1.1 --- a/src/HOL/Lubs.thy Tue Nov 05 09:45:00 2013 +0100
1.2 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000
1.3 @@ -1,103 +0,0 @@
1.4 -(* Title: HOL/Lubs.thy
1.5 - Author: Jacques D. Fleuriot, University of Cambridge
1.6 -*)
1.7 -
1.8 -header {* Definitions of Upper Bounds and Least Upper Bounds *}
1.9 -
1.10 -theory Lubs
1.11 -imports Main
1.12 -begin
1.13 -
1.14 -text {* Thanks to suggestions by James Margetson *}
1.15 -
1.16 -definition setle :: "'a set \<Rightarrow> 'a::ord \<Rightarrow> bool" (infixl "*<=" 70)
1.17 - where "S *<= x = (ALL y: S. y \<le> x)"
1.18 -
1.19 -definition setge :: "'a::ord \<Rightarrow> 'a set \<Rightarrow> bool" (infixl "<=*" 70)
1.20 - where "x <=* S = (ALL y: S. x \<le> y)"
1.21 -
1.22 -definition leastP :: "('a \<Rightarrow> bool) \<Rightarrow> 'a::ord \<Rightarrow> bool"
1.23 - where "leastP P x = (P x \<and> x <=* Collect P)"
1.24 -
1.25 -definition isUb :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"
1.26 - where "isUb R S x = (S *<= x \<and> x: R)"
1.27 -
1.28 -definition isLub :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"
1.29 - where "isLub R S x = leastP (isUb R S) x"
1.30 -
1.31 -definition ubs :: "'a set \<Rightarrow> 'a::ord set \<Rightarrow> 'a set"
1.32 - where "ubs R S = Collect (isUb R S)"
1.33 -
1.34 -
1.35 -subsection {* Rules for the Relations @{text "*<="} and @{text "<=*"} *}
1.36 -
1.37 -lemma setleI: "ALL y: S. y \<le> x \<Longrightarrow> S *<= x"
1.38 - by (simp add: setle_def)
1.39 -
1.40 -lemma setleD: "S *<= x \<Longrightarrow> y: S \<Longrightarrow> y \<le> x"
1.41 - by (simp add: setle_def)
1.42 -
1.43 -lemma setgeI: "ALL y: S. x \<le> y \<Longrightarrow> x <=* S"
1.44 - by (simp add: setge_def)
1.45 -
1.46 -lemma setgeD: "x <=* S \<Longrightarrow> y: S \<Longrightarrow> x \<le> y"
1.47 - by (simp add: setge_def)
1.48 -
1.49 -
1.50 -subsection {* Rules about the Operators @{term leastP}, @{term ub} and @{term lub} *}
1.51 -
1.52 -lemma leastPD1: "leastP P x \<Longrightarrow> P x"
1.53 - by (simp add: leastP_def)
1.54 -
1.55 -lemma leastPD2: "leastP P x \<Longrightarrow> x <=* Collect P"
1.56 - by (simp add: leastP_def)
1.57 -
1.58 -lemma leastPD3: "leastP P x \<Longrightarrow> y: Collect P \<Longrightarrow> x \<le> y"
1.59 - by (blast dest!: leastPD2 setgeD)
1.60 -
1.61 -lemma isLubD1: "isLub R S x \<Longrightarrow> S *<= x"
1.62 - by (simp add: isLub_def isUb_def leastP_def)
1.63 -
1.64 -lemma isLubD1a: "isLub R S x \<Longrightarrow> x: R"
1.65 - by (simp add: isLub_def isUb_def leastP_def)
1.66 -
1.67 -lemma isLub_isUb: "isLub R S x \<Longrightarrow> isUb R S x"
1.68 - unfolding isUb_def by (blast dest: isLubD1 isLubD1a)
1.69 -
1.70 -lemma isLubD2: "isLub R S x \<Longrightarrow> y : S \<Longrightarrow> y \<le> x"
1.71 - by (blast dest!: isLubD1 setleD)
1.72 -
1.73 -lemma isLubD3: "isLub R S x \<Longrightarrow> leastP (isUb R S) x"
1.74 - by (simp add: isLub_def)
1.75 -
1.76 -lemma isLubI1: "leastP(isUb R S) x \<Longrightarrow> isLub R S x"
1.77 - by (simp add: isLub_def)
1.78 -
1.79 -lemma isLubI2: "isUb R S x \<Longrightarrow> x <=* Collect (isUb R S) \<Longrightarrow> isLub R S x"
1.80 - by (simp add: isLub_def leastP_def)
1.81 -
1.82 -lemma isUbD: "isUb R S x \<Longrightarrow> y : S \<Longrightarrow> y \<le> x"
1.83 - by (simp add: isUb_def setle_def)
1.84 -
1.85 -lemma isUbD2: "isUb R S x \<Longrightarrow> S *<= x"
1.86 - by (simp add: isUb_def)
1.87 -
1.88 -lemma isUbD2a: "isUb R S x \<Longrightarrow> x: R"
1.89 - by (simp add: isUb_def)
1.90 -
1.91 -lemma isUbI: "S *<= x \<Longrightarrow> x: R \<Longrightarrow> isUb R S x"
1.92 - by (simp add: isUb_def)
1.93 -
1.94 -lemma isLub_le_isUb: "isLub R S x \<Longrightarrow> isUb R S y \<Longrightarrow> x \<le> y"
1.95 - unfolding isLub_def by (blast intro!: leastPD3)
1.96 -
1.97 -lemma isLub_ubs: "isLub R S x \<Longrightarrow> x <=* ubs R S"
1.98 - unfolding ubs_def isLub_def by (rule leastPD2)
1.99 -
1.100 -lemma isLub_unique: "[| isLub R S x; isLub R S y |] ==> x = (y::'a::linorder)"
1.101 - apply (frule isLub_isUb)
1.102 - apply (frule_tac x = y in isLub_isUb)
1.103 - apply (blast intro!: order_antisym dest!: isLub_le_isUb)
1.104 - done
1.105 -
1.106 -end