**
** Copyright (c) 2000-2015, Toshimi Sawada. All rights reserved.
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**
sys:mod! NAT
      principal-sort Nat
{
  imports {
    protecting (NZNAT)
    protecting (NAT-VALUE)
  }
  signature {
    op s _ : Nat -> NzNat { demod prec: 15 }
    pred _ >= _ : Nat Nat  { demod prec: 51 }
    pred _ > _ : Nat Nat  { demod prec: 51 }
    pred _ <= _ : Nat Nat  { demod prec: 51 }
    pred _ < _ : Nat Nat  { demod prec: 51 }
    op _ * _ : Nat Nat -> Nat { assoc comm idr: 1 demod prec: 31 }
    op _ + _ : Nat Nat -> Nat { assoc comm idr: 0 demod prec: 33 }
    op sd : Nat Nat -> Nat { comm demod prec: 0 }
    op _ quo _ : Nat NzNat -> Nat { demod prec: 31 }
    op _ rem _ : Nat NzNat -> Nat { demod prec: 31 l-assoc }
    pred _ divides _ : NzNat Nat  { demod prec: 51 }
    op p _ : NzNat -> Nat { demod prec: 15 }
  }
  axioms {
    var M : Nat
    var N : Nat
    var NN : NzNat
    eq [:BDEMOD]: sd(M,N) = #! (abs (- m n)) .
    eq [:BDEMOD]: (M + N) = #! (+ m n) .
    eq [:BDEMOD]: (N * 0) = 0 .
    eq [:BDEMOD]: (M quo NN) = #! (truncate m nn) .
    eq [:BDEMOD]: (M rem NN) = #! (rem m nn) .
    eq [:BDEMOD]: (NN divides M) = #! (= 0 (rem m nn)) .
    eq [:BDEMOD]: (N < 0) = false .
    eq [:BDEMOD]: (0 < NN) = true .
    eq [:BDEMOD]: (NN <= 0) = false .
    eq [:BDEMOD]: (0 <= N) = true .
    eq [:BDEMOD]: (0 > N) = false .
    eq [:BDEMOD]: (NN > 0) = true .
    eq [:BDEMOD]: (0 >= NN) = false .
    eq [:BDEMOD]: (N >= 0) = true .
    eq [:BDEMOD]: (s 0) = 1 .
    eq [:BDEMOD]: (p NN) = #! (- nn 1) .
 }
}

lispq
(defun chaos-nat-tram-interface ()
  (setq *z-nat* (get-z-module-or-panic "NAT"))
  (push *z-nat* *tram-builtin-modules*))

lispq
(eval-when (:execute :load-toplevel)
  (chaos-nat-tram-interface))

protect NAT
provide nat
--
eof
