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
"x ^+ n"	:= (exp x n) : ring_scope.	Notation	x ^+ n	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" ]	[ "exp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\prod_ ( i <- r \| P ) F"	:= (\big[*%R/1]_(i <- r \| P) F).	Notation	\prod_ ( i <- r \| P ) F	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
"\prod_ ( i \| P ) F"	:= (\big[*%R/1]_(i \| P) F).	Notation	\prod_ ( i \| P ) F	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
"\prod_ ( i 'in' A ) F"	:= (\big[*%R/1]_(i in A) F).	Notation	\prod_ ( i 'in' A ) F	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
"\prod_ ( m <= i < n ) F"	:= (\big[*%R/1%R]_(m <= i < n) F%R).	Notation	\prod_ ( m <= i < n ) F	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
pchar (R : nzSemiRingType) : nat_pred	:= [pred p \| prime p & p%:R == 0 :> R].	Definition	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" ]	[ "nat_pred", "prime" ]	results for a non commutative ring in the proof of Gorenstein 2.6.3.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
has_pchar0 L	:= (pchar L =i pred0).	Notation	has_pchar0	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" ]	[ "pchar" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
converse R : Type	:= R.	Definition	converse	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" ]	[]	Converse ring tag.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"R ^c"	:= (converse R) : type_scope.	Notation	R ^c	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" ]	[ "converse" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulr_suml I r P (F : I -> R) x : (\sum_(i <- r \| P i) F i) * x = \sum_(i <- r \| P i) F i * x.	Proof. exact: big_distrl. Qed.	Lemma	mulr_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_distrl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulr_sumr I r P (F : I -> R) x : x * (\sum_(i <- r \| P i) F i) = \sum_(i <- r \| P i) x * F i.	Proof. exact: big_distrr. Qed.	Lemma	mulr_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_distrr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulrnAl x y n : (x + n) y = (x * y) *+ n.	Proof. by elim: n => [\|n IHn]; rewrite ?mul0r // !mulrS mulrDl IHn. Qed.	Lemma	mulrnAl	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" ]	[ "mul0r", "mulrDl", "mulrS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulrnAr x y n : x * (y + n) = (x y) *+ n.	Proof. by elim: n => [\|n IHn]; rewrite ?mulr0 // !mulrS mulrDr IHn. Qed.	Lemma	mulrnAr	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", "mulrDr", "mulrS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulr_natl x n : n%:R * x = x *+ n.	Proof. by rewrite mulrnAl mul1r. Qed.	Lemma	mulr_natl	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", "mulrnAl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulr_natr x n : x * n%:R = x *+ n.	Proof. by rewrite mulrnAr mulr1. Qed.	Lemma	mulr_natr	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", "mulrnAr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
natrD m n : (m + n)%:R = m%:R + n%:R :> R.	Proof. exact: mulrnDr. Qed.	Lemma	natrD	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" ]	[ "mulrnDr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
natr1 n : n%:R + 1 = n.+1%:R :> R.	Proof. by rewrite mulrSr. Qed.	Lemma	natr1	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" ]	[ "mulrSr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nat1r n : 1 + n%:R = n.+1%:R :> R.	Proof. by rewrite mulrS. Qed.	Lemma	nat1r	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
natr_sum	:= big_morph (natmul 1) natrD (mulr0n 1).	Definition	natr_sum	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", "mulr0n", "natmul", "natrD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
natrM m n : (m * n)%:R = m%:R * n%:R :> R.	Proof. by rewrite mulrnA mulr_natr. Qed.	Lemma	natrM	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_natr", "mulrnA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expr0 x : x ^+ 0 = 1.	Proof. by []. Qed.	Lemma	expr0	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
expr1 x : x ^+ 1 = x.	Proof. by []. Qed.	Lemma	expr1	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
expr2 x : x ^+ 2 = x * x.	Proof. by []. Qed.	Lemma	expr2	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
exprS x n : x ^+ n.+1 = x * x ^+ n.	Proof. by case: n => //; rewrite mulr1. Qed.	Lemma	exprS	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expr0n n : 0 ^+ n = (n == 0%N)%:R :> R.	Proof. by case: n => // n; rewrite exprS mul0r. Qed.	Lemma	expr0n	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", "mul0r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expr1n n : 1 ^+ n = 1 :> R.	Proof. by elim: n => // n IHn; rewrite exprS mul1r. Qed.	Lemma	expr1n	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", "mul1r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exprD x m n : x ^+ (m + n) = x ^+ m * x ^+ n.	Proof. by elim: m => [\|m IHm]; rewrite ?mul1r // !exprS -mulrA -IHm. Qed.	Lemma	exprD	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", "mul1r", "mulrA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exprSr x n : x ^+ n.+1 = x ^+ n * x.	Proof. by rewrite -addn1 exprD expr1. Qed.	Lemma	exprSr	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" ]	[ "addn1", "expr1", "exprD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expr_sum x (I : Type) (s : seq I) (P : pred I) F : x ^+ (\sum_(i <- s \| P i) F i) = \prod_(i <- s \| P i) x ^+ F i :> R.	Proof. exact: (big_morph _ (exprD _)). Qed.	Lemma	expr_sum	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", "exprD", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commr_sym x y : comm x y -> comm y x.	Proof. by []. Qed.	Lemma	commr_sym	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commr_refl x : comm x x.	Proof. by []. Qed.	Lemma	commr_refl	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" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commr0 x : comm x 0.	Proof. by rewrite /comm mulr0 mul0r. Qed.	Lemma	commr0	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", "mul0r", "mulr0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commr1 x : comm x 1.	Proof. by rewrite /comm mulr1 mul1r. Qed.	Lemma	commr1	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", "mul1r", "mulr1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commrD x y z : comm x y -> comm x z -> comm x (y + z).	Proof. by rewrite /comm mulrDl mulrDr => -> ->. Qed.	Lemma	commrD	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", "mulrDl", "mulrDr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commr_sum (I : Type) (s : seq I) (P : pred I) (F : I -> R) x : (forall i, P i -> comm x (F i)) -> comm x (\sum_(i <- s \| P i) F i).	Proof. move=> comm_x_F; rewrite /comm mulr_suml mulr_sumr. by apply: eq_bigr => i /comm_x_F. Qed.	Lemma	commr_sum	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", "eq_bigr", "mulr_suml", "mulr_sumr", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commrMn x y n : comm x y -> comm x (y *+ n).	Proof. rewrite /comm => com_xy. by elim: n => [\|n IHn]; rewrite ?commr0 // mulrS commrD. Qed.	Lemma	commrMn	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", "commr0", "commrD", "mulrS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commrM x y z : comm x y -> comm x z -> comm x (y * z).	Proof. by move=> com_xy; rewrite /comm mulrA com_xy -!mulrA => ->. Qed.	Lemma	commrM	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", "mulrA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commr_prod (I : Type) (s : seq I) (P : pred I) (F : I -> R) x : (forall i, P i -> comm x (F i)) -> comm x (\prod_(i <- s \| P i) F i).	Proof. exact: (big_ind _ (commr1 x) (@commrM x)). Qed.	Lemma	commr_prod	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_ind", "comm", "commr1", "commrM", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commr_nat x n : comm x n%:R.	Proof. exact/commrMn/commr1. Qed.	Lemma	commr_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" ]	[ "comm", "commr1", "commrMn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
commrX x y n : comm x y -> comm x (y ^+ n).	Proof. rewrite /comm => com_xy. by elim: n => [\|n IHn]; rewrite ?commr1 // exprS commrM. Qed.	Lemma	commrX	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", "commrM", "exprS" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exprMn_comm x y n : comm x y -> (x * y) ^+ n = x ^+ n * y ^+ n.	Proof. move=> com_xy; elim: n => /= [\|n IHn]; first by rewrite mulr1. by rewrite !exprS IHn !mulrA; congr (_ * _); rewrite -!mulrA -commrX. Qed.	Lemma	exprMn_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", "commrX", "exprS", "mulr1", "mulrA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exprMn_n x m n : (x + m) ^+ n = x ^+ n + (m ^ n) :> R.	Proof. elim: n => [\|n IHn]; first by rewrite mulr1n. by rewrite exprS IHn mulrnAl mulrnAr -mulrnA exprS expnSr. Qed.	Lemma	exprMn_n	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" ]	[ "expnSr", "exprS", "mulr1n", "mulrnA", "mulrnAl", "mulrnAr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exprM x m n : x ^+ (m * n) = x ^+ m ^+ n.	Proof. elim: m => [\|m IHm]; first by rewrite expr1n. by rewrite mulSn exprD IHm exprS exprMn_comm //; apply: commrX. Qed.	Lemma	exprM	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", "commrX", "expr1n", "exprD", "exprMn_comm", "exprS", "mulSn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exprAC x m n : (x ^+ m) ^+ n = (x ^+ n) ^+ m.	Proof. by rewrite -!exprM mulnC. Qed.	Lemma	exprAC	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" ]	[ "exprM", "mulnC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expr_mod n x i : x ^+ n = 1 -> x ^+ (i %% n) = x ^+ i.	Proof. move=> xn1; rewrite {2}(divn_eq i n) exprD mulnC exprM xn1. by rewrite expr1n mul1r. Qed.	Lemma	expr_mod	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" ]	[ "divn_eq", "expr1n", "exprD", "exprM", "mul1r", "mulnC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expr_dvd n x i : x ^+ n = 1 -> n %\| i -> x ^+ i = 1.	Proof. by move=> xn1 dvd_n_i; rewrite -(expr_mod i xn1) (eqnP dvd_n_i). Qed.	Lemma	expr_dvd	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" ]	[ "eqnP", "expr_mod" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
natrX n k : (n ^ k)%:R = n%:R ^+ k :> R.	Proof. by rewrite exprMn_n expr1n. Qed.	Lemma	natrX	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", "exprMn_n" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulrI_eq0 x y : lreg x -> (x * y == 0) = (y == 0).	Proof. by move=> reg_x; rewrite -{1}(mulr0 x) (inj_eq reg_x). Qed.	Lemma	mulrI_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" ]	[ "inj_eq", "lreg", "mulr0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lreg1 : lreg (1 : R).	Proof. by move=> x y; rewrite !mul1r. Qed.	Lemma	lreg1	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", "mul1r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lregM x y : lreg x -> lreg y -> lreg (x * y).	Proof. by move=> reg_x reg_y z t; rewrite -!mulrA => /reg_x/reg_y. Qed.	Lemma	lregM	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", "mulrA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lregMl (a b: R) : lreg (a * b) -> lreg b.	Proof. by move=> rab c c' eq_bc; apply/rab; rewrite -!mulrA eq_bc. Qed.	Lemma	lregMl	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", "mulrA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rregMr (a b: R) : rreg (a * b) -> rreg a.	Proof. by move=> rab c c' eq_ca; apply/rab; rewrite !mulrA eq_ca. Qed.	Lemma	rregMr	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", "mulrA", "rreg" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lregX x n : lreg x -> lreg (x ^+ n).	Proof. by move=> reg_x; elim: n => [\|n]; [apply: lreg1 \| rewrite exprS; apply: lregM]. Qed.	Lemma	lregX	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", "exprS", "lreg", "lreg1", "lregM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
iter_mulr n x y : iter n ( %R x) y = x ^+ n y.	Proof. by elim: n => [\|n ih]; rewrite ?expr0 ?mul1r //= ih exprS -mulrA. Qed.	Lemma	iter_mulr	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" ]	[ "expr0", "exprS", "iter", "mul1r", "mulrA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
iter_mulr_1 n x : iter n ( *%R x) 1 = x ^+ n.	Proof. by rewrite iter_mulr mulr1. Qed.	Lemma	iter_mulr_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" ]	[ "iter", "iter_mulr", "mulr1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodr_const (I : finType) (A : pred I) x : \prod_(i in A) x = x ^+ #\|A\|.	Proof. by rewrite big_const -iteropE. Qed.	Lemma	prodr_const	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_const", "iteropE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodr_const_nat n m x : \prod_(n <= i < m) x = x ^+ (m - n).	Proof. by rewrite big_const_nat -iteropE. Qed.	Lemma	prodr_const_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" ]	[ "big_const_nat", "iteropE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodrXr x I r P (F : I -> nat) : \prod_(i <- r \| P i) x ^+ F i = x ^+ (\sum_(i <- r \| P i) F i).	Proof. by rewrite (big_morph _ (exprD _) (erefl _)). Qed.	Lemma	prodrXr	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", "exprD", "nat" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodrM_comm {I : eqType} r (P : pred I) (F G : I -> R) : (forall i j, P i -> P j -> comm (F i) (G j)) -> \prod_(i <- r \| P i) (F i * G i) = \prod_(i <- r \| P i) F i * \prod_(i <- r \| P i) G i.	Proof. move=> FG; elim: r => [\|i r IHr]; rewrite !(big_nil, big_cons) ?mulr1//. case: ifPn => // Pi; rewrite IHr !mulrA; congr (_ * _); rewrite -!mulrA. by rewrite commr_prod // => j Pj; apply/commr_sym/FG. Qed.	Lemma	prodrM_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", "big_cons", "big_nil", "comm", "commr_prod", "commr_sym", "mulr1", "mulrA" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodrMl_comm {I : finType} (A : pred I) (x : R) F : (forall i, A i -> comm x (F i)) -> \prod_(i in A) (x * F i) = x ^+ #\|A\| * \prod_(i in A) F i.	Proof. by move=> xF; rewrite prodrM_comm ?prodr_const// => i j _ /xF. Qed.	Lemma	prodrMl_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", "prodrM_comm", "prodr_const" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodrMr_comm {I : finType} (A : pred I) (x : R) F : (forall i, A i -> comm x (F i)) -> \prod_(i in A) (F i * x) = \prod_(i in A) F i * x ^+ #\|A\|.	Proof. by move=> xF; rewrite prodrM_comm ?prodr_const// => i j /xF. Qed.	Lemma	prodrMr_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", "prodrM_comm", "prodr_const" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodrMn (I : Type) (s : seq I) (P : pred I) (F : I -> R) (g : I -> nat) : \prod_(i <- s \| P i) (F i + g i) = \prod_(i <- s \| P i) (F i) + \prod_(i <- s \| P i) g i.	Proof. by elim/big_rec3: _ => // i y1 y2 y3 _ ->; rewrite mulrnAr mulrnAl -mulrnA. Qed.	Lemma	prodrMn	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_rec3", "mulrnA", "mulrnAl", "mulrnAr", "nat", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodrMn_const n (I : finType) (A : pred I) (F : I -> R) : \prod_(i in A) (F i + n) = \prod_(i in A) F i + n ^ #\|A\|.	Proof. by rewrite prodrMn prod_nat_const. Qed.	Lemma	prodrMn_const	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" ]	[ "prod_nat_const", "prodrMn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
natr_prod I r P (F : I -> nat) : (\prod_(i <- r \| P i) F i)%:R = \prod_(i <- r \| P i) (F i)%:R :> R.	Proof. exact: (big_morph _ natrM). Qed.	Lemma	natr_prod	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", "nat", "natrM" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exprDn_comm x y n (cxy : comm x y) : (x + y) ^+ n = \sum_(i < n.+1) (x ^+ (n - i) * y ^+ i) *+ 'C(n, i).	Proof. elim: n => [\|n IHn]; rewrite big_ord_recl mulr1 ?big_ord0 ?addr0 //=. rewrite exprS {}IHn /= mulrDl !big_distrr /= big_ord_recl mulr1 subn0. rewrite !big_ord_recr /= !binn !subnn !mul1r !subn0 bin0 !exprS -addrA. congr (_ + _); rewrite addrA -big_split /=; congr (_ + _). apply: eq_bigr => i _; rewrite !mulrnAr !...	Lemma	exprDn_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" ]	[ "addr0", "addrA", "apply", "big_distrr", "big_ord0", "big_ord_recl", "big_ord_recr", "big_split", "bin0", "binn", "comm", "commrX", "commr_sym", "eq_bigr", "exprS", "mul1r", "mulr1", "mulrA", "mulrDl", "mulrnAr", "mulrnDr", "subSS", "subSn", "subn0", "subnn", "valP"...		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
exprD1n x n : (x + 1) ^+ n = \sum_(i < n.+1) x ^+ i *+ 'C(n, i).	Proof. rewrite addrC (exprDn_comm n (commr_sym (commr1 x))). by apply: eq_bigr => i _; rewrite expr1n mul1r. Qed.	Lemma	exprD1n	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", "commr1", "commr_sym", "eq_bigr", "expr1n", "exprDn_comm", "mul1r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sqrrD1 x : (x + 1) ^+ 2 = x ^+ 2 + x *+ 2 + 1.	Proof. rewrite exprD1n !big_ord_recr big_ord0 /= add0r. by rewrite addrC addrA addrAC. Qed.	Lemma	sqrrD1	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", "addrAC", "addrC", "big_ord0", "big_ord_recr", "exprD1n" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulr_2closed	:= {in S &, forall u v, u * v \in S}.	Definition	mulr_2closed	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_closed	:= 1 \in S /\ mulr_2closed.	Definition	mulr_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
semiring_closed	:= nmod_closed S /\ mulr_closed.	Definition	semiring_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_closed", "nmod_closed" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
semiring_closedD : semiring_closed -> nmod_closed S.	Proof. by case. Qed.	Lemma	semiring_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", "semiring_closed" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
semiring_closedM : semiring_closed -> mulr_closed.	Proof. by case. Qed.	Lemma	semiring_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" ]	[ "mulr_closed", "semiring_closed" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
oner_eq0 : (1 == 0 :> R) = false.	Proof. exact: negbTE oner_neq0. Qed.	Lemma	oner_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" ]	[ "oner_neq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lastr_eq0 (s : seq R) x : x != 0 -> (last x s == 0) = (last 1 s == 0).	Proof. by case: s => [\|y s] /negPf // ->; rewrite oner_eq0. Qed.	Lemma	lastr_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" ]	[ "last", "oner_eq0", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lreg_neq0 x : lreg x -> x != 0.	Proof. by move=> reg_x; rewrite -[x]mulr1 mulrI_eq0 ?oner_eq0. Qed.	Lemma	lreg_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", "mulr1", "mulrI_eq0", "oner_eq0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pFrobenius_aut p & p \in pchar R	:= fun x => x ^+ p.	Definition	pFrobenius_aut	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" ]	[ "pchar" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcharFp : p \in pchar R.		Hypothesis	pcharFp	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" ]	[ "pchar" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcharf0 : p%:R = 0 :> R.	Proof. by apply/eqP; case/andP: pcharFp. Qed.	Lemma	pcharf0	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", "pcharFp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcharf_prime : prime p.	Proof. by case/andP: pcharFp. Qed.	Lemma	pcharf_prime	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" ]	[ "pcharFp", "prime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulrn_pchar x : x *+ p = 0.	Proof. by rewrite -mulr_natl pcharf0 mul0r. Qed.	Lemma	mulrn_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" ]	[ "mul0r", "mulr_natl", "pcharf0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
natr_mod_pchar n : (n %% p)%:R = n%:R :> R.	Proof. by rewrite {2}(divn_eq n p) natrD mulrnA mulrn_pchar add0r. Qed.	Lemma	natr_mod_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" ]	[ "add0r", "divn_eq", "mulrnA", "mulrn_pchar", "natrD" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
dvdn_pcharf n : (p %\| n)%N = (n%:R == 0 :> R).	Proof. apply/idP/eqP=> [/dvdnP[n' ->]\|n0]; first by rewrite natrM pcharf0 mulr0. apply/idPn; rewrite -prime_coprime // => /eqnP pn1. have [a _ /dvdnP[b]] := Bezoutl n (prime_gt0 pcharf_prime). move/(congr1 (fun m => m%:R : R))/eqP. by rewrite natrD !natrM pcharf0 n0 !mulr0 pn1 addr0 oner_eq0. Qed.	Lemma	dvdn_pcharf	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" ]	[ "Bezoutl", "addr0", "apply", "dvdnP", "eqnP", "mulr0", "n'", "natrD", "natrM", "oner_eq0", "pcharf0", "pcharf_prime", "prime_coprime", "prime_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcharf_eq : pchar R =i (p : nat_pred).	Proof. move=> q; apply/andP/eqP=> [[q_pr q0] \| ->]; last by rewrite pcharf0. by apply/eqP; rewrite eq_sym -dvdn_prime2 // dvdn_pcharf. Qed.	Lemma	pcharf_eq	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", "dvdn_pcharf", "dvdn_prime2", "eq_sym", "last", "nat_pred", "pchar", "pcharf0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
bin_lt_pcharf_0 k : 0 < k < p -> 'C(p, k)%:R = 0 :> R.	Proof. by move=> lt0kp; apply/eqP; rewrite -dvdn_pcharf prime_dvd_bin. Qed.	Lemma	bin_lt_pcharf_0	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", "dvdn_pcharf", "prime_dvd_bin" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pFrobenius_autE x : x^f = x ^+ p.	Proof. by []. Qed.	Lemma	pFrobenius_autE	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
f'E	:= pFrobenius_autE.	Notation	f'E	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" ]	[ "pFrobenius_autE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pFrobenius_aut0 : 0^f = 0.	Proof. by rewrite f'E -(prednK (prime_gt0 pcharf_prime)) exprS mul0r. Qed.	Lemma	pFrobenius_aut0	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", "f'E", "mul0r", "pcharf_prime", "prednK", "prime_gt0" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pFrobenius_aut1 : 1^f = 1.	Proof. by rewrite f'E expr1n. Qed.	Lemma	pFrobenius_aut1	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", "f'E" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pFrobenius_autD_comm x y (cxy : comm x y) : (x + y)^f = x^f + y^f.	Proof. have defp := prednK (prime_gt0 pcharf_prime). rewrite !f'E exprDn_comm // big_ord_recr subnn -defp big_ord_recl /= defp. rewrite subn0 mulr1 mul1r bin0 binn big1 ?addr0 // => i _. by rewrite -mulr_natl bin_lt_pcharf_0 ?mul0r //= -{2}defp ltnS (valP i). Qed.	Lemma	pFrobenius_autD_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" ]	[ "addr0", "big1", "big_ord_recl", "big_ord_recr", "bin0", "bin_lt_pcharf_0", "binn", "comm", "exprDn_comm", "f'E", "ltnS", "mul0r", "mul1r", "mulr1", "mulr_natl", "pcharf_prime", "prednK", "prime_gt0", "subn0", "subnn", "valP" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pFrobenius_autMn x n : (x + n)^f = x^f + n.	Proof. elim: n => [\|n IHn]; first exact: pFrobenius_aut0. by rewrite !mulrS pFrobenius_autD_comm ?IHn //; apply: commrMn. Qed.	Lemma	pFrobenius_autMn	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", "commrMn", "mulrS", "pFrobenius_aut0", "pFrobenius_autD_comm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pFrobenius_aut_nat n : (n%:R)^f = n%:R.	Proof. by rewrite pFrobenius_autMn pFrobenius_aut1. Qed.	Lemma	pFrobenius_aut_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" ]	[ "pFrobenius_aut1", "pFrobenius_autMn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pFrobenius_autM_comm x y : comm x y -> (x * y)^f = x^f * y^f.	Proof. exact: exprMn_comm. Qed.	Lemma	pFrobenius_autM_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", "exprMn_comm" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pFrobenius_autX x n : (x ^+ n)^f = x^f ^+ n.	Proof. by rewrite !f'E -!exprM mulnC. Qed.	Lemma	pFrobenius_autX	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" ]	[ "exprM", "f'E", "mulnC" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcharR2 : 2 \in pchar R.		Hypothesis	pcharR2	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" ]	[ "pchar" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addrr_pchar2 x : x + x = 0.	Proof. by rewrite -mulr2n mulrn_pchar. Qed.	Lemma	addrr_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" ]	[ "mulr2n", "mulrn_pchar" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"1"	:= one.	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" ]	[ "one" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x * y"	:= (mul x y).	Notation	x * y	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" ]	[ "mul" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul0r : @left_zero R R 0 mul.	Proof. by move=> x; apply: (addIr (1 * x)); rewrite -mulrDl !add0r mul1r. Qed.	Lemma	mul0r	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", "mul", "mul1r", "mulrDl" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulr0 : @right_zero R R 0 mul.	Proof. by move=> x; apply: (addIr (x * 1)); rewrite -mulrDr !add0r mulr1. Qed.	Lemma	mulr0	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", "mul", "mulr1", "mulrDr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sign R b	:= (exp (- @one R) (nat_of_bool b)) (only parsing).	Notation	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" ]	[ "exp", "nat_of_bool", "one" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x ^+ n"

:= (exp x n) : ring_scope.

Notation

x ^+ n

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" ]

[ "exp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\prod_ ( i <- r | P ) F"

:= (\big[*%R/1]_(i <- r | P) F).

Notation

\prod_ ( i <- r | P ) F

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

"\prod_ ( i | P ) F"

:= (\big[*%R/1]_(i | P) F).

Notation

\prod_ ( i | P ) F

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

"\prod_ ( i 'in' A ) F"

:= (\big[*%R/1]_(i in A) F).

Notation

\prod_ ( i 'in' A ) F

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

"\prod_ ( m <= i < n ) F"

:= (\big[*%R/1%R]_(m <= i < n) F%R).

Notation

\prod_ ( m <= i < n ) F

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

pchar (R : nzSemiRingType) : nat_pred

:= [pred p | prime p & p%:R == 0 :> R].

Definition

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" ]

[ "nat_pred", "prime" ]

results for a non commutative ring in the proof of Gorenstein 2.6.3.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

has_pchar0 L

:= (pchar L =i pred0).

Notation

has_pchar0

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" ]

[ "pchar" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

converse R : Type

:= R.

Definition

converse

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" ]

[]

Converse ring tag.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"R ^c"

:= (converse R) : type_scope.

Notation

R ^c

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" ]

[ "converse" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulr_suml I r P (F : I -> R) x : (\sum_(i <- r | P i) F i) * x = \sum_(i <- r | P i) F i * x.

Proof. exact: big_distrl. Qed.

Lemma

mulr_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_distrl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulr_sumr I r P (F : I -> R) x : x * (\sum_(i <- r | P i) F i) = \sum_(i <- r | P i) x * F i.

Proof. exact: big_distrr. Qed.

Lemma

mulr_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_distrr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulrnAl x y n : (x *+ n) * y = (x * y) *+ n.

Proof. by elim: n => [|n IHn]; rewrite ?mul0r // !mulrS mulrDl IHn. Qed.

Lemma

mulrnAl

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" ]

[ "mul0r", "mulrDl", "mulrS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulrnAr x y n : x * (y *+ n) = (x * y) *+ n.

Proof. by elim: n => [|n IHn]; rewrite ?mulr0 // !mulrS mulrDr IHn. Qed.

Lemma

mulrnAr

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", "mulrDr", "mulrS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulr_natl x n : n%:R * x = x *+ n.

Proof. by rewrite mulrnAl mul1r. Qed.

Lemma

mulr_natl

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", "mulrnAl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulr_natr x n : x * n%:R = x *+ n.

Proof. by rewrite mulrnAr mulr1. Qed.

Lemma

mulr_natr

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", "mulrnAr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

natrD m n : (m + n)%:R = m%:R + n%:R :> R.

Proof. exact: mulrnDr. Qed.

Lemma

natrD

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" ]

[ "mulrnDr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

natr1 n : n%:R + 1 = n.+1%:R :> R.

Proof. by rewrite mulrSr. Qed.

Lemma

natr1

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" ]

[ "mulrSr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nat1r n : 1 + n%:R = n.+1%:R :> R.

Proof. by rewrite mulrS. Qed.

Lemma

nat1r

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

natr_sum

:= big_morph (natmul 1) natrD (mulr0n 1).

Definition

natr_sum

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", "mulr0n", "natmul", "natrD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

natrM m n : (m * n)%:R = m%:R * n%:R :> R.

Proof. by rewrite mulrnA mulr_natr. Qed.

Lemma

natrM

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_natr", "mulrnA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expr0 x : x ^+ 0 = 1.

Proof. by []. Qed.

Lemma

expr0

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

expr1 x : x ^+ 1 = x.

Proof. by []. Qed.

Lemma

expr1

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

expr2 x : x ^+ 2 = x * x.

Proof. by []. Qed.

Lemma

expr2

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

exprS x n : x ^+ n.+1 = x * x ^+ n.

Proof. by case: n => //; rewrite mulr1. Qed.

Lemma

exprS

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expr0n n : 0 ^+ n = (n == 0%N)%:R :> R.

Proof. by case: n => // n; rewrite exprS mul0r. Qed.

Lemma

expr0n

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", "mul0r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expr1n n : 1 ^+ n = 1 :> R.

Proof. by elim: n => // n IHn; rewrite exprS mul1r. Qed.

Lemma

expr1n

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", "mul1r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exprD x m n : x ^+ (m + n) = x ^+ m * x ^+ n.

Proof. by elim: m => [|m IHm]; rewrite ?mul1r // !exprS -mulrA -IHm. Qed.

Lemma

exprD

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", "mul1r", "mulrA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exprSr x n : x ^+ n.+1 = x ^+ n * x.

Proof. by rewrite -addn1 exprD expr1. Qed.

Lemma

exprSr

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" ]

[ "addn1", "expr1", "exprD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expr_sum x (I : Type) (s : seq I) (P : pred I) F : x ^+ (\sum_(i <- s | P i) F i) = \prod_(i <- s | P i) x ^+ F i :> R.

Proof. exact: (big_morph _ (exprD _)). Qed.

Lemma

expr_sum

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", "exprD", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commr_sym x y : comm x y -> comm y x.

Proof. by []. Qed.

Lemma

commr_sym

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commr_refl x : comm x x.

Proof. by []. Qed.

Lemma

commr_refl

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" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commr0 x : comm x 0.

Proof. by rewrite /comm mulr0 mul0r. Qed.

Lemma

commr0

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", "mul0r", "mulr0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commr1 x : comm x 1.

Proof. by rewrite /comm mulr1 mul1r. Qed.

Lemma

commr1

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", "mul1r", "mulr1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commrD x y z : comm x y -> comm x z -> comm x (y + z).

Proof. by rewrite /comm mulrDl mulrDr => -> ->. Qed.

Lemma

commrD

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", "mulrDl", "mulrDr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commr_sum (I : Type) (s : seq I) (P : pred I) (F : I -> R) x : (forall i, P i -> comm x (F i)) -> comm x (\sum_(i <- s | P i) F i).

Proof. move=> comm_x_F; rewrite /comm mulr_suml mulr_sumr. by apply: eq_bigr => i /comm_x_F. Qed.

Lemma

commr_sum

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", "eq_bigr", "mulr_suml", "mulr_sumr", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commrMn x y n : comm x y -> comm x (y *+ n).

Proof. rewrite /comm => com_xy. by elim: n => [|n IHn]; rewrite ?commr0 // mulrS commrD. Qed.

Lemma

commrMn

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", "commr0", "commrD", "mulrS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commrM x y z : comm x y -> comm x z -> comm x (y * z).

Proof. by move=> com_xy; rewrite /comm mulrA com_xy -!mulrA => ->. Qed.

Lemma

commrM

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", "mulrA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commr_prod (I : Type) (s : seq I) (P : pred I) (F : I -> R) x : (forall i, P i -> comm x (F i)) -> comm x (\prod_(i <- s | P i) F i).

Proof. exact: (big_ind _ (commr1 x) (@commrM x)). Qed.

Lemma

commr_prod

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_ind", "comm", "commr1", "commrM", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commr_nat x n : comm x n%:R.

Proof. exact/commrMn/commr1. Qed.

Lemma

commr_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" ]

[ "comm", "commr1", "commrMn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

commrX x y n : comm x y -> comm x (y ^+ n).

Proof. rewrite /comm => com_xy. by elim: n => [|n IHn]; rewrite ?commr1 // exprS commrM. Qed.

Lemma

commrX

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", "commrM", "exprS" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exprMn_comm x y n : comm x y -> (x * y) ^+ n = x ^+ n * y ^+ n.

Proof. move=> com_xy; elim: n => /= [|n IHn]; first by rewrite mulr1. by rewrite !exprS IHn !mulrA; congr (_ * _); rewrite -!mulrA -commrX. Qed.

Lemma

exprMn_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", "commrX", "exprS", "mulr1", "mulrA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exprMn_n x m n : (x *+ m) ^+ n = x ^+ n *+ (m ^ n) :> R.

Proof. elim: n => [|n IHn]; first by rewrite mulr1n. by rewrite exprS IHn mulrnAl mulrnAr -mulrnA exprS expnSr. Qed.

Lemma

exprMn_n

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" ]

[ "expnSr", "exprS", "mulr1n", "mulrnA", "mulrnAl", "mulrnAr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exprM x m n : x ^+ (m * n) = x ^+ m ^+ n.

Proof. elim: m => [|m IHm]; first by rewrite expr1n. by rewrite mulSn exprD IHm exprS exprMn_comm //; apply: commrX. Qed.

Lemma

exprM

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", "commrX", "expr1n", "exprD", "exprMn_comm", "exprS", "mulSn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exprAC x m n : (x ^+ m) ^+ n = (x ^+ n) ^+ m.

Proof. by rewrite -!exprM mulnC. Qed.

Lemma

exprAC

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" ]

[ "exprM", "mulnC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expr_mod n x i : x ^+ n = 1 -> x ^+ (i %% n) = x ^+ i.

Proof. move=> xn1; rewrite {2}(divn_eq i n) exprD mulnC exprM xn1. by rewrite expr1n mul1r. Qed.

Lemma

expr_mod

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" ]

[ "divn_eq", "expr1n", "exprD", "exprM", "mul1r", "mulnC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expr_dvd n x i : x ^+ n = 1 -> n %| i -> x ^+ i = 1.

Proof. by move=> xn1 dvd_n_i; rewrite -(expr_mod i xn1) (eqnP dvd_n_i). Qed.

Lemma

expr_dvd

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" ]

[ "eqnP", "expr_mod" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

natrX n k : (n ^ k)%:R = n%:R ^+ k :> R.

Proof. by rewrite exprMn_n expr1n. Qed.

Lemma

natrX

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", "exprMn_n" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulrI_eq0 x y : lreg x -> (x * y == 0) = (y == 0).

Proof. by move=> reg_x; rewrite -{1}(mulr0 x) (inj_eq reg_x). Qed.

Lemma

mulrI_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" ]

[ "inj_eq", "lreg", "mulr0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lreg1 : lreg (1 : R).

Proof. by move=> x y; rewrite !mul1r. Qed.

Lemma

lreg1

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", "mul1r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lregM x y : lreg x -> lreg y -> lreg (x * y).

Proof. by move=> reg_x reg_y z t; rewrite -!mulrA => /reg_x/reg_y. Qed.

Lemma

lregM

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", "mulrA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lregMl (a b: R) : lreg (a * b) -> lreg b.

Proof. by move=> rab c c' eq_bc; apply/rab; rewrite -!mulrA eq_bc. Qed.

Lemma

lregMl

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", "mulrA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rregMr (a b: R) : rreg (a * b) -> rreg a.

Proof. by move=> rab c c' eq_ca; apply/rab; rewrite !mulrA eq_ca. Qed.

Lemma

rregMr

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", "mulrA", "rreg" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lregX x n : lreg x -> lreg (x ^+ n).

Proof. by move=> reg_x; elim: n => [|n]; [apply: lreg1 | rewrite exprS; apply: lregM]. Qed.

Lemma

lregX

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", "exprS", "lreg", "lreg1", "lregM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

iter_mulr n x y : iter n ( *%R x) y = x ^+ n * y.

Proof. by elim: n => [|n ih]; rewrite ?expr0 ?mul1r //= ih exprS -mulrA. Qed.

Lemma

iter_mulr

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" ]

[ "expr0", "exprS", "iter", "mul1r", "mulrA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

iter_mulr_1 n x : iter n ( *%R x) 1 = x ^+ n.

Proof. by rewrite iter_mulr mulr1. Qed.

Lemma

iter_mulr_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" ]

[ "iter", "iter_mulr", "mulr1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodr_const (I : finType) (A : pred I) x : \prod_(i in A) x = x ^+ #|A|.

Proof. by rewrite big_const -iteropE. Qed.

Lemma

prodr_const

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_const", "iteropE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodr_const_nat n m x : \prod_(n <= i < m) x = x ^+ (m - n).

Proof. by rewrite big_const_nat -iteropE. Qed.

Lemma

prodr_const_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" ]

[ "big_const_nat", "iteropE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodrXr x I r P (F : I -> nat) : \prod_(i <- r | P i) x ^+ F i = x ^+ (\sum_(i <- r | P i) F i).

Proof. by rewrite (big_morph _ (exprD _) (erefl _)). Qed.

Lemma

prodrXr

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", "exprD", "nat" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodrM_comm {I : eqType} r (P : pred I) (F G : I -> R) : (forall i j, P i -> P j -> comm (F i) (G j)) -> \prod_(i <- r | P i) (F i * G i) = \prod_(i <- r | P i) F i * \prod_(i <- r | P i) G i.

Proof. move=> FG; elim: r => [|i r IHr]; rewrite !(big_nil, big_cons) ?mulr1//. case: ifPn => // Pi; rewrite IHr !mulrA; congr (_ * _); rewrite -!mulrA. by rewrite commr_prod // => j Pj; apply/commr_sym/FG. Qed.

Lemma

prodrM_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", "big_cons", "big_nil", "comm", "commr_prod", "commr_sym", "mulr1", "mulrA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodrMl_comm {I : finType} (A : pred I) (x : R) F : (forall i, A i -> comm x (F i)) -> \prod_(i in A) (x * F i) = x ^+ #|A| * \prod_(i in A) F i.

Proof. by move=> xF; rewrite prodrM_comm ?prodr_const// => i j _ /xF. Qed.

Lemma

prodrMl_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", "prodrM_comm", "prodr_const" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodrMr_comm {I : finType} (A : pred I) (x : R) F : (forall i, A i -> comm x (F i)) -> \prod_(i in A) (F i * x) = \prod_(i in A) F i * x ^+ #|A|.

Proof. by move=> xF; rewrite prodrM_comm ?prodr_const// => i j /xF. Qed.

Lemma

prodrMr_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", "prodrM_comm", "prodr_const" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodrMn (I : Type) (s : seq I) (P : pred I) (F : I -> R) (g : I -> nat) : \prod_(i <- s | P i) (F i *+ g i) = \prod_(i <- s | P i) (F i) *+ \prod_(i <- s | P i) g i.

Proof. by elim/big_rec3: _ => // i y1 y2 y3 _ ->; rewrite mulrnAr mulrnAl -mulrnA. Qed.

Lemma

prodrMn

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_rec3", "mulrnA", "mulrnAl", "mulrnAr", "nat", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. by rewrite prodrMn prod_nat_const. Qed.

Lemma

prodrMn_const

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" ]

[ "prod_nat_const", "prodrMn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

natr_prod I r P (F : I -> nat) : (\prod_(i <- r | P i) F i)%:R = \prod_(i <- r | P i) (F i)%:R :> R.

Proof. exact: (big_morph _ natrM). Qed.

Lemma

natr_prod

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", "nat", "natrM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. elim: n => [|n IHn]; rewrite big_ord_recl mulr1 ?big_ord0 ?addr0 //=. rewrite exprS {}IHn /= mulrDl !big_distrr /= big_ord_recl mulr1 subn0. rewrite !big_ord_recr /= !binn !subnn !mul1r !subn0 bin0 !exprS -addrA. congr (_ + _); rewrite addrA -big_split /=; congr (_ + _). apply: eq_bigr => i _; rewrite !mulrnAr !...

Lemma

exprDn_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" ]

[ "addr0", "addrA", "apply", "big_distrr", "big_ord0", "big_ord_recl", "big_ord_recr", "big_split", "bin0", "binn", "comm", "commrX", "commr_sym", "eq_bigr", "exprS", "mul1r", "mulr1", "mulrA", "mulrDl", "mulrnAr", "mulrnDr", "subSS", "subSn", "subn0", "subnn", "valP"...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

exprD1n x n : (x + 1) ^+ n = \sum_(i < n.+1) x ^+ i *+ 'C(n, i).

Proof. rewrite addrC (exprDn_comm n (commr_sym (commr1 x))). by apply: eq_bigr => i _; rewrite expr1n mul1r. Qed.

Lemma

exprD1n

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", "commr1", "commr_sym", "eq_bigr", "expr1n", "exprDn_comm", "mul1r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sqrrD1 x : (x + 1) ^+ 2 = x ^+ 2 + x *+ 2 + 1.

Proof. rewrite exprD1n !big_ord_recr big_ord0 /= add0r. by rewrite addrC addrA addrAC. Qed.

Lemma

sqrrD1

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", "addrAC", "addrC", "big_ord0", "big_ord_recr", "exprD1n" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulr_2closed

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

Definition

mulr_2closed

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_closed

:= 1 \in S /\ mulr_2closed.

Definition

mulr_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

semiring_closed

:= nmod_closed S /\ mulr_closed.

Definition

semiring_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_closed", "nmod_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

semiring_closedD : semiring_closed -> nmod_closed S.

Proof. by case. Qed.

Lemma

semiring_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", "semiring_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

semiring_closedM : semiring_closed -> mulr_closed.

Proof. by case. Qed.

Lemma

semiring_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" ]

[ "mulr_closed", "semiring_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

oner_eq0 : (1 == 0 :> R) = false.

Proof. exact: negbTE oner_neq0. Qed.

Lemma

oner_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" ]

[ "oner_neq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lastr_eq0 (s : seq R) x : x != 0 -> (last x s == 0) = (last 1 s == 0).

Proof. by case: s => [|y s] /negPf // ->; rewrite oner_eq0. Qed.

Lemma

lastr_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" ]

[ "last", "oner_eq0", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lreg_neq0 x : lreg x -> x != 0.

Proof. by move=> reg_x; rewrite -[x]mulr1 mulrI_eq0 ?oner_eq0. Qed.

Lemma

lreg_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", "mulr1", "mulrI_eq0", "oner_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pFrobenius_aut p & p \in pchar R

:= fun x => x ^+ p.

Definition

pFrobenius_aut

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" ]

[ "pchar" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcharFp : p \in pchar R.

Hypothesis

pcharFp

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" ]

[ "pchar" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcharf0 : p%:R = 0 :> R.

Proof. by apply/eqP; case/andP: pcharFp. Qed.

Lemma

pcharf0

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", "pcharFp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcharf_prime : prime p.

Proof. by case/andP: pcharFp. Qed.

Lemma

pcharf_prime

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" ]

[ "pcharFp", "prime" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulrn_pchar x : x *+ p = 0.

Proof. by rewrite -mulr_natl pcharf0 mul0r. Qed.

Lemma

mulrn_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" ]

[ "mul0r", "mulr_natl", "pcharf0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

natr_mod_pchar n : (n %% p)%:R = n%:R :> R.

Proof. by rewrite {2}(divn_eq n p) natrD mulrnA mulrn_pchar add0r. Qed.

Lemma

natr_mod_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" ]

[ "add0r", "divn_eq", "mulrnA", "mulrn_pchar", "natrD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

dvdn_pcharf n : (p %| n)%N = (n%:R == 0 :> R).

Proof. apply/idP/eqP=> [/dvdnP[n' ->]|n0]; first by rewrite natrM pcharf0 mulr0. apply/idPn; rewrite -prime_coprime // => /eqnP pn1. have [a _ /dvdnP[b]] := Bezoutl n (prime_gt0 pcharf_prime). move/(congr1 (fun m => m%:R : R))/eqP. by rewrite natrD !natrM pcharf0 n0 !mulr0 pn1 addr0 oner_eq0. Qed.

Lemma

dvdn_pcharf

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" ]

[ "Bezoutl", "addr0", "apply", "dvdnP", "eqnP", "mulr0", "n'", "natrD", "natrM", "oner_eq0", "pcharf0", "pcharf_prime", "prime_coprime", "prime_gt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcharf_eq : pchar R =i (p : nat_pred).

Proof. move=> q; apply/andP/eqP=> [[q_pr q0] | ->]; last by rewrite pcharf0. by apply/eqP; rewrite eq_sym -dvdn_prime2 // dvdn_pcharf. Qed.

Lemma

pcharf_eq

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", "dvdn_pcharf", "dvdn_prime2", "eq_sym", "last", "nat_pred", "pchar", "pcharf0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

bin_lt_pcharf_0 k : 0 < k < p -> 'C(p, k)%:R = 0 :> R.

Proof. by move=> lt0kp; apply/eqP; rewrite -dvdn_pcharf prime_dvd_bin. Qed.

Lemma

bin_lt_pcharf_0

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", "dvdn_pcharf", "prime_dvd_bin" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pFrobenius_autE x : x^f = x ^+ p.

Proof. by []. Qed.

Lemma

pFrobenius_autE

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

f'E

:= pFrobenius_autE.

Notation

f'E

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" ]

[ "pFrobenius_autE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pFrobenius_aut0 : 0^f = 0.

Proof. by rewrite f'E -(prednK (prime_gt0 pcharf_prime)) exprS mul0r. Qed.

Lemma

pFrobenius_aut0

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", "f'E", "mul0r", "pcharf_prime", "prednK", "prime_gt0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pFrobenius_aut1 : 1^f = 1.

Proof. by rewrite f'E expr1n. Qed.

Lemma

pFrobenius_aut1

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", "f'E" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pFrobenius_autD_comm x y (cxy : comm x y) : (x + y)^f = x^f + y^f.

Proof. have defp := prednK (prime_gt0 pcharf_prime). rewrite !f'E exprDn_comm // big_ord_recr subnn -defp big_ord_recl /= defp. rewrite subn0 mulr1 mul1r bin0 binn big1 ?addr0 // => i _. by rewrite -mulr_natl bin_lt_pcharf_0 ?mul0r //= -{2}defp ltnS (valP i). Qed.

Lemma

pFrobenius_autD_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" ]

[ "addr0", "big1", "big_ord_recl", "big_ord_recr", "bin0", "bin_lt_pcharf_0", "binn", "comm", "exprDn_comm", "f'E", "ltnS", "mul0r", "mul1r", "mulr1", "mulr_natl", "pcharf_prime", "prednK", "prime_gt0", "subn0", "subnn", "valP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pFrobenius_autMn x n : (x *+ n)^f = x^f *+ n.

Proof. elim: n => [|n IHn]; first exact: pFrobenius_aut0. by rewrite !mulrS pFrobenius_autD_comm ?IHn //; apply: commrMn. Qed.

Lemma

pFrobenius_autMn

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", "commrMn", "mulrS", "pFrobenius_aut0", "pFrobenius_autD_comm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pFrobenius_aut_nat n : (n%:R)^f = n%:R.

Proof. by rewrite pFrobenius_autMn pFrobenius_aut1. Qed.

Lemma

pFrobenius_aut_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" ]

[ "pFrobenius_aut1", "pFrobenius_autMn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

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

Proof. exact: exprMn_comm. Qed.

Lemma

pFrobenius_autM_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", "exprMn_comm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pFrobenius_autX x n : (x ^+ n)^f = x^f ^+ n.

Proof. by rewrite !f'E -!exprM mulnC. Qed.

Lemma

pFrobenius_autX

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" ]

[ "exprM", "f'E", "mulnC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcharR2 : 2 \in pchar R.

Hypothesis

pcharR2

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" ]

[ "pchar" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addrr_pchar2 x : x + x = 0.

Proof. by rewrite -mulr2n mulrn_pchar. Qed.

Lemma

addrr_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" ]

[ "mulr2n", "mulrn_pchar" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"1"

:= one.

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" ]

[ "one" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x * y"

:= (mul x y).

Notation

x * y

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" ]

[ "mul" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mul0r : @left_zero R R 0 mul.

Proof. by move=> x; apply: (addIr (1 * x)); rewrite -mulrDl !add0r mul1r. Qed.

Lemma

mul0r

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", "mul", "mul1r", "mulrDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulr0 : @right_zero R R 0 mul.

Proof. by move=> x; apply: (addIr (x * 1)); rewrite -mulrDr !add0r mulr1. Qed.

Lemma

mulr0

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", "mul", "mulr1", "mulrDr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sign R b

:= (exp (- @one R) (nat_of_bool b)) (only parsing).

Notation

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" ]

[ "exp", "nat_of_bool", "one" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d