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 |
|---|---|---|---|---|---|---|
int_spec(x : int) : int -> Type :=
| ZintNull of x = 0 : int_spec x 0
| ZintPos n of x = n.+1 : int_spec x n.+1
| ZintNeg n of x = - (n.+1)%:Z : int_spec x (- n.+1).
|
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_spec
| |
intPx : int_spec x x. Proof. by move: x=> [] []; constructor. 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
|
intP
| |
addzC: commutative addz.
Proof. by move=> [] m [] n //=; rewrite addnC. 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
|
addzC
| |
add0z: left_id 0 addz. Proof. by do 2?case. 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
|
add0z
| |
oppzK: involutive oppz. Proof. by do 2?case. 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
|
oppzK
| |
oppzD: {morph oppz : m n / m + n}.
Proof.
by move=> [[|n]|n] [[|m]|m] /=; rewrite ?addn0 ?subn0 ?addnS //;
rewrite !NegzE !ltnS !subSS; case: ltngtP => [?|?|->];
rewrite ?subnn // ?oppzK ?subnS ?prednK // subn_gt0.
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
|
oppzD
| |
add1Pz(n : int) : 1 + (n - 1) = n.
Proof. by case: (intP n)=> // n' /= _; rewrite ?(subn1, addn0). 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
|
add1Pz
| |
subSz1(n : int) : 1 + n - 1 = n.
Proof.
by apply: (inv_inj oppzK); rewrite addzC !oppzD oppzK [_ - n]addzC add1Pz.
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
|
subSz1
| |
addSnz(m : nat) (n : int) : m.+1%N + n = 1 + (m + n).
Proof.
move: m n=> [|m] [] [|n] //=; rewrite ?add1n ?subn1 // !(ltnS, subSS).
case: ltngtP=> hnm /=; rewrite ?hnm ?subnn //.
by rewrite subnS add1n prednK ?subn_gt0.
by rewrite ltnS leqn0 subn_eq0 leqNgt hnm /= subnS subn1.
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
|
addSnz
| |
addSz(m n : int) : (1 + m) + n = 1 + (m + n).
Proof.
case: m => [] m; first by rewrite -PoszD add1n addSnz.
rewrite !NegzE; apply: (inv_inj oppzK).
rewrite !oppzD !oppzK addSnz [-1%:Z + _]addzC addSnz add1Pz.
by rewrite [-1%:Z + _]addzC subSz1.
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
|
addSz
| |
addPz(m n : int) : (m - 1) + n = (m + n) - 1.
Proof.
by apply: (inv_inj oppzK); rewrite !oppzD oppzK [_ + 1]addzC addSz addzC.
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
|
addPz
| |
addzA: associative addz.
Proof.
elim=> [|m ihm|m ihm] n p; first by rewrite !add0z.
by rewrite -add1n PoszD !addSz ihm.
by rewrite -add1n addnC PoszD oppzD !addPz ihm.
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
|
addzA
| |
addNz: left_inverse (0:int) oppz addz. Proof. by do 3?elim. 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
|
addNz
| |
predn_int(n : nat) : 0 < n -> n.-1%:Z = n - 1.
Proof. by case: n => //= n _; rewrite subn1. 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
|
predn_int
| |
Mixin:= GRing.isZmodule.Build int addzA addzC add0z addNz.
|
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
|
Mixin
| |
Definition_ := intZmod.Mixin.
HB.instance Definition _ := GRing.isNmodMorphism.Build nat int Posz
(erefl, intZmod.PoszD).
Local Open Scope ring_scope.
|
HB.instance
|
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
|
Definition
| |
PoszD: {morph Posz : n m / (n + m)%N >-> n + m}. 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
|
PoszD
| |
NegzE(n : nat) : Negz n = -(n.+1)%:Z. 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)%N)
-> (forall n : nat, P (- (n%:Z)) -> P (- (n.+1%N%:Z)))
-> 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
| |
int_spec(x : int) : int -> Type :=
| ZintNull : int_spec x 0
| ZintPos n : int_spec x n.+1
| ZintNeg n : int_spec x (- (n.+1)%:Z).
|
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_spec
| |
intPx : int_spec x x.
Proof. by move: x=> [] [] *; rewrite ?NegzE; constructor. 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
|
intP
| |
oppzD:= @opprD int.
|
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
|
oppzD
| |
subzn(m n : nat) : (n <= m)%N -> m%:Z - n%:Z = (m - n)%N.
Proof.
elim: n=> //= [|n ihn] hmn; first by rewrite subr0 subn0.
rewrite subnS -addn1 !PoszD opprD addrA ihn 1?ltnW //.
by rewrite intZmod.predn_int // subn_gt0.
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
|
subzn
| |
subzSS(m n : nat) : m.+1%:Z - n.+1%:Z = m%:Z - n%:Z.
Proof. by elim: n m=> [|n ihn] m //; rewrite !subzn. 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
|
subzSS
| |
mulz(m n : int) :=
match m, n with
| Posz m', Posz n' => (m' * n')%N%:Z
| Negz m', Negz n' => (m'.+1%N * n'.+1%N)%N%:Z
| Posz m', Negz n' => - (m' * (n'.+1%N))%N%:Z
| Negz n', Posz m' => - (m' * (n'.+1%N))%N%:Z
end.
Local Notation "*%Z" := (@mulz) : int_scope.
Local Notation "x * y" := (mulz x y) : 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
|
mulz
| |
mul0z: left_zero 0 *%Z.
Proof. by case=> [n|[|n]] //=; rewrite muln0. 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
|
mul0z
| |
mulzC: commutative mulz.
Proof. by move=> [] m [] n //=; rewrite mulnC. 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
|
mulzC
| |
mulz0: right_zero 0 *%Z.
Proof. by move=> x; rewrite mulzC mul0z. 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
|
mulz0
| |
mulzN(m n : int) : (m * (- n))%Z = - (m * n)%Z.
Proof.
by case: (intP m)=> {m} [|m|m]; rewrite ?mul0z //;
case: (intP n)=> {n} [|n|n]; rewrite ?mulz0 //= mulnC.
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
|
mulzN
| |
mulNz(m n : int) : ((- m) * n)%Z = - (m * n)%Z.
Proof. by rewrite mulzC mulzN mulzC. 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
|
mulNz
| |
mulzA: associative mulz.
Proof.
by move=> [] m [] n [] p; rewrite ?NegzE ?(mulnA,mulNz,mulzN,opprK) //= ?mulnA.
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
|
mulzA
| |
mul1z: left_id 1%Z mulz.
Proof. by case=> [[|n]|n] //=; rewrite ?mul1n// plusE addn0. 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
|
mul1z
| |
mulzS(x : int) (n : nat) : (x * n.+1%:Z)%Z = x + (x * n)%Z.
Proof.
by case: (intP x)=> [|m'|m'] //=; [rewrite mulnS|rewrite mulSn -opprD].
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
|
mulzS
| |
mulz_addl: left_distributive mulz (+%R).
Proof.
move=> x y z; elim: z=> [|n|n]; first by rewrite !(mul0z,mulzC).
by rewrite !mulzS=> ->; rewrite !addrA [X in X + _]addrAC.
rewrite !mulzN !mulzS -!opprD=> /oppr_inj->.
by rewrite !addrA [X in X + _]addrAC.
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
|
mulz_addl
| |
nonzero1z: 1%Z != 0. 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
|
nonzero1z
| |
comMixin:= GRing.Zmodule_isComNzRing.Build int
mulzA mulzC mul1z mulz_addl nonzero1z.
|
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
|
comMixin
| |
Definition_ := intRing.comMixin.
|
HB.instance
|
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
|
Definition
| |
PoszM: {morph Posz : n m / (n * m)%N >-> n * m}. 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
|
PoszM
| |
intS(n : nat) : n.+1%:Z = 1 + n%:Z. Proof. by rewrite -PoszD. 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
|
intS
| |
predn_int(n : nat) : (0 < n)%N -> n.-1%:Z = n%:Z - 1.
Proof. exact: intZmod.predn_int. 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
|
predn_int
| |
Definition_ := GRing.isMonoidMorphism.Build nat int Posz
(erefl, PoszM).
|
HB.instance
|
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
|
Definition
| |
unitz:= [qualify a n : int | (n == 1) || (n == -1)].
|
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
|
unitz
| |
invzn : int := n.
|
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
|
invz
| |
mulVz: {in unitz, left_inverse 1%R invz *%R}.
Proof. by move=> n /pred2P[] ->. 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
|
mulVz
| |
mulzn_eq1m (n : nat) : (m * n == 1) = (m == 1) && (n == 1).
Proof. by case: m => m /=; [rewrite -PoszM [_==_]muln_eq1 | case: n]. 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
|
mulzn_eq1
| |
unitzPlm n : n * m = 1 -> m \is a unitz.
Proof.
rewrite qualifE => /eqP.
by case: m => m; rewrite ?NegzE ?mulrN -?mulNr mulzn_eq1 => /andP[_ /eqP->].
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
|
unitzPl
| |
invz_out: {in [predC unitz], invz =1 id}.
Proof. exact. 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
|
invz_out
| |
idomain_axiomzm n : m * n = 0 -> (m == 0) || (n == 0).
Proof. by case: m n => [[|m]|m] [[|n]|n]. 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
|
idomain_axiomz
| |
comMixin:= GRing.ComNzRing_hasMulInverse.Build int
mulVz unitzPl invz_out.
|
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
|
comMixin
| |
Definition_ := intUnitRing.comMixin.
HB.instance Definition _ := GRing.ComUnitRing_isIntegral.Build int
intUnitRing.idomain_axiomz.
|
HB.instance
|
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
|
Definition
| |
abszm := match m with Posz p => p | Negz n => n.+1 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
|
absz
| |
lezm n :=
match m, n with
| Posz m', Posz n' => (m' <= n')%N
| Posz m', Negz n' => false
| Negz m', Posz n' => true
| Negz m', Negz n' => (n' <= m')%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
|
lez
| |
ltzm n :=
match m, n with
| Posz m', Posz n' => (m' < n')%N
| Posz m', Negz n' => false
| Negz m', Posz n' => true
| Negz m', Negz n' => (n' < m')%N
end.
Fact lez_add m n : lez 0 m -> lez 0 n -> lez 0 (m + n).
Proof. by case: m n => [] m [] n. Qed.
Fact lez_mul m n : lez 0 m -> lez 0 n -> lez 0 (m * n).
Proof. by case: m n => [] m [] n. Qed.
Fact lez_anti m : lez 0 m -> lez m 0 -> m = 0.
Proof. by case: m; first case. Qed.
|
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
|
ltz
| |
subz_ge0m n : lez 0 (n - m) = lez m n.
Proof.
case: (intP m); case: (intP n)=> // {}m {}n /=;
rewrite ?ltnS -?opprD ?opprB ?subzSS; case: leqP=> // hmn;
by [ rewrite subzn //
| rewrite -opprB subzn ?(ltnW hmn) //;
move: hmn; rewrite -subn_gt0; case: (_ - _)%N].
Qed.
Fact lez_total m n : lez m n || lez n m.
Proof. by move: m n => [] m [] n //=; apply: leq_total. Qed.
Fact normzN m : normz (- m) = normz m.
Proof. by case: m => // -[]. Qed.
Fact gez0_norm m : lez 0 m -> normz m = m.
Proof. by case: m. Qed.
Fact ltz_def m n : (ltz m n) = (n != m) && (lez m n).
Proof.
by move: m n => [] m [] n //=; rewrite (ltn_neqAle, leq_eqVlt) // eq_sym.
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
|
subz_ge0
| |
Mixin:= Num.IntegralDomain_isLeReal.Build int
lez_add lez_mul lez_anti subz_ge0 (lez_total 0) normzN gez0_norm ltz_def.
|
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
|
Mixin
| |
Definition_ := intOrdered.Mixin.
|
HB.instance
|
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
|
Definition
| |
lez_natm n : (m <= n :> int) = (m <= n)%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
|
lez_nat
| |
ltz_natm n : (m < n :> int) = (m < n)%N.
Proof. by rewrite ltnNge ltNge lez_nat. 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
|
ltz_nat
| |
ltez_nat:= (lez_nat, ltz_nat).
|
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
|
ltez_nat
| |
leNz_natm n : (- m%:Z <= n). Proof. by case: m. 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
|
leNz_nat
| |
ltNz_natm n : (- m%:Z < n) = (m != 0) || (n != 0).
Proof. by move: m n=> [|?] []. 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
|
ltNz_nat
| |
lteNz_nat:= (leNz_nat, ltNz_nat).
|
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
|
lteNz_nat
| |
lezN_natm n : (m%:Z <= - n%:Z) = (m == 0) && (n == 0).
Proof. by move: m n=> [|?] []. 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
|
lezN_nat
| |
ltzN_natm n : (m%:Z < - n%:Z) = false.
Proof. by move: m n=> [|?] []. 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
|
ltzN_nat
| |
le0z_natn : 0 <= n :> int. 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
|
le0z_nat
| |
lez0_natn : n <= 0 :> int = (n == 0 :> nat). Proof. by elim: n. 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
|
lez0_nat
| |
ltezN_nat:= (lezN_nat, ltzN_nat).
|
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
|
ltezN_nat
| |
ltez_natE:= (ltez_nat, lteNz_nat, ltezN_nat, le0z_nat, lez0_nat).
|
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
|
ltez_natE
| |
gtz0_ge1x : (0 < x) = (1 <= x). Proof. by case: (intP x). 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
|
gtz0_ge1
| |
lez1Dx y : (1 + x <= y) = (x < y).
Proof. by rewrite -subr_gt0 gtz0_ge1 lterBDr. 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
|
lez1D
| |
lezD1x y : (x + 1 <= y) = (x < y).
Proof. by rewrite addrC lez1D. 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
|
lezD1
| |
ltz1Dx y : (x < 1 + y) = (x <= y).
Proof. by rewrite -lez1D lerD2l. 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
|
ltz1D
| |
ltzD1x y : (x < y + 1) = (x <= y).
Proof. by rewrite -lezD1 lerD2r. 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
|
ltzD1
| |
intmul(R : zmodType) (x : R) (n : int) :=
match n with
| Posz n => (x *+ n)%R
| Negz n => (x *- (n.+1))%R
end.
Arguments intmul : simpl never.
|
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
|
intmul
| |
intr:= ( *~%R 1).
|
Notation
|
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
|
intr
| |
pmulrn(R : zmodType) (x : R) (n : nat) : x *+ n = x *~ n%:Z.
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
|
pmulrn
| |
nmulrn(R : zmodType) (x : R) (n : nat) : x *- n = x *~ - n%:Z.
Proof. by case: n; rewrite // oppr0. 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
|
nmulrn
| |
zmodule(M : Type) : Type := M.
Local Notation "M ^z" := (zmodule M) (format "M ^z") : type_scope.
Local Coercion Posz : nat >-> int.
Variable M : zmodType.
Implicit Types m n : int.
Implicit Types x y z : M.
Fact mulrzA_C m n x : (x *~ n) *~ m = x *~ (m * n).
Proof.
elim: m=> [|m _|m _]; elim: n=> [|n _|n _]; rewrite /intmul //=;
rewrite ?(muln0, mulr0n, mul0rn, oppr0, mulNrn, opprK) //;
do ?by rewrite mulnC mulrnA.
* by rewrite -mulrnA mulnC.
* by rewrite -mulrnA.
Qed.
Fact mulrzAC m n x : (x *~ n) *~ m = (x *~ m) *~ n.
Proof. by rewrite !mulrzA_C mulrC. Qed.
Fact mulr1z (x : M) : x *~ 1 = x. Proof. by []. Qed.
Fact mulrzDl m : {morph ( *~%R^~ m : M -> M) : x y / x + y}.
Proof. by case: m => m x y; rewrite /intmul mulrnDl // opprD. Qed.
|
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
|
zmodule
| |
mulrzBl_nat(m n : nat) x : x *~ (m%:Z - n%:Z) = x *~ m - x *~ n.
Proof.
wlog/subnK <-: m n / (n <= m)%N; last by rewrite -!pmulrn PoszD mulrnDr !addrK.
have [hmn|/ltnW hmn] := leqP n m; first exact.
by rewrite -[in LHS]opprB -[RHS]opprB subzn // -nmulrn pmulrn -subzn // => ->.
Qed.
Fact mulrzDr x : {morph *~%R x : m n / m + n}.
Proof.
by case=> []m []n; rewrite ?NegzE /intmul /= -/(intmul _ _) -?opprD;
rewrite -?[- _ + _]addrC ?mulrzBl_nat // -mulrnDr // addnS.
Qed.
HB.instance Definition _ := GRing.Zmodule.on M^z.
HB.instance Definition _ := @GRing.Zmodule_isLmodule.Build _ M^z
(fun n x => x *~ n) mulrzA_C mulr1z mulrzDl mulrzDr.
|
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
|
mulrzBl_nat
| |
scalezrEn x : n *: (x : M^z) = x *~ 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
|
scalezrE
| |
mulrzAx m n : x *~ (m * n) = x *~ m *~ n.
Proof. by rewrite -!scalezrE scalerA mulrC. 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
|
mulrzA
| |
mulr0zx : x *~ 0 = 0. 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
|
mulr0z
| |
mul0rzn : 0 *~ n = 0 :> M.
Proof. by rewrite -scalezrE scaler0. 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
|
mul0rz
| |
mulrNzx n : x *~ (- n) = - (x *~ n).
Proof. by rewrite -scalezrE scaleNr. 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
|
mulrNz
| |
mulrN1zx : x *~ (- 1) = - x. Proof. by rewrite -scalezrE scaleN1r. 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
|
mulrN1z
| |
mulNrzx n : (- x) *~ n = - (x *~ n).
Proof. by rewrite -scalezrE scalerN. 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
|
mulNrz
| |
mulrzBrx m n : x *~ (m - n) = x *~ m - x *~ n.
Proof. by rewrite -scalezrE scalerBl. 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
|
mulrzBr
| |
mulrzBlx y n : (x - y) *~ n = x *~ n - y *~ n.
Proof. by rewrite -scalezrE scalerBr. 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
|
mulrzBl
| |
mulrz_nat(n : nat) x : x *~ n%:R = x *+ n.
Proof. by rewrite -scalezrE scaler_nat. 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
|
mulrz_nat
| |
mulrz_sumr: forall x I r (P : pred I) F,
x *~ (\sum_(i <- r | P i) F i) = \sum_(i <- r | P i) x *~ F i.
Proof. by rewrite -/M^z; apply: scaler_suml. 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
|
mulrz_sumr
| |
mulrz_suml: forall n I r (P : pred I) (F : I -> M),
(\sum_(i <- r | P i) F i) *~ n= \sum_(i <- r | P i) F i *~ n.
Proof. by rewrite -/M^z; apply: scaler_sumr. Qed.
HB.instance Definition _ (x : M) := GRing.isZmodMorphism.Build int M ( *~%R x)
(@mulrzBr x).
|
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
|
mulrz_suml
| |
mulrzDl_tmp:= mulrzDl.
#[deprecated(since="mathcomp 2.3.0", note="Use mulrzDr instead.")]
|
Notation
|
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
|
mulrzDl_tmp
| |
mulrzDr_tmp:= mulrzDr.
|
Notation
|
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
|
mulrzDr_tmp
| |
ffunMzE(I : finType) (M : zmodType) (f : {ffun I -> M}) z x :
(f *~ z) x = f x *~ z.
Proof. by case: z => n; rewrite ?ffunE ffunMnE. 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
|
ffunMzE
| |
intz(n : int) : n%:~R = n.
Proof. by case: n => n; rewrite ?NegzE /intmul/= -(rmorphMn Posz)/= natn. 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
|
intz
| |
natz(n : nat) : n%:R = n%:Z :> int.
Proof. by rewrite pmulrn intz. 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
|
natz
| |
mulrzAln x y : (x *~ n) * y = (x * y) *~ n.
Proof. by case: n => n; rewrite ?mulNr mulrnAl. 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
|
mulrzAl
| |
mulrzArn x y : x * (y *~ n) = (x * y) *~ n.
Proof. by case: n => n; rewrite ?mulrN mulrnAr. 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
|
mulrzAr
|
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