--- a/src/HOL/IMPP/Hoare.thy Sat Sep 17 19:17:35 2005 +0200
+++ b/src/HOL/IMPP/Hoare.thy Sat Sep 17 20:14:30 2005 +0200
@@ -2,98 +2,106 @@
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 +
+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"
+ "a assn" <= (type)"a => state => bool"
constdefs
state_not_singleton :: bool
- "state_not_singleton == ? s t::state. s ~= t" (* at least two elements *)
+ "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"
+ "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)
-
+ 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"
+ 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)
+ 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')"
+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"
+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)
+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"
+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 *)
+ 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 *)
+ cut: "[| G'||-ts; G||-G' |] ==> G||-ts" (* for convenience and efficiency *)
- weaken"[| G||-ts' ; ts <= ts' |] ==> G||-ts"
+ 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}"
+ 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}"
+ Skip: "G|-{P}. SKIP .{P}"
- Ass "G|-{%Z s. P Z (s[X::=a s])}. X:==a .{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}"
+ 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}"
+ 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}"
+ 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)}"
+ 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}"
+ 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"
+ 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}"
+ 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