# Library Coq.setoid_ring.Ring_theory

Require Import Setoid.
Require Import BinPos.
Require Import BinNat.

Module RingSyntax.
Reserved Notation "x ?=! y" (at level 70, no associativity).
Reserved Notation "x +! y " (at level 50, left associativity).
Reserved Notation "x -! y" (at level 50, left associativity).
Reserved Notation "x *! y" (at level 40, left associativity).
Reserved Notation "-! x" (at level 35, right associativity).

Reserved Notation "[ x ]" (at level 0).

Reserved Notation "x ?== y" (at level 70, no associativity).
Reserved Notation "x -- y" (at level 50, left associativity).
Reserved Notation "x ** y" (at level 40, left associativity).
Reserved Notation "-- x" (at level 35, right associativity).

Reserved Notation "x == y" (at level 70, no associativity).
End RingSyntax.
Import RingSyntax.

Section Power.
Variable R:Type.
Variable rI : R.
Variable rmul : R -> R -> R.
Variable req : R -> R -> Prop.
Variable Rsth : Setoid_Theory R req.
Notation "x * y " := (rmul x y).
Notation "x == y" := (req x y).

Hypothesis mul_ext :
forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 * y1 == x2 * y2.
Hypothesis mul_comm : forall x y, x * y == y * x.
Hypothesis mul_assoc : forall x y z, x * (y * z) == (x * y) * z.
Add Setoid R req Rsth as R_set_Power.

Fixpoint pow_pos (x:R) (i:positive) {struct i}: R :=
match i with
| xH => x
| xO i => let p := pow_pos x i in rmul p p
| xI i => let p := pow_pos x i in rmul x (rmul p p)
end.

Lemma pow_pos_Psucc : forall x j, pow_pos x (Psucc j) == x * pow_pos x j.

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

Definition pow_N (x:R) (p:N) :=
match p with
| N0 => rI
| Npos p => pow_pos x p
end.

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.

End Power.

Section DEFINITIONS.
Variable R : Type.
Variable (rO rI : R) (radd rmul rsub: R->R->R) (ropp : R -> R).
Variable req : R -> R -> Prop.
Notation "0" := rO. Notation "1" := rI.
Notation "x + y" := (radd x y). Notation "x * y " := (rmul x y).
Notation "x - y " := (rsub x y). Notation "- x" := (ropp x).
Notation "x == y" := (req x y).

Semi Ring
Record semi_ring_theory : Prop := mk_srt {
SRadd_0_l : forall n, 0 + n == n;
SRadd_comm : forall n m, n + m == m + n ;
SRadd_assoc : forall n m p, n + (m + p) == (n + m) + p;
SRmul_1_l : forall n, 1*n == n;
SRmul_0_l : forall n, 0*n == 0;
SRmul_comm : forall n m, n*m == m*n;
SRmul_assoc : forall n m p, n*(m*p) == (n*m)*p;
SRdistr_l : forall n m p, (n + m)*p == n*p + m*p
}.

Almost Ring
Record almost_ring_theory : Prop := mk_art {
ARadd_0_l : forall x, 0 + x == x;
ARadd_comm : forall x y, x + y == y + x;
ARadd_assoc : forall x y z, x + (y + z) == (x + y) + z;
ARmul_1_l : forall x, 1 * x == x;
ARmul_0_l : forall x, 0 * x == 0;
ARmul_comm : forall x y, x * y == y * x;
ARmul_assoc : forall x y z, x * (y * z) == (x * y) * z;
ARdistr_l : forall x y z, (x + y) * z == (x * z) + (y * z);
ARopp_mul_l : forall x y, -(x * y) == -x * y;
ARopp_add : forall x y, -(x + y) == -x + -y;
ARsub_def : forall x y, x - y == x + -y
}.

Ring
Record ring_theory : Prop := mk_rt {
Radd_0_l : forall x, 0 + x == x;
Radd_comm : forall x y, x + y == y + x;
Radd_assoc : forall x y z, x + (y + z) == (x + y) + z;
Rmul_1_l : forall x, 1 * x == x;
Rmul_comm : forall x y, x * y == y * x;
Rmul_assoc : forall x y z, x * (y * z) == (x * y) * z;
Rdistr_l : forall x y z, (x + y) * z == (x * z) + (y * z);
Rsub_def : forall x y, x - y == x + -y;
Ropp_def : forall x, x + (- x) == 0
}.

Equality is extensional

Record sring_eq_ext : Prop := mk_seqe {

forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 + y1 == x2 + y2;
SRmul_ext :
forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 * y1 == x2 * y2
}.

Record ring_eq_ext : Prop := mk_reqe {

forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 + y1 == x2 + y2;
Rmul_ext :
forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 * y1 == x2 * y2;
Ropp_ext : forall x1 x2, x1 == x2 -> -x1 == -x2
}.

Interpretation morphisms definition
Section MORPHISM.
Variable C:Type.
Variable (cO cI : C) (cadd cmul csub : C->C->C) (copp : C->C).
Variable ceqb : C->C->bool.
Variable phi : C -> R.
Notation "x +! y" := (cadd x y). Notation "x -! y " := (csub x y).
Notation "x *! y " := (cmul x y). Notation "-! x" := (copp x).
Notation "x ?=! y" := (ceqb x y). Notation "[ x ]" := (phi x).

Record semi_morph : Prop := mkRmorph {
Smorph0 : [cO] == 0;
Smorph1 : [cI] == 1;
Smorph_add : forall x y, [x +! y] == [x]+[y];
Smorph_mul : forall x y, [x *! y] == [x]*[y];
Smorph_eq : forall x y, x?=!y = true -> [x] == [y]
}.

Record ring_morph : Prop := mkmorph {
morph0 : [cO] == 0;
morph1 : [cI] == 1;
morph_add : forall x y, [x +! y] == [x]+[y];
morph_sub : forall x y, [x -! y] == [x]-[y];
morph_mul : forall x y, [x *! y] == [x]*[y];
morph_opp : forall x, [-!x] == -[x];
morph_eq : forall x y, x?=!y = true -> [x] == [y]
}.

Section SIGN.
Variable get_sign : C -> option C.
Record sign_theory : Prop := mksign_th {
sign_spec : forall c c', get_sign c = Some c' -> c ?=! -! c' = true
}.
End SIGN.

Definition get_sign_None (c:C) := @None C.

Lemma get_sign_None_th : sign_theory get_sign_None.

Section DIV.
Variable cdiv: C -> C -> C*C.
Record div_theory : Prop := mkdiv_th {
div_eucl_th : forall a b, let (q,r) := cdiv a b in [a] == [b *! q +! r]
}.
End DIV.

End MORPHISM.

Identity is a morphism
Variable Rsth : Setoid_Theory R req.
Add Setoid R req Rsth as R_setoid1.
Variable reqb : R->R->bool.
Hypothesis morph_req : forall x y, (reqb x y) = true -> x == y.
Definition IDphi (x:R) := x.
Lemma IDmorph : ring_morph rO rI radd rmul rsub ropp reqb IDphi.

Specification of the power function
Section POWER.
Variable Cpow : Set.
Variable Cp_phi : N -> Cpow.
Variable rpow : R -> Cpow -> R.

Record power_theory : Prop := mkpow_th {
rpow_pow_N : forall r n, req (rpow r (Cp_phi n)) (pow_N rI rmul r n)
}.

End POWER.

Definition pow_N_th := mkpow_th id_phi_N (pow_N rI rmul) (pow_N_pow_N rI rmul Rsth).

End DEFINITIONS.

Section ALMOST_RING.
Variable R : Type.
Variable (rO rI : R) (radd rmul rsub: R->R->R) (ropp : R -> R).
Variable req : R -> R -> Prop.
Notation "0" := rO. Notation "1" := rI.
Notation "x + y" := (radd x y). Notation "x * y " := (rmul x y).
Notation "x - y " := (rsub x y). Notation "- x" := (ropp x).
Notation "x == y" := (req x y).

Leibniz equality leads to a setoid theory and is extensional
Lemma Eqsth : Setoid_Theory R (@eq R).

Lemma Eq_s_ext : sring_eq_ext radd rmul (@eq R).

Lemma Eq_ext : ring_eq_ext radd rmul ropp (@eq R).

Variable Rsth : Setoid_Theory R req.
Add Setoid R req Rsth as R_setoid2.
Ltac sreflexivity := apply (Seq_refl _ _ Rsth).

Section SEMI_RING.
Variable SReqe : sring_eq_ext radd rmul req.
Variable SRth : semi_ring_theory 0 1 radd rmul req.

Every semi ring can be seen as an almost ring, by taking :
• x = x and x - y = x + y
Definition SRopp (x:R) := x. Notation "- x" := (SRopp x).

Definition SRsub x y := x + -y. Notation "x - y " := (SRsub x y).

Lemma SRopp_ext : forall x y, x == y -> -x == -y.

Lemma SReqe_Reqe : ring_eq_ext radd rmul SRopp req.

Lemma SRopp_mul_l : forall x y, -(x * y) == -x * y.

Lemma SRopp_add : forall x y, -(x + y) == -x + -y.

Lemma SRsub_def : forall x y, x - y == x + -y.

Lemma SRth_ARth : almost_ring_theory 0 1 radd rmul SRsub SRopp req.
Identity morphism for semi-ring equipped with their almost-ring structure
Variable reqb : R->R->bool.

Hypothesis morph_req : forall x y, (reqb x y) = true -> x == y.

Definition SRIDmorph : ring_morph 0 1 radd rmul SRsub SRopp req
0 1 radd rmul SRsub SRopp reqb (@IDphi R).

Variable C : Type.
Variable (cO cI : C) (cadd cmul: C->C->C).
Variable (ceqb : C -> C -> bool).
Variable phi : C -> R.
Variable Smorph : semi_morph rO rI radd rmul req cO cI cadd cmul ceqb phi.

Lemma SRmorph_Rmorph :
ring_morph rO rI radd rmul SRsub SRopp req

End SEMI_RING.

Variable Reqe : ring_eq_ext radd rmul ropp req.

Section RING.
Variable Rth : ring_theory 0 1 radd rmul rsub ropp req.

Rings are almost rings
Lemma Rmul_0_l : forall x, 0 * x == 0.

Lemma Ropp_mul_l : forall x y, -(x * y) == -x * y.

Lemma Ropp_add : forall x y, -(x + y) == -x + -y.

Lemma Ropp_opp : forall x, - -x == x.

Lemma Rth_ARth : almost_ring_theory 0 1 radd rmul rsub ropp req.
Every semi morphism between two rings is a morphism
Variable C : Type.
Variable (cO cI : C) (cadd cmul csub: C->C->C) (copp : C -> C).
Variable (ceq : C -> C -> Prop) (ceqb : C -> C -> bool).
Variable phi : C -> R.
Notation "x +! y" := (cadd x y). Notation "x *! y " := (cmul x y).
Notation "x -! y " := (csub x y). Notation "-! x" := (copp x).
Notation "x ?=! y" := (ceqb x y). Notation "[ x ]" := (phi x).
Variable Csth : Setoid_Theory C ceq.
Variable Ceqe : ring_eq_ext cadd cmul copp ceq.
Add Setoid C ceq Csth as C_setoid.
Variable Cth : ring_theory cO cI cadd cmul csub copp ceq.
Variable Smorph : semi_morph 0 1 radd rmul req cO cI cadd cmul ceqb phi.
Variable phi_ext : forall x y, ceq x y -> [x] == [y].
Lemma Smorph_opp : forall x, [-!x] == -[x].

Lemma Smorph_sub : forall x y, [x -! y] == [x] - [y].

Lemma Smorph_morph : ring_morph 0 1 radd rmul rsub ropp req
cO cI cadd cmul csub copp ceqb phi.
End RING.

Useful lemmas on almost ring
Variable ARth : almost_ring_theory 0 1 radd rmul rsub ropp req.

Lemma ARth_SRth : semi_ring_theory 0 1 radd rmul req.

Lemma ARsub_ext :
forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 - y1 == x2 - y2.

Ltac mrewrite :=
repeat first
| rewrite <- ((ARadd_comm ARth) 0)
| rewrite (ARmul_1_l ARth)
| rewrite <- ((ARmul_comm ARth) 1)
| rewrite (ARmul_0_l ARth)
| rewrite <- ((ARmul_comm ARth) 0)
| rewrite (ARdistr_l ARth)
| sreflexivity
| match goal with
| |- context [?z * (?x + ?y)] => rewrite ((ARmul_comm ARth) z (x+y))
end].

Lemma ARadd_0_r : forall x, (x + 0) == x.

Lemma ARmul_1_r : forall x, x * 1 == x.

Lemma ARmul_0_r : forall x, x * 0 == 0.

Lemma ARdistr_r : forall x y z, z * (x + y) == z*x + z*y.

Lemma ARadd_assoc1 : forall x y z, (x + y) + z == (y + z) + x.

Lemma ARadd_assoc2 : forall x y z, (y + x) + z == (y + z) + x.

Lemma ARmul_assoc1 : forall x y z, (x * y) * z == (y * z) * x.

Lemma ARmul_assoc2 : forall x y z, (y * x) * z == (y * z) * x.

Lemma ARopp_mul_r : forall x y, - (x * y) == x * -y.

Lemma ARopp_zero : -0 == 0.

End ALMOST_RING.

Inductive ring_kind : Type :=
| Abstract
| Computational
(R:Type)
(req : R -> R -> Prop)
(reqb : R -> R -> bool)
(_ : forall x y, (reqb x y) = true -> req x y)
| Morphism
(R : Type)
(rO rI : R) (radd rmul rsub: R->R->R) (ropp : R -> R)
(req : R -> R -> Prop)
(C : Type)
(cO cI : C) (cadd cmul csub : C->C->C) (copp : C->C)
(ceqb : C->C->bool)
phi
(_ : ring_morph rO rI radd rmul rsub ropp req
cO cI cadd cmul csub copp ceqb phi).

Some simplification tactics
Ltac gen_reflexivity Rsth := apply (Seq_refl _ _ Rsth).

Ltac gen_srewrite Rsth Reqe ARth :=
repeat first
[ gen_reflexivity Rsth
| progress rewrite (ARopp_zero Rsth Reqe ARth)
| rewrite (ARmul_1_l ARth)
| rewrite (ARmul_1_r Rsth ARth)
| rewrite (ARmul_0_l ARth)
| rewrite (ARmul_0_r Rsth ARth)
| rewrite (ARdistr_l ARth)
| rewrite (ARdistr_r Rsth Reqe ARth)
| rewrite (ARmul_assoc ARth)
| progress rewrite (ARsub_def ARth)
| progress rewrite <- (ARopp_mul_l ARth)
| progress rewrite <- (ARopp_mul_r Rsth Reqe ARth) ].

repeat (match goal with