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(************************************************************************)
(* * The Coq Proof Assistant / The Coq Development Team *)
(* v * INRIA, CNRS and contributors - Copyright 1999-2018 *)
(* <O___,, * (see CREDITS file for the list of authors) *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(* * (see LICENSE file for the text of the license) *)
(************************************************************************)
Require Import Mult.
Require Import BinNat.
Require Import Nnat.
Require Export Ring.
Set Implicit Arguments.
Lemma natSRth : semi_ring_theory O (S O) plus mult (@eq nat).
Proof.
constructor. exact plus_0_l. exact plus_comm. exact plus_assoc.
exact mult_1_l. exact mult_0_l. exact mult_comm. exact mult_assoc.
exact mult_plus_distr_r.
Qed.
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.
Proof.
constructor;trivial.
exact N2Nat.inj_add.
exact N2Nat.inj_mul.
intros x y H. apply N.eqb_eq in H. now subst.
Qed.
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.
(* For internal use only *)
Local Definition protected_to_nat := N.to_nat.
Ltac natprering :=
match goal with
|- context C [S ?p] =>
match p with
O => fail 1 (* avoid replacing 1 with 1+0 ! *)
| 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]).
|
