# Library Coq.setoid_ring.ArithRing

Require Import Mult.
Require Import BinNat.
Require Import Nnat.
Require Export Ring.
Lemma natSRth : semi_ring_theory O (S O) plus mult (@eq nat).
Lemma nat_morph_N :
semi_morph 0 1 plus mult (eq (A:=nat))
0%N 1%N Nplus Nmult Neq_bool nat_of_N.
Ltac natcst t :=
match isnatcst t with
true => constr:(N_of_nat t)
| _ => constr:InitialRing.NotConstant
end.
Ltac Ss_to_add f acc :=
match f with
| S ?f1 => Ss_to_add f1 (S acc)
| _ => constr:(acc + f)%nat
end.
Ltac natprering :=
match goal with
|- context C [S ?p] =>
match p with
O => fail 1
| p => match isnatcst p with
| true => fail 1
| false => let v := Ss_to_add p (S 0) in
fold v; natprering
end
end
| _ => idtac
end.
Add Ring natr : natSRth
(morphism nat_morph_N, constants [natcst], preprocess [natprering]).