src/HOL/Induct/Com.thy

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(*  Title:      HOL/Induct/Com
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1997  University of Cambridge

Example of Mutual Induction via Iteratived Inductive Definitions: Commands
*)

Com = Main +

types loc
      state = "loc => nat"
      n2n2n = "nat => nat => nat"

arities loc :: term

datatype
  exp = N nat
      | X loc
      | Op n2n2n exp exp
      | valOf com exp          ("VALOF _ RESULTIS _"  60)
and
  com = SKIP
      | ":="  loc exp          (infixl  60)
      | Semi  com com          ("_;;_"  [60, 60] 60)
      | Cond  exp com com      ("IF _ THEN _ ELSE _"  60)
      | While exp com          ("WHILE _ DO _"  60)

(** Execution of commands **)
consts  exec    :: "((exp*state) * (nat*state)) set => ((com*state)*state)set"
       "@exec"  :: "((exp*state) * (nat*state)) set => 
                    [com*state,state] => bool"     ("_/ -[_]-> _" [50,0,50] 50)

translations  "csig -[eval]-> s" == "(csig,s) : exec eval"

syntax  eval'   :: "[exp*state,nat*state] => 
		    ((exp*state) * (nat*state)) set => bool"
						   ("_/ -|[_]-> _" [50,0,50] 50)
translations
    "esig -|[eval]-> ns" => "(esig,ns) : eval"

(*Command execution.  Natural numbers represent Booleans: 0=True, 1=False*)
inductive "exec eval"
  intrs
    Skip    "(SKIP,s) -[eval]-> s"

    Assign  "(e,s) -|[eval]-> (v,s') ==> (x := e, s) -[eval]-> s'(x:=v)"

    Semi    "[| (c0,s) -[eval]-> s2; (c1,s2) -[eval]-> s1 |] 
            ==> (c0 ;; c1, s) -[eval]-> s1"

    IfTrue "[| (e,s) -|[eval]-> (0,s');  (c0,s') -[eval]-> s1 |] 
            ==> (IF e THEN c0 ELSE c1, s) -[eval]-> s1"

    IfFalse "[| (e,s) -|[eval]->  (1,s');  (c1,s') -[eval]-> s1 |] 
             ==> (IF e THEN c0 ELSE c1, s) -[eval]-> s1"

    WhileFalse "(e,s) -|[eval]-> (1,s1) ==> (WHILE e DO c, s) -[eval]-> s1"

    WhileTrue  "[| (e,s) -|[eval]-> (0,s1);
                (c,s1) -[eval]-> s2;  (WHILE e DO c, s2) -[eval]-> s3 |] 
                ==> (WHILE e DO c, s) -[eval]-> s3"
end