(* Title: HOL/IMP/Hoare.thy
ID: $Id$
Author: Tobias Nipkow
Copyright 1995 TUM
Inductive definition of Hoare logic
*)
Hoare = Denotation +
types assn = state => bool
consts
hoare :: "(assn * com * assn) set"
spec :: [state=>bool,com,state=>bool] => bool
defs
spec_def "spec P c Q == !s t. (s,t) : C(c) --> P s --> Q t"
syntax "@hoare" :: [bool,com,bool] => bool ("{{(1_)}}/ (_)/ {{(1_)}}" 10)
translations "{{P}}c{{Q}}" == "(P,c,Q) : hoare"
inductive "hoare"
intrs
hoare_skip "{{P}}skip{{P}}"
hoare_ass "{{%s.P(s[A a s/x])}} x:=a {{P}}"
hoare_semi "[| {{P}}c{{Q}}; {{Q}}d{{R}} |] ==> {{P}} c;d {{R}}"
hoare_if "[| {{%s. P s & B b s}}c{{Q}}; {{%s. P s & ~B b s}}d{{Q}} |] ==>
{{P}} ifc b then c else d {{Q}}"
hoare_while "[| {{%s. P s & B b s}} c {{P}} |] ==>
{{P}} while b do c {{%s. P s & ~B b s}}"
hoare_conseq "[| !s. P' s --> P s; {{P}}c{{Q}}; !s. Q s --> Q' s |] ==>
{{P'}}c{{Q'}}"
end