# Theory Nat

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theory Nat
imports FOLP
uses [Nat.ML]
begin

    (*  Title:      FOLP/ex/nat.thy
        ID:         $Id: Nat.thy,v 1.5 2005/09/18 12:25:49 wenzelm Exp $
        Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
        Copyright   1992  University of Cambridge
    *)

    header {* Theory of the natural numbers: Peano's axioms, primitive recursion *}

    theory Nat
    imports FOLP
    begin

    typedecl nat
    arities nat         :: "term"
    consts  "0"         :: "nat"    ("0")
            Suc         :: "nat=>nat"
            rec         :: "[nat, 'a, [nat,'a]=>'a] => 'a"
            "+"         :: "[nat, nat] => nat"              (infixl 60)

      (*Proof terms*)
           nrec         :: "[nat,p,[nat,p]=>p]=>p"
           ninj         :: "p=>p"
           nneq         :: "p=>p"
           rec0         :: "p"
           recSuc       :: "p"

    axioms
      induct:     "[| b:P(0); !!x u. u:P(x) ==> c(x,u):P(Suc(x))
                  |] ==> nrec(n,b,c):P(n)"

      Suc_inject: "p:Suc(m)=Suc(n) ==> ninj(p) : m=n"
      Suc_neq_0:  "p:Suc(m)=0      ==> nneq(p) : R"
      rec_0:      "rec0 : rec(0,a,f) = a"
      rec_Suc:    "recSuc : rec(Suc(m), a, f) = f(m, rec(m,a,f))"
      add_def:    "m+n == rec(m, n, %x y. Suc(y))"

      nrecB0:     "b: A ==> nrec(0,b,c) = b : A"
      nrecBSuc:   "c(n,nrec(n,b,c)) : A ==> nrec(Suc(n),b,c) = c(n,nrec(n,b,c)) : A"

    ML {* use_legacy_bindings (the_context ()) *}

    end