
locale distrib_lattice = lattice +
assumes sup_inf_distrib1: "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
lemma (in distrib_lattice) sup_inf_distrib2:
"(y \<sqinter> z) \<squnion> x = (y \<squnion> x) \<sqinter> (z \<squnion> x)"
by(simp add:ACI sup_inf_distrib1)
lemma (in distrib_lattice) inf_sup_distrib1:
"x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
by(rule distrib_imp2[OF sup_inf_distrib1])
lemma (in distrib_lattice) inf_sup_distrib2:
"(y \<squnion> z) \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
by(simp add:ACI inf_sup_distrib1)
lemmas (in distrib_lattice) distrib =
sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2
end