Theory Lemmas

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theory Lemmas
imports Main

(*  Title:      HOL/IOA/NTP/Lemmas.thy
Author: Tobias Nipkow & Konrad Slind
*)


theory Lemmas
imports Main
begin


subsubsection {* Logic *}

lemma neg_flip: "(X = (~ Y)) = ((~X) = Y)"
by blast


subsection {* Sets *}

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 {* Arithmetic *}

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