# Theory Nat

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theory Nat
imports Wfd
uses [Nat.ML]
begin

    (*  Title:      CCL/ex/Nat.thy
        ID:         $Id: Nat.thy,v 1.5 2005/09/17 15:35:32 wenzelm Exp $
        Author:     Martin Coen, Cambridge University Computer Laboratory
        Copyright   1993  University of Cambridge
    *)

    header {* Programs defined over the natural numbers *}

    theory Nat
    imports Wfd
    begin

    consts

      not   :: "i=>i"
      "#+"  :: "[i,i]=>i"            (infixr 60)
      "#*"  :: "[i,i]=>i"            (infixr 60)
      "#-"  :: "[i,i]=>i"            (infixr 60)
      "##"  :: "[i,i]=>i"            (infixr 60)
      "#<"  :: "[i,i]=>i"            (infixr 60)
      "#<=" :: "[i,i]=>i"            (infixr 60)
      ackermann :: "[i,i]=>i"

    defs

      not_def:     "not(b) == if b then false else true"

      add_def:     "a #+ b == nrec(a,b,%x g. succ(g))"
      mult_def:    "a #* b == nrec(a,zero,%x g. b #+ g)"
      sub_def:     "a #- b == letrec sub x y be ncase(y,x,%yy. ncase(x,zero,%xx. sub(xx,yy)))
                            in sub(a,b)"
      le_def:     "a #<= b == letrec le x y be ncase(x,true,%xx. ncase(y,false,%yy. le(xx,yy)))
                            in le(a,b)"
      lt_def:     "a #< b == not(b #<= a)"

      div_def:    "a ## b == letrec div x y be if x #< y then zero else succ(div(x#-y,y))
                           in div(a,b)"
      ack_def:
      "ackermann(a,b) == letrec ack n m be ncase(n,succ(m),%x.
                              ncase(m,ack(x,succ(zero)),%y. ack(x,ack(succ(x),y))))
                        in ack(a,b)"

    ML {* use_legacy_bindings (the_context ()) *}

    end

theorem napply_f:

      n : Nat ==> f ^ n ` f(a) = f ^ succ(n) ` a

theorem addT:

      [| a : Nat; b : Nat |] ==> a #+ b : Nat

theorem multT:

      [| a : Nat; b : Nat |] ==> a #* b : Nat

theorem subT:

      [| a : Nat; b : Nat |] ==> a #- b : Nat

theorem leT:

      [| a : Nat; b : Nat |] ==> a #<= b : Bool

theorem ltT:

      [| a : Nat; b : Nat |] ==> a #< b : Bool