(* Title: HOL/HOLCF/IOA/NTP/Lemmas.thy
Author: Tobias Nipkow & Konrad Slind
*)
theory Lemmas
imports Main
begin
subsubsection \<open>Logic\<close>
lemma neg_flip: "(X = (~ Y)) = ((~X) = Y)"
by blast
subsection \<open>Sets\<close>
lemma set_lemmas:
"f(x) : (UN x. {f(x)})"
"f x y : (UN x y. {f x y})"
"!!a. (!x. a ~= f(x)) ==> a ~: (UN x. {f(x)})"
"!!a. (!x y. a ~= f x y) ==> a ~: (UN x y. {f x y})"
by auto
subsection \<open>Arithmetic\<close>
lemma pred_suc: "0<x ==> (x - 1 = y) = (x = Suc(y))"
by (simp add: diff_Suc split add: nat.split)
lemmas [simp] = hd_append set_lemmas
end