The Coq Proof Assistant

# Library Coq.setoid_ring.ArithRing

Require Import PeanoNat.
Require Import BinNat.
Require Import Nnat.
Require Export Ring.
Set Implicit Arguments.
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 N.add N.mul N.eqb N.to_nat.
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.
Local Definition protected_to_nat := N.to_nat.
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
| _ => change N.to_nat with protected_to_nat
end.
Ltac natpostring :=
match goal with
| |- context [N.to_nat ?x] =>
let v := eval cbv in (N.to_nat x) in
change (N.to_nat x) with v;
natpostring
| _ => change protected_to_nat with N.to_nat
end.
Add Ring natr : natSRth
(morphism nat_morph_N, constants [natcst],
preprocess [natprering], postprocess [natpostring]).