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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) *)
(************************************************************************)
(* non commutative rings *)
Require Import Setoid.
Require Import BinPos.
Require Import BinNat.
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}.
Instance zero_notation(T:Type)`{Ring_ops T}:Zero T:= ring0.
Instance one_notation(T:Type)`{Ring_ops T}:One T:= ring1.
Instance add_notation(T:Type)`{Ring_ops T}:Addition T:= add.
Instance mul_notation(T:Type)`{Ring_ops T}:@Multiplication T T:= mul.
Instance sub_notation(T:Type)`{Ring_ops T}:Subtraction T:= sub.
Instance opp_notation(T:Type)`{Ring_ops T}:Opposite T:= opp.
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.
*)
Existing Instance ring_setoid.
Existing Instance ring_plus_comp.
Existing Instance ring_mult_comp.
Existing Instance ring_sub_comp.
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.
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_comm : forall x j, x * pow_pos x j == pow_pos x j * x.
Proof.
induction j; simpl. rewrite <- ring_mul_assoc.
rewrite <- ring_mul_assoc.
rewrite <- IHj. rewrite (ring_mul_assoc (pow_pos x j) x (pow_pos x j)).
rewrite <- IHj. rewrite <- ring_mul_assoc. reflexivity.
rewrite <- ring_mul_assoc. rewrite <- IHj.
rewrite ring_mul_assoc. rewrite IHj.
rewrite <- ring_mul_assoc. rewrite IHj. reflexivity. reflexivity.
Qed.
Lemma pow_pos_succ : forall x j, pow_pos x (Pos.succ j) == x * pow_pos x j.
Proof.
induction j; simpl.
rewrite IHj.
rewrite <- (ring_mul_assoc x (pow_pos x j) (x * pow_pos x j)).
rewrite (ring_mul_assoc (pow_pos x j) x (pow_pos x j)).
rewrite <- pow_pos_comm.
rewrite <- ring_mul_assoc. reflexivity.
reflexivity. reflexivity.
Qed.
Lemma pow_pos_add : forall x i j,
pow_pos x (i + j) == pow_pos x i * pow_pos x j.
Proof.
intro x;induction i;intros.
rewrite Pos.xI_succ_xO;rewrite <- Pos.add_1_r.
rewrite <- Pos.add_diag;repeat rewrite <- Pos.add_assoc.
repeat rewrite IHi.
rewrite Pos.add_comm;rewrite Pos.add_1_r;
rewrite pow_pos_succ.
simpl;repeat rewrite ring_mul_assoc. reflexivity.
rewrite <- Pos.add_diag;repeat rewrite <- Pos.add_assoc.
repeat rewrite IHi. rewrite ring_mul_assoc. reflexivity.
rewrite Pos.add_comm;rewrite Pos.add_1_r;rewrite pow_pos_succ.
simpl. reflexivity.
Qed.
Definition id_phi_N (x:N) : N := x.
Lemma pow_N_pow_N : forall x n, pow_N x (id_phi_N n) == pow_N x n.
Proof.
intros; reflexivity.
Qed.
(** Identity is a morphism *)
(*
Instance IDmorph : Ring_morphism _ _ _ (fun x => x).
Proof.
apply (Build_Ring_morphism H6 H6 (fun x => x));intros;
try reflexivity. trivial.
Qed.
*)
(** rings are almost rings*)
Lemma ring_mul_0_l : forall x, 0 * x == 0.
Proof.
intro x. setoid_replace (0*x) with ((0+1)*x + -x).
rewrite ring_add_0_l. rewrite ring_mul_1_l .
rewrite ring_opp_def . fold zero. reflexivity.
rewrite ring_distr_l . rewrite ring_mul_1_l .
rewrite <- ring_add_assoc ; rewrite ring_opp_def .
rewrite ring_add_comm ; rewrite ring_add_0_l ;reflexivity.
Qed.
Lemma ring_mul_0_r : forall x, x * 0 == 0.
Proof.
intro x; setoid_replace (x*0) with (x*(0+1) + -x).
rewrite ring_add_0_l ; rewrite ring_mul_1_r .
rewrite ring_opp_def ; fold zero; reflexivity.
rewrite ring_distr_r ;rewrite ring_mul_1_r .
rewrite <- ring_add_assoc ; rewrite ring_opp_def .
rewrite ring_add_comm ; rewrite ring_add_0_l ;reflexivity.
Qed.
Lemma ring_opp_mul_l : forall x y, -(x * y) == -x * y.
Proof.
intros x y;rewrite <- (ring_add_0_l (- x * y)).
rewrite ring_add_comm .
rewrite <- (ring_opp_def (x*y)).
rewrite ring_add_assoc .
rewrite <- ring_distr_l.
rewrite (ring_add_comm (-x));rewrite ring_opp_def .
rewrite ring_mul_0_l;rewrite ring_add_0_l ;reflexivity.
Qed.
Lemma ring_opp_mul_r : forall x y, -(x * y) == x * -y.
Proof.
intros x y;rewrite <- (ring_add_0_l (x * - y)).
rewrite ring_add_comm .
rewrite <- (ring_opp_def (x*y)).
rewrite ring_add_assoc .
rewrite <- ring_distr_r .
rewrite (ring_add_comm (-y));rewrite ring_opp_def .
rewrite ring_mul_0_r;rewrite ring_add_0_l ;reflexivity.
Qed.
Lemma ring_opp_add : forall x y, -(x + y) == -x + -y.
Proof.
intros x y;rewrite <- (ring_add_0_l (-(x+y))).
rewrite <- (ring_opp_def x).
rewrite <- (ring_add_0_l (x + - x + - (x + y))).
rewrite <- (ring_opp_def y).
rewrite (ring_add_comm x).
rewrite (ring_add_comm y).
rewrite <- (ring_add_assoc (-y)).
rewrite <- (ring_add_assoc (- x)).
rewrite (ring_add_assoc y).
rewrite (ring_add_comm y).
rewrite <- (ring_add_assoc (- x)).
rewrite (ring_add_assoc y).
rewrite (ring_add_comm y);rewrite ring_opp_def .
rewrite (ring_add_comm (-x) 0);rewrite ring_add_0_l .
rewrite ring_add_comm; reflexivity.
Qed.
Lemma ring_opp_opp : forall x, - -x == x.
Proof.
intros x; rewrite <- (ring_add_0_l (- -x)).
rewrite <- (ring_opp_def x).
rewrite <- ring_add_assoc ; rewrite ring_opp_def .
rewrite (ring_add_comm x); rewrite ring_add_0_l . reflexivity.
Qed.
Lemma ring_sub_ext :
forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 - y1 == x2 - y2.
Proof.
intros.
setoid_replace (x1 - y1) with (x1 + -y1).
setoid_replace (x2 - y2) with (x2 + -y2).
rewrite H;rewrite H0;reflexivity.
rewrite ring_sub_def. reflexivity.
rewrite ring_sub_def. reflexivity.
Qed.
Ltac mrewrite :=
repeat first
[ rewrite ring_add_0_l
| rewrite <- (ring_add_comm 0)
| rewrite ring_mul_1_l
| rewrite ring_mul_0_l
| rewrite ring_distr_l
| reflexivity
].
Lemma ring_add_0_r : forall x, (x + 0) == x.
Proof. intros; mrewrite. Qed.
Lemma ring_add_assoc1 : forall x y z, (x + y) + z == (y + z) + x.
Proof.
intros;rewrite <- (ring_add_assoc x).
rewrite (ring_add_comm x);reflexivity.
Qed.
Lemma ring_add_assoc2 : forall x y z, (y + x) + z == (y + z) + x.
Proof.
intros; repeat rewrite <- ring_add_assoc.
rewrite (ring_add_comm x); reflexivity.
Qed.
Lemma ring_opp_zero : -0 == 0.
Proof.
rewrite <- (ring_mul_0_r 0). rewrite ring_opp_mul_l.
repeat rewrite ring_mul_0_r. reflexivity.
Qed.
End Ring.
(** Some simplification tactics*)
Ltac gen_reflexivity := reflexivity.
Ltac gen_rewrite :=
repeat first
[ reflexivity
| progress rewrite ring_opp_zero
| rewrite ring_add_0_l
| rewrite ring_add_0_r
| rewrite ring_mul_1_l
| rewrite ring_mul_1_r
| rewrite ring_mul_0_l
| rewrite ring_mul_0_r
| rewrite ring_distr_l
| rewrite ring_distr_r
| rewrite ring_add_assoc
| rewrite ring_mul_assoc
| progress rewrite ring_opp_add
| progress rewrite ring_sub_def
| progress rewrite <- ring_opp_mul_l
| progress rewrite <- ring_opp_mul_r ].
Ltac gen_add_push x :=
repeat (match goal with
| |- context [(?y + x) + ?z] =>
progress rewrite (ring_add_assoc2 x y z)
| |- context [(x + ?y) + ?z] =>
progress rewrite (ring_add_assoc1 x y z)
end).
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