src/HOL/Library/Glbs.thy
 changeset 54263 c4159fe6fa46 parent 54262 326fd7103cb4 child 54264 27501a51d847
```     1.1 --- a/src/HOL/Library/Glbs.thy	Tue Nov 05 09:45:00 2013 +0100
1.2 +++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
1.3 @@ -1,79 +0,0 @@
1.4 -(* Author: Amine Chaieb, University of Cambridge *)
1.5 -
1.6 -header {* Definitions of Lower Bounds and Greatest Lower Bounds, analogous to Lubs *}
1.7 -
1.8 -theory Glbs
1.9 -imports Lubs
1.10 -begin
1.11 -
1.12 -definition greatestP :: "('a \<Rightarrow> bool) \<Rightarrow> 'a::ord \<Rightarrow> bool"
1.13 -  where "greatestP P x = (P x \<and> Collect P *<=  x)"
1.14 -
1.15 -definition isLb :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"
1.16 -  where "isLb R S x = (x <=* S \<and> x: R)"
1.17 -
1.18 -definition isGlb :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"
1.19 -  where "isGlb R S x = greatestP (isLb R S) x"
1.20 -
1.21 -definition lbs :: "'a set \<Rightarrow> 'a::ord set \<Rightarrow> 'a set"
1.22 -  where "lbs R S = Collect (isLb R S)"
1.23 -
1.24 -
1.25 -subsection {* Rules about the Operators @{term greatestP}, @{term isLb}
1.26 -  and @{term isGlb} *}
1.27 -
1.28 -lemma leastPD1: "greatestP P x \<Longrightarrow> P x"
1.29 -  by (simp add: greatestP_def)
1.30 -
1.31 -lemma greatestPD2: "greatestP P x \<Longrightarrow> Collect P *<= x"
1.32 -  by (simp add: greatestP_def)
1.33 -
1.34 -lemma greatestPD3: "greatestP P x \<Longrightarrow> y: Collect P \<Longrightarrow> x \<ge> y"
1.35 -  by (blast dest!: greatestPD2 setleD)
1.36 -
1.37 -lemma isGlbD1: "isGlb R S x \<Longrightarrow> x <=* S"
1.38 -  by (simp add: isGlb_def isLb_def greatestP_def)
1.39 -
1.40 -lemma isGlbD1a: "isGlb R S x \<Longrightarrow> x: R"
1.41 -  by (simp add: isGlb_def isLb_def greatestP_def)
1.42 -
1.43 -lemma isGlb_isLb: "isGlb R S x \<Longrightarrow> isLb R S x"
1.44 -  unfolding isLb_def by (blast dest: isGlbD1 isGlbD1a)
1.45 -
1.46 -lemma isGlbD2: "isGlb R S x \<Longrightarrow> y : S \<Longrightarrow> y \<ge> x"
1.47 -  by (blast dest!: isGlbD1 setgeD)
1.48 -
1.49 -lemma isGlbD3: "isGlb R S x \<Longrightarrow> greatestP (isLb R S) x"
1.50 -  by (simp add: isGlb_def)
1.51 -
1.52 -lemma isGlbI1: "greatestP (isLb R S) x \<Longrightarrow> isGlb R S x"
1.53 -  by (simp add: isGlb_def)
1.54 -
1.55 -lemma isGlbI2: "isLb R S x \<Longrightarrow> Collect (isLb R S) *<= x \<Longrightarrow> isGlb R S x"
1.56 -  by (simp add: isGlb_def greatestP_def)
1.57 -
1.58 -lemma isLbD: "isLb R S x \<Longrightarrow> y : S \<Longrightarrow> y \<ge> x"
1.59 -  by (simp add: isLb_def setge_def)
1.60 -
1.61 -lemma isLbD2: "isLb R S x \<Longrightarrow> x <=* S "
1.62 -  by (simp add: isLb_def)
1.63 -
1.64 -lemma isLbD2a: "isLb R S x \<Longrightarrow> x: R"
1.65 -  by (simp add: isLb_def)
1.66 -
1.67 -lemma isLbI: "x <=* S \<Longrightarrow> x: R \<Longrightarrow> isLb R S x"
1.68 -  by (simp add: isLb_def)
1.69 -
1.70 -lemma isGlb_le_isLb: "isGlb R S x \<Longrightarrow> isLb R S y \<Longrightarrow> x \<ge> y"
1.71 -  unfolding isGlb_def by (blast intro!: greatestPD3)
1.72 -
1.73 -lemma isGlb_ubs: "isGlb R S x \<Longrightarrow> lbs R S *<= x"
1.74 -  unfolding lbs_def isGlb_def by (rule greatestPD2)
1.75 -
1.76 -lemma isGlb_unique: "[| isGlb R S x; isGlb R S y |] ==> x = (y::'a::linorder)"
1.77 -  apply (frule isGlb_isLb)
1.78 -  apply (frule_tac x = y in isGlb_isLb)
1.79 -  apply (blast intro!: order_antisym dest!: isGlb_le_isLb)
1.80 -  done
1.81 -
1.82 -end
```