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From Corelib Require Import ssreflect ssrfun.
From HB Require Import structures.
HB.mixin Record AddComoid_of_TYPE A := {
zero : A;
add : A -> A -> A;
addrA : associative add;
addrC : commutative add;
add0r : left_id zero add;
}.
HB.structure Definition AddComoid := { A of AddComoid_of_TYPE A }.
(* Begin change *)
HB.mixin Record AddAG_of_AddComoid A of AddComoid A := {
opp : A -> A;
addNr : left_inverse zero opp add;
}.
HB.factory Record AddAG_of_TYPE A := {
zero : A;
add : A -> A -> A;
opp : A -> A;
addrA : associative add;
addrC : commutative add;
add0r : left_id zero add;
addNr : left_inverse zero opp add;
}.
HB.builders Context A (a : AddAG_of_TYPE A).
HB.instance
Definition to_AddComoid_of_TYPE :=
AddComoid_of_TYPE.Build A zero add addrA addrC add0r.
HB.instance
Definition to_AddAG_of_AddComoid :=
AddAG_of_AddComoid.Build A _ addNr.
HB.end.
HB.structure Definition AddAG := { A of AddAG_of_TYPE A }.
HB.mixin Record Ring_of_AddAG A of AddAG A := {
one : A;
mul : A -> A -> A;
mulrA : associative mul;
mul1r : left_id one mul;
mulr1 : right_id one mul;
mulrDl : left_distributive mul add;
mulrDr : right_distributive mul add;
}.
HB.factory Record Ring_of_AddComoid A of AddComoid A := {
opp : A -> A;
one : A;
mul : A -> A -> A;
addNr : left_inverse zero opp add;
mulrA : associative mul;
mul1r : left_id one mul;
mulr1 : right_id one mul;
mulrDl : left_distributive mul add;
mulrDr : right_distributive mul add;
}.
HB.builders Context A (a : Ring_of_AddComoid A).
HB.instance
Definition to_AddAG_of_AddComoid := AddAG_of_AddComoid.Build A _ addNr.
HB.instance
Definition to_Ring_of_AddAG := Ring_of_AddAG.Build A
_ _ mulrA mul1r mulr1 mulrDl mulrDr.
HB.end.
(* End change *)
HB.factory Record Ring_of_TYPE A := {
zero : A;
one : A;
add : A -> A -> A;
opp : A -> A;
mul : A -> A -> A;
addrA : associative add;
addrC : commutative add;
add0r : left_id zero add;
addNr : left_inverse zero opp add;
mulrA : associative mul;
mul1r : left_id one mul;
mulr1 : right_id one mul;
mulrDl : left_distributive mul add;
mulrDr : right_distributive mul add;
}.
HB.builders Context A (a : Ring_of_TYPE A).
HB.instance
Definition to_AddComoid_of_TYPE := AddComoid_of_TYPE.Build A
zero add addrA addrC add0r.
HB.instance
Definition to_Ring_of_AddComoid := Ring_of_AddComoid.Build A
_ _ _ addNr mulrA mul1r mulr1 mulrDl mulrDr.
HB.end.
HB.structure Definition Ring := { A of Ring_of_TYPE A }.
(* Notations *)
Declare Scope hb_scope.
Delimit Scope hb_scope with G.
Local Open Scope hb_scope.
Notation "0" := zero : hb_scope.
Notation "1" := one : hb_scope.
Infix "+" := (@add _) : hb_scope.
Notation "- x" := (@opp _ x) : hb_scope.
Infix "*" := (@mul _) : hb_scope.
Notation "x - y" := (x + - y) : hb_scope.
(* Theory *)
Section Theory.
Variable R : Ring.type.
Implicit Type (x : R).
Lemma addr0 : right_id (@zero R) add.
Proof. by move=> x; rewrite addrC add0r. Qed.
Lemma addrN : right_inverse (@zero R) opp add.
Proof. by move=> x; rewrite addrC addNr. Qed.
Lemma subrr x : x - x = 0.
Proof. by rewrite addrN. Qed.
Lemma addrNK x y : x + y - y = x.
Proof. by rewrite -addrA subrr addr0. Qed.
End Theory.
HB.mixin Record LModule_of_AG (R : Ring.type) (M : Type) of AddAG M := {
scale : Ring.sort R -> M -> M; (* TODO: insert coercions automatically *)
scaleDl : forall v, {morph scale^~ v: a b / a + b};
scaleDr : right_distributive scale add;
scaleA : forall a b v, scale a (scale b v) = scale (a * b) v;
scale1r : forall m, scale 1 m = m;
}.
HB.structure Definition LModule (R : Ring.type) :=
{ M of LModule_of_AG R M & }.
Infix "*:" := (@scale _ _) (at level 30) : hb_scope.
Definition regular (R : Type) := R.
HB.instance Definition regular_AG (R : AddAG.type) :=
AddAG_of_TYPE.Build (regular (AddAG.sort R)) zero add opp addrA addrC add0r addNr.
HB.instance Definition regular_LModule (R : Ring.type) :=
LModule_of_AG.Build R (regular (Ring.sort R)) mul
(fun _ _ _ => mulrDl _ _ _) mulrDr mulrA mul1r.
From Corelib Require Import BinNums IntDef.
Axiom Z_add_assoc : forall x y z, Z.add x (Z.add y z) = Z.add (Z.add x y) z.
Axiom Z_add_comm : forall x y, Z.add x y = Z.add y x.
Axiom Z_add_0_l : forall x, Z.add Z0 x = x.
Axiom Z_add_0_r : forall x, Z.add x Z0 = x.
Axiom Z_add_opp_diag_l : forall x, Z.add (Z.opp x) x = Z0.
Axiom Z_mul_add_distr_l :
forall x y z, Z.mul x (Z.add y z) = Z.add (Z.mul x y) (Z.mul x z).
Axiom Z_mul_add_distr_r :
forall x y z, Z.mul (Z.add x y) z = Z.add (Z.mul x z) (Z.mul y z).
Axiom Z_mul_assoc : forall x y z, Z.mul x (Z.mul y z) = Z.mul (Z.mul x y) z.
Axiom Z_mul_1_l : forall x, Z.mul (Zpos xH) x = x.
Axiom Z_mul_1_r : forall x, Z.mul x (Zpos xH) = x.
HB.instance Definition Z_ring_axioms :=
Ring_of_TYPE.Build Z Z0 (Zpos xH) Z.add Z.opp Z.mul
Z_add_assoc Z_add_comm Z_add_0_l Z_add_opp_diag_l
Z_mul_assoc Z_mul_1_l Z_mul_1_r
Z_mul_add_distr_r Z_mul_add_distr_l.
Lemma test (m : Z) (n : regular Z) : m *: n = m * n.
Proof. by []. Qed.
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