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 |
|---|---|---|---|---|---|---|
mulrnDrx m n : x *+ (m + n) = x *+ m + x *+ n.
Proof. exact: (@expgnDr G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | mulrnDr | |
mulrnAx m n : x *+ (m * n) = x *+ m *+ n.
Proof. exact: (@expgnA G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | mulrnA | |
mulrnACx m n : x *+ m *+ n = x *+ n *+ m.
Proof. exact: (@expgnAC G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | mulrnAC | |
iter_addrn x y : iter n (+%R x) y = x *+ n + y.
Proof. exact: (@iter_mulg G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | iter_addr | |
iter_addr_0n x : iter n (+%R x) 0 = x *+ n.
Proof. exact: (@iter_mulg_1 G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | iter_addr_0 | |
sumrMnlI r P (F : I -> V) n :
\sum_(i <- r | P i) F i *+ n = (\sum_(i <- r | P i) F i) *+ n.
Proof. by rewrite (big_morph _ (mulrnDl n) (mul0rn _)). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | sumrMnl | |
sumrMnrx I r P (F : I -> nat) :
\sum_(i <- r | P i) x *+ F i = x *+ (\sum_(i <- r | P i) F i).
Proof. by rewrite (big_morph _ (mulrnDr x) (erefl _)). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | sumrMnr | |
sumr_const(I : finType) (A : pred I) x : \sum_(i in A) x = x *+ #|A|.
Proof. by rewrite big_const -iteropE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | sumr_const | |
sumr_const_natm n x : \sum_(n <= i < m) x = x *+ (m - n).
Proof. by rewrite big_const_nat iter_addr_0. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | sumr_const_nat | |
nmod_closed:= addumagma_closed.
HB.mixin Record hasOpp V := {
opp : V -> V
}.
#[short(type="baseZmodType")]
HB.structure Definition BaseZmodule := {V of hasOpp V & BaseAddUMagma V}. | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | nmod_closed | |
oppr_closed:= {in S, forall u, - u \in S}. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | oppr_closed | |
subr_closed:= {in S &, forall u v, u - v \in S}. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | subr_closed | |
zmod_closed:= 0 \in S /\ subr_closed. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | zmod_closed | |
RecordBaseZmoduleNmodule_isZmodule V of BaseZmodule V := {
addNr : left_inverse zero opp (@add V)
}.
#[short(type="zmodType")]
HB.structure Definition Zmodule :=
{V of BaseZmoduleNmodule_isZmodule V & BaseZmodule V & Nmodule V}.
HB.factory Record Nmodule_isZmodule V of Nmodule V := {
opp : V -> V;
addNr : left_inverse zero opp add
}.
HB.builders Context V of Nmodule_isZmodule V.
HB.instance Definition _ := hasOpp.Build V opp.
HB.instance Definition _ := BaseZmoduleNmodule_isZmodule.Build V addNr.
HB.end.
HB.factory Record isZmodule V of Choice V := {
zero : V;
opp : V -> V;
add : V -> V -> V;
addrA : associative add;
addrC : commutative add;
add0r : left_id zero add;
addNr : left_inverse zero opp add
}.
HB.builders Context V of isZmodule V.
HB.instance Definition _ := isNmodule.Build V addrA addrC add0r.
HB.instance Definition _ := Nmodule_isZmodule.Build V addNr.
HB.end. | HB.mixin | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Record | |
addrN(V : zmodType) : @right_inverse V V V 0 -%R +%R.
Proof. by move=> x; rewrite addrC addNr. Qed.
#[export]
HB.instance Definition _ (V : baseZmodType) :=
hasInv.Build (to_multiplicative V) (@opp V).
#[export]
HB.instance Definition _ (V : zmodType) :=
Monoid_isGroup.Build (to_multiplicative V) addNr (@addrN V). | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | addrN | |
subrr:= addrN. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | subrr | |
addKr: @left_loop V V -%R +%R.
Proof. exact: (@mulKg G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | addKr | |
addNKr: @rev_left_loop V V -%R +%R.
Proof. exact: (@mulVKg G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | addNKr | |
addrK: @right_loop V V -%R +%R.
Proof. exact: (@mulgK G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | addrK | |
addrNK: @rev_right_loop V V -%R +%R.
Proof. exact: (@mulgVK G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | addrNK | |
subrK:= addrNK. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | subrK | |
subKrx : involutive (fun y => x - y).
Proof. by move=> y; exact/(@divKg G)/commuteT. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | subKr | |
addrI: @right_injective V V V +%R.
Proof. exact: (@mulgI G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | addrI | |
addIr: @left_injective V V V +%R.
Proof. exact: (@mulIg G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | addIr | |
subrI: right_injective (fun x y => x - y).
Proof. exact: (@divgI G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | subrI | |
subIr: left_injective (fun x y => x - y).
Proof. exact: (@divIg G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | subIr | |
opprK: @involutive V -%R.
Proof. exact: (@invgK G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | opprK | |
oppr_inj: @injective V V -%R.
Proof. exact: (@invg_inj G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | oppr_inj | |
oppr0: -0 = 0 :> V.
Proof. exact: (@invg1 G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | oppr0 | |
oppr_eq0x : (- x == 0) = (x == 0).
Proof. exact: (@invg_eq1 G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | oppr_eq0 | |
subr0x : x - 0 = x. Proof. exact: (@divg1 G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | subr0 | |
sub0rx : 0 - x = - x. Proof. exact: (@div1g G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | sub0r | |
opprBx y : - (x - y) = y - x.
Proof. exact: (@invgF G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | opprB | |
opprD: {morph -%R: x y / x + y : V}.
Proof. by move=> x y; rewrite -[y in LHS]opprK opprB addrC. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | opprD | |
addrKAz x y : (x + z) - (z + y) = x - y.
Proof. by rewrite opprD addrA addrK. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | addrKA | |
subrKAz x y : (x - z) + (z + y) = x + y.
Proof. exact: (@divgKA G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | subrKA | |
addr0_eqx y : x + y = 0 -> - x = y.
Proof. exact: (@mulg1_eq G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | addr0_eq | |
subr0_eqx y : x - y = 0 -> x = y.
Proof. exact: (@divg1_eq G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | subr0_eq | |
subr_eqx y z : (x - z == y) = (x == y + z).
Proof. exact: (@divg_eq G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | subr_eq | |
subr_eq0x y : (x - y == 0) = (x == y).
Proof. exact: (@divg_eq1 G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | subr_eq0 | |
addr_eq0x y : (x + y == 0) = (x == - y).
Proof. exact: (@mulg_eq1 G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | addr_eq0 | |
eqr_oppx y : (- x == - y) = (x == y).
Proof. exact: (@eqg_inv G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | eqr_opp | |
eqr_oppLRx y : (- x == y) = (x == - y).
Proof. exact: (@eqg_invLR G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | eqr_oppLR | |
mulNrnx n : (- x) *+ n = x *- n.
Proof. exact: (@expVgn G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | mulNrn | |
mulrnBln : {morph (fun x => x *+ n) : x y / x - y}.
Proof. by move=> x y; exact/(@expgnFl G)/commuteT. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | mulrnBl | |
mulrnBrx m n : n <= m -> x *+ (m - n) = x *+ m - x *+ n.
Proof. exact: (@expgnFr G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | mulrnBr | |
sumrNI r P (F : I -> V) :
(\sum_(i <- r | P i) - F i = - (\sum_(i <- r | P i) F i)).
Proof. by rewrite (big_morph _ opprD oppr0). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | sumrN | |
sumrBI r (P : pred I) (F1 F2 : I -> V) :
\sum_(i <- r | P i) (F1 i - F2 i)
= \sum_(i <- r | P i) F1 i - \sum_(i <- r | P i) F2 i.
Proof. by rewrite -sumrN -big_split /=. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | sumrB | |
telescope_sumrn m (f : nat -> V) : n <= m ->
\sum_(n <= k < m) (f k.+1 - f k) = f m - f n.
Proof.
move=> nm; rewrite (telescope_big (fun i j => f j - f i)).
by case: ltngtP nm => // ->; rewrite subrr.
by move=> k /andP[nk km]/=; rewrite addrC subrKA.
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | telescope_sumr | |
telescope_sumr_eqn m (f u : nat -> V) : n <= m ->
(forall k, (n <= k < m)%N -> u k = f k.+1 - f k) ->
\sum_(n <= k < m) u k = f m - f n.
Proof.
by move=> ? uE; under eq_big_nat do rewrite uE //=; exact: telescope_sumr.
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | telescope_sumr_eq | |
zmod_closedN: zmod_closed S -> oppr_closed S.
Proof. exact: (@group_closedV G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | zmod_closedN | |
zmod_closedD: zmod_closed S -> addr_closed S.
Proof. exact: (@group_closedM G). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | zmod_closedD | |
zmod_closed0D: zmod_closed S -> nmod_closed S.
Proof. by move=> z; split; [case: z|apply: zmod_closedD]. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | zmod_closed0D | |
nmod_morphism(U V : baseAddUMagmaType) (f : U -> V) : Prop :=
(f 0 = 0) * {morph f : x y / x + y}.
#[deprecated(since="mathcomp 2.5.0", note="use `nmod_morphism` instead")] | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | nmod_morphism | |
semi_additive:= nmod_morphism.
HB.mixin Record isNmodMorphism (U V : baseAddUMagmaType) (apply : U -> V) := {
nmod_morphism_subproof : nmod_morphism apply;
}. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | semi_additive | |
BuildU V apply := (isNmodMorphism.Build U V apply) (only parsing). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Build | |
DefinitionAdditive (U V : baseAddUMagmaType) :=
{f of isNmodMorphism U V f}. | HB.structure | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Definition | |
zmod_morphism(U V : zmodType) (f : U -> V) :=
{morph f : x y / x - y}.
#[deprecated(since="mathcomp 2.5.0", note="use `zmod_morphism` instead")] | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | zmod_morphism | |
additive:= zmod_morphism.
HB.factory Record isZmodMorphism (U V : zmodType) (apply : U -> V) := {
zmod_morphism_subproof : zmod_morphism apply;
}. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | additive | |
BuildU V apply := (isZmodMorphism.Build U V apply) (only parsing). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Build | |
Definition_ := isNmodMorphism.Build U V apply (conj raddf0 raddfD).
HB.end. | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Definition | |
raddf0: f 0 = 0.
Proof. exact: nmod_morphism_subproof.1. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | raddf0 | |
raddfD:
{morph f : x y / x + y}.
Proof. exact: nmod_morphism_subproof.2. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | raddfD | |
to_fmultiplicativeU V :=
@id (to_multiplicative U -> to_multiplicative V).
#[export]
HB.instance Definition _ U V (f : {additive U -> V}) :=
isMultiplicative.Build (to_multiplicative U) (to_multiplicative V)
(to_fmultiplicative f) (@raddfD _ _ f).
#[export]
HB.instance Definition _ (U V : baseAddUMagmaType) (f : {additive U -> V}) :=
Multiplicative_isUMagmaMorphism.Build
(to_multiplicative U) (to_multiplicative V) (to_fmultiplicative f)
(@raddf0 _ _ f). | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | to_fmultiplicative | |
add_fun(f g : U -> V) x := f x + g x. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | add_fun | |
null_funof U : V := 0. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | null_fun | |
opp_fun(f : U -> V) x := - f x. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | opp_fun | |
sub_fun(f g : U -> V) x := f x - g x. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | sub_fun | |
raddf_eq0x : injective f -> (f x == 0) = (x == 0).
Proof. exact: (@gmulf_eq1 _ _ g). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | raddf_eq0 | |
raddfMnn : {morph f : x / x *+ n}.
Proof. exact: (@gmulfXn _ _ g). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | raddfMn | |
raddf_sumI r (P : pred I) E :
f (\sum_(i <- r | P i) E i) = \sum_(i <- r | P i) f (E i).
Proof. exact: (@gmulf_prod _ _ g). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | raddf_sum | |
can2_nmod_morphismf' : cancel f f' -> cancel f' f -> nmod_morphism f'.
Proof.
split; first exact/(@can2_gmulf1 _ _ g).
exact/(@can2_gmulfM _ _ g).
Qed.
#[deprecated(since="mathcomp 2.5.0", note="use `can2_nmod_morphism` instead")] | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | can2_nmod_morphism | |
can2_semi_additive:= can2_nmod_morphism. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | can2_semi_additive | |
raddfN: {morph f : x / - x}.
Proof. exact: (@gmulfV _ _ g). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | raddfN | |
raddfB: {morph f : x y / x - y}.
Proof. exact: (@gmulfF _ _ g). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | raddfB | |
raddf_inj: (forall x, f x = 0 -> x = 0) -> injective f.
Proof. exact: (@gmulf_inj _ _ g). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | raddf_inj | |
raddfMNnn : {morph f : x / x *- n}.
Proof. exact: (@gmulfXVn _ _ g). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | raddfMNn | |
can2_zmod_morphismf' : cancel f f' -> cancel f' f -> zmod_morphism f'.
Proof. by move=> fK f'K x y /=; apply: (canLR fK); rewrite raddfB !f'K. Qed.
#[warning="-deprecated-since-mathcomp-2.5.0",
deprecated(since="mathcomp 2.5.0", note="use `can2_zmod_morphism` instead")] | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | can2_zmod_morphism | |
can2_additive:= can2_zmod_morphism. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | can2_additive | |
Definition_ f g :=
isNmodMorphism.Build U V (add_fun f g) (add_fun_nmod_morphism f g). | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Definition | |
Definition_ := isNmodMorphism.Build U U idfun
idfun_is_nmod_morphism.
Fact comp_is_nmod_morphism : nmod_morphism (f \o g).
Proof. by split=> [|x y]; rewrite /= ?raddf0// !raddfD. Qed.
#[export]
HB.instance Definition _ := isNmodMorphism.Build U W (f \o g)
comp_is_nmod_morphism. | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Definition | |
Definition_ :=
isNmodMorphism.Build U V (\0 : U -> V)
null_fun_is_nmod_morphism. | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Definition | |
Definition_ :=
isZmodMorphism.Build V V -%R opp_is_zmod_morphism.
Fact opp_fun_is_zmod_morphism : nmod_morphism (\- f).
Proof.
split=> [|x y]; first by rewrite -[LHS]/(- (f 0)) raddf0 oppr0.
by rewrite -[LHS]/(- (f (x + y))) !raddfD/=.
Qed.
#[export]
HB.instance Definition _ :=
isNmodMorphism.Build U V (opp_fun f) opp_fun_is_zmod_morphism.
Fact sub_fun_is_zmod_morphism :
nmod_morphism (f \- g).
Proof.
split=> [|x y]/=; first by rewrite !raddf0 addr0.
by rewrite !raddfD/= addrACA.
Qed.
#[export]
HB.instance Definition _ :=
isNmodMorphism.Build U V (f \- g) sub_fun_is_zmod_morphism. | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Definition | |
RecordisAddClosed (V : baseAddUMagmaType) (S : {pred V}) := {
nmod_closed_subproof : addumagma_closed S
}.
HB.mixin Record isOppClosed (V : zmodType) (S : {pred V}) := {
oppr_closed_subproof : oppr_closed S
}. | HB.mixin | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Record | |
DefinitionAddClosed V := {S of isAddClosed V S}.
#[short(type="opprClosed")]
HB.structure Definition OppClosed V := {S of isOppClosed V S}.
#[short(type="zmodClosed")]
HB.structure Definition ZmodClosed V := {S of OppClosed V S & AddClosed V S}. | HB.structure | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Definition | |
RecordisZmodClosed (V : zmodType) (S : V -> bool) := {
zmod_closed_subproof : zmod_closed S
}.
HB.builders Context V S of isZmodClosed V S.
HB.instance Definition _ := isOppClosed.Build V S
(zmod_closedN zmod_closed_subproof).
HB.instance Definition _ := isAddClosed.Build V S
(zmod_closed0D zmod_closed_subproof).
HB.end. | HB.factory | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Record | |
to_pmultiplicative(T : Type) := @id {pred to_multiplicative T}.
#[export]
HB.instance Definition _ (U : baseAddUMagmaType) (S : addrClosed U) :=
isMulClosed.Build (to_multiplicative U) (to_pmultiplicative S)
(snd nmod_closed_subproof).
#[export]
HB.instance Definition _ (U : baseAddUMagmaType) (S : addrClosed U) :=
isMul1Closed.Build (to_multiplicative U) (to_pmultiplicative S)
(fst nmod_closed_subproof).
#[export]
HB.instance Definition _ (U : zmodType) (S : opprClosed U) :=
isInvClosed.Build (to_multiplicative U) (to_pmultiplicative S)
oppr_closed_subproof. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | to_pmultiplicative | |
Definition_ (U : zmodType) (S : zmodClosed U) :=
InvClosed.on (to_pmultiplicative S). | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | Definition | |
rpred0: 0 \in S.
Proof. by case: (@nmod_closed_subproof V S). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | rpred0 | |
rpredD: {in S &, forall u v, u + v \in S}.
Proof. by case: (@nmod_closed_subproof V S). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | rpredD | |
rpred0D: addumagma_closed S.
Proof. exact: nmod_closed_subproof. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | rpred0D | |
rpredMnn : {in S, forall u, u *+ n \in S}.
Proof. exact: (@gpredXn _ (to_pmultiplicative S)). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | rpredMn | |
rpred_sumI r (P : pred I) F :
(forall i, P i -> F i \in S) -> \sum_(i <- r | P i) F i \in S.
Proof. by move=> IH; elim/big_ind: _; [apply: rpred0 | apply: rpredD |]. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | rpred_sum | |
rpredNr: {in S, forall u, - u \in S}.
Proof. exact: oppr_closed_subproof. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | rpredNr | |
rpredN: {mono -%R: u / u \in S}.
Proof. exact: (gpredV (to_pmultiplicative S)). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | rpredN | |
rpredB: {in S &, forall u v, u - v \in S}.
Proof. exact: (@gpredF _ T). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | rpredB | |
rpredBCu v : u - v \in S = (v - u \in S).
Proof. exact: (@gpredFC _ T). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | rpredBC | |
rpredMNnn: {in S, forall u, u *- n \in S}.
Proof. exact: (@gpredXNn _ T). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | rpredMNn | |
rpredDrx y : x \in S -> (y + x \in S) = (y \in S).
Proof. exact: (@gpredMr _ T). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | rpredDr | |
rpredDlx y : x \in S -> (x + y \in S) = (y \in S).
Proof. exact: (@gpredMl _ T). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun monoid"
] | boot/nmodule.v | rpredDl |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.