Datasets:

phanerozoic

/

Coq-HoTT

like 2

(R : Ring) (m n p q : nat) : HeteroAssociative (@matrix_mult R m n q) (@matrix_mult R n p q) (@matrix_mult R m p q) (@matrix_mult R m n p). Proof. intros M N P; nrapply path_matrix; intros i j Hi Hj. rewrite 2 entry_Build_Matrix. lhs nrapply path_ab_sum. { intros k Hk. rewrite entry_Build_Matrix. apply rng_sum_dist_l. } lhs nrapply ab_sum_sum. rhs nrapply path_ab_sum. 2: intros k Hk; by rewrite entry_Build_Matrix. nrapply path_ab_sum. intros k Hk. rhs nrapply rng_sum_dist_r. nrapply path_ab_sum. intros l Hl. apply associativity. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

associative_matrix_mult

(R : Ring) (m n p : nat) : LeftHeteroDistribute (@matrix_mult R m n p) matrix_plus matrix_plus. Proof. intros M N P; apply path_matrix; intros i j Hi Hj. rewrite !entry_Build_Matrix, !entry_Build_Vector. change (?x = ?y + ?z) with (x = sg_op y z). rewrite <- ab_sum_plus. nrapply path_ab_sum. intros k Hk. rewrite entry_Build_Matrix. apply rng_dist_l. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

left_distribute_matrix_mult

(R : Ring) (m n p : nat) : RightHeteroDistribute (@matrix_mult R m n p) matrix_plus matrix_plus. Proof. intros M N P; apply path_matrix; intros i j Hi Hj. rewrite !entry_Build_Matrix, !entry_Build_Vector. change (?x = ?y + ?z) with (x = sg_op y z). rewrite <- ab_sum_plus. nrapply path_ab_sum. intros k Hk. rewrite entry_Build_Matrix. apply rng_dist_r. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

right_distribute_matrix_mult

(R : Ring) (m n: nat) : LeftIdentity (@matrix_mult R m m n) (identity_matrix R m). Proof. intros M; apply path_matrix; intros i j Hi Hj. rewrite entry_Build_Matrix. lhs nrapply path_ab_sum. 1: intros k Hk; by rewrite entry_Build_Matrix. nrapply rng_sum_kronecker_delta_l. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

left_identity_matrix_mult

(R : Ring) (m n : nat) : RightIdentity (@matrix_mult R m n n) (identity_matrix R n). Proof. intros M; apply path_matrix; intros i j Hi Hj. rewrite entry_Build_Matrix. lhs nrapply path_ab_sum. 1: intros k Hk; by rewrite entry_Build_Matrix. nrapply rng_sum_kronecker_delta_r'. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

right_identity_matrix_mult

(R : Ring@{i}) (n : nat) : Ring. Proof. snrapply Build_Ring. - exact (abgroup_matrix R n n). - exact matrix_mult. - exact (identity_matrix R n). - exact (associative_matrix_mult R n n n n). - exact (left_distribute_matrix_mult R n n n). - exact (right_distribute_matrix_mult R n n n). - exact (left_identity_matrix_mult R n n). - exact (right_identity_matrix_mult R n n). Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_ring

{R : Ring} {m n p : nat} : HeteroAssociative (@matrix_lact R m p) (@matrix_mult R m n p) (@matrix_mult R m n p) (@matrix_lact R m n). Proof. intros r M N. snrapply path_matrix. intros i j Hi Hj. rewrite !entry_Build_Matrix, !entry_Build_Vector. lhs nrapply rng_sum_dist_l. snrapply path_ab_sum. intros k Hk; cbn. rewrite !entry_Build_Matrix. snrapply rng_mult_assoc. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_mult_lact_l

{R : CRing} {m n p} (M : Matrix R m n) (N : Matrix R n p) : matrix_mult (R:=rng_op R) M N = matrix_mult M N. Proof. apply path_matrix; intros i j Hi Hj. rewrite 2 entry_Build_Matrix. apply path_ab_sum; intros k Hk. apply rng_mult_comm. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_mult_rng_op

{R : Type} {m n} : Matrix R m n -> Matrix R n m := fun M => Build_Matrix R n m (fun i j H1 H2 => entry M j i).

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_transpose

{R : Type} {m n} (M : Matrix R m n) : matrix_transpose (matrix_transpose M) = M. Proof. apply path_matrix. intros i j Hi Hj. lhs nrapply entry_Build_Matrix. nrapply entry_Build_Matrix. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_transpose_transpose

{R : Ring} {m n} (M N : Matrix R m n) : matrix_transpose (matrix_plus M N) = matrix_plus (matrix_transpose M) (matrix_transpose N). Proof. apply path_matrix. intros i j Hi Hj. by rewrite !entry_Build_Matrix, !entry_Build_Vector. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_transpose_plus

{R : Ring} {m n} (r : R) (M : Matrix R m n) : matrix_transpose (matrix_lact r M) = matrix_lact r (matrix_transpose M). Proof. apply path_matrix. intros i j Hi Hj. by rewrite !entry_Build_Matrix, !entry_Build_Vector. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_transpose_lact

{R : Ring} {m n} (M : Matrix R m n) : matrix_transpose (matrix_negate M) = matrix_negate (matrix_transpose M). Proof. apply path_matrix. intros i j Hi Hj. by rewrite !entry_Build_Matrix, !entry_Build_Vector. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_transpose_negate

{R : Ring} {m n p} (M : Matrix R m n) (N : Matrix R n p) : matrix_transpose (matrix_mult M N) = matrix_mult (R:=rng_op R) (matrix_transpose N) (matrix_transpose M). Proof. apply path_matrix. intros i j Hi Hj. rewrite 3 entry_Build_Matrix. apply path_ab_sum. intros k Hk. rewrite 2 entry_Build_Matrix. reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_transpose_mult

{R : CRing} {m n p} (M : Matrix R m n) (N : Matrix R n p) : matrix_transpose (matrix_mult M N) = matrix_mult (matrix_transpose N) (matrix_transpose M). Proof. lhs nrapply matrix_transpose_mult. apply matrix_mult_rng_op. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_transpose_mult_comm

{R : Ring} {m n} : matrix_transpose (matrix_zero R m n) = matrix_zero R n m. Proof. apply path_matrix. intros i j Hi Hj. by rewrite !entry_Build_Matrix. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_transpose_zero

matrix_transpose_identity@{i} {R : Ring@{i}} {n} : matrix_transpose (identity_matrix R n) = identity_matrix R n. Proof. apply path_matrix. intros i j Hi Hj. rewrite 3 entry_Build_Matrix. apply kronecker_delta_symm. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_transpose_identity@

{R : Ring@{i}} {n : nat} (v : Vector R n) : Matrix R n n. Proof. snrapply Build_Matrix. intros i j H1 H2. exact (kronecker_delta i j * Vector.entry v i). Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_diag

{R : Ring@{i}} {n : nat} (v w : Vector R n) : matrix_plus (matrix_diag v) (matrix_diag w) = matrix_diag (vector_plus v w). Proof. symmetry. snrapply path_matrix. intros i j Hi Hj. rewrite 2 entry_Build_Matrix, 5 entry_Build_Vector. nrapply rng_dist_l. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_diag_plus

{R : Ring} {n : nat} (v w : Vector R n) : matrix_mult (matrix_diag v) (matrix_diag w) = matrix_diag (vector_map2 (.*.) v w). Proof. snrapply path_matrix. intros i j Hi Hj. rewrite 2 entry_Build_Matrix. lhs snrapply path_ab_sum. { intros k Hk. rewrite 2 entry_Build_Matrix. rewrite rng_mult_assoc. rewrite <- (rng_mult_assoc (kronecker_delta _ _)). rewrite kronecker_delta_comm. rewrite <- 2 rng_mult_assoc. reflexivity. } rewrite (rng_sum_kronecker_delta_l _ _ Hi). by rewrite entry_Build_Vector. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_diag_mult

{R : Ring@{i}} {n : nat} (v : Vector R n) : matrix_transpose (matrix_diag v) = matrix_diag v. Proof. snrapply path_matrix. intros i j Hi Hj. rewrite 3 entry_Build_Matrix. rewrite kronecker_delta_symm. unfold kronecker_delta. destruct (dec (i = j)) as [p|np]. 1: f_ap; symmetry; by apply path_entry_vector. by rewrite !rng_mult_zero_l. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_transpose_diag

{R : Ring@{i}} {n : nat} {M : Matrix R n n} : _ <~> IsDiagonal M := ltac:(issig). Global Instance ishprop_isdiagonal {R : Ring@{i}} {n : nat} (M : Matrix R n n) : IsHProp (IsDiagonal M). Proof. snrapply hprop_allpath. intros x y. snrapply ((equiv_ap' ^-1%equiv _ _ )^-1%equiv). rapply path_sigma_hprop; cbn. apply isinj_matrix_diag. exact ((isdiagonal_diag M)^ @ isdiagonal_diag M). Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

issig_IsDiagonal

{R : Ring} {n : nat} (M : Matrix R n n) : Vector R n := Build_Vector R n (fun i _ => entry M i i).

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_diag_vector

matrix_diag_ring@{i} (R : Ring@{i}) (n : nat) : Subring@{i i} (matrix_ring R n). Proof. snrapply (Build_Subring' (fun M : matrix_ring R n => IsDiagonal M) _); hnf. - intros; exact _. - intros x y dx dy. nrapply isdiagonal_matrix_plus; trivial. by nrapply isdiagonal_matrix_negate. - nrapply isdiagonal_matrix_mult. - nrapply isdiagonal_identity_matrix. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_diag_ring@

{R : Ring} {n} (M : Matrix R n n) : R := ab_sum n (fun i Hi => entry M i i).

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_trace

{R : Ring} {n} (M N : Matrix R n n) : matrix_trace (matrix_plus M N) = (matrix_trace M) + (matrix_trace N). Proof. unfold matrix_trace. lhs nrapply path_ab_sum. { intros i Hi. by rewrite entry_Build_Matrix. } by rewrite ab_sum_plus. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_trace_plus

{R : Ring} {n} (r : R) (M : Matrix R n n) : matrix_trace (matrix_lact r M) = r * matrix_trace M. Proof. unfold matrix_trace. rewrite rng_sum_dist_l. apply path_ab_sum. intros i Hi. by rewrite entry_Build_Matrix. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_trace_lact

{R : CRing} {m n : nat} (M : Matrix R m n) (N : Matrix R n m) : matrix_trace (matrix_mult M N) = matrix_trace (matrix_mult N M). Proof. lhs nrapply path_ab_sum. { intros i Hi. lhs nrapply entry_Build_Matrix. nrapply path_ab_sum. intros j Hj. apply rng_mult_comm. } lhs nrapply ab_sum_sum. apply path_ab_sum. intros i Hi. rhs nrapply entry_Build_Matrix. reflexivity. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_trace_mult

{R : Ring} {n} (M : Matrix R n n) : matrix_trace (matrix_transpose M) = matrix_trace M. Proof. apply path_ab_sum. intros i Hi. nrapply entry_Build_Matrix. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

trace_transpose

(n : nat) : nat -> nat := fun i => if dec (i < n)%nat then i else i.+1%nat. Global Instance isinjective_skip n : IsInjective ( n). Proof. hnf. intros x y p. unfold in p. destruct (dec (x < n)%nat) as [H|H], (dec (y < n)%nat) as [H'|H']. - exact p. - destruct p^. contradiction (H' (leq_trans _ H)). - destruct p. contradiction (H (leq_trans _ H')). - by apply path_nat_succ. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

skip

{R : Ring@{i}} {n : nat} (i j : nat) {Hi : (i < n.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_minor

{R : Ring@{i}} {n : nat} (i j : nat) (Hi : (i < n.+1)%nat) (Hj : (j < n.+1)%nat) : matrix_minor i j (matrix_zero R n.+1 n.+1) = matrix_zero R n n. Proof. apply path_matrix. intros k l Hk Hl. by rewrite !entry_Build_Matrix. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_minor_zero

{R : Ring@{i}} {n : nat} (i : nat) (Hi : (i < n.+1)%nat) : matrix_minor i i (identity_matrix R n.+1) = identity_matrix R n. Proof. apply path_matrix. intros j k Hj Hk. rewrite 3 entry_Build_Matrix. rapply kronecker_delta_map_inj. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_minor_identity

{R : Ring@{i}} {n : nat} (i j : nat) (Hi : (i < n.+1)%nat) (Hj : (j < n.+1)%nat) (M N : Matrix R n.+1 n.+1) : matrix_minor i j (matrix_plus M N) = matrix_plus (matrix_minor i j M) (matrix_minor i j N). Proof. apply path_matrix. intros k l Hk Hl. by rewrite !entry_Build_Matrix, !entry_Build_Vector. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_minor_plus

{R : Ring@{i}} {n : nat} (i j : nat) (Hi : (i < n.+1)%nat) (Hj : (j < n.+1)%nat) (r : R) (M : Matrix R n.+1 n.+1) : matrix_minor i j (matrix_lact r M) = matrix_lact r (matrix_minor i j M). Proof. apply path_matrix. intros k l Hk Hl. by rewrite !entry_Build_Matrix, !entry_Build_Vector. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_minor_scale

{R : Ring@{i}} {n : nat} (i j : nat) (Hi : (i < n.+1)%nat) (Hj : (j < n.+1)%nat) (M : Matrix R n.+1 n.+1) : matrix_minor j i (matrix_transpose M) = matrix_transpose (matrix_minor i j M). Proof. apply path_matrix. intros k l Hk Hl. by rewrite 4 entry_Build_Matrix. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

matrix_minor_transpose

upper_triangular_matrix_ring@{i} (R : Ring@{i}) (n : nat) : Subring@{i i} (matrix_ring@{i} R n). Proof. nrapply (Build_Subring' (fun M : matrix_ring R n => IsUpperTriangular M)). - exact _. - intros x y ? ?; exact (upper_triangular_plus x (-y)). - exact upper_triangular_mult. - exact upper_triangular_identity. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

upper_triangular_matrix_ring@

lower_triangular_matrix_ring@{i} (R : Ring@{i}) (n : nat) : Subring@{i i} (matrix_ring R n). Proof. nrapply (Build_Subring'@{i i} (fun M : matrix_ring R n => IsLowerTriangular M)). - exact _. - intros x y ? ?; exact (lower_triangular_plus x (-y)). - exact lower_triangular_mult. - exact lower_triangular_identity. Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

lower_triangular_matrix_ring@

(R : Ring) := nat. Global Instance isgraph_matrixcat {R : Ring} : IsGraph ( R) := {| Hom := Matrix R |}. Global Instance is01cat_matrixcat {R : Ring} : Is01Cat ( R). Proof. snrapply Build_Is01Cat. - exact (identity_matrix R). - intros l m n M N. exact (matrix_mult N M). Defined.

Definition

Require Import Basics.Overture Basics.Trunc Basics.Tactics Basics.Decidable. Require Import Types.Sigma. Require Import Spaces.List.Core Spaces.List.Theory Spaces.List.Paths. Require Import Algebra.Rings.Ring Algebra.Rings.Module Algebra.Rings.CRing Require Import abstract_algebra. Require Import WildCat.Core WildCat.Paths. Require Import Modalities.ReflectiveSubuniverse.

Algebra\Rings\Matrix.v

MatrixCat

(R : Ring) := { lm_carrier :> AbGroup; lm_lact :: IsLeftModule R lm_carrier; }.

Record

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

LeftModule

r *L (m + n) = r *L m + r *L n := lact_left_dist r m n.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

lm_dist_l

(r + s) *L m = r *L m + s *L m := lact_right_dist r s m.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

lm_dist_r

r *L (s *L m) = (r * s) *L m := lact_assoc r s m.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

lm_assoc

1 *L m = m := lact_unit m.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

lm_unit

0 *L m = 0. Proof. apply (grp_cancelL1 (z := lact 0 m)). lhs_V nrapply lm_dist_r. f_ap. apply rng_plus_zero_r. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

lm_zero_l

r *L (0 : M) = 0. Proof. apply (grp_cancelL1 (z := lact r 0)). lhs_V nrapply lm_dist_l. f_ap. apply grp_unit_l. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

lm_zero_r

-1 *L m = -m. Proof. apply grp_moveL_1V. lhs nrapply (ap (_ +) (lm_unit m)^). lhs_V nrapply lm_dist_r. rhs_V nrapply lm_zero_l. f_ap. apply grp_inv_l. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

lm_minus_one

r *L -m = - (r *L m). Proof. apply grp_moveL_1V. lhs_V nrapply lm_dist_l. rhs_V nrapply lm_zero_r. f_ap. apply grp_inv_l. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

lm_neg

{R : Ring} {M : AbGroup} `{!IsRightModule R M} : M -> R -> M := fun m r => lact (R:=rng_op R) r m.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

ract

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

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

RightModule

(m + n) *R r = m *R r + n *R r := lm_dist_l (R:=rng_op R) r m n.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

rm_dist_r

m *R (r + s) = m *R r + m *R s := lm_dist_r (R:=rng_op R) r s m.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

rm_dist_l

(m *R r) *R s = m *R (r * s) := lm_assoc (R:=rng_op R) s r m.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

rm_assoc

m *R 1 = m := lm_unit (R:=rng_op R) m.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

rm_unit

(0 : M) *R r = 0 := lm_zero_r (R:=rng_op R) r.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

rm_zero_l

m *R 0 = 0 := lm_zero_l (R:=rng_op R) m.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

rm_zero_r

m *R -1 = -m := lm_minus_one (R:=rng_op R) m.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

rm_minus_one

-m *R r = - (m *R r) := lm_neg (R:=rng_op R) r m.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

rm_neg

{R : Ring} (M : LeftModule R) := { lsm_carrier :> M -> Type; lsm_submodule :: IsLeftSubmodule lsm_carrier; }.

Record

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

LeftSubmodule

{R : Ring} (M : RightModule R) := LeftSubmodule (R:=rng_op R) M.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

RightSubmodule

{R : Ring} {M : LeftModule R} : LeftSubmodule M -> Subgroup M := fun N => Build_Subgroup M N _.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

subgroup_leftsubmodule

{R : Ring} {M : RightModule R} : RightSubmodule M -> Subgroup M := idmap. Coercion : RightSubmodule >-> Subgroup. Global Instance isleftmodule_leftsubmodule {R : Ring} {M : LeftModule R} (N : LeftSubmodule M) : IsLeftModule R N. Proof. snrapply Build_IsLeftModule. - intros r [n n_in_N]. exists (r *L n). by apply lsm_submodule. - intros r [n] [m]; apply path_sigma_hprop. apply lact_left_dist. - intros r s [n]; apply path_sigma_hprop. apply lact_right_dist. - intros r s [n]; apply path_sigma_hprop. apply lact_assoc. - intros [n]; apply path_sigma_hprop. apply lact_unit. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

subgroup_rightsubmodule

{R : Ring} {M : LeftModule R} (N : LeftSubmodule M) : LeftModule R := Build_LeftModule R N _.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

leftmodule_leftsubmodule

{R : Ring} {M : RightModule R} (N : RightSubmodule M) : RightModule R := N.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

rightmodule_rightsubmodule

Build_IsLeftSubmodule' {R : Ring} {M : LeftModule R} (H : M -> Type) `{forall x, IsHProp (H x)} (z : H zero) (c : forall r n m, H n -> H m -> H (n + r *L m)) : IsLeftSubmodule H. Proof. snrapply Build_IsLeftSubmodule. - snrapply Build_IsSubgroup'. + exact _. + exact z. + intros x y hx hy. change (sg_op ?x ?y) with (x + y). pose proof (p := c (-1) x y hx hy). rewrite lm_minus_one in p. exact p. - intros r m hm. rewrite <- (grp_unit_l). by apply c. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

Build_IsLeftSubmodule'

Build_IsRightSubmodule' {R : Ring} {M : RightModule R} (H : M -> Type) `{forall x, IsHProp (H x)} (z : H zero) (c : forall r n m, H n -> H m -> H (n + ract m r)) : IsRightSubmodule H := Build_IsLeftSubmodule' (R:=rng_op R) H z c.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

Build_IsRightSubmodule'

Build_LeftSubmodule' {R : Ring} {M : LeftModule R} (H : M -> Type) `{forall x, IsHProp (H x)} (z : H zero) (c : forall r n m, H n -> H m -> H (n + r *L m)) : LeftSubmodule M. Proof. pose (p := Build_IsLeftSubmodule' H z c). snrapply Build_LeftSubmodule. 1: snrapply (Build_Subgroup _ H). 2: exact p. rapply ils_issubgroup. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

Build_LeftSubmodule'

{R : Ring} {M : RightModule R} (H : M -> Type) `{forall x, IsHProp (H x)} (z : H zero) (c : forall r n m, H n -> H m -> H (n + m *R r)) : RightSubmodule M := Build_LeftSubmodule' (R:=rng_op R) H z c.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

Build_RightSubmodule

{R : Ring} (M N : LeftModule R) := { lm_homo_map :> GroupHomomorphism M N; lm_homo_lact : forall r m, lm_homo_map (r *L m) = r *L lm_homo_map m; }.

Record

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

LeftModuleHomomorphism

{R : Ring} (M N : RightModule R) := LeftModuleHomomorphism (R:=rng_op R) M N.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

RightModuleHomomorphism

{R : Ring} {M N : RightModule R} : RightModuleHomomorphism M N -> GroupHomomorphism M N := lm_homo_map (R:=rng_op R) M N.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

rm_homo_map

{R : Ring} {M N : RightModule R} (f : RightModuleHomomorphism M N) : forall m r, f (ract m r) = ract (f m) r := fun m r => lm_homo_lact (R:=rng_op R) M N f r m.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

rm_homo_ract

{R : Ring} (M : LeftModule R) : LeftModuleHomomorphism M M. Proof. snrapply Build_LeftModuleHomomorphism. - exact grp_homo_id. - reflexivity. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

lm_homo_id

{R : Ring} (M : RightModule R) : RightModuleHomomorphism M M := lm_homo_id (R:=rng_op R) M.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

rm_homo_id

{R : Ring} {M N L : LeftModule R} : LeftModuleHomomorphism N L -> LeftModuleHomomorphism M N -> LeftModuleHomomorphism M L. Proof. intros f g. snrapply Build_LeftModuleHomomorphism. - exact (grp_homo_compose f g). - intros r m. rhs_V nrapply lm_homo_lact. apply (ap f). apply lm_homo_lact. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

lm_homo_compose

{R : Ring} {M N L : RightModule R} : RightModuleHomomorphism N L -> RightModuleHomomorphism M N -> RightModuleHomomorphism M L := lm_homo_compose (R:=rng_op R).

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

rm_homo_compose

Build_LeftModuleHomomorphism' {R : Ring} {M N : LeftModule R} (f : M -> N) (p : forall r x y, f (r *L x + y) = r *L f x + f y) : LeftModuleHomomorphism M N. Proof. snrapply Build_LeftModuleHomomorphism. - snrapply Build_GroupHomomorphism. + exact f. + intros x y. rewrite <- (lm_unit (f x)). set (lact 1 (f x)). rewrite <- (lm_unit x). apply p. - intros r m. simpl. rewrite <- (grp_unit_r (lact r m)). rewrite p. rhs_V nrapply grp_unit_r. apply grp_cancelL. specialize (p 1 0 0). rewrite 2 lm_unit in p. apply (grp_cancelL1 (z := f 0)). lhs_V nrapply p. apply ap. apply grp_unit_l. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

Build_LeftModuleHomomorphism'

Build_RightModuleHomomorphism' {R :Ring} {M N : RightModule R} (f : M -> N) (p : forall r x y, f (x *R r + y) = f x *R r + f y) : RightModuleHomomorphism M N := Build_LeftModuleHomomorphism' (R:=rng_op R) f p.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

Build_RightModuleHomomorphism'

{R : Ring} (M N : LeftModule R) := { lm_iso_map :> LeftModuleHomomorphism M N; isequiv_lm_iso_map :: IsEquiv lm_iso_map; }.

Record

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

LeftModuleIsomorphism

{R : Ring} (M N : RightModule R) := LeftModuleIsomorphism (R:=rng_op R) M N.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

RightModuleIsomorphism

Build_LeftModuleIsomorphism' {R : Ring} (M N : LeftModule R) (f : GroupIsomorphism M N) (p : forall r x, f (r *L x) = r *L f x) : LeftModuleIsomorphism M N. Proof. snrapply Build_LeftModuleIsomorphism. - snrapply Build_LeftModuleHomomorphism. + exact f. + exact p. - exact _. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

Build_LeftModuleIsomorphism'

Build_RightModuleIsomorphism' {R : Ring} (M N : RightModule R) (f : GroupIsomorphism M N) (p : forall r x, f (ract x r) = ract (f x) r) : RightModuleIsomorphism M N := Build_LeftModuleIsomorphism' (R:=rng_op R) M N f p.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

Build_RightModuleIsomorphism'

{R : Ring} {M N : LeftModule R} : LeftModuleIsomorphism M N -> LeftModuleIsomorphism N M. Proof. intros f. snrapply Build_LeftModuleIsomorphism. - snrapply Build_LeftModuleHomomorphism'. + exact f^-1. + intros r m n. apply moveR_equiv_V. rhs nrapply grp_homo_op. symmetry. f_ap. 2: apply eisretr. lhs nrapply lm_homo_lact. apply ap. apply eisretr. - exact _. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

lm_iso_inverse

{R : Ring} {M N : RightModule R} : RightModuleIsomorphism M N -> RightModuleIsomorphism N M := lm_iso_inverse (R:=rng_op R). Global Instance isgraph_leftmodule {R : Ring} : IsGraph (LeftModule R) := Build_IsGraph _ LeftModuleHomomorphism. Global Instance is01cat_leftmodule {R : Ring} : Is01Cat (LeftModule R) := Build_Is01Cat _ _ lm_homo_id (@lm_homo_compose R). Global Instance is2graph_leftmodule {R : Ring} : Is2Graph (LeftModule R) := fun M N => isgraph_induced (@lm_homo_map R M N). Global Instance is1cat_leftmodule {R : Ring} : Is1Cat (LeftModule R). Proof. snrapply Build_Is1Cat'. - intros M N; rapply is01cat_induced. - intros M N; rapply is0gpd_induced. - intros M N L h. snrapply Build_Is0Functor. intros f g p m. exact (ap h (p m)). - intros M N L f. snrapply Build_Is0Functor. intros g h p m. exact (p (f m)). - simpl; reflexivity. - simpl; reflexivity. - simpl; reflexivity. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

rm_iso_inverse

{R : Ring} {M N : LeftModule R} (f : M $-> N) : LeftSubmodule M := Build_LeftSubmodule _ _ (grp_kernel f) _.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

lm_kernel

{R : Ring} {M N : RightModule R} (f : M $-> N) : RightSubmodule M := lm_kernel (R:=rng_op R) f. Global Instance isleftsubmodule_grp_image {R : Ring} {M N : LeftModule R} (f : M $-> N) : IsLeftSubmodule (grp_image f). Proof. srapply Build_IsLeftSubmodule. intros r m; apply Trunc_functor; intros [n p]. exists (r *L n). lhs nrapply lm_homo_lact. apply ap. exact p. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

rm_kernel

{R : Ring} {M N : LeftModule R} (f : M $-> N) : LeftSubmodule N := Build_LeftSubmodule _ _ (grp_image f) _.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

lm_image

{R : Ring} {M N : RightModule R} (f : M $-> N) : RightSubmodule N := lm_image (R:=rng_op R) f. Global Instance isleftmodule_quotientabgroup {R : Ring} (M : LeftModule R) (N : LeftSubmodule M) : IsLeftModule R (QuotientAbGroup M N). Proof. snrapply Build_IsLeftModule. - intros r. snrapply quotient_abgroup_rec. + refine (grp_quotient_map $o _). snrapply Build_GroupHomomorphism. * exact (lact r). * intros x y. apply lm_dist_l. + intros n Nn; simpl. apply qglue. apply issubgroup_in_inv_op. 2: apply issubgroup_in_unit. by apply is_left_submodule. - intros r m n; revert m. snrapply Quotient_ind_hprop; [exact _ | intros m; revert n]. snrapply Quotient_ind_hprop; [exact _ | intros n; simpl]. rapply ap. apply lm_dist_l. - intros r s. snrapply Quotient_ind_hprop; [exact _| intros m; simpl]. rapply ap. apply lm_dist_r. - intros r s. snrapply Quotient_ind_hprop; [exact _| intros m; simpl]. rapply ap. apply lm_assoc. - snrapply Quotient_ind_hprop; [exact _| intros m; simpl]. rapply ap. apply lm_unit. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

rm_image

{R : Ring} (M : LeftModule R) (N : LeftSubmodule M) : LeftModule R := Build_LeftModule R (QuotientAbGroup M N) _.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

QuotientLeftModule

{R : Ring} (M : RightModule R) (N : RightSubmodule M) : RightModule R := QuotientLeftModule (R:=rng_op R) M N.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

QuotientRightModule

`{Funext} {R : Ring} {M N : LeftModule R} (f : M $-> N) : M / lm_kernel f ≅ lm_image f. Proof. snrapply Build_LeftModuleIsomorphism'. 1: rapply abgroup_first_iso. intros r. srapply Quotient_ind_hprop; intros m. apply path_sigma_hprop; simpl. apply lm_homo_lact. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

lm_first_iso

`{Funext} {R : Ring} {M N : RightModule R} (f : M $-> N) : QuotientRightModule M (rm_kernel f) ≅ rm_image f := lm_first_iso (R:=rng_op R) f.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

rm_first_iso

{R : Ring} : LeftModule R -> LeftModule R -> LeftModule R. Proof. intros M N. snrapply (Build_LeftModule R (ab_biprod M N)). snrapply Build_IsLeftModule. - intros r. apply functor_prod; exact (lact r). - intros r m n. apply path_prod; apply lm_dist_l. - intros r m n. apply path_prod; apply lm_dist_r. - intros r s m. apply path_prod; apply lm_assoc. - intros r. apply path_prod; apply lm_unit. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

lm_prod

{R : Ring} : RightModule R -> RightModule R -> RightModule R := lm_prod (R:=rng_op R).

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

rm_prod

{R : Ring} {M N : LeftModule R} : lm_prod M N $-> M. Proof. snrapply Build_LeftModuleHomomorphism. - apply grp_prod_pr1. - reflexivity. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

lm_prod_fst

{R : Ring} {M N : RightModule R} : rm_prod M N $-> M := lm_prod_fst (R:=rng_op R).

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

rm_prod_fst

{R : Ring} {M N : LeftModule R} : lm_prod M N $-> N. Proof. snrapply Build_LeftModuleHomomorphism. - apply grp_prod_pr2. - reflexivity. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

lm_prod_snd

{R : Ring} {M N : RightModule R} : rm_prod M N $-> N := lm_prod_snd (R:=rng_op R).

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

rm_prod_snd

{R : Ring} {M N : LeftModule R} (L : LeftModule R) (f : L $-> M) (g : L $-> N) : L $-> lm_prod M N. Proof. snrapply Build_LeftModuleHomomorphism. - apply (grp_prod_corec f g). - intros r l. apply path_prod; apply lm_homo_lact. Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

lm_prod_corec

{R : Ring} {M N : RightModule R} (R' : RightModule R) (f : R' $-> M) (g : R' $-> N) : R' $-> rm_prod M N := lm_prod_corec (R:=rng_op R) R' f g. Global Instance hasbinaryproducts_leftmodule {R : Ring} : HasBinaryProducts (LeftModule R). Proof. intros M N. snrapply Build_BinaryProduct. - exact (lm_prod M N). - exact lm_prod_fst. - exact lm_prod_snd. - exact lm_prod_corec. - cbn; reflexivity. - cbn; reflexivity. - intros L f g p q a. exact (path_prod' (p a) (q a)). Defined.

Definition

Require Import WildCat. Require Import Spaces.Nat.Core. Require Import Classes.interfaces.canonical_names. Require Import Algebra.Groups.Kernel Algebra.Groups.Image Algebra.Groups.QuotientGroup. Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct. Require Import Rings.Ring.

Algebra\Rings\Module.v

rm_prod_corec