fact stringlengths 8 1.54k | type stringclasses 19 values | library stringclasses 8 values | imports listlengths 1 10 | filename stringclasses 98 values | symbolic_name stringlengths 1 42 | docstring stringclasses 1 value |
|---|---|---|---|---|---|---|
Zmodule_isComRingR := (Zmodule_isComNzRing R) (only parsing).
HB.builders Context R of Zmodule_isComNzRing R. | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | Zmodule_isComRing | |
mulr1:= Monoid.mulC_id mulrC mul1r. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulr1 | |
mulrDr:= Monoid.mulC_dist mulrC mulrDl.
HB.instance Definition _ := Zmodule_isNzRing.Build R
mulrA mul1r mulr1 mulrDl mulrDr oner_neq0.
HB.instance Definition _ := PzRing_hasCommutativeMul.Build R mulrC.
HB.end. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulrDr | |
exprBnx y n :
(x - y) ^+ n =
\sum_(i < n.+1) ((-1) ^+ i * x ^+ (n - i) * y ^+ i) *+ 'C(n, i).
Proof. by rewrite exprBn_comm //; apply: mulrC. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | exprBn | |
subrXXx y n :
x ^+ n - y ^+ n = (x - y) * (\sum_(i < n) x ^+ (n.-1 - i) * y ^+ i).
Proof. by rewrite -subrXX_comm //; apply: mulrC. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subrXX | |
sqrrBx y : (x - y) ^+ 2 = x ^+ 2 - x * y *+ 2 + y ^+ 2.
Proof. by rewrite sqrrD mulrN mulNrn sqrrN. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | sqrrB | |
subr_sqrx y : x ^+ 2 - y ^+ 2 = (x - y) * (x + y).
Proof. by rewrite subrXX !big_ord_recr big_ord0 /= add0r mulr1 mul1r. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subr_sqr | |
subr_sqrDBx y : (x + y) ^+ 2 - (x - y) ^+ 2 = x * y *+ 4.
Proof.
rewrite sqrrD sqrrB -!(addrAC _ (y ^+ 2)) opprB.
by rewrite [LHS]addrC addrA subrK -mulrnDr.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | subr_sqrDB | |
RecordLSemiAlgebra_isSemiAlgebra R V of LSemiAlgebra R V := {
scalerAr : forall k (x y : V), k *: (x * y) = x * (k *: y);
}.
#[short(type="semiAlgType")]
HB.structure Definition SemiAlgebra (R : pzSemiRingType) :=
{A of LSemiAlgebra_isSemiAlgebra R A & LSemiAlgebra R A}. | HB.mixin | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | Record | |
RecordLSemiAlgebra_isComSemiAlgebra R V
of ComPzSemiRing V & LSemiAlgebra R V := {}.
HB.builders Context R V of LSemiAlgebra_isComSemiAlgebra R V. | HB.factory | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | Record | |
scalarArk (x y : V) : k *: (x * y) = x * (k *: y).
Proof. by rewrite mulrC scalerAl mulrC. Qed.
HB.instance Definition _ := LSemiAlgebra_isSemiAlgebra.Build R V scalarAr.
HB.end.
#[short(type="algType")]
HB.structure Definition Algebra (R : pzRingType) :=
{A of LSemiAlgebra_isSemiAlgebra R A & Lalgebra R A}. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | scalarAr | |
RecordLalgebra_isAlgebra (R : pzRingType) V of Lalgebra R V := {
scalerAr : forall k (x y : V), k *: (x * y) = x * (k *: y);
}.
HB.builders Context R V of Lalgebra_isAlgebra R V.
HB.instance Definition _ := LSemiAlgebra_isSemiAlgebra.Build R V scalerAr.
HB.end.
HB.factory Record Lalgebra_isComAlgebra R V of ComPzRing V & Lalgebra R V := {}.
HB.builders Context R V of Lalgebra_isComAlgebra R V.
HB.instance Definition _ := LSemiAlgebra_isComSemiAlgebra.Build R V.
HB.end.
#[short(type="comSemiAlgType")]
HB.structure Definition ComSemiAlgebra R :=
{V of ComNzSemiRing V & SemiAlgebra R V}. | HB.factory | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | Record | |
Definition_ (R : comPzSemiRingType) :=
PzSemiRing_hasCommutativeMul.Build R^c (fun _ _ => mulrC _ _).
#[export]
HB.instance Definition _ (R : comPzSemiRingType) := ComPzSemiRing.on R^o.
#[export]
HB.instance Definition _ (R : comNzSemiRingType) := ComNzSemiRing.on R^c.
#[export]
HB.instance Definition _ (R : comNzSemiRingType) := ComNzSemiRing.on R^o.
#[export]
HB.instance Definition _ (R : comNzSemiRingType) :=
LSemiAlgebra_isComSemiAlgebra.Build R R^o. | HB.instance | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | Definition | |
DefinitionComAlgebra R := {V of ComNzRing V & Algebra R V}. | HB.structure | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | Definition | |
Definition_ (R : comPzRingType) := ComPzRing.on R^c.
#[export]
HB.instance Definition _ (R : comPzRingType) := ComPzRing.on R^o.
#[export]
HB.instance Definition _ (R : comNzRingType) := ComNzRing.on R^c.
#[export]
HB.instance Definition _ (R : comNzRingType) := ComNzRing.on R^o. | HB.instance | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | Definition | |
scalerCAk x y : k *: x * y = x * (k *: y).
Proof. by rewrite -scalerAl scalerAr. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | scalerCA | |
mulr_algra x : x * a%:A = a *: x.
Proof. by rewrite -scalerAr mulr1. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulr_algr | |
comm_alga x : comm a%:A x.
Proof. by rewrite /comm mulr_algr mulr_algl. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | comm_alg | |
exprZnk x n : (k *: x) ^+ n = k ^+ n *: x ^+ n.
Proof.
elim: n => [|n IHn]; first by rewrite !expr0 scale1r.
by rewrite !exprS IHn -scalerA scalerAr scalerAl.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | exprZn | |
scaler_prodI r (P : pred I) (F : I -> R) (G : I -> A) :
\prod_(i <- r | P i) (F i *: G i) =
\prod_(i <- r | P i) F i *: \prod_(i <- r | P i) G i.
Proof.
elim/big_rec3: _ => [|i x a _ _ ->]; first by rewrite scale1r.
by rewrite -scalerAl -scalerAr scalerA.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | scaler_prod | |
scaler_prodl(I : finType) (S : pred I) (F : I -> A) k :
\prod_(i in S) (k *: F i) = k ^+ #|S| *: \prod_(i in S) F i.
Proof. by rewrite scaler_prod prodr_const. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | scaler_prodl | |
scaler_prodr(I : finType) (S : pred I) (F : I -> R) x :
\prod_(i in S) (F i *: x) = \prod_(i in S) F i *: x ^+ #|S|.
Proof. by rewrite scaler_prod prodr_const. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | scaler_prodr | |
mull_fun_is_scalable: scalable (a \*o f).
Proof. by move=> k x /=; rewrite linearZ scalerAr. Qed.
#[export]
HB.instance Definition _ := isScalable.Build R U A *:%R (a \*o f)
mull_fun_is_scalable. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mull_fun_is_scalable | |
RecordNzRing_hasMulInverse R of NzRing R := {
unit_subdef : pred R;
inv : R -> R;
mulVr_subproof : {in unit_subdef, left_inverse 1 inv *%R};
divrr_subproof : {in unit_subdef, right_inverse 1 inv *%R};
unitrP_subproof : forall x y, y * x = 1 /\ x * y = 1 -> unit_subdef x;
invr_out_subproof : {in [predC unit_subdef], inv =1 id}
}. | HB.mixin | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | Record | |
BuildR := (NzRing_hasMulInverse.Build R) (only parsing). | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | Build | |
Ring_hasMulInverseR := (NzRing_hasMulInverse R) (only parsing).
#[short(type="unitRingType")]
HB.structure Definition UnitRing := {R of NzRing_hasMulInverse R & NzRing R}. | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | Ring_hasMulInverse | |
unit_pred{R : unitRingType} :=
Eval cbv [ unit_subdef NzRing_hasMulInverse.unit_subdef ] in
(fun u : R => unit_subdef u).
Arguments unit_pred _ _ /. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | unit_pred | |
unit{R : unitRingType} := [qualify a u : R | unit_pred u].
Local Notation "x ^-1" := (inv x).
Local Notation "x / y" := (x * y^-1).
Local Notation "x ^- n" := ((x ^+ n)^-1). | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | unit | |
divrr: {in unit, right_inverse 1 (@inv R) *%R}.
Proof. exact: divrr_subproof. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divrr | |
mulrV:= divrr. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulrV | |
mulVr: {in unit, left_inverse 1 (@inv R) *%R}.
Proof. exact: mulVr_subproof. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulVr | |
invr_outx : x \isn't a unit -> x^-1 = x.
Proof. exact: invr_out_subproof. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | invr_out | |
unitrPx : reflect (exists y, y * x = 1 /\ x * y = 1) (x \is a unit).
Proof.
apply: (iffP idP) => [Ux | []]; last exact: unitrP_subproof.
by exists x^-1; rewrite divrr ?mulVr.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | unitrP | |
mulKr: {in unit, left_loop (@inv R) *%R}.
Proof. by move=> x Ux y; rewrite mulrA mulVr ?mul1r. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulKr | |
mulVKr: {in unit, rev_left_loop (@inv R) *%R}.
Proof. by move=> x Ux y; rewrite mulrA mulrV ?mul1r. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulVKr | |
mulrK: {in unit, right_loop (@inv R) *%R}.
Proof. by move=> x Ux y; rewrite -mulrA divrr ?mulr1. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulrK | |
mulrVK: {in unit, rev_right_loop (@inv R) *%R}.
Proof. by move=> x Ux y; rewrite -mulrA mulVr ?mulr1. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulrVK | |
divrK:= mulrVK. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divrK | |
mulrI: {in @unit R, right_injective *%R}.
Proof. by move=> x Ux; apply: can_inj (mulKr Ux). Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulrI | |
mulIr: {in @unit R, left_injective *%R}.
Proof. by move=> x Ux; apply: can_inj (mulrK Ux). Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulIr | |
telescope_prodrn m (f : nat -> R) :
(forall k, n < k < m -> f k \is a unit) -> n < m ->
\prod_(n <= k < m) (f k / f k.+1) = f n / f m.
Proof.
move=> Uf ltnm; rewrite (telescope_big (fun i j => f i / f j)) ?ltnm//.
by move=> k ltnkm /=; rewrite mulrA divrK// Uf.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | telescope_prodr | |
telescope_prodr_eqn m (f u : nat -> R) : n < m ->
(forall k, n < k < m -> f k \is a unit) ->
(forall k, (n <= k < m)%N -> u k = f k / f k.+1) ->
\prod_(n <= k < m) u k = f n / f m.
Proof.
by move=> ? ? uE; under eq_big_nat do rewrite uE //=; exact: telescope_prodr.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | telescope_prodr_eq | |
commrVx y : comm x y -> comm x y^-1.
Proof.
have [Uy cxy | /invr_out-> //] := boolP (y \in unit).
by apply: (canLR (mulrK Uy)); rewrite -mulrA cxy mulKr.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | commrV | |
unitrEx : (x \is a unit) = (x / x == 1).
Proof.
apply/idP/eqP=> [Ux | xx1]; first exact: divrr.
by apply/unitrP; exists x^-1; rewrite -commrV.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | unitrE | |
invrK: involutive (@inv R).
Proof.
move=> x; case Ux: (x \in unit); last by rewrite !invr_out ?Ux.
rewrite -(mulrK Ux _^-1) -mulrA commrV ?mulKr //.
by apply/unitrP; exists x; rewrite divrr ?mulVr.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | invrK | |
invr_inj: injective (@inv R). Proof. exact: inv_inj invrK. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | invr_inj | |
unitrVx : (x^-1 \in unit) = (x \in unit).
Proof. by rewrite !unitrE invrK commrV. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | unitrV | |
unitr1: 1 \in @unit R.
Proof. by apply/unitrP; exists 1; rewrite mulr1. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | unitr1 | |
invr1: 1^-1 = 1 :> R.
Proof. by rewrite -{2}(mulVr unitr1) mulr1. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | invr1 | |
div1rx : 1 / x = x^-1. Proof. by rewrite mul1r. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | div1r | |
divr1x : x / 1 = x. Proof. by rewrite invr1 mulr1. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divr1 | |
natr_divm d :
d %| m -> d%:R \is a @unit R -> (m %/ d)%:R = m%:R / d%:R :> R.
Proof.
by rewrite dvdn_eq => /eqP def_m unit_d; rewrite -{2}def_m natrM mulrK.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | natr_div | |
divrI: {in unit, right_injective (fun x y => x / y)}.
Proof. by move=> x /mulrI/inj_comp; apply; apply: invr_inj. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divrI | |
divIr: {in unit, left_injective (fun x y => x / y)}.
Proof. by move=> x; rewrite -unitrV => /mulIr. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divIr | |
unitr0: (0 \is a @unit R) = false.
Proof. by apply/unitrP=> [[x [_ /esym/eqP]]]; rewrite mul0r oner_eq0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | unitr0 | |
invr0: 0^-1 = 0 :> R.
Proof. by rewrite invr_out ?unitr0. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | invr0 | |
unitrN1: -1 \is a @unit R.
Proof. by apply/unitrP; exists (-1); rewrite mulrNN mulr1. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | unitrN1 | |
invrN1: (-1)^-1 = -1 :> R.
Proof. by rewrite -{2}(divrr unitrN1) mulN1r opprK. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | invrN1 | |
invr_signn : ((-1) ^- n) = (-1) ^+ n :> R.
Proof. by rewrite -signr_odd; case: (odd n); rewrite (invr1, invrN1). Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | invr_sign | |
unitrMlx y : y \is a unit -> (x * y \is a unit) = (x \is a unit).
Proof.
move=> Uy; wlog Ux: x y Uy / x \is a unit => [WHxy|].
by apply/idP/idP=> Ux; first rewrite -(mulrK Uy x); rewrite WHxy ?unitrV.
rewrite Ux; apply/unitrP; exists (y^-1 * x^-1).
by rewrite -!mulrA mulKr ?mulrA ?mulrK ?divrr ?mulVr.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | unitrMl | |
unitrMrx y : x \is a unit -> (x * y \is a unit) = (y \is a unit).
Proof.
move=> Ux; apply/idP/idP=> [Uxy | Uy]; last by rewrite unitrMl.
by rewrite -(mulKr Ux y) unitrMl ?unitrV.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | unitrMr | |
unitr_prod{I : Type} (P : pred I) (E : I -> R) (r : seq I) :
(forall i, P i -> E i \is a GRing.unit) ->
(\prod_(i <- r | P i) E i \is a GRing.unit).
Proof.
by move=> Eunit; elim/big_rec: _ => [/[!unitr1] |i x /Eunit/unitrMr->].
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | unitr_prod | |
unitr_prod_in{I : eqType} (P : pred I) (E : I -> R) (r : seq I) :
{in r, forall i, P i -> E i \is a GRing.unit} ->
(\prod_(i <- r | P i) E i \is a GRing.unit).
Proof.
by rewrite big_seq_cond => H; apply: unitr_prod => i /andP[]; exact: H.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | unitr_prod_in | |
invrM: {in unit &, forall x y, (x * y)^-1 = y^-1 * x^-1}.
Proof.
move=> x y Ux Uy; have Uxy: (x * y \in unit) by rewrite unitrMl.
by apply: (mulrI Uxy); rewrite divrr ?mulrA ?mulrK ?divrr.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | invrM | |
unitrM_commx y :
comm x y -> (x * y \is a unit) = (x \is a unit) && (y \is a unit).
Proof.
move=> cxy; apply/idP/andP=> [Uxy | [Ux Uy]]; last by rewrite unitrMl.
suffices Ux: x \in unit by rewrite unitrMr in Uxy.
apply/unitrP; case/unitrP: Uxy => z [zxy xyz]; exists (y * z).
rewrite mulrA xyz -{1}[y]mul1r -{1}zxy cxy -!mulrA (mulrA x) (mulrA _ z) xyz.
by rewrite mul1r -cxy.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | unitrM_comm | |
unitrXx n : x \is a unit -> x ^+ n \is a unit.
Proof.
by move=> Ux; elim: n => [|n IHn]; rewrite ?unitr1 // exprS unitrMl.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | unitrX | |
unitrX_posx n : n > 0 -> (x ^+ n \in unit) = (x \in unit).
Proof.
case: n => // n _; rewrite exprS unitrM_comm; last exact: commrX.
by case Ux: (x \is a unit); rewrite // unitrX.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | unitrX_pos | |
exprVnx n : x^-1 ^+ n = x ^- n.
Proof.
elim: n => [|n IHn]; first by rewrite !expr0 ?invr1.
case Ux: (x \is a unit); first by rewrite exprSr exprS IHn -invrM // unitrX.
by rewrite !invr_out ?unitrX_pos ?Ux.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | exprVn | |
exprBm n x : n <= m -> x \is a unit -> x ^+ (m - n) = x ^+ m / x ^+ n.
Proof. by move/subnK=> {2}<- Ux; rewrite exprD mulrK ?unitrX. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | exprB | |
invr_neq0x : x != 0 -> x^-1 != 0.
Proof.
move=> nx0; case Ux: (x \is a unit); last by rewrite invr_out ?Ux.
by apply/eqP=> x'0; rewrite -unitrV x'0 unitr0 in Ux.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | invr_neq0 | |
invr_eq0x : (x^-1 == 0) = (x == 0).
Proof. by apply: negb_inj; apply/idP/idP; move/invr_neq0; rewrite ?invrK. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | invr_eq0 | |
invr_eq1x : (x^-1 == 1) = (x == 1).
Proof. by rewrite (inv_eq invrK) invr1. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | invr_eq1 | |
rev_unitrP(x y : R^c) : y * x = 1 /\ x * y = 1 -> x \is a unit.
Proof. by case=> [yx1 xy1]; apply/unitrP; exists y. Qed.
#[export]
HB.instance Definition _ :=
NzRing_hasMulInverse.Build R^c mulrV mulVr rev_unitrP invr_out.
#[export]
HB.instance Definition _ := UnitRing.on R^o. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | rev_unitrP | |
rev_prodrV(R : unitRingType)
(I : Type) (r : seq I) (P : pred I) (E : I -> R) :
(forall i, P i -> E i \is a GRing.unit) ->
\prod_(i <- r | P i) (E i)^-1 = ((\prod_(i <- r | P i) (E i : R^c))^-1).
Proof.
move=> Eunit; symmetry.
apply: (big_morph_in GRing.unit _ _ (unitr1 R^c) (@invrM _) (invr1 _)) Eunit.
by move=> x y xunit; rewrite unitrMr.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | rev_prodrV | |
invr_closed:= {in S, forall x, x^-1 \in S}. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | invr_closed | |
divr_2closed:= {in S &, forall x y, x / y \in S}. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divr_2closed | |
divr_closed:= 1 \in S /\ divr_2closed. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divr_closed | |
sdivr_closed:= -1 \in S /\ divr_2closed. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | sdivr_closed | |
divring_closed:= [/\ 1 \in S, subr_2closed S & divr_2closed]. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divring_closed | |
divr_closedV: divr_closed -> invr_closed.
Proof. by case=> S1 Sdiv x Sx; rewrite -[x^-1]mul1r Sdiv. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divr_closedV | |
divr_closedM: divr_closed -> mulr_closed S.
Proof.
by case=> S1 Sdiv; split=> // x y Sx Sy; rewrite -[y]invrK -[y^-1]mul1r !Sdiv.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divr_closedM | |
sdivr_closed_div: sdivr_closed -> divr_closed.
Proof. by case=> SN1 Sdiv; split; rewrite // -(divrr (@unitrN1 _)) Sdiv. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | sdivr_closed_div | |
sdivr_closedM: sdivr_closed -> smulr_closed S.
Proof.
by move=> Sdiv; have [_ SM] := divr_closedM (sdivr_closed_div Sdiv); case: Sdiv.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | sdivr_closedM | |
divring_closedBM: divring_closed -> subring_closed S.
Proof. by case=> S1 SB Sdiv; split=> //; case: divr_closedM. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divring_closedBM | |
divring_closed_div: divring_closed -> sdivr_closed.
Proof.
case=> S1 SB Sdiv; split; rewrite ?zmod_closedN //.
exact/subring_closedB/divring_closedBM.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divring_closed_div | |
rmorph_unitx : x \in unit -> f x \in unit.
Proof.
case/unitrP=> y [yx1 xy1]; apply/unitrP.
by exists (f y); rewrite -!rmorphM // yx1 xy1 rmorph1.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | rmorph_unit | |
rmorphV: {in unit, {morph f: x / x^-1}}.
Proof.
move=> x Ux; rewrite /= -[(f x)^-1]mul1r.
by apply: (canRL (mulrK (rmorph_unit Ux))); rewrite -rmorphM mulVr ?rmorph1.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | rmorphV | |
rmorph_divx y : y \in unit -> f (x / y) = f x / f y.
Proof. by move=> Uy; rewrite rmorphM /= rmorphV. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | rmorph_div | |
DefinitionComUnitRing := {R of ComNzRing R & UnitRing R}. | HB.structure | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | Definition | |
RecordComNzRing_hasMulInverse R of ComNzRing R := {
unit : {pred R};
inv : R -> R;
mulVx : {in unit, left_inverse 1 inv *%R};
unitPl : forall x y, y * x = 1 -> unit x;
invr_out : {in [predC unit], inv =1 id}
}. | HB.factory | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | Record | |
BuildR := (ComNzRing_hasMulInverse.Build R) (only parsing). | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | Build | |
ComRing_hasMulInverseR := (ComNzRing_hasMulInverse R) (only parsing).
HB.builders Context R of ComNzRing_hasMulInverse R.
Fact mulC_mulrV : {in unit, right_inverse 1 inv *%R}.
Proof. by move=> x Ux /=; rewrite mulrC mulVx. Qed.
Fact mulC_unitP x y : y * x = 1 /\ x * y = 1 -> unit x.
Proof. by case=> yx _; apply: unitPl yx. Qed.
HB.instance Definition _ :=
NzRing_hasMulInverse.Build R mulVx mulC_mulrV mulC_unitP invr_out.
HB.end.
#[short(type="unitAlgType")]
HB.structure Definition UnitAlgebra R := {V of Algebra R V & UnitRing V}. | Notation | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | ComRing_hasMulInverse | |
DefinitionComUnitAlgebra R := {V of ComAlgebra R V & UnitRing V}. | HB.structure | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | Definition | |
unitrMx y : (x * y \in unit) = (x \in unit) && (y \in unit).
Proof. exact/unitrM_comm/mulrC. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | unitrM | |
unitrPrx : reflect (exists y, x * y = 1) (x \in unit).
Proof.
by apply: (iffP (unitrP x)) => [[y []] | [y]]; exists y; rewrite // mulrC.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | unitrPr | |
mulr1_eqx y : x * y = 1 -> x^-1 = y.
Proof.
by move=> xy_eq1; rewrite -[LHS]mulr1 -xy_eq1; apply/mulKr/unitrPr; exists y.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | mulr1_eq | |
divr1_eqx y : x / y = 1 -> x = y. Proof. by move/mulr1_eq/invr_inj. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divr1_eq | |
divKrx : x \is a unit -> {in unit, involutive (fun y => x / y)}.
Proof. by move=> Ux y Uy; rewrite /= invrM ?unitrV // invrK mulrC divrK. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | divKr | |
expr_div_nx y n : (x / y) ^+ n = x ^+ n / y ^+ n.
Proof. by rewrite exprMn exprVn. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | expr_div_n | |
unitr_prodP(I : eqType) (r : seq I) (P : pred I) (E : I -> R) :
reflect {in r, forall i, P i -> E i \is a GRing.unit}
(\prod_(i <- r | P i) E i \is a GRing.unit).
Proof.
rewrite (big_morph [in unit] unitrM (@unitr1 _) ) big_all_cond.
exact: 'all_implyP.
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat div seq",
"From mathcomp Require Import choice fintype finfun bigop prime binomial",
"From mathcomp Require Export nmodule"
] | algebra/ssralg.v | unitr_prodP |
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