End ZmoduleTheory.
Arguments addrI {V} y [x1 x2].
Arguments addIr {V} x [x1 x2].
Arguments opprK {V}.
Arguments oppr_inj {V} [x1 x2].
Module Ring.
Record mixin_of (R : zmodType) : Type := Mixin {
one : R;
mul : R -> R -> R;
_ : associative mul;
_ : left_id one mul;
_ : right_id one mul;
_ : left_distributive mul +%R;
_ : right_distributive mul +%R;
_ : one != 0
}.
Definition EtaMixin R one mul mulA mul1x mulx1 mul_addl mul_addr nz1 :=
let _ := @Mixin R one mul mulA mul1x mulx1 mul_addl mul_addr nz1 in
@Mixin (Zmodule.Pack (Zmodule.class R)) _ _
mulA mul1x mulx1 mul_addl mul_addr nz1.
Section ClassDef.
Record class_of (R : Type) : Type := Class {
base : Zmodule.class_of R;
mixin : mixin_of (Zmodule.Pack base)
}.
Local Coercion base : class_of >-> Zmodule.class_of.
Structure type := Pack {sort; _ : class_of sort}.
Local Coercion sort : type >-> Sortclass.
Variables (T : Type) (cT : type).
Definition class := let: Pack _ c as cT' := cT return class_of cT' in c.
Definition clone c of phant_id class c := @Pack T c.
Let xT := let: Pack T _ := cT in T.
Notation xclass := (class : class_of xT).
Definition pack b0 (m0 : mixin_of (@Zmodule.Pack T b0)) :=
fun bT b & phant_id (Zmodule.class bT) b =>
fun    m & phant_id m0 m => Pack (@Class T b m).
Definition eqType := @Equality.Pack cT xclass.
Definition choiceType := @Choice.Pack cT xclass.
Definition zmodType := @Zmodule.Pack cT xclass.
End ClassDef.
Module Exports.
Coercion base : class_of >-> Zmodule.class_of.
Coercion mixin : class_of >-> mixin_of.
Coercion sort : type >-> Sortclass.
Bind Scope ring_scope with sort.
Coercion eqType : type >-> Equality.type.
Canonical eqType.
Coercion choiceType : type >-> Choice.type.
Canonical choiceType.
Coercion zmodType : type >-> Zmodule.type.
Canonical zmodType.
Notation ringType := type.
Notation RingType T m := (@pack T _ m _ _ id _ id).
Notation RingMixin := Mixin.
Notation "[ 'ringType' 'of' T 'for' cT ]" := (@clone T cT _ idfun)
(at level 0, format "[ 'ringType'  'of'  T  'for'  cT ]") : form_scope.
Notation "[ 'ringType' 'of' T ]" := (@clone T _ _ id)
(at level 0, format "[ 'ringType'  'of'  T ]") : form_scope.
End Exports.
End Ring.
Definition one (R : ringType) : R := Ring.one (Ring.class R).
Definition mul (R : ringType) : R -> R -> R := Ring.mul (Ring.class R).
Definition exp R x n := nosimpl iterop _ n (@mul R) x (one R).
Notation sign R b := (exp (- one R) (nat_of_bool b)) (only parsing).
Definition comm R x y := @mul R x y = mul y x.
Definition lreg R x := injective (@mul R x).
Definition rreg R x := injective ((@mul R)^~ x).
Local Notation "1" := (one _) : ring_scope.
Local Notation "- 1" := (- (1)) : ring_scope.
Local Notation "n %:R" := (1 *+ n) : ring_scope.
Local Notation "*%R" := (@mul _).
Local Notation "x * y" := (mul x y) : ring_scope.
Local Notation "x ^+ n" := (exp x n) : ring_scope.
Local Notation "\prod_ ( i <- r | P ) F" := (\big[*%R/1]_(i <- r | P) F).
Local Notation "\prod_ ( i | P ) F" := (\big[*%R/1]_(i | P) F).
Local Notation "\prod_ ( i 'in' A ) F" := (\big[*%R/1]_(i in A) F).
Local Notation "\prod_ ( m <= i < n ) F" := (\big[*%R/1%R]_(m <= i < n) F%R).
(* The ``field'' characteristic; the definition, and many of the theorems,   *)
(* has to apply to rings as well; indeed, we need the Frobenius automorphism *)
(* results for a non commutative ring in the proof of Gorenstein 2.6.3.      *)
Definition char (R : Ring.type) of phant R : nat_pred :=
[pred p | prime p & p%:R == 0 :> R].
Local Notation "[ 'char' R ]" := (char (Phant R)) : ring_scope.
(* Converse ring tag. *)
Definition converse R : Type := R.
Local Notation "R ^c" := (converse R) (at level 2, format "R ^c") : type_scope.
Section RingTheory.
Variable R : ringType.
Implicit Types x y : R.

Lemma exprBn_comm x y n (cxy : comm x y) :
(x - y) ^+ n =
\sum_(i < n.+1) ((-1) ^+ i * x ^+ (n - i) * y ^+ i) *+ 'C(n, i).