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 |
|---|---|---|---|---|---|---|
linear_sum:= linear_sum. | 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 | linear_sum | |
linearMn:= linearMn. | 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 | linearMn | |
linearMNn:= linearMNn. | 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 | linearMNn | |
semilinearP:= semilinearP. | 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 | semilinearP | |
linearP:= linearP. | 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 | linearP | |
linearZ_LR:= linearZ_LR. | 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 | linearZ_LR | |
linearZ:= linearZ. | 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 | linearZ | |
semilinearPZ:= semilinearPZ. | 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 | semilinearPZ | |
linearPZ:= linearPZ. | 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 | linearPZ | |
linearZZ:= linearZZ. | 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 | linearZZ | |
semiscalarP:= semiscalarP. | 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 | semiscalarP | |
scalarP:= scalarP. | 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 | scalarP | |
scalarZ:= scalarZ. | 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 | scalarZ | |
can2_scalable:= can2_scalable. | 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 | can2_scalable | |
can2_linear:= can2_linear. | 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 | can2_linear | |
can2_semilinear:= can2_semilinear. | 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 | can2_semilinear | |
rmorph_alg:= rmorph_alg. | 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 | rmorph_alg | |
imaginary_exists:= imaginary_exists. | 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 | imaginary_exists | |
raddf:= (raddf0, raddfN, raddfD, raddfMn). | 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 | raddf | |
rmorphE:=
(rmorphD, rmorph0, rmorphB, rmorphN, rmorphMNn, rmorphMn, rmorph1, rmorphXn). | 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 | rmorphE | |
linearE:=
(linearD, linear0, linearB, linearMNn, linearMn, linearZ). | 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 | linearE | |
null_funV := (null_fun V) (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 | null_fun | |
in_algA := (in_alg A) (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 | in_alg | |
semiRingType:= (nzSemiRingType) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Try pzRingType (the potentially-zero counterpart) first, or use nzRingType instead.")] | 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 | semiRingType | |
ringType:= (nzRingType) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Try comPzSemiRingType (the potentially-zero counterpart) first, or use comNzSemiRingType instead.")] | 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 | ringType | |
comSemiRingType:= (comNzSemiRingType) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Try comPzRingType (the potentially-zero counterpart) first, or use comNzRingType instead.")] | 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 | comSemiRingType | |
comRingType:= (comNzRingType) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Try subPzSemiRingType (the potentially-zero counterpart) first, or use subNzSemiRingType instead.")] | 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 | comRingType | |
subSemiRingType:= (subNzSemiRingType) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Try subComPzSemiRingType (the potentially-zero counterpart) first, or use subComNzSemiRingType instead.")] | 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 | subSemiRingType | |
subComSemiRingType:= (subComNzSemiRingType) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Try subPzRingType (the potentially-zero counterpart) first, or use subNzRingType instead.")] | 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 | subComSemiRingType | |
subRingType:= (subNzRingType) (only parsing).
#[deprecated(since="mathcomp 2.4.0",
note="Try subComPzRingType (the potentially-zero counterpart) first, or use subComNzRingType instead.")] | 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 | subRingType | |
subComNzRingType:= (subComNzRingType) (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 | subComNzRingType | |
addrClosed:= addrClosed. | 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 | addrClosed | |
opprClosed:= opprClosed. | 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 | opprClosed | |
Ione:= IOne : Ione. | Variant | 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 | Ione | |
Inatmul:=
| INatmul : Ione -> nat -> Inatmul
| IOpp : Inatmul -> Inatmul. | Inductive | 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 | Inatmul | |
Idummy_placeholder:=. | Variant | 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 | Idummy_placeholder | |
parse(x : Number.int) : Inatmul :=
match x with
| Number.IntDecimal (Decimal.Pos u) => INatmul IOne (Nat.of_uint u)
| Number.IntDecimal (Decimal.Neg u) => IOpp (INatmul IOne (Nat.of_uint u))
| Number.IntHexadecimal (Hexadecimal.Pos u) =>
INatmul IOne (Nat.of_hex_uint u)
| Number.IntHexadecimal (Hexadeci... | 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 | parse | |
print(x : Inatmul) : option Number.int :=
match x with
| INatmul IOne n =>
Some (Number.IntDecimal (Decimal.Pos (Nat.to_uint n)))
| IOpp (INatmul IOne n) =>
Some (Number.IntDecimal (Decimal.Neg (Nat.to_uint n)))
| _ => None
end.
Arguments GRing.one {_}.
Set Warnings "-via-type-remapping,-via-type-... | 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 | print | |
support:= 0.-support. | 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 | support | |
has_pchar0R := (GRing.pchar R =i pred0).
#[deprecated(since="mathcomp 2.4.0", note="Use has_pchar0 instead.")] | 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 | has_pchar0 | |
has_char0R := (GRing.pchar R =i pred0). | 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 | has_char0 | |
pFrobenius_autchRp := (pFrobenius_aut chRp).
#[deprecated(since="mathcomp 2.4.0", note="Use pFrobenius_aut instead.")] | 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 | pFrobenius_aut | |
Frobenius_autchRp := (pFrobenius_aut chRp). | 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 | Frobenius_aut | |
sum_ffunEx : (\sum_(i <- r | P i) F i) x = \sum_(i <- r | P i) F i x.
Proof. by elim/big_rec2: _ => // [|i _ y _ <-]; rewrite !ffunE. 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 | sum_ffunE | |
sum_ffun:
\sum_(i <- r | P i) F i = [ffun x => \sum_(i <- r | P i) F i x].
Proof. by apply/ffunP=> i; rewrite sum_ffunE ffunE. 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 | sum_ffun | |
ffun_one: {ffun aT -> R} := [ffun => 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 | ffun_one | |
ffun_mul(f g : {ffun aT -> R}) := [ffun x => f x * g x].
Fact ffun_mulA : associative ffun_mul.
Proof. by move=> f1 f2 f3; apply/ffunP=> i; rewrite !ffunE mulrA. Qed.
Fact ffun_mul_1l : left_id ffun_one ffun_mul.
Proof. by move=> f; apply/ffunP=> i; rewrite !ffunE mul1r. Qed.
Fact ffun_mul_1r : right_id ffun_one ffun_m... | 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 | ffun_mul | |
ffun_semiring: pzSemiRingType := {ffun aT -> R}. | 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 | ffun_semiring | |
Definition_ := PzSemiRing_isNonZero.Build {ffun aT -> R}
ffun1_nonzero.
*) | 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 | |
Definition_ (aT : finType) (R : pzRingType) :=
Zmodule_isPzRing.Build {ffun aT -> R}
(@ffun_mulA _ _) (@ffun_mul_1l _ _) (@ffun_mul_1r _ _)
(@ffun_mul_addl _ _) (@ffun_mul_addr _ _). | 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 | |
Definition_ :=
PzSemiRing_isNonZero.Build {ffun aT -> R} (@ffun1_nonzero _ _ a). | 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 | |
ffun_ring: nzRingType := {ffun aT -> R}. | 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 | ffun_ring | |
Definition_ :=
Ring_hasCommutativeMul.Build (ffun_ring _ a) ffun_mulC.
*) | 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 | |
ffun_scalek f := [ffun a => k *: f a].
Fact ffun_scaleA k1 k2 f :
ffun_scale k1 (ffun_scale k2 f) = ffun_scale (k1 * k2) f.
Proof. by apply/ffunP=> a; rewrite !ffunE scalerA. Qed.
Fact ffun_scale0r f : ffun_scale 0 f = 0.
Proof. by apply/ffunP=> a; rewrite !ffunE scale0r. Qed.
Fact ffun_scale1 : left_id 1 ffun_scale.... | 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 | ffun_scale | |
Definition_ (R : pzRingType) (aT : finType) (rT : lmodType R) :=
LSemiModule.on {ffun aT -> rT}. | 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 | |
mul_pair(x y : R1 * R2) := (x.1 * y.1, x.2 * y.2).
Fact pair_mulA : associative mul_pair.
Proof. by move=> x y z; congr (_, _); apply: mulrA. Qed.
Fact pair_mul1l : left_id (1, 1) mul_pair.
Proof. by case=> x1 x2; congr (_, _); apply: mul1r. Qed.
Fact pair_mul1r : right_id (1, 1) mul_pair.
Proof. by case=> x1 x2; congr... | 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 | mul_pair | |
Definition_ := PzSemiRing_isNonZero.Build (R1 * R2)%type
pair_one_neq0. | 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 | |
Definition_ := PzSemiRing_hasCommutativeMul.Build (R1 * R2)%type
pair_mulC. | 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 | |
Definition_ (R1 R2 : comNzSemiRingType) :=
NzSemiRing.on (R1 * R2)%type.
#[export]
HB.instance Definition _ (R1 R2 : pzRingType) := PzSemiRing.on (R1 * R2)%type.
#[export]
HB.instance Definition _ (R1 R2 : nzRingType) := NzSemiRing.on (R1 * R2)%type.
#[export]
HB.instance Definition _ (R1 R2 : comPzRingType) := PzRin... | 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 | |
scale_paira (v : V1 * V2) : V1 * V2 := (a *: v.1, a *: v.2).
Fact pair_scaleA a b u : scale_pair a (scale_pair b u) = scale_pair (a * b) u.
Proof. by congr (_, _); apply: scalerA. Qed.
Fact pair_scale0 u : scale_pair 0 u = 0.
Proof. by case: u => u1 u2; congr (_, _); apply: scale0r. Qed.
Fact pair_scale1 u : scale_pair... | 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 | scale_pair | |
Definition_ := LSemiModule_isLSemiAlgebra.Build R (A1 * A2)%type
pair_scaleAl. | 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 | |
Definition_ := RMorphism.on (@fst A1 A2).
#[export]
HB.instance Definition _ := RMorphism.on (@snd A1 A2). | 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 | |
Definition_ := LSemiAlgebra_isSemiAlgebra.Build R (A1 * A2)%type
pair_scaleAr. | 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 | |
pair_unitr:=
[qualify a x : R1 * R2 | (x.1 \is a GRing.unit) && (x.2 \is a GRing.unit)]. | 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 | pair_unitr | |
pair_invrx :=
if x \is a pair_unitr then (x.1^-1, x.2^-1) else x. | 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 | pair_invr | |
pair_mulVl: {in pair_unitr, left_inverse 1 pair_invr *%R}.
Proof.
rewrite /pair_invr=> x; case: ifP => // /andP[Ux1 Ux2] _.
by congr (_, _); apply: 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 | pair_mulVl | |
pair_mulVr: {in pair_unitr, right_inverse 1 pair_invr *%R}.
Proof.
rewrite /pair_invr=> x; case: ifP => // /andP[Ux1 Ux2] _.
by congr (_, _); apply: mulrV.
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 | pair_mulVr | |
pair_unitPx y : y * x = 1 /\ x * y = 1 -> x \is a pair_unitr.
Proof.
case=> [[y1x y2x] [x1y x2y]]; apply/andP.
by split; apply/unitrP; [exists y.1 | exists y.2].
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 | pair_unitP | |
pair_invr_out: {in [predC pair_unitr], pair_invr =1 id}.
Proof. by rewrite /pair_invr => x /negPf/= ->. Qed.
#[export]
HB.instance Definition _ := NzRing_hasMulInverse.Build (R1 * R2)%type
pair_mulVl pair_mulVr pair_unitP pair_invr_out. | 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 | pair_invr_out | |
Definition_ (R1 R2 : comUnitRingType) :=
UnitRing.on (R1 * R2)%type.
#[export]
HB.instance Definition _ (R : pzSemiRingType) (A1 A2 : comSemiAlgType R) :=
SemiAlgebra.on (A1 * A2)%type.
#[export]
HB.instance Definition _ (R : pzRingType) (V1 V2 : lmodType R) :=
LSemiModule.on (V1 * V2)%type.
#[export]
HB.instance... | 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 | |
pairMnE(M1 M2 : zmodType) (x : M1 * M2) n :
x *+ n = (x.1 *+ n, x.2 *+ n).
Proof. by case: x => x y; elim: n => //= n; rewrite !mulrS => ->. 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 | pairMnE | |
B:= mkB x & x \in S. | Inductive | 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 | B | |
vBu := let: mkB x _ := u in x.
HB.instance Definition _ := [isSub for vB].
HB.instance Definition _ := [Choice of B by <:]. | 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 | vB | |
Definition_ := [SubChoice_isSubUnitRing of B S by <:]. | 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 | |
Definition_ := [SubZmodule_isSubLmodule of B S by <:].
HB.instance Definition _ := [SubNzRing_SubLmodule_isSubLalgebra of B S by <:].
HB.instance Definition _ := [SubLalgebra_isSubAlgebra of B S by <:]. | 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 | |
Definition_ := [SubRing_isSubComNzRing of B S by <:].
HB.instance Definition _ := [SubComUnitRing_isSubIntegralDomain of B S by <:].
HB.instance Definition _ := [SubIntegralDomain_isSubField of B S by <:]. | 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 | |
Definition_ := Zmodule_isComNzRing.Build bool
andbA andbC andTb andb_addl isT.
Fact mulVb (b : bool) : b != 0 -> b * b = 1.
Proof. by case: b. Qed.
Fact invb_out (x y : bool) : y * x = 1 -> x != 0.
Proof. by case: x; case: y. Qed.
HB.instance Definition _ := ComNzRing_hasMulInverse.Build bool
mulVb invb_out (fun x ... | 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 | |
bool_fieldP: Field.axiom bool. Proof. by []. Qed.
HB.instance Definition _ := ComUnitRing_isField.Build bool bool_fieldP. | 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 | bool_fieldP | |
Definition_ := Nmodule_isComNzSemiRing.Build nat
mulnA mulnC mul1n mulnDl mul0n erefl.
HB.instance Definition _ (R : pzSemiRingType) :=
isMonoidMorphism.Build nat R (natmul 1) (mulr1n 1, natrM R). | 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 | |
natr0E: 0 = 0%N. Proof. by []. 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 | natr0E | |
natr1E: 1 = 1%N. Proof. by []. 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 | natr1E | |
natnn : n%:R = n.
Proof. by elim: n => [//|n IHn]; rewrite -nat1r IHn. 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 | natn | |
natrDEn m : n + m = (n + m)%N. Proof. by []. 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 | natrDE | |
natrMEn m : n * m = (n * m)%N. Proof. by []. 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 | natrME | |
natrXEn m : n ^+ m = (n ^ m)%N. Proof. by []. 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 | natrXE | |
natrE:= (natr0E, natr1E, natn, natrDE, natrME, natrXE). | 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 | natrE | |
int: Set := Posz of nat | Negz of nat. | Variant | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | int | |
parse_int(x : Number.int) : int :=
match x with
| Number.IntDecimal (Decimal.Pos u) => Posz (Nat.of_uint u)
| Number.IntDecimal (Decimal.Neg u) => Negz (Nat.of_uint u).-1
| Number.IntHexadecimal (Hexadecimal.Pos u) => Posz (Nat.of_hex_uint u)
| Number.IntHexadecimal (Hexadecimal.Neg u) => Negz (Nat.of_hex_uin... | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | parse_int | |
print_int(x : int) : Number.int :=
match x with
| Posz n => Number.IntDecimal (Decimal.Pos (Nat.to_uint n))
| Negz n => Number.IntDecimal (Decimal.Neg (Nat.to_uint n.+1))
end.
Number Notation int parse_int print_int : int_scope. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | print_int | |
natsum_of_int(m : int) : nat + nat :=
match m with Posz p => inl _ p | Negz n => inr _ n end. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | natsum_of_int | |
int_of_natsum(m : nat + nat) :=
match m with inl p => Posz p | inr n => Negz n end. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | int_of_natsum | |
natsum_of_intK: cancel natsum_of_int int_of_natsum.
Proof. by case. Qed.
HB.instance Definition _ := Countable.copy int (can_type natsum_of_intK). | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | natsum_of_intK | |
eqz_nat(m n : nat) : (m%:Z == n%:Z) = (m == n). Proof. by []. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | eqz_nat | |
addz(m n : int) :=
match m, n with
| Posz m', Posz n' => Posz (m' + n')
| Negz m', Negz n' => Negz (m' + n').+1
| Posz m', Negz n' => if n' < m' then Posz (m' - n'.+1) else Negz (n' - m')
| Negz n', Posz m' => if n' < m' then Posz (m' - n'.+1) else Negz (n' - m')
end. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | addz | |
oppzm :=
match m with
| Posz n => if n is (n'.+1)%N then Negz n' else Posz 0
| Negz n => Posz (n.+1)%N
end.
Arguments oppz : simpl never.
Local Notation "-%Z" := (@oppz) : int_scope.
Local Notation "- x" := (oppz x) : int_scope.
Local Notation "+%Z" := (@addz) : int_scope.
Local Notation "x + y" := (addz x ... | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | oppz | |
PoszD: {morph Posz : m n / (m + n)%N >-> m + n}. Proof. by []. Qed.
Local Coercion Posz : nat >-> int. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | PoszD | |
NegzE(n : nat) : Negz n = - n.+1. Proof. by []. Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | NegzE | |
int_rect(P : int -> Type) :
P 0 -> (forall n : nat, P n -> P (n.+1))
-> (forall n : nat, P (- n) -> P (- (n.+1)))
-> forall n : int, P n.
Proof.
by move=> P0 hPp hPn []; elim=> [|n ihn]//; do ?[apply: hPn | apply: hPp].
Qed. | Lemma | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | int_rect | |
int_rec:= int_rect. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | int_rec | |
int_ind:= int_rect. | Definition | algebra | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice seq",
"From mathcomp Require Import fintype finfun bigop order ssralg countalg ssrnum",
"From mathcomp Require Import poly"
] | algebra/ssrint.v | int_ind |
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