Theory Natural

theory Natural
imports Com
(*  Title:      HOL/IMPP/Natural.thy
Author: David von Oheimb (based on a theory by Tobias Nipkow et al), 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)

abbreviation
loc :: "state => locals" ("_<_>" [75,0] 75) where
"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 (auto elim!: Suc_le_D_lemma)
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' [dtac @{thm evaln_max2}, assume_tac, 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