Definition natModel : Model LNT :=
  model LNT nat
    (fun f : Functions LNT =>
     match f return (naryFunc nat (arity LNT (inr (Relations LNT) f))) with
     | Languages.Plus => fun x y : nat => y + x
     | Languages.Times => fun x y : nat => y * x
     | Languages.Succ => S
     | Languages.Zero => 0
     end) (fun r : Relations LNT => match r with
                                    end).

Theorem PAconsistent : Consistent LNT PA.