Require Import WildCat.
[Loading ML file number_string_notation_plugin.cmxs (using legacy method) ... done]
Require Import Spaces.Nat.Core.
(* Some of the material in abstract_algebra and canonical names could be selectively exported to the user, as is done in Groups/Group.v. *)
Require Import Classes.interfaces.canonical_names.
Require Import Algebra.Groups.QuotientGroup.
Require Import Algebra.AbGroups.AbelianGroup Algebra.AbGroups.Biproduct.
Require Import Rings.Ring.

Declare Scope module_scope.
Local Open Scope module_scope.

(** * Modules over a ring. *)

(** ** Left Modules *)

(** An abelian group [M] is a left [R]-module when equipped with the following data: *) 
Class IsLeftModule (R : Ring) (M : AbGroup) := {
  (** A function [lact] (left-action) that takes an element [r : R] and an element [m : M] and returns an element [lact r m : M], which we also denote [r *L m]. *)
  lact : R -> M -> M;
  (** Actions distribute on the left over addition in the abelian group. That is [r *L (m + n) = r *L m + r *L n]. *)
  lact_left_dist :: LeftHeteroDistribute lact (+) (+);
  (** Actions distribute on the right over addition in the ring. That is [(r + s) *L m = r *L m + s *L m]. *)
  lact_right_dist :: RightHeteroDistribute lact (+) (+);
  (** Actions are associative. That is [(r * s) *L m = r *L (s *L m)]. *)
  lact_assoc :: HeteroAssociative lact lact lact (.*.);
  (** Actions preserve the multiplicative identity. That is [1 *L m = m]. *)
  lact_unit :: LeftIdentity lact 1;
}.

Infix "*L" := lact : module_scope.

(** A left R-module is an abelian group equipped with a left R-module structure. *)
Record LeftModule (R : Ring) := {
  lm_carrier :> AbGroup;
  lm_lact :: IsLeftModule R lm_carrier;
}.

Section LeftModuleAxioms.
  Context {R : Ring} {M : LeftModule R} (r s : R) (m n : M).
  (** Here we state the module axioms in a readable form for direct use. *)

  Definition lm_dist_l : r *L (m + n) = r *L m + r *L n := lact_left_dist r m n.
  Definition lm_dist_r : (r + s) *L m = r *L m + s *L m := lact_right_dist r s m.
  Definition lm_assoc : r *L (s *L m) = (r * s) *L m := lact_assoc r s m.
  Definition lm_unit : 1 *L m = m := lact_unit m.

End LeftModuleAxioms.

(** ** Facts about left modules *)

Section LeftModuleFacts.
  Context {R : Ring} {M : LeftModule R} (r : R) (m : M).

  (** Here are some quick facts that hold in modules. *) 

  (** The left action of zero is zero. *) 
  Definition lm_zero_l : 0 *L m = 0.
R: Ring
M: LeftModule R
r: R
m: M
0 *L m = 0
  Proof.
R: Ring
M: LeftModule R
r: R
m: M
0 *L m = 0
    apply (grp_cancelL1 (z := lact 0 m)).
R: Ring
M: LeftModule R
r: R
m: M
sg_op (0 *L m) (0 *L m) = 0 *L m
    lhs_V napply lm_dist_r.
R: Ring
M: LeftModule R
r: R
m: M
(0 + 0) *L m = 0 *L m
    f_ap.
R: Ring
M: LeftModule R
r: R
m: M
0 + 0 = 0
    apply rng_plus_zero_r.
  Defined.

  (** The left action on zero is zero. *)
  Definition lm_zero_r : r *L (0 : M) = 0.
R: Ring
M: LeftModule R
r: R
m: M
r *L (0 : M) = 0
  Proof.
R: Ring
M: LeftModule R
r: R
m: M
r *L (0 : M) = 0
    apply (grp_cancelL1 (z := lact r 0)).
R: Ring
M: LeftModule R
r: R
m: M
sg_op (r *L 0) (r *L 0) = r *L 0
    lhs_V napply lm_dist_l.
R: Ring
M: LeftModule R
r: R
m: M
r *L (0 + 0) = r *L 0
    f_ap.
R: Ring
M: LeftModule R
r: R
m: M
0 + 0 = 0
    apply grp_unit_l.
  Defined.

  (** The left action of [-1] is the additive inverse. *)
  Definition lm_minus_one : -1 *L m = -m.
R: Ring
M: LeftModule R
r: R
m: M
-1 *L m = - m
  Proof.
R: Ring
M: LeftModule R
r: R
m: M
-1 *L m = - m
     apply grp_moveL_1V.
R: Ring
M: LeftModule R
r: R
m: M
sg_op (-1 *L m) m = mon_unit
     lhs napply (ap (_ +) (lm_unit m)^).
R: Ring
M: LeftModule R
r: R
m: M
-1 *L m + 1 *L m = mon_unit
     lhs_V napply lm_dist_r.
R: Ring
M: LeftModule R
r: R
m: M
(-1 + 1) *L m = mon_unit
     rhs_V napply lm_zero_l.
R: Ring
M: LeftModule R
r: R
m: M
(-1 + 1) *L m = 0 *L m
     f_ap.
R: Ring
M: LeftModule R
r: R
m: M
-1 + 1 = 0
     apply grp_inv_l.
  Defined.

  (** The left action of [r] on the additive inverse of [m] is the additive inverse of the left action of [r] on [m]. *)
  Definition lm_neg : r *L -m = - (r *L m).
R: Ring
M: LeftModule R
r: R
m: M
r *L - m = - (r *L m)
  Proof.
R: Ring
M: LeftModule R
r: R
m: M
r *L - m = - (r *L m)
    apply grp_moveL_1V.
R: Ring
M: LeftModule R
r: R
m: M
sg_op (r *L - m) (r *L m) = mon_unit
    lhs_V napply lm_dist_l.
R: Ring
M: LeftModule R
r: R
m: M
r *L (- m + m) = mon_unit
    rhs_V napply lm_zero_r.
R: Ring
M: LeftModule R
r: R
m: M
r *L (- m + m) = r *L 0
    f_ap.
R: Ring
M: LeftModule R
r: R
m: M
- m + m = 0
    apply grp_inv_l.
  Defined.

End LeftModuleFacts.

(** Every ring [R] is a left [R]-module over itself. *)
Instance isleftmodule_ring (R : Ring) : IsLeftModule R R.
R: Ring
IsLeftModule R R
Proof.
R: Ring
IsLeftModule R R
  rapply Build_IsLeftModule.
Defined.

(** ** Right Modules *)

(** An abelian group [M] is a right [R]-module when it is a left [R^op]-module. *)
Class IsRightModule (R : Ring) (M : AbGroup)
  := isleftmodule_op_isrightmodule :: IsLeftModule (rng_op R) M.

(** [ract] (right-action) that takes an element [m : M] and an element [r : R] and returns an element [ract m r : M] which we also denote [m *R r]. *)
Definition ract {R : Ring} {M : AbGroup} `{!IsRightModule R M}
  : M -> R -> M
  := fun m r => lact (R:=rng_op R) r m.

Infix "*R" := ract.

(** A right module is a left module over the opposite ring. *)
Definition RightModule (R : Ring) := LeftModule (rng_op R).

(** Right modules are right modules. *)
Instance rm_ract {R : Ring} {M : RightModule R} : IsRightModule R M
  := lm_lact (rng_op R) M.

Section RightModuleAxioms.
  Context {R : Ring} {M : RightModule R} (m n : M) (r s : R).
  (** Here we state the module axioms in a readable form for direct use. *)

  Definition rm_dist_r : (m + n) *R r = m *R r + n *R r
    := lm_dist_l (R:=rng_op R) r m n.
  Definition rm_dist_l : m *R (r + s) = m *R r + m *R s
    := lm_dist_r (R:=rng_op R) r s m.
  Definition rm_assoc : (m *R r) *R s = m *R (r * s)
    := lm_assoc (R:=rng_op R) s r m.
  Definition rm_unit : m *R 1 = m
    := lm_unit (R:=rng_op R) m.

End RightModuleAxioms.

(** ** Facts about right modules *)

Section RightModuleFacts.
  Context {R : Ring} {M : RightModule R} (m : M) (r : R).

  (** The right action on zero is zero. *)
  Definition rm_zero_l : (0 : M) *R r = 0
    := lm_zero_r (R:=rng_op R) r.

  (** The right action of zero is zero. *)
  Definition rm_zero_r : m *R 0 = 0
    := lm_zero_l (R:=rng_op R) m.

  (** The right action of [-1] is the additive inverse. *)
  Definition rm_minus_one : m *R -1 = -m
    := lm_minus_one (R:=rng_op R) m. 

  (** The right action of [r] on the additive inverse of [m] is the additive inverse of the right action of [r] on [m]. *)
  Definition rm_neg : -m *R r = - (m *R r)
    := lm_neg (R:=rng_op R) r m. 

End RightModuleFacts.

(** Every ring [R] is a right [R]-module over itself. *)
Instance isrightmodule_ring (R : Ring) : IsRightModule R R
  := isleftmodule_ring (rng_op R).

(** ** Submodules *)

(** A subgroup of a left R-module is a left submodule if it is closed under the action of R. *)
Class IsLeftSubmodule {R : Ring} {M : LeftModule R} (N : M -> Type) := {
  ils_issubgroup :: IsSubgroup N;
  is_left_submodule : forall r m, N m -> N (r *L m);
}.

(** A subgroup of a right R-module is a right submodule if it is a left submodule over the opposite ring. *)
Class IsRightSubmodule {R : Ring} {M : RightModule R} (N : M -> Type)
  := isleftsubmodule_op_isrightsubmodule :: IsLeftSubmodule (R:=rng_op R) N.

(** A left submodule is a subgroup of the abelian group closed under the left action of R. *)
Record LeftSubmodule {R : Ring} (M : LeftModule R) := {
  lsm_carrier :> M -> Type;
  lsm_submodule :: IsLeftSubmodule lsm_carrier;
}.

(** A right submodule is a subgroup of the abelian group closed under the right action of R. *)
Definition RightSubmodule {R : Ring} (M : RightModule R)
  := LeftSubmodule (R:=rng_op R) M.

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

Definition subgroup_rightsubmodule {R : Ring} {M : RightModule R}
  : RightSubmodule M -> Subgroup M
  := idmap.
Coercion subgroup_rightsubmodule : RightSubmodule >-> Subgroup.

(** Left submodules inherit the left R-module structure of their parent. *)
Instance isleftmodule_leftsubmodule {R : Ring}
  {M : LeftModule R} (N : LeftSubmodule M)
  : IsLeftModule R N.
R: Ring
M: LeftModule R
N: LeftSubmodule M
IsLeftModule R N
Proof.
R: Ring
M: LeftModule R
N: LeftSubmodule M
IsLeftModule R N
  snapply Build_IsLeftModule.
R: Ring
M: LeftModule R
N: LeftSubmodule M
R -> N -> N
R: Ring
M: LeftModule R
N: LeftSubmodule M
LeftHeteroDistribute ?lact plus plus
R: Ring
M: LeftModule R
N: LeftSubmodule M
RightHeteroDistribute ?lact plus plus
R: Ring
M: LeftModule R
N: LeftSubmodule M
HeteroAssociative ?lact ?lact ?lact mult
R: Ring
M: LeftModule R
N: LeftSubmodule M
LeftIdentity ?lact 1
  -
R: Ring
M: LeftModule R
N: LeftSubmodule M
R -> N -> N intros r [n n_in_N].
R: Ring
M: LeftModule R
N: LeftSubmodule M
r: R
n: M
n_in_N: N n
N
    exists (r *L n).
R: Ring
M: LeftModule R
N: LeftSubmodule M
r: R
n: M
n_in_N: N n
N (r *L n)
    by apply lsm_submodule.
  -
R: Ring
M: LeftModule R
N: LeftSubmodule M
LeftHeteroDistribute
  (fun (r : R) (X : N) =>
   (fun (n : M) (n_in_N : N n) =>
    (r *L n;
    let X0 :=
      fun (M : LeftModule R) (l : LeftSubmodule M) =>
      is_left_submodule in
    X0 M N r n n_in_N)) X.1 X.2) plus plus intros r [n] [m]; apply path_sigma_hprop.
R: Ring
M: LeftModule R
N: LeftSubmodule M
r: R
n: M
proj2: N n
m: M
proj0: N m
(r *L ((n; proj2) + (m; proj0)).1;
let X :=
  fun (M : LeftModule R) (l : LeftSubmodule M) =>
  is_left_submodule in
X M N r ((n; proj2) + (m; proj0)).1
  ((n; proj2) + (m; proj0)).2).1 =
((r *L n;
 let X :=
   fun (M : LeftModule R) (l : LeftSubmodule M) =>
   is_left_submodule in
 X M N r n proj2) +
 (r *L m;
 let X :=
   fun (M : LeftModule R) (l : LeftSubmodule M) =>
   is_left_submodule in
 X M N r m proj0)).1
    apply lact_left_dist.
  -
R: Ring
M: LeftModule R
N: LeftSubmodule M
RightHeteroDistribute
  (fun (r : R) (X : N) =>
   (fun (n : M) (n_in_N : N n) =>
    (r *L n;
    let X0 :=
      fun (M : LeftModule R) (l : LeftSubmodule M) =>
      is_left_submodule in
    X0 M N r n n_in_N)) X.1 X.2) plus plus intros r s [n]; apply path_sigma_hprop.
R: Ring
M: LeftModule R
N: LeftSubmodule M
r, s: R
n: M
proj2: N n
((r + s) *L n;
let X :=
  fun (M : LeftModule R) (l : LeftSubmodule M) =>
  is_left_submodule in
X M N (r + s) n proj2).1 =
((r *L n;
 let X :=
   fun (M : LeftModule R) (l : LeftSubmodule M) =>
   is_left_submodule in
 X M N r n proj2) +
 (s *L n;
 let X :=
   fun (M : LeftModule R) (l : LeftSubmodule M) =>
   is_left_submodule in
 X M N s n proj2)).1
    apply lact_right_dist.
  -
R: Ring
M: LeftModule R
N: LeftSubmodule M
HeteroAssociative
  (fun (r : R) (X : N) =>
   (fun (n : M) (n_in_N : N n) =>
    (r *L n;
    let X0 :=
      fun (M : LeftModule R) (l : LeftSubmodule M) =>
      is_left_submodule in
    X0 M N r n n_in_N)) X.1 X.2)
  (fun (r : R) (X : N) =>
   (fun (n : M) (n_in_N : N n) =>
    (r *L n;
    let X0 :=
      fun (M : LeftModule R) (l : LeftSubmodule M) =>
      is_left_submodule in
    X0 M N r n n_in_N)) X.1 X.2)
  (fun (r : R) (X : N) =>
   (fun (n : M) (n_in_N : N n) =>
    (r *L n;
    let X0 :=
      fun (M : LeftModule R) (l : LeftSubmodule M) =>
      is_left_submodule in
    X0 M N r n n_in_N)) X.1 X.2) mult intros r s [n]; apply path_sigma_hprop.
R: Ring
M: LeftModule R
N: LeftSubmodule M
r, s: R
n: M
proj2: N n
(r *L (s *L n);
let X :=
  fun (M : LeftModule R) (l : LeftSubmodule M) =>
  is_left_submodule in
X M N r (s *L n)
  (let X0 :=
     fun (M : LeftModule R) (l : LeftSubmodule M) =>
     is_left_submodule in
   X0 M N s n proj2)).1 =
((r * s) *L n;
let X :=
  fun (M : LeftModule R) (l : LeftSubmodule M) =>
  is_left_submodule in
X M N (r * s) n proj2).1
    apply lact_assoc.
  -
R: Ring
M: LeftModule R
N: LeftSubmodule M
LeftIdentity
  (fun (r : R) (X : N) =>
   (fun (n : M) (n_in_N : N n) =>
    (r *L n;
    let X0 :=
      fun (M : LeftModule R) (l : LeftSubmodule M) =>
      is_left_submodule in
    X0 M N r n n_in_N)) X.1 X.2) 1 intros [n]; apply path_sigma_hprop.
R: Ring
M: LeftModule R
N: LeftSubmodule M
n: M
proj2: N n
(1 *L n;
let X :=
  fun (M : LeftModule R) (l : LeftSubmodule M) =>
  is_left_submodule in
X M N 1 n proj2).1 = (n; proj2).1
    apply lact_unit.
Defined.

(** Right submodules inherit the right R-module structure of their parent. *)
Instance isrightmodule_rightsubmodule {R : Ring}
  {M : RightModule R} (N : RightSubmodule M)
  : IsRightModule R N
  := isleftmodule_leftsubmodule (R:=rng_op R) N.

(** Any left submodule of a left R-module is a left R-module. *)
Definition leftmodule_leftsubmodule {R : Ring}
  {M : LeftModule R} (N : LeftSubmodule M)
  : LeftModule R
  := Build_LeftModule R N _.
Coercion leftmodule_leftsubmodule : LeftSubmodule >-> LeftModule.

(** Any right submodule of a right R-module is a right R-module. *)
Definition rightmodule_rightsubmodule {R : Ring}
  {M : RightModule R} (N : RightSubmodule M)
  : RightModule R
  := N.
Coercion rightmodule_rightsubmodule : RightSubmodule >-> RightModule.

(** The submodule criterion. This is a convenient way to build submodules. *)
Definition 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.
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
IsLeftSubmodule H
Proof.
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
IsLeftSubmodule H
  snapply Build_IsLeftSubmodule.
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
IsSubgroup H
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
forall (r : R) (m : M), H m -> H (r *L m)
  -
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
IsSubgroup H snapply Build_IsSubgroup'.
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
forall x : M, IsHProp (H x)
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
H mon_unit
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
forall x y : M, H x -> H y -> H (sg_op x (inv y))
    +
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
forall x : M, IsHProp (H x) exact _.
    +
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
H mon_unit exact z.
    +
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
forall x y : M, H x -> H y -> H (sg_op x (inv y)) intros x y hx hy.
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
x, y: M
hx: H x
hy: H y
H (sg_op x (inv y))
      change (sg_op ?x ?y) with (x + y).
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
x, y: M
hx: H x
hy: H y
H (x + inv y)
      pose proof (p := c (-1) x y hx hy).
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
x, y: M
hx: H x
hy: H y
p: H (x + -1 *L y)
H (x + inv y)
      rewrite lm_minus_one in p.
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
x, y: M
hx: H x
hy: H y
p: H (x - y)
H (x + inv y)
      exact p.
  -
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
forall (r : R) (m : M), H m -> H (r *L m) intros r m hm.
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
r: R
m: M
hm: H m
H (r *L m)
    rewrite <- (grp_unit_l).
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
r: R
m: M
hm: H m
H (sg_op mon_unit (r *L m))
    by apply c.
Defined.

Definition 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 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.
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
LeftSubmodule M
Proof.
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
LeftSubmodule M
  pose (p := Build_IsLeftSubmodule' H z c).
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
p:= Build_IsLeftSubmodule' H z c: IsLeftSubmodule H
LeftSubmodule M
  snapply Build_LeftSubmodule.
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
p:= Build_IsLeftSubmodule' H z c: IsLeftSubmodule H
M -> Type
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
p:= Build_IsLeftSubmodule' H z c: IsLeftSubmodule H
IsLeftSubmodule ?lsm_carrier
  1: snapply (Build_Subgroup _ H).
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
p:= Build_IsLeftSubmodule' H z c: IsLeftSubmodule H
IsSubgroup H
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
p:= Build_IsLeftSubmodule' H z c: IsLeftSubmodule H
IsLeftSubmodule
  {|
    subgroup_pred := H;
    subgroup_issubgroup := ?subgroup_issubgroup
  |}
  2: exact p.
R: Ring
M: LeftModule R
H: M -> Type
H0: forall x : M, IsHProp (H x)
z: H 0
c: forall (r : R) (n m : M),
H n -> H m -> H (n + r *L m)
p:= Build_IsLeftSubmodule' H z c: IsLeftSubmodule H
IsSubgroup H
  rapply ils_issubgroup.
Defined.

Definition Build_RightSubmodule {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.

(** ** R-module homomorphisms *)

(** A left module homomorphism is a group homomorphism that commutes with the left action of R. *)
Record LeftModuleHomomorphism {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;
}.

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

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

Definition rm_homo_ract {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 lm_homo_id {R : Ring} (M : LeftModule R) : LeftModuleHomomorphism M M.
R: Ring
M: LeftModule R
LeftModuleHomomorphism M M
Proof.
R: Ring
M: LeftModule R
LeftModuleHomomorphism M M
  snapply Build_LeftModuleHomomorphism.
R: Ring
M: LeftModule R
GroupHomomorphism M M
R: Ring
M: LeftModule R
forall (r : R) (m : M),
?lm_homo_map (r *L m) = r *L ?lm_homo_map m
  -
R: Ring
M: LeftModule R
GroupHomomorphism M M exact grp_homo_id.
  -
R: Ring
M: LeftModule R
forall (r : R) (m : M),
grp_homo_id (r *L m) = r *L grp_homo_id m reflexivity.
Defined.

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

Definition lm_homo_compose {R : Ring} {M N L : LeftModule R}
  : LeftModuleHomomorphism N L -> LeftModuleHomomorphism M N
  -> LeftModuleHomomorphism M L.
R: Ring
M, N, L: LeftModule R
LeftModuleHomomorphism N L ->
LeftModuleHomomorphism M N ->
LeftModuleHomomorphism M L
Proof.
R: Ring
M, N, L: LeftModule R
LeftModuleHomomorphism N L ->
LeftModuleHomomorphism M N ->
LeftModuleHomomorphism M L
  intros f g.
R: Ring
M, N, L: LeftModule R
f: LeftModuleHomomorphism N L
g: LeftModuleHomomorphism M N
LeftModuleHomomorphism M L
  snapply Build_LeftModuleHomomorphism.
R: Ring
M, N, L: LeftModule R
f: LeftModuleHomomorphism N L
g: LeftModuleHomomorphism M N
GroupHomomorphism M L
R: Ring
M, N, L: LeftModule R
f: LeftModuleHomomorphism N L
g: LeftModuleHomomorphism M N
forall (r : R) (m : M),
?lm_homo_map (r *L m) = r *L ?lm_homo_map m
  -
R: Ring
M, N, L: LeftModule R
f: LeftModuleHomomorphism N L
g: LeftModuleHomomorphism M N
GroupHomomorphism M L exact (grp_homo_compose f g).
  -
R: Ring
M, N, L: LeftModule R
f: LeftModuleHomomorphism N L
g: LeftModuleHomomorphism M N
forall (r : R) (m : M),
grp_homo_compose f g (r *L m) =
r *L grp_homo_compose f g m intros r m.
R: Ring
M, N, L: LeftModule R
f: LeftModuleHomomorphism N L
g: LeftModuleHomomorphism M N
r: R
m: M
grp_homo_compose f g (r *L m) =
r *L grp_homo_compose f g m
    rhs_V napply lm_homo_lact.
R: Ring
M, N, L: LeftModule R
f: LeftModuleHomomorphism N L
g: LeftModuleHomomorphism M N
r: R
m: M
grp_homo_compose f g (r *L m) = f (r *L g m)
    apply (ap f).
R: Ring
M, N, L: LeftModule R
f: LeftModuleHomomorphism N L
g: LeftModuleHomomorphism M N
r: R
m: M
g (r *L m) = r *L g m
    apply lm_homo_lact.
Defined.

Definition rm_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).

(** Smart constructor for building left module homomorphisms from a map. *)
Definition 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.
R: Ring
M, N: LeftModule R
f: M -> N
p: forall (r : R) (x y : M),
f (r *L x + y) = r *L f x + f y
LeftModuleHomomorphism M N
Proof.
R: Ring
M, N: LeftModule R
f: M -> N
p: forall (r : R) (x y : M),
f (r *L x + y) = r *L f x + f y
LeftModuleHomomorphism M N
  snapply Build_LeftModuleHomomorphism.
R: Ring
M, N: LeftModule R
f: M -> N
p: forall (r : R) (x y : M),
f (r *L x + y) = r *L f x + f y
GroupHomomorphism M N
R: Ring
M, N: LeftModule R
f: M -> N
p: forall (r : R) (x y : M),
f (r *L x + y) = r *L f x + f y
forall (r : R) (m : M),
?lm_homo_map (r *L m) = r *L ?lm_homo_map m
  -
R: Ring
M, N: LeftModule R
f: M -> N
p: forall (r : R) (x y : M),
f (r *L x + y) = r *L f x + f y
GroupHomomorphism M N snapply Build_GroupHomomorphism.
R: Ring
M, N: LeftModule R
f: M -> N
p: forall (r : R) (x y : M),
f (r *L x + y) = r *L f x + f y
M -> N
R: Ring
M, N: LeftModule R
f: M -> N
p: forall (r : R) (x y : M),
f (r *L x + y) = r *L f x + f y
IsSemiGroupPreserving ?grp_homo_map
    +
R: Ring
M, N: LeftModule R
f: M -> N
p: forall (r : R) (x y : M),
f (r *L x + y) = r *L f x + f y
M -> N exact f.
    +
R: Ring
M, N: LeftModule R
f: M -> N
p: forall (r : R) (x y : M),
f (r *L x + y) = r *L f x + f y
IsSemiGroupPreserving f intros x y.
R: Ring
M, N: LeftModule R
f: M -> N
p: forall (r : R) (x y : M),
f (r *L x + y) = r *L f x + f y
x, y: M
f (sg_op x y) = sg_op (f x) (f y)
      rewrite <- (lm_unit (f x)).
R: Ring
M, N: LeftModule R
f: M -> N
p: forall (r : R) (x y : M),
f (r *L x + y) = r *L f x + f y
x, y: M
f (sg_op x y) = sg_op (1 *L f x) (f y)
      set (lact 1 (f x)).
R: Ring
M, N: LeftModule R
f: M -> N
p: forall (r : R) (x y : M),
f (r *L x + y) = r *L f x + f y
x, y: M
g:= 1 *L f x: N
f (sg_op x y) = sg_op g (f y)
      rewrite <- (lm_unit x).
R: Ring
M, N: LeftModule R
f: M -> N
p: forall (r : R) (x y : M),
f (r *L x + y) = r *L f x + f y
x, y: M
g:= 1 *L f x: N
f (sg_op (1 *L x) y) = sg_op g (f y)
      apply p.
  -
R: Ring
M, N: LeftModule R
f: M -> N
p: forall (r : R) (x y : M),
f (r *L x + y) = r *L f x + f y
forall (r : R) (m : M),
{|
  grp_homo_map := f;
  issemigrouppreserving_grp_homo :=
    (fun x y : M =>
     internal_paths_rew
       (fun g : N => f (sg_op x y) = sg_op g (f y))
       (let g := 1 *L f x in
        internal_paths_rew
          (fun x0 : M =>
           f (sg_op x0 y) = sg_op g (f y)) (p 1 x y)
          (lm_unit x)) (lm_unit (f x)))
    :
    IsSemiGroupPreserving f
|} (r *L m) =
r *L
{|
  grp_homo_map := f;
  issemigrouppreserving_grp_homo :=
    (fun x y : M =>
     internal_paths_rew
       (fun g : N => f (sg_op x y) = sg_op g (f y))
       (let g := 1 *L f x in
        internal_paths_rew
          (fun x0 : M =>
           f (sg_op x0 y) = sg_op g (f y)) (p 1 x y)
          (lm_unit x)) (lm_unit (f x)))
    :
    IsSemiGroupPreserving f
|} m intros r m.
R: Ring
M, N: LeftModule R
f: M -> N
p: forall (r : R) (x y : M),
f (r *L x + y) = r *L f x + f y
r: R
m: M
{|
  grp_homo_map := f;
  issemigrouppreserving_grp_homo :=
    (fun x y : M =>
     internal_paths_rew
       (fun g : N => f (sg_op x y) = sg_op g (f y))
       (let g := 1 *L f x in
        internal_paths_rew
          (fun x0 : M =>
           f (sg_op x0 y) = sg_op g (f y)) (p 1 x y)
          (lm_unit x)) (lm_unit (f x)))
    :
    IsSemiGroupPreserving f
|} (r *L m) =
r *L
{|
  grp_homo_map := f;
  issemigrouppreserving_grp_homo :=
    (fun x y : M =>
     internal_paths_rew
       (fun g : N => f (sg_op x y) = sg_op g (f y))
       (let g := 1 *L f x in
        internal_paths_rew
          (fun x0 : M =>
           f (sg_op x0 y) = sg_op g (f y)) (p 1 x y)
          (lm_unit x)) (lm_unit (f x)))
    :
    IsSemiGroupPreserving f
|} m
    simpl.
R: Ring
M, N: LeftModule R
f: M -> N
p: forall (r : R) (x y : M),
f (r *L x + y) = r *L f x + f y
r: R
m: M
f (r *L m) = r *L f m
    rewrite <- (grp_unit_r (lact r m)).
R: Ring
M, N: LeftModule R
f: M -> N
p: forall (r : R) (x y : M),
f (r *L x + y) = r *L f x + f y
r: R
m: M
f (sg_op (r *L m) mon_unit) = r *L f m
    rewrite p.
R: Ring
M, N: LeftModule R
f: M -> N
p: forall (r : R) (x y : M),
f (r *L x + y) = r *L f x + f y
r: R
m: M
r *L f m + f mon_unit = r *L f m
    rhs_V napply grp_unit_r.
R: Ring
M, N: LeftModule R
f: M -> N
p: forall (r : R) (x y : M),
f (r *L x + y) = r *L f x + f y
r: R
m: M
r *L f m + f mon_unit = sg_op (r *L f m) mon_unit
    apply grp_cancelL.
R: Ring
M, N: LeftModule R
f: M -> N
p: forall (r : R) (x y : M),
f (r *L x + y) = r *L f x + f y
r: R
m: M
f mon_unit = mon_unit
    specialize (p 1 0 0).
R: Ring
M, N: LeftModule R
f: M -> N
p: f (1 *L 0 + 0) = 1 *L f 0 + f 0
r: R
m: M
f mon_unit = mon_unit
    rewrite 2 lm_unit in p.
R: Ring
M, N: LeftModule R
f: M -> N
p: f (0 + 0) = f 0 + f 0
r: R
m: M
f mon_unit = mon_unit
    apply (grp_cancelL1 (z := f 0)).
R: Ring
M, N: LeftModule R
f: M -> N
p: f (0 + 0) = f 0 + f 0
r: R
m: M
sg_op (f 0) (f mon_unit) = f 0
    lhs_V napply p.
R: Ring
M, N: LeftModule R
f: M -> N
p: f (0 + 0) = f 0 + f 0
r: R
m: M
f (0 + 0) = f 0
    apply ap.
R: Ring
M, N: LeftModule R
f: M -> N
p: f (0 + 0) = f 0 + f 0
r: R
m: M
0 + 0 = 0
    apply grp_unit_l.
Defined.

Definition 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.

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

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

Definition 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.
R: Ring
M, N: LeftModule R
f: GroupIsomorphism M N
p: forall (r : R) (x : M), f (r *L x) = r *L f x
LeftModuleIsomorphism M N
Proof.
R: Ring
M, N: LeftModule R
f: GroupIsomorphism M N
p: forall (r : R) (x : M), f (r *L x) = r *L f x
LeftModuleIsomorphism M N
  snapply Build_LeftModuleIsomorphism.
R: Ring
M, N: LeftModule R
f: GroupIsomorphism M N
p: forall (r : R) (x : M), f (r *L x) = r *L f x
LeftModuleHomomorphism M N
R: Ring
M, N: LeftModule R
f: GroupIsomorphism M N
p: forall (r : R) (x : M), f (r *L x) = r *L f x
IsEquiv ?lm_iso_map
  -
R: Ring
M, N: LeftModule R
f: GroupIsomorphism M N
p: forall (r : R) (x : M), f (r *L x) = r *L f x
LeftModuleHomomorphism M N snapply Build_LeftModuleHomomorphism.
R: Ring
M, N: LeftModule R
f: GroupIsomorphism M N
p: forall (r : R) (x : M), f (r *L x) = r *L f x
GroupHomomorphism M N
R: Ring
M, N: LeftModule R
f: GroupIsomorphism M N
p: forall (r : R) (x : M), f (r *L x) = r *L f x
forall (r : R) (m : M),
?lm_homo_map (r *L m) = r *L ?lm_homo_map m
    +
R: Ring
M, N: LeftModule R
f: GroupIsomorphism M N
p: forall (r : R) (x : M), f (r *L x) = r *L f x
GroupHomomorphism M N exact f.
    +
R: Ring
M, N: LeftModule R
f: GroupIsomorphism M N
p: forall (r : R) (x : M), f (r *L x) = r *L f x
forall (r : R) (m : M), f (r *L m) = r *L f m exact p.
  -
R: Ring
M, N: LeftModule R
f: GroupIsomorphism M N
p: forall (r : R) (x : M), f (r *L x) = r *L f x
IsEquiv {| lm_homo_map := f; lm_homo_lact := p |} exact _.
Defined.

Definition 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 lm_iso_inverse {R : Ring} {M N : LeftModule R}
  : LeftModuleIsomorphism M N -> LeftModuleIsomorphism N M.
R: Ring
M, N: LeftModule R
LeftModuleIsomorphism M N -> LeftModuleIsomorphism N M
Proof.
R: Ring
M, N: LeftModule R
LeftModuleIsomorphism M N -> LeftModuleIsomorphism N M
  intros f.
R: Ring
M, N: LeftModule R
f: LeftModuleIsomorphism M N
LeftModuleIsomorphism N M
  snapply Build_LeftModuleIsomorphism.
R: Ring
M, N: LeftModule R
f: LeftModuleIsomorphism M N
LeftModuleHomomorphism N M
R: Ring
M, N: LeftModule R
f: LeftModuleIsomorphism M N
IsEquiv ?lm_iso_map
  -
R: Ring
M, N: LeftModule R
f: LeftModuleIsomorphism M N
LeftModuleHomomorphism N M snapply Build_LeftModuleHomomorphism'.
R: Ring
M, N: LeftModule R
f: LeftModuleIsomorphism M N
N -> M
R: Ring
M, N: LeftModule R
f: LeftModuleIsomorphism M N
forall (r : R) (x y : N),
?f (r *L x + y) = r *L ?f x + ?f y
    +
R: Ring
M, N: LeftModule R
f: LeftModuleIsomorphism M N
N -> M exact f^-1.
    +
R: Ring
M, N: LeftModule R
f: LeftModuleIsomorphism M N
forall (r : R) (x y : N),
f^-1 (r *L x + y) = r *L f^-1 x + f^-1 y intros r m n.
R: Ring
M, N: LeftModule R
f: LeftModuleIsomorphism M N
r: R
m, n: N
f^-1 (r *L m + n) = r *L f^-1 m + f^-1 n
      apply moveR_equiv_V.
R: Ring
M, N: LeftModule R
f: LeftModuleIsomorphism M N
r: R
m, n: N
r *L m + n = f (r *L f^-1 m + f^-1 n) 
      rhs napply grp_homo_op.
R: Ring
M, N: LeftModule R
f: LeftModuleIsomorphism M N
r: R
m, n: N
r *L m + n = sg_op (f (r *L f^-1 m)) (f (f^-1 n))
      symmetry.
R: Ring
M, N: LeftModule R
f: LeftModuleIsomorphism M N
r: R
m, n: N
sg_op (f (r *L f^-1 m)) (f (f^-1 n)) = r *L m + n
      f_ap.
R: Ring
M, N: LeftModule R
f: LeftModuleIsomorphism M N
r: R
m, n: N
f (r *L f^-1 m) = r *L m
R: Ring
M, N: LeftModule R
f: LeftModuleIsomorphism M N
r: R
m, n: N
f (f^-1 n) = n
      2: apply eisretr.
R: Ring
M, N: LeftModule R
f: LeftModuleIsomorphism M N
r: R
m, n: N
f (r *L f^-1 m) = r *L m
      lhs napply lm_homo_lact.
R: Ring
M, N: LeftModule R
f: LeftModuleIsomorphism M N
r: R
m, n: N
r *L f (f^-1 m) = r *L m
      apply ap.
R: Ring
M, N: LeftModule R
f: LeftModuleIsomorphism M N
r: R
m, n: N
f (f^-1 m) = m
      apply eisretr.
  -
R: Ring
M, N: LeftModule R
f: LeftModuleIsomorphism M N
IsEquiv
  (Build_LeftModuleHomomorphism' f^-1
     (fun (r : R) (m n : N) =>
      moveR_equiv_V (r *L m + n)
        (r *L f^-1 m + f^-1 n)
        ((ap11
            (ap11 1
               (lm_homo_lact M N f r (f^-1 m) @
                ap (lact r) (eisretr f m)))
            (eisretr f n))^ @
         (grp_homo_op f (r *L f^-1 m) (f^-1 n))^))) exact _.
Defined.

Definition rm_iso_inverse {R : Ring} {M N : RightModule R}
  : RightModuleIsomorphism M N -> RightModuleIsomorphism N M
  := lm_iso_inverse (R:=rng_op R).

(** ** Category of left and right R-modules *)

(** TODO: define as a displayed category over Ring *)

(** *** Category of left R-modules *)

Instance isgraph_leftmodule {R : Ring} : IsGraph (LeftModule R)
  := Build_IsGraph _ LeftModuleHomomorphism.

Instance is01cat_leftmodule {R : Ring} : Is01Cat (LeftModule R)
  := Build_Is01Cat _ _ lm_homo_id (@lm_homo_compose R).

Instance is2graph_leftmodule {R : Ring} : Is2Graph (LeftModule R)
  := fun M N => isgraph_induced (@lm_homo_map R M N).

Instance is1cat_leftmodule {R : Ring} : Is1Cat (LeftModule R).
R: Ring
Is1Cat (LeftModule R)
Proof.
R: Ring
Is1Cat (LeftModule R)
  snapply Build_Is1Cat'.
R: Ring
forall a b : LeftModule R, Is01Cat (a $-> b)
R: Ring
forall a b : LeftModule R, Is0Gpd (a $-> b)
R: Ring
forall (a b c : LeftModule R) (g : b $-> c),
Is0Functor (cat_postcomp a g)
R: Ring
forall (a b c : LeftModule R) (f : a $-> b),
Is0Functor (cat_precomp c f)
R: Ring
forall (a b c d : LeftModule R) (f : a $-> b)
(g : b $-> c) (h : c $-> d),
h $o g $o f $== h $o (g $o f)
R: Ring
forall (a b : LeftModule R) (f : a $-> b),
Id b $o f $== f
R: Ring
forall (a b : LeftModule R) (f : a $-> b),
f $o Id a $== f
  -
R: Ring
forall a b : LeftModule R, Is01Cat (a $-> b) intros M N; rapply is01cat_induced.
  -
R: Ring
forall a b : LeftModule R, Is0Gpd (a $-> b) intros M N; rapply is0gpd_induced.
  -
R: Ring
forall (a b c : LeftModule R) (g : b $-> c),
Is0Functor (cat_postcomp a g) intros M N L h.
R: Ring
M, N, L: LeftModule R
h: N $-> L
Is0Functor (cat_postcomp M h)
    snapply Build_Is0Functor.
R: Ring
M, N, L: LeftModule R
h: N $-> L
forall a b : M $-> N,
(a $-> b) -> cat_postcomp M h a $-> cat_postcomp M h b
    intros f g p m.
R: Ring
M, N, L: LeftModule R
h: N $-> L
f, g: M $-> N
p: f $-> g
m: M
cat_postcomp M h f m = cat_postcomp M h g m
    exact (ap h (p m)).
  -
R: Ring
forall (a b c : LeftModule R) (f : a $-> b),
Is0Functor (cat_precomp c f) intros M N L f.
R: Ring
M, N, L: LeftModule R
f: M $-> N
Is0Functor (cat_precomp L f)
    snapply Build_Is0Functor.
R: Ring
M, N, L: LeftModule R
f: M $-> N
forall a b : N $-> L,
(a $-> b) -> cat_precomp L f a $-> cat_precomp L f b
    intros g h p m.
R: Ring
M, N, L: LeftModule R
f: M $-> N
g, h: N $-> L
p: g $-> h
m: M
cat_precomp L f g m = cat_precomp L f h m
    exact (p (f m)).
  -
R: Ring
forall (a b c d : LeftModule R) (f : a $-> b)
(g : b $-> c) (h : c $-> d),
h $o g $o f $== h $o (g $o f) simpl; reflexivity.
  -
R: Ring
forall (a b : LeftModule R) (f : a $-> b),
Id b $o f $== f simpl; reflexivity.
  -
R: Ring
forall (a b : LeftModule R) (f : a $-> b),
f $o Id a $== f simpl; reflexivity.
Defined.

Instance hasequivs_leftmodule {R : Ring} : HasEquivs (LeftModule R).
R: Ring
HasEquivs (LeftModule R)
Proof.
R: Ring
HasEquivs (LeftModule R)
  snapply Build_HasEquivs.
R: Ring
LeftModule R -> LeftModule R -> Type
R: Ring
forall a b : LeftModule R, (a $-> b) -> Type
R: Ring
forall a b : LeftModule R, ?CatEquiv' a b -> a $-> b
R: Ring
forall (a b : LeftModule R) (f : ?CatEquiv' a b),
?CatIsEquiv' a b (?cate_fun' a b f)
R: Ring
forall (a b : LeftModule R) (f : a $-> b),
?CatIsEquiv' a b f -> ?CatEquiv' a b
R: Ring
forall (a b : LeftModule R) (f : a $-> b)
(fe : ?CatIsEquiv' a b f),
?cate_fun' a b (?cate_buildequiv' a b f fe) $== f
R: Ring
forall a b : LeftModule R, ?CatEquiv' a b -> b $-> a
R: Ring
forall (a b : LeftModule R) (f : ?CatEquiv' a b),
?cate_inv' a b f $o ?cate_fun' a b f $== Id a
R: Ring
forall (a b : LeftModule R) (f : ?CatEquiv' a b),
?cate_fun' a b f $o ?cate_inv' a b f $== Id b
R: Ring
forall (a b : LeftModule R) (f : a $-> b)
(g : b $-> a),
f $o g $== Id b ->
g $o f $== Id a -> ?CatIsEquiv' a b f
  -
R: Ring
LeftModule R -> LeftModule R -> Type exact LeftModuleIsomorphism.
  -
R: Ring
forall a b : LeftModule R, (a $-> b) -> Type intros M N; exact IsEquiv.
  -
R: Ring
forall a b : LeftModule R,
LeftModuleIsomorphism a b -> a $-> b intros M N f; exact f.
  -
R: Ring
forall (a b : LeftModule R)
(f : LeftModuleIsomorphism a b),
(fun (M N : LeftModule R) (f0 : M $-> N) => IsEquiv f0)
  a b
  ((fun (M N : LeftModule R)
      (f0 : LeftModuleIsomorphism M N) =>
    lm_iso_map M N f0) a b f) simpl; exact _.
  -
R: Ring
forall (a b : LeftModule R) (f : a $-> b),
(fun (M N : LeftModule R) (f0 : M $-> N) => IsEquiv f0)
  a b f -> LeftModuleIsomorphism a b apply Build_LeftModuleIsomorphism.
  -
R: Ring
forall (a b : LeftModule R) (f : a $-> b)
(fe : (fun (M N : LeftModule R) (f0 : M $-> N) =>
       IsEquiv f0) a b f),
(fun (M N : LeftModule R)
   (f0 : LeftModuleIsomorphism M N) =>
 lm_iso_map M N f0) a b
  {| lm_iso_map := f; isequiv_lm_iso_map := fe |} $==
f reflexivity.
  -
R: Ring
forall a b : LeftModule R,
LeftModuleIsomorphism a b -> b $-> a intros M N; exact lm_iso_inverse.
  -
R: Ring
forall (a b : LeftModule R)
(f : LeftModuleIsomorphism a b),
(fun (M N : LeftModule R)
   (x : LeftModuleIsomorphism M N) =>
 lm_iso_map N M (lm_iso_inverse x)) a b f $o
(fun (M N : LeftModule R)
   (f0 : LeftModuleIsomorphism M N) =>
 lm_iso_map M N f0) a b f $== Id a intros M N f; apply eissect.
  -
R: Ring
forall (a b : LeftModule R)
(f : LeftModuleIsomorphism a b),
(fun (M N : LeftModule R)
   (f0 : LeftModuleIsomorphism M N) =>
 lm_iso_map M N f0) a b f $o
(fun (M N : LeftModule R)
   (x : LeftModuleIsomorphism M N) =>
 lm_iso_map N M (lm_iso_inverse x)) a b f $== Id b intros M N f; apply eisretr.
  -
R: Ring
forall (a b : LeftModule R) (f : a $-> b)
(g : b $-> a),
f $o g $== Id b ->
g $o f $== Id a ->
(fun (M N : LeftModule R) (f0 : M $-> N) => IsEquiv f0)
  a b f intros M N f g fg gf.
R: Ring
M, N: LeftModule R
f: M $-> N
g: N $-> M
fg: f $o g $== Id N
gf: g $o f $== Id M
(fun (M N : LeftModule R) (f : M $-> N) => IsEquiv f)
  M N f
    exact (isequiv_adjointify f g fg gf).
Defined.

(** *** Category of right R-modules *)

Instance isgraph_rightmodule {R : Ring} : IsGraph (RightModule R)
  := isgraph_leftmodule (R:=rng_op R).

Instance is01cat_rightmodule {R : Ring} : Is01Cat (RightModule R)
  := is01cat_leftmodule (R:=rng_op R).

Instance is2graph_rightmodule {R : Ring} : Is2Graph (RightModule R)
  := is2graph_leftmodule (R:=rng_op R).

Instance is1cat_rightmodule {R : Ring} : Is1Cat (RightModule R)
  := is1cat_leftmodule (R:=rng_op R).

Instance hasequivs_rightmodule {R : Ring} : HasEquivs (RightModule R)
  := hasequivs_leftmodule (R:=rng_op R).

(** ** Kernel of module homomorphism *)

Instance isleftsubmodule_grp_kernel {R : Ring}
  {M N : LeftModule R} (f : M $-> N)
  : IsLeftSubmodule (grp_kernel f).
R: Ring
M, N: LeftModule R
f: M $-> N
IsLeftSubmodule (grp_kernel f)
Proof.
R: Ring
M, N: LeftModule R
f: M $-> N
IsLeftSubmodule (grp_kernel f)
  srapply Build_IsLeftSubmodule.
R: Ring
M, N: LeftModule R
f: M $-> N
forall (r : R) (m : M),
grp_kernel f m -> grp_kernel f (r *L m)
  intros r m n.
R: Ring
M, N: LeftModule R
f: M $-> N
r: R
m: M
n: grp_kernel f m
grp_kernel f (r *L m)
  lhs napply lm_homo_lact.
R: Ring
M, N: LeftModule R
f: M $-> N
r: R
m: M
n: grp_kernel f m
r *L f m = mon_unit
  rhs_V napply (lm_zero_r r).
R: Ring
M, N: LeftModule R
f: M $-> N
r: R
m: M
n: grp_kernel f m
r *L f m = r *L 0
  apply ap.
R: Ring
M, N: LeftModule R
f: M $-> N
r: R
m: M
n: grp_kernel f m
f m = 0
  exact n.
Defined.

Instance isrightsubmodule_grp_kernel {R : Ring}
  {M N : RightModule R} (f : M $-> N)
  : IsRightSubmodule (grp_kernel f)
  := isleftsubmodule_grp_kernel (R:=rng_op R) f.

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

Definition rm_kernel {R : Ring} {M N : RightModule R} (f : M $-> N)
  : RightSubmodule M
  := lm_kernel (R:=rng_op R) f.

(** ** Image of module homomorphism *)

Instance isleftsubmodule_grp_image {R : Ring}
  {M N : LeftModule R} (f : M $-> N)
  : IsLeftSubmodule (grp_image f).
R: Ring
M, N: LeftModule R
f: M $-> N
IsLeftSubmodule (grp_image f)
Proof.
R: Ring
M, N: LeftModule R
f: M $-> N
IsLeftSubmodule (grp_image f)
  srapply Build_IsLeftSubmodule.
R: Ring
M, N: LeftModule R
f: M $-> N
forall (r : R) (m : N),
grp_image f m -> grp_image f (r *L m)
  intros r m; apply Trunc_functor; intros [n p].
R: Ring
M, N: LeftModule R
f: M $-> N
r: R
m: N
n: M
p: f n = m
{x : M & f x = r *L m}
  exists (r *L n).
R: Ring
M, N: LeftModule R
f: M $-> N
r: R
m: N
n: M
p: f n = m
f (r *L n) = r *L m
  lhs napply lm_homo_lact.
R: Ring
M, N: LeftModule R
f: M $-> N
r: R
m: N
n: M
p: f n = m
r *L f n = r *L m
  apply ap.
R: Ring
M, N: LeftModule R
f: M $-> N
r: R
m: N
n: M
p: f n = m
f n = m
  exact p.
Defined.

Instance isrightsubmodule_grp_image {R : Ring}
  {M N : RightModule R} (f : M $-> N)
  : IsRightSubmodule (grp_image f)
  := isleftsubmodule_grp_image (R:=rng_op R) f.

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

Definition rm_image {R : Ring} {M N : RightModule R} (f : M $-> N)
  : RightSubmodule N
  := lm_image (R:=rng_op R) f.

(** ** Quotient Modules *)

(** The quotient abelian group of a module and a submodule has a natural ring action. *)
Instance isleftmodule_quotientabgroup {R : Ring}
  (M : LeftModule R) (N : LeftSubmodule M)
  : IsLeftModule R (QuotientAbGroup M N).
R: Ring
M: LeftModule R
N: LeftSubmodule M
IsLeftModule R (QuotientAbGroup M N)
Proof.
R: Ring
M: LeftModule R
N: LeftSubmodule M
IsLeftModule R (QuotientAbGroup M N)
  snapply Build_IsLeftModule.
R: Ring
M: LeftModule R
N: LeftSubmodule M
R -> QuotientAbGroup M N -> QuotientAbGroup M N
R: Ring
M: LeftModule R
N: LeftSubmodule M
LeftHeteroDistribute ?lact plus plus
R: Ring
M: LeftModule R
N: LeftSubmodule M
RightHeteroDistribute ?lact plus plus
R: Ring
M: LeftModule R
N: LeftSubmodule M
HeteroAssociative ?lact ?lact ?lact mult
R: Ring
M: LeftModule R
N: LeftSubmodule M
LeftIdentity ?lact 1
  -
R: Ring
M: LeftModule R
N: LeftSubmodule M
R -> QuotientAbGroup M N -> QuotientAbGroup M N intros r.
R: Ring
M: LeftModule R
N: LeftSubmodule M
r: R
QuotientAbGroup M N -> QuotientAbGroup M N
    snapply quotient_abgroup_rec.
R: Ring
M: LeftModule R
N: LeftSubmodule M
r: R
GroupHomomorphism M (QuotientAbGroup M N)
R: Ring
M: LeftModule R
N: LeftSubmodule M
r: R
forall n : M, N n -> ?f n = mon_unit
    +
R: Ring
M: LeftModule R
N: LeftSubmodule M
r: R
GroupHomomorphism M (QuotientAbGroup M N) refine (grp_quotient_map $o _).
R: Ring
M: LeftModule R
N: LeftSubmodule M
r: R
M $-> M 
      snapply Build_GroupHomomorphism.
R: Ring
M: LeftModule R
N: LeftSubmodule M
r: R
M -> M
R: Ring
M: LeftModule R
N: LeftSubmodule M
r: R
IsSemiGroupPreserving ?grp_homo_map
      *
R: Ring
M: LeftModule R
N: LeftSubmodule M
r: R
M -> M exact (lact r).
      *
R: Ring
M: LeftModule R
N: LeftSubmodule M
r: R
IsSemiGroupPreserving (lact r) intros x y.
R: Ring
M: LeftModule R
N: LeftSubmodule M
r: R
x, y: M
r *L sg_op x y = sg_op (r *L x) (r *L y)
        apply lm_dist_l.
    +
R: Ring
M: LeftModule R
N: LeftSubmodule M
r: R
forall n : M,
N n ->
(grp_quotient_map $o
 {|
   grp_homo_map := lact r;
   issemigrouppreserving_grp_homo :=
     (fun x y : M => lm_dist_l r x y)
     :
     IsSemiGroupPreserving (lact r)
 |}) n = mon_unit intros n Nn; simpl.
R: Ring
M: LeftModule R
N: LeftSubmodule M
r: R
n: M
Nn: N n
class_of (fun x y : M => N (sg_op (inv x) y)) (r *L n) =
mon_unit
      apply qglue.
R: Ring
M: LeftModule R
N: LeftSubmodule M
r: R
n: M
Nn: N n
in_cosetL N (r *L n) mon_unit
      apply issubgroup_in_inv_op.
R: Ring
M: LeftModule R
N: LeftSubmodule M
r: R
n: M
Nn: N n
N (r *L n)
R: Ring
M: LeftModule R
N: LeftSubmodule M
r: R
n: M
Nn: N n
N mon_unit
      2: exact issubgroup_in_unit.
R: Ring
M: LeftModule R
N: LeftSubmodule M
r: R
n: M
Nn: N n
N (r *L n)
      by apply is_left_submodule.
  -
R: Ring
M: LeftModule R
N: LeftSubmodule M
LeftHeteroDistribute
  (fun r : R =>
   quotient_abgroup_rec N (QuotientAbGroup M N)
     (grp_quotient_map $o
      {|
        grp_homo_map := lact r;
        issemigrouppreserving_grp_homo :=
          (fun x y : M => lm_dist_l r x y)
          :
          IsSemiGroupPreserving (lact r)
      |})
     (fun (n : M) (Nn : N n) =>
      qglue
        (issubgroup_in_inv_op (r *L n) mon_unit
           (is_left_submodule r n Nn)
           issubgroup_in_unit)
      :
      (grp_quotient_map $o
       {|
         grp_homo_map := lact r;
         issemigrouppreserving_grp_homo :=
           (fun x y : M => lm_dist_l r x y)
           :
           IsSemiGroupPreserving (lact r)
       |}) n = mon_unit)) plus plus intros r m n; revert m.
R: Ring
M: LeftModule R
N: LeftSubmodule M
r: R
n: QuotientAbGroup M N
forall m : QuotientAbGroup M N,
quotient_abgroup_rec N (QuotientAbGroup M N)
  (grp_quotient_map $o
   {|
     grp_homo_map := lact r;
     issemigrouppreserving_grp_homo :=
       fun x y : M => lm_dist_l r x y
   |})
  (fun (n : M) (Nn : N n) =>
   qglue
     (issubgroup_in_inv_op (r *L n) mon_unit
        (is_left_submodule r n Nn) issubgroup_in_unit))
  (m + n) =
quotient_abgroup_rec N (QuotientAbGroup M N)
  (grp_quotient_map $o
   {|
     grp_homo_map := lact r;
     issemigrouppreserving_grp_homo :=
       fun x y : M => lm_dist_l r x y
   |})
  (fun (n : M) (Nn : N n) =>
   qglue
     (issubgroup_in_inv_op (r *L n) mon_unit
        (is_left_submodule r n Nn) issubgroup_in_unit))
  m +
quotient_abgroup_rec N (QuotientAbGroup M N)
  (grp_quotient_map $o
   {|
     grp_homo_map := lact r;
     issemigrouppreserving_grp_homo :=
       fun x y : M => lm_dist_l r x y
   |})
  (fun (n : M) (Nn : N n) =>
   qglue
     (issubgroup_in_inv_op (r *L n) mon_unit
        (is_left_submodule r n Nn) issubgroup_in_unit))
  n
    snapply Quotient_ind_hprop; [exact _ | intros m; revert n].
R: Ring
M: LeftModule R
N: LeftSubmodule M
r: R
m: M
forall n : QuotientAbGroup M N,
(fun
   x : M /
       in_cosetL
         {|
           normalsubgroup_subgroup := N;
           normalsubgroup_isnormal :=
             isnormal_ab_subgroup M N
         |} =>
 quotient_abgroup_rec N (QuotientAbGroup M N)
   (grp_quotient_map $o
    {|
      grp_homo_map := lact r;
      issemigrouppreserving_grp_homo :=
        fun x0 y : M => lm_dist_l r x0 y
    |})
   (fun (n0 : M) (Nn : N n0) =>
    qglue
      (issubgroup_in_inv_op (r *L n0) mon_unit
         (is_left_submodule r n0 Nn)
         issubgroup_in_unit)) (x + n) =
 quotient_abgroup_rec N (QuotientAbGroup M N)
   (grp_quotient_map $o
    {|
      grp_homo_map := lact r;
      issemigrouppreserving_grp_homo :=
        fun x0 y : M => lm_dist_l r x0 y
    |})
   (fun (n0 : M) (Nn : N n0) =>
    qglue
      (issubgroup_in_inv_op (r *L n0) mon_unit
         (is_left_submodule r n0 Nn)
         issubgroup_in_unit)) x +
 quotient_abgroup_rec N (QuotientAbGroup M N)
   (grp_quotient_map $o
    {|
      grp_homo_map := lact r;
      issemigrouppreserving_grp_homo :=
        fun x0 y : M => lm_dist_l r x0 y
    |})
   (fun (n0 : M) (Nn : N n0) =>
    qglue
      (issubgroup_in_inv_op (r *L n0) mon_unit
         (is_left_submodule r n0 Nn)
         issubgroup_in_unit)) n)
  (class_of
     (in_cosetL
        {|
          normalsubgroup_subgroup := N;
          normalsubgroup_isnormal :=
            isnormal_ab_subgroup M N
        |}) m)
    snapply Quotient_ind_hprop; [exact _ | intros n; simpl].
R: Ring
M: LeftModule R
N: LeftSubmodule M
r: R
m, n: M
class_of (fun x y : M => N (sg_op (inv x) y))
  (r *L sg_op m n) =
class_of (fun x y : M => N (sg_op (inv x) y)) (r *L m) +
class_of (fun x y : M => N (sg_op (inv x) y)) (r *L n)
    rapply ap.
R: Ring
M: LeftModule R
N: LeftSubmodule M
r: R
m, n: M
r *L sg_op m n = sg_op (r *L m) (r *L n)
    apply lm_dist_l.
  -
R: Ring
M: LeftModule R
N: LeftSubmodule M
RightHeteroDistribute
  (fun r : R =>
   quotient_abgroup_rec N (QuotientAbGroup M N)
     (grp_quotient_map $o
      {|
        grp_homo_map := lact r;
        issemigrouppreserving_grp_homo :=
          (fun x y : M => lm_dist_l r x y)
          :
          IsSemiGroupPreserving (lact r)
      |})
     (fun (n : M) (Nn : N n) =>
      qglue
        (issubgroup_in_inv_op (r *L n) mon_unit
           (is_left_submodule r n Nn)
           issubgroup_in_unit)
      :
      (grp_quotient_map $o
       {|
         grp_homo_map := lact r;
         issemigrouppreserving_grp_homo :=
           (fun x y : M => lm_dist_l r x y)
           :
           IsSemiGroupPreserving (lact r)
       |}) n = mon_unit)) plus plus intros r s.
R: Ring
M: LeftModule R
N: LeftSubmodule M
r, s: R
forall c : QuotientAbGroup M N,
quotient_abgroup_rec N (QuotientAbGroup M N)
  (grp_quotient_map $o
   {|
     grp_homo_map := lact (r + s);
     issemigrouppreserving_grp_homo :=
       fun x y : M => lm_dist_l (r + s) x y
   |})
  (fun (n : M) (Nn : N n) =>
   qglue
     (issubgroup_in_inv_op ((r + s) *L n) mon_unit
        (is_left_submodule (r + s) n Nn)
        issubgroup_in_unit)) c =
quotient_abgroup_rec N (QuotientAbGroup M N)
  (grp_quotient_map $o
   {|
     grp_homo_map := lact r;
     issemigrouppreserving_grp_homo :=
       fun x y : M => lm_dist_l r x y
   |})
  (fun (n : M) (Nn : N n) =>
   qglue
     (issubgroup_in_inv_op (r *L n) mon_unit
        (is_left_submodule r n Nn) issubgroup_in_unit))
  c +
quotient_abgroup_rec N (QuotientAbGroup M N)
  (grp_quotient_map $o
   {|
     grp_homo_map := lact s;
     issemigrouppreserving_grp_homo :=
       fun x y : M => lm_dist_l s x y
   |})
  (fun (n : M) (Nn : N n) =>
   qglue
     (issubgroup_in_inv_op (s *L n) mon_unit
        (is_left_submodule s n Nn) issubgroup_in_unit))
  c
    snapply Quotient_ind_hprop; [exact _| intros m; simpl].
R: Ring
M: LeftModule R
N: LeftSubmodule M
r, s: R
m: M
class_of (fun x y : M => N (sg_op (inv x) y))
  ((r + s) *L m) =
class_of (fun x y : M => N (sg_op (inv x) y)) (r *L m) +
class_of (fun x y : M => N (sg_op (inv x) y)) (s *L m)
    rapply ap.
R: Ring
M: LeftModule R
N: LeftSubmodule M
r, s: R
m: M
(r + s) *L m = sg_op (r *L m) (s *L m)
    apply lm_dist_r.
  -
R: Ring
M: LeftModule R
N: LeftSubmodule M
HeteroAssociative
  (fun r : R =>
   quotient_abgroup_rec N (QuotientAbGroup M N)
     (grp_quotient_map $o
      {|
        grp_homo_map := lact r;
        issemigrouppreserving_grp_homo :=
          (fun x y : M => lm_dist_l r x y)
          :
          IsSemiGroupPreserving (lact r)
      |})
     (fun (n : M) (Nn : N n) =>
      qglue
        (issubgroup_in_inv_op (r *L n) mon_unit
           (is_left_submodule r n Nn)
           issubgroup_in_unit)
      :
      (grp_quotient_map $o
       {|
         grp_homo_map := lact r;
         issemigrouppreserving_grp_homo :=
           (fun x y : M => lm_dist_l r x y)
           :
           IsSemiGroupPreserving (lact r)
       |}) n = mon_unit))
  (fun r : R =>
   quotient_abgroup_rec N (QuotientAbGroup M N)
     (grp_quotient_map $o
      {|
        grp_homo_map := lact r;
        issemigrouppreserving_grp_homo :=
          (fun x y : M => lm_dist_l r x y)
          :
          IsSemiGroupPreserving (lact r)
      |})
     (fun (n : M) (Nn : N n) =>
      qglue
        (issubgroup_in_inv_op (r *L n) mon_unit
           (is_left_submodule r n Nn)
           issubgroup_in_unit)
      :
      (grp_quotient_map $o
       {|
         grp_homo_map := lact r;
         issemigrouppreserving_grp_homo :=
           (fun x y : M => lm_dist_l r x y)
           :
           IsSemiGroupPreserving (lact r)
       |}) n = mon_unit))
  (fun r : R =>
   quotient_abgroup_rec N (QuotientAbGroup M N)
     (grp_quotient_map $o
      {|
        grp_homo_map := lact r;
        issemigrouppreserving_grp_homo :=
          (fun x y : M => lm_dist_l r x y)
          :
          IsSemiGroupPreserving (lact r)
      |})
     (fun (n : M) (Nn : N n) =>
      qglue
        (issubgroup_in_inv_op (r *L n) mon_unit
           (is_left_submodule r n Nn)
           issubgroup_in_unit)
      :
      (grp_quotient_map $o
       {|
         grp_homo_map := lact r;
         issemigrouppreserving_grp_homo :=
           (fun x y : M => lm_dist_l r x y)
           :
           IsSemiGroupPreserving (lact r)
       |}) n = mon_unit)) mult intros r s.
R: Ring
M: LeftModule R
N: LeftSubmodule M
r, s: R
forall z : QuotientAbGroup M N,
quotient_abgroup_rec N (QuotientAbGroup M N)
  (grp_quotient_map $o
   {|
     grp_homo_map := lact r;
     issemigrouppreserving_grp_homo :=
       fun x y : M => lm_dist_l r x y
   |})
  (fun (n : M) (Nn : N n) =>
   qglue
     (issubgroup_in_inv_op (r *L n) mon_unit
        (is_left_submodule r n Nn) issubgroup_in_unit))
  (quotient_abgroup_rec N (QuotientAbGroup M N)
     (grp_quotient_map $o
      {|
        grp_homo_map := lact s;
        issemigrouppreserving_grp_homo :=
          fun x y : M => lm_dist_l s x y
      |})
     (fun (n : M) (Nn : N n) =>
      qglue
        (issubgroup_in_inv_op (s *L n) mon_unit
           (is_left_submodule s n Nn)
           issubgroup_in_unit)) z) =
quotient_abgroup_rec N (QuotientAbGroup M N)
  (grp_quotient_map $o
   {|
     grp_homo_map := lact (r * s);
     issemigrouppreserving_grp_homo :=
       fun x y : M => lm_dist_l (r * s) x y
   |})
  (fun (n : M) (Nn : N n) =>
   qglue
     (issubgroup_in_inv_op ((r * s) *L n) mon_unit
        (is_left_submodule (r * s) n Nn)
        issubgroup_in_unit)) z
    snapply Quotient_ind_hprop; [exact _| intros m; simpl].
R: Ring
M: LeftModule R
N: LeftSubmodule M
r, s: R
m: M
class_of (fun x y : M => N (sg_op (inv x) y))
  (r *L (s *L m)) =
class_of (fun x y : M => N (sg_op (inv x) y))
  ((r * s) *L m)
    rapply ap.
R: Ring
M: LeftModule R
N: LeftSubmodule M
r, s: R
m: M
r *L (s *L m) = (r * s) *L m
    apply lm_assoc.
  -
R: Ring
M: LeftModule R
N: LeftSubmodule M
LeftIdentity
  (fun r : R =>
   quotient_abgroup_rec N (QuotientAbGroup M N)
     (grp_quotient_map $o
      {|
        grp_homo_map := lact r;
        issemigrouppreserving_grp_homo :=
          (fun x y : M => lm_dist_l r x y)
          :
          IsSemiGroupPreserving (lact r)
      |})
     (fun (n : M) (Nn : N n) =>
      qglue
        (issubgroup_in_inv_op (r *L n) mon_unit
           (is_left_submodule r n Nn)
           issubgroup_in_unit)
      :
      (grp_quotient_map $o
       {|
         grp_homo_map := lact r;
         issemigrouppreserving_grp_homo :=
           (fun x y : M => lm_dist_l r x y)
           :
           IsSemiGroupPreserving (lact r)
       |}) n = mon_unit)) 1 snapply Quotient_ind_hprop; [exact _| intros m; simpl].
R: Ring
M: LeftModule R
N: LeftSubmodule M
m: M
class_of (fun x y : M => N (sg_op (inv x) y)) (1 *L m) =
class_of (fun x y : M => N (sg_op (inv x) y)) m
    rapply ap.
R: Ring
M: LeftModule R
N: LeftSubmodule M
m: M
1 *L m = m
    apply lm_unit.
Defined.

Instance isrightmodule_quotientabgroup {R : Ring}
  (M : RightModule R) (N : RightSubmodule M)
  : IsRightModule R (QuotientAbGroup M N)
  := isleftmodule_quotientabgroup (R:=rng_op R) M N.

(** We can therefore form the quotient module of a module by its submodule. *)
Definition QuotientLeftModule {R : Ring} (M : LeftModule R) (N : LeftSubmodule M)
  : LeftModule R
  := Build_LeftModule R (QuotientAbGroup M N) _.

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

Infix "/" := QuotientLeftModule : module_scope.

(** TODO: Notation for right module quotient? *)

(** ** First Isomorphism Theorem *)

Local Open Scope module_scope.
Local Open Scope wc_iso_scope.

Definition lm_first_iso `{Funext} {R : Ring} {M N : LeftModule R} (f : M $-> N)
  : M / lm_kernel f ≅ lm_image f.
H: Funext
R: Ring
M, N: LeftModule R
f: M $-> N
M / lm_kernel f ≅ lm_image f
Proof.
H: Funext
R: Ring
M, N: LeftModule R
f: M $-> N
M / lm_kernel f ≅ lm_image f
  snapply Build_LeftModuleIsomorphism'.
H: Funext
R: Ring
M, N: LeftModule R
f: M $-> N
GroupIsomorphism (M / lm_kernel f) (lm_image f)
H: Funext
R: Ring
M, N: LeftModule R
f: M $-> N
forall (r : R) (x : M / lm_kernel f),
?f (r *L x) = r *L ?f x
  1: rapply abgroup_first_iso.
H: Funext
R: Ring
M, N: LeftModule R
f: M $-> N
forall (r : R) (x : M / lm_kernel f),
abgroup_first_iso f (r *L x) =
r *L abgroup_first_iso f x
  intros r.
H: Funext
R: Ring
M, N: LeftModule R
f: M $-> N
r: R
forall x : M / lm_kernel f,
abgroup_first_iso f (r *L x) =
r *L abgroup_first_iso f x
  srapply Quotient_ind_hprop; intros m.
H: Funext
R: Ring
M, N: LeftModule R
f: M $-> N
r: R
m: M
(fun
   x : Quotient
         (in_cosetL
            {|
              normalsubgroup_subgroup := lm_kernel f;
              normalsubgroup_isnormal :=
                isnormal_ab_subgroup M (lm_kernel f)
            |}) =>
 abgroup_first_iso f (r *L x) =
 r *L abgroup_first_iso f x)
  (class_of
     (in_cosetL
        {|
          normalsubgroup_subgroup := lm_kernel f;
          normalsubgroup_isnormal :=
            isnormal_ab_subgroup M (lm_kernel f)
        |}) m)
  apply path_sigma_hprop; simpl.
H: Funext
R: Ring
M, N: LeftModule R
f: M $-> N
r: R
m: M
f (r *L m) = r *L f m
  apply lm_homo_lact.
Defined.

Definition rm_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.

(** ** Direct products *)

(** TODO: generalise to biproducts *)
(** The direct product of modules *)
Definition lm_prod {R : Ring} : LeftModule R -> LeftModule R -> LeftModule R.
R: Ring
LeftModule R -> LeftModule R -> LeftModule R
Proof.
R: Ring
LeftModule R -> LeftModule R -> LeftModule R
  intros M N.
R: Ring
M, N: LeftModule R
LeftModule R
  snapply (Build_LeftModule R (ab_biprod M N)).
R: Ring
M, N: LeftModule R
IsLeftModule R (ab_biprod M N)
  snapply Build_IsLeftModule.
R: Ring
M, N: LeftModule R
R -> ab_biprod M N -> ab_biprod M N
R: Ring
M, N: LeftModule R
LeftHeteroDistribute ?lact plus plus
R: Ring
M, N: LeftModule R
RightHeteroDistribute ?lact plus plus
R: Ring
M, N: LeftModule R
HeteroAssociative ?lact ?lact ?lact mult
R: Ring
M, N: LeftModule R
LeftIdentity ?lact 1
  -
R: Ring
M, N: LeftModule R
R -> ab_biprod M N -> ab_biprod M N intros r.
R: Ring
M, N: LeftModule R
r: R
ab_biprod M N -> ab_biprod M N
    apply functor_prod; exact (lact r).
  -
R: Ring
M, N: LeftModule R
LeftHeteroDistribute
  (fun r : R => functor_prod (lact r) (lact r)) plus
  plus intros r m n.
R: Ring
M, N: LeftModule R
r: R
m, n: ab_biprod M N
functor_prod (lact r) (lact r) (m + n) =
functor_prod (lact r) (lact r) m +
functor_prod (lact r) (lact r) n
    apply path_prod; apply lm_dist_l.
  -
R: Ring
M, N: LeftModule R
RightHeteroDistribute
  (fun r : R => functor_prod (lact r) (lact r)) plus
  plus intros r m n.
R: Ring
M, N: LeftModule R
r, m: R
n: ab_biprod M N
functor_prod (lact (r + m)) (lact (r + m)) n =
functor_prod (lact r) (lact r) n +
functor_prod (lact m) (lact m) n
    apply path_prod; apply lm_dist_r.
  -
R: Ring
M, N: LeftModule R
HeteroAssociative
  (fun r : R => functor_prod (lact r) (lact r))
  (fun r : R => functor_prod (lact r) (lact r))
  (fun r : R => functor_prod (lact r) (lact r)) mult intros r s m.
R: Ring
M, N: LeftModule R
r, s: R
m: ab_biprod M N
functor_prod (lact r) (lact r)
  (functor_prod (lact s) (lact s) m) =
functor_prod (lact (r * s)) (lact (r * s)) m
    apply path_prod; apply lm_assoc.
  -
R: Ring
M, N: LeftModule R
LeftIdentity
  (fun r : R => functor_prod (lact r) (lact r)) 1 intros r.
R: Ring
M, N: LeftModule R
r: ab_biprod M N
functor_prod (lact 1) (lact 1) r = r
    apply path_prod; apply lm_unit.
Defined.

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

Definition lm_prod_fst {R : Ring} {M N : LeftModule R} : lm_prod M N $-> M.
R: Ring
M, N: LeftModule R
lm_prod M N $-> M
Proof.
R: Ring
M, N: LeftModule R
lm_prod M N $-> M
  snapply Build_LeftModuleHomomorphism.
R: Ring
M, N: LeftModule R
GroupHomomorphism (lm_prod M N) M
R: Ring
M, N: LeftModule R
forall (r : R) (m : lm_prod M N),
?lm_homo_map (r *L m) = r *L ?lm_homo_map m
  -
R: Ring
M, N: LeftModule R
GroupHomomorphism (lm_prod M N) M exact grp_prod_pr1.
  -
R: Ring
M, N: LeftModule R
forall (r : R) (m : lm_prod M N),
grp_prod_pr1 (r *L m) = r *L grp_prod_pr1 m reflexivity.
Defined.

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

Definition lm_prod_snd {R : Ring} {M N : LeftModule R} : lm_prod M N $-> N.
R: Ring
M, N: LeftModule R
lm_prod M N $-> N
Proof.
R: Ring
M, N: LeftModule R
lm_prod M N $-> N
  snapply Build_LeftModuleHomomorphism.
R: Ring
M, N: LeftModule R
GroupHomomorphism (lm_prod M N) N
R: Ring
M, N: LeftModule R
forall (r : R) (m : lm_prod M N),
?lm_homo_map (r *L m) = r *L ?lm_homo_map m
  -
R: Ring
M, N: LeftModule R
GroupHomomorphism (lm_prod M N) N exact grp_prod_pr2.
  -
R: Ring
M, N: LeftModule R
forall (r : R) (m : lm_prod M N),
grp_prod_pr2 (r *L m) = r *L grp_prod_pr2 m reflexivity.
Defined.

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

Definition lm_prod_corec {R : Ring} {M N : LeftModule R} (L : LeftModule R)
  (f : L $-> M) (g : L $-> N)
  : L $-> lm_prod M N.
R: Ring
M, N, L: LeftModule R
f: L $-> M
g: L $-> N
L $-> lm_prod M N
Proof.
R: Ring
M, N, L: LeftModule R
f: L $-> M
g: L $-> N
L $-> lm_prod M N
  snapply Build_LeftModuleHomomorphism.
R: Ring
M, N, L: LeftModule R
f: L $-> M
g: L $-> N
GroupHomomorphism L (lm_prod M N)
R: Ring
M, N, L: LeftModule R
f: L $-> M
g: L $-> N
forall (r : R) (m : L),
?lm_homo_map (r *L m) = r *L ?lm_homo_map m
  -
R: Ring
M, N, L: LeftModule R
f: L $-> M
g: L $-> N
GroupHomomorphism L (lm_prod M N) exact (grp_prod_corec f g).
  -
R: Ring
M, N, L: LeftModule R
f: L $-> M
g: L $-> N
forall (r : R) (m : L),
grp_prod_corec f g (r *L m) =
r *L grp_prod_corec f g m intros r l.
R: Ring
M, N, L: LeftModule R
f: L $-> M
g: L $-> N
r: R
l: L
grp_prod_corec f g (r *L l) =
r *L grp_prod_corec f g l
    apply path_prod; apply lm_homo_lact.
Defined.

Definition rm_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.

Instance hasbinaryproducts_leftmodule {R : Ring}
  : HasBinaryProducts (LeftModule R).
R: Ring
HasBinaryProducts (LeftModule R)
Proof.
R: Ring
HasBinaryProducts (LeftModule R)
  intros M N.
R: Ring
M, N: LeftModule R
BinaryProduct M N
  snapply Build_BinaryProduct.
R: Ring
M, N: LeftModule R
LeftModule R
R: Ring
M, N: LeftModule R
?cat_binprod' $-> M
R: Ring
M, N: LeftModule R
?cat_binprod' $-> N
R: Ring
M, N: LeftModule R
forall z : LeftModule R,
(z $-> M) -> (z $-> N) -> z $-> ?cat_binprod'
R: Ring
M, N: LeftModule R
forall (z : LeftModule R) (f : z $-> M) (g : z $-> N),
?cat_pr1 $o ?cat_binprod_corec z f g $== f
R: Ring
M, N: LeftModule R
forall (z : LeftModule R) (f : z $-> M) (g : z $-> N),
?cat_pr2 $o ?cat_binprod_corec z f g $== g
R: Ring
M, N: LeftModule R
forall (z : LeftModule R) (f g : z $-> ?cat_binprod'),
?cat_pr1 $o f $== ?cat_pr1 $o g ->
?cat_pr2 $o f $== ?cat_pr2 $o g -> f $== g
  -
R: Ring
M, N: LeftModule R
LeftModule R exact (lm_prod M N).
  -
R: Ring
M, N: LeftModule R
lm_prod M N $-> M exact lm_prod_fst.
  -
R: Ring
M, N: LeftModule R
lm_prod M N $-> N exact lm_prod_snd.
  -
R: Ring
M, N: LeftModule R
forall z : LeftModule R,
(z $-> M) -> (z $-> N) -> z $-> lm_prod M N exact lm_prod_corec.
  -
R: Ring
M, N: LeftModule R
forall (z : LeftModule R) (f : z $-> M) (g : z $-> N),
lm_prod_fst $o lm_prod_corec z f g $== f cbn; reflexivity.
  -
R: Ring
M, N: LeftModule R
forall (z : LeftModule R) (f : z $-> M) (g : z $-> N),
lm_prod_snd $o lm_prod_corec z f g $== g cbn; reflexivity.
  -
R: Ring
M, N: LeftModule R
forall (z : LeftModule R) (f g : z $-> lm_prod M N),
lm_prod_fst $o f $== lm_prod_fst $o g ->
lm_prod_snd $o f $== lm_prod_snd $o g -> f $== g intros L f g p q a.
R: Ring
M, N, L: LeftModule R
f, g: L $-> lm_prod M N
p: lm_prod_fst $o f $== lm_prod_fst $o g
q: lm_prod_snd $o f $== lm_prod_snd $o g
a: L
f a = g a
    exact (path_prod' (p a) (q a)).
Defined.

Instance hasbinaryproducts_rightmodule {R : Ring}
  : HasBinaryProducts (RightModule R)
  := hasbinaryproducts_leftmodule (R:=rng_op R).

(** ** Finite Sums *)

(** Left scalar multiplication distributes over finite sums of left module elements. *)
Definition lm_sum_dist_l {R : Ring} (M : LeftModule R) (n : nat)
  (f : forall k, (k < n)%nat -> M) (r : R)
  : r *L ab_sum n f = ab_sum n (fun k Hk => r *L f k Hk).
R: Ring
M: LeftModule R
n: nat
f: forall k : nat, (k < n)%nat -> M
r: R
r *L ab_sum n f =
ab_sum n
  (fun (k : nat) (Hk : (k < n)%nat) => r *L f k Hk)
Proof.
R: Ring
M: LeftModule R
n: nat
f: forall k : nat, (k < n)%nat -> M
r: R
r *L ab_sum n f =
ab_sum n
  (fun (k : nat) (Hk : (k < n)%nat) => r *L f k Hk)
  induction n as [|n IHn].
R: Ring
M: LeftModule R
f: forall k : nat, (k < 0)%nat -> M
r: R
r *L ab_sum 0 f =
ab_sum 0
  (fun (k : nat) (Hk : (k < 0)%nat) => r *L f k Hk)
R: Ring
M: LeftModule R
n: nat
f: forall k : nat, (k < n.+1)%nat -> M
r: R
IHn: forall f : forall k : nat, (k < n)%nat -> M,
r *L ab_sum n f = ab_sum n (fun (k : nat) (Hk : (k < n)%nat) => r *L f k Hk)
r *L ab_sum n.+1 f =
ab_sum n.+1
  (fun (k : nat) (Hk : (k < n.+1)%nat) => r *L f k Hk)
  1: apply lm_zero_r.
R: Ring
M: LeftModule R
n: nat
f: forall k : nat, (k < n.+1)%nat -> M
r: R
IHn: forall f : forall k : nat, (k < n)%nat -> M,
r *L ab_sum n f = ab_sum n (fun (k : nat) (Hk : (k < n)%nat) => r *L f k Hk)
r *L ab_sum n.+1 f =
ab_sum n.+1
  (fun (k : nat) (Hk : (k < n.+1)%nat) => r *L f k Hk)
  lhs napply lm_dist_l; simpl; f_ap.
Defined.

(** Right scalar multiplication distributes over finite sums of right module elements. *)
Definition rm_sum_dist_r {R : Ring} (M : RightModule R) (n : nat)
  (f : forall k, (k < n)%nat -> M) (r : R)
  : ab_sum n f *R r = ab_sum n (fun k Hk => f k Hk *R r)
  := lm_sum_dist_l (R:=rng_op R) M n f r.

(** Left module elements distribute over finite sums of scalars. *)
Definition lm_sum_dist_r {R : Ring} (M : LeftModule R) (n : nat)
  (f : forall k, (k < n)%nat -> R) (x : M)
  : ab_sum n f *L x = ab_sum n (fun k Hk => f k Hk *L x).
R: Ring
M: LeftModule R
n: nat
f: forall k : nat, (k < n)%nat -> R
x: M
ab_sum n f *L x =
ab_sum n
  (fun (k : nat) (Hk : (k < n)%nat) => f k Hk *L x)
Proof.
R: Ring
M: LeftModule R
n: nat
f: forall k : nat, (k < n)%nat -> R
x: M
ab_sum n f *L x =
ab_sum n
  (fun (k : nat) (Hk : (k < n)%nat) => f k Hk *L x)
  induction n as [|n IHn].
R: Ring
M: LeftModule R
f: forall k : nat, (k < 0)%nat -> R
x: M
ab_sum 0 f *L x =
ab_sum 0
  (fun (k : nat) (Hk : (k < 0)%nat) => f k Hk *L x)
R: Ring
M: LeftModule R
n: nat
f: forall k : nat, (k < n.+1)%nat -> R
x: M
IHn: forall f : forall k : nat, (k < n)%nat -> R,
ab_sum n f *L x = ab_sum n (fun (k : nat) (Hk : (k < n)%nat) => f k Hk *L x)
ab_sum n.+1 f *L x =
ab_sum n.+1
  (fun (k : nat) (Hk : (k < n.+1)%nat) => f k Hk *L x)
  1: apply lm_zero_l.
R: Ring
M: LeftModule R
n: nat
f: forall k : nat, (k < n.+1)%nat -> R
x: M
IHn: forall f : forall k : nat, (k < n)%nat -> R,
ab_sum n f *L x = ab_sum n (fun (k : nat) (Hk : (k < n)%nat) => f k Hk *L x)
ab_sum n.+1 f *L x =
ab_sum n.+1
  (fun (k : nat) (Hk : (k < n.+1)%nat) => f k Hk *L x)
  lhs napply lm_dist_r; simpl; f_ap.
Defined.

(** Right module elements distribute over finite sums of scalar. *)
Definition rm_sum_dist_l {R : Ring} (M : RightModule R) (n : nat)
  (f : forall k, (k < n)%nat -> R) (x : M)
  : x *R ab_sum n f = ab_sum n (fun k Hk => x *R f k Hk)
  := lm_sum_dist_r (R:=rng_op R) M n f x.