Datasets:

phanerozoic

/

Coq-HoTT

like 2

{M : Monoid} : MonoidHomomorphism M M := Build_MonoidHomomorphism idmap _.

Definition

Require Import Basics.Overture Basics.Tactics Basics.Equivalences Basics.Trunc. Require Import Types.Sigma Types.Forall Types.Prod. Require Import WildCat.Core WildCat.Induced WildCat.Equiv WildCat.Universe Require Import (notations) Classes.interfaces.canonical_names.

Algebra\Monoids\Monoid.v

mnd_homo_id

{M N P : Monoid} : MonoidHomomorphism N P -> MonoidHomomorphism M N -> MonoidHomomorphism M P := fun f g => Build_MonoidHomomorphism (f o g) _.

Definition

Require Import Basics.Overture Basics.Tactics Basics.Equivalences Basics.Trunc. Require Import Types.Sigma Types.Forall Types.Prod. Require Import WildCat.Core WildCat.Induced WildCat.Equiv WildCat.Universe Require Import (notations) Classes.interfaces.canonical_names.

Algebra\Monoids\Monoid.v

mnd_homo_compose

(M N : Monoid) := { mnd_iso_homo :> MonoidHomomorphism M N; isequiv_mnd_iso :: IsEquiv mnd_iso_homo; }.

Record

Require Import Basics.Overture Basics.Tactics Basics.Equivalences Basics.Trunc. Require Import Types.Sigma Types.Forall Types.Prod. Require Import WildCat.Core WildCat.Induced WildCat.Equiv WildCat.Universe Require Import (notations) Classes.interfaces.canonical_names.

Algebra\Monoids\Monoid.v

MonoidIsomorphism

Build_MonoidIsomorphism' {M N : Monoid} (f : M <~> N) (h : IsMonoidPreserving f) : MonoidHomomorphism M N. Proof. snrapply Build_MonoidIsomorphism. 1: srapply Build_MonoidHomomorphism. exact _. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Basics.Equivalences Basics.Trunc. Require Import Types.Sigma Types.Forall Types.Prod. Require Import WildCat.Core WildCat.Induced WildCat.Equiv WildCat.Universe Require Import (notations) Classes.interfaces.canonical_names.

Algebra\Monoids\Monoid.v

Build_MonoidIsomorphism'

(M N : Monoid) : _ <~> MonoidIsomorphism M N := ltac:(issig).

Definition

Require Import Basics.Overture Basics.Tactics Basics.Equivalences Basics.Trunc. Require Import Types.Sigma Types.Forall Types.Prod. Require Import WildCat.Core WildCat.Induced WildCat.Equiv WildCat.Universe Require Import (notations) Classes.interfaces.canonical_names.

Algebra\Monoids\Monoid.v

issig_MonoidIsomorphism

{M : Monoid} : MonoidIsomorphism M M := Build_MonoidIsomorphism _ _ mnd_homo_id _.

Definition

Require Import Basics.Overture Basics.Tactics Basics.Equivalences Basics.Trunc. Require Import Types.Sigma Types.Forall Types.Prod. Require Import WildCat.Core WildCat.Induced WildCat.Equiv WildCat.Universe Require Import (notations) Classes.interfaces.canonical_names.

Algebra\Monoids\Monoid.v

mnd_iso_id

{M N P : Monoid} : MonoidIsomorphism N P -> MonoidIsomorphism M N -> MonoidIsomorphism M P := fun g f => Build_MonoidIsomorphism _ _ (mnd_homo_compose g f) _.

Definition

Require Import Basics.Overture Basics.Tactics Basics.Equivalences Basics.Trunc. Require Import Types.Sigma Types.Forall Types.Prod. Require Import WildCat.Core WildCat.Induced WildCat.Equiv WildCat.Universe Require Import (notations) Classes.interfaces.canonical_names.

Algebra\Monoids\Monoid.v

mnd_iso_compose

{M N : Monoid} : MonoidIsomorphism M N -> MonoidIsomorphism N M := fun f => Build_MonoidIsomorphism _ _ (Build_MonoidHomomorphism f^-1 _) _. Global Instance reflexive_monoidisomorphism : Reflexive MonoidIsomorphism := fun M => mnd_iso_id. Global Instance symmetric_monoidisomorphism : Symmetric MonoidIsomorphism := fun M N => . Global Instance transitive_monoidisomorphism : Transitive MonoidIsomorphism := fun M N P f g => mnd_iso_compose g f. Global Instance isgraph_monoid : IsGraph Monoid := Build_IsGraph Monoid MonoidHomomorphism. Global Instance is01cat_monoid : Is01Cat Monoid := Build_Is01Cat Monoid _ (@mnd_homo_id) (@mnd_homo_compose). Local Notation mnd_homo_map' M N := (@mnd_homo_map M N : _ -> (monoid_type M $-> _)). Global Instance is2graph_monoid : Is2Graph Monoid := fun M N => isgraph_induced (mnd_homo_map' M N). Global Instance isgraph_monoidhomomorphism {M N : Monoid} : IsGraph (M $-> N) := isgraph_induced (mnd_homo_map' M N). Global Instance is01cat_monoidhomomorphism {M N : Monoid} : Is01Cat (M $-> N) := is01cat_induced (mnd_homo_map' M N). Global Instance is0gpd_monoidhomomorphism {M N : Monoid} : Is0Gpd (M $-> N) := is0gpd_induced (mnd_homo_map' M N). Global Instance is0functor_postcomp_monoidhomomorphism {M N P : Monoid} (h : N $-> P) : Is0Functor (@cat_postcomp Monoid _ _ M N P h). Proof. apply Build_Is0Functor. intros ? ? p a; exact (ap h (p a)). Defined.

Definition

Require Import Basics.Overture Basics.Tactics Basics.Equivalences Basics.Trunc. Require Import Types.Sigma Types.Forall Types.Prod. Require Import WildCat.Core WildCat.Induced WildCat.Equiv WildCat.Universe Require Import (notations) Classes.interfaces.canonical_names.

Algebra\Monoids\Monoid.v

mnd_iso_inverse

Monoid -> Monoid -> Monoid. Proof. intros M N. snrapply (Build_Monoid (M * N)). 3: repeat split. - intros [m1 n1] [m2 n2]. exact (m1 * m2, n1 * n2). - exact (mon_unit, mon_unit). - exact _. - intros x y z; snrapply path_prod; nrapply mnd_assoc. - intros x; snrapply path_prod; nrapply mnd_unit_l. - intros x; snrapply path_prod; nrapply mnd_unit_r. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Basics.Equivalences Basics.Trunc. Require Import Types.Sigma Types.Forall Types.Prod. Require Import WildCat.Core WildCat.Induced WildCat.Equiv WildCat.Universe Require Import (notations) Classes.interfaces.canonical_names.

Algebra\Monoids\Monoid.v

mnd_prod

{M N : Monoid} : MonoidHomomorphism (mnd_prod M N) M. Proof. snrapply Build_MonoidHomomorphism. 1: exact fst. split; hnf; reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Basics.Equivalences Basics.Trunc. Require Import Types.Sigma Types.Forall Types.Prod. Require Import WildCat.Core WildCat.Induced WildCat.Equiv WildCat.Universe Require Import (notations) Classes.interfaces.canonical_names.

Algebra\Monoids\Monoid.v

mnd_prod_pr1

{M N : Monoid} : MonoidHomomorphism (mnd_prod M N) N. Proof. snrapply Build_MonoidHomomorphism. 1: exact snd. split; hnf; reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Basics.Equivalences Basics.Trunc. Require Import Types.Sigma Types.Forall Types.Prod. Require Import WildCat.Core WildCat.Induced WildCat.Equiv WildCat.Universe Require Import (notations) Classes.interfaces.canonical_names.

Algebra\Monoids\Monoid.v

mnd_prod_pr2

{M N P : Monoid} (f : MonoidHomomorphism P M) (g : MonoidHomomorphism P N) : MonoidHomomorphism P (mnd_prod M N). Proof. snrapply Build_MonoidHomomorphism. 2: split. - exact (fun x => (f x, g x)). - intros x y; snrapply path_prod; nrapply mnd_homo_op. - snrapply path_prod; nrapply mnd_homo_unit. Defined.

Definition

Require Import Basics.Overture Basics.Tactics Basics.Equivalences Basics.Trunc. Require Import Types.Sigma Types.Forall Types.Prod. Require Import WildCat.Core WildCat.Induced WildCat.Equiv WildCat.Universe Require Import (notations) Classes.interfaces.canonical_names.

Algebra\Monoids\Monoid.v

mnd_prod_corec

R $-> (R / I) × (R / J). Proof. apply ring_product_corec. 1,2: apply rng_quotient_map. Defined.

Definition

Require Import Classes.interfaces.canonical_names. Require Import WildCat. Require Import Modalities.ReflectiveSubuniverse. Require Import Algebra.AbGroups. Require Import Algebra.Rings.Ring. Require Import Algebra.Rings.Ideal. Require Import Algebra.Rings.QuotientRing. Require Import Algebra.Rings.CRing. Import Ideal.Notation.

Algebra\Rings\ChineseRemainder.v

rng_homo_crt

issurjection_rng_homo_crt' (x y : R) (q1 : rng_homo_crt x = (0, 1)) (q2 : rng_homo_crt y = (1, 0)) : IsSurjection rng_homo_crt. Proof. intros [a b]. revert a; srapply QuotientRing_ind_hprop; intro a. revert b; srapply QuotientRing_ind_hprop; intro b. snrapply Build_Contr; [|intros z; strip_truncations; apply path_ishprop]. apply tr; exists (b * x + a * y). rewrite rng_homo_plus. rewrite 2 rng_homo_mult. rewrite q1, q2. apply path_prod. + change ([b] * 0 + [a] * 1 = [a] :> R / I). by rewrite rng_mult_one_r, rng_mult_zero_r, rng_plus_zero_l. + change ([b] * 1 + [a] * 0 = [b] :> R / J). by rewrite rng_mult_one_r, rng_mult_zero_r, rng_plus_zero_r. Defined.

Lemma

Require Import Classes.interfaces.canonical_names. Require Import WildCat. Require Import Modalities.ReflectiveSubuniverse. Require Import Algebra.AbGroups. Require Import Algebra.Rings.Ring. Require Import Algebra.Rings.Ideal. Require Import Algebra.Rings.QuotientRing. Require Import Algebra.Rings.CRing. Import Ideal.Notation.

Algebra\Rings\ChineseRemainder.v

issurjection_rng_homo_crt'

rng_homo_crt x = (0, 1). Proof. apply rng_moveR_Mr in p. rewrite rng_plus_comm in p. apply path_prod; apply qglue. - change (I (-x + 0)). apply ideal_in_negate_plus. 1: assumption. apply ideal_in_zero. - change (J (-x + 1)). rewrite rng_plus_comm. by rewrite <- p. Defined.

Lemma

Require Import Classes.interfaces.canonical_names. Require Import WildCat. Require Import Modalities.ReflectiveSubuniverse. Require Import Algebra.AbGroups. Require Import Algebra.Rings.Ring. Require Import Algebra.Rings.Ideal. Require Import Algebra.Rings.QuotientRing. Require Import Algebra.Rings.CRing. Import Ideal.Notation.

Algebra\Rings\ChineseRemainder.v

rng_homo_crt_beta_left

rng_homo_crt y = (1, 0). Proof. apply rng_moveR_rM in p. rewrite rng_plus_comm in p. apply path_prod; apply qglue. - change (I (-y + 1)). by rewrite <- p. - change (J (-y + 0)). apply ideal_in_negate_plus. 1: assumption. apply ideal_in_zero. Defined.

Lemma

Require Import Classes.interfaces.canonical_names. Require Import WildCat. Require Import Modalities.ReflectiveSubuniverse. Require Import Algebra.AbGroups. Require Import Algebra.Rings.Ring. Require Import Algebra.Rings.Ideal. Require Import Algebra.Rings.QuotientRing. Require Import Algebra.Rings.CRing. Import Ideal.Notation.

Algebra\Rings\ChineseRemainder.v

rng_homo_crt_beta_right

R / (I ∩ J)%ideal ≅ (R / I) × (R / J). Proof. snrapply rng_first_iso'. 1: rapply rng_homo_crt. 1: exact _. apply ideal_subset_antisymm. - intros r [i j]. apply path_prod; apply qglue. 1: change (I (- r + 0)). 2: change (J (- r + 0)). 1,2: rewrite rng_plus_comm. 1,2: apply ideal_in_plus_negate. 1,3: apply ideal_in_zero. 1,2: assumption. - intros i p. apply equiv_path_prod in p. destruct p as [p q]. apply ideal_in_negate'. rewrite <- rng_plus_zero_r. split. 1: exact (related_quotient_paths _ _ _ p). 1: exact (related_quotient_paths _ _ _ q). Defined.

Theorem

Require Import Classes.interfaces.canonical_names. Require Import WildCat. Require Import Modalities.ReflectiveSubuniverse. Require Import Algebra.AbGroups. Require Import Algebra.Rings.Ring. Require Import Algebra.Rings.Ideal. Require Import Algebra.Rings.QuotientRing. Require Import Algebra.Rings.CRing. Import Ideal.Notation.

Algebra\Rings\ChineseRemainder.v

chinese_remainder

`{Univalence} {R : CRing} (I J : Ideal R) (c : Coprime I J) : R / (I ⋅ J)%ideal ≅ (R / I) × (R / J). Proof. etransitivity. { rapply rng_quotient_invar. symmetry. rapply ideal_intersection_is_product. } rapply chinese_remainder. Defined.

Theorem

Require Import Classes.interfaces.canonical_names. Require Import WildCat. Require Import Modalities.ReflectiveSubuniverse. Require Import Algebra.AbGroups. Require Import Algebra.Rings.Ring. Require Import Algebra.Rings.Ideal. Require Import Algebra.Rings.QuotientRing. Require Import Algebra.Rings.CRing. Import Ideal.Notation.

Algebra\Rings\ChineseRemainder.v

chinese_remainder_prod

{ cring_ring :> Ring; cring_commutative :: Commutative (A:=cring_ring) (.*.); }.

Record

Require Import WildCat. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.AbGroups. Import Ideal.Notation.

Algebra\Rings\CRing.v

CRing

_ <~> CRing := ltac:(issig).

Definition

Require Import WildCat. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.AbGroups. Import Ideal.Notation.

Algebra\Rings\CRing.v

issig_CRing

Build_CRing' (R : AbGroup) `(!One R, !Mult R) (comm : Commutative (.*.)) (assoc : Associative (.*.)) (dist_l : LeftDistribute (.*.) (+)) (unit_l : LeftIdentity (.*.) 1) : CRing. Proof. snrapply Build_CRing. - rapply (Build_Ring R); only 1,2,4: exact _. + intros x y z. lhs nrapply comm. lhs rapply dist_l. f_ap. + intros x. lhs rapply comm. apply unit_l. - exact _. Defined.

Definition

Require Import WildCat. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.AbGroups. Import Ideal.Notation.

Algebra\Rings\CRing.v

Build_CRing'

{R : CRing} (x y : R) : x * y = y * x := commutativity x y.

Definition

Require Import WildCat. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.AbGroups. Import Ideal.Notation.

Algebra\Rings\CRing.v

rng_mult_comm

{R : CRing} (x y : R) (n : nat) : rng_power (R:=R) (x * y) n = rng_power (R:=R) x n * rng_power (R:=R) y n. Proof. induction n. 1: symmetry; rapply rng_mult_one_l. simpl. rewrite (rng_mult_assoc (A:=R)). rewrite <- (rng_mult_assoc (A:=R) x _ y). rewrite (rng_mult_comm (rng_power (R:=R) x n) y). rewrite rng_mult_assoc. rewrite <- (rng_mult_assoc _ (rng_power (R:=R) x n)). f_ap. Defined.

Lemma

Require Import WildCat. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.AbGroups. Import Ideal.Notation.

Algebra\Rings\CRing.v

rng_power_mult

{R : CRing} (x y z : R) : x * y * z = x * z * y. Proof. lhs_V nrapply rng_mult_assoc. rhs_V nrapply rng_mult_assoc. apply ap, rng_mult_comm. Defined.

Definition

Require Import WildCat. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.AbGroups. Import Ideal.Notation.

Algebra\Rings\CRing.v

rng_mult_permute_2_3

{R : CRing} (x y z : R) : x * y * z = y * x * z. Proof. f_ap; apply rng_mult_comm. Defined.

Definition

Require Import WildCat. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.AbGroups. Import Ideal.Notation.

Algebra\Rings\CRing.v

rng_mult_move_left_assoc

{R : CRing} (x y z : R) : x * (y * z) = y * (x * z). Proof. refine (rng_mult_assoc _ _ _ @ _ @ (rng_mult_assoc _ _ _)^). apply rng_mult_move_left_assoc. Defined.

Definition

Require Import WildCat. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.AbGroups. Import Ideal.Notation.

Algebra\Rings\CRing.v

rng_mult_move_right_assoc

(R : CRing) (x : R) (inv : R) (inv_l : inv * x = 1) : IsInvertible R x. Proof. snrapply Build_IsInvertible. - exact inv. - exact inv_l. - lhs nrapply rng_mult_comm. exact inv_l. Defined.

Definition

Require Import WildCat. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.AbGroups. Import Ideal.Notation.

Algebra\Rings\CRing.v

isinvertible_cring

(I J : Ideal R) : I ⋅ J ⊆ J ⋅ I. Proof. intros r p. strip_truncations. induction p as [r p | |]. 2: apply ideal_in_zero. 2: by apply ideal_in_plus_negate. destruct p as [s t p q]. rewrite rng_mult_comm. apply tr. apply sgt_in. by rapply ipn_in. Defined.

Lemma

Require Import WildCat. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.AbGroups. Import Ideal.Notation.

Algebra\Rings\CRing.v

ideal_product_subset_product_commutative

(I J : Ideal R) : I ⋅ J ↔ J ⋅ I. Proof. apply ideal_subset_antisymm; apply ideal_product_subset_product_commutative. Defined.

Lemma

Require Import WildCat. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.AbGroups. Import Ideal.Notation.

Algebra\Rings\CRing.v

ideal_product_comm

ideal_product_intersection_sum_subset' (I J : Ideal R) : (I ∩ J) ⋅ (I + J) ⊆ I ⋅ J. Proof. etransitivity. 2: rapply ideal_sum_self. etransitivity. 2: rapply ideal_sum_subset_pres_r. 2: rapply ideal_product_comm. apply ideal_product_intersection_sum_subset. Defined.

Lemma

Require Import WildCat. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.AbGroups. Import Ideal.Notation.

Algebra\Rings\CRing.v

ideal_product_intersection_sum_subset'

(I J : Ideal R) : ideal_unit R ⊆ (I + J) -> I ∩ J ⊆ I ⋅ J. Proof. intros p. etransitivity. { apply ideal_eq_subset. symmetry. apply ideal_product_unit_r. } etransitivity. 1: rapply (ideal_product_subset_pres_r _ _ _ p). rapply ideal_product_intersection_sum_subset'. Defined.

Lemma

Require Import WildCat. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.AbGroups. Import Ideal.Notation.

Algebra\Rings\CRing.v

ideal_intersection_subset_product

(I J : Ideal R) : Coprime I J -> I ∩ J ↔ I ⋅ J. Proof. intros p. apply ideal_subset_antisymm. - apply ideal_intersection_subset_product. unfold Coprime in p. apply symmetry in p. rapply p. - apply ideal_product_subset_intersection. Defined.

Lemma

Require Import WildCat. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.AbGroups. Import Ideal.Notation.

Algebra\Rings\CRing.v

ideal_intersection_is_product

(I J K : Ideal R) : (I :: J) :: K ↔ (I :: (J ⋅ K)). Proof. apply ideal_subset_antisymm. - intros x [p q]; strip_truncations; split; apply tr; intros r; rapply Trunc_rec; intros jk. + induction jk as [y [z z' j k] | | ? ? ? ? ? ? ]. * rewrite (rng_mult_comm z z'). rewrite rng_mult_assoc. destruct (p z' k) as [p' ?]. revert p'; apply Trunc_rec; intros p'. exact (p' z j). * change (I (x * 0)). rewrite rng_mult_zero_r. apply ideal_in_zero. * change (I (x * (g - h))). rewrite rng_dist_l. rewrite rng_mult_negate_r. by apply ideal_in_plus_negate. + induction jk as [y [z z' j k] | | ? ? ? ? ? ? ]. * change (I (z * z' * x)). rewrite <- rng_mult_assoc. rewrite (rng_mult_comm z). destruct (q z' k) as [q' ?]. revert q'; apply Trunc_rec; intros q'. exact (q' z j). * change (I (0 * x)). rewrite rng_mult_zero_l. apply ideal_in_zero. * change (I ((g - h) * x)). rewrite rng_dist_r. rewrite rng_mult_negate_l. by apply ideal_in_plus_negate. - intros x [p q]; strip_truncations; split; apply tr; intros r k; split; apply tr; intros z j. + rewrite <- rng_mult_assoc. rewrite (rng_mult_comm r z). by apply p, tr, sgt_in, ipn_in. + cbn in z. change (I (z * (x * r))). rewrite (rng_mult_comm x). rewrite rng_mult_assoc. by apply q, tr, sgt_in, ipn_in. + cbn in r. change (I (r * x * z)). rewrite <- rng_mult_assoc. rewrite (rng_mult_comm r). rewrite <- rng_mult_assoc. by apply p, tr, sgt_in, ipn_in. + cbn in r, z. change (I (z * (r * x))). rewrite rng_mult_assoc. rewrite rng_mult_comm. by apply p, tr, sgt_in, ipn_in. Defined.

Lemma

Require Import WildCat. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.AbGroups. Import Ideal.Notation.

Algebra\Rings\CRing.v

ideal_quotient_product

(I J K : Ideal R) : I ⋅ J ⊆ K <-> I ⊆ (K :: J). Proof. split. - intros p r i; split; apply tr; intros s j; cbn in s, r. + by apply p, tr, sgt_in, ipn_in. + change (K (s * r)). rewrite (rng_mult_comm s r). by apply p, tr, sgt_in; rapply ipn_in. - intros p x. apply Trunc_rec. intros q. induction q as [r x | | ]. { destruct x. specialize (p x s); destruct p as [p q]. revert p; apply Trunc_rec; intros p. by apply p. } 1: apply ideal_in_zero. by apply ideal_in_plus_negate. Defined.

Lemma

Require Import WildCat. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.AbGroups. Import Ideal.Notation.

Algebra\Rings\CRing.v

ideal_quotient_subset_prod

(I J : Ideal R) : (I :: J) ⋅ J ⊆ I. Proof. by apply ideal_quotient_subset_prod. Defined.

Lemma

Require Import WildCat. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.AbGroups. Import Ideal.Notation.

Algebra\Rings\CRing.v

ideal_quotient_product_left

(R : CRing) (I : Ideal R) : CRing := Build_CRing (QuotientRing R I) _.

Definition

Require Import WildCat. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.AbGroups. Import Ideal.Notation.

Algebra\Rings\CRing.v

cring_quotient

{R : CRing} (I : Ideal R) : R $-> cring_quotient R I := rng_quotient_map I.

Definition

Require Import WildCat. Require Import Classes.interfaces.abstract_algebra. Require Import Algebra.AbGroups. Import Ideal.Notation.

Algebra\Rings\CRing.v

cring_quotient_map

{R : Ring} (I : Subgroup R) (x r : R) : IsRightIdeal I -> I x -> I (x * r) := fun _ => isleftideal (R := rng_op R) r x.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

isrightideal

{R : Ring} (I : Subgroup R) : _ <~> IsIdeal I := ltac:(issig). Hint Immediate Build_IsIdeal : typeclass_instances. Global Instance isideal_op {R : Ring} (I : Subgroup R) : IsIdeal I -> IsIdeal (R := rng_op R) I. Proof. intros [? ?]; exact _. Defined.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

issig_IsIdeal

(R : Ring) := { leftideal_subgroup :> Subgroup R; leftideal_isleftideal :: IsLeftIdeal leftideal_subgroup; }.

Record

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

LeftIdeal

(R : Ring) : _ <~> LeftIdeal R := ltac:(issig).

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

issig_LeftIdeal

(R : Ring) := LeftIdeal (rng_op R).

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

RightIdeal

(R : Ring) (I : Subgroup R) (H : IsRightIdeal I) : RightIdeal R := Build_LeftIdeal (rng_op R) I H.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

Build_RightIdeal

(R : Ring) : {I : Subgroup R& IsRightIdeal (R:=R) I} <~> RightIdeal R := ltac:(issig).

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

issig_RightIdeal

(R : Ring) := { ideal_subgroup :> Subgroup R; ideal_isideal :: IsIdeal ideal_subgroup; }.

Record

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

Ideal

(R : Ring) : _ <~> Ideal R := ltac:(issig).

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

issig_Ideal

(R : Ring) : Ideal R -> Ideal (rng_op R) := fun I => Build_Ideal (rng_op R) I _.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_op

{R : Ring} : Ideal R -> LeftIdeal R := fun I => Build_LeftIdeal R I _.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

leftideal_of_ideal

{R : Ring} : Ideal R -> RightIdeal R := fun I => Build_RightIdeal R I _.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

rightideal_of_ideal

I ring_zero := subgroup_in_unit I.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_in_zero

I a -> I b -> I (a + b) := subgroup_in_op I a b.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_in_plus

I a -> I (- a) := subgroup_in_inv I a.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_in_negate

ideal_in_negate' : I (- a) -> I a := subgroup_in_inv' I a.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_in_negate'

I a -> I b -> I (a - b) := subgroup_in_op_inv I a b.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_in_plus_negate

I a -> I b -> I (-a + b) := subgroup_in_inv_op I a b.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_in_negate_plus

I (a + b) -> I b -> I a := subgroup_in_op_l I a b.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_in_plus_l

I (a + b) -> I a -> I b := subgroup_in_op_r I a b. End IdealElements. Global Instance isleftideal_trivial_subgroup (R : Ring) : IsLeftIdeal (R := R) trivial_subgroup. Proof. intros r x p. rhs_V nrapply (rng_mult_zero_r). f_ap. Defined.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_in_plus_r

(R : Ring) : Ideal R := Build_Ideal R _ _.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_zero

(R : Ring) : Ideal R := Build_Ideal R _ (isideal_maximal_subgroup R). Global Instance isleftideal_subgroup_intersection (R : Ring) (I J : Subgroup R) `{IsLeftIdeal R I, IsLeftIdeal R J} : IsLeftIdeal (subgroup_intersection I J). Proof. intros r x [a b]; split; by apply isleftideal. Defined.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_unit

{R : Ring} : LeftIdeal R -> LeftIdeal R -> LeftIdeal R := fun I J => Build_LeftIdeal R (subgroup_intersection I J) _.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

leftideal_intersection

{R : Ring} : RightIdeal R -> RightIdeal R -> RightIdeal R := leftideal_intersection.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

rightideal_intersection

{R : Ring} : Ideal R -> Ideal R -> Ideal R := fun I J => Build_Ideal R (subgroup_intersection I J) _. Global Instance isleftideal_subgroup_product (R : Ring) (I J : Subgroup R) `{IsLeftIdeal R I, IsLeftIdeal R J} : IsLeftIdeal (subgroup_product I J). Proof. intros r. nrapply subgroup_product_ind. - intros x p. apply tr, sgt_in. left; by apply isleftideal. - intros x p. apply tr, sgt_in. right; by apply isleftideal. - apply tr, sgt_in. left; apply isleftideal. apply ideal_in_zero. - intros x y p q IHp IHq; cbn beta. rewrite rng_dist_l. rewrite rng_mult_negate_r. by apply subgroup_in_op_inv. - exact _. Defined.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_intersection

{R : Ring} : LeftIdeal R -> LeftIdeal R -> LeftIdeal R := fun I J => Build_LeftIdeal R (subgroup_product I J) _.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

leftideal_sum

{R : Ring} : RightIdeal R -> RightIdeal R -> RightIdeal R := leftideal_sum.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

rightideal_sum

{R : Ring} : Ideal R -> Ideal R -> Ideal R := fun I J => Build_Ideal R (subgroup_product I J) _.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_sum

{R : Ring} (I J : Ideal R) (P : forall x, ideal_sum I J x -> Type) (P_I_in : forall x y, P x (tr (sgt_in (inl y)))) (P_J_in : forall x y, P x (tr (sgt_in (inr y)))) (P_unit : P mon_unit (tr sgt_unit)) (P_op : forall x y h k, P x (tr h) -> P y (tr k) -> P (x - y) (tr (sgt_op h k))) `{forall x y, IsHProp (P x y)} : forall x (p : ideal_sum I J x), P x p := subgroup_product_ind I J P P_I_in P_J_in P_unit P_op.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_sum_ind

{R : Ring} (I J : Subgroup R) : R -> Type := | ipn_in : forall x y, I x -> J y -> I J (x * y).

Inductive

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_product_naive_type

{R : Ring} (I J : Subgroup R) : Subgroup R := subgroup_generated (G := R) (ideal_product_naive_type I J). Global Instance isleftideal_ideal_product_type {R : Ring} (I J : Subgroup R) `{IsLeftIdeal R I, IsLeftIdeal R J} : IsLeftIdeal ( I J). Proof. intro r. nrapply (functor_subgroup_generated _ _ (grp_homo_rng_left_mult r)). intros s [s1 s2 p1 p2]; cbn. rewrite simple_associativity. nrefine (ipn_in I J (r * s1) s2 _ p2). by apply isleftideal. Defined.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_product_type

{R : Ring} : LeftIdeal R -> LeftIdeal R -> LeftIdeal R := fun I J => Build_LeftIdeal R (ideal_product_type I J) _.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

leftideal_product

{R : Ring} : RightIdeal R -> RightIdeal R -> RightIdeal R := leftideal_product.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

rightideal_product

{R : Ring} : Ideal R -> Ideal R -> Ideal R := fun I J => Build_Ideal R (ideal_product_type I J) _. Global Instance isleftideal_grp_kernel {R S : Ring} (f : RingHomomorphism R S) : IsLeftIdeal (grp_kernel f). Proof. intros r x p. lhs nrapply rng_homo_mult. rhs_V nrapply (rng_mult_zero_r (f r)). by apply ap. Defined.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_product

{R S : Ring} (f : RingHomomorphism R S) : Ideal R := Build_Ideal R (grp_kernel f) _.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_kernel

(R : Ring) (X : R -> Type) : R -> Type := (** It should contain all elements of the original family.

Inductive

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

leftideal_generated_type

leftideal_generated@{u v} {R : Ring@{u}} (X : R -> Type@{v}) : LeftIdeal@{u v} R. Proof. snrapply Build_LeftIdeal. - snrapply Build_Subgroup'. + exact (fun x => merely (leftideal_generated_type X x)). + exact _. + apply tr, ligt_zero. + intros x y p q; strip_truncations. by apply tr, ligt_add_neg. - intros r x; apply Trunc_functor. apply ligt_mul. Defined.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

leftideal_generated@

rightideal_generated@{u v} {R : Ring@{u}} (X : R -> Type@{v}) : RightIdeal@{u v} R := Build_RightIdeal R (leftideal_generated (R:=rng_op R) X) _.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

rightideal_generated@

(R : Ring) (X : R -> Type) : R -> Type := (** It should contain all elements of the original family.

Inductive

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_generated_type

{R : Ring} (X : R -> Type) : Ideal R. Proof. snrapply Build_Ideal; [|split]. - snrapply Build_Subgroup'. + exact (fun x => merely (ideal_generated_type X x)). + exact _. + apply tr, igt_zero. + intros x y p q; strip_truncations. by apply tr, igt_add_neg. - intros r x; apply Trunc_functor. nrapply igt_mul_l. - intros x r; apply Trunc_functor. nrapply igt_mul_r. Defined.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_generated

{R : Ring} {n : nat} (X : Fin n -> R) : Ideal R. Proof. apply ideal_generated. exact (hfiber X). Defined.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_generated_finite

{R : Ring} (x : R) : Ideal R := ideal_generated (fun r => x = r).

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_principal

{R : Ring} (I J : Subgroup R) := forall x, I x <-> J x.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_eq

`{Univalence} {R : Ring} {I J : Ideal R} : ideal_eq I J <~> I = J. Proof. refine ((equiv_ap' (issig_Ideal R)^-1 _ _)^-1 oE _). refine (equiv_path_sigma_hprop _ _ oE _). rapply equiv_path_subgroup'. Defined.

Lemma

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

equiv_path_ideal

{R : Ring} (I J : Subgroup R) := (forall x, I x -> J x). Global Instance reflexive_ideal_subset {R : Ring} : Reflexive (@ R) := fun _ _ => idmap. Global Instance transitive_ideal_subset {R : Ring} : Transitive (@ R). Proof. intros x y z p q a. exact (q a o p a). Defined.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_subset

{R : Ring} (I J : Subgroup R) : Subgroup R. Proof. snrapply Build_Subgroup'. - exact (fun r => merely (forall x, J x -> I (r * x))). - exact _. - apply tr. intros r p. rewrite rng_mult_zero_l. apply ideal_in_zero. - intros x y p q. strip_truncations; apply tr. hnf; intros s j. rewrite rng_dist_r. rewrite rng_mult_negate_l. apply ideal_in_plus_negate. + by apply p. + by apply q. Defined.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

subgroup_leftideal_quotient

{R : Ring} : LeftIdeal R -> Subgroup R -> LeftIdeal R := fun I J => Build_LeftIdeal R (subgroup_leftideal_quotient I J) _.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

leftideal_quotient

{R : Ring} (I J : Subgroup R) : Subgroup R := subgroup_leftideal_quotient (R:=rng_op R) I J. Global Instance isrightideal_subgroup_rightideal_quotient {R : Ring} (I J : Subgroup R) `{IsRightIdeal R I} : IsRightIdeal ( I J) := isleftideal_subgroup_leftideal_quotient (R:=rng_op R) I J. Global Instance isleftideal_subgroup_rightideal_quotient {R : Ring} (I J : Subgroup R) `{H : IsLeftIdeal R I, IsRightIdeal R J} : IsLeftIdeal ( I J). Proof. snrapply (isrightideal_subgroup_leftideal_quotient (R:=rng_op R) I J). - exact H. - exact _. Defined.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

subgroup_rightideal_quotient

{R : Ring} : RightIdeal R -> Subgroup R -> RightIdeal R := fun I J => Build_RightIdeal R (subgroup_rightideal_quotient (R:=R) I J) _.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

rightideal_quotient

{R : Ring} : Ideal R -> Ideal R -> Ideal R := fun I J => Build_Ideal R (subgroup_intersection (leftideal_quotient I J) (rightideal_quotient I J)) (Build_IsIdeal _ _ _ _).

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_quotient

{R : Ring} (S : R -> Type) : Subgroup R. Proof. snrapply Build_Subgroup'. - exact (fun r => merely (forall x, S x -> r * x = ring_zero)). - exact _. - apply tr. intros r p. apply rng_mult_zero_l. - intros x y p q. strip_truncations; apply tr. intros r s. lhs rapply rng_dist_r. rewrite (p r s). rewrite rng_mult_negate_l. rewrite (q r s). rewrite <- rng_mult_negate. rewrite rng_mult_zero_r. apply left_identity. Defined.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

subgroup_ideal_left_annihilator

{R : Ring} (I : LeftIdeal R) : Ideal R := Build_Ideal R (subgroup_ideal_left_annihilator I) _.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_left_annihilator

{R : Ring} (I : R -> Type) : Subgroup R := subgroup_ideal_left_annihilator (R:=rng_op R) I.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

subgroup_ideal_right_annihilator

{R : Ring} (I : RightIdeal R) : Ideal R := Build_Ideal R (subgroup_ideal_right_annihilator (R:=R) I) (isideal_ideal_right_annihilator (R:=R) I).

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_right_annihilator

{R : Ring} (I : Ideal R) : Ideal R := ideal_intersection (ideal_left_annihilator I) (ideal_right_annihilator I).

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_annihilator

{R : Ring} (I J : Ideal R) : Type := ideal_eq (ideal_sum I J) (ideal_unit R). Existing Class . Global Instance ishprop_coprime `{Funext} {R : Ring} (I J : Ideal R) : IsHProp ( I J). Proof. unfold . exact _. Defined.

Definition

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

Coprime

`{Funext} {R : Ring} (I J : Ideal R) : Coprime I J <~> hexists (fun '(((i ; p) , (j ; q)) : sig I * sig J) => i + j = ring_one). Proof. simpl. srapply equiv_iff_hprop. - intros c. pose (snd (c ring_one) tt) as d; clearbody d; clear c. strip_truncations. apply tr. induction d. + destruct x. * exists ((g ; s), (ring_zero; ideal_in_zero _)). apply rng_plus_zero_r. * exists ((ring_zero; ideal_in_zero _), (g ; s)). apply rng_plus_zero_l. + exists ((ring_zero; ideal_in_zero _), (ring_zero; ideal_in_zero _)). apply rng_plus_zero_l. + destruct IHd1 as [[[x xi] [y yj]] p]. destruct IHd2 as [[[w wi] [z zj]] q]. srefine (((_;_),(_;_));_). * exact (x - w). * by apply ideal_in_plus_negate. * exact (y - z). * by apply ideal_in_plus_negate. * cbn. refine (_ @ ap011 (fun x y => x - y) p q). rewrite <- 2 rng_plus_assoc. f_ap. rewrite negate_sg_op. rewrite rng_plus_comm. rewrite rng_plus_assoc. reflexivity. - intro x. strip_truncations. intros r. split;[intro; exact tt|]. intros _. destruct x as [[[x xi] [y yj]] p]. rewrite <- rng_mult_one_r. change (x + y = 1) in p. rewrite <- p. rewrite rng_dist_l. apply tr. rapply sgt_op'. + apply sgt_in. left. by apply isleftideal. + apply sgt_in. right. by apply isleftideal. Defined.

Lemma

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

equiv_coprime_sum

(I J : Subgroup R) : I ⊆ J -> J ⊆ I -> I ↔ J. Proof. intros p q x; split; by revert x. Defined.

Lemma

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_subset_antisymm

(I : Subgroup R) : ideal_zero R ⊆ I. Proof. intros x p; rewrite p; apply ideal_in_zero. Defined.

Lemma

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_zero_subset

(I : Subgroup R) : I ⊆ ideal_unit R. Proof. hnf; cbn; trivial. Defined.

Lemma

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_unit_subset

(I J : Ideal R) : I ∩ J ⊆ I. Proof. intro; exact fst. Defined.

Lemma

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_intersection_subset_l

(I J : Ideal R) : I ∩ J ⊆ J. Proof. intro; exact snd. Defined.

Lemma

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_intersection_subset_r

(I J K : Ideal R) : K ⊆ I -> K ⊆ J -> K ⊆ I ∩ J. Proof. intros p q x r; specialize (p x r); specialize (q x r); by split. Defined.

Lemma

Require Import Basics Types. Require Import Spaces.Finite.Fin. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Rings.Ring. Require Import Algebra.Groups.Subgroup. Require Import Algebra.AbGroups. Require Import WildCat.Core. Module Import Notation.

Algebra\Rings\Ideal.v

ideal_intersection_subset