(************************************************************************)
(*         *      The Rocq Prover / The Rocq Development Team           *)
(*  v      *         Copyright INRIA, CNRS and contributors             *)
(* <O___,, * (see version control and CREDITS file for authors & dates) *)
(*   \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)         *)
(************************************************************************)
(* non commutative rings *)
Require Export Morphisms Setoid Bool.
Require Export ZArith_base.
Require Export Algebra_syntax.
Set Implicit Arguments.
Class Ring_ops(T:Type)
{ring0:T}
{ring1:T}
{add:T->T->T}
{mul:T->T->T}
{sub:T->T->T}
{opp:T->T}
{ring_eq:T->T->Prop}.
#[global]
Instance zero_notation(T:Type)`{Ring_ops T}:Zero T:= ring0.
#[global]
Instance one_notation(T:Type)`{Ring_ops T}:One T:= ring1.
#[global]
Instance add_notation(T:Type)`{Ring_ops T}:Addition T:= add.
#[global]
Instance mul_notation(T:Type)`{Ring_ops T}:@Multiplication T T:= mul.
#[global]
Instance sub_notation(T:Type)`{Ring_ops T}:Subtraction T:= sub.
#[global]
Instance opp_notation(T:Type)`{Ring_ops T}:Opposite T:= opp.
#[global]
Instance eq_notation(T:Type)`{Ring_ops T}:@Equality T:= ring_eq.
Class Ring `{Ro:Ring_ops}:={
ring_setoid: Equivalence _==_;
ring_plus_comp: Proper (_==_ ==> _==_ ==>_==_) _+_;
ring_mult_comp: Proper (_==_ ==> _==_ ==>_==_) _*_;
ring_sub_comp: Proper (_==_ ==> _==_ ==>_==_) _-_;
ring_opp_comp: Proper (_==_==>_==_) -_;
ring_add_0_l    : forall x, 0 + x == x;
ring_add_comm   : forall x y, x + y == y + x;
ring_add_assoc  : forall x y z, x + (y + z) == (x + y) + z;
ring_mul_1_l    : forall x, 1 * x == x;
ring_mul_1_r    : forall x, x * 1 == x;
ring_mul_assoc  : forall x y z, x * (y * z) == (x * y) * z;
ring_distr_l    : forall x y z, (x + y) * z == x * z + y * z;
ring_distr_r    : forall x y z, z * ( x + y) == z * x + z * y;
ring_sub_def    : forall x y, x - y == x + -y;
ring_opp_def    : forall x, x + -x == 0
}.
(* inutile! je sais plus pourquoi j'ai mis ca... Instance ring_Ring_ops(R:Type)`{Ring R} :@Ring_ops R 0 1 addition multiplication subtraction opposite equality. *)
#[global]
Existing Instance ring_setoid.
#[global]
Existing Instance ring_plus_comp.
#[global]
Existing Instance ring_mult_comp.
#[global]
Existing Instance ring_sub_comp.
#[global]
Existing Instance ring_opp_comp.
Section Ring_power.
Context {R:Type}`{Ring R}.
Fixpoint pow_pos (x:R) (i:positive) {struct i}: R :=
match i with
| xH => x
| xO i => let p := pow_pos x i in p * p
| xI i => let p := pow_pos x i in x * (p * p)
end.
Definition pow_N (x:R) (p:N) :=
match p with
| N0 => 1
| Npos p => pow_pos x p
end.
End Ring_power.
Definition ZN(x:Z):=
match x with
Z0 => N0
|Zpos p | Zneg p => Npos p
end.
#[global]
Instance power_ring {R:Type}`{Ring R} : Power:=
{power x y := pow_N x (ZN y)}.
(** Interpretation morphisms definition*)
Class Ring_morphism (C R:Type)`{Cr:Ring C} `{Rr:Ring R}`{Rh:Bracket C R}:= {
ring_morphism0    : [0] == 0;
ring_morphism1    : [1] == 1;
ring_morphism_add : forall x y, [x + y] == [x] + [y];
ring_morphism_sub : forall x y, [x - y] == [x] - [y];
ring_morphism_mul : forall x y, [x * y] == [x] * [y];
ring_morphism_opp : forall  x, [-x] == -[x];
ring_morphism_eq  : forall x y, x == y -> [x] == [y]}.
Section Ring.
Context {R:Type}`{Rr:Ring R}.
(* Powers *)

Lemma pow_pos_add : forall  x i j,
pow_pos x (i + j) == pow_pos x i * pow_pos x j.