--- a/src/HOL/Lubs.thy Thu Feb 16 22:53:56 2012 +0100
+++ b/src/HOL/Lubs.thy Thu Feb 16 22:54:40 2012 +0100
@@ -1,112 +1,97 @@
-(* Title : Lubs.thy
- Author : Jacques D. Fleuriot
- Copyright : 1998 University of Cambridge
+(* Title: HOL/Lubs.thy
+ Author: Jacques D. Fleuriot, University of Cambridge
*)
-header{*Definitions of Upper Bounds and Least Upper Bounds*}
+header {* Definitions of Upper Bounds and Least Upper Bounds *}
theory Lubs
imports Main
begin
-text{*Thanks to suggestions by James Margetson*}
+text {* Thanks to suggestions by James Margetson *}
-definition
- setle :: "['a set, 'a::ord] => bool" (infixl "*<=" 70) where
- "S *<= x = (ALL y: S. y <= x)"
+definition setle :: "'a set \<Rightarrow> 'a::ord \<Rightarrow> bool" (infixl "*<=" 70)
+ where "S *<= x = (ALL y: S. y \<le> x)"
-definition
- setge :: "['a::ord, 'a set] => bool" (infixl "<=*" 70) where
- "x <=* S = (ALL y: S. x <= y)"
+definition setge :: "'a::ord \<Rightarrow> 'a set \<Rightarrow> bool" (infixl "<=*" 70)
+ where "x <=* S = (ALL y: S. x \<le> y)"
-definition
- leastP :: "['a =>bool,'a::ord] => bool" where
- "leastP P x = (P x & x <=* Collect P)"
+definition leastP :: "('a \<Rightarrow> bool) \<Rightarrow> 'a::ord \<Rightarrow> bool"
+ where "leastP P x = (P x \<and> x <=* Collect P)"
-definition
- isUb :: "['a set, 'a set, 'a::ord] => bool" where
- "isUb R S x = (S *<= x & x: R)"
+definition isUb :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"
+ where "isUb R S x = (S *<= x \<and> x: R)"
-definition
- isLub :: "['a set, 'a set, 'a::ord] => bool" where
- "isLub R S x = leastP (isUb R S) x"
+definition isLub :: "'a set \<Rightarrow> 'a set \<Rightarrow> 'a::ord \<Rightarrow> bool"
+ where "isLub R S x = leastP (isUb R S) x"
-definition
- ubs :: "['a set, 'a::ord set] => 'a set" where
- "ubs R S = Collect (isUb R S)"
+definition ubs :: "'a set \<Rightarrow> 'a::ord set \<Rightarrow> 'a set"
+ where "ubs R S = Collect (isUb R S)"
-
-subsection{*Rules for the Relations @{text "*<="} and @{text "<=*"}*}
+subsection {* Rules for the Relations @{text "*<="} and @{text "<=*"} *}
-lemma setleI: "ALL y: S. y <= x ==> S *<= x"
-by (simp add: setle_def)
+lemma setleI: "ALL y: S. y \<le> x \<Longrightarrow> S *<= x"
+ by (simp add: setle_def)
-lemma setleD: "[| S *<= x; y: S |] ==> y <= x"
-by (simp add: setle_def)
+lemma setleD: "S *<= x \<Longrightarrow> y: S \<Longrightarrow> y \<le> x"
+ by (simp add: setle_def)
-lemma setgeI: "ALL y: S. x<= y ==> x <=* S"
-by (simp add: setge_def)
+lemma setgeI: "ALL y: S. x \<le> y \<Longrightarrow> x <=* S"
+ by (simp add: setge_def)
-lemma setgeD: "[| x <=* S; y: S |] ==> x <= y"
-by (simp add: setge_def)
+lemma setgeD: "x <=* S \<Longrightarrow> y: S \<Longrightarrow> x \<le> y"
+ by (simp add: setge_def)
-subsection{*Rules about the Operators @{term leastP}, @{term ub}
- and @{term lub}*}
+subsection {* Rules about the Operators @{term leastP}, @{term ub} and @{term lub} *}
-lemma leastPD1: "leastP P x ==> P x"
-by (simp add: leastP_def)
+lemma leastPD1: "leastP P x \<Longrightarrow> P x"
+ by (simp add: leastP_def)
-lemma leastPD2: "leastP P x ==> x <=* Collect P"
-by (simp add: leastP_def)
+lemma leastPD2: "leastP P x \<Longrightarrow> x <=* Collect P"
+ by (simp add: leastP_def)
-lemma leastPD3: "[| leastP P x; y: Collect P |] ==> x <= y"
-by (blast dest!: leastPD2 setgeD)
+lemma leastPD3: "leastP P x \<Longrightarrow> y: Collect P \<Longrightarrow> x \<le> y"
+ by (blast dest!: leastPD2 setgeD)
-lemma isLubD1: "isLub R S x ==> S *<= x"
-by (simp add: isLub_def isUb_def leastP_def)
+lemma isLubD1: "isLub R S x \<Longrightarrow> S *<= x"
+ by (simp add: isLub_def isUb_def leastP_def)
-lemma isLubD1a: "isLub R S x ==> x: R"
-by (simp add: isLub_def isUb_def leastP_def)
+lemma isLubD1a: "isLub R S x \<Longrightarrow> x: R"
+ by (simp add: isLub_def isUb_def leastP_def)
-lemma isLub_isUb: "isLub R S x ==> isUb R S x"
-apply (simp add: isUb_def)
-apply (blast dest: isLubD1 isLubD1a)
-done
+lemma isLub_isUb: "isLub R S x \<Longrightarrow> isUb R S x"
+ unfolding isUb_def by (blast dest: isLubD1 isLubD1a)
-lemma isLubD2: "[| isLub R S x; y : S |] ==> y <= x"
-by (blast dest!: isLubD1 setleD)
+lemma isLubD2: "isLub R S x \<Longrightarrow> y : S \<Longrightarrow> y \<le> x"
+ by (blast dest!: isLubD1 setleD)
-lemma isLubD3: "isLub R S x ==> leastP(isUb R S) x"
-by (simp add: isLub_def)
+lemma isLubD3: "isLub R S x \<Longrightarrow> leastP (isUb R S) x"
+ by (simp add: isLub_def)
-lemma isLubI1: "leastP(isUb R S) x ==> isLub R S x"
-by (simp add: isLub_def)
+lemma isLubI1: "leastP(isUb R S) x \<Longrightarrow> isLub R S x"
+ by (simp add: isLub_def)
-lemma isLubI2: "[| isUb R S x; x <=* Collect (isUb R S) |] ==> isLub R S x"
-by (simp add: isLub_def leastP_def)
+lemma isLubI2: "isUb R S x \<Longrightarrow> x <=* Collect (isUb R S) \<Longrightarrow> isLub R S x"
+ by (simp add: isLub_def leastP_def)
-lemma isUbD: "[| isUb R S x; y : S |] ==> y <= x"
-by (simp add: isUb_def setle_def)
-
-lemma isUbD2: "isUb R S x ==> S *<= x"
-by (simp add: isUb_def)
+lemma isUbD: "isUb R S x \<Longrightarrow> y : S \<Longrightarrow> y \<le> x"
+ by (simp add: isUb_def setle_def)
-lemma isUbD2a: "isUb R S x ==> x: R"
-by (simp add: isUb_def)
+lemma isUbD2: "isUb R S x \<Longrightarrow> S *<= x"
+ by (simp add: isUb_def)
-lemma isUbI: "[| S *<= x; x: R |] ==> isUb R S x"
-by (simp add: isUb_def)
+lemma isUbD2a: "isUb R S x \<Longrightarrow> x: R"
+ by (simp add: isUb_def)
-lemma isLub_le_isUb: "[| isLub R S x; isUb R S y |] ==> x <= y"
-apply (simp add: isLub_def)
-apply (blast intro!: leastPD3)
-done
+lemma isUbI: "S *<= x \<Longrightarrow> x: R \<Longrightarrow> isUb R S x"
+ by (simp add: isUb_def)
-lemma isLub_ubs: "isLub R S x ==> x <=* ubs R S"
-apply (simp add: ubs_def isLub_def)
-apply (erule leastPD2)
-done
+lemma isLub_le_isUb: "isLub R S x \<Longrightarrow> isUb R S y \<Longrightarrow> x \<le> y"
+ unfolding isLub_def by (blast intro!: leastPD3)
+
+lemma isLub_ubs: "isLub R S x \<Longrightarrow> x <=* ubs R S"
+ unfolding ubs_def isLub_def by (rule leastPD2)
end