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.