(* Title: HOL/IMPP/Natural.thy ID: $Id: Natural.thy,v 1.5 2007/08/07 18:19:55 wenzelm Exp $ 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 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 :: "[com,state, state] => bool" ("<_,_>/ -c-> _" [0,0, 51] 51) where 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 :: "[com,state,nat,state] => bool" ("<_,_>/ -_-> _" [0,0,0,51] 51) where 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 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
lemma com_det:
[| <c,s> -c-> t; <c,s> -c-> u |] ==> u = t
lemma evaln_evalc:
<c,s> -n-> t ==> <c,s> -c-> t
lemma Suc_le_D_lemma:
[| Suc n ≤ m'; !!m. n ≤ m ==> P (Suc m) |] ==> P m'
lemma evaln_nonstrict:
[| <c,s> -n-> t; n ≤ m |] ==> <c,s> -m-> t
lemma evaln_Suc:
<c,s> -n-> s' ==> <c,s> -Suc n-> s'
lemma evaln_max2:
[| <c1.0,s1.0> -n1.0-> t1.0; <c2.0,s2.0> -n2.0-> t2.0 |]
==> ∃n. <c1.0,s1.0> -n-> t1.0 ∧ <c2.0,s2.0> -n-> t2.0
lemma evalc_evaln:
<c,s> -c-> t ==> ∃n. <c,s> -n-> t
lemma eval_eq:
<c,s> -c-> t = (∃n. <c,s> -n-> t)