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From Corelib Require Import ssreflect ssrfun.
From HB Require Import structures.
(**************************************************************************)
(* Stage 2: AddComoid -> +AddAG+ -> Ring *)
(**************************************************************************)
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;
mulr1 : left_id one mul;
mul1r : right_id one mul;
mulrDl : left_distributive mul add;
mulrDr : right_distributive mul add;
}.
HB.structure Definition Ring := { A of Ring_of_AddAG A }.
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.
#[verbose]
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.
(* 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.
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