Theory Com

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


header {* Semantics of arithmetic and boolean expressions, Syntax of commands *}

theory Com
imports Main
begin

type_synonym val = nat
(* for the meta theory, this may be anything, but types cannot be refined later *)

typedecl glb
typedecl loc

axiomatization
Arg :: loc and
Res :: loc

datatype vname = Glb glb | Loc loc
type_synonym globs = "glb => val"
type_synonym locals = "loc => val"
datatype state = st globs locals
(* for the meta theory, the following would be sufficient:
typedecl state
consts st :: "[globs , locals] => state"
*)

type_synonym aexp = "state => val"
type_synonym bexp = "state => bool"

typedecl pname

datatype com
= SKIP
| Ass vname aexp ("_:==_" [65, 65 ] 60)
| Local loc aexp com ("LOCAL _:=_ IN _" [65, 0, 61] 60)
| Semi com com ("_;; _" [59, 60 ] 59)
| Cond bexp com com ("IF _ THEN _ ELSE _" [65, 60, 61] 60)
| While bexp com ("WHILE _ DO _" [65, 61] 60)
| BODY pname
| Call vname pname aexp ("_:=CALL _'(_')" [65, 65, 0] 60)

consts bodies :: "(pname * com) list"(* finitely many procedure definitions *)
definition
body :: " pname ~=> com" where
"body = map_of bodies"


(* Well-typedness: all procedures called must exist *)

inductive WT :: "com => bool" where

Skip: "WT SKIP"

| Assign: "WT (X :== a)"

| Local: "WT c ==>
WT (LOCAL Y := a IN c)"


| Semi: "[| WT c0; WT c1 |] ==>
WT (c0;; c1)"


| If: "[| WT c0; WT c1 |] ==>
WT (IF b THEN c0 ELSE c1)"


| While: "WT c ==>
WT (WHILE b DO c)"


| Body: "body pn ~= None ==>
WT (BODY pn)"


| Call: "WT (BODY pn) ==>
WT (X:=CALL pn(a))"


inductive_cases WTs_elim_cases:
"WT SKIP" "WT (X:==a)" "WT (LOCAL Y:=a IN c)"
"WT (c1;;c2)" "WT (IF b THEN c1 ELSE c2)" "WT (WHILE b DO c)"
"WT (BODY P)" "WT (X:=CALL P(a))"

definition
WT_bodies :: bool where
"WT_bodies = (!(pn,b):set bodies. WT b)"


ML {* val make_imp_tac = EVERY'[rtac mp, fn i => atac (i+1), etac thin_rl] *}

lemma finite_dom_body: "finite (dom body)"
apply (unfold body_def)
apply (rule finite_dom_map_of)
done

lemma WT_bodiesD: "[| WT_bodies; body pn = Some b |] ==> WT b"
apply (unfold WT_bodies_def body_def)
apply (drule map_of_SomeD)
apply fast
done

declare WTs_elim_cases [elim!]

end