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CohenCyrilCohenCyril
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moving category theory to theories
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‎theories/algebra.v‎

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Require Import ssreflect ssrfun.
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From HB Require Import structures cat.
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Set Implicit Arguments.
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Unset Strict Implicit.
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Unset Printing Implicit Defensive.
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Add Search Blacklist "__canonical__".
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Local Open Scope algebra_scope.
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Local Open Scope cat_scope.
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HB.mixin Record IsRing A := {
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zero : A;
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one : A;
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add : A -> A -> A;
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opp : A -> A;
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mul : A -> A -> A;
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addrA : associative add;
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addrC : commutative add;
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add0r : left_id zero add;
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addNr : left_inverse zero opp add;
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mulrA : associative mul;
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mul1r : left_id one mul;
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mulr1 : right_id one mul;
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mulrDl : left_distributive mul add;
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mulrDr : right_distributive mul add;
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}.
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#[short(type="ring")]
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HB.structure Definition Ring := { A of IsRing A }.
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Bind Scope algebra_scope with Ring.sort.
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Notation "0" := zero : algebra_scope.
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Notation "1" := one : algebra_scope.
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Infix "+" := (@add _) : algebra_scope.
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Notation "- x" := (@opp _ x) : algebra_scope.
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Infix "*" := (@mul _) : algebra_scope.
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Notation "x - y" := (x + - y) : algebra_scope.
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Lemma addr0 (R : ring) : right_id (@zero R) add.
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Proof. by move=> x; rewrite addrC add0r. Qed.
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Lemma addrN (R : ring) : right_inverse (@zero R) opp add.
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Proof. by move=> x; rewrite addrC addNr. Qed.
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Lemma addKr (R : ring) (x : R) : cancel (add x) (add (- x)).
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Proof. by move=> y; rewrite addrA addNr add0r. Qed.
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Lemma addrI (R : ring) (x : R) : injective (add x).
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Proof. exact: can_inj (addKr _). Qed.
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Lemma opprK (R : ring) : involutive (@opp R).
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Proof. by move=> x; apply: (@addrI _ (- x)); rewrite addNr addrN. Qed.
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HB.mixin Record IsRingHom (A B : ring) (f : A -> B) := {
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hom1_subproof : f 1%A = 1%A;
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homB_subproof : {morph f : x y / x - y};
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homM_subproof : {morph f : x y / (x * y)%A};
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}.
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HB.structure Definition RingHom A B := { f of IsRingHom A B f }.
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Lemma id_IsRingHom A : IsRingHom A A idfun. Proof. by []. Defined.
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HB.instance Definition _ A := id_IsRingHom A.
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Lemma comp_IsRingHom (A B C : ring)
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(f : RingHom.type A B) (g : RingHom.type B C) :
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IsRingHom A C (f \; g :> U).
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Proof.
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by constructor => [|x y|x y];
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rewrite /comp/= ?hom1_subproof ?homB_subproof ?homM_subproof.
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Qed.
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HB.instance Definition _ A B C f g := @comp_IsRingHom A B C f g.
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HB.instance Definition _ := IsQuiver.Build ring RingHom.type.
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HB.instance Definition _ :=
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Quiver_IsPreCat.Build ring (fun _ => idfun) (fun _ _ _ f g => f \; g :> U).
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HB.instance Definition _ := Quiver_IsPreConcrete.Build ring (fun _ _ => id).
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Lemma ring_precat : PreConcrete_IsConcrete ring.
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Proof.
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constructor => A B [f cf] [g cg]//=; rewrite -[_ = _]/(f = g) => fg.
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case: _ / fg in cg *; congr {|RingHom.sort := _ ; RingHom.class := _|}.
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case: cf cg => [[? ? ?]] [[? ? ?]].
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by congr RingHom.Class; congr IsRingHom.phant_Build => //=; apply: Prop_irrelevance.
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Qed.
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HB.instance Definition _ := ring_precat.
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Lemma ring_IsCat : PreCat_IsCat ring.
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Proof.
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by constructor=> [A B f|A B f|A B C D f g h]; exact: concrete_fun_inj.
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Qed.
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HB.instance Definition _ := ring_IsCat.
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Lemma hom1 (R S : ring) (f : R ~> S) : f 1%A = 1%A.
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Proof. exact: hom1_subproof. Qed.
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Lemma homB (R S : ring) (f : R ~> S) : {morph f : x y / x - y}.
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Proof. exact: homB_subproof. Qed.
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Lemma homM (R S : ring) (f : R ~> S) : {morph f : x y / (x * y)%A}.
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Proof. exact: homM_subproof. Qed.
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Lemma hom0 (R S : ring) (f : R ~> S) : f 0%A = 0%A.
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Proof. by rewrite -(addrN 1%A) homB addrN. Qed.
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Lemma homN (R S : ring) (f : R ~> S) : {morph f : x / - x}.
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Proof. by move=> x; rewrite -[- x]add0r homB hom0 add0r. Qed.
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Lemma homD (R S : ring) (f : R ~> S) : {morph f : x y / x + y}.
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Proof. by move=> x y; rewrite -[y]opprK !homB !homN. Qed.
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HB.mixin Record IsIdeal (R : ring) (I : R -> Prop) := {
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ideal0 : I 0;
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idealD : forall x y, I x -> I y -> I (x + y);
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idealM : forall x y, I y -> I (x * y)%A;
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}.
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HB.structure Definition Ideal (R : ring) := { I of IsIdeal R I }.
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HB.mixin Record Ideal_IsPrime (R : ring) (I : R -> Prop) of IsIdeal R I := {
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ideal_prime : forall x y : R, I (x * y)%A -> I x \/ I y
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}.
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#[short(type="spectrum")]
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HB.structure Definition PrimeIdeal (R : ring) :=
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{ I of Ideal_IsPrime R I & Ideal R I }.
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‎examples/cat/cat.v‎ renamed to ‎theories/cat.v‎

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@@ -881,113 +881,4 @@ HB.structure Definition Monoidal : Set :=
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{ C of PreMonoidal_IsMonoidal C & PreMonoidal C }.
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Set Universe Checking.
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HB.mixin Record IsRing A := {
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zero : A;
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one : A;
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add : A -> A -> A;
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opp : A -> A;
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mul : A -> A -> A;
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addrA : associative add;
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addrC : commutative add;
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add0r : left_id zero add;
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addNr : left_inverse zero opp add;
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mulrA : associative mul;
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mul1r : left_id one mul;
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mulr1 : right_id one mul;
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mulrDl : left_distributive mul add;
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mulrDr : right_distributive mul add;
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}.
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#[short(type="ring")]
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HB.structure Definition Ring := { A of IsRing A }.
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Bind Scope algebra_scope with Ring.sort.
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Notation "0" := zero : algebra_scope.
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Notation "1" := one : algebra_scope.
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Infix "+" := (@add _) : algebra_scope.
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Notation "- x" := (@opp _ x) : algebra_scope.
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Infix "*" := (@mul _) : algebra_scope.
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Notation "x - y" := (x + - y) : algebra_scope.
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Lemma addr0 (R : ring) : right_id (@zero R) add.
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Proof. by move=> x; rewrite addrC add0r. Qed.
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Lemma addrN (R : ring) : right_inverse (@zero R) opp add.
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Proof. by move=> x; rewrite addrC addNr. Qed.
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Lemma addKr (R : ring) (x : R) : cancel (add x) (add (- x)).
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Proof. by move=> y; rewrite addrA addNr add0r. Qed.
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Lemma addrI (R : ring) (x : R) : injective (add x).
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Proof. exact: can_inj (addKr _). Qed.
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Lemma opprK (R : ring) : involutive (@opp R).
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Proof. by move=> x; apply: (@addrI _ (- x)); rewrite addNr addrN. Qed.
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HB.mixin Record IsRingHom (A B : ring) (f : A -> B) := {
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hom1_subproof : f 1%A = 1%A;
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homB_subproof : {morph f : x y / x - y};
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homM_subproof : {morph f : x y / (x * y)%A};
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}.
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HB.structure Definition RingHom A B := { f of IsRingHom A B f }.
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Lemma id_IsRingHom A : IsRingHom A A idfun. Proof. by []. Defined.
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HB.instance Definition _ A := id_IsRingHom A.
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Lemma comp_IsRingHom (A B C : ring)
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(f : RingHom.type A B) (g : RingHom.type B C) :
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IsRingHom A C (f \; g :> U).
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Proof.
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by constructor => [|x y|x y];
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rewrite /comp/= ?hom1_subproof ?homB_subproof ?homM_subproof.
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Qed.
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HB.instance Definition _ A B C f g := @comp_IsRingHom A B C f g.
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HB.instance Definition _ := IsQuiver.Build ring RingHom.type.
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HB.instance Definition _ :=
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Quiver_IsPreCat.Build ring (fun _ => idfun) (fun _ _ _ f g => f \; g :> U).
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HB.instance Definition _ := Quiver_IsPreConcrete.Build ring (fun _ _ => id).
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Lemma ring_precat : PreConcrete_IsConcrete ring.
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Proof.
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constructor => A B [f cf] [g cg]//=; rewrite -[_ = _]/(f = g) => fg.
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case: _ / fg in cg *; congr {|RingHom.sort := _ ; RingHom.class := _|}.
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case: cf cg => [[? ? ?]] [[? ? ?]].
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by congr RingHom.Class; congr IsRingHom.phant_Build => //=; apply: Prop_irrelevance.
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Qed.
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HB.instance Definition _ := ring_precat.
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Lemma ring_IsCat : PreCat_IsCat ring.
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Proof.
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by constructor=> [A B f|A B f|A B C D f g h]; exact: concrete_fun_inj.
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Qed.
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HB.instance Definition _ := ring_IsCat.
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Lemma hom1 (R S : ring) (f : R ~> S) : f 1%A = 1%A.
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Proof. exact: hom1_subproof. Qed.
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Lemma homB (R S : ring) (f : R ~> S) : {morph f : x y / x - y}.
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Proof. exact: homB_subproof. Qed.
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Lemma homM (R S : ring) (f : R ~> S) : {morph f : x y / (x * y)%A}.
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Proof. exact: homM_subproof. Qed.
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Lemma hom0 (R S : ring) (f : R ~> S) : f 0%A = 0%A.
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Proof. by rewrite -(addrN 1%A) homB addrN. Qed.
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Lemma homN (R S : ring) (f : R ~> S) : {morph f : x / - x}.
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Proof. by move=> x; rewrite -[- x]add0r homB hom0 add0r. Qed.
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Lemma homD (R S : ring) (f : R ~> S) : {morph f : x y / x + y}.
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Proof. by move=> x y; rewrite -[y]opprK !homB !homN. Qed.
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HB.mixin Record IsIdeal (R : ring) (I : R -> Prop) := {
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ideal0 : I 0;
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idealD : forall x y, I x -> I y -> I (x + y);
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idealM : forall x y, I y -> I (x * y)%A;
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}.
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HB.structure Definition Ideal (R : ring) := { I of IsIdeal R I }.
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HB.mixin Record Ideal_IsPrime (R : ring) (I : R -> Prop) of IsIdeal R I := {
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ideal_prime : forall x y : R, I (x * y)%A -> I x \/ I y
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}.
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#[short(type="spectrum")]
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HB.structure Definition PrimeIdeal (R : ring) :=
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{ I of Ideal_IsPrime R I & Ideal R I }.
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