src/HOL/IMPP/Hoare.thy
 changeset 17477 ceb42ea2f223 parent 10834 a7897aebbffc child 19803 aa2581752afb
```     1.1 --- a/src/HOL/IMPP/Hoare.thy	Sat Sep 17 19:17:35 2005 +0200
1.2 +++ b/src/HOL/IMPP/Hoare.thy	Sat Sep 17 20:14:30 2005 +0200
1.3 @@ -2,98 +2,106 @@
1.4      ID:         \$Id\$
1.5      Author:     David von Oheimb
1.7 -
1.8 -Inductive definition of Hoare logic for partial correctness
1.9 -Completeness is taken relative to completeness of the underlying logic
1.10 -Two versions of completeness proof:
1.11 -  nested single recursion vs. simultaneous recursion in call rule
1.12  *)
1.13
1.14 -Hoare = Natural +
1.15 +header {* Inductive definition of Hoare logic for partial correctness *}
1.16 +
1.17 +theory Hoare
1.18 +imports Natural
1.19 +begin
1.20 +
1.21 +text {*
1.22 +  Completeness is taken relative to completeness of the underlying logic.
1.23 +
1.24 +  Two versions of completeness proof: nested single recursion
1.25 +  vs. simultaneous recursion in call rule
1.26 +*}
1.27
1.28  types 'a assn = "'a => state => bool"
1.29  translations
1.30 -      "a assn"   <= (type)"a => state => bool"
1.31 +  "a assn"   <= (type)"a => state => bool"
1.32
1.33  constdefs
1.34    state_not_singleton :: bool
1.35 - "state_not_singleton == ? s t::state. s ~= t" (* at least two elements *)
1.36 +  "state_not_singleton == \<exists>s t::state. s ~= t" (* at least two elements *)
1.37
1.38    peek_and    :: "'a assn => (state => bool) => 'a assn" (infixr "&>" 35)
1.39 - "peek_and P p == %Z s. P Z s & p s"
1.40 +  "peek_and P p == %Z s. P Z s & p s"
1.41
1.42  datatype 'a triple =
1.43 -    triple ('a assn) com ('a assn)         ("{(1_)}./ (_)/ .{(1_)}" [3,60,3] 58)
1.44 -
1.45 +  triple "'a assn"  com  "'a assn"       ("{(1_)}./ (_)/ .{(1_)}" [3,60,3] 58)
1.46 +
1.47  consts
1.48 -  triple_valid ::            nat => 'a triple     => bool ( "|=_:_" [0 , 58] 57)
1.49 -  hoare_valids ::  'a triple set => 'a triple set => bool ("_||=_"  [58, 58] 57)
1.50 -  hoare_derivs ::"('a triple set *  'a triple set)   set"
1.51 +  triple_valid ::            "nat => 'a triple     => bool" ( "|=_:_" [0 , 58] 57)
1.52 +  hoare_valids ::  "'a triple set => 'a triple set => bool" ("_||=_"  [58, 58] 57)
1.53 +  hoare_derivs :: "('a triple set *  'a triple set)   set"
1.54  syntax
1.55 -  triples_valid::            nat => 'a triple set => bool ("||=_:_" [0 , 58] 57)
1.56 -  hoare_valid  ::  'a triple set => 'a triple     => bool ("_|=_"   [58, 58] 57)
1.57 -"@hoare_derivs"::  'a triple set => 'a triple set => bool ("_||-_"  [58, 58] 57)
1.58 -"@hoare_deriv" ::  'a triple set => 'a triple     => bool ("_|-_"   [58, 58] 57)
1.59 +  triples_valid::            "nat => 'a triple set => bool" ("||=_:_" [0 , 58] 57)
1.60 +  hoare_valid  ::  "'a triple set => 'a triple     => bool" ("_|=_"   [58, 58] 57)
1.61 +"@hoare_derivs"::  "'a triple set => 'a triple set => bool" ("_||-_"  [58, 58] 57)
1.62 +"@hoare_deriv" ::  "'a triple set => 'a triple     => bool" ("_|-_"   [58, 58] 57)
1.63
1.64 -defs triple_valid_def  "|=n:t  ==  case t of {P}.c.{Q} =>
1.65 -		                !Z s. P Z s --> (!s'. <c,s> -n-> s' --> Q Z s')"
1.66 +defs triple_valid_def: "|=n:t  ==  case t of {P}.c.{Q} =>
1.67 +                                !Z s. P Z s --> (!s'. <c,s> -n-> s' --> Q Z s')"
1.68  translations          "||=n:G" == "Ball G (triple_valid n)"
1.69 -defs hoare_valids_def"G||=ts   ==  !n. ||=n:G --> ||=n:ts"
1.70 +defs hoare_valids_def: "G||=ts   ==  !n. ||=n:G --> ||=n:ts"
1.71  translations         "G |=t  " == " G||={t}"
1.72                       "G||-ts"  == "(G,ts) : hoare_derivs"
1.73                       "G |-t"   == " G||-{t}"
1.74
1.75  (* Most General Triples *)
1.76 -constdefs MGT    :: com => state triple              ("{=}._.{->}" [60] 58)
1.77 +constdefs MGT    :: "com => state triple"            ("{=}._.{->}" [60] 58)
1.78           "{=}.c.{->} == {%Z s0. Z = s0}. c .{%Z s1. <c,Z> -c-> s1}"
1.79
1.80 -inductive hoare_derivs intrs
1.81 -
1.82 -  empty    "G||-{}"
1.83 -  insert"[| G |-t;  G||-ts |]
1.84 -	==> G||-insert t ts"
1.85 +inductive hoare_derivs intros
1.86 +
1.87 +  empty:    "G||-{}"
1.88 +  insert: "[| G |-t;  G||-ts |]
1.89 +        ==> G||-insert t ts"
1.90
1.91 -  asm	   "ts <= G ==>
1.92 -	    G||-ts" (* {P}.BODY pn.{Q} instead of (general) t for SkipD_lemma *)
1.93 +  asm:      "ts <= G ==>
1.94 +             G||-ts" (* {P}.BODY pn.{Q} instead of (general) t for SkipD_lemma *)
1.95
1.96 -  cut   "[| G'||-ts; G||-G' |] ==> G||-ts" (* for convenience and efficiency *)
1.97 +  cut:   "[| G'||-ts; G||-G' |] ==> G||-ts" (* for convenience and efficiency *)
1.98
1.99 -  weaken"[| G||-ts' ; ts <= ts' |] ==> G||-ts"
1.100 +  weaken: "[| G||-ts' ; ts <= ts' |] ==> G||-ts"
1.101
1.102 -  conseq"!Z s. P  Z  s --> (? P' Q'. G|-{P'}.c.{Q'} &
1.103 -                                  (!s'. (!Z'. P' Z' s --> Q' Z' s') --> Q Z s'))
1.104 -         ==> G|-{P}.c.{Q}"
1.105 +  conseq: "!Z s. P  Z  s --> (? P' Q'. G|-{P'}.c.{Q'} &
1.106 +                                   (!s'. (!Z'. P' Z' s --> Q' Z' s') --> Q Z s'))
1.107 +          ==> G|-{P}.c.{Q}"
1.108
1.109
1.110 -  Skip	"G|-{P}. SKIP .{P}"
1.111 +  Skip:  "G|-{P}. SKIP .{P}"
1.112
1.113 -  Ass	"G|-{%Z s. P Z (s[X::=a s])}. X:==a .{P}"
1.114 +  Ass:   "G|-{%Z s. P Z (s[X::=a s])}. X:==a .{P}"
1.115
1.116 -  Local	"G|-{P}. c .{%Z s. Q Z (s[Loc X::=s'<X>])}
1.117 -     ==> G|-{%Z s. s'=s & P Z (s[Loc X::=a s])}. LOCAL X:=a IN c .{Q}"
1.118 +  Local: "G|-{P}. c .{%Z s. Q Z (s[Loc X::=s'<X>])}
1.119 +      ==> G|-{%Z s. s'=s & P Z (s[Loc X::=a s])}. LOCAL X:=a IN c .{Q}"
1.120
1.121 -  Comp	"[| G|-{P}.c.{Q};
1.122 -	    G|-{Q}.d.{R} |]
1.123 -	==> G|-{P}. (c;;d) .{R}"
1.124 +  Comp:  "[| G|-{P}.c.{Q};
1.125 +             G|-{Q}.d.{R} |]
1.126 +         ==> G|-{P}. (c;;d) .{R}"
1.127
1.128 -  If	"[| G|-{P &>        b }.c.{Q};
1.129 -	    G|-{P &> (Not o b)}.d.{Q} |]
1.130 -	==> G|-{P}. IF b THEN c ELSE d .{Q}"
1.131 +  If:    "[| G|-{P &>        b }.c.{Q};
1.132 +             G|-{P &> (Not o b)}.d.{Q} |]
1.133 +         ==> G|-{P}. IF b THEN c ELSE d .{Q}"
1.134
1.135 -  Loop  "G|-{P &> b}.c.{P} ==>
1.136 -	 G|-{P}. WHILE b DO c .{P &> (Not o b)}"
1.137 +  Loop:  "G|-{P &> b}.c.{P} ==>
1.138 +          G|-{P}. WHILE b DO c .{P &> (Not o b)}"
1.139
1.140  (*
1.141 -  BodyN	"(insert ({P}. BODY pn  .{Q}) G)
1.142 -	  |-{P}.  the (body pn) .{Q} ==>
1.143 -	 G|-{P}.       BODY pn  .{Q}"
1.144 +  BodyN: "(insert ({P}. BODY pn  .{Q}) G)
1.145 +           |-{P}.  the (body pn) .{Q} ==>
1.146 +          G|-{P}.       BODY pn  .{Q}"
1.147  *)
1.148 -  Body	"[| G Un (%p. {P p}.      BODY p  .{Q p})`Procs
1.149 -	      ||-(%p. {P p}. the (body p) .{Q p})`Procs |]
1.150 -	==>  G||-(%p. {P p}.      BODY p  .{Q p})`Procs"
1.151 +  Body:  "[| G Un (%p. {P p}.      BODY p  .{Q p})`Procs
1.152 +               ||-(%p. {P p}. the (body p) .{Q p})`Procs |]
1.153 +         ==>  G||-(%p. {P p}.      BODY p  .{Q p})`Procs"
1.154
1.155 -  Call	   "G|-{P}. BODY pn .{%Z s. Q Z (setlocs s (getlocs s')[X::=s<Res>])}
1.156 -	==> G|-{%Z s. s'=s & P Z (setlocs s newlocs[Loc Arg::=a s])}.
1.157 -	    X:=CALL pn(a) .{Q}"
1.158 +  Call:     "G|-{P}. BODY pn .{%Z s. Q Z (setlocs s (getlocs s')[X::=s<Res>])}
1.159 +         ==> G|-{%Z s. s'=s & P Z (setlocs s newlocs[Loc Arg::=a s])}.
1.160 +             X:=CALL pn(a) .{Q}"
1.161 +
1.162 +ML {* use_legacy_bindings (the_context ()) *}
1.163
1.164  end
```