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
"- 1"	:= (- (1)) : ring_scope.	Notation	- 1	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulrN x y : x * (- y) = - (x * y).	Proof. by apply: (addrI (x * y)); rewrite -mulrDr !subrr mulr0. Qed.	Lemma	mulrN	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "addrI", "apply", "mulr0", "mulrDr", "subrr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulNr x y : (- x) * y = - (x * y).	Proof. by apply: (addrI (x * y)); rewrite -mulrDl !subrr mul0r. Qed.	Lemma	mulNr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "addrI", "apply", "mul0r", "mulrDl", "subrr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulrNN x y : (- x) * (- y) = x * y.	Proof. by rewrite mulrN mulNr opprK. Qed.	Lemma	mulrNN	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mulNr", "mulrN", "opprK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulN1r x : -1 * x = - x.	Proof. by rewrite mulNr mul1r. Qed.	Lemma	mulN1r	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mul1r", "mulNr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulrN1 x : x * -1 = - x.	Proof. by rewrite mulrN mulr1. Qed.	Lemma	mulrN1	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mulr1", "mulrN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulrBl x y z : (y - z) * x = y * x - z * x.	Proof. by rewrite mulrDl mulNr. Qed.	Lemma	mulrBl	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mulNr", "mulrDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulrBr x y z : x * (y - z) = x * y - x * z.	Proof. by rewrite mulrDr mulrN. Qed.	Lemma	mulrBr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mulrDr", "mulrN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
natrB m n : n <= m -> (m - n)%:R = m%:R - n%:R :> R.	Proof. exact: mulrnBr. Qed.	Lemma	natrB	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mulrnBr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commrN x y : comm x y -> comm x (- y).	Proof. by move=> com_xy; rewrite /comm mulrN com_xy mulNr. Qed.	Lemma	commrN	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "comm", "mulNr", "mulrN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commrN1 x : comm x (-1).	Proof. exact/commrN/commr1. Qed.	Lemma	commrN1	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "comm", "commr1", "commrN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commrB x y z : comm x y -> comm x z -> comm x (y - z).	Proof. by move=> com_xy com_xz; apply: commrD => //; apply: commrN. Qed.	Lemma	commrB	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "comm", "commrD", "commrN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commr_sign x n : comm x ((-1) ^+ n).	Proof. exact: (commrX n (commrN1 x)). Qed.	Lemma	commr_sign	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "comm", "commrN1", "commrX" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
signr_odd n : (-1) ^+ (odd n) = (-1) ^+ n :> R.	Proof. elim: n => //= n IHn; rewrite exprS -{}IHn. by case/odd: n; rewrite !mulN1r ?opprK. Qed.	Lemma	signr_odd	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "exprS", "mulN1r", "odd", "opprK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulr_sign (b : bool) x : (-1) ^+ b * x = (if b then - x else x).	Proof. by case: b; rewrite ?mulNr mul1r. Qed.	Lemma	mulr_sign	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mul1r", "mulNr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
signr_addb b1 b2 : (-1) ^+ (b1 (+) b2) = (-1) ^+ b1 * (-1) ^+ b2 :> R.	Proof. by rewrite mulr_sign; case: b1 b2 => [] []; rewrite ?opprK. Qed.	Lemma	signr_addb	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mulr_sign", "opprK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
signrE (b : bool) : (-1) ^+ b = 1 - b.*2%:R :> R.	Proof. by case: b; rewrite ?subr0 // opprD addNKr. Qed.	Lemma	signrE	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "addNKr", "opprD", "subr0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
signrN b : (-1) ^+ (~~ b) = - (-1) ^+ b :> R.	Proof. by case: b; rewrite ?opprK. Qed.	Lemma	signrN	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "opprK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulr_signM (b1 b2 : bool) x1 x2 : ((-1) ^+ b1 * x1) * ((-1) ^+ b2 * x2) = (-1) ^+ (b1 (+) b2) * (x1 * x2).	Proof. by rewrite signr_addb -!mulrA; congr (_ * _); rewrite !mulrA commr_sign. Qed.	Lemma	mulr_signM	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "commr_sign", "mulrA", "signr_addb" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exprNn x n : (- x) ^+ n = (-1) ^+ n * x ^+ n :> R.	Proof. by rewrite -mulN1r exprMn_comm // /comm mulN1r mulrN mulr1. Qed.	Lemma	exprNn	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "comm", "exprMn_comm", "mulN1r", "mulr1", "mulrN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sqrrN x : (- x) ^+ 2 = x ^+ 2.	Proof. exact: mulrNN. Qed.	Lemma	sqrrN	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mulrNN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sqrr_sign n : ((-1) ^+ n) ^+ 2 = 1 :> R.	Proof. by rewrite exprAC sqrrN !expr1n. Qed.	Lemma	sqrr_sign	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "expr1n", "exprAC", "sqrrN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
signrMK n : @involutive R ( *%R ((-1) ^+ n)).	Proof. by move=> x; rewrite mulrA -expr2 sqrr_sign mul1r. Qed.	Lemma	signrMK	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "expr2", "mul1r", "mulrA", "sqrr_sign" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulrI0_lreg x : (forall y, x * y = 0 -> y = 0) -> lreg x.	Proof. move=> reg_x y z eq_xy_xz; apply/eqP; rewrite -subr_eq0 [y - z]reg_x //. by rewrite mulrBr eq_xy_xz subrr. Qed.	Lemma	mulrI0_lreg	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "lreg", "mulrBr", "subr_eq0", "subrr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lregN x : lreg x -> lreg (- x).	Proof. by move=> reg_x y z; rewrite !mulNr => /oppr_inj/reg_x. Qed.	Lemma	lregN	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "lreg", "mulNr", "oppr_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lreg_sign n : lreg ((-1) ^+ n : R).	Proof. exact/lregX/lregN/lreg1. Qed.	Lemma	lreg_sign	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "lreg", "lreg1", "lregN", "lregX" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodrN (I : finType) (A : pred I) (F : I -> R) : \prod_(i in A) - F i = (- 1) ^+ #\|A\| * \prod_(i in A) F i.	Proof. rewrite -sum1_card; elim/big_rec3: _ => [\|i x n _ _ ->]; first by rewrite mulr1. by rewrite exprS !mulrA mulN1r !mulNr commrX //; apply: commrN1. Qed.	Lemma	prodrN	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "big_rec3", "commrN1", "commrX", "exprS", "mulN1r", "mulNr", "mulr1", "mulrA", "sum1_card" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exprBn_comm x y n (cxy : comm x y) : (x - y) ^+ n = \sum_(i < n.+1) ((-1) ^+ i * x ^+ (n - i) * y ^+ i) *+ 'C(n, i).	Proof. rewrite exprDn_comm; first exact: commrN. by apply: eq_bigr => i _; congr (_ *+ _); rewrite -commr_sign -mulrA -exprNn. Qed.	Lemma	exprBn_comm	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "comm", "commrN", "commr_sign", "eq_bigr", "exprDn_comm", "exprNn", "mulrA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subrXX_comm x y n (cxy : comm x y) : x ^+ n - y ^+ n = (x - y) * (\sum_(i < n) x ^+ (n.-1 - i) * y ^+ i).	Proof. case: n => [\|n]; first by rewrite big_ord0 mulr0 subrr. rewrite mulrBl !big_distrr big_ord_recl big_ord_recr /= subnn mulr1 mul1r. rewrite subn0 -!exprS opprD -!addrA; congr (_ + _); rewrite addrA -sumrB. rewrite big1 ?add0r // => i _; rewrite !mulrA -exprS -subSn ?(valP i) //. by rewrite subSS (commrX _ (commr_...	Lemma	subrXX_comm	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "add0r", "addrA", "big1", "big_distrr", "big_ord0", "big_ord_recl", "big_ord_recr", "comm", "commrX", "commr_sym", "exprS", "mul1r", "mulr0", "mulr1", "mulrA", "mulrBl", "opprD", "subSS", "subSn", "subn0", "subnn", "subrr", "sumrB", "valP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subrX1 x n : x ^+ n - 1 = (x - 1) * (\sum_(i < n) x ^+ i).	Proof. rewrite -!(opprB 1) mulNr -{1}(expr1n _ n). rewrite (subrXX_comm _ (commr_sym (commr1 x))); congr (- (_ * _)). by apply: eq_bigr => i _; rewrite expr1n mul1r. Qed.	Lemma	subrX1	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "commr1", "commr_sym", "eq_bigr", "expr1n", "mul1r", "mulNr", "opprB", "subrXX_comm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sqrrB1 x : (x - 1) ^+ 2 = x ^+ 2 - x *+ 2 + 1.	Proof. by rewrite -sqrrN opprB addrC sqrrD1 sqrrN mulNrn. Qed.	Lemma	sqrrB1	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "addrC", "mulNrn", "opprB", "sqrrD1", "sqrrN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subr_sqr_1 x : x ^+ 2 - 1 = (x - 1) * (x + 1).	Proof. by rewrite subrX1 !big_ord_recr big_ord0 /= addrAC add0r. Qed.	Lemma	subr_sqr_1	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "add0r", "addrAC", "big_ord0", "big_ord_recr", "subrX1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
smulr_closed	:= -1 \in S /\ mulr_2closed S.	Definition	smulr_closed	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mulr_2closed" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subring_closed	:= [/\ 1 \in S, subr_closed S & mulr_2closed S].	Definition	subring_closed	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mulr_2closed", "subr_closed" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
smulr_closedM : smulr_closed -> mulr_closed S.	Proof. by case=> SN1 SM; split=> //; rewrite -[1]mulr1 -mulrNN SM. Qed.	Lemma	smulr_closedM	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mulr1", "mulrNN", "mulr_closed", "smulr_closed", "split" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
smulr_closedN : smulr_closed -> oppr_closed S.	Proof. by case=> SN1 SM x Sx; rewrite -mulN1r SM. Qed.	Lemma	smulr_closedN	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mulN1r", "oppr_closed", "smulr_closed" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subring_closedB : subring_closed -> zmod_closed S.	Proof. by case=> S1 SB _; split; rewrite // -(subrr 1) SB. Qed.	Lemma	subring_closedB	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "S1", "split", "subring_closed", "subrr", "zmod_closed" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subring_closedM : subring_closed -> smulr_closed.	Proof. by case=> S1 SB SM; split; rewrite ?(zmod_closedN (subring_closedB _)). Qed.	Lemma	subring_closedM	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "S1", "smulr_closed", "split", "subring_closed", "subring_closedB", "zmod_closedN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subring_closed_semi : subring_closed -> semiring_closed S.	Proof. by move=> ringS; split; [apply/zmod_closed0D/subring_closedB \| case: ringS]. Qed.	Lemma	subring_closed_semi	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "semiring_closed", "split", "subring_closed", "subring_closedB", "zmod_closed0D" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
signr_eq0 n : ((-1) ^+ n == 0 :> R) = false.	Proof. by rewrite -signr_odd; case: odd; rewrite ?oppr_eq0 oner_eq0. Qed.	Lemma	signr_eq0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "odd", "oner_eq0", "oppr_eq0", "signr_odd" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pFrobenius_autN x : (- x)^f = - x^f.	Proof. apply/eqP; rewrite -subr_eq0 opprK addrC. by rewrite -(pFrobenius_autD_comm _ (commrN _)) // subrr pFrobenius_aut0. Qed.	Lemma	pFrobenius_autN	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "addrC", "apply", "commrN", "opprK", "pFrobenius_aut0", "pFrobenius_autD_comm", "subr_eq0", "subrr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pFrobenius_autB_comm x y : comm x y -> (x - y)^f = x^f - y^f.	Proof. by move/commrN/pFrobenius_autD_comm->; rewrite pFrobenius_autN. Qed.	Lemma	pFrobenius_autB_comm	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "comm", "commrN", "pFrobenius_autD_comm", "pFrobenius_autN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exprNn_pchar x n : (pchar R).-nat n -> (- x) ^+ n = - (x ^+ n).	Proof. pose p := pdiv n; have [\|n_gt1 pcharRn] := leqP n 1; first by case: (n) => [\|[]]. have pcharRp: p \in pchar R by rewrite (pnatPpi pcharRn) // pi_pdiv. have /p_natP[e ->]: p.-nat n by rewrite -(eq_pnat _ (pcharf_eq pcharRp)). elim: e => // e IHe; rewrite expnSr !exprM {}IHe. by rewrite -pFrobenius_autE pFrobenius...	Lemma	exprNn_pchar	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "eq_pnat", "expnSr", "exprM", "leqP", "nat", "pFrobenius_autE", "pFrobenius_autN", "p_natP", "pchar", "pcharRp", "pcharf_eq", "pdiv", "pi_pdiv", "pnatPpi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
oppr_pchar2 x : - x = x.	Proof. by apply/esym/eqP; rewrite -addr_eq0 addrr_pchar2. Qed.	Lemma	oppr_pchar2	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "addr_eq0", "addrr_pchar2", "apply" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subr_pchar2 x y : x - y = x + y.	Proof. by rewrite oppr_pchar2. Qed.	Lemma	subr_pchar2	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "oppr_pchar2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addrK_pchar2 x : involutive (+%R^~ x).	Proof. by move=> y; rewrite /= -subr_pchar2 addrK. Qed.	Lemma	addrK_pchar2	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "addrK", "subr_pchar2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addKr_pchar2 x : involutive (+%R x).	Proof. by move=> y; rewrite -{1}[x]oppr_pchar2 addKr. Qed.	Lemma	addKr_pchar2	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "addKr", "oppr_pchar2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rev_prodr (R : pzSemiRingType) (I : Type) (r : seq I) (P : pred I) (E : I -> R) : \prod_(i <- r \| P i) (E i : R^c) = \prod_(i <- rev r \| P i) E i.	Proof. by rewrite rev_big_rev. Qed.	Lemma	rev_prodr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "rev", "rev_big_rev", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulIr_eq0 x y : rreg x -> (y * x == 0) = (y == 0).	Proof. exact: (@mulrI_eq0 R^c). Qed.	Lemma	mulIr_eq0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mulrI_eq0", "rreg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rreg1 : rreg (1 : R).	Proof. exact: (@lreg1 R^c). Qed.	Lemma	rreg1	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "lreg1", "rreg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rregM x y : rreg x -> rreg y -> rreg (x * y).	Proof. by move=> reg_x reg_y; apply: (@lregM R^c). Qed.	Lemma	rregM	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "apply", "lregM", "rreg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
revrX x n : (x : R^c) ^+ n = (x : R) ^+ n.	Proof. by elim: n => // n IHn; rewrite exprS exprSr IHn. Qed.	Lemma	revrX	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "exprS", "exprSr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rregX x n : rreg x -> rreg (x ^+ n).	Proof. by move/(@lregX R^c x n); rewrite revrX. Qed.	Lemma	rregX	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "lregX", "revrX", "rreg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rreg_neq0 (R : nzSemiRingType) (x : R) : rreg x -> x != 0.	Proof. exact: (@lreg_neq0 R^c). Qed.	Lemma	rreg_neq0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "lreg_neq0", "rreg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulIr0_rreg x : (forall y, y * x = 0 -> y = 0) -> rreg x.	Proof. exact: (@mulrI0_lreg R^c). Qed.	Lemma	mulIr0_rreg	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mulrI0_lreg", "rreg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rregN x : rreg x -> rreg (- x).	Proof. exact: (@lregN R^c). Qed.	Lemma	rregN	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "lregN", "rreg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"*:%R"	:= (@scale _ _) : function_scope.	Notation	*:%R	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"a *: v"	:= (scale a v) : ring_scope.	Notation	a *: v	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
scale0r v : scale 0 v = 0.	Proof. by apply: (addIr (scale 1 v)); rewrite -scalerDl !add0r. Qed.	Lemma	scale0r	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "add0r", "addIr", "apply", "scalerDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
opp : V -> V	:= scale (- 1).	Definition	opp	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addNr : left_inverse 0 opp +%R.	Proof. move=> v; suff : scale (-1 + 1) v = 0 by rewrite scalerDl scale1r. by rewrite addNr scale0r. Qed.	Lemma	addNr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "opp", "scale0r", "scale1r", "scalerDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
scaler0 a : a *: 0 = 0 :> V.	Proof. by rewrite -[0 in LHS](scale0r 0) scalerA mulr0 scale0r. Qed.	Lemma	scaler0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mulr0", "scale0r", "scalerA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
scaler_nat n v : n%:R : v = v + n.	Proof. elim: n => /= [\|n]; first by rewrite scale0r. by rewrite !mulrS scalerDl ?scale1r => ->. Qed.	Lemma	scaler_nat	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mulrS", "scale0r", "scale1r", "scalerDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
scalerMnl a v n : a : v + n = (a + n) : v.	Proof. elim: n => [\|n IHn]; first by rewrite !mulr0n scale0r. by rewrite !mulrSr IHn scalerDl. Qed.	Lemma	scalerMnl	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mulr0n", "mulrSr", "scale0r", "scalerDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
scalerMnr a v n : a : v + n = a : (v + n).	Proof. elim: n => [\|n IHn]; first by rewrite !mulr0n scaler0. by rewrite !mulrSr IHn scalerDr. Qed.	Lemma	scalerMnr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mulr0n", "mulrSr", "scaler0", "scalerDr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
scaler_suml v I r (P : pred I) F : (\sum_(i <- r \| P i) F i) : v = \sum_(i <- r \| P i) F i : v.	Proof. exact: (big_morph _ (scalerDl v) (scale0r v)). Qed.	Lemma	scaler_suml	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "big_morph", "scale0r", "scalerDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
scaler_sumr a I r (P : pred I) (F : I -> V) : a : (\sum_(i <- r \| P i) F i) = \sum_(i <- r \| P i) a : F i.	Proof. exact: big_endo (scalerDr a) (scaler0 a) I r P F. Qed.	Lemma	scaler_sumr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "big_endo", "scaler0", "scalerDr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
scaler_closed	:= forall a, {in S, forall v, a *: v \in S}.	Definition	scaler_closed	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsemimod_closed	:= nmod_closed S /\ scaler_closed.	Definition	subsemimod_closed	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "nmod_closed", "scaler_closed" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsemimod_closedD : subsemimod_closed -> nmod_closed S.	Proof. by case. Qed.	Lemma	subsemimod_closedD	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "nmod_closed", "subsemimod_closed" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsemimod_closedZ : subsemimod_closed -> scaler_closed.	Proof. by case. Qed.	Lemma	subsemimod_closedZ	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "scaler_closed", "subsemimod_closed" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
scaleNr a v : - a : v = - (a : v).	Proof. by apply: (addIr (a *: v)); rewrite -scalerDl !addNr scale0r. Qed.	Lemma	scaleNr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "addIr", "addNr", "apply", "scale0r", "scalerDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
scaleN1r v : - 1 *: v = - v.	Proof. by rewrite scaleNr scale1r. Qed.	Lemma	scaleN1r	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "scale1r", "scaleNr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
scalerN a v : a : - v = - (a : v).	Proof. by apply: (addIr (a *: v)); rewrite -scalerDr !addNr scaler0. Qed.	Lemma	scalerN	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "addIr", "addNr", "apply", "scaler0", "scalerDr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
scalerBl a b v : (a - b) : v = a : v - b *: v.	Proof. by rewrite scalerDl scaleNr. Qed.	Lemma	scalerBl	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "scaleNr", "scalerDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
scalerBr a u v : a : (u - v) = a : u - a *: v.	Proof. by rewrite scalerDr scalerN. Qed.	Lemma	scalerBr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "scalerDr", "scalerN" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
scaler_sign (b : bool) v : (-1) ^+ b *: v = (if b then - v else v).	Proof. by case: b; rewrite ?scaleNr scale1r. Qed.	Lemma	scaler_sign	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "scale1r", "scaleNr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
signrZK n : @involutive V ( *:%R ((-1) ^+ n)).	Proof. by move=> u; rewrite scalerA -expr2 sqrr_sign scale1r. Qed.	Lemma	signrZK	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "expr2", "scale1r", "scalerA", "sqrr_sign" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
linear_closed	:= forall a, {in S &, forall u v, a *: u + v \in S}.	Definition	linear_closed	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
submod_closed	:= 0 \in S /\ linear_closed.	Definition	submod_closed	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "linear_closed" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
linear_closedB : linear_closed -> subr_closed S.	Proof. by move=> Slin u v Su Sv; rewrite addrC -scaleN1r Slin. Qed.	Lemma	linear_closedB	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "Sv", "addrC", "linear_closed", "scaleN1r", "subr_closed" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
submod_closedB : submod_closed -> zmod_closed S.	Proof. by case=> S0 /linear_closedB. Qed.	Lemma	submod_closedB	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "S0", "linear_closedB", "submod_closed", "zmod_closed" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
submod_closed_semi : submod_closed -> subsemimod_closed S.	Proof. move=> /[dup] /submod_closedB /zmod_closed0D SD [S0 Slin]; split => // a v Sv. by rewrite -[a *: v]addr0 Slin. Qed.	Lemma	submod_closed_semi	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "S0", "Sv", "addr0", "split", "submod_closed", "submod_closedB", "subsemimod_closed", "zmod_closed0D" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsemimod_closed_submod : subsemimod_closed S -> submod_closed.	Proof. by case=> [[S0 SD] SZ]; split => // a u v Su Sv; apply/SD/Sv/SZ. Qed.	Lemma	subsemimod_closed_submod	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "S0", "Sv", "apply", "split", "submod_closed", "subsemimod_closed" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsemimod_closedB : subsemimod_closed S -> zmod_closed S.	Proof. by move/subsemimod_closed_submod/submod_closedB. Qed.	Lemma	subsemimod_closedB	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "submod_closedB", "subsemimod_closed", "subsemimod_closed_submod", "zmod_closed" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"k %:A"	:= (k *: 1) : ring_scope.	Notation	k %:A	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]	Scalar injection (see the definition of in_alg A below).	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
regular R : Type	:= R.	Definition	regular	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]	Regular ring algebra tag.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"R ^o"	:= (regular R) : type_scope.	Notation	R ^o	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "regular" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulr_algl (a : R) (x : A) : (a : 1) x = a *: x.	Proof. by rewrite -scalerAl mul1r. Qed.	Lemma	mulr_algl	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mul1r", "scalerAl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsemialg_closed	:= [/\ 1 \in S, nmod_closed S, scaler_closed S & mulr_2closed S].	Definition	subsemialg_closed	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "mulr_2closed", "nmod_closed", "scaler_closed" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsemialg_closedZ : subsemialg_closed -> subsemimod_closed S.	Proof. by case. Qed.	Lemma	subsemialg_closedZ	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "subsemialg_closed", "subsemimod_closed" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsemialg_closedM : subsemialg_closed -> semiring_closed S.	Proof. by case. Qed.	Lemma	subsemialg_closedM	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "semiring_closed", "subsemialg_closed" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subalg_closed	:= [/\ 1 \in S, linear_closed S & mulr_2closed S].	Definition	subalg_closed	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "linear_closed", "mulr_2closed" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subalg_closedZ : subalg_closed -> submod_closed S.	Proof. by case=> S1 Slin _; split; rewrite // -(subrr 1) linear_closedB. Qed.	Lemma	subalg_closedZ	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "S1", "linear_closedB", "split", "subalg_closed", "submod_closed", "subrr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subalg_closedBM : subalg_closed -> subring_closed S.	Proof. by case=> S1 Slin SM; split=> //; apply: linear_closedB. Qed.	Lemma	subalg_closedBM	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "S1", "apply", "linear_closedB", "split", "subalg_closed", "subring_closed" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subalg_closed_semi : subalg_closed -> subsemialg_closed S.	Proof. move=> /[dup] /subalg_closedZ /submod_closedB /zmod_closed0D. by move=> [S0 SD] [S1 Slin SM]; split => // a u Su; rewrite -[a *: u]addr0 Slin. Qed.	Lemma	subalg_closed_semi	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "S0", "S1", "addr0", "split", "subalg_closed", "subalg_closedZ", "submod_closedB", "subsemialg_closed", "zmod_closed0D" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsemialg_closed_subalg : subsemialg_closed S -> subalg_closed.	Proof. by case=> S1 [S0 SD] SZ SM; split => // a u v Su Sv; apply/SD/Sv/SZ. Qed.	Lemma	subsemialg_closed_subalg	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "S0", "S1", "Sv", "apply", "split", "subalg_closed", "subsemialg_closed" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
subsemialg_closedBM : subsemialg_closed S -> subring_closed S.	Proof. by move/subsemialg_closed_subalg/subalg_closedBM. Qed.	Lemma	subsemialg_closedBM	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[ "subalg_closedBM", "subring_closed", "subsemialg_closed", "subsemialg_closed_subalg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mull_fun a f x	:= a * f x.	Definition	mull_fun	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulr_fun a f x	:= f x * a.	Definition	mulr_fun	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/rings_modules_and_algebras.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

"- 1"

:= (- (1)) : ring_scope.

Notation

- 1

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulrN x y : x * (- y) = - (x * y).

Proof. by apply: (addrI (x * y)); rewrite -mulrDr !subrr mulr0. Qed.

Lemma

mulrN

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "addrI", "apply", "mulr0", "mulrDr", "subrr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulNr x y : (- x) * y = - (x * y).

Proof. by apply: (addrI (x * y)); rewrite -mulrDl !subrr mul0r. Qed.

Lemma

mulNr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "addrI", "apply", "mul0r", "mulrDl", "subrr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulrNN x y : (- x) * (- y) = x * y.

Proof. by rewrite mulrN mulNr opprK. Qed.

Lemma

mulrNN

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mulNr", "mulrN", "opprK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulN1r x : -1 * x = - x.

Proof. by rewrite mulNr mul1r. Qed.

Lemma

mulN1r

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mul1r", "mulNr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulrN1 x : x * -1 = - x.

Proof. by rewrite mulrN mulr1. Qed.

Lemma

mulrN1

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mulr1", "mulrN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulrBl x y z : (y - z) * x = y * x - z * x.

Proof. by rewrite mulrDl mulNr. Qed.

Lemma

mulrBl

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mulNr", "mulrDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulrBr x y z : x * (y - z) = x * y - x * z.

Proof. by rewrite mulrDr mulrN. Qed.

Lemma

mulrBr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mulrDr", "mulrN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

natrB m n : n <= m -> (m - n)%:R = m%:R - n%:R :> R.

Proof. exact: mulrnBr. Qed.

Lemma

natrB

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mulrnBr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commrN x y : comm x y -> comm x (- y).

Proof. by move=> com_xy; rewrite /comm mulrN com_xy mulNr. Qed.

Lemma

commrN

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "comm", "mulNr", "mulrN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commrN1 x : comm x (-1).

Proof. exact/commrN/commr1. Qed.

Lemma

commrN1

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "comm", "commr1", "commrN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commrB x y z : comm x y -> comm x z -> comm x (y - z).

Proof. by move=> com_xy com_xz; apply: commrD => //; apply: commrN. Qed.

Lemma

commrB

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "comm", "commrD", "commrN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commr_sign x n : comm x ((-1) ^+ n).

Proof. exact: (commrX n (commrN1 x)). Qed.

Lemma

commr_sign

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "comm", "commrN1", "commrX" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

signr_odd n : (-1) ^+ (odd n) = (-1) ^+ n :> R.

Proof. elim: n => //= n IHn; rewrite exprS -{}IHn. by case/odd: n; rewrite !mulN1r ?opprK. Qed.

Lemma

signr_odd

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "exprS", "mulN1r", "odd", "opprK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulr_sign (b : bool) x : (-1) ^+ b * x = (if b then - x else x).

Proof. by case: b; rewrite ?mulNr mul1r. Qed.

Lemma

mulr_sign

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mul1r", "mulNr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

signr_addb b1 b2 : (-1) ^+ (b1 (+) b2) = (-1) ^+ b1 * (-1) ^+ b2 :> R.

Proof. by rewrite mulr_sign; case: b1 b2 => [] []; rewrite ?opprK. Qed.

Lemma

signr_addb

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mulr_sign", "opprK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

signrE (b : bool) : (-1) ^+ b = 1 - b.*2%:R :> R.

Proof. by case: b; rewrite ?subr0 // opprD addNKr. Qed.

Lemma

signrE

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "addNKr", "opprD", "subr0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

signrN b : (-1) ^+ (~~ b) = - (-1) ^+ b :> R.

Proof. by case: b; rewrite ?opprK. Qed.

Lemma

signrN

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "opprK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulr_signM (b1 b2 : bool) x1 x2 : ((-1) ^+ b1 * x1) * ((-1) ^+ b2 * x2) = (-1) ^+ (b1 (+) b2) * (x1 * x2).

Proof. by rewrite signr_addb -!mulrA; congr (_ * _); rewrite !mulrA commr_sign. Qed.

Lemma

mulr_signM

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "commr_sign", "mulrA", "signr_addb" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exprNn x n : (- x) ^+ n = (-1) ^+ n * x ^+ n :> R.

Proof. by rewrite -mulN1r exprMn_comm // /comm mulN1r mulrN mulr1. Qed.

Lemma

exprNn

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "comm", "exprMn_comm", "mulN1r", "mulr1", "mulrN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sqrrN x : (- x) ^+ 2 = x ^+ 2.

Proof. exact: mulrNN. Qed.

Lemma

sqrrN

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mulrNN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sqrr_sign n : ((-1) ^+ n) ^+ 2 = 1 :> R.

Proof. by rewrite exprAC sqrrN !expr1n. Qed.

Lemma

sqrr_sign

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "expr1n", "exprAC", "sqrrN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

signrMK n : @involutive R ( *%R ((-1) ^+ n)).

Proof. by move=> x; rewrite mulrA -expr2 sqrr_sign mul1r. Qed.

Lemma

signrMK

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "expr2", "mul1r", "mulrA", "sqrr_sign" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulrI0_lreg x : (forall y, x * y = 0 -> y = 0) -> lreg x.

Proof. move=> reg_x y z eq_xy_xz; apply/eqP; rewrite -subr_eq0 [y - z]reg_x //. by rewrite mulrBr eq_xy_xz subrr. Qed.

Lemma

mulrI0_lreg

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "lreg", "mulrBr", "subr_eq0", "subrr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lregN x : lreg x -> lreg (- x).

Proof. by move=> reg_x y z; rewrite !mulNr => /oppr_inj/reg_x. Qed.

Lemma

lregN

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "lreg", "mulNr", "oppr_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lreg_sign n : lreg ((-1) ^+ n : R).

Proof. exact/lregX/lregN/lreg1. Qed.

Lemma

lreg_sign

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "lreg", "lreg1", "lregN", "lregX" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodrN (I : finType) (A : pred I) (F : I -> R) : \prod_(i in A) - F i = (- 1) ^+ #|A| * \prod_(i in A) F i.

Proof. rewrite -sum1_card; elim/big_rec3: _ => [|i x n _ _ ->]; first by rewrite mulr1. by rewrite exprS !mulrA mulN1r !mulNr commrX //; apply: commrN1. Qed.

Lemma

prodrN

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "big_rec3", "commrN1", "commrX", "exprS", "mulN1r", "mulNr", "mulr1", "mulrA", "sum1_card" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exprBn_comm x y n (cxy : comm x y) : (x - y) ^+ n = \sum_(i < n.+1) ((-1) ^+ i * x ^+ (n - i) * y ^+ i) *+ 'C(n, i).

Proof. rewrite exprDn_comm; first exact: commrN. by apply: eq_bigr => i _; congr (_ *+ _); rewrite -commr_sign -mulrA -exprNn. Qed.

Lemma

exprBn_comm

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "comm", "commrN", "commr_sign", "eq_bigr", "exprDn_comm", "exprNn", "mulrA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subrXX_comm x y n (cxy : comm x y) : x ^+ n - y ^+ n = (x - y) * (\sum_(i < n) x ^+ (n.-1 - i) * y ^+ i).

Proof. case: n => [|n]; first by rewrite big_ord0 mulr0 subrr. rewrite mulrBl !big_distrr big_ord_recl big_ord_recr /= subnn mulr1 mul1r. rewrite subn0 -!exprS opprD -!addrA; congr (_ + _); rewrite addrA -sumrB. rewrite big1 ?add0r // => i _; rewrite !mulrA -exprS -subSn ?(valP i) //. by rewrite subSS (commrX _ (commr_...

Lemma

subrXX_comm

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "add0r", "addrA", "big1", "big_distrr", "big_ord0", "big_ord_recl", "big_ord_recr", "comm", "commrX", "commr_sym", "exprS", "mul1r", "mulr0", "mulr1", "mulrA", "mulrBl", "opprD", "subSS", "subSn", "subn0", "subnn", "subrr", "sumrB", "valP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subrX1 x n : x ^+ n - 1 = (x - 1) * (\sum_(i < n) x ^+ i).

Proof. rewrite -!(opprB 1) mulNr -{1}(expr1n _ n). rewrite (subrXX_comm _ (commr_sym (commr1 x))); congr (- (_ * _)). by apply: eq_bigr => i _; rewrite expr1n mul1r. Qed.

Lemma

subrX1

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "commr1", "commr_sym", "eq_bigr", "expr1n", "mul1r", "mulNr", "opprB", "subrXX_comm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sqrrB1 x : (x - 1) ^+ 2 = x ^+ 2 - x *+ 2 + 1.

Proof. by rewrite -sqrrN opprB addrC sqrrD1 sqrrN mulNrn. Qed.

Lemma

sqrrB1

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "addrC", "mulNrn", "opprB", "sqrrD1", "sqrrN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subr_sqr_1 x : x ^+ 2 - 1 = (x - 1) * (x + 1).

Proof. by rewrite subrX1 !big_ord_recr big_ord0 /= addrAC add0r. Qed.

Lemma

subr_sqr_1

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "add0r", "addrAC", "big_ord0", "big_ord_recr", "subrX1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

smulr_closed

:= -1 \in S /\ mulr_2closed S.

Definition

smulr_closed

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mulr_2closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subring_closed

:= [/\ 1 \in S, subr_closed S & mulr_2closed S].

Definition

subring_closed

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mulr_2closed", "subr_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

smulr_closedM : smulr_closed -> mulr_closed S.

Proof. by case=> SN1 SM; split=> //; rewrite -[1]mulr1 -mulrNN SM. Qed.

Lemma

smulr_closedM

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mulr1", "mulrNN", "mulr_closed", "smulr_closed", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

smulr_closedN : smulr_closed -> oppr_closed S.

Proof. by case=> SN1 SM x Sx; rewrite -mulN1r SM. Qed.

Lemma

smulr_closedN

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mulN1r", "oppr_closed", "smulr_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subring_closedB : subring_closed -> zmod_closed S.

Proof. by case=> S1 SB _; split; rewrite // -(subrr 1) SB. Qed.

Lemma

subring_closedB

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "S1", "split", "subring_closed", "subrr", "zmod_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subring_closedM : subring_closed -> smulr_closed.

Proof. by case=> S1 SB SM; split; rewrite ?(zmod_closedN (subring_closedB _)). Qed.

Lemma

subring_closedM

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "S1", "smulr_closed", "split", "subring_closed", "subring_closedB", "zmod_closedN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subring_closed_semi : subring_closed -> semiring_closed S.

Proof. by move=> ringS; split; [apply/zmod_closed0D/subring_closedB | case: ringS]. Qed.

Lemma

subring_closed_semi

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "semiring_closed", "split", "subring_closed", "subring_closedB", "zmod_closed0D" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

signr_eq0 n : ((-1) ^+ n == 0 :> R) = false.

Proof. by rewrite -signr_odd; case: odd; rewrite ?oppr_eq0 oner_eq0. Qed.

Lemma

signr_eq0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "odd", "oner_eq0", "oppr_eq0", "signr_odd" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pFrobenius_autN x : (- x)^f = - x^f.

Proof. apply/eqP; rewrite -subr_eq0 opprK addrC. by rewrite -(pFrobenius_autD_comm _ (commrN _)) // subrr pFrobenius_aut0. Qed.

Lemma

pFrobenius_autN

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "addrC", "apply", "commrN", "opprK", "pFrobenius_aut0", "pFrobenius_autD_comm", "subr_eq0", "subrr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pFrobenius_autB_comm x y : comm x y -> (x - y)^f = x^f - y^f.

Proof. by move/commrN/pFrobenius_autD_comm->; rewrite pFrobenius_autN. Qed.

Lemma

pFrobenius_autB_comm

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "comm", "commrN", "pFrobenius_autD_comm", "pFrobenius_autN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exprNn_pchar x n : (pchar R).-nat n -> (- x) ^+ n = - (x ^+ n).

Proof. pose p := pdiv n; have [|n_gt1 pcharRn] := leqP n 1; first by case: (n) => [|[]]. have pcharRp: p \in pchar R by rewrite (pnatPpi pcharRn) // pi_pdiv. have /p_natP[e ->]: p.-nat n by rewrite -(eq_pnat _ (pcharf_eq pcharRp)). elim: e => // e IHe; rewrite expnSr !exprM {}IHe. by rewrite -pFrobenius_autE pFrobenius...

Lemma

exprNn_pchar

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "eq_pnat", "expnSr", "exprM", "leqP", "nat", "pFrobenius_autE", "pFrobenius_autN", "p_natP", "pchar", "pcharRp", "pcharf_eq", "pdiv", "pi_pdiv", "pnatPpi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

oppr_pchar2 x : - x = x.

Proof. by apply/esym/eqP; rewrite -addr_eq0 addrr_pchar2. Qed.

Lemma

oppr_pchar2

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "addr_eq0", "addrr_pchar2", "apply" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subr_pchar2 x y : x - y = x + y.

Proof. by rewrite oppr_pchar2. Qed.

Lemma

subr_pchar2

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "oppr_pchar2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addrK_pchar2 x : involutive (+%R^~ x).

Proof. by move=> y; rewrite /= -subr_pchar2 addrK. Qed.

Lemma

addrK_pchar2

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "addrK", "subr_pchar2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addKr_pchar2 x : involutive (+%R x).

Proof. by move=> y; rewrite -{1}[x]oppr_pchar2 addKr. Qed.

Lemma

addKr_pchar2

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "addKr", "oppr_pchar2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rev_prodr (R : pzSemiRingType) (I : Type) (r : seq I) (P : pred I) (E : I -> R) : \prod_(i <- r | P i) (E i : R^c) = \prod_(i <- rev r | P i) E i.

Proof. by rewrite rev_big_rev. Qed.

Lemma

rev_prodr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "rev", "rev_big_rev", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulIr_eq0 x y : rreg x -> (y * x == 0) = (y == 0).

Proof. exact: (@mulrI_eq0 R^c). Qed.

Lemma

mulIr_eq0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mulrI_eq0", "rreg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rreg1 : rreg (1 : R).

Proof. exact: (@lreg1 R^c). Qed.

Lemma

rreg1

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "lreg1", "rreg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rregM x y : rreg x -> rreg y -> rreg (x * y).

Proof. by move=> reg_x reg_y; apply: (@lregM R^c). Qed.

Lemma

rregM

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "apply", "lregM", "rreg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

revrX x n : (x : R^c) ^+ n = (x : R) ^+ n.

Proof. by elim: n => // n IHn; rewrite exprS exprSr IHn. Qed.

Lemma

revrX

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "exprS", "exprSr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rregX x n : rreg x -> rreg (x ^+ n).

Proof. by move/(@lregX R^c x n); rewrite revrX. Qed.

Lemma

rregX

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "lregX", "revrX", "rreg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rreg_neq0 (R : nzSemiRingType) (x : R) : rreg x -> x != 0.

Proof. exact: (@lreg_neq0 R^c). Qed.

Lemma

rreg_neq0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "lreg_neq0", "rreg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulIr0_rreg x : (forall y, y * x = 0 -> y = 0) -> rreg x.

Proof. exact: (@mulrI0_lreg R^c). Qed.

Lemma

mulIr0_rreg

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mulrI0_lreg", "rreg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rregN x : rreg x -> rreg (- x).

Proof. exact: (@lregN R^c). Qed.

Lemma

rregN

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "lregN", "rreg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"*:%R"

:= (@scale _ _) : function_scope.

Notation

*:%R

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"a *: v"

:= (scale a v) : ring_scope.

Notation

a *: v

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

scale0r v : scale 0 v = 0.

Proof. by apply: (addIr (scale 1 v)); rewrite -scalerDl !add0r. Qed.

Lemma

scale0r

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "add0r", "addIr", "apply", "scalerDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

opp : V -> V

:= scale (- 1).

Definition

opp

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addNr : left_inverse 0 opp +%R.

Proof. move=> v; suff : scale (-1 + 1) v = 0 by rewrite scalerDl scale1r. by rewrite addNr scale0r. Qed.

Lemma

addNr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "opp", "scale0r", "scale1r", "scalerDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

scaler0 a : a *: 0 = 0 :> V.

Proof. by rewrite -[0 in LHS](scale0r 0) scalerA mulr0 scale0r. Qed.

Lemma

scaler0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mulr0", "scale0r", "scalerA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

scaler_nat n v : n%:R *: v = v *+ n.

Proof. elim: n => /= [|n]; first by rewrite scale0r. by rewrite !mulrS scalerDl ?scale1r => ->. Qed.

Lemma

scaler_nat

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mulrS", "scale0r", "scale1r", "scalerDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

scalerMnl a v n : a *: v *+ n = (a *+ n) *: v.

Proof. elim: n => [|n IHn]; first by rewrite !mulr0n scale0r. by rewrite !mulrSr IHn scalerDl. Qed.

Lemma

scalerMnl

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mulr0n", "mulrSr", "scale0r", "scalerDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

scalerMnr a v n : a *: v *+ n = a *: (v *+ n).

Proof. elim: n => [|n IHn]; first by rewrite !mulr0n scaler0. by rewrite !mulrSr IHn scalerDr. Qed.

Lemma

scalerMnr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mulr0n", "mulrSr", "scaler0", "scalerDr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

scaler_suml v I r (P : pred I) F : (\sum_(i <- r | P i) F i) *: v = \sum_(i <- r | P i) F i *: v.

Proof. exact: (big_morph _ (scalerDl v) (scale0r v)). Qed.

Lemma

scaler_suml

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "big_morph", "scale0r", "scalerDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

scaler_sumr a I r (P : pred I) (F : I -> V) : a *: (\sum_(i <- r | P i) F i) = \sum_(i <- r | P i) a *: F i.

Proof. exact: big_endo (scalerDr a) (scaler0 a) I r P F. Qed.

Lemma

scaler_sumr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "big_endo", "scaler0", "scalerDr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

scaler_closed

:= forall a, {in S, forall v, a *: v \in S}.

Definition

scaler_closed

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsemimod_closed

:= nmod_closed S /\ scaler_closed.

Definition

subsemimod_closed

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "nmod_closed", "scaler_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsemimod_closedD : subsemimod_closed -> nmod_closed S.

Proof. by case. Qed.

Lemma

subsemimod_closedD

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "nmod_closed", "subsemimod_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsemimod_closedZ : subsemimod_closed -> scaler_closed.

Proof. by case. Qed.

Lemma

subsemimod_closedZ

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "scaler_closed", "subsemimod_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

scaleNr a v : - a *: v = - (a *: v).

Proof. by apply: (addIr (a *: v)); rewrite -scalerDl !addNr scale0r. Qed.

Lemma

scaleNr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "addIr", "addNr", "apply", "scale0r", "scalerDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

scaleN1r v : - 1 *: v = - v.

Proof. by rewrite scaleNr scale1r. Qed.

Lemma

scaleN1r

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "scale1r", "scaleNr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

scalerN a v : a *: - v = - (a *: v).

Proof. by apply: (addIr (a *: v)); rewrite -scalerDr !addNr scaler0. Qed.

Lemma

scalerN

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "addIr", "addNr", "apply", "scaler0", "scalerDr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

scalerBl a b v : (a - b) *: v = a *: v - b *: v.

Proof. by rewrite scalerDl scaleNr. Qed.

Lemma

scalerBl

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "scaleNr", "scalerDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

scalerBr a u v : a *: (u - v) = a *: u - a *: v.

Proof. by rewrite scalerDr scalerN. Qed.

Lemma

scalerBr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "scalerDr", "scalerN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

scaler_sign (b : bool) v : (-1) ^+ b *: v = (if b then - v else v).

Proof. by case: b; rewrite ?scaleNr scale1r. Qed.

Lemma

scaler_sign

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "scale1r", "scaleNr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

signrZK n : @involutive V ( *:%R ((-1) ^+ n)).

Proof. by move=> u; rewrite scalerA -expr2 sqrr_sign scale1r. Qed.

Lemma

signrZK

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "expr2", "scale1r", "scalerA", "sqrr_sign" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

linear_closed

:= forall a, {in S &, forall u v, a *: u + v \in S}.

Definition

linear_closed

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

submod_closed

:= 0 \in S /\ linear_closed.

Definition

submod_closed

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "linear_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

linear_closedB : linear_closed -> subr_closed S.

Proof. by move=> Slin u v Su Sv; rewrite addrC -scaleN1r Slin. Qed.

Lemma

linear_closedB

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "Sv", "addrC", "linear_closed", "scaleN1r", "subr_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

submod_closedB : submod_closed -> zmod_closed S.

Proof. by case=> S0 /linear_closedB. Qed.

Lemma

submod_closedB

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "S0", "linear_closedB", "submod_closed", "zmod_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

submod_closed_semi : submod_closed -> subsemimod_closed S.

Proof. move=> /[dup] /submod_closedB /zmod_closed0D SD [S0 Slin]; split => // a v Sv. by rewrite -[a *: v]addr0 Slin. Qed.

Lemma

submod_closed_semi

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "S0", "Sv", "addr0", "split", "submod_closed", "submod_closedB", "subsemimod_closed", "zmod_closed0D" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsemimod_closed_submod : subsemimod_closed S -> submod_closed.

Proof. by case=> [[S0 SD] SZ]; split => // a u v Su Sv; apply/SD/Sv/SZ. Qed.

Lemma

subsemimod_closed_submod

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "S0", "Sv", "apply", "split", "submod_closed", "subsemimod_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsemimod_closedB : subsemimod_closed S -> zmod_closed S.

Proof. by move/subsemimod_closed_submod/submod_closedB. Qed.

Lemma

subsemimod_closedB

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "submod_closedB", "subsemimod_closed", "subsemimod_closed_submod", "zmod_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"k %:A"

:= (k *: 1) : ring_scope.

Notation

k %:A

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

Scalar injection (see the definition of in_alg A below).

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

regular R : Type

:= R.

Definition

regular

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

Regular ring algebra tag.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"R ^o"

:= (regular R) : type_scope.

Notation

R ^o

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "regular" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulr_algl (a : R) (x : A) : (a *: 1) * x = a *: x.

Proof. by rewrite -scalerAl mul1r. Qed.

Lemma

mulr_algl

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mul1r", "scalerAl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsemialg_closed

:= [/\ 1 \in S, nmod_closed S, scaler_closed S & mulr_2closed S].

Definition

subsemialg_closed

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "mulr_2closed", "nmod_closed", "scaler_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsemialg_closedZ : subsemialg_closed -> subsemimod_closed S.

Proof. by case. Qed.

Lemma

subsemialg_closedZ

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "subsemialg_closed", "subsemimod_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsemialg_closedM : subsemialg_closed -> semiring_closed S.

Proof. by case. Qed.

Lemma

subsemialg_closedM

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "semiring_closed", "subsemialg_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subalg_closed

:= [/\ 1 \in S, linear_closed S & mulr_2closed S].

Definition

subalg_closed

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "linear_closed", "mulr_2closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subalg_closedZ : subalg_closed -> submod_closed S.

Proof. by case=> S1 Slin _; split; rewrite // -(subrr 1) linear_closedB. Qed.

Lemma

subalg_closedZ

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "S1", "linear_closedB", "split", "subalg_closed", "submod_closed", "subrr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subalg_closedBM : subalg_closed -> subring_closed S.

Proof. by case=> S1 Slin SM; split=> //; apply: linear_closedB. Qed.

Lemma

subalg_closedBM

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "S1", "apply", "linear_closedB", "split", "subalg_closed", "subring_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subalg_closed_semi : subalg_closed -> subsemialg_closed S.

Proof. move=> /[dup] /subalg_closedZ /submod_closedB /zmod_closed0D. by move=> [S0 SD] [S1 Slin SM]; split => // a u Su; rewrite -[a *: u]addr0 Slin. Qed.

Lemma

subalg_closed_semi

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "S0", "S1", "addr0", "split", "subalg_closed", "subalg_closedZ", "submod_closedB", "subsemialg_closed", "zmod_closed0D" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsemialg_closed_subalg : subsemialg_closed S -> subalg_closed.

Proof. by case=> S1 [S0 SD] SZ SM; split => // a u v Su Sv; apply/SD/Sv/SZ. Qed.

Lemma

subsemialg_closed_subalg

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "S0", "S1", "Sv", "apply", "split", "subalg_closed", "subsemialg_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

subsemialg_closedBM : subsemialg_closed S -> subring_closed S.

Proof. by move/subsemialg_closed_subalg/subalg_closedBM. Qed.

Lemma

subsemialg_closedBM

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[ "subalg_closedBM", "subring_closed", "subsemialg_closed", "subsemialg_closed_subalg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mull_fun a f x

:= a * f x.

Definition

mull_fun

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulr_fun a f x

:= f x * a.

Definition

mulr_fun

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/rings_modules_and_algebras.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "div", "seq", "choice", "fintype", "finfun", "bigop", "prime", "binomial", "nmodule", "Algebra", "Monoid.Theory", "Scale.Exports", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d