library theories for debugging and parallel computing using code generation towards Isabelle/ML
1 (* Author: Amine Chaieb, University of Cambridge *)
3 header {* Definitions of Lower Bounds and Greatest Lower Bounds, analogous to Lubs *}
9 definition greatestP :: "('a \<Rightarrow> bool) \<Rightarrow> 'a::ord \<Rightarrow> bool"
10 where "greatestP P x = (P x \<and> Collect P *<= x)"
12 definition isLb :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"
13 where "isLb R S x = (x <=* S \<and> x: R)"
15 definition isGlb :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"
16 where "isGlb R S x = greatestP (isLb R S) x"
18 definition lbs :: "'a set \<Rightarrow> 'a::ord set \<Rightarrow> 'a set"
19 where "lbs R S = Collect (isLb R S)"
22 subsection {* Rules about the Operators @{term greatestP}, @{term isLb}
25 lemma leastPD1: "greatestP P x \<Longrightarrow> P x"
26 by (simp add: greatestP_def)
28 lemma greatestPD2: "greatestP P x \<Longrightarrow> Collect P *<= x"
29 by (simp add: greatestP_def)
31 lemma greatestPD3: "greatestP P x \<Longrightarrow> y: Collect P \<Longrightarrow> x \<ge> y"
32 by (blast dest!: greatestPD2 setleD)
34 lemma isGlbD1: "isGlb R S x \<Longrightarrow> x <=* S"
35 by (simp add: isGlb_def isLb_def greatestP_def)
37 lemma isGlbD1a: "isGlb R S x \<Longrightarrow> x: R"
38 by (simp add: isGlb_def isLb_def greatestP_def)
40 lemma isGlb_isLb: "isGlb R S x \<Longrightarrow> isLb R S x"
41 unfolding isLb_def by (blast dest: isGlbD1 isGlbD1a)
43 lemma isGlbD2: "isGlb R S x \<Longrightarrow> y : S \<Longrightarrow> y \<ge> x"
44 by (blast dest!: isGlbD1 setgeD)
46 lemma isGlbD3: "isGlb R S x \<Longrightarrow> greatestP (isLb R S) x"
47 by (simp add: isGlb_def)
49 lemma isGlbI1: "greatestP (isLb R S) x \<Longrightarrow> isGlb R S x"
50 by (simp add: isGlb_def)
52 lemma isGlbI2: "isLb R S x \<Longrightarrow> Collect (isLb R S) *<= x \<Longrightarrow> isGlb R S x"
53 by (simp add: isGlb_def greatestP_def)
55 lemma isLbD: "isLb R S x \<Longrightarrow> y : S \<Longrightarrow> y \<ge> x"
56 by (simp add: isLb_def setge_def)
58 lemma isLbD2: "isLb R S x \<Longrightarrow> x <=* S "
59 by (simp add: isLb_def)
61 lemma isLbD2a: "isLb R S x \<Longrightarrow> x: R"
62 by (simp add: isLb_def)
64 lemma isLbI: "x <=* S \<Longrightarrow> x: R \<Longrightarrow> isLb R S x"
65 by (simp add: isLb_def)
67 lemma isGlb_le_isLb: "isGlb R S x \<Longrightarrow> isLb R S y \<Longrightarrow> x \<ge> y"
68 unfolding isGlb_def by (blast intro!: greatestPD3)
70 lemma isGlb_ubs: "isGlb R S x \<Longrightarrow> lbs R S *<= x"
71 unfolding lbs_def isGlb_def by (rule greatestPD2)