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(* $Id$ *)
(* Instantiation of the Ring tactic for the naturals of Arith $*)
Require Export Ring.
Require Export Arith.
Require Eqdep_dec.
Fixpoint nateq [n,m:nat] : bool :=
Cases n m of
| O O => true
| (S n') (S m') => (nateq n' m')
| _ _ => false
end.
Lemma nateq_prop : (n,m:nat)(Is_true (nateq n m))->n==m.
Proof.
Induction n; Induction m; Intros; Try Contradiction.
Trivial.
Unfold Is_true in H1.
Rewrite (H n1 H1).
Trivial.
Save.
Hints Resolve nateq_prop eq2eqT : arithring.
Definition NatTheory : (Semi_Ring_Theory plus mult (1) (0) nateq).
Split; Intros; Auto with arith arithring.
Apply eq2eqT; Apply simpl_plus_l with n.
Apply eqT2eq; Trivial.
Defined.
Add Semi Ring nat plus mult (1) (0) nateq NatTheory [O S].
Goal (n:nat)(S n)=(plus (S O) n).
Intro; Reflexivity.
Save S_to_plus_one.
Tactic Definition NatRing :=
[<:tactic:<(Repeat Rewrite S_to_plus_one); Ring>>].
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