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algid_subproof U : {e | e \in U & has_algid U ==> (U <= lker (amull e - 1) :&: lker (amulr e - 1))%VS}.
Proof. apply: sig2W; case: has_algidP => [[e]|]; last by exists 0; rewrite ?mem0v. case=> Ae _ idAe; exists e => //; apply/subvP=> u /idAe[eu_u ue_u]. by rewrite memv_cap !memv_ker !lfun_simp /= eu_u ue_u subrr eqxx. Qed.
Lemma
algid_subproof
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "amull", "amulr", "apply", "eqxx", "has_algid", "has_algidP", "last", "lfun_simp", "lker", "mem0v", "memv_cap", "memv_ker", "sig2W", "subrr", "subvP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
algid U
:= s2val (algid_subproof U).
Definition
algid
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "algid_subproof" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
memv_algid U : algid U \in U.
Proof. by rewrite /algid; case: algid_subproof. Qed.
Lemma
memv_algid
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "algid", "algid_subproof" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
algidl A : {in A, left_id (algid A) *%R}.
Proof. rewrite /algid; case: algid_subproof => e _ /=; have /andP[-> _] := valP A. move/subvP=> idAe u /idAe/memv_capP[]. by rewrite memv_ker !lfun_simp /= subr_eq0 => /eqP. Qed.
Lemma
algidl
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "algid", "algid_subproof", "lfun_simp", "memv_capP", "memv_ker", "subr_eq0", "subvP", "valP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
algidr A : {in A, right_id (algid A) *%R}.
Proof. rewrite /algid; case: algid_subproof => e _ /=; have /andP[-> _] := valP A. move/subvP=> idAe u /idAe/memv_capP[_]. by rewrite memv_ker !lfun_simp /= subr_eq0 => /eqP. Qed.
Lemma
algidr
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "algid", "algid_subproof", "lfun_simp", "memv_capP", "memv_ker", "subr_eq0", "subvP", "valP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
unitr_algid1 A u : u \in A -> u \is a GRing.unit -> algid A = 1.
Proof. by move=> Eu /mulrI; apply; rewrite mulr1 algidr. Qed.
Lemma
unitr_algid1
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "algid", "algidr", "apply", "mulr1", "mulrI", "unit" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
algid_eq1 A : (algid A == 1) = (1 \in A).
Proof. by apply/eqP/idP=> [<- | /algidr <-]; rewrite ?memv_algid ?mul1r. Qed.
Lemma
algid_eq1
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "algid", "algidr", "apply", "memv_algid", "mul1r" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
algid_neq0 A : algid A != 0.
Proof. have /andP[/has_algidP[u [Au nz_u _]] _] := valP A. by apply: contraNneq nz_u => e0; rewrite -(algidr Au) e0 mulr0. Qed.
Lemma
algid_neq0
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "algid", "algidr", "apply", "contraNneq", "e0", "has_algidP", "mulr0", "nz_u", "valP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
dim_algid A : \dim <[algid A]> = 1%N.
Proof. by rewrite dim_vline algid_neq0. Qed.
Lemma
dim_algid
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "algid", "algid_neq0", "dim", "dim_vline" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
adim_gt0 A : (0 < \dim A)%N.
Proof. by rewrite -(dim_algid A) dimvS // -memvE ?memv_algid. Qed.
Lemma
adim_gt0
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "dim", "dim_algid", "dimvS", "memvE", "memv_algid" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
not_asubv0 A : ~~ (A <= 0)%VS.
Proof. by rewrite subv0 -dimv_eq0 -lt0n adim_gt0. Qed.
Lemma
not_asubv0
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "adim_gt0", "dimv_eq0", "lt0n", "subv0" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
adim1P {A} : reflect (A = <[algid A]>%VS :> {vspace aT}) (\dim A == 1%N).
Proof. rewrite eqn_leq adim_gt0 -(memv_algid A) andbC -(dim_algid A) -eqEdim eq_sym. exact: eqP. Qed.
Lemma
adim1P
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "aT", "adim_gt0", "algid", "dim", "dim_algid", "eqEdim", "eq_sym", "eqn_leq", "memv_algid" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
asubv A : (A * A <= A)%VS.
Proof. by have /andP[] := valP A. Qed.
Lemma
asubv
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "valP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
memvM A : {in A &, forall u v, u * v \in A}.
Proof. exact/prodvP/asubv. Qed.
Lemma
memvM
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "asubv", "prodvP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodv_id A : (A * A)%VS = A.
Proof. apply/eqP; rewrite eqEsubv asubv; apply/subvP=> u Au. by rewrite -(algidl Au) memv_mul // memv_algid. Qed.
Lemma
prodv_id
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "algidl", "apply", "asubv", "eqEsubv", "memv_algid", "memv_mul", "subvP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodv_sub U V A : (U <= A -> V <= A -> U * V <= A)%VS.
Proof. by move=> sUA sVA; rewrite -prodv_id prodvS. Qed.
Lemma
prodv_sub
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "prodvS", "prodv_id" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
expv_id A n : (A ^+ n.+1)%VS = A.
Proof. by elim: n => // n IHn; rewrite !expvSl prodvA prodv_id -expvSl. Qed.
Lemma
expv_id
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "expvSl", "prodvA", "prodv_id" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
limg_amulr U v : (amulr v @: U = U * <[v]>)%VS.
Proof. rewrite -(span_basis (vbasisP U)) limg_span !span_def big_distrl /= big_map. by apply: eq_bigr => u; rewrite prodv_line lfunE. Qed.
Lemma
limg_amulr
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "amulr", "apply", "big_distrl", "big_map", "eq_bigr", "lfunE", "limg_span", "prodv_line", "span_basis", "span_def", "vbasisP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
memv_cosetP {U v w} : reflect (exists2 u, u\in U & w = u * v) (w \in U * <[v]>)%VS.
Proof. rewrite -limg_amulr. by apply: (iffP memv_imgP) => [] [u] Uu ->; exists u; rewrite ?lfunE. Qed.
Lemma
memv_cosetP
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "Uu", "apply", "lfunE", "limg_amulr", "memv_imgP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
dim_cosetv_unit V u : u \is a GRing.unit -> \dim (V * <[u]>) = \dim V.
Proof. by move/lker0_amulr/eqP=> Uu; rewrite -limg_amulr limg_dim_eq // Uu capv0. Qed.
Lemma
dim_cosetv_unit
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "Uu", "capv0", "dim", "limg_amulr", "limg_dim_eq", "lker0_amulr", "unit" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
memvV A u : (u^-1 \in A) = (u \in A).
Proof. suffices{u} invA: invr_closed A by apply/idP/idP=> /invA; rewrite ?invrK. move=> u Au; have [Uu | /invr_out-> //] := boolP (u \is a GRing.unit). rewrite memvE -(limg_ker0 _ _ (lker0_amulr Uu)) limg_line lfunE /= mulVr //. suff ->: (amulr u @: A)%VS = A by rewrite -memvE -algid_eq1 (unitr_algid1 Au). by apply/eqP...
Lemma
memvV
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "Uu", "algid_eq1", "amulr", "apply", "dim_cosetv_unit", "dimv_leqif_eq", "invrK", "invr_closed", "invr_out", "lfunE", "limg_amulr", "limg_ker0", "limg_line", "lker0_amulr", "memvE", "mulVr", "prodv_sub", "unit", "unitr_algid1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
aspace_cap_subproof A B : algid A \in B -> is_aspace (A :&: B).
Proof. move=> BeA; apply/andP. split; [apply/has_algidP | by rewrite subv_cap !prodv_sub ?capvSl ?capvSr]. exists (algid A); rewrite /is_algid algid_neq0 memv_cap memv_algid. by split=> // u /memv_capP[Au _]; rewrite ?algidl ?algidr. Qed.
Fact
aspace_cap_subproof
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "algid", "algid_neq0", "algidl", "algidr", "apply", "capvSl", "capvSr", "has_algidP", "is_algid", "is_aspace", "memv_algid", "memv_cap", "memv_capP", "prodv_sub", "split", "subv_cap" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
aspace_cap A B BeA
:= ASpace (@aspace_cap_subproof A B BeA).
Definition
aspace_cap
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "aspace_cap_subproof" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
centraliser1_is_aspace u : is_aspace 'C[u].
Proof. rewrite /is_aspace has_algid1 ?cent1v1 //=. apply/prodvP=> v w /cent1vP-cuv /cent1vP-cuw. by apply/cent1vP; rewrite -mulrA cuw !mulrA cuv. Qed.
Fact
centraliser1_is_aspace
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "apply", "cent1v1", "cent1vP", "has_algid1", "is_aspace", "mulrA", "prodvP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
centraliser1_aspace u
:= ASpace (centraliser1_is_aspace u).
Canonical
centraliser1_aspace
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "centraliser1_is_aspace" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
centraliser_is_aspace V : is_aspace 'C(V).
Proof. rewrite /is_aspace has_algid1 ?centv1 //=. apply/prodvP=> u w /centvP-cVu /centvP-cVw. by apply/centvP=> v Vv; rewrite /= -mulrA cVw // !mulrA cVu. Qed.
Fact
centraliser_is_aspace
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "apply", "centv1", "centvP", "has_algid1", "is_aspace", "mulrA", "prodvP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
centraliser_aspace V
:= ASpace (centraliser_is_aspace V).
Canonical
centraliser_aspace
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "centraliser_is_aspace" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
centv_algid A : algid A \in 'C(A)%VS.
Proof. by apply/centvP=> u Au; rewrite algidl ?algidr. Qed.
Lemma
centv_algid
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "algid", "algidl", "algidr", "apply", "centvP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
center_aspace A
:= [aspace of 'Z(A) for aspace_cap (centv_algid A)].
Canonical
center_aspace
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "aspace", "aspace_cap", "centv_algid" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
algid_center A : algid 'Z(A) = algid A.
Proof. rewrite -(algidl (subvP (centerv_sub A) _ (memv_algid _))) algidr //=. by rewrite memv_cap memv_algid centv_algid. Qed.
Lemma
algid_center
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "algid", "algidl", "algidr", "centerv_sub", "centv_algid", "memv_algid", "memv_cap", "subvP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
Falgebra_FieldMixin : GRing.integral_domain_axiom aT -> GRing.field_axiom aT.
Proof. move=> domT u nz_u; apply/unitrP. have kerMu: lker (amulr u) == 0%VS. rewrite eqEsubv sub0v andbT; apply/subvP=> v; rewrite memv_ker lfunE /=. by move/eqP/domT; rewrite (negPf nz_u) orbF memv0. have /memv_imgP[v _ vu1]: 1 \in limg (amulr u); last rewrite lfunE /= in vu1. suffices /eqP->: limg (amulr u) == ...
Lemma
Falgebra_FieldMixin
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "aT", "amulr", "apply", "capv0", "dimv_leqif_eq", "eqEsubv", "field_axiom", "fullv", "integral_domain_axiom", "last", "lfunE", "limg", "limg_dim_eq", "lker", "lker0P", "memv0", "memv_imgP", "memv_ker", "memvf", "mul1r", "mulr1", "mulrA", "nz_u", "split", "sub0v", "s...
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
fieldT : GRing.field_axiom aT.
Hypothesis
fieldT
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "aT", "field_axiom" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
skew_field_algid1 A : algid A = 1.
Proof. by rewrite (unitr_algid1 (memv_algid A)) ?fieldT ?algid_neq0. Qed.
Lemma
skew_field_algid1
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "algid", "algid_neq0", "fieldT", "memv_algid", "unitr_algid1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
skew_field_module_semisimple A M : let sumA X := (\sum_(x <- X) A * <[x]>)%VS in (A * M <= M)%VS -> {X | [/\ sumA X = M, directv (sumA X) & 0 \notin X]}.
Proof. move=> sumA sAM_M; pose X := Nil aT; pose k := (\dim (A * M) - \dim (sumA X))%N. have: (\dim (A * M) - \dim (sumA X) < k.+1)%N by []. have: [/\ (sumA X <= A * M)%VS, directv (sumA X) & 0 \notin X]. by rewrite /sumA directvE /= !big_nil sub0v dimv0. elim: {X k}k.+1 (X) => // k IHk X [sAX_AM dxAX nzX]; rewrite l...
Lemma
skew_field_module_semisimple
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "My", "Nil", "aT", "add1n", "adim_gt0", "apply", "big_cons", "big_distrr", "big_nil", "dim", "dim_cosetv_unit", "dimv0", "dimvS", "directv", "directvE", "directvP", "directv_addE", "eqEsubv", "eq_sym", "eqxx", "fieldT", "inE", "leq_add2r", "leq_sub2r", "leq_subLR", ...
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
skew_field_module_dimS A M : (A * M <= M)%VS -> \dim A %| \dim M.
Proof. case/skew_field_module_semisimple=> X [<- /directvP-> nzX] /=. rewrite big_seq prime.dvdn_sum // => x /(memPn nzX)nz_x. by rewrite dim_cosetv_unit ?fieldT. Qed.
Lemma
skew_field_module_dimS
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "big_seq", "dim", "dim_cosetv_unit", "directvP", "dvdn_sum", "fieldT", "memPn", "prime", "skew_field_module_semisimple" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
skew_field_dimS A B : (A <= B)%VS -> \dim A %| \dim B.
Proof. by move=> sAB; rewrite skew_field_module_dimS ?prodv_sub. Qed.
Lemma
skew_field_dimS
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "dim", "prodv_sub", "skew_field_module_dimS" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'C [ u ]"
:= (centraliser1_aspace u) : aspace_scope.
Notation
'C [ u ]
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "centraliser1_aspace" ]
Note that local centraliser might not be proper sub-algebras.
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'C ( V )"
:= (centraliser_aspace V) : aspace_scope.
Notation
'C ( V )
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "centraliser_aspace" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"'Z ( A )"
:= (center_aspace A) : aspace_scope.
Notation
'Z ( A )
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "center_aspace" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
agenv U
:= (\sum_(i < \dim {:aT}) U ^+ i)%VS.
Definition
agenv
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "aT", "dim" ]
Subspaces of an F-algebra form a Kleene algebra
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<< U & vs >>"
:= (agenv (U + <<vs>>)) : vspace_scope.
Notation
<< U & vs >>
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "agenv" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<< U ; x >>"
:= (agenv (U + <[x]>)) : vspace_scope.
Notation
<< U ; x >>
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "agenv" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
agenvEl U : agenv U = (1 + U * agenv U)%VS.
Proof. pose f V := (1 + U * V)%VS; rewrite -/(f _); pose n := \dim {:aT}. have ->: agenv U = iter n f 0%VS. rewrite /agenv -/n; elim: n => [|n IHn]; first by rewrite big_ord0. rewrite big_ord_recl /= -{}IHn; congr (1 + _)%VS; rewrite big_distrr /=. by apply: eq_bigr => i; rewrite expvSl. have fS i j: i <= j -> (i...
Lemma
agenvEl
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "aT", "addvS", "agenv", "apply", "big_distrr", "big_ord0", "big_ord_recl", "dim", "dimvS", "eqEdim", "eqEsubv", "eq_bigr", "eq_sym", "expvSl", "inE", "iter", "iterS", "leqW", "leq_ltn_trans", "looping", "looping_uniq", "ltnNge", "mem_rcons", "prodvSr", "rcons_uniq", ...
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
agenvEr U : agenv U = (1 + agenv U * U)%VS.
Proof. rewrite [lhs in lhs = _]agenvEl big_distrr big_distrl /=; congr (_ + _)%VS. by apply: eq_bigr => i _ /=; rewrite -expvSr -expvSl. Qed.
Lemma
agenvEr
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "agenv", "agenvEl", "apply", "big_distrl", "big_distrr", "eq_bigr", "expvSl", "expvSr" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
agenv_modl U V : (U * V <= V -> agenv U * V <= V)%VS.
Proof. rewrite big_distrl /= => idlU_V; apply/subv_sumP=> [[i _] /= _]. elim: i => [|i]; first by rewrite expv0 prod1v. by apply: subv_trans; rewrite expvSr -prodvA prodvSr. Qed.
Lemma
agenv_modl
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "agenv", "apply", "big_distrl", "expv0", "expvSr", "prod1v", "prodvA", "prodvSr", "subv_sumP", "subv_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
agenv_modr U V : (V * U <= V -> V * agenv U <= V)%VS.
Proof. rewrite big_distrr /= => idrU_V; apply/subv_sumP=> [[i _] /= _]. elim: i => [|i]; first by rewrite expv0 prodv1. by apply: subv_trans; rewrite expvSl prodvA prodvSl. Qed.
Lemma
agenv_modr
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "agenv", "apply", "big_distrr", "expv0", "expvSl", "prodv1", "prodvA", "prodvSl", "subv_sumP", "subv_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
agenv_is_aspace U : is_aspace (agenv U).
Proof. rewrite /is_aspace has_algid1; first by rewrite memvE agenvEl addvSl. by rewrite agenv_modl // [V in (_ <= V)%VS]agenvEl addvSr. Qed.
Fact
agenv_is_aspace
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "addvSl", "addvSr", "agenv", "agenvEl", "agenv_modl", "has_algid1", "is_aspace", "memvE" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
agenv_aspace U : {aspace aT}
:= ASpace (agenv_is_aspace U).
Canonical
agenv_aspace
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "aT", "agenv_is_aspace", "aspace" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
agenvE U : agenv U = agenv_aspace U.
Proof. by []. Qed.
Lemma
agenvE
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "agenv", "agenv_aspace" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
agenvM U : (agenv U * agenv U)%VS = agenv U.
Proof. exact: prodv_id. Qed.
Lemma
agenvM
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "agenv", "prodv_id" ]
Kleene algebra properties
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
agenvX n U : (agenv U ^+ n.+1)%VS = agenv U.
Proof. exact: expv_id. Qed.
Lemma
agenvX
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "agenv", "expv_id" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub1_agenv U : (1 <= agenv U)%VS.
Proof. by rewrite agenvEl addvSl. Qed.
Lemma
sub1_agenv
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "addvSl", "agenv", "agenvEl" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_agenv U : (U <= agenv U)%VS.
Proof. by rewrite 2!agenvEl addvC prodvDr prodv1 -addvA addvSl. Qed.
Lemma
sub_agenv
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "addvA", "addvC", "addvSl", "agenv", "agenvEl", "prodv1", "prodvDr" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
subX_agenv U n : (U ^+ n <= agenv U)%VS.
Proof. by case: n => [|n]; rewrite ?sub1_agenv // -(agenvX n) expvS // sub_agenv. Qed.
Lemma
subX_agenv
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "agenv", "agenvX", "expvS", "sub1_agenv", "sub_agenv" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
agenv_sub_modl U V : (1 <= V -> U * V <= V -> agenv U <= V)%VS.
Proof. move=> s1V /agenv_modl; apply: subv_trans. by rewrite -[Us in (Us <= _)%VS]prodv1 prodvSr. Qed.
Lemma
agenv_sub_modl
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "agenv", "agenv_modl", "apply", "prodv1", "prodvSr", "subv_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
agenv_sub_modr U V : (1 <= V -> V * U <= V -> agenv U <= V)%VS.
Proof. move=> s1V /agenv_modr; apply: subv_trans. by rewrite -[Us in (Us <= _)%VS]prod1v prodvSl. Qed.
Lemma
agenv_sub_modr
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "agenv", "agenv_modr", "apply", "prod1v", "prodvSl", "subv_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
agenv_id U : agenv (agenv U) = agenv U.
Proof. apply/eqP; rewrite eqEsubv sub_agenv andbT. by rewrite agenv_sub_modl ?sub1_agenv ?agenvM. Qed.
Lemma
agenv_id
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "agenv", "agenvM", "agenv_sub_modl", "apply", "eqEsubv", "sub1_agenv", "sub_agenv" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
agenvS U V : (U <= V -> agenv U <= agenv V)%VS.
Proof. move=> sUV; rewrite agenv_sub_modl ?sub1_agenv //. by rewrite -[Vs in (_ <= Vs)%VS]agenvM prodvSl ?(subv_trans sUV) ?sub_agenv. Qed.
Lemma
agenvS
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "agenv", "agenvM", "agenv_sub_modl", "prodvSl", "sub1_agenv", "sub_agenv", "subv_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
agenv_add_id U V : agenv (agenv U + V) = agenv (U + V).
Proof. apply/eqP; rewrite eqEsubv andbC agenvS ?addvS ?sub_agenv //=. rewrite agenv_sub_modl ?sub1_agenv //. rewrite -[rhs in (_ <= rhs)%VS]agenvM prodvSl // subv_add agenvS ?addvSl //=. exact: subv_trans (addvSr U V) (sub_agenv _). Qed.
Lemma
agenv_add_id
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "addvS", "addvSl", "addvSr", "agenv", "agenvM", "agenvS", "agenv_sub_modl", "apply", "eqEsubv", "prodvSl", "rhs", "sub1_agenv", "sub_agenv", "subv_add", "subv_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
subv_adjoin U x : (U <= <<U; x>>)%VS.
Proof. by rewrite (subv_trans (sub_agenv _)) ?agenvS ?addvSl. Qed.
Lemma
subv_adjoin
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "addvSl", "agenvS", "sub_agenv", "subv_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
subv_adjoin_seq U xs : (U <= <<U & xs>>)%VS.
Proof. by rewrite (subv_trans (sub_agenv _)) // ?agenvS ?addvSl. Qed.
Lemma
subv_adjoin_seq
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "addvSl", "agenvS", "sub_agenv", "subv_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
memv_adjoin U x : x \in <<U; x>>%VS.
Proof. by rewrite memvE (subv_trans (sub_agenv _)) ?agenvS ?addvSr. Qed.
Lemma
memv_adjoin
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "addvSr", "agenvS", "memvE", "sub_agenv", "subv_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
seqv_sub_adjoin U xs : {subset xs <= <<U & xs>>%VS}.
Proof. by apply/span_subvP; rewrite (subv_trans (sub_agenv _)) ?agenvS ?addvSr. Qed.
Lemma
seqv_sub_adjoin
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "addvSr", "agenvS", "apply", "span_subvP", "sub_agenv", "subv_trans" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
subvP_adjoin U x y : y \in U -> y \in <<U; x>>%VS.
Proof. exact/subvP/subv_adjoin. Qed.
Lemma
subvP_adjoin
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "subvP", "subv_adjoin" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
adjoin_nil V : <<V & [::]>>%VS = agenv V.
Proof. by rewrite span_nil addv0. Qed.
Lemma
adjoin_nil
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "addv0", "agenv", "span_nil" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
adjoin_cons V x rs : <<V & x :: rs>>%VS = << <<V; x>> & rs>>%VS.
Proof. by rewrite span_cons addvA agenv_add_id. Qed.
Lemma
adjoin_cons
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "addvA", "agenv_add_id", "span_cons" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
adjoin_rcons V rs x : <<V & rcons rs x>>%VS = << <<V & rs>>%VS; x>>%VS.
Proof. by rewrite -cats1 span_cat addvA span_seq1 agenv_add_id. Qed.
Lemma
adjoin_rcons
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "addvA", "agenv_add_id", "cats1", "rcons", "span_cat", "span_seq1" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
adjoin_seq1 V x : <<V & [:: x]>>%VS = <<V; x>>%VS.
Proof. by rewrite adjoin_cons adjoin_nil agenv_id. Qed.
Lemma
adjoin_seq1
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "adjoin_cons", "adjoin_nil", "agenv_id" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
adjoinC V x y : << <<V; x>>; y>>%VS = << <<V; y>>; x>>%VS.
Proof. by rewrite !agenv_add_id -!addvA (addvC <[x]>%VS). Qed.
Lemma
adjoinC
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "addvA", "addvC", "agenv_add_id" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
adjoinSl U V x : (U <= V -> <<U; x>> <= <<V; x>>)%VS.
Proof. by move=> sUV; rewrite agenvS ?addvS. Qed.
Lemma
adjoinSl
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "addvS", "agenvS" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
adjoin_seqSl U V rs : (U <= V -> <<U & rs>> <= <<V & rs>>)%VS.
Proof. by move=> sUV; rewrite agenvS ?addvS. Qed.
Lemma
adjoin_seqSl
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "addvS", "agenvS" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
adjoin_seqSr U rs1 rs2 : {subset rs1 <= rs2} -> (<<U & rs1>> <= <<U & rs2>>)%VS.
Proof. by move/sub_span=> s_rs12; rewrite agenvS ?addvS. Qed.
Lemma
adjoin_seqSr
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "addvS", "agenvS", "sub_span" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<< U >>"
:= (agenv_aspace U) : aspace_scope.
Notation
<< U >>
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "agenv_aspace" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<< U & vs >>"
:= << U + <<vs>> >>%AS : aspace_scope.
Notation
<< U & vs >>
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
"<< U ; x >>"
:= << U + <[x]> >>%AS : aspace_scope.
Notation
<< U ; x >>
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
subvs_one
:= Subvs (memv_algid A).
Definition
subvs_one
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "memv_algid" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
subvs_mul (u v : subvs_of A)
:= Subvs (subv_trans (memv_mul (subvsP u) (subvsP v)) (asubv _)).
Definition
subvs_mul
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "asubv", "memv_mul", "subv_trans", "subvsP", "subvs_of" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
subvs_mulA : associative subvs_mul.
Proof. by move=> x y z; apply/val_inj/mulrA. Qed.
Fact
subvs_mulA
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "apply", "mulrA", "subvs_mul", "val_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
subvs_mu1l : left_id subvs_one subvs_mul.
Proof. by move=> x; apply/val_inj/algidl/(valP x). Qed.
Fact
subvs_mu1l
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "algidl", "apply", "subvs_mul", "subvs_one", "valP", "val_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
subvs_mul1 : right_id subvs_one subvs_mul.
Proof. by move=> x; apply/val_inj/algidr/(valP x). Qed.
Fact
subvs_mul1
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "algidr", "apply", "subvs_mul", "subvs_one", "valP", "val_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
subvs_mulDl : left_distributive subvs_mul +%R.
Proof. by move=> x y z; apply/val_inj/mulrDl. Qed.
Fact
subvs_mulDl
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "apply", "mulrDl", "subvs_mul", "val_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
subvs_mulDr : right_distributive subvs_mul +%R.
Proof. by move=> x y z; apply/val_inj/mulrDr. Qed.
Fact
subvs_mulDr
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "apply", "mulrDr", "subvs_mul", "val_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
subvs_scaleAl k (x y : subvs_of A) : k *: (x * y) = (k *: x) * y.
Proof. exact/val_inj/scalerAl. Qed.
Lemma
subvs_scaleAl
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "scalerAl", "subvs_of", "val_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
subvs_scaleAr k (x y : subvs_of A) : k *: (x * y) = x * (k *: y).
Proof. exact/val_inj/scalerAr. Qed.
Lemma
subvs_scaleAr
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "scalerAr", "subvs_of", "val_inj" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
vsval_unitr w : vsval w \is a GRing.unit -> w \is a GRing.unit.
Proof. case: w => /= u Au Uu; have Au1: u^-1 \in A by rewrite memvV. apply/unitrP; exists (Subvs Au1). by split; apply: val_inj; rewrite /= ?mulrV ?mulVr ?(unitr_algid1 Au). Qed.
Lemma
vsval_unitr
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "Uu", "apply", "memvV", "mulVr", "mulrV", "split", "unit", "unitrP", "unitr_algid1", "val_inj", "vsval" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
vsval_invr w : vsval w \is a GRing.unit -> val w^-1 = (val w)^-1.
Proof. move=> Uu; have def_w: w / w * w = w by rewrite divrK ?vsval_unitr. by apply: (mulrI Uu); rewrite -[in u in u / _]def_w ?mulrK. Qed.
Lemma
vsval_invr
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "Uu", "apply", "divrK", "mulrI", "mulrK", "unit", "val", "vsval", "vsval_unitr" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ahom_in (U : {vspace aT}) (f : 'Hom(aT, rT))
:= all2rel (fun x y : aT => f (x * y) == f x * f y) (vbasis U) && (f 1 == 1).
Definition
ahom_in
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "aT", "all2rel", "vbasis" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ahom_inP {f : 'Hom(aT, rT)} {U : {vspace aT}} : reflect ({in U &, {morph f : x y / x * y >-> x * y}} * (f 1 = 1)) (ahom_in U f).
Proof. apply: (iffP andP) => [[/allrelP fM /eqP f1] | [fM f1]]; last first. rewrite f1; split=> //; apply/allrelP => x y Ax Ay. by rewrite fM // vbasis_mem. split=> // x y /coord_vbasis -> /coord_vbasis ->. rewrite !mulr_suml ![f _]linear_sum mulr_suml; apply: eq_bigr => i _ /=. rewrite !mulr_sumr linear_sum; apply...
Lemma
ahom_inP
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "aT", "ahom_in", "allrelP", "apply", "coord_vbasis", "eq_bigr", "f1", "fM", "last", "linearZ", "linear_sum", "memt_nth", "mulr_suml", "mulr_sumr", "scalerAl", "scalerAr", "split", "vbasis_mem" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ahomP_tmp {f : 'Hom(aT, rT)} : reflect (monoid_morphism f) (ahom_in {:aT} f).
Proof. apply: (iffP ahom_inP) => [[fM f1] | fRM_P]; last first. by split=> [x y|]; [rewrite fRM_P.2|rewrite fRM_P.1]. by split=> // x y; rewrite fM ?memvf. Qed.
Lemma
ahomP_tmp
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "aT", "ahom_in", "ahom_inP", "apply", "f1", "fM", "last", "memvf", "monoid_morphism", "split" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ahomP {f : 'Hom(aT, rT)} : reflect (multiplicative f) (ahom_in {:aT} f).
Proof. by apply: (iffP ahomP_tmp) => [][]. Qed.
Lemma
ahomP
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "aT", "ahomP_tmp", "ahom_in", "apply", "multiplicative" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ahom
:= AHom {ahval :> 'Hom(aT, rT); _ : ahom_in {:aT} ahval}.
Structure
ahom
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "aT", "ahom_in" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
linfun_is_ahom (f : {lrmorphism aT -> rT}) : ahom_in {:aT} (linfun f).
Proof. by apply/ahom_inP; split=> [x y|]; rewrite !lfunE ?rmorphM ?rmorph1. Qed.
Fact
linfun_is_ahom
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "aT", "ahom_in", "ahom_inP", "apply", "lfunE", "linfun", "rmorph1", "rmorphM", "split" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
linfun_ahom f
:= AHom (linfun_is_ahom f).
Canonical
linfun_ahom
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "linfun_is_ahom" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ahom_is_monoid_morphism (f : ahom aT rT) : monoid_morphism f.
Proof. by apply/ahomP_tmp; case: f. Qed.
Fact
ahom_is_monoid_morphism
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "aT", "ahom", "ahomP_tmp", "apply", "monoid_morphism" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ahom_is_multiplicative (f : ahom aT rT) : multiplicative f
:= (fun p => (p.2, p.1)) (ahom_is_monoid_morphism f).
Definition
ahom_is_multiplicative
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "aT", "ahom", "ahom_is_monoid_morphism", "multiplicative" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
ahomWin (f : ahom aT rT) U : ahom_in U f.
Proof. by apply/ahom_inP; split; [apply: in2W (rmorphM _) | apply: rmorph1]. Qed.
Lemma
ahomWin
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "aT", "ahom", "ahom_in", "ahom_inP", "apply", "rmorph1", "rmorphM", "split" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
id_is_ahom (V : {vspace aT}) : ahom_in V \1.
Proof. by apply/ahom_inP; split=> [x y|] /=; rewrite !id_lfunE. Qed.
Lemma
id_is_ahom
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "aT", "ahom_in", "ahom_inP", "apply", "id_lfunE", "split" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
id_ahom
:= AHom (id_is_ahom (aspacef aT)).
Canonical
id_ahom
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "aT", "aspacef", "id_is_ahom" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_is_ahom (V : {vspace aT}) (f : 'Hom(rT, sT)) (g : 'Hom(aT, rT)) : ahom_in {:rT} f -> ahom_in V g -> ahom_in V (f \o g).
Proof. move=> /ahom_inP fM /ahom_inP gM; apply/ahom_inP. by split=> [x y Vx Vy|] /=; rewrite !comp_lfunE gM // fM ?memvf. Qed.
Lemma
comp_is_ahom
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "aT", "ahom_in", "ahom_inP", "apply", "comp_lfunE", "fM", "memvf", "sT", "split" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d
comp_ahom (f : ahom rT sT) (g : ahom aT rT)
:= AHom (comp_is_ahom (valP f) (valP g)).
Canonical
comp_ahom
field
field/falgebra.v
[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "path", "choice", "fintype", "div", "tuple", "finfun", "bigop", "ssralg", "finalg", "zmodp", "matrix", "vector", "poly", "GRing.Theory", "VectorInternalTheory", "FalgLfun" ]
[ "aT", "ahom", "comp_is_ahom", "sT", "valP" ]
https://github.com/math-comp/math-comp
91d97df9cf3204b4dab84f4e24bc633e84b6473d