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subrKC x y : x + (y - x) = y.
Proof. by rewrite addrC subrK. Qed.
Lemma
subrKC
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "addrC", "subrK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
subKr x : involutive (fun y => x - y).
Proof. by move=> y; exact/(@divKg G)/commuteT. Qed.
Lemma
subKr
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "commuteT", "divKg" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
addrI : @right_injective V V V +%R.
Proof. exact: (@mulgI G). Qed.
Lemma
addrI
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "mulgI" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
addIr : @left_injective V V V +%R.
Proof. exact: (@mulIg G). Qed.
Lemma
addIr
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "mulIg" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
subrI : right_injective (fun x y => x - y).
Proof. exact: (@divgI G). Qed.
Lemma
subrI
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "divgI" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
subIr : left_injective (fun x y => x - y).
Proof. exact: (@divIg G). Qed.
Lemma
subIr
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "divIg" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
opprK : @involutive V -%R.
Proof. exact: (@invgK G). Qed.
Lemma
opprK
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "invgK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
oppr_inj : @injective V V -%R.
Proof. exact: (@invg_inj G). Qed.
Lemma
oppr_inj
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "invg_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
oppr0 : -0 = 0 :> V.
Proof. exact: (@invg1 G). Qed.
Lemma
oppr0
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "invg1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
oppr_eq0 x : (- x == 0) = (x == 0).
Proof. exact: (@invg_eq1 G). Qed.
Lemma
oppr_eq0
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "invg_eq1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
subr0 x : x - 0 = x.
Proof. exact: (@divg1 G). Qed.
Lemma
subr0
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "divg1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub0r x : 0 - x = - x.
Proof. exact: (@div1g G). Qed.
Lemma
sub0r
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "div1g" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
opprB x y : - (x - y) = y - x.
Proof. exact: (@invgF G). Qed.
Lemma
opprB
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "invgF" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
opprD : {morph -%R: x y / x + y : V}.
Proof. by move=> x y; rewrite -[y in LHS]opprK opprB addrC. Qed.
Lemma
opprD
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "addrC", "opprB", "opprK" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
addrKA z x y : (x + z) - (z + y) = x - y.
Proof. by rewrite opprD addrA addrK. Qed.
Lemma
addrKA
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "addrA", "addrK", "opprD" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
subrKA z x y : (x - z) + (z + y) = x + y.
Proof. exact: (@divgKA G). Qed.
Lemma
subrKA
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "divgKA" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
addr0_eq x y : x + y = 0 -> - x = y.
Proof. exact: (@mulg1_eq G). Qed.
Lemma
addr0_eq
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "mulg1_eq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
subr0_eq x y : x - y = 0 -> x = y.
Proof. exact: (@divg1_eq G). Qed.
Lemma
subr0_eq
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "divg1_eq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
subr_eq x y z : (x - z == y) = (x == y + z).
Proof. exact: (@divg_eq G). Qed.
Lemma
subr_eq
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "divg_eq" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
subr_eq0 x y : (x - y == 0) = (x == y).
Proof. exact: (@divg_eq1 G). Qed.
Lemma
subr_eq0
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "divg_eq1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
addr_eq0 x y : (x + y == 0) = (x == - y).
Proof. exact: (@mulg_eq1 G). Qed.
Lemma
addr_eq0
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "mulg_eq1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
eqr_opp x y : (- x == - y) = (x == y).
Proof. exact: (@eqg_inv G). Qed.
Lemma
eqr_opp
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "eqg_inv" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
eqr_oppLR x y : (- x == y) = (x == - y).
Proof. exact: (@eqg_invLR G). Qed.
Lemma
eqr_oppLR
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "eqg_invLR" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulNrn x n : (- x) *+ n = x *- n.
Proof. exact: (@expVgn G). Qed.
Lemma
mulNrn
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "expVgn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulrnBl n : {morph (fun x => x *+ n) : x y / x - y}.
Proof. by move=> x y; exact/(@expgnFl G)/commuteT. Qed.
Lemma
mulrnBl
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "commuteT", "expgnFl" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulrnBr x m n : n <= m -> x *+ (m - n) = x *+ m - x *+ n.
Proof. exact: (@expgnFr G). Qed.
Lemma
mulrnBr
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "expgnFr" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sumrN I 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
sumrN
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "big_morph", "oppr0", "opprD" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sumrB I 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
sumrB
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "F1", "F2", "big_split", "sumrN" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
telescope_sumr n 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 move=> k /andP[nk km]/=; rewrite addrC subrKA. by case: ltngtP nm => // ->; rewrite subrr. Qed.
Lemma
telescope_sumr
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "addrC", "ltngtP", "nat", "subrKA", "subrr", "telescope_big" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
telescope_sumr_eq n 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
telescope_sumr_eq
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "eq_big_nat", "nat", "telescope_sumr" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
zmod_closed
:= 0 \in S /\ subr_closed S.
Definition
zmod_closed
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "subr_closed" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
zmod_closedN : zmod_closed -> oppr_closed S.
Proof. exact: (@group_closedV G). Qed.
Lemma
zmod_closedN
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "group_closedV", "oppr_closed", "zmod_closed" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
zmod_closedD : zmod_closed -> addr_closed S.
Proof. exact: (@group_closedM G). Qed.
Lemma
zmod_closedD
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "addr_closed", "group_closedM", "zmod_closed" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
zmod_closed0D : zmod_closed -> nmod_closed S.
Proof. by move=> z; split; [case: z|apply: zmod_closedD]. Qed.
Lemma
zmod_closed0D
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "apply", "nmod_closed", "split", "zmod_closed", "zmod_closedD" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
nmod_morphism (U V : baseAddUMagmaType) (f : U -> V) : Prop
:= (f 0 = 0) * {morph f : x y / x + y}.
Definition
nmod_morphism
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
semi_additive
:= nmod_morphism.
Definition
semi_additive
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "nmod_morphism" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Build U V apply
:= (isNmodMorphism.Build U V apply) (only parsing).
Notation
Build
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "apply" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
zmod_morphism (U V : zmodType) (f : U -> V)
:= {morph f : x y / x - y}.
Definition
zmod_morphism
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
additive
:= zmod_morphism.
Definition
additive
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "zmod_morphism" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Build U V apply
:= (isZmodMorphism.Build U V apply) (only parsing).
Notation
Build
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "apply" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
raddf0 : apply 0 = 0.
Proof. by rewrite -[0]subr0 zmod_morphism_subproof subrr. Qed.
Lemma
raddf0
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "apply", "subr0", "subrr" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
raddfD : {morph apply : x y / x + y}.
Proof. move=> x y; rewrite -[y in LHS]opprK -[- y]add0r. by rewrite !zmod_morphism_subproof raddf0 sub0r opprK. Qed.
Lemma
raddfD
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "add0r", "apply", "opprK", "raddf0", "sub0r" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'additive' U -> V }"
:= (Additive.type U%type V%type) : type_scope.
Notation
{ 'additive' U -> V }
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "type" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
raddf0 : f 0 = 0.
Proof. exact: nmod_morphism_subproof.1. Qed.
Lemma
raddf0
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
raddfD : {morph f : x y / x + y}.
Proof. exact: nmod_morphism_subproof.2. Qed.
Lemma
raddfD
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
to_fmultiplicative U V
:= @id (to_multiplicative U -> to_multiplicative V).
Definition
to_fmultiplicative
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "id", "to_multiplicative" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
add_fun (f g : U -> V) x
:= f x + g x.
Definition
add_fun
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
null_fun & U : V
:= 0.
Definition
null_fun
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
opp_fun (f : U -> V) x
:= - f x.
Definition
opp_fun
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_fun (f g : U -> V) x
:= f x - g x.
Definition
sub_fun
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
g
:= to_fmultiplicative f.
Let
g
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "to_fmultiplicative" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
raddf_eq0 x : injective f -> (f x == 0) = (x == 0).
Proof. exact: (@gmulf_eq1 _ _ g). Qed.
Lemma
raddf_eq0
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "gmulf_eq1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
raddfMn n : {morph f : x / x *+ n}.
Proof. exact: (@gmulfXn _ _ g). Qed.
Lemma
raddfMn
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "gmulfXn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
raddf_sum I 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
raddf_sum
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "gmulf_prod" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
can2_nmod_morphism f' : cancel f f' -> cancel f' f -> nmod_morphism f'.
Proof. split; first exact/(@can2_gmulf1 _ _ g). exact/(@can2_gmulfM _ _ g). Qed.
Lemma
can2_nmod_morphism
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "can2_gmulf1", "can2_gmulfM", "nmod_morphism", "split" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
can2_semi_additive
:= can2_nmod_morphism.
Definition
can2_semi_additive
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "can2_nmod_morphism" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
raddfN : {morph f : x / - x}.
Proof. exact: (@gmulfV _ _ g). Qed.
Lemma
raddfN
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "gmulfV" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
raddfB : {morph f : x y / x - y}.
Proof. exact: (@gmulfF _ _ g). Qed.
Lemma
raddfB
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "gmulfF" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
raddf_inj : (forall x, f x = 0 -> x = 0) -> injective f.
Proof. exact: (@gmulf_inj _ _ g). Qed.
Lemma
raddf_inj
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "gmulf_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
raddfMNn n : {morph f : x / x *- n}.
Proof. exact: (@gmulfXVn _ _ g). Qed.
Lemma
raddfMNn
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "gmulfXVn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
can2_zmod_morphism f' : 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.
Lemma
can2_zmod_morphism
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "apply", "fK", "raddfB", "zmod_morphism" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
can2_additive
:= can2_zmod_morphism.
Definition
can2_additive
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "can2_zmod_morphism" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
add_fun_nmod_morphism f g : nmod_morphism (f \+ g).
Proof. by split=> [|x y]; rewrite /= ?raddf0 ?addr0// !raddfD addrACA. Qed.
Fact
add_fun_nmod_morphism
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "addr0", "addrACA", "nmod_morphism", "raddf0", "raddfD", "split" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
idfun_is_nmod_morphism : nmod_morphism (@idfun U).
Proof. by []. Qed.
Fact
idfun_is_nmod_morphism
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "nmod_morphism" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_is_nmod_morphism : nmod_morphism (f \o g).
Proof. by split=> [|x y]; rewrite /= ?raddf0// !raddfD. Qed.
Fact
comp_is_nmod_morphism
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "nmod_morphism", "raddf0", "raddfD", "split" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
null_fun_is_nmod_morphism : nmod_morphism (\0 : U -> V).
Proof. by split=> // x y /=; rewrite addr0. Qed.
Fact
null_fun_is_nmod_morphism
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "addr0", "nmod_morphism", "split" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
opp_is_zmod_morphism : zmod_morphism (-%R : V -> V).
Proof. by move=> x y; rewrite /= opprD. Qed.
Fact
opp_is_zmod_morphism
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "opprD", "zmod_morphism" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
to_pmultiplicative (T : Type)
:= @id {pred to_multiplicative T}.
Definition
to_pmultiplicative
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "id", "to_multiplicative" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpred0 : 0 \in S.
Proof. by case: (@nmod_closed_subproof V S). Qed.
Lemma
rpred0
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpredD : {in S &, forall u v, u + v \in S}.
Proof. by case: (@nmod_closed_subproof V S). Qed.
Lemma
rpredD
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpred0D : addumagma_closed S.
Proof. exact: nmod_closed_subproof. Qed.
Lemma
rpred0D
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "addumagma_closed" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpredMn n : {in S, forall u, u *+ n \in S}.
Proof. exact: (@gpredXn _ (to_pmultiplicative S)). Qed.
Lemma
rpredMn
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "gpredXn", "to_pmultiplicative" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpred_sum I 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
rpred_sum
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "apply", "big_ind", "rpred0", "rpredD" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpredNr : {in S, forall u, - u \in S}.
Proof. exact: oppr_closed_subproof. Qed.
Lemma
rpredNr
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpredN : {mono -%R: u / u \in S}.
Proof. exact: (gpredV (to_pmultiplicative S)). Qed.
Lemma
rpredN
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "gpredV", "to_pmultiplicative" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
T
:= to_pmultiplicative S.
Let
T
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "to_pmultiplicative" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpredB : {in S &, forall u v, u - v \in S}.
Proof. exact: (@gpredF _ T). Qed.
Lemma
rpredB
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "gpredF" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpredBC u v : (u - v \in S) = (v - u \in S).
Proof. exact: (@gpredFC _ T). Qed.
Lemma
rpredBC
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "gpredFC" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpredMNn n: {in S, forall u, u *- n \in S}.
Proof. exact: (@gpredXNn _ T). Qed.
Lemma
rpredMNn
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "gpredXNn" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpredDr x y : x \in S -> (y + x \in S) = (y \in S).
Proof. exact: (@gpredMr _ T). Qed.
Lemma
rpredDr
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "gpredMr" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpredDl x y : x \in S -> (x + y \in S) = (y \in S).
Proof. exact: (@gpredMl _ T). Qed.
Lemma
rpredDl
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "gpredMl" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpredBr x y : x \in S -> (y - x \in S) = (y \in S).
Proof. exact: (@gpredFr _ T). Qed.
Lemma
rpredBr
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "gpredFr" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
rpredBl x y : x \in S -> (x - y \in S) = (y \in S).
Proof. exact: (@gpredFl _ T). Qed.
Lemma
rpredBl
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "gpredFl" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
zmodClosedP : zmod_closed S.
Proof. split; [ exact: rpred0 | exact: rpredB ]. Qed.
Lemma
zmodClosedP
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "rpred0", "rpredB", "split", "zmod_closed" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
val
:= (val : U -> V).
Notation
val
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
valD : {morph val : x y / x + y}.
Proof. exact: raddfD. Qed.
Lemma
valD
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "raddfD", "val" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
val0 : val 0 = 0.
Proof. exact: raddf0. Qed.
Lemma
val0
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "raddf0", "val" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
addU (u1 u2 : U)
:= inU (rpredD (valP u1) (valP u2)).
Let
addU
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "inU", "rpredD", "valP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
oneU
:= inU (fst addumagma_closed_subproof).
Let
oneU
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "inU" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
addrC : commutative addU.
Proof. by move=> x y; apply/val_inj; rewrite !SubK addrC. Qed.
Lemma
addrC
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "SubK", "addU", "apply", "val_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
add0r : left_id oneU addU.
Proof. by move=> x; apply/val_inj; rewrite !SubK add0r. Qed.
Lemma
add0r
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "SubK", "addU", "apply", "oneU", "val_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
valD0 : nmod_morphism (val : U -> V).
Proof. by split=> [|x y]; rewrite !SubK. Qed.
Lemma
valD0
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "SubK", "nmod_morphism", "split", "val" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
addrA : associative (@add U).
Proof. by move=> x y z; apply/val_inj; rewrite !SubK addrA. Qed.
Lemma
addrA
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "SubK", "add", "apply", "val_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
valB : {morph val : x y / x - y}.
Proof. exact: raddfB. Qed.
Lemma
valB
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "raddfB", "val" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
valN : {morph val : x / - x}.
Proof. exact: raddfN. Qed.
Lemma
valN
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "raddfN", "val" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
valD0 : nmod_morphism (val : U -> V).
Proof. have val0: (val : U -> V) 0 = 0. by rewrite -[X in val X](subr0 0) valB_subproof subrr. split=> // x y; apply/(@subIr _ (val y)). by rewrite -valB_subproof -!addrA !subrr !addr0. Qed.
Fact
valD0
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "addr0", "addrA", "apply", "nmod_morphism", "split", "subIr", "subr0", "subrr", "val", "val0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
oppU (u : U)
:= inU (rpredNr (valP u)).
Let
oppU
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "inU", "rpredNr", "valP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
addNr : left_inverse 0 oppU (@add U).
Proof. by move=> x; apply/val_inj; rewrite raddf0 raddfD/= SubK addNr. Qed.
Lemma
addNr
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "SubK", "add", "apply", "oppU", "raddf0", "raddfD", "val_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubChoice_isSubNmodule' 'of' U 'by' <: ]"
:= (SubChoice_isSubNmodule.Build _ _ U rpred0D) (at level 0, format "[ 'SubChoice_isSubNmodule' 'of' U 'by' <: ]") : form_scope.
Notation
[ 'SubChoice_isSubNmodule' 'of' U 'by' <: ]
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "Build", "rpred0D" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'SubNmodule_isSubZmodule' 'of' U 'by' <: ]"
:= (SubNmodule_isSubZmodule.Build _ _ U (@rpredNr _ _)) (at level 0, format "[ 'SubNmodule_isSubZmodule' 'of' U 'by' <: ]") : form_scope.
Notation
[ 'SubNmodule_isSubZmodule' 'of' U 'by' <: ]
boot
boot/nmodule.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "bigop", "fintype", "finfun", "monoid", "div", "AllExports" ]
[ "Build", "rpredNr" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d