diff -r 91c05f4b509e -r 5f9138bcb3d7 src/HOL/FixedPoint.thy --- a/src/HOL/FixedPoint.thy Sun May 06 21:49:44 2007 +0200 +++ b/src/HOL/FixedPoint.thy Sun May 06 21:50:17 2007 +0200 @@ -14,18 +14,19 @@ subsection {* Complete lattices *} class complete_lattice = lattice + - fixes Inf :: "'a set \ 'a" - assumes Inf_lower: "x \ A \ Inf A \ x" - assumes Inf_greatest: "(\x. x \ A \ z \ x) \ z \ Inf A" + fixes Inf :: "'a set \ 'a" ("\_" [90] 90) + assumes Inf_lower: "x \ A \ \A \ x" + assumes Inf_greatest: "(\x. x \ A \ z \ x) \ z \ \A" definition - Sup :: "'a\complete_lattice set \ 'a" where + Sup :: "'a\complete_lattice set \ 'a" +where "Sup A = Inf {b. \a \ A. a \ b}" -theorem Sup_upper: "(x::'a::complete_lattice) \ A \ x <= Sup A" +theorem Sup_upper: "(x::'a::complete_lattice) \ A \ x \ Sup A" by (auto simp: Sup_def intro: Inf_greatest) -theorem Sup_least: "(\x::'a::complete_lattice. x \ A \ x <= z) \ Sup A <= z" +theorem Sup_least: "(\x::'a::complete_lattice. x \ A \ x \ z) \ Sup A \ z" by (auto simp: Sup_def intro: Inf_lower) definition @@ -198,7 +199,7 @@ apply (iprover elim: le_funE) done -lemmas [code nofunc] = Inf_fun_def +lemmas [code func del] = Inf_fun_def theorem Sup_fun_eq: "Sup A = (\x. Sup {y. \f\A. y = f x})" apply (rule order_antisym) @@ -221,7 +222,7 @@ Inf_set_def: "Inf S \ \S" by intro_classes (auto simp add: Inf_set_def) -lemmas [code nofunc] = Inf_set_def +lemmas [code func del] = Inf_set_def theorem Sup_set_eq: "Sup S = \S" apply (rule subset_antisym)