(* Title: HOL/HOLCF/IOA/Pred.thy
Author: Olaf Müller
*)
section \<open>Logical Connectives lifted to predicates\<close>
theory Pred
imports Main
begin
default_sort type
type_synonym 'a predicate = "'a \<Rightarrow> bool"
definition satisfies :: "'a \<Rightarrow> 'a predicate \<Rightarrow> bool" ("_ \<Turnstile> _" [100,9] 8)
where "(s \<Turnstile> P) \<longleftrightarrow> P s"
definition valid :: "'a predicate \<Rightarrow> bool" (* ("|-") *)
where "valid P \<longleftrightarrow> (\<forall>s. (s \<Turnstile> P))"
definition NOT :: "'a predicate \<Rightarrow> 'a predicate" ("\<^bold>\<not> _" [40] 40)
where "NOT P s \<longleftrightarrow> ~ (P s)"
definition AND :: "'a predicate \<Rightarrow> 'a predicate \<Rightarrow> 'a predicate" (infixr "\<^bold>\<and>" 35)
where "(P \<^bold>\<and> Q) s \<longleftrightarrow> P s \<and> Q s"
definition OR :: "'a predicate \<Rightarrow> 'a predicate \<Rightarrow> 'a predicate" (infixr "\<^bold>\<or>" 30)
where "(P \<^bold>\<or> Q) s \<longleftrightarrow> P s \<or> Q s"
definition IMPLIES :: "'a predicate \<Rightarrow> 'a predicate \<Rightarrow> 'a predicate" (infixr "\<^bold>\<longrightarrow>" 25)
where "(P \<^bold>\<longrightarrow> Q) s \<longleftrightarrow> P s \<longrightarrow> Q s"
end