tuned proofs -- clarified flow of facts wrt. calculation;
(* Author: Amine Chaieb, University of Cambridge *)
header {* Definitions of Lower Bounds and Greatest Lower Bounds, analogous to Lubs *}
theory Glbs
imports Lubs
begin
definition greatestP :: "('a \<Rightarrow> bool) \<Rightarrow> 'a::ord \<Rightarrow> bool"
where "greatestP P x = (P x \<and> Collect P *<= x)"
definition isLb :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"
where "isLb R S x = (x <=* S \<and> x: R)"
definition isGlb :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"
where "isGlb R S x = greatestP (isLb R S) x"
definition lbs :: "'a set \<Rightarrow> 'a::ord set \<Rightarrow> 'a set"
where "lbs R S = Collect (isLb R S)"
subsection {* Rules about the Operators @{term greatestP}, @{term isLb}
and @{term isGlb} *}
lemma leastPD1: "greatestP P x \<Longrightarrow> P x"
by (simp add: greatestP_def)
lemma greatestPD2: "greatestP P x \<Longrightarrow> Collect P *<= x"
by (simp add: greatestP_def)
lemma greatestPD3: "greatestP P x \<Longrightarrow> y: Collect P \<Longrightarrow> x \<ge> y"
by (blast dest!: greatestPD2 setleD)
lemma isGlbD1: "isGlb R S x \<Longrightarrow> x <=* S"
by (simp add: isGlb_def isLb_def greatestP_def)
lemma isGlbD1a: "isGlb R S x \<Longrightarrow> x: R"
by (simp add: isGlb_def isLb_def greatestP_def)
lemma isGlb_isLb: "isGlb R S x \<Longrightarrow> isLb R S x"
unfolding isLb_def by (blast dest: isGlbD1 isGlbD1a)
lemma isGlbD2: "isGlb R S x \<Longrightarrow> y : S \<Longrightarrow> y \<ge> x"
by (blast dest!: isGlbD1 setgeD)
lemma isGlbD3: "isGlb R S x \<Longrightarrow> greatestP (isLb R S) x"
by (simp add: isGlb_def)
lemma isGlbI1: "greatestP (isLb R S) x \<Longrightarrow> isGlb R S x"
by (simp add: isGlb_def)
lemma isGlbI2: "isLb R S x \<Longrightarrow> Collect (isLb R S) *<= x \<Longrightarrow> isGlb R S x"
by (simp add: isGlb_def greatestP_def)
lemma isLbD: "isLb R S x \<Longrightarrow> y : S \<Longrightarrow> y \<ge> x"
by (simp add: isLb_def setge_def)
lemma isLbD2: "isLb R S x \<Longrightarrow> x <=* S "
by (simp add: isLb_def)
lemma isLbD2a: "isLb R S x \<Longrightarrow> x: R"
by (simp add: isLb_def)
lemma isLbI: "x <=* S \<Longrightarrow> x: R \<Longrightarrow> isLb R S x"
by (simp add: isLb_def)
lemma isGlb_le_isLb: "isGlb R S x \<Longrightarrow> isLb R S y \<Longrightarrow> x \<ge> y"
unfolding isGlb_def by (blast intro!: greatestPD3)
lemma isGlb_ubs: "isGlb R S x \<Longrightarrow> lbs R S *<= x"
unfolding lbs_def isGlb_def by (rule greatestPD2)
lemma isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::'a::linorder)"
apply (frule isGlb_isLb)
apply (frule_tac x = y in isGlb_isLb)
apply (blast intro!: order_antisym dest!: isGlb_le_isLb)
done
end