--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/IMPP/Hoare.thy Mon Jan 31 18:30:35 2000 +0100
@@ -0,0 +1,99 @@
+(* Title: HOL/IMPP/Hoare.thy
+ ID: $Id$
+ Author: David von Oheimb
+ Copyright 1999 TUM
+
+Inductive definition of Hoare logic for partial correctness
+Completeness is taken relative to completeness of the underlying logic
+Two versions of completeness proof:
+ nested single recursion vs. simultaneous recursion in call rule
+*)
+
+Hoare = Natural +
+
+types 'a assn = "'a => state => bool"
+translations
+ "a assn" <= (type)"a => state => bool"
+
+constdefs
+ state_not_singleton :: bool
+ "state_not_singleton == ? 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 intrs
+
+ 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}"
+
+end