(* Title: HOL/IMPP/Hoare.thy
ID: $Id$
Author: David von Oheimb
Copyright 1999 TUM
*)
header {* Inductive definition of Hoare logic for partial correctness *}
theory Hoare
imports Natural
begin
text {*
Completeness is taken relative to completeness of the underlying logic.
Two versions of completeness proof: nested single recursion
vs. simultaneous recursion in call rule
*}
types 'a assn = "'a => state => bool"
translations
"a assn" <= (type)"a => state => bool"
constdefs
state_not_singleton :: bool
"state_not_singleton == \<exists>s t::state. s ~= t" (* at least two elements *)
peek_and :: "'a assn => (state => bool) => 'a assn" (infixr "&>" 35)
"peek_and P p == %Z s. P Z s & p s"
datatype 'a triple =
triple "'a assn" com "'a assn" ("{(1_)}./ (_)/ .{(1_)}" [3,60,3] 58)
consts
triple_valid :: "nat => 'a triple => bool" ( "|=_:_" [0 , 58] 57)
hoare_valids :: "'a triple set => 'a triple set => bool" ("_||=_" [58, 58] 57)
hoare_derivs :: "('a triple set * 'a triple set) set"
syntax
triples_valid:: "nat => 'a triple set => bool" ("||=_:_" [0 , 58] 57)
hoare_valid :: "'a triple set => 'a triple => bool" ("_|=_" [58, 58] 57)
"@hoare_derivs":: "'a triple set => 'a triple set => bool" ("_||-_" [58, 58] 57)
"@hoare_deriv" :: "'a triple set => 'a triple => bool" ("_|-_" [58, 58] 57)
defs triple_valid_def: "|=n:t == case t of {P}.c.{Q} =>
!Z s. P Z s --> (!s'. <c,s> -n-> s' --> Q Z s')"
translations "||=n:G" == "Ball G (triple_valid n)"
defs hoare_valids_def: "G||=ts == !n. ||=n:G --> ||=n:ts"
translations "G |=t " == " G||={t}"
"G||-ts" == "(G,ts) : hoare_derivs"
"G |-t" == " G||-{t}"
(* Most General Triples *)
constdefs MGT :: "com => state triple" ("{=}._.{->}" [60] 58)
"{=}.c.{->} == {%Z s0. Z = s0}. c .{%Z s1. <c,Z> -c-> s1}"
inductive hoare_derivs intros
empty: "G||-{}"
insert: "[| G |-t; G||-ts |]
==> G||-insert t ts"
asm: "ts <= G ==>
G||-ts" (* {P}.BODY pn.{Q} instead of (general) t for SkipD_lemma *)
cut: "[| G'||-ts; G||-G' |] ==> G||-ts" (* for convenience and efficiency *)
weaken: "[| G||-ts' ; ts <= ts' |] ==> G||-ts"
conseq: "!Z s. P Z s --> (? P' Q'. G|-{P'}.c.{Q'} &
(!s'. (!Z'. P' Z' s --> Q' Z' s') --> Q Z s'))
==> G|-{P}.c.{Q}"
Skip: "G|-{P}. SKIP .{P}"
Ass: "G|-{%Z s. P Z (s[X::=a s])}. X:==a .{P}"
Local: "G|-{P}. c .{%Z s. Q Z (s[Loc X::=s'<X>])}
==> G|-{%Z s. s'=s & P Z (s[Loc X::=a s])}. LOCAL X:=a IN c .{Q}"
Comp: "[| G|-{P}.c.{Q};
G|-{Q}.d.{R} |]
==> G|-{P}. (c;;d) .{R}"
If: "[| G|-{P &> b }.c.{Q};
G|-{P &> (Not o b)}.d.{Q} |]
==> G|-{P}. IF b THEN c ELSE d .{Q}"
Loop: "G|-{P &> b}.c.{P} ==>
G|-{P}. WHILE b DO c .{P &> (Not o b)}"
(*
BodyN: "(insert ({P}. BODY pn .{Q}) G)
|-{P}. the (body pn) .{Q} ==>
G|-{P}. BODY pn .{Q}"
*)
Body: "[| G Un (%p. {P p}. BODY p .{Q p})`Procs
||-(%p. {P p}. the (body p) .{Q p})`Procs |]
==> G||-(%p. {P p}. BODY p .{Q p})`Procs"
Call: "G|-{P}. BODY pn .{%Z s. Q Z (setlocs s (getlocs s')[X::=s<Res>])}
==> G|-{%Z s. s'=s & P Z (setlocs s newlocs[Loc Arg::=a s])}.
X:=CALL pn(a) .{Q}"
ML {* use_legacy_bindings (the_context ()) *}
end