phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	docstring stringclasses 798 values	source_url stringclasses 1 value	commit stringclasses 1 value
memv_projC U w : w - projv U w \in (U^C)%VS.	Proof. rewrite -{1}[w](daddv_pi_add (capv_compl U)) ?addv_complf ?memvf //. by rewrite addrC addKr memv_pi. Qed.	Lemma	memv_projC	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addKr", "addrC", "addv_complf", "capv_compl", "daddv_pi_add", "memv_pi", "memvf", "projv" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
limg_proj U : limg (projv U) = U.	Proof. apply/vspaceP=> u; apply/memv_imgP/idP=> [[u1 _ ->] \| ]; first exact: memv_proj. by exists (projv U u); rewrite ?projv_id ?memv_img ?memvf. Qed.	Lemma	limg_proj	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "limg", "memv_img", "memv_imgP", "memv_proj", "memvf", "projv", "projv_id", "vspaceP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lker_proj U : lker (projv U) = (U^C)%VS.	Proof. apply/eqP; rewrite eqEdim andbC; apply/andP; split. by rewrite dimv_compl -(limg_ker_dim (projv U) fullv) limg_proj addnK capfv. by apply/subvP=> v; rewrite memv_ker -{2}[v]subr0 => /eqP <-; apply: memv_projC. Qed.	Lemma	lker_proj	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addnK", "apply", "capfv", "dimv_compl", "eqEdim", "fullv", "limg_ker_dim", "limg_proj", "lker", "memv_ker", "memv_projC", "projv", "split", "subr0", "subvP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addv_pi1_proj U V w (pi1 := addv_pi1 U V) : pi1 (pi1 w) = pi1 w.	Proof. by rewrite daddv_pi_proj // capv_diff. Qed.	Lemma	addv_pi1_proj	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addv_pi1", "capv_diff", "daddv_pi_proj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addv_pi2_id U V v : v \in V -> addv_pi2 U V v = v.	Proof. by apply: daddv_pi_id; rewrite capvC capv_diff. Qed.	Lemma	addv_pi2_id	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addv_pi2", "apply", "capvC", "capv_diff", "daddv_pi_id" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addv_pi2_proj U V w (pi2 := addv_pi2 U V) : pi2 (pi2 w) = pi2 w.	Proof. by rewrite addv_pi2_id ?memv_pi2. Qed.	Lemma	addv_pi2_proj	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addv_pi2", "addv_pi2_id", "memv_pi2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addv_pi1_pi2 U V w : w \in (U + V)%VS -> addv_pi1 U V w + addv_pi2 U V w = w.	Proof. by rewrite -addv_diff; exact/daddv_pi_add/capv_diff. Qed.	Lemma	addv_pi1_pi2	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addv_diff", "addv_pi1", "addv_pi2", "capv_diff", "daddv_pi_add" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sumv_pi_rec i	:= fix loop r := if r is j :: r1 then let V1 := (\sum_(k <- r1) Vs k)%VS in if j == i then addv_pi1 (Vs j) V1 else (loop r1 \o addv_pi2 (Vs j) V1)%VF else 0.	Let	sumv_pi_rec	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addv_pi1", "addv_pi2", "r1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sumV	:= (\sum_(i <- r0 \| P i) Vs i)%VS.	Notation	sumV	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sumv_pi_for V & V = sumV	:= fun i => sumv_pi_rec i (filter P r0).	Definition	sumv_pi_for	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "filter", "sumV", "sumv_pi_rec" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
memv_sum_pi i v : sumv_pi_for defV i v \in Vs i.	Proof. rewrite /sumv_pi_for. elim: (filter P r0) v => [\|j r IHr] v /=; first by rewrite lfunE mem0v. by case: eqP => [->\|_]; rewrite ?lfunE ?memv_pi1 /=. Qed.	Lemma	memv_sum_pi	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "filter", "lfunE", "mem0v", "memv_pi1", "sumv_pi_for" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sumv_pi_uniq_sum v : uniq (filter P r0) -> v \in V -> \sum_(i <- r0 \| P i) sumv_pi_for defV i v = v.	Proof. rewrite /sumv_pi_for defV -!(big_filter r0 P). elim: (filter P r0) v => [\|i r IHr] v /= => [_ \| /andP[r'i /IHr{}IHr]]. by rewrite !big_nil memv0 => /eqP. rewrite !big_cons eqxx => /addv_pi1_pi2; congr (_ + _ = v). rewrite -[_ v]IHr ?memv_pi2 //; apply: eq_big_seq => j /=. by case: eqP => [<- /idPn \| _]; rewrit...	Lemma	sumv_pi_uniq_sum	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addv_pi1_pi2", "apply", "big_cons", "big_filter", "big_nil", "eq_big_seq", "eqxx", "filter", "lfunE", "memv0", "memv_pi2", "sumv_pi_for", "uniq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sumv_pi V	:= (sumv_pi_for (erefl V)).	Notation	sumv_pi	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "sumv_pi_for" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sumv_pi_sum (I : finType) (P : pred I) Vs v (V : {vspace vT}) (defV : V = (\sum_(i \| P i) Vs i)%VS) : v \in V -> \sum_(i \| P i) sumv_pi_for defV i v = v :> vT.	Proof. by apply: sumv_pi_uniq_sum; have [e _ []] := big_enumP. Qed.	Lemma	sumv_pi_sum	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "big_enumP", "sumv_pi_for", "sumv_pi_uniq_sum", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sumv_pi_nat_sum m n (P : pred nat) Vs v (V : {vspace vT}) (defV : V = (\sum_(m <= i < n \| P i) Vs i)%VS) : v \in V -> \sum_(m <= i < n \| P i) sumv_pi_for defV i v = v :> vT.	Proof. by apply: sumv_pi_uniq_sum; apply/filter_uniq/iota_uniq. Qed.	Lemma	sumv_pi_nat_sum	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "filter_uniq", "iota_uniq", "nat", "sumv_pi_for", "sumv_pi_uniq_sum", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subvs_of : predArgType	:= Subvs u & u \in U.	Inductive	subvs_of	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vsval w : vT	:= let: Subvs u _ := w in u.	Definition	vsval	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subvsP w : vsval w \in U.	Proof. exact: valP. Qed.	Lemma	subvsP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "valP", "vsval" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subvs_inj : injective vsval.	Proof. exact: val_inj. Qed.	Lemma	subvs_inj	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "val_inj", "vsval" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
congr_subvs u v : u = v -> vsval u = vsval v.	Proof. exact: congr1. Qed.	Lemma	congr_subvs	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "vsval" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vsval_is_linear : linear vsval.	Proof. by []. Qed.	Lemma	vsval_is_linear	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "linear", "vsval" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vsproj_key : unit.	Proof. by []. Qed.	Fact	vsproj_key	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "unit" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vsproj_def u	:= Subvs (memv_proj U u).	Definition	vsproj_def	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "memv_proj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vsproj	:= locked_with vsproj_key vsproj_def.	Definition	vsproj	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "vsproj_def", "vsproj_key" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vsproj_unlockable	:= [unlockable fun vsproj].	Canonical	vsproj_unlockable	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "vsproj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vsprojK : {in U, cancel vsproj vsval}.	Proof. by rewrite unlock; apply: projv_id. Qed.	Lemma	vsprojK	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "projv_id", "vsproj", "vsval" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vsvalK : cancel vsval vsproj.	Proof. by move=> w; apply/val_inj/vsprojK/subvsP. Qed.	Lemma	vsvalK	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "subvsP", "val_inj", "vsproj", "vsprojK", "vsval" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vsproj_is_linear : linear vsproj.	Proof. by move=> k w1 w2; apply: val_inj; rewrite unlock /= linearP. Qed.	Lemma	vsproj_is_linear	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "linear", "linearP", "val_inj", "vsproj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subvs_vect_iso : Vector.axiom (\dim U) subvs_of.	Proof. exists (fun w => \row_i coord (vbasis U) i (vsval w)). by move=> k w1 w2; apply/rowP=> i; rewrite !mxE linearP. exists (fun rw : 'rV_(\dim U) => vsproj (\sum_i rw 0 i *: (vbasis U)`_i)). move=> w /=; congr (vsproj _ = w): (vsvalK w). by rewrite {1}(coord_vbasis (subvsP w)); apply: eq_bigr => i _; rewrite m...	Fact	subvs_vect_iso	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "axiom", "basis_free", "coord", "coord_sum_free", "coord_vbasis", "dim", "eq_bigr", "linearP", "memt_nth", "mxE", "rowP", "rpredZ", "rpred_sum", "subvsP", "subvs_of", "vbasis", "vbasisP", "vbasis_mem", "vsproj", "vsprojK", "vsval", "vsvalK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
SubvsE x (xU : x \in U) : Subvs xU = vsproj x.	Proof. by apply/val_inj; rewrite /= vsprojK. Qed.	Lemma	SubvsE	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "val_inj", "vsproj", "vsprojK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
matrix_vect_iso : SemiVector.axiom (m * n) 'M[R]_(m, n).	Proof. exists mxvec; first exact: semilinearP. by exists vec_mx; [apply: mxvecK \| apply: vec_mxK]. Qed.	Fact	matrix_vect_iso	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "axiom", "mxvec", "mxvecK", "semilinearP", "vec_mx", "vec_mxK" ]	expansions that the Ltac interpretation of exists is incapable of doing.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dim_matrix : dim 'M[R]_(m, n) = m * n.	Proof. by []. Qed.	Lemma	dim_matrix	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "dim" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
regular_vect_iso : SemiVector.axiom 1 R^o.	Proof. exists (fun a => a%:M). by split => [a b\|c d]; rewrite 1?rmorphD 1?scale_scalar_mx. by exists (fun A : 'M_1 => A 0 0) => [a \| A]; rewrite ?mxE // -mx11_scalar. Qed.	Fact	regular_vect_iso	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "axiom", "mx11_scalar", "mxE", "rmorphD", "scale_scalar_mx", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pair_vect_iso : SemiVector.axiom (dim vT1 + dim vT2) (vT1 * vT2).	Proof. pose p2r (u : vT1 * vT2) := row_mx (v2r u.1) (v2r u.2). pose r2p w := (r2v (lsubmx w) : vT1, r2v (rsubmx w) : vT2). have r2pK : cancel r2p p2r by move=> w; rewrite /p2r !r2vK hsubmxK. have p2rK : cancel p2r r2p by case=> u v; rewrite /r2p row_mxKl row_mxKr !v2rK. have r2p_lin: semilinear r2p. by split=> [a u\|u...	Fact	pair_vect_iso	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "Build", "apply", "axiom", "can2_semilinear", "dim", "hsubmxK", "linear", "lsubmx", "r2v", "r2vK", "row_mx", "row_mxKl", "row_mxKr", "rsubmx", "semilinear", "semilinearP", "split", "v2r", "v2rK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ffun_vect_iso : SemiVector.axiom (#\|I\| * dim vT) {ffun I -> vT}.	Proof. pose fr (f : {ffun I -> vT}) := mxvec (\matrix_(i < #\|I\|) v2r (f (enum_val i))). exists fr. by split=> [k f\|f g]; rewrite -semilinearP; congr mxvec; apply/matrixP=> i j; rewrite mxE ffunE semilinearP !mxE. exists (fun r => [ffun i => r2v (row (enum_rank i) (vec_mx r)) : vT]) => [g\|r]. by apply/ffunP=> i;...	Fact	ffun_vect_iso	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "axiom", "dim", "enum_rank", "enum_rankK", "enum_val", "enum_valK", "ffunE", "ffunP", "matrixP", "mxE", "mxvec", "mxvecK", "r2v", "r2vK", "row", "rowK", "semilinearP", "split", "v2r", "v2rK", "vT", "vec_mx", "vec_mxK" ]	Type unification with exist is again a problem in this proof.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lhsf u	:= finfun ((tnth lhs)^~ u).	Let	lhsf	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "tnth" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vsolve_eq U	:= finfun (tnth rhs) \in (linfun lhsf @: U)%VS.	Definition	vsolve_eq	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "lhsf", "linfun", "rhs", "tnth" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vsolve_eqP (U : {vspace vT}) : reflect (exists2 u, u \in U & forall i, tnth lhs i u = tnth rhs i) (vsolve_eq U).	Proof. have lhsZ: linear lhsf by move=> a u v; apply/ffunP=> i; rewrite !ffunE linearP. pose lhslM := GRing.isLinear.Build _ _ _ _ lhsf lhsZ. pose lhsL : {linear _ -> _} := HB.pack lhsf lhslM. apply: (iffP memv_imgP) => [] [u Uu sol_u]; exists u => //. by move=> i; rewrite -[tnth rhs i]ffunE sol_u (lfunE lhsL) ffunE....	Lemma	vsolve_eqP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "Build", "Uu", "apply", "ffunE", "ffunP", "lfunE", "lhsf", "linear", "linearP", "memv_imgP", "rhs", "tnth", "vT", "vsolve_eq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
m	:= (\dim {:uT}).	Notation	m	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "dim" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
n	:= (\dim {:vT}).	Notation	n	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "dim", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
span_lfunP (U : seq uT) (phi psi : 'Hom(uT,vT)) : {in <<U>>%VS, phi =1 psi} <-> {in U, phi =1 psi}.	Proof. split=> eq_phi_psi u uU; first by rewrite eq_phi_psi ?memv_span. rewrite [u](@coord_span _ _ _ (in_tuple U))// !linear_sum/=. by apply: eq_bigr=> i _; rewrite 2!linearZ/= eq_phi_psi// ?mem_nth. Qed.	Lemma	span_lfunP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "coord_span", "eq_bigr", "in_tuple", "linearZ", "linear_sum", "mem_nth", "memv_span", "seq", "split", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fullv_lfunP (U : seq uT) (phi psi : 'Hom(uT,vT)) : <<U>>%VS = fullv -> phi = psi <-> {in U, phi =1 psi}.	Proof. by move=> Uf; split=> [->//\|/span_lfunP]; rewrite Uf=> /(_ _ (memvf _))-/lfunP. Qed.	Lemma	fullv_lfunP	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "fullv", "lfunP", "memvf", "seq", "span_lfunP", "split", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rVof (v : vT)	:= \row_i coord e i v.	Definition	rVof	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "coord", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rVof_linear : linear rVof.	Proof. by move=> x v1 v2; apply/rowP=> i; rewrite !mxE linearP. Qed.	Lemma	rVof_linear	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "linear", "linearP", "mxE", "rVof", "rowP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coord_rVof i v : coord e i v = rVof v 0 i.	Proof. by rewrite !mxE. Qed.	Lemma	coord_rVof	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "coord", "mxE", "rVof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vecof (v : 'rV_n)	:= \sum_i v 0 i *: e`_i.	Definition	vecof	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vecof_delta i : vecof (delta_mx 0 i) = e`_i.	Proof. rewrite /vecof (bigD1 i)//= mxE !eqxx scale1r big1 ?addr0// => j neq_ji. by rewrite mxE (negPf neq_ji) andbF scale0r. Qed.	Lemma	vecof_delta	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "addr0", "big1", "bigD1", "delta_mx", "eqxx", "mxE", "scale0r", "scale1r", "vecof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vecof_linear : linear vecof.	Proof. move=> x v1 v2; rewrite linear_sum -big_split/=. by apply: eq_bigr => i _/=; rewrite !mxE scalerDl scalerA. Qed.	Lemma	vecof_linear	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "big_split", "eq_bigr", "linear", "linear_sum", "mxE", "scalerA", "scalerDl", "vecof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rVofK : cancel rVof vecof.	Proof. move=> v; rewrite [v in RHS](coord_basis e_basis) ?memvf//. by apply: eq_bigr => i; rewrite !mxE. Qed.	Lemma	rVofK	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "coord_basis", "e_basis", "eq_bigr", "memvf", "mxE", "rVof", "vecof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vecofK : cancel vecof rVof.	Proof. move=> v; apply/rowP=> i; rewrite !(lfunE, mxE). by rewrite coord_sum_free ?(basis_free e_basis). Qed.	Lemma	vecofK	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "basis_free", "coord_sum_free", "e_basis", "lfunE", "mxE", "rVof", "rowP", "vecof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rVofE (i : 'I_n) : rVof e`_i = delta_mx 0 i.	Proof. apply/rowP=> k; rewrite !mxE. by rewrite eqxx coord_free ?(basis_free e_basis)// eq_sym. Qed.	Lemma	rVofE	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "basis_free", "coord_free", "delta_mx", "e_basis", "eq_sym", "eqxx", "mxE", "rVof", "rowP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
coord_vecof i v : coord e i (vecof v) = v 0 i.	Proof. by rewrite coord_rVof vecofK. Qed.	Lemma	coord_vecof	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "coord", "coord_rVof", "vecof", "vecofK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rVof_eq0 v : (rVof v == 0) = (v == 0).	Proof. by rewrite -(inj_eq (can_inj vecofK)) rVofK linear0. Qed.	Lemma	rVof_eq0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "inj_eq", "linear0", "rVof", "rVofK", "vecofK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vecof_eq0 v : (vecof v == 0) = (v == 0).	Proof. by rewrite -(inj_eq (can_inj rVofK)) vecofK linear0. Qed.	Lemma	vecof_eq0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "inj_eq", "linear0", "rVofK", "vecof", "vecofK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mxof (h : 'Hom(uT, vT))	:= lin1_mx (rVof e' \o h \o vecof e).	Definition	mxof	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "e'", "lin1_mx", "rVof", "vT", "vecof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mxof_linear : linear mxof.	Proof. move=> x h1 h2; apply/matrixP=> i j; do !rewrite ?lfunE/= ?mxE. by rewrite linearP. Qed.	Lemma	mxof_linear	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "lfunE", "linear", "linearP", "matrixP", "mxE", "mxof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
funmx (M : 'M[F]_(m, n)) u	:= vecof e' (rVof e u *m M).	Definition	funmx	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "e'", "rVof", "vecof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
funmx_linear M : linear (funmx M).	Proof. by rewrite /funmx => x u v; rewrite linearP mulmxDl -scalemxAl linearP. Qed.	Lemma	funmx_linear	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "funmx", "linear", "linearP", "mulmxDl", "scalemxAl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hommx M : 'Hom(uT, vT)	:= linfun (funmx M).	Definition	hommx	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "funmx", "linfun", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hommx_linear : linear hommx.	Proof. rewrite /hommx; move=> x A B; apply/lfunP=> u; do !rewrite lfunE/=. by rewrite /funmx mulmxDr -scalemxAr linearP. Qed.	Lemma	hommx_linear	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "funmx", "hommx", "lfunE", "lfunP", "linear", "linearP", "mulmxDr", "scalemxAr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
e_basis: basis_of {:uT} e.		Hypothesis	e_basis	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "basis_of" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
f_basis: basis_of {:vT} e'.		Hypothesis	f_basis	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "basis_of", "e'", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mxofK : cancel mxof hommx.	Proof. by move=> h; apply/lfunP=> u; rewrite lfunE/= /funmx mul_rV_lin1/= !rVofK. Qed.	Lemma	mxofK	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "funmx", "hommx", "lfunE", "lfunP", "mul_rV_lin1", "mxof", "rVofK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hommxK : cancel hommx mxof.	Proof. move=> M; apply/matrixP => i j; rewrite !mxE/= lfunE/=. by rewrite /funmx vecofK// -rowE coord_vecof// mxE. Qed.	Lemma	hommxK	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "coord_vecof", "funmx", "hommx", "lfunE", "matrixP", "mxE", "mxof", "rowE", "vecofK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul_mxof phi u : u *m mxof phi = rVof e' (phi (vecof e u)).	Proof. by rewrite mul_rV_lin1/=. Qed.	Lemma	mul_mxof	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "e'", "mul_rV_lin1", "mxof", "rVof", "vecof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hommxE M u : hommx M u = vecof e' (rVof e u *m M).	Proof. by rewrite -[M in RHS]hommxK mul_mxof !rVofK//. Qed.	Lemma	hommxE	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "e'", "hommx", "hommxK", "mul_mxof", "rVof", "rVofK", "vecof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rVof_mul M u : rVof e u *m M = rVof e' (hommx M u).	Proof. by rewrite hommxE vecofK. Qed.	Lemma	rVof_mul	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "e'", "hommx", "hommxE", "rVof", "vecofK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hom_vecof (phi : 'Hom(uT, vT)) u : phi (vecof e u) = vecof e' (u *m mxof phi).	Proof. by rewrite mul_mxof rVofK. Qed.	Lemma	hom_vecof	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "e'", "mul_mxof", "mxof", "rVofK", "vT", "vecof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rVof_app (phi : 'Hom(uT, vT)) u : rVof e' (phi u) = rVof e u *m mxof phi.	Proof. by rewrite mul_mxof !rVofK. Qed.	Lemma	rVof_app	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "e'", "mul_mxof", "mxof", "rVof", "rVofK", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vecof_mul M u : vecof e' (u *m M) = hommx M (vecof e u).	Proof. by rewrite hommxE vecofK. Qed.	Lemma	vecof_mul	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "e'", "hommx", "hommxE", "vecof", "vecofK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mxof_eq0 phi : (mxof phi == 0) = (phi == 0).	Proof. by rewrite -(inj_eq (can_inj hommxK)) mxofK linear0. Qed.	Lemma	mxof_eq0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "hommxK", "inj_eq", "linear0", "mxof", "mxofK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hommx_eq0 M : (hommx M == 0) = (M == 0).	Proof. by rewrite -(inj_eq (can_inj mxofK)) hommxK linear0. Qed.	Lemma	hommx_eq0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "hommx", "hommxK", "inj_eq", "linear0", "mxofK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
p	:= (\dim {:wT}).	Notation	p	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "dim" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
f_basis: basis_of {:vT} f.		Hypothesis	f_basis	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "basis_of", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
g_basis: basis_of {:wT} g.		Hypothesis	g_basis	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "basis_of" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mxof_comp (phi : 'Hom(uT, vT)) (psi : 'Hom(vT, wT)) : mxof e g (psi \o phi)%VF = mxof e f phi *m mxof f g psi.	Proof. apply/matrixP => i k; rewrite !(mxE, comp_lfunE, lfunE) /=. rewrite [phi _](coord_basis f_basis) ?memvf// 2!linear_sum/=. by apply: eq_bigr => j _ /=; rewrite !mxE !linearZ/= !vecof_delta. Qed.	Lemma	mxof_comp	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "comp_lfunE", "coord_basis", "eq_bigr", "f_basis", "lfunE", "linearZ", "linear_sum", "matrixP", "memvf", "mxE", "mxof", "vT", "vecof_delta" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hommx_mul (A : 'M_(m,n)) (B : 'M_(n, p)) : hommx e g (A *m B) = (hommx f g B \o hommx e f A)%VF.	Proof. by apply: (can_inj (mxofK e_basis g_basis)); rewrite mxof_comp !hommxK. Qed.	Lemma	hommx_mul	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "e_basis", "g_basis", "hommx", "hommxK", "mxofK", "mxof_comp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
msof (V : {vspace vT}) : 'M_n	:= mxof e e (projv V).	Definition	msof	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "mxof", "projv", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vsof (M : 'M[F]_n)	:= limg (hommx e e M).	Definition	vsof	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "hommx", "limg" ]	(\sum_(v <- vbasis V) <<rVof e v>>)%MS.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mxof1 : free e -> mxof e e \1 = 1%:M.	Proof. by move=> eF; apply/matrixP=> i j; rewrite !mxE vecof_delta lfunE coord_free. Qed.	Lemma	mxof1	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "coord_free", "free", "lfunE", "matrixP", "mxE", "mxof", "vecof_delta" ]	<<[seq vecof e (row i M) \| i : 'I_n]>>%VS.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
e_basis : basis_of {:vT} e.		Hypothesis	e_basis	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "basis_of", "vT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
hommx1 : hommx e e 1%:M = \1%VF.	Proof. by rewrite -mxof1 ?(basis_free e_basis)// mxofK. Qed.	Lemma	hommx1	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "basis_free", "e_basis", "hommx", "mxof1", "mxofK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
msofK : cancel msof vsof.	Proof. by rewrite /msof /vsof; move=> V; rewrite mxofK// limg_proj. Qed.	Lemma	msofK	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "limg_proj", "msof", "mxofK", "vsof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_vecof u (V : {vspace vT}) : (vecof e u \in V) = (u <= msof V)%MS.	Proof. apply/idP/submxP=> [\|[v ->{u}]]; last by rewrite -hom_vecof// memv_proj. rewrite -[V in X in X -> _]msofK => /memv_imgP[v _]. by move=> /(canRL (vecofK _)) ->//; rewrite -rVof_mul//; eexists. Qed.	Lemma	mem_vecof	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "hom_vecof", "last", "memv_imgP", "memv_proj", "msof", "msofK", "rVof_mul", "submxP", "vT", "vecof", "vecofK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rVof_sub u M : (rVof e u <= M)%MS = (u \in vsof M).	Proof. apply/submxP/memv_imgP => [[v /(canRL (rVofK _)) ->//]\|[v _ ->]]{u}. by exists (vecof e v); rewrite ?memvf// -vecof_mul. by exists (rVof e v); rewrite -rVof_mul. Qed.	Lemma	rVof_sub	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "memv_imgP", "memvf", "rVof", "rVofK", "rVof_mul", "submxP", "vecof", "vecof_mul", "vsof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vsof_sub M V : (vsof M <= V)%VS = (M <= msof V)%MS.	Proof. apply/subvP/rV_subP => [MsubV _/submxP[u ->]\|VsubM _/memv_imgP[u _ ->]]. by rewrite -mem_vecof MsubV// -rVof_sub vecofK// submxMl. by rewrite -[V]msofK -rVof_sub VsubM// -rVof_mul// submxMl. Qed.	Lemma	vsof_sub	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "mem_vecof", "memv_imgP", "msof", "msofK", "rV_subP", "rVof_mul", "rVof_sub", "submxMl", "submxP", "subvP", "vecofK", "vsof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
msof_sub V M : (msof V <= M)%MS = (V <= vsof M)%VS.	Proof. apply/rV_subP/subvP => [VsubM v vV\|MsubV _/submxP[u ->]]. by rewrite -rVof_sub VsubM// -mem_vecof rVofK. by rewrite mul_mxof rVof_sub MsubV// memv_proj. Qed.	Lemma	msof_sub	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "mem_vecof", "memv_proj", "msof", "mul_mxof", "rV_subP", "rVofK", "rVof_sub", "submxP", "subvP", "vsof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vsofK M : (msof (vsof M) == M)%MS.	Proof. by rewrite msof_sub -vsof_sub subvv. Qed.	Lemma	vsofK	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "msof", "msof_sub", "subvv", "vsof", "vsof_sub" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_msof : {mono msof : V V' / (V <= V')%VS >-> (V <= V')%MS}.	Proof. by move=> V V'; rewrite msof_sub msofK. Qed.	Lemma	sub_msof	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "msof", "msofK", "msof_sub" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sub_vsof : {mono vsof : M M' / (M <= M')%MS >-> (M <= M')%VS}.	Proof. by move=> M M'; rewrite vsof_sub (eqmxP (vsofK _)). Qed.	Lemma	sub_vsof	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "eqmxP", "vsof", "vsofK", "vsof_sub" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
msof0 : msof 0 = 0.	Proof. apply/eqP; rewrite -submx0; apply/rV_subP => v. by rewrite -mem_vecof memv0 vecof_eq0// => /eqP->; rewrite sub0mx. Qed.	Lemma	msof0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "mem_vecof", "memv0", "msof", "rV_subP", "sub0mx", "submx0", "vecof_eq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vsof0 : vsof 0 = 0%VS.	Proof. by apply/vspaceP=> v; rewrite memv0 -rVof_sub submx0 rVof_eq0. Qed.	Lemma	vsof0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "memv0", "rVof_eq0", "rVof_sub", "submx0", "vsof", "vspaceP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
msof_eq0 V : (msof V == 0) = (V == 0%VS).	Proof. by rewrite -(inj_eq (can_inj msofK)) msof0. Qed.	Lemma	msof_eq0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "inj_eq", "msof", "msof0", "msofK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
vsof_eq0 M : (vsof M == 0%VS) = (M == 0).	Proof. rewrite (sameP eqP eqmx0P) -!(eqmxP (vsofK M)) (sameP eqmx0P eqP) -msof0. by rewrite (inj_eq (can_inj msofK)). Qed.	Lemma	vsof_eq0	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "eqmx0P", "eqmxP", "inj_eq", "msof0", "msofK", "vsof", "vsofK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leigenspace (phi : 'End(uT)) a	:= lker (phi - a *: \1%VF).	Definition	leigenspace	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "lker" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leigenvalue phi a	:= leigenspace phi a != 0%VS.	Definition	leigenvalue	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "leigenspace" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
e_free	:= basis_free e_basis.	Let	e_free	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "basis_free", "e_basis" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lker_ker phi : lker phi = vsof e (kermx (mxof e e phi)).	Proof. apply/vspaceP => v; rewrite memv_ker -rVof_sub// (sameP sub_kermxP eqP). by rewrite -rVof_app// rVof_eq0. Qed.	Lemma	lker_ker	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "kermx", "lker", "memv_ker", "mxof", "rVof_app", "rVof_eq0", "rVof_sub", "sub_kermxP", "vsof", "vspaceP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
limgE phi : limg phi = vsof e (mxof e e phi).	Proof. apply/vspaceP => v; rewrite -rVof_sub//. apply/memv_imgP/submxP => [[u _ ->]\|[u /(canRL (rVofK _)) ->//]]. by exists (rVof e u); rewrite -rVof_app. by exists (vecof e u); rewrite ?memvf// -hom_vecof. Qed.	Lemma	limgE	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "apply", "hom_vecof", "limg", "memv_imgP", "memvf", "mxof", "rVof", "rVofK", "rVof_app", "rVof_sub", "submxP", "vecof", "vsof", "vspaceP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
leigenspaceE f a : leigenspace f a = vsof e (eigenspace (mxof e e f) a).	Proof. by rewrite [LHS]lker_ker linearB linearZ/= mxof1// scalemx1. Qed.	Lemma	leigenspaceE	algebra	algebra/vector.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]	[ "eigenspace", "leigenspace", "linearB", "linearZ", "lker_ker", "mxof", "mxof1", "scalemx1", "vsof" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

memv_projC U w : w - projv U w \in (U^C)%VS.

Proof. rewrite -{1}[w](daddv_pi_add (capv_compl U)) ?addv_complf ?memvf //. by rewrite addrC addKr memv_pi. Qed.

Lemma

memv_projC

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addKr", "addrC", "addv_complf", "capv_compl", "daddv_pi_add", "memv_pi", "memvf", "projv" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

limg_proj U : limg (projv U) = U.

Proof. apply/vspaceP=> u; apply/memv_imgP/idP=> [[u1 _ ->] | ]; first exact: memv_proj. by exists (projv U u); rewrite ?projv_id ?memv_img ?memvf. Qed.

Lemma

limg_proj

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "limg", "memv_img", "memv_imgP", "memv_proj", "memvf", "projv", "projv_id", "vspaceP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lker_proj U : lker (projv U) = (U^C)%VS.

Proof. apply/eqP; rewrite eqEdim andbC; apply/andP; split. by rewrite dimv_compl -(limg_ker_dim (projv U) fullv) limg_proj addnK capfv. by apply/subvP=> v; rewrite memv_ker -{2}[v]subr0 => /eqP <-; apply: memv_projC. Qed.

Lemma

lker_proj

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addnK", "apply", "capfv", "dimv_compl", "eqEdim", "fullv", "limg_ker_dim", "limg_proj", "lker", "memv_ker", "memv_projC", "projv", "split", "subr0", "subvP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addv_pi1_proj U V w (pi1 := addv_pi1 U V) : pi1 (pi1 w) = pi1 w.

Proof. by rewrite daddv_pi_proj // capv_diff. Qed.

Lemma

addv_pi1_proj

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addv_pi1", "capv_diff", "daddv_pi_proj" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addv_pi2_id U V v : v \in V -> addv_pi2 U V v = v.

Proof. by apply: daddv_pi_id; rewrite capvC capv_diff. Qed.

Lemma

addv_pi2_id

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addv_pi2", "apply", "capvC", "capv_diff", "daddv_pi_id" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addv_pi2_proj U V w (pi2 := addv_pi2 U V) : pi2 (pi2 w) = pi2 w.

Proof. by rewrite addv_pi2_id ?memv_pi2. Qed.

Lemma

addv_pi2_proj

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addv_pi2", "addv_pi2_id", "memv_pi2" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addv_pi1_pi2 U V w : w \in (U + V)%VS -> addv_pi1 U V w + addv_pi2 U V w = w.

Proof. by rewrite -addv_diff; exact/daddv_pi_add/capv_diff. Qed.

Lemma

addv_pi1_pi2

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addv_diff", "addv_pi1", "addv_pi2", "capv_diff", "daddv_pi_add" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sumv_pi_rec i

:= fix loop r := if r is j :: r1 then let V1 := (\sum_(k <- r1) Vs k)%VS in if j == i then addv_pi1 (Vs j) V1 else (loop r1 \o addv_pi2 (Vs j) V1)%VF else 0.

Let

sumv_pi_rec

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addv_pi1", "addv_pi2", "r1" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sumV

:= (\sum_(i <- r0 | P i) Vs i)%VS.

Notation

sumV

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sumv_pi_for V & V = sumV

:= fun i => sumv_pi_rec i (filter P r0).

Definition

sumv_pi_for

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "filter", "sumV", "sumv_pi_rec" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

memv_sum_pi i v : sumv_pi_for defV i v \in Vs i.

Proof. rewrite /sumv_pi_for. elim: (filter P r0) v => [|j r IHr] v /=; first by rewrite lfunE mem0v. by case: eqP => [->|_]; rewrite ?lfunE ?memv_pi1 /=. Qed.

Lemma

memv_sum_pi

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "filter", "lfunE", "mem0v", "memv_pi1", "sumv_pi_for" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sumv_pi_uniq_sum v : uniq (filter P r0) -> v \in V -> \sum_(i <- r0 | P i) sumv_pi_for defV i v = v.

Proof. rewrite /sumv_pi_for defV -!(big_filter r0 P). elim: (filter P r0) v => [|i r IHr] v /= => [_ | /andP[r'i /IHr{}IHr]]. by rewrite !big_nil memv0 => /eqP. rewrite !big_cons eqxx => /addv_pi1_pi2; congr (_ + _ = v). rewrite -[_ v]IHr ?memv_pi2 //; apply: eq_big_seq => j /=. by case: eqP => [<- /idPn | _]; rewrit...

Lemma

sumv_pi_uniq_sum

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addv_pi1_pi2", "apply", "big_cons", "big_filter", "big_nil", "eq_big_seq", "eqxx", "filter", "lfunE", "memv0", "memv_pi2", "sumv_pi_for", "uniq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sumv_pi V

:= (sumv_pi_for (erefl V)).

Notation

sumv_pi

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "sumv_pi_for" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sumv_pi_sum (I : finType) (P : pred I) Vs v (V : {vspace vT}) (defV : V = (\sum_(i | P i) Vs i)%VS) : v \in V -> \sum_(i | P i) sumv_pi_for defV i v = v :> vT.

Proof. by apply: sumv_pi_uniq_sum; have [e _ []] := big_enumP. Qed.

Lemma

sumv_pi_sum

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "big_enumP", "sumv_pi_for", "sumv_pi_uniq_sum", "vT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sumv_pi_nat_sum m n (P : pred nat) Vs v (V : {vspace vT}) (defV : V = (\sum_(m <= i < n | P i) Vs i)%VS) : v \in V -> \sum_(m <= i < n | P i) sumv_pi_for defV i v = v :> vT.

Proof. by apply: sumv_pi_uniq_sum; apply/filter_uniq/iota_uniq. Qed.

Lemma

sumv_pi_nat_sum

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "filter_uniq", "iota_uniq", "nat", "sumv_pi_for", "sumv_pi_uniq_sum", "vT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subvs_of : predArgType

:= Subvs u & u \in U.

Inductive

subvs_of

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vsval w : vT

:= let: Subvs u _ := w in u.

Definition

vsval

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "vT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subvsP w : vsval w \in U.

Proof. exact: valP. Qed.

Lemma

subvsP

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "valP", "vsval" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subvs_inj : injective vsval.

Proof. exact: val_inj. Qed.

Lemma

subvs_inj

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "val_inj", "vsval" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

congr_subvs u v : u = v -> vsval u = vsval v.

Proof. exact: congr1. Qed.

Lemma

congr_subvs

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "vsval" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vsval_is_linear : linear vsval.

Proof. by []. Qed.

Lemma

vsval_is_linear

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "linear", "vsval" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vsproj_key : unit.

Proof. by []. Qed.

Fact

vsproj_key

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "unit" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vsproj_def u

:= Subvs (memv_proj U u).

Definition

vsproj_def

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "memv_proj" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vsproj

:= locked_with vsproj_key vsproj_def.

Definition

vsproj

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "vsproj_def", "vsproj_key" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vsproj_unlockable

:= [unlockable fun vsproj].

Canonical

vsproj_unlockable

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "vsproj" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vsprojK : {in U, cancel vsproj vsval}.

Proof. by rewrite unlock; apply: projv_id. Qed.

Lemma

vsprojK

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "projv_id", "vsproj", "vsval" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vsvalK : cancel vsval vsproj.

Proof. by move=> w; apply/val_inj/vsprojK/subvsP. Qed.

Lemma

vsvalK

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "subvsP", "val_inj", "vsproj", "vsprojK", "vsval" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vsproj_is_linear : linear vsproj.

Proof. by move=> k w1 w2; apply: val_inj; rewrite unlock /= linearP. Qed.

Lemma

vsproj_is_linear

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "linear", "linearP", "val_inj", "vsproj" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subvs_vect_iso : Vector.axiom (\dim U) subvs_of.

Proof. exists (fun w => \row_i coord (vbasis U) i (vsval w)). by move=> k w1 w2; apply/rowP=> i; rewrite !mxE linearP. exists (fun rw : 'rV_(\dim U) => vsproj (\sum_i rw 0 i *: (vbasis U)`_i)). move=> w /=; congr (vsproj _ = w): (vsvalK w). by rewrite {1}(coord_vbasis (subvsP w)); apply: eq_bigr => i _; rewrite m...

Fact

subvs_vect_iso

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "axiom", "basis_free", "coord", "coord_sum_free", "coord_vbasis", "dim", "eq_bigr", "linearP", "memt_nth", "mxE", "rowP", "rpredZ", "rpred_sum", "subvsP", "subvs_of", "vbasis", "vbasisP", "vbasis_mem", "vsproj", "vsprojK", "vsval", "vsvalK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

SubvsE x (xU : x \in U) : Subvs xU = vsproj x.

Proof. by apply/val_inj; rewrite /= vsprojK. Qed.

Lemma

SubvsE

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "val_inj", "vsproj", "vsprojK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

matrix_vect_iso : SemiVector.axiom (m * n) 'M[R]_(m, n).

Proof. exists mxvec; first exact: semilinearP. by exists vec_mx; [apply: mxvecK | apply: vec_mxK]. Qed.

Fact

matrix_vect_iso

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "axiom", "mxvec", "mxvecK", "semilinearP", "vec_mx", "vec_mxK" ]

expansions that the Ltac interpretation of exists is incapable of doing.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dim_matrix : dim 'M[R]_(m, n) = m * n.

Proof. by []. Qed.

Lemma

dim_matrix

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "dim" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

regular_vect_iso : SemiVector.axiom 1 R^o.

Proof. exists (fun a => a%:M). by split => [a b|c d]; rewrite 1?rmorphD 1?scale_scalar_mx. by exists (fun A : 'M_1 => A 0 0) => [a | A]; rewrite ?mxE // -mx11_scalar. Qed.

Fact

regular_vect_iso

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "axiom", "mx11_scalar", "mxE", "rmorphD", "scale_scalar_mx", "split" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pair_vect_iso : SemiVector.axiom (dim vT1 + dim vT2) (vT1 * vT2).

Proof. pose p2r (u : vT1 * vT2) := row_mx (v2r u.1) (v2r u.2). pose r2p w := (r2v (lsubmx w) : vT1, r2v (rsubmx w) : vT2). have r2pK : cancel r2p p2r by move=> w; rewrite /p2r !r2vK hsubmxK. have p2rK : cancel p2r r2p by case=> u v; rewrite /r2p row_mxKl row_mxKr !v2rK. have r2p_lin: semilinear r2p. by split=> [a u|u...

Fact

pair_vect_iso

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "Build", "apply", "axiom", "can2_semilinear", "dim", "hsubmxK", "linear", "lsubmx", "r2v", "r2vK", "row_mx", "row_mxKl", "row_mxKr", "rsubmx", "semilinear", "semilinearP", "split", "v2r", "v2rK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ffun_vect_iso : SemiVector.axiom (#|I| * dim vT) {ffun I -> vT}.

Proof. pose fr (f : {ffun I -> vT}) := mxvec (\matrix_(i < #|I|) v2r (f (enum_val i))). exists fr. by split=> [k f|f g]; rewrite -semilinearP; congr mxvec; apply/matrixP=> i j; rewrite mxE ffunE semilinearP !mxE. exists (fun r => [ffun i => r2v (row (enum_rank i) (vec_mx r)) : vT]) => [g|r]. by apply/ffunP=> i;...

Fact

ffun_vect_iso

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "axiom", "dim", "enum_rank", "enum_rankK", "enum_val", "enum_valK", "ffunE", "ffunP", "matrixP", "mxE", "mxvec", "mxvecK", "r2v", "r2vK", "row", "rowK", "semilinearP", "split", "v2r", "v2rK", "vT", "vec_mx", "vec_mxK" ]

Type unification with exist is again a problem in this proof.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lhsf u

:= finfun ((tnth lhs)^~ u).

Let

lhsf

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "tnth" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vsolve_eq U

:= finfun (tnth rhs) \in (linfun lhsf @: U)%VS.

Definition

vsolve_eq

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "lhsf", "linfun", "rhs", "tnth" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vsolve_eqP (U : {vspace vT}) : reflect (exists2 u, u \in U & forall i, tnth lhs i u = tnth rhs i) (vsolve_eq U).

Proof. have lhsZ: linear lhsf by move=> a u v; apply/ffunP=> i; rewrite !ffunE linearP. pose lhslM := GRing.isLinear.Build _ _ _ _ lhsf lhsZ. pose lhsL : {linear _ -> _} := HB.pack lhsf lhslM. apply: (iffP memv_imgP) => [] [u Uu sol_u]; exists u => //. by move=> i; rewrite -[tnth rhs i]ffunE sol_u (lfunE lhsL) ffunE....

Lemma

vsolve_eqP

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "Build", "Uu", "apply", "ffunE", "ffunP", "lfunE", "lhsf", "linear", "linearP", "memv_imgP", "rhs", "tnth", "vT", "vsolve_eq" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

m

:= (\dim {:uT}).

Notation

m

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "dim" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

n

:= (\dim {:vT}).

Notation

n

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "dim", "vT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

span_lfunP (U : seq uT) (phi psi : 'Hom(uT,vT)) : {in <<U>>%VS, phi =1 psi} <-> {in U, phi =1 psi}.

Proof. split=> eq_phi_psi u uU; first by rewrite eq_phi_psi ?memv_span. rewrite [u](@coord_span _ _ _ (in_tuple U))// !linear_sum/=. by apply: eq_bigr=> i _; rewrite 2!linearZ/= eq_phi_psi// ?mem_nth. Qed.

Lemma

span_lfunP

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "coord_span", "eq_bigr", "in_tuple", "linearZ", "linear_sum", "mem_nth", "memv_span", "seq", "split", "vT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fullv_lfunP (U : seq uT) (phi psi : 'Hom(uT,vT)) : <<U>>%VS = fullv -> phi = psi <-> {in U, phi =1 psi}.

Proof. by move=> Uf; split=> [->//|/span_lfunP]; rewrite Uf=> /(_ _ (memvf _))-/lfunP. Qed.

Lemma

fullv_lfunP

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "fullv", "lfunP", "memvf", "seq", "span_lfunP", "split", "vT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rVof (v : vT)

:= \row_i coord e i v.

Definition

rVof

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "coord", "vT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rVof_linear : linear rVof.

Proof. by move=> x v1 v2; apply/rowP=> i; rewrite !mxE linearP. Qed.

Lemma

rVof_linear

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "linear", "linearP", "mxE", "rVof", "rowP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coord_rVof i v : coord e i v = rVof v 0 i.

Proof. by rewrite !mxE. Qed.

Lemma

coord_rVof

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "coord", "mxE", "rVof" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vecof (v : 'rV_n)

:= \sum_i v 0 i *: e`_i.

Definition

vecof

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vecof_delta i : vecof (delta_mx 0 i) = e`_i.

Proof. rewrite /vecof (bigD1 i)//= mxE !eqxx scale1r big1 ?addr0// => j neq_ji. by rewrite mxE (negPf neq_ji) andbF scale0r. Qed.

Lemma

vecof_delta

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "addr0", "big1", "bigD1", "delta_mx", "eqxx", "mxE", "scale0r", "scale1r", "vecof" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vecof_linear : linear vecof.

Proof. move=> x v1 v2; rewrite linear_sum -big_split/=. by apply: eq_bigr => i _/=; rewrite !mxE scalerDl scalerA. Qed.

Lemma

vecof_linear

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "big_split", "eq_bigr", "linear", "linear_sum", "mxE", "scalerA", "scalerDl", "vecof" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rVofK : cancel rVof vecof.

Proof. move=> v; rewrite [v in RHS](coord_basis e_basis) ?memvf//. by apply: eq_bigr => i; rewrite !mxE. Qed.

Lemma

rVofK

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "coord_basis", "e_basis", "eq_bigr", "memvf", "mxE", "rVof", "vecof" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vecofK : cancel vecof rVof.

Proof. move=> v; apply/rowP=> i; rewrite !(lfunE, mxE). by rewrite coord_sum_free ?(basis_free e_basis). Qed.

Lemma

vecofK

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "basis_free", "coord_sum_free", "e_basis", "lfunE", "mxE", "rVof", "rowP", "vecof" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rVofE (i : 'I_n) : rVof e`_i = delta_mx 0 i.

Proof. apply/rowP=> k; rewrite !mxE. by rewrite eqxx coord_free ?(basis_free e_basis)// eq_sym. Qed.

Lemma

rVofE

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "basis_free", "coord_free", "delta_mx", "e_basis", "eq_sym", "eqxx", "mxE", "rVof", "rowP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

coord_vecof i v : coord e i (vecof v) = v 0 i.

Proof. by rewrite coord_rVof vecofK. Qed.

Lemma

coord_vecof

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "coord", "coord_rVof", "vecof", "vecofK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rVof_eq0 v : (rVof v == 0) = (v == 0).

Proof. by rewrite -(inj_eq (can_inj vecofK)) rVofK linear0. Qed.

Lemma

rVof_eq0

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "inj_eq", "linear0", "rVof", "rVofK", "vecofK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vecof_eq0 v : (vecof v == 0) = (v == 0).

Proof. by rewrite -(inj_eq (can_inj rVofK)) vecofK linear0. Qed.

Lemma

vecof_eq0

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "inj_eq", "linear0", "rVofK", "vecof", "vecofK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mxof (h : 'Hom(uT, vT))

:= lin1_mx (rVof e' \o h \o vecof e).

Definition

mxof

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "e'", "lin1_mx", "rVof", "vT", "vecof" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mxof_linear : linear mxof.

Proof. move=> x h1 h2; apply/matrixP=> i j; do !rewrite ?lfunE/= ?mxE. by rewrite linearP. Qed.

Lemma

mxof_linear

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "lfunE", "linear", "linearP", "matrixP", "mxE", "mxof" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

funmx (M : 'M[F]_(m, n)) u

:= vecof e' (rVof e u *m M).

Definition

funmx

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "e'", "rVof", "vecof" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

funmx_linear M : linear (funmx M).

Proof. by rewrite /funmx => x u v; rewrite linearP mulmxDl -scalemxAl linearP. Qed.

Lemma

funmx_linear

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "funmx", "linear", "linearP", "mulmxDl", "scalemxAl" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hommx M : 'Hom(uT, vT)

:= linfun (funmx M).

Definition

hommx

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "funmx", "linfun", "vT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hommx_linear : linear hommx.

Proof. rewrite /hommx; move=> x A B; apply/lfunP=> u; do !rewrite lfunE/=. by rewrite /funmx mulmxDr -scalemxAr linearP. Qed.

Lemma

hommx_linear

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "funmx", "hommx", "lfunE", "lfunP", "linear", "linearP", "mulmxDr", "scalemxAr" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

e_basis: basis_of {:uT} e.

Hypothesis

e_basis

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "basis_of" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

f_basis: basis_of {:vT} e'.

Hypothesis

f_basis

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "basis_of", "e'", "vT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mxofK : cancel mxof hommx.

Proof. by move=> h; apply/lfunP=> u; rewrite lfunE/= /funmx mul_rV_lin1/= !rVofK. Qed.

Lemma

mxofK

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "funmx", "hommx", "lfunE", "lfunP", "mul_rV_lin1", "mxof", "rVofK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hommxK : cancel hommx mxof.

Proof. move=> M; apply/matrixP => i j; rewrite !mxE/= lfunE/=. by rewrite /funmx vecofK// -rowE coord_vecof// mxE. Qed.

Lemma

hommxK

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "coord_vecof", "funmx", "hommx", "lfunE", "matrixP", "mxE", "mxof", "rowE", "vecofK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mul_mxof phi u : u *m mxof phi = rVof e' (phi (vecof e u)).

Proof. by rewrite mul_rV_lin1/=. Qed.

Lemma

mul_mxof

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "e'", "mul_rV_lin1", "mxof", "rVof", "vecof" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hommxE M u : hommx M u = vecof e' (rVof e u *m M).

Proof. by rewrite -[M in RHS]hommxK mul_mxof !rVofK//. Qed.

Lemma

hommxE

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "e'", "hommx", "hommxK", "mul_mxof", "rVof", "rVofK", "vecof" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rVof_mul M u : rVof e u *m M = rVof e' (hommx M u).

Proof. by rewrite hommxE vecofK. Qed.

Lemma

rVof_mul

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "e'", "hommx", "hommxE", "rVof", "vecofK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hom_vecof (phi : 'Hom(uT, vT)) u : phi (vecof e u) = vecof e' (u *m mxof phi).

Proof. by rewrite mul_mxof rVofK. Qed.

Lemma

hom_vecof

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "e'", "mul_mxof", "mxof", "rVofK", "vT", "vecof" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rVof_app (phi : 'Hom(uT, vT)) u : rVof e' (phi u) = rVof e u *m mxof phi.

Proof. by rewrite mul_mxof !rVofK. Qed.

Lemma

rVof_app

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "e'", "mul_mxof", "mxof", "rVof", "rVofK", "vT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vecof_mul M u : vecof e' (u *m M) = hommx M (vecof e u).

Proof. by rewrite hommxE vecofK. Qed.

Lemma

vecof_mul

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "e'", "hommx", "hommxE", "vecof", "vecofK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mxof_eq0 phi : (mxof phi == 0) = (phi == 0).

Proof. by rewrite -(inj_eq (can_inj hommxK)) mxofK linear0. Qed.

Lemma

mxof_eq0

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "hommxK", "inj_eq", "linear0", "mxof", "mxofK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hommx_eq0 M : (hommx M == 0) = (M == 0).

Proof. by rewrite -(inj_eq (can_inj mxofK)) hommxK linear0. Qed.

Lemma

hommx_eq0

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "hommx", "hommxK", "inj_eq", "linear0", "mxofK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

p

:= (\dim {:wT}).

Notation

p

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "dim" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

f_basis: basis_of {:vT} f.

Hypothesis

f_basis

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "basis_of", "vT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

g_basis: basis_of {:wT} g.

Hypothesis

g_basis

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "basis_of" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mxof_comp (phi : 'Hom(uT, vT)) (psi : 'Hom(vT, wT)) : mxof e g (psi \o phi)%VF = mxof e f phi *m mxof f g psi.

Proof. apply/matrixP => i k; rewrite !(mxE, comp_lfunE, lfunE) /=. rewrite [phi _](coord_basis f_basis) ?memvf// 2!linear_sum/=. by apply: eq_bigr => j _ /=; rewrite !mxE !linearZ/= !vecof_delta. Qed.

Lemma

mxof_comp

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "comp_lfunE", "coord_basis", "eq_bigr", "f_basis", "lfunE", "linearZ", "linear_sum", "matrixP", "memvf", "mxE", "mxof", "vT", "vecof_delta" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hommx_mul (A : 'M_(m,n)) (B : 'M_(n, p)) : hommx e g (A *m B) = (hommx f g B \o hommx e f A)%VF.

Proof. by apply: (can_inj (mxofK e_basis g_basis)); rewrite mxof_comp !hommxK. Qed.

Lemma

hommx_mul

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "e_basis", "g_basis", "hommx", "hommxK", "mxofK", "mxof_comp" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

msof (V : {vspace vT}) : 'M_n

:= mxof e e (projv V).

Definition

msof

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "mxof", "projv", "vT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vsof (M : 'M[F]_n)

:= limg (hommx e e M).

Definition

vsof

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "hommx", "limg" ]

(\sum_(v <- vbasis V) <<rVof e v>>)%MS.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mxof1 : free e -> mxof e e \1 = 1%:M.

Proof. by move=> eF; apply/matrixP=> i j; rewrite !mxE vecof_delta lfunE coord_free. Qed.

Lemma

mxof1

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "coord_free", "free", "lfunE", "matrixP", "mxE", "mxof", "vecof_delta" ]

<<[seq vecof e (row i M) | i : 'I_n]>>%VS.

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

e_basis : basis_of {:vT} e.

Hypothesis

e_basis

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "basis_of", "vT" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

hommx1 : hommx e e 1%:M = \1%VF.

Proof. by rewrite -mxof1 ?(basis_free e_basis)// mxofK. Qed.

Lemma

hommx1

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "basis_free", "e_basis", "hommx", "mxof1", "mxofK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

msofK : cancel msof vsof.

Proof. by rewrite /msof /vsof; move=> V; rewrite mxofK// limg_proj. Qed.

Lemma

msofK

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "limg_proj", "msof", "mxofK", "vsof" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_vecof u (V : {vspace vT}) : (vecof e u \in V) = (u <= msof V)%MS.

Proof. apply/idP/submxP=> [|[v ->{u}]]; last by rewrite -hom_vecof// memv_proj. rewrite -[V in X in X -> _]msofK => /memv_imgP[v _]. by move=> /(canRL (vecofK _)) ->//; rewrite -rVof_mul//; eexists. Qed.

Lemma

mem_vecof

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "hom_vecof", "last", "memv_imgP", "memv_proj", "msof", "msofK", "rVof_mul", "submxP", "vT", "vecof", "vecofK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rVof_sub u M : (rVof e u <= M)%MS = (u \in vsof M).

Proof. apply/submxP/memv_imgP => [[v /(canRL (rVofK _)) ->//]|[v _ ->]]{u}. by exists (vecof e v); rewrite ?memvf// -vecof_mul. by exists (rVof e v); rewrite -rVof_mul. Qed.

Lemma

rVof_sub

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "memv_imgP", "memvf", "rVof", "rVofK", "rVof_mul", "submxP", "vecof", "vecof_mul", "vsof" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vsof_sub M V : (vsof M <= V)%VS = (M <= msof V)%MS.

Proof. apply/subvP/rV_subP => [MsubV _/submxP[u ->]|VsubM _/memv_imgP[u _ ->]]. by rewrite -mem_vecof MsubV// -rVof_sub vecofK// submxMl. by rewrite -[V]msofK -rVof_sub VsubM// -rVof_mul// submxMl. Qed.

Lemma

vsof_sub

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "mem_vecof", "memv_imgP", "msof", "msofK", "rV_subP", "rVof_mul", "rVof_sub", "submxMl", "submxP", "subvP", "vecofK", "vsof" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

msof_sub V M : (msof V <= M)%MS = (V <= vsof M)%VS.

Proof. apply/rV_subP/subvP => [VsubM v vV|MsubV _/submxP[u ->]]. by rewrite -rVof_sub VsubM// -mem_vecof rVofK. by rewrite mul_mxof rVof_sub MsubV// memv_proj. Qed.

Lemma

msof_sub

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "mem_vecof", "memv_proj", "msof", "mul_mxof", "rV_subP", "rVofK", "rVof_sub", "submxP", "subvP", "vsof" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vsofK M : (msof (vsof M) == M)%MS.

Proof. by rewrite msof_sub -vsof_sub subvv. Qed.

Lemma

vsofK

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "msof", "msof_sub", "subvv", "vsof", "vsof_sub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_msof : {mono msof : V V' / (V <= V')%VS >-> (V <= V')%MS}.

Proof. by move=> V V'; rewrite msof_sub msofK. Qed.

Lemma

sub_msof

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "msof", "msofK", "msof_sub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sub_vsof : {mono vsof : M M' / (M <= M')%MS >-> (M <= M')%VS}.

Proof. by move=> M M'; rewrite vsof_sub (eqmxP (vsofK _)). Qed.

Lemma

sub_vsof

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "eqmxP", "vsof", "vsofK", "vsof_sub" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

msof0 : msof 0 = 0.

Proof. apply/eqP; rewrite -submx0; apply/rV_subP => v. by rewrite -mem_vecof memv0 vecof_eq0// => /eqP->; rewrite sub0mx. Qed.

Lemma

msof0

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "mem_vecof", "memv0", "msof", "rV_subP", "sub0mx", "submx0", "vecof_eq0" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vsof0 : vsof 0 = 0%VS.

Proof. by apply/vspaceP=> v; rewrite memv0 -rVof_sub submx0 rVof_eq0. Qed.

Lemma

vsof0

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "memv0", "rVof_eq0", "rVof_sub", "submx0", "vsof", "vspaceP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

msof_eq0 V : (msof V == 0) = (V == 0%VS).

Proof. by rewrite -(inj_eq (can_inj msofK)) msof0. Qed.

Lemma

msof_eq0

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "inj_eq", "msof", "msof0", "msofK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

vsof_eq0 M : (vsof M == 0%VS) = (M == 0).

Proof. rewrite (sameP eqP eqmx0P) -!(eqmxP (vsofK M)) (sameP eqmx0P eqP) -msof0. by rewrite (inj_eq (can_inj msofK)). Qed.

Lemma

vsof_eq0

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "eqmx0P", "eqmxP", "inj_eq", "msof0", "msofK", "vsof", "vsofK" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leigenspace (phi : 'End(uT)) a

:= lker (phi - a *: \1%VF).

Definition

leigenspace

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "lker" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leigenvalue phi a

:= leigenspace phi a != 0%VS.

Definition

leigenvalue

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "leigenspace" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

e_free

:= basis_free e_basis.

Let

e_free

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "basis_free", "e_basis" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lker_ker phi : lker phi = vsof e (kermx (mxof e e phi)).

Proof. apply/vspaceP => v; rewrite memv_ker -rVof_sub// (sameP sub_kermxP eqP). by rewrite -rVof_app// rVof_eq0. Qed.

Lemma

lker_ker

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "kermx", "lker", "memv_ker", "mxof", "rVof_app", "rVof_eq0", "rVof_sub", "sub_kermxP", "vsof", "vspaceP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

limgE phi : limg phi = vsof e (mxof e e phi).

Proof. apply/vspaceP => v; rewrite -rVof_sub//. apply/memv_imgP/submxP => [[u _ ->]|[u /(canRL (rVofK _)) ->//]]. by exists (rVof e u); rewrite -rVof_app. by exists (vecof e u); rewrite ?memvf// -hom_vecof. Qed.

Lemma

limgE

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "apply", "hom_vecof", "limg", "memv_imgP", "memvf", "mxof", "rVof", "rVofK", "rVof_app", "rVof_sub", "submxP", "vecof", "vsof", "vspaceP" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d

leigenspaceE f a : leigenspace f a = vsof e (eigenspace (mxof e e f) a).

Proof. by rewrite [LHS]lker_ker linearB linearZ/= mxof1// scalemx1. Qed.

Lemma

leigenspaceE

algebra

algebra/vector.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "choice", "fintype", "tuple", "finfun", "bigop", "nmodule", "rings_modules_and_algebras", "divalg", "matrix", "mxalgebra", "GRing.Theory", "VectorExports", "VectorInternalTheory" ]

[ "eigenspace", "leigenspace", "linearB", "linearZ", "lker_ker", "mxof", "mxof1", "scalemx1", "vsof" ]

https://github.com/math-comp/math-comp

91d97df9cf3204b4dab84f4e24bc633e84b6473d