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From Coq Require Import ssreflect ssrfun.
From HB Require Import structures.
HB.mixin Record MulMonoid_of_Type A := {
one : A;
mul : A -> A -> A;
mulrA : associative mul;
mul1r : left_id one mul;
mulr1 : right_id one mul;
}.
HB.structure Definition MulMonoid := { A of MulMonoid_of_Type A }.
HB.mixin Record AddMonoid_of_Type A := {
zero : A;
add : A -> A -> A;
addrA : associative add;
add0r : left_id zero add;
addr0 : right_id zero add;
}.
HB.structure Definition AddMonoid := { A of AddMonoid_of_Type A }.
HB.mixin Record AbGroup_of_AddMonoid A of AddMonoid A := {
opp : A -> A;
addrC : commutative (add : A -> A -> A);
addNr : left_inverse zero opp add;
}.
HB.structure Definition AbGroup := { A of AddMonoid A & AbGroup_of_AddMonoid A }.
HB.mixin Record Ring_of_AbGroupMulMonoid A of MulMonoid A & AbGroup A := {
mulrDl : left_distributive mul (add : A -> A -> A);
mulrDr : right_distributive mul (add : A -> A -> A);
}.
HB.structure Definition Ring := { A of MulMonoid A & AbGroup A & Ring_of_AbGroupMulMonoid A }.
HB.factory Record Ring_of_AddMulMonoid A of MulMonoid A & AddMonoid A := {
opp : A -> A;
addrC : commutative (add : A -> A -> A);
addNr : left_inverse zero opp add;
mulrDl : left_distributive mul (add : A -> A -> A);
mulrDr : right_distributive mul (add : A -> A -> A);
}.
HB.builders Context A (a : Ring_of_AddMulMonoid A).
HB.instance
Definition to_AbGroup_of_AddMonoid :=
AbGroup_of_AddMonoid.Build A opp addrC addNr.
HB.instance
Definition to_Ring_of_AbGroupMulMonoid :=
Ring_of_AbGroupMulMonoid.Build A mulrDl mulrDr.
HB.end.
HB.factory Record Ring_of_MulMonoid A of MulMonoid A := {
zero : A;
add : A -> A -> A;
addrA : associative add;
add0r : left_id zero add;
addr0 : right_id zero add;
opp : A -> A;
addrC : commutative (add : A -> A -> A);
addNr : left_inverse zero opp add;
mulrDl : left_distributive mul add;
mulrDr : right_distributive mul add;
}.
HB.builders Context A (a : Ring_of_MulMonoid A).
HB.instance
Definition to_AddMonoid_of_Type :=
AddMonoid_of_Type.Build A zero add addrA add0r addr0.
HB.instance
Definition to_AbGroup_of_AddMonoid :=
AbGroup_of_AddMonoid.Build A opp addrC addNr.
HB.instance
Definition to_Ring_of_AddMulMonoid :=
Ring_of_AddMulMonoid.Build A opp addrC addNr mulrDl mulrDr.
HB.end.
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