src/HOL/IMPP/Natural.thy
author wenzelm
Wed, 07 Jun 2006 01:51:22 +0200
changeset 19803 aa2581752afb
parent 17477 ceb42ea2f223
child 23746 a455e69c31cc
permissions -rw-r--r--
removed obsolete ML files;

(*  Title:      HOL/IMPP/Natural.thy
    ID:         $Id$
    Author:     David von Oheimb (based on a theory by Tobias Nipkow et al), TUM
    Copyright   1999 TUM
*)

header {* Natural semantics of commands *}

theory Natural
imports Com
begin

(** Execution of commands **)
consts
  evalc :: "(com * state *       state) set"
  evaln :: "(com * state * nat * state) set"

syntax
  "@evalc":: "[com,state,    state] => bool"  ("<_,_>/ -c-> _" [0,0,  51] 51)
  "@evaln":: "[com,state,nat,state] => bool"  ("<_,_>/ -_-> _" [0,0,0,51] 51)

translations
  "<c,s> -c-> s'" == "(c,s,  s') : evalc"
  "<c,s> -n-> s'" == "(c,s,n,s') : evaln"

consts
  newlocs :: locals
  setlocs :: "state => locals => state"
  getlocs :: "state => locals"
  update  :: "state => vname => val => state"     ("_/[_/::=/_]" [900,0,0] 900)
syntax (* IN Natural.thy *)
  loc :: "state => locals"    ("_<_>" [75,0] 75)
translations
  "s<X>" == "getlocs s X"

inductive evalc
  intros
    Skip:    "<SKIP,s> -c-> s"

    Assign:  "<X :== a,s> -c-> s[X::=a s]"

    Local:   "<c, s0[Loc Y::= a s0]> -c-> s1 ==>
              <LOCAL Y := a IN c, s0> -c-> s1[Loc Y::=s0<Y>]"

    Semi:    "[| <c0,s0> -c-> s1; <c1,s1> -c-> s2 |] ==>
              <c0;; c1, s0> -c-> s2"

    IfTrue:  "[| b s; <c0,s> -c-> s1 |] ==>
              <IF b THEN c0 ELSE c1, s> -c-> s1"

    IfFalse: "[| ~b s; <c1,s> -c-> s1 |] ==>
              <IF b THEN c0 ELSE c1, s> -c-> s1"

    WhileFalse: "~b s ==> <WHILE b DO c,s> -c-> s"

    WhileTrue:  "[| b s0;  <c,s0> -c-> s1;  <WHILE b DO c, s1> -c-> s2 |] ==>
                 <WHILE b DO c, s0> -c-> s2"

    Body:       "<the (body pn), s0> -c-> s1 ==>
                 <BODY pn, s0> -c-> s1"

    Call:       "<BODY pn, (setlocs s0 newlocs)[Loc Arg::=a s0]> -c-> s1 ==>
                 <X:=CALL pn(a), s0> -c-> (setlocs s1 (getlocs s0))
                                          [X::=s1<Res>]"

inductive evaln
  intros
    Skip:    "<SKIP,s> -n-> s"

    Assign:  "<X :== a,s> -n-> s[X::=a s]"

    Local:   "<c, s0[Loc Y::= a s0]> -n-> s1 ==>
              <LOCAL Y := a IN c, s0> -n-> s1[Loc Y::=s0<Y>]"

    Semi:    "[| <c0,s0> -n-> s1; <c1,s1> -n-> s2 |] ==>
              <c0;; c1, s0> -n-> s2"

    IfTrue:  "[| b s; <c0,s> -n-> s1 |] ==>
              <IF b THEN c0 ELSE c1, s> -n-> s1"

    IfFalse: "[| ~b s; <c1,s> -n-> s1 |] ==>
              <IF b THEN c0 ELSE c1, s> -n-> s1"

    WhileFalse: "~b s ==> <WHILE b DO c,s> -n-> s"

    WhileTrue:  "[| b s0;  <c,s0> -n-> s1;  <WHILE b DO c, s1> -n-> s2 |] ==>
                 <WHILE b DO c, s0> -n-> s2"

    Body:       "<the (body pn), s0> -    n-> s1 ==>
                 <BODY pn, s0> -Suc n-> s1"

    Call:       "<BODY pn, (setlocs s0 newlocs)[Loc Arg::=a s0]> -n-> s1 ==>
                 <X:=CALL pn(a), s0> -n-> (setlocs s1 (getlocs s0))
                                          [X::=s1<Res>]"


inductive_cases evalc_elim_cases:
  "<SKIP,s> -c-> t"  "<X:==a,s> -c-> t"  "<LOCAL Y:=a IN c,s> -c-> t"
  "<c1;;c2,s> -c-> t"  "<IF b THEN c1 ELSE c2,s> -c-> t"
  "<BODY P,s> -c-> s1"  "<X:=CALL P(a),s> -c-> s1"

inductive_cases evaln_elim_cases:
  "<SKIP,s> -n-> t"  "<X:==a,s> -n-> t"  "<LOCAL Y:=a IN c,s> -n-> t"
  "<c1;;c2,s> -n-> t"  "<IF b THEN c1 ELSE c2,s> -n-> t"
  "<BODY P,s> -n-> s1"  "<X:=CALL P(a),s> -n-> s1"

inductive_cases evalc_WHILE_case: "<WHILE b DO c,s> -c-> t"
inductive_cases evaln_WHILE_case: "<WHILE b DO c,s> -n-> t"

declare evalc.intros [intro]
declare evaln.intros [intro]

declare evalc_elim_cases [elim!]
declare evaln_elim_cases [elim!]

(* evaluation of com is deterministic *)
lemma com_det [rule_format (no_asm)]: "<c,s> -c-> t ==> (!u. <c,s> -c-> u --> u=t)"
apply (erule evalc.induct)
apply (erule_tac [8] V = "<?c,s1> -c-> s2" in thin_rl)
(*blast_tac needs Unify.search_bound := 40*)
apply (best elim: evalc_WHILE_case)+
done

lemma evaln_evalc: "<c,s> -n-> t ==> <c,s> -c-> t"
apply (erule evaln.induct)
apply (tactic {* ALLGOALS (resolve_tac (thms "evalc.intros") THEN_ALL_NEW atac) *})
done

lemma Suc_le_D_lemma: "[| Suc n <= m'; (!!m. n <= m ==> P (Suc m)) |] ==> P m'"
apply (frule Suc_le_D)
apply blast
done

lemma evaln_nonstrict [rule_format]: "<c,s> -n-> t ==> !m. n<=m --> <c,s> -m-> t"
apply (erule evaln.induct)
apply (tactic {* ALLGOALS (EVERY'[strip_tac,TRY o etac (thm "Suc_le_D_lemma"), REPEAT o smp_tac 1]) *})
apply (tactic {* ALLGOALS (resolve_tac (thms "evaln.intros") THEN_ALL_NEW atac) *})
done

lemma evaln_Suc: "<c,s> -n-> s' ==> <c,s> -Suc n-> s'"
apply (erule evaln_nonstrict)
apply auto
done

lemma evaln_max2: "[| <c1,s1> -n1-> t1;  <c2,s2> -n2-> t2 |] ==>  
    ? n. <c1,s1> -n -> t1 & <c2,s2> -n -> t2"
apply (cut_tac m = "n1" and n = "n2" in nat_le_linear)
apply (blast dest: evaln_nonstrict)
done

lemma evalc_evaln: "<c,s> -c-> t ==> ? n. <c,s> -n-> t"
apply (erule evalc.induct)
apply (tactic {* ALLGOALS (REPEAT o etac exE) *})
apply (tactic {* TRYALL (EVERY'[datac (thm "evaln_max2") 1, REPEAT o eresolve_tac [exE, conjE]]) *})
apply (tactic {* ALLGOALS (rtac exI THEN' resolve_tac (thms "evaln.intros") THEN_ALL_NEW atac) *})
done

lemma eval_eq: "<c,s> -c-> t = (? n. <c,s> -n-> t)"
apply (fast elim: evalc_evaln evaln_evalc)
done

end