Theory Lemmas

(*  Title:      HOL/HOLCF/IOA/ABP/Lemmas.thy
    Author:     Olaf Müller
*)

theory Lemmas
imports Main
begin

subsection ‹Logic›

lemma and_de_morgan_and_absorbe: "(¬(A∧B)) = ((¬A)∧B∨ ¬B)"
  by blast

lemma bool_if_impl_or: "(if C then A else B) ⟶ (A∨B)"
  by auto

lemma exis_elim: "(∃x. x=P ∧ Q(x)) = Q(P)"
  by blast


subsection ‹Sets›

lemma set_lemmas:
    "f(x) ∈ (⋃x. {f(x)})"
    "f x y ∈ (⋃x y. {f x y})"
    "⋀a. (∀x. a ≠ f(x)) ⟹ a ∉ (⋃x. {f(x)})"
    "⋀a. (∀x y. a ≠ f x y) ==> a ∉ (⋃x y. {f x y})"
  by auto

text ‹2 Lemmas to add to ‹set_lemmas›, used also for action handling, 
   namely for Intersections and the empty list (compatibility of IOA!).›
lemma singleton_set: "(⋃b.{x. x=f(b)}) = (⋃b.{f(b)})"
  by blast

lemma de_morgan: "((A∨B)=False) = ((¬A)∧(¬B))"
  by blast


subsection ‹Lists›

lemma cons_not_nil: "l ≠ [] ⟶ (∃x xs. l = (x#xs))"
  by (induct l) simp_all

end