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	source_url stringclasses 1 value	commit stringclasses 1 value
invr_closed	:= {in S, forall x, x^-1 \in S}.	Definition	invr_closed	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divr_2closed	:= {in S &, forall x y, x / y \in S}.	Definition	divr_2closed	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divr_closed	:= 1 \in S /\ divr_2closed.	Definition	divr_closed	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "divr_2closed" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdivr_closed	:= -1 \in S /\ divr_2closed.	Definition	sdivr_closed	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "divr_2closed" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divring_closed	:= [/\ 1 \in S, subr_closed S & divr_2closed].	Definition	divring_closed	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "divr_2closed", "subr_closed" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divr_closedV : divr_closed -> invr_closed.	Proof. by case=> S1 Sdiv x Sx; rewrite -[x^-1]mul1r Sdiv. Qed.	Lemma	divr_closedV	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "S1", "divr_closed", "invr_closed", "mul1r" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divr_closedM : divr_closed -> mulr_closed S.	Proof. by case=> S1 Sdiv; split=> // x y Sx Sy; rewrite -[y]invrK -[y^-1]mul1r !Sdiv. Qed.	Lemma	divr_closedM	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "S1", "divr_closed", "invrK", "mul1r", "mulr_closed", "split" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdivr_closed_div : sdivr_closed -> divr_closed.	Proof. by case=> SN1 Sdiv; split; rewrite // -(divrr (@unitrN1 _)) Sdiv. Qed.	Lemma	sdivr_closed_div	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "divr_closed", "divrr", "sdivr_closed", "split", "unitrN1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdivr_closedM : sdivr_closed -> smulr_closed S.	Proof. by move=> Sdiv; have [_ SM] := divr_closedM (sdivr_closed_div Sdiv); case: Sdiv. Qed.	Lemma	sdivr_closedM	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "divr_closedM", "sdivr_closed", "sdivr_closed_div", "smulr_closed" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divring_closedBM : divring_closed -> subring_closed S.	Proof. by case=> S1 SB Sdiv; split=> //; case: divr_closedM. Qed.	Lemma	divring_closedBM	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "S1", "divr_closedM", "divring_closed", "split", "subring_closed" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divring_closed_div : divring_closed -> sdivr_closed.	Proof. case=> S1 SB Sdiv; split; rewrite ?zmod_closedN //. exact/subring_closedB/divring_closedBM. Qed.	Lemma	divring_closed_div	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "S1", "divring_closed", "divring_closedBM", "sdivr_closed", "split", "subring_closedB", "zmod_closedN" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmorph_unit x : x \in unit -> f x \in unit.	Proof. case/unitrP=> y [yx1 xy1]; apply/unitrP. by exists (f y); rewrite -!rmorphM // yx1 xy1 rmorph1. Qed.	Lemma	rmorph_unit	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "rmorph1", "rmorphM", "unit", "unitrP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmorphV : {in unit, {morph f: x / x^-1}}.	Proof. move=> x Ux; rewrite /= -[(f x)^-1]mul1r. by apply: (canRL (mulrK (rmorph_unit Ux))); rewrite -rmorphM mulVr ?rmorph1. Qed.	Lemma	rmorphV	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "mul1r", "mulVr", "mulrK", "rmorph1", "rmorphM", "rmorph_unit", "unit" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rmorph_div x y : y \in unit -> f (x / y) = f x / f y.	Proof. by move=> Uy; rewrite rmorphM /= rmorphV. Qed.	Lemma	rmorph_div	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "rmorphM", "rmorphV", "unit" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulC_mulrV : {in unit, right_inverse 1 inv *%R}.	Proof. by move=> x Ux /=; rewrite mulrC mulVx. Qed.	Fact	mulC_mulrV	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "inv", "mulrC", "unit" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulC_unitP x y : y * x = 1 /\ x * y = 1 -> unit x.	Proof. by case=> yx _; apply: unitPl yx. Qed.	Fact	mulC_unitP	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "unit" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unitrM x y : (x * y \in unit) = (x \in unit) && (y \in unit).	Proof. exact/unitrM_comm/mulrC. Qed.	Lemma	unitrM	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "mulrC", "unit", "unitrM_comm" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unitrPr x : reflect (exists y, x * y = 1) (x \in unit).	Proof. by apply: (iffP (unitrP x)) => [[y []] \| [y]]; exists y; rewrite // mulrC. Qed.	Lemma	unitrPr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "mulrC", "unit", "unitrP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulr1_eq x y : x * y = 1 -> x^-1 = y.	Proof. by move=> xy_eq1; rewrite -[LHS]mulr1 -xy_eq1; apply/mulKr/unitrPr; exists y. Qed.	Lemma	mulr1_eq	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "mulKr", "mulr1", "unitrPr" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divr1_eq x y : x / y = 1 -> x = y.	Proof. by move/mulr1_eq/invr_inj. Qed.	Lemma	divr1_eq	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "invr_inj", "mulr1_eq" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divKr x : x \is a unit -> {in unit, involutive (fun y => x / y)}.	Proof. by move=> Ux y Uy; rewrite /= invrM ?unitrV // invrK mulrC divrK. Qed.	Lemma	divKr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "divrK", "invrK", "invrM", "mulrC", "unit", "unitrV" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expr_div_n x y n : (x / y) ^+ n = x ^+ n / y ^+ n.	Proof. by rewrite exprMn exprVn. Qed.	Lemma	expr_div_n	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "exprMn", "exprVn" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unitr_prodP (I : eqType) (r : seq I) (P : pred I) (E : I -> R) : reflect {in r, forall i, P i -> E i \is a GRing.unit} (\prod_(i <- r \| P i) E i \is a GRing.unit).	Proof. rewrite (big_morph [in unit] unitrM (@unitr1 _) ) big_all_cond. exact: 'all_implyP. Qed.	Lemma	unitr_prodP	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "big_all_cond", "big_morph", "seq", "unit", "unitr1", "unitrM" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodrV (I : eqType) (r : seq I) (P : pred I) (E : I -> R) : (forall i, P i -> E i \is a GRing.unit) -> \prod_(i <- r \| P i) (E i)^-1 = (\prod_(i <- r \| P i) E i)^-1.	Proof. by move=> /rev_prodrV->; rewrite rev_prodr (perm_big r)// perm_rev. Qed.	Lemma	prodrV	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "perm_big", "perm_rev", "rev_prodr", "rev_prodrV", "seq", "unit" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
scaler_injl : {in unit, @right_injective R A A *:%R}.	Proof. move=> k Uk x1 x2 Hx1x2. by rewrite -[x1]scale1r -(mulVr Uk) -scalerA Hx1x2 scalerA mulVr // scale1r. Qed.	Lemma	scaler_injl	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "mulVr", "scale1r", "scalerA", "unit" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
scaler_unit k x : k \in unit -> (k *: x \in unit) = (x \in unit).	Proof. move=> Uk; apply/idP/idP=> [Ukx \| Ux]; apply/unitrP; last first. exists (k^-1 : x^-1). by rewrite -!scalerAl -!scalerAr !scalerA !mulVr // !mulrV // scale1r. exists (k : (k *: x)^-1); split. apply: (mulrI Ukx). by rewrite mulr1 mulrA -scalerAr mulrV // -scalerAl mul1r. apply: (mulIr Ukx). by rewrite mu...	Lemma	scaler_unit	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "last", "mul1r", "mulIr", "mulVr", "mulr1", "mulrA", "mulrI", "mulrV", "scale1r", "scalerA", "scalerAl", "scalerAr", "split", "unit", "unitrP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invrZ k x : k \in unit -> x \in unit -> (k : x)^-1 = k^-1 : x^-1.	Proof. move=> Uk Ux; have Ukx: (k *: x \in unit) by rewrite scaler_unit. apply: (mulIr Ukx). by rewrite mulVr // -scalerAl -scalerAr scalerA !mulVr // scale1r. Qed.	Lemma	invrZ	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "mulIr", "mulVr", "scale1r", "scalerA", "scalerAl", "scalerAr", "scaler_unit", "unit" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divalg_closed	:= [/\ 1 \in S, linear_closed S & divr_2closed S].	Definition	divalg_closed	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "divr_2closed", "linear_closed" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divalg_closedBdiv : divalg_closed -> divring_closed S.	Proof. by case=> S1 /linear_closedB. Qed.	Lemma	divalg_closedBdiv	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "S1", "divalg_closed", "divring_closed", "linear_closedB" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divalg_closedZ : divalg_closed -> subalg_closed S.	Proof. by case=> S1 Slin Sdiv; split=> //; have [] := @divr_closedM A S. Qed.	Lemma	divalg_closedZ	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "S1", "divalg_closed", "divr_closedM", "split", "subalg_closed" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invr_closed	:= invr_closed.	Notation	invr_closed	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divr_2closed	:= divr_2closed.	Notation	divr_2closed	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divr_closed	:= divr_closed.	Notation	divr_closed	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdivr_closed	:= sdivr_closed.	Notation	sdivr_closed	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divring_closed	:= divring_closed.	Notation	divring_closed	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divalg_closed	:= divalg_closed.	Notation	divalg_closed	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divr_closedV : divr_closed >-> invr_closed.		Coercion	divr_closedV	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "divr_closed", "invr_closed" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divr_closedM : divr_closed >-> mulr_closed.		Coercion	divr_closedM	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "divr_closed", "mulr_closed" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdivr_closed_div : sdivr_closed >-> divr_closed.		Coercion	sdivr_closed_div	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "divr_closed", "sdivr_closed" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sdivr_closedM : sdivr_closed >-> smulr_closed.		Coercion	sdivr_closedM	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "sdivr_closed", "smulr_closed" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divring_closedBM : divring_closed >-> subring_closed.		Coercion	divring_closedBM	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "divring_closed", "subring_closed" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divring_closed_div : divring_closed >-> sdivr_closed.		Coercion	divring_closed_div	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "divring_closed", "sdivr_closed" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divalg_closedBdiv : divalg_closed >-> divring_closed.		Coercion	divalg_closedBdiv	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "divalg_closed", "divring_closed" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divalg_closedZ : divalg_closed >-> subalg_closed.		Coercion	divalg_closedZ	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "divalg_closed", "subalg_closed" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
integral_domain_axiom (R : pzRingType)	:= forall x y : R, x * y = 0 -> (x == 0) \|\| (y == 0).	Definition	integral_domain_axiom	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulf_eq0 x y : (x * y == 0) = (x == 0) \|\| (y == 0).	Proof. apply/eqP/idP; first exact: mulf_eq0_subproof. by case/pred2P=> ->; rewrite (mulr0, mul0r). Qed.	Lemma	mulf_eq0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "mul0r", "mulr0", "pred2P" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodf_eq0 (I : finType) (P : pred I) (F : I -> R) : reflect (exists2 i, P i & (F i == 0)) (\prod_(i \| P i) F i == 0).	Proof. apply: (iffP idP) => [\|[i Pi /eqP Fi0]]; last first. by rewrite (bigD1 i) //= Fi0 mul0r. elim: (index_enum _) => [\|i r IHr]; first by rewrite big_nil oner_eq0. rewrite big_cons /=; have [Pi \| _] := ifP; last exact: IHr. by rewrite mulf_eq0; case/orP=> // Fi0; exists i. Qed.	Lemma	prodf_eq0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "bigD1", "big_cons", "big_nil", "index_enum", "last", "mul0r", "mulf_eq0", "oner_eq0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodf_seq_eq0 I r (P : pred I) (F : I -> R) : (\prod_(i <- r \| P i) F i == 0) = has (fun i => P i && (F i == 0)) r.	Proof. by rewrite (big_morph _ mulf_eq0 (oner_eq0 _)) big_has_cond. Qed.	Lemma	prodf_seq_eq0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "big_has_cond", "big_morph", "has", "mulf_eq0", "oner_eq0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulf_neq0 x y : x != 0 -> y != 0 -> x * y != 0.	Proof. by move=> x0 y0; rewrite mulf_eq0; apply/norP. Qed.	Lemma	mulf_neq0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "mulf_eq0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodf_neq0 (I : finType) (P : pred I) (F : I -> R) : reflect (forall i, P i -> (F i != 0)) (\prod_(i \| P i) F i != 0).	Proof. by rewrite (sameP (prodf_eq0 _ _) exists_inP); apply: exists_inPn. Qed.	Lemma	prodf_neq0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "exists_inP", "exists_inPn", "prodf_eq0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodf_seq_neq0 I r (P : pred I) (F : I -> R) : (\prod_(i <- r \| P i) F i != 0) = all (fun i => P i ==> (F i != 0)) r.	Proof. rewrite prodf_seq_eq0 -all_predC; apply: eq_all => i /=. by rewrite implybE negb_and. Qed.	Lemma	prodf_seq_neq0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "all", "all_predC", "apply", "eq_all", "prodf_seq_eq0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expf_eq0 x n : (x ^+ n == 0) = (n > 0) && (x == 0).	Proof. elim: n => [\|n IHn]; first by rewrite oner_eq0. by rewrite exprS mulf_eq0 IHn andKb. Qed.	Lemma	expf_eq0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "andKb", "exprS", "mulf_eq0", "oner_eq0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sqrf_eq0 x : (x ^+ 2 == 0) = (x == 0).	Proof. exact: expf_eq0. Qed.	Lemma	sqrf_eq0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "expf_eq0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expf_neq0 x m : x != 0 -> x ^+ m != 0.	Proof. by move=> x_nz; rewrite expf_eq0; apply/nandP; right. Qed.	Lemma	expf_neq0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "expf_eq0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
natf_neq0_pchar n : (n%:R != 0 :> R) = (pchar R)^'.-nat n.	Proof. have [-> \| /prod_prime_decomp->] := posnP n; first by rewrite eqxx. rewrite !big_seq; elim/big_rec: _ => [\|[p e] s /=]; first by rewrite oner_eq0. case/mem_prime_decomp=> p_pr _ _; rewrite pnatM pnatX eqn0Ngt orbC => <-. by rewrite natrM natrX mulf_eq0 expf_eq0 negb_or negb_and pnatE ?inE p_pr. Qed.	Lemma	natf_neq0_pchar	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "big_rec", "big_seq", "eqn0Ngt", "eqxx", "expf_eq0", "inE", "mem_prime_decomp", "mulf_eq0", "nat", "natrM", "natrX", "oner_eq0", "p_pr", "pchar", "pnatE", "pnatM", "pnatX", "posnP", "prod_prime_decomp" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
natf0_pchar n : n > 0 -> n%:R == 0 :> R -> exists p, p \in pchar R.	Proof. move=> n_gt0 nR_0; exists (pdiv n`_(pchar R)). apply: pnatP (pdiv_dvd _); rewrite ?part_pnat // ?pdiv_prime //. by rewrite ltn_neqAle eq_sym partn_eq1 // -natf_neq0_pchar nR_0 /=. Qed.	Lemma	natf0_pchar	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "eq_sym", "ltn_neqAle", "n_gt0", "natf_neq0_pchar", "part_pnat", "partn_eq1", "pchar", "pdiv", "pdiv_dvd", "pdiv_prime", "pnatP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcharf'_nat n : (pchar R)^'.-nat n = (n%:R != 0 :> R).	Proof. have [-> \| n_gt0] := posnP n; first by rewrite eqxx. apply/idP/idP => [\|nz_n]; last first. by apply/pnatP=> // p p_pr p_dvd_n; apply: contra nz_n => /dvdn_pcharf <-. apply: contraL => n0; have [// \| p pcharRp] := natf0_pchar _ n0. have [p_pr _] := andP pcharRp; rewrite (eq_pnat _ (eq_negn (pcharf_eq pcharRp)))...	Lemma	pcharf'_nat	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "dvdn_pcharf", "eq_negn", "eq_pnat", "eqxx", "last", "n_gt0", "nat", "natf0_pchar", "p'natE", "p_pr", "pchar", "pcharRp", "pcharf_eq", "pnatP", "posnP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pcharf0P : pchar R =i pred0 <-> (forall n, (n%:R == 0 :> R) = (n == 0)%N).	Proof. split=> pcharF0 n; last by rewrite !inE pcharF0 andbC; case: eqP => // ->. have [-> \| n_gt0] := posnP; first exact: eqxx. by apply/negP; case/natf0_pchar=> // p; rewrite pcharF0. Qed.	Lemma	pcharf0P	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "eqxx", "inE", "last", "n_gt0", "natf0_pchar", "pchar", "posnP", "split" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eqf_sqr x y : (x ^+ 2 == y ^+ 2) = (x == y) \|\| (x == - y).	Proof. by rewrite -subr_eq0 subr_sqr mulf_eq0 subr_eq0 addr_eq0. Qed.	Lemma	eqf_sqr	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "addr_eq0", "mulf_eq0", "subr_eq0", "subr_sqr" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulfI x : x != 0 -> injective ( *%R x).	Proof. move=> nz_x y z; apply: contra_eq => neq_yz. by rewrite -subr_eq0 -mulrBr mulf_neq0 ?subr_eq0. Qed.	Lemma	mulfI	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "contra_eq", "mulf_neq0", "mulrBr", "subr_eq0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulIf x : x != 0 -> injective ( *%R^~ x).	Proof. by move=> nz_x y z; rewrite -!(mulrC x); apply: mulfI. Qed.	Lemma	mulIf	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "mulfI", "mulrC" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divfI x : x != 0 -> injective (fun y => x / y).	Proof. by move/mulfI/inj_comp; apply; apply: invr_inj. Qed.	Lemma	divfI	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "invr_inj", "mulfI" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divIf y : y != 0 -> injective (fun x => x / y).	Proof. by rewrite -invr_eq0; apply: mulIf. Qed.	Lemma	divIf	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "invr_eq0", "mulIf" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sqrf_eq1 x : (x ^+ 2 == 1) = (x == 1) \|\| (x == -1).	Proof. by rewrite -subr_eq0 subr_sqr_1 mulf_eq0 subr_eq0 addr_eq0. Qed.	Lemma	sqrf_eq1	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "addr_eq0", "mulf_eq0", "subr_eq0", "subr_sqr_1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expfS_eq1 x n : (x ^+ n.+1 == 1) = (x == 1) \|\| (\sum_(i < n.+1) x ^+ i == 0).	Proof. by rewrite -![_ == 1]subr_eq0 subrX1 mulf_eq0. Qed.	Lemma	expfS_eq1	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "mulf_eq0", "subrX1", "subr_eq0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lregP x : reflect (lreg x) (x != 0).	Proof. by apply: (iffP idP) => [/mulfI \| /lreg_neq0]. Qed.	Lemma	lregP	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "lreg", "lreg_neq0", "mulfI" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rregP x : reflect (rreg x) (x != 0).	Proof. by apply: (iffP idP) => [/mulIf \| /rreg_neq0]. Qed.	Lemma	rregP	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "mulIf", "rreg", "rreg_neq0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
field_axiom (R : unitRingType)	:= forall x : R, x != 0 -> x \in unit.	Definition	field_axiom	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "unit" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
IdomainMixin (R : unitRingType): Field.axiom R -> IntegralDomain.axiom R.	Proof. move=> m x y xy0; apply/norP=> [[]] /m Ux /m. by rewrite -(unitrMr _ Ux) xy0 unitr0. Qed.	Lemma	IdomainMixin	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "axiom", "unitr0", "unitrMr" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
intro_unit (x y : R) : y * x = 1 -> x != 0.	Proof. move=> yx1; apply: contraNneq (@oner_neq0 R) => x0. by rewrite -yx1 x0 mulr0. Qed.	Fact	intro_unit	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "contraNneq", "mulr0", "oner_neq0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
inv_out : {in predC (predC1 0), inv =1 id}.	Proof. by move=> x /negbNE/eqP->; exact: invr0. Qed.	Fact	inv_out	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "id", "inv", "invr0", "predC1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
unitfE x : (x \in unit) = (x != 0).	Proof. by apply/idP/idP=> [/(memPn _)-> \| /fieldP]; rewrite ?unitr0. Qed.	Lemma	unitfE	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "fieldP", "memPn", "unit", "unitr0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulVf x : x != 0 -> x^-1 * x = 1.	Proof. by rewrite -unitfE; apply: mulVr. Qed.	Lemma	mulVf	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "mulVr", "unitfE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divff x : x != 0 -> x / x = 1.	Proof. by rewrite -unitfE; apply: divrr. Qed.	Lemma	divff	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "divrr", "unitfE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulfV	:= divff.	Definition	mulfV	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "divff" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulKf x : x != 0 -> cancel ( %R x) ( %R x^-1).	Proof. by rewrite -unitfE; apply: mulKr. Qed.	Lemma	mulKf	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "mulKr", "unitfE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulVKf x : x != 0 -> cancel ( %R x^-1) ( %R x).	Proof. by rewrite -unitfE; apply: mulVKr. Qed.	Lemma	mulVKf	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "mulVKr", "unitfE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulfK x : x != 0 -> cancel ( %R^~ x) ( %R^~ x^-1).	Proof. by rewrite -unitfE; apply: mulrK. Qed.	Lemma	mulfK	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "mulrK", "unitfE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulfVK x : x != 0 -> cancel ( %R^~ x^-1) ( %R^~ x).	Proof. by rewrite -unitfE; apply: divrK. Qed.	Lemma	mulfVK	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "divrK", "unitfE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divfK	:= mulfVK.	Definition	divfK	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "mulfVK" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invfM : {morph @inv F : x y / x * y}.	Proof. move=> x y; have [->\|nzx] := eqVneq x 0; first by rewrite !(mul0r, invr0). have [->\|nzy] := eqVneq y 0; first by rewrite !(mulr0, invr0). by rewrite mulrC invrM ?unitfE. Qed.	Lemma	invfM	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "eqVneq", "inv", "invr0", "invrM", "mul0r", "mulr0", "mulrC", "unitfE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
invf_div x y : (x / y)^-1 = y / x.	Proof. by rewrite invfM invrK mulrC. Qed.	Lemma	invf_div	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "invfM", "invrK", "mulrC" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
divKf x : x != 0 -> involutive (fun y => x / y).	Proof. by move=> nz_x y; rewrite invf_div mulrC divfK. Qed.	Lemma	divKf	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "divfK", "invf_div", "mulrC" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expfB_cond m n x : (x == 0) + n <= m -> x ^+ (m - n) = x ^+ m / x ^+ n.	Proof. move/subnK=> <-; rewrite addnA addnK !exprD. have [-> \| nz_x] := eqVneq; first by rewrite !mulr0 !mul0r. by rewrite mulfK ?expf_neq0. Qed.	Lemma	expfB_cond	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "addnA", "addnK", "eqVneq", "expf_neq0", "exprD", "mul0r", "mulfK", "mulr0", "subnK" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
expfB m n x : n < m -> x ^+ (m - n) = x ^+ m / x ^+ n.	Proof. by move=> lt_n_m; apply: expfB_cond; case: eqP => // _; apply: ltnW. Qed.	Lemma	expfB	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "expfB_cond", "ltnW" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodfV I r (P : pred I) (E : I -> F) : \prod_(i <- r \| P i) (E i)^-1 = (\prod_(i <- r \| P i) E i)^-1.	Proof. by rewrite (big_morph _ invfM (invr1 F)). Qed.	Lemma	prodfV	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "big_morph", "invfM", "invr1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prodf_div I r (P : pred I) (E D : I -> F) : \prod_(i <- r \| P i) (E i / D i) = \prod_(i <- r \| P i) E i / \prod_(i <- r \| P i) D i.	Proof. by rewrite big_split prodfV. Qed.	Lemma	prodf_div	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "big_split", "prodfV" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
telescope_prodf n m (f : nat -> F) : (forall k, n < k < m -> f k != 0) -> n < m -> \prod_(n <= k < m) (f k.+1 / f k) = f m / f n.	Proof. move=> nz_f ltnm; apply: invr_inj; rewrite prodf_div !invf_div -prodf_div. by apply: telescope_prodr => // k /nz_f; rewrite unitfE. Qed.	Lemma	telescope_prodf	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "invf_div", "invr_inj", "nat", "prodf_div", "telescope_prodr", "unitfE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
telescope_prodf_eq n m (f u : nat -> F) : (forall k, n < k < m -> f k != 0) -> n < m -> (forall k, n <= k < m -> u k = f k.+1 / f k) -> \prod_(n <= k < m) u k = f m / f n.	Proof. by move=> ? ? uE; under eq_big_nat do rewrite uE //=; exact: telescope_prodf. Qed.	Lemma	telescope_prodf_eq	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "eq_big_nat", "nat", "telescope_prodf" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addf_div x1 y1 x2 y2 : y1 != 0 -> y2 != 0 -> x1 / y1 + x2 / y2 = (x1 * y2 + x2 * y1) / (y1 * y2).	Proof. by move=> nzy1 nzy2; rewrite invfM mulrDl !mulrA mulrAC !mulfK. Qed.	Lemma	addf_div	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "invfM", "mulfK", "mulrA", "mulrAC", "mulrDl" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulf_div x1 y1 x2 y2 : (x1 / y1) * (x2 / y2) = (x1 * x2) / (y1 * y2).	Proof. by rewrite mulrACA -invfM. Qed.	Lemma	mulf_div	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "invfM", "mulrACA" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eqr_div x y z t : y != 0 -> t != 0 -> (x / y == z / t) = (x * t == z * y).	Proof. move=> yD0 tD0; rewrite -[x in RHS](divfK yD0) -[z in RHS](divfK tD0) mulrAC. by apply/eqP/eqP => [->\|/(mulIf yD0)/(mulIf tD0)]. Qed.	Lemma	eqr_div	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "divfK", "mulIf", "mulrAC" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eqr_sum_div I r P (f : I -> F) c a : c != 0 -> (\big[+%R/0]_(x <- r \| P x) (f x / c) == a) = (\big[+%R/0]_(x <- r \| P x) f x == a * c).	Proof. by move=> ?; rewrite -mulr_suml -(divr1 a) eqr_div ?oner_eq0// mulr1 divr1. Qed.	Lemma	eqr_sum_div	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "divr1", "eqr_div", "mulr1", "mulr_suml", "oner_eq0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pchar0_natf_div : pchar F =i pred0 -> forall m d, d %\| m -> (m %/ d)%:R = m%:R / d%:R :> F.	Proof. move/pcharf0P=> pchar0F m [\|d] d_dv_m; first by rewrite divn0 invr0 mulr0. by rewrite natr_div // unitfE pchar0F. Qed.	Lemma	pchar0_natf_div	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "divn0", "invr0", "mulr0", "natr_div", "pchar", "pcharf0P", "unitfE" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fmorph_eq0 x : (f x == 0) = (x == 0).	Proof. have [-> \| nz_x] := eqVneq x; first by rewrite rmorph0 eqxx. apply/eqP; move/(congr1 ( *%R (f x^-1)))/eqP. by rewrite -rmorphM mulVf // mulr0 rmorph1 ?oner_eq0. Qed.	Lemma	fmorph_eq0	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "eqVneq", "eqxx", "mulVf", "mulr0", "oner_eq0", "rmorph0", "rmorph1", "rmorphM" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fmorph_inj : injective f.	Proof. by apply/raddf_inj => x /eqP; rewrite fmorph_eq0 => /eqP. Qed.	Lemma	fmorph_inj	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "fmorph_eq0", "raddf_inj" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fmorph_eq : {mono f : x y / x == y}.	Proof. exact: inj_eq fmorph_inj. Qed.	Lemma	fmorph_eq	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "fmorph_inj", "inj_eq" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fmorph_eq1 x : (f x == 1) = (x == 1).	Proof. by rewrite -(inj_eq fmorph_inj) rmorph1. Qed.	Lemma	fmorph_eq1	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "fmorph_inj", "inj_eq", "rmorph1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fmorph_pchar : pchar R =i pchar F.	Proof. by move=> p; rewrite !inE -fmorph_eq0 rmorph_nat. Qed.	Lemma	fmorph_pchar	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "fmorph_eq0", "inE", "pchar", "rmorph_nat" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
fmorph_unit x : (f x \in unit) = (x != 0).	Proof. have [-> \|] := eqVneq x; first by rewrite rmorph0 unitr0. by rewrite -unitfE; apply: rmorph_unit. Qed.	Lemma	fmorph_unit	algebra.algebraic_hierarchy	algebra/algebraic_hierarchy/divalg.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]	[ "apply", "eqVneq", "rmorph0", "rmorph_unit", "unit", "unitfE", "unitr0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

invr_closed

:= {in S, forall x, x^-1 \in S}.

Definition

invr_closed

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divr_2closed

:= {in S &, forall x y, x / y \in S}.

Definition

divr_2closed

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divr_closed

:= 1 \in S /\ divr_2closed.

Definition

divr_closed

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "divr_2closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdivr_closed

:= -1 \in S /\ divr_2closed.

Definition

sdivr_closed

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "divr_2closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divring_closed

:= [/\ 1 \in S, subr_closed S & divr_2closed].

Definition

divring_closed

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "divr_2closed", "subr_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divr_closedV : divr_closed -> invr_closed.

Proof. by case=> S1 Sdiv x Sx; rewrite -[x^-1]mul1r Sdiv. Qed.

Lemma

divr_closedV

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "S1", "divr_closed", "invr_closed", "mul1r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divr_closedM : divr_closed -> mulr_closed S.

Proof. by case=> S1 Sdiv; split=> // x y Sx Sy; rewrite -[y]invrK -[y^-1]mul1r !Sdiv. Qed.

Lemma

divr_closedM

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "S1", "divr_closed", "invrK", "mul1r", "mulr_closed", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdivr_closed_div : sdivr_closed -> divr_closed.

Proof. by case=> SN1 Sdiv; split; rewrite // -(divrr (@unitrN1 _)) Sdiv. Qed.

Lemma

sdivr_closed_div

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "divr_closed", "divrr", "sdivr_closed", "split", "unitrN1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdivr_closedM : sdivr_closed -> smulr_closed S.

Proof. by move=> Sdiv; have [_ SM] := divr_closedM (sdivr_closed_div Sdiv); case: Sdiv. Qed.

Lemma

sdivr_closedM

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "divr_closedM", "sdivr_closed", "sdivr_closed_div", "smulr_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divring_closedBM : divring_closed -> subring_closed S.

Proof. by case=> S1 SB Sdiv; split=> //; case: divr_closedM. Qed.

Lemma

divring_closedBM

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "S1", "divr_closedM", "divring_closed", "split", "subring_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divring_closed_div : divring_closed -> sdivr_closed.

Proof. case=> S1 SB Sdiv; split; rewrite ?zmod_closedN //. exact/subring_closedB/divring_closedBM. Qed.

Lemma

divring_closed_div

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "S1", "divring_closed", "divring_closedBM", "sdivr_closed", "split", "subring_closedB", "zmod_closedN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rmorph_unit x : x \in unit -> f x \in unit.

Proof. case/unitrP=> y [yx1 xy1]; apply/unitrP. by exists (f y); rewrite -!rmorphM // yx1 xy1 rmorph1. Qed.

Lemma

rmorph_unit

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "rmorph1", "rmorphM", "unit", "unitrP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rmorphV : {in unit, {morph f: x / x^-1}}.

Proof. move=> x Ux; rewrite /= -[(f x)^-1]mul1r. by apply: (canRL (mulrK (rmorph_unit Ux))); rewrite -rmorphM mulVr ?rmorph1. Qed.

Lemma

rmorphV

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "mul1r", "mulVr", "mulrK", "rmorph1", "rmorphM", "rmorph_unit", "unit" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rmorph_div x y : y \in unit -> f (x / y) = f x / f y.

Proof. by move=> Uy; rewrite rmorphM /= rmorphV. Qed.

Lemma

rmorph_div

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "rmorphM", "rmorphV", "unit" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulC_mulrV : {in unit, right_inverse 1 inv *%R}.

Proof. by move=> x Ux /=; rewrite mulrC mulVx. Qed.

Fact

mulC_mulrV

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "inv", "mulrC", "unit" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulC_unitP x y : y * x = 1 /\ x * y = 1 -> unit x.

Proof. by case=> yx _; apply: unitPl yx. Qed.

Fact

mulC_unitP

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "unit" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unitrM x y : (x * y \in unit) = (x \in unit) && (y \in unit).

Proof. exact/unitrM_comm/mulrC. Qed.

Lemma

unitrM

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "mulrC", "unit", "unitrM_comm" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unitrPr x : reflect (exists y, x * y = 1) (x \in unit).

Proof. by apply: (iffP (unitrP x)) => [[y []] | [y]]; exists y; rewrite // mulrC. Qed.

Lemma

unitrPr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "mulrC", "unit", "unitrP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulr1_eq x y : x * y = 1 -> x^-1 = y.

Proof. by move=> xy_eq1; rewrite -[LHS]mulr1 -xy_eq1; apply/mulKr/unitrPr; exists y. Qed.

Lemma

mulr1_eq

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "mulKr", "mulr1", "unitrPr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divr1_eq x y : x / y = 1 -> x = y.

Proof. by move/mulr1_eq/invr_inj. Qed.

Lemma

divr1_eq

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "invr_inj", "mulr1_eq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divKr x : x \is a unit -> {in unit, involutive (fun y => x / y)}.

Proof. by move=> Ux y Uy; rewrite /= invrM ?unitrV // invrK mulrC divrK. Qed.

Lemma

divKr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "divrK", "invrK", "invrM", "mulrC", "unit", "unitrV" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expr_div_n x y n : (x / y) ^+ n = x ^+ n / y ^+ n.

Proof. by rewrite exprMn exprVn. Qed.

Lemma

expr_div_n

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "exprMn", "exprVn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unitr_prodP (I : eqType) (r : seq I) (P : pred I) (E : I -> R) : reflect {in r, forall i, P i -> E i \is a GRing.unit} (\prod_(i <- r | P i) E i \is a GRing.unit).

Proof. rewrite (big_morph [in unit] unitrM (@unitr1 _) ) big_all_cond. exact: 'all_implyP. Qed.

Lemma

unitr_prodP

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "big_all_cond", "big_morph", "seq", "unit", "unitr1", "unitrM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodrV (I : eqType) (r : seq I) (P : pred I) (E : I -> R) : (forall i, P i -> E i \is a GRing.unit) -> \prod_(i <- r | P i) (E i)^-1 = (\prod_(i <- r | P i) E i)^-1.

Proof. by move=> /rev_prodrV->; rewrite rev_prodr (perm_big r)// perm_rev. Qed.

Lemma

prodrV

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "perm_big", "perm_rev", "rev_prodr", "rev_prodrV", "seq", "unit" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

scaler_injl : {in unit, @right_injective R A A *:%R}.

Proof. move=> k Uk x1 x2 Hx1x2. by rewrite -[x1]scale1r -(mulVr Uk) -scalerA Hx1x2 scalerA mulVr // scale1r. Qed.

Lemma

scaler_injl

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "mulVr", "scale1r", "scalerA", "unit" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

scaler_unit k x : k \in unit -> (k *: x \in unit) = (x \in unit).

Proof. move=> Uk; apply/idP/idP=> [Ukx | Ux]; apply/unitrP; last first. exists (k^-1 *: x^-1). by rewrite -!scalerAl -!scalerAr !scalerA !mulVr // !mulrV // scale1r. exists (k *: (k *: x)^-1); split. apply: (mulrI Ukx). by rewrite mulr1 mulrA -scalerAr mulrV // -scalerAl mul1r. apply: (mulIr Ukx). by rewrite mu...

Lemma

scaler_unit

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "last", "mul1r", "mulIr", "mulVr", "mulr1", "mulrA", "mulrI", "mulrV", "scale1r", "scalerA", "scalerAl", "scalerAr", "split", "unit", "unitrP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invrZ k x : k \in unit -> x \in unit -> (k *: x)^-1 = k^-1 *: x^-1.

Proof. move=> Uk Ux; have Ukx: (k *: x \in unit) by rewrite scaler_unit. apply: (mulIr Ukx). by rewrite mulVr // -scalerAl -scalerAr scalerA !mulVr // scale1r. Qed.

Lemma

invrZ

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "mulIr", "mulVr", "scale1r", "scalerA", "scalerAl", "scalerAr", "scaler_unit", "unit" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divalg_closed

:= [/\ 1 \in S, linear_closed S & divr_2closed S].

Definition

divalg_closed

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "divr_2closed", "linear_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divalg_closedBdiv : divalg_closed -> divring_closed S.

Proof. by case=> S1 /linear_closedB. Qed.

Lemma

divalg_closedBdiv

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "S1", "divalg_closed", "divring_closed", "linear_closedB" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divalg_closedZ : divalg_closed -> subalg_closed S.

Proof. by case=> S1 Slin Sdiv; split=> //; have [] := @divr_closedM A S. Qed.

Lemma

divalg_closedZ

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "S1", "divalg_closed", "divr_closedM", "split", "subalg_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invr_closed

:= invr_closed.

Notation

invr_closed

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divr_2closed

:= divr_2closed.

Notation

divr_2closed

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divr_closed

:= divr_closed.

Notation

divr_closed

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdivr_closed

:= sdivr_closed.

Notation

sdivr_closed

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divring_closed

:= divring_closed.

Notation

divring_closed

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divalg_closed

:= divalg_closed.

Notation

divalg_closed

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divr_closedV : divr_closed >-> invr_closed.

Coercion

divr_closedV

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "divr_closed", "invr_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divr_closedM : divr_closed >-> mulr_closed.

Coercion

divr_closedM

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "divr_closed", "mulr_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdivr_closed_div : sdivr_closed >-> divr_closed.

Coercion

sdivr_closed_div

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "divr_closed", "sdivr_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sdivr_closedM : sdivr_closed >-> smulr_closed.

Coercion

sdivr_closedM

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "sdivr_closed", "smulr_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divring_closedBM : divring_closed >-> subring_closed.

Coercion

divring_closedBM

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "divring_closed", "subring_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divring_closed_div : divring_closed >-> sdivr_closed.

Coercion

divring_closed_div

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "divring_closed", "sdivr_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divalg_closedBdiv : divalg_closed >-> divring_closed.

Coercion

divalg_closedBdiv

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "divalg_closed", "divring_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divalg_closedZ : divalg_closed >-> subalg_closed.

Coercion

divalg_closedZ

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "divalg_closed", "subalg_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

integral_domain_axiom (R : pzRingType)

:= forall x y : R, x * y = 0 -> (x == 0) || (y == 0).

Definition

integral_domain_axiom

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulf_eq0 x y : (x * y == 0) = (x == 0) || (y == 0).

Proof. apply/eqP/idP; first exact: mulf_eq0_subproof. by case/pred2P=> ->; rewrite (mulr0, mul0r). Qed.

Lemma

mulf_eq0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "mul0r", "mulr0", "pred2P" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodf_eq0 (I : finType) (P : pred I) (F : I -> R) : reflect (exists2 i, P i & (F i == 0)) (\prod_(i | P i) F i == 0).

Proof. apply: (iffP idP) => [|[i Pi /eqP Fi0]]; last first. by rewrite (bigD1 i) //= Fi0 mul0r. elim: (index_enum _) => [|i r IHr]; first by rewrite big_nil oner_eq0. rewrite big_cons /=; have [Pi | _] := ifP; last exact: IHr. by rewrite mulf_eq0; case/orP=> // Fi0; exists i. Qed.

Lemma

prodf_eq0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "bigD1", "big_cons", "big_nil", "index_enum", "last", "mul0r", "mulf_eq0", "oner_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodf_seq_eq0 I r (P : pred I) (F : I -> R) : (\prod_(i <- r | P i) F i == 0) = has (fun i => P i && (F i == 0)) r.

Proof. by rewrite (big_morph _ mulf_eq0 (oner_eq0 _)) big_has_cond. Qed.

Lemma

prodf_seq_eq0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "big_has_cond", "big_morph", "has", "mulf_eq0", "oner_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulf_neq0 x y : x != 0 -> y != 0 -> x * y != 0.

Proof. by move=> x0 y0; rewrite mulf_eq0; apply/norP. Qed.

Lemma

mulf_neq0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "mulf_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodf_neq0 (I : finType) (P : pred I) (F : I -> R) : reflect (forall i, P i -> (F i != 0)) (\prod_(i | P i) F i != 0).

Proof. by rewrite (sameP (prodf_eq0 _ _) exists_inP); apply: exists_inPn. Qed.

Lemma

prodf_neq0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "exists_inP", "exists_inPn", "prodf_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodf_seq_neq0 I r (P : pred I) (F : I -> R) : (\prod_(i <- r | P i) F i != 0) = all (fun i => P i ==> (F i != 0)) r.

Proof. rewrite prodf_seq_eq0 -all_predC; apply: eq_all => i /=. by rewrite implybE negb_and. Qed.

Lemma

prodf_seq_neq0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "all", "all_predC", "apply", "eq_all", "prodf_seq_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expf_eq0 x n : (x ^+ n == 0) = (n > 0) && (x == 0).

Proof. elim: n => [|n IHn]; first by rewrite oner_eq0. by rewrite exprS mulf_eq0 IHn andKb. Qed.

Lemma

expf_eq0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "andKb", "exprS", "mulf_eq0", "oner_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sqrf_eq0 x : (x ^+ 2 == 0) = (x == 0).

Proof. exact: expf_eq0. Qed.

Lemma

sqrf_eq0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "expf_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expf_neq0 x m : x != 0 -> x ^+ m != 0.

Proof. by move=> x_nz; rewrite expf_eq0; apply/nandP; right. Qed.

Lemma

expf_neq0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "expf_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

natf_neq0_pchar n : (n%:R != 0 :> R) = (pchar R)^'.-nat n.

Proof. have [-> | /prod_prime_decomp->] := posnP n; first by rewrite eqxx. rewrite !big_seq; elim/big_rec: _ => [|[p e] s /=]; first by rewrite oner_eq0. case/mem_prime_decomp=> p_pr _ _; rewrite pnatM pnatX eqn0Ngt orbC => <-. by rewrite natrM natrX mulf_eq0 expf_eq0 negb_or negb_and pnatE ?inE p_pr. Qed.

Lemma

natf_neq0_pchar

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "big_rec", "big_seq", "eqn0Ngt", "eqxx", "expf_eq0", "inE", "mem_prime_decomp", "mulf_eq0", "nat", "natrM", "natrX", "oner_eq0", "p_pr", "pchar", "pnatE", "pnatM", "pnatX", "posnP", "prod_prime_decomp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

natf0_pchar n : n > 0 -> n%:R == 0 :> R -> exists p, p \in pchar R.

Proof. move=> n_gt0 nR_0; exists (pdiv n`_(pchar R)). apply: pnatP (pdiv_dvd _); rewrite ?part_pnat // ?pdiv_prime //. by rewrite ltn_neqAle eq_sym partn_eq1 // -natf_neq0_pchar nR_0 /=. Qed.

Lemma

natf0_pchar

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "eq_sym", "ltn_neqAle", "n_gt0", "natf_neq0_pchar", "part_pnat", "partn_eq1", "pchar", "pdiv", "pdiv_dvd", "pdiv_prime", "pnatP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcharf'_nat n : (pchar R)^'.-nat n = (n%:R != 0 :> R).

Proof. have [-> | n_gt0] := posnP n; first by rewrite eqxx. apply/idP/idP => [|nz_n]; last first. by apply/pnatP=> // p p_pr p_dvd_n; apply: contra nz_n => /dvdn_pcharf <-. apply: contraL => n0; have [// | p pcharRp] := natf0_pchar _ n0. have [p_pr _] := andP pcharRp; rewrite (eq_pnat _ (eq_negn (pcharf_eq pcharRp)))...

Lemma

pcharf'_nat

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "dvdn_pcharf", "eq_negn", "eq_pnat", "eqxx", "last", "n_gt0", "nat", "natf0_pchar", "p'natE", "p_pr", "pchar", "pcharRp", "pcharf_eq", "pnatP", "posnP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pcharf0P : pchar R =i pred0 <-> (forall n, (n%:R == 0 :> R) = (n == 0)%N).

Proof. split=> pcharF0 n; last by rewrite !inE pcharF0 andbC; case: eqP => // ->. have [-> | n_gt0] := posnP; first exact: eqxx. by apply/negP; case/natf0_pchar=> // p; rewrite pcharF0. Qed.

Lemma

pcharf0P

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "eqxx", "inE", "last", "n_gt0", "natf0_pchar", "pchar", "posnP", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eqf_sqr x y : (x ^+ 2 == y ^+ 2) = (x == y) || (x == - y).

Proof. by rewrite -subr_eq0 subr_sqr mulf_eq0 subr_eq0 addr_eq0. Qed.

Lemma

eqf_sqr

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "addr_eq0", "mulf_eq0", "subr_eq0", "subr_sqr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulfI x : x != 0 -> injective ( *%R x).

Proof. move=> nz_x y z; apply: contra_eq => neq_yz. by rewrite -subr_eq0 -mulrBr mulf_neq0 ?subr_eq0. Qed.

Lemma

mulfI

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "contra_eq", "mulf_neq0", "mulrBr", "subr_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulIf x : x != 0 -> injective ( *%R^~ x).

Proof. by move=> nz_x y z; rewrite -!(mulrC x); apply: mulfI. Qed.

Lemma

mulIf

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "mulfI", "mulrC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divfI x : x != 0 -> injective (fun y => x / y).

Proof. by move/mulfI/inj_comp; apply; apply: invr_inj. Qed.

Lemma

divfI

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "invr_inj", "mulfI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divIf y : y != 0 -> injective (fun x => x / y).

Proof. by rewrite -invr_eq0; apply: mulIf. Qed.

Lemma

divIf

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "invr_eq0", "mulIf" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sqrf_eq1 x : (x ^+ 2 == 1) = (x == 1) || (x == -1).

Proof. by rewrite -subr_eq0 subr_sqr_1 mulf_eq0 subr_eq0 addr_eq0. Qed.

Lemma

sqrf_eq1

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "addr_eq0", "mulf_eq0", "subr_eq0", "subr_sqr_1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expfS_eq1 x n : (x ^+ n.+1 == 1) = (x == 1) || (\sum_(i < n.+1) x ^+ i == 0).

Proof. by rewrite -![_ == 1]subr_eq0 subrX1 mulf_eq0. Qed.

Lemma

expfS_eq1

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "mulf_eq0", "subrX1", "subr_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lregP x : reflect (lreg x) (x != 0).

Proof. by apply: (iffP idP) => [/mulfI | /lreg_neq0]. Qed.

Lemma

lregP

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "lreg", "lreg_neq0", "mulfI" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rregP x : reflect (rreg x) (x != 0).

Proof. by apply: (iffP idP) => [/mulIf | /rreg_neq0]. Qed.

Lemma

rregP

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "mulIf", "rreg", "rreg_neq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

field_axiom (R : unitRingType)

:= forall x : R, x != 0 -> x \in unit.

Definition

field_axiom

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "unit" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

IdomainMixin (R : unitRingType): Field.axiom R -> IntegralDomain.axiom R.

Proof. move=> m x y xy0; apply/norP=> [[]] /m Ux /m. by rewrite -(unitrMr _ Ux) xy0 unitr0. Qed.

Lemma

IdomainMixin

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "axiom", "unitr0", "unitrMr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

intro_unit (x y : R) : y * x = 1 -> x != 0.

Proof. move=> yx1; apply: contraNneq (@oner_neq0 R) => x0. by rewrite -yx1 x0 mulr0. Qed.

Fact

intro_unit

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "contraNneq", "mulr0", "oner_neq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

inv_out : {in predC (predC1 0), inv =1 id}.

Proof. by move=> x /negbNE/eqP->; exact: invr0. Qed.

Fact

inv_out

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "id", "inv", "invr0", "predC1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

unitfE x : (x \in unit) = (x != 0).

Proof. by apply/idP/idP=> [/(memPn _)-> | /fieldP]; rewrite ?unitr0. Qed.

Lemma

unitfE

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "fieldP", "memPn", "unit", "unitr0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulVf x : x != 0 -> x^-1 * x = 1.

Proof. by rewrite -unitfE; apply: mulVr. Qed.

Lemma

mulVf

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "mulVr", "unitfE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divff x : x != 0 -> x / x = 1.

Proof. by rewrite -unitfE; apply: divrr. Qed.

Lemma

divff

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "divrr", "unitfE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulfV

:= divff.

Definition

mulfV

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "divff" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulKf x : x != 0 -> cancel ( *%R x) ( *%R x^-1).

Proof. by rewrite -unitfE; apply: mulKr. Qed.

Lemma

mulKf

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "mulKr", "unitfE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulVKf x : x != 0 -> cancel ( *%R x^-1) ( *%R x).

Proof. by rewrite -unitfE; apply: mulVKr. Qed.

Lemma

mulVKf

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "mulVKr", "unitfE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulfK x : x != 0 -> cancel ( *%R^~ x) ( *%R^~ x^-1).

Proof. by rewrite -unitfE; apply: mulrK. Qed.

Lemma

mulfK

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "mulrK", "unitfE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulfVK x : x != 0 -> cancel ( *%R^~ x^-1) ( *%R^~ x).

Proof. by rewrite -unitfE; apply: divrK. Qed.

Lemma

mulfVK

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "divrK", "unitfE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divfK

:= mulfVK.

Definition

divfK

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "mulfVK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invfM : {morph @inv F : x y / x * y}.

Proof. move=> x y; have [->|nzx] := eqVneq x 0; first by rewrite !(mul0r, invr0). have [->|nzy] := eqVneq y 0; first by rewrite !(mulr0, invr0). by rewrite mulrC invrM ?unitfE. Qed.

Lemma

invfM

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "eqVneq", "inv", "invr0", "invrM", "mul0r", "mulr0", "mulrC", "unitfE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

invf_div x y : (x / y)^-1 = y / x.

Proof. by rewrite invfM invrK mulrC. Qed.

Lemma

invf_div

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "invfM", "invrK", "mulrC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

divKf x : x != 0 -> involutive (fun y => x / y).

Proof. by move=> nz_x y; rewrite invf_div mulrC divfK. Qed.

Lemma

divKf

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "divfK", "invf_div", "mulrC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expfB_cond m n x : (x == 0) + n <= m -> x ^+ (m - n) = x ^+ m / x ^+ n.

Proof. move/subnK=> <-; rewrite addnA addnK !exprD. have [-> | nz_x] := eqVneq; first by rewrite !mulr0 !mul0r. by rewrite mulfK ?expf_neq0. Qed.

Lemma

expfB_cond

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "addnA", "addnK", "eqVneq", "expf_neq0", "exprD", "mul0r", "mulfK", "mulr0", "subnK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

expfB m n x : n < m -> x ^+ (m - n) = x ^+ m / x ^+ n.

Proof. by move=> lt_n_m; apply: expfB_cond; case: eqP => // _; apply: ltnW. Qed.

Lemma

expfB

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "expfB_cond", "ltnW" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodfV I r (P : pred I) (E : I -> F) : \prod_(i <- r | P i) (E i)^-1 = (\prod_(i <- r | P i) E i)^-1.

Proof. by rewrite (big_morph _ invfM (invr1 F)). Qed.

Lemma

prodfV

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "big_morph", "invfM", "invr1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prodf_div I r (P : pred I) (E D : I -> F) : \prod_(i <- r | P i) (E i / D i) = \prod_(i <- r | P i) E i / \prod_(i <- r | P i) D i.

Proof. by rewrite big_split prodfV. Qed.

Lemma

prodf_div

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "big_split", "prodfV" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

telescope_prodf n m (f : nat -> F) : (forall k, n < k < m -> f k != 0) -> n < m -> \prod_(n <= k < m) (f k.+1 / f k) = f m / f n.

Proof. move=> nz_f ltnm; apply: invr_inj; rewrite prodf_div !invf_div -prodf_div. by apply: telescope_prodr => // k /nz_f; rewrite unitfE. Qed.

Lemma

telescope_prodf

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "invf_div", "invr_inj", "nat", "prodf_div", "telescope_prodr", "unitfE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

telescope_prodf_eq n m (f u : nat -> F) : (forall k, n < k < m -> f k != 0) -> n < m -> (forall k, n <= k < m -> u k = f k.+1 / f k) -> \prod_(n <= k < m) u k = f m / f n.

Proof. by move=> ? ? uE; under eq_big_nat do rewrite uE //=; exact: telescope_prodf. Qed.

Lemma

telescope_prodf_eq

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "eq_big_nat", "nat", "telescope_prodf" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addf_div x1 y1 x2 y2 : y1 != 0 -> y2 != 0 -> x1 / y1 + x2 / y2 = (x1 * y2 + x2 * y1) / (y1 * y2).

Proof. by move=> nzy1 nzy2; rewrite invfM mulrDl !mulrA mulrAC !mulfK. Qed.

Lemma

addf_div

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "invfM", "mulfK", "mulrA", "mulrAC", "mulrDl" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulf_div x1 y1 x2 y2 : (x1 / y1) * (x2 / y2) = (x1 * x2) / (y1 * y2).

Proof. by rewrite mulrACA -invfM. Qed.

Lemma

mulf_div

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "invfM", "mulrACA" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eqr_div x y z t : y != 0 -> t != 0 -> (x / y == z / t) = (x * t == z * y).

Proof. move=> yD0 tD0; rewrite -[x in RHS](divfK yD0) -[z in RHS](divfK tD0) mulrAC. by apply/eqP/eqP => [->|/(mulIf yD0)/(mulIf tD0)]. Qed.

Lemma

eqr_div

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "divfK", "mulIf", "mulrAC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eqr_sum_div I r P (f : I -> F) c a : c != 0 -> (\big[+%R/0]_(x <- r | P x) (f x / c) == a) = (\big[+%R/0]_(x <- r | P x) f x == a * c).

Proof. by move=> ?; rewrite -mulr_suml -(divr1 a) eqr_div ?oner_eq0// mulr1 divr1. Qed.

Lemma

eqr_sum_div

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "divr1", "eqr_div", "mulr1", "mulr_suml", "oner_eq0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pchar0_natf_div : pchar F =i pred0 -> forall m d, d %| m -> (m %/ d)%:R = m%:R / d%:R :> F.

Proof. move/pcharf0P=> pchar0F m [|d] d_dv_m; first by rewrite divn0 invr0 mulr0. by rewrite natr_div // unitfE pchar0F. Qed.

Lemma

pchar0_natf_div

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "divn0", "invr0", "mulr0", "natr_div", "pchar", "pcharf0P", "unitfE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fmorph_eq0 x : (f x == 0) = (x == 0).

Proof. have [-> | nz_x] := eqVneq x; first by rewrite rmorph0 eqxx. apply/eqP; move/(congr1 ( *%R (f x^-1)))/eqP. by rewrite -rmorphM mulVf // mulr0 rmorph1 ?oner_eq0. Qed.

Lemma

fmorph_eq0

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "eqVneq", "eqxx", "mulVf", "mulr0", "oner_eq0", "rmorph0", "rmorph1", "rmorphM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fmorph_inj : injective f.

Proof. by apply/raddf_inj => x /eqP; rewrite fmorph_eq0 => /eqP. Qed.

Lemma

fmorph_inj

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "fmorph_eq0", "raddf_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fmorph_eq : {mono f : x y / x == y}.

Proof. exact: inj_eq fmorph_inj. Qed.

Lemma

fmorph_eq

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "fmorph_inj", "inj_eq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fmorph_eq1 x : (f x == 1) = (x == 1).

Proof. by rewrite -(inj_eq fmorph_inj) rmorph1. Qed.

Lemma

fmorph_eq1

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "fmorph_inj", "inj_eq", "rmorph1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fmorph_pchar : pchar R =i pchar F.

Proof. by move=> p; rewrite !inE -fmorph_eq0 rmorph_nat. Qed.

Lemma

fmorph_pchar

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "fmorph_eq0", "inE", "pchar", "rmorph_nat" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

fmorph_unit x : (f x \in unit) = (x != 0).

Proof. have [-> |] := eqVneq x; first by rewrite rmorph0 unitr0. by rewrite -unitfE; apply: rmorph_unit. Qed.

Lemma

fmorph_unit

algebra.algebraic_hierarchy

algebra/algebraic_hierarchy/divalg.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "div", "choice", "fintype", "bigop", "prime", "nmodule", "rings_modules_and_algebras", "GRing", "GRing.Theory", "AllExports", "ClosedExports" ]

[ "apply", "eqVneq", "rmorph0", "rmorph_unit", "unit", "unitfE", "unitr0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d