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
pi_is_additive	:= pi_is_zmod_morphism.	Definition	pi_is_additive	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "pi_is_zmod_morphism" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Build T eqT zeroT oppT addT oneT mulT Q	:= (isNzRingQuotient.Build T eqT zeroT oppT addT oneT mulT Q) (only parsing).	Notation	Build	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "mulT" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
isRingQuotient T eqT zeroT oppT addT oneT mulT Q	:= (isNzRingQuotient T eqT zeroT oppT addT oneT mulT Q) (only parsing).	Notation	isRingQuotient	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "mulT" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
ringQuotType	:= (nzRingQuotType) (only parsing).	Notation	ringQuotType	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_one_quot_morph rqT	:= PiMorph (@pi_oner _ _ _ _ _ _ _ rqT).	Canonical	pi_one_quot_morph	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "PiMorph" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_mul_quot_morph rqT	:= PiMorph2 (@pi_mulr _ _ _ _ _ _ _ rqT).	Canonical	pi_mul_quot_morph	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "PiMorph2" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_is_monoid_morphism : monoid_morphism \pi_Q.	Proof. by split; do ?move=> x y /=; rewrite !piE. Qed.	Lemma	pi_is_monoid_morphism	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "monoid_morphism", "piE", "split" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_is_multiplicative	:= (fun g => (g.2,g.1)) pi_is_monoid_morphism.	Definition	pi_is_multiplicative	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "pi_is_monoid_morphism" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_unit_quot_morph urqT	:= PiMono1 (@pi_unitr _ _ _ _ _ _ _ _ _ urqT).	Canonical	pi_unit_quot_morph	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "PiMono1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_inv_quot_morph urqT	:= PiMorph1 (@pi_invr _ _ _ _ _ _ _ _ _ urqT).	Canonical	pi_inv_quot_morph	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "PiMorph1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
proper_ideal (R : nzRingType) (S : {pred R}) : Prop	:= 1 \notin S /\ forall a, {in S, forall u, a * u \in S}.	Definition	proper_ideal	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prime_idealr_closed (R : nzRingType) (S : {pred R}) : Prop	:= forall u v, u * v \in S -> (u \in S) \|\| (v \in S).	Definition	prime_idealr_closed	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
idealr_closed (R : nzRingType) (S : {pred R})	:= [/\ 0 \in S, 1 \notin S & forall a, {in S &, forall u v, a * u + v \in S}].	Definition	idealr_closed	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
idealr_closed_nontrivial R S : @idealr_closed R S -> proper_ideal S.	Proof. by case=> S0 S1 hS; split => // a x xS; rewrite -[_ * _]addr0 hS. Qed.	Lemma	idealr_closed_nontrivial	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "S0", "S1", "addr0", "idealr_closed", "proper_ideal", "split" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
idealr_closedB R S : @idealr_closed R S -> zmod_closed S.	Proof. by case=> S0 _ hS; split=> // x y xS yS; rewrite -mulN1r addrC hS. Qed.	Lemma	idealr_closedB	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "S0", "addrC", "idealr_closed", "mulN1r", "split", "zmod_closed" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
I	:= (idealrI : pred R).	Notation	I	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
idealr1 : (1 \in I) = false.	Proof. apply: negPf; exact: proper_ideal_subproof.1. Qed.	Lemma	idealr1	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "apply" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
idealMr a u : u \in I -> a * u \in I.	Proof. exact: proper_ideal_subproof.2. Qed.	Lemma	idealMr	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
idealr0 : 0 \in I.	Proof. exact: rpred0. Qed.	Lemma	idealr0	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "rpred0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
I	:= (pidealI : pred R).	Notation	I	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prime_idealrM u v : (u * v \in I) = (u \in I) \|\| (v \in I).	Proof. apply/idP/idP; last by case/orP => /idealMr hI; rewrite // mulrC. exact: prime_idealr_closed_subproof. Qed.	Lemma	prime_idealrM	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "apply", "hI", "idealMr", "last", "mulrC" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
equiv (x y : R)	:= (x - y) \in I.	Definition	equiv	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
equivE x y : (equiv x y) = (x - y \in I).	Proof. by []. Qed.	Lemma	equivE	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "equiv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
equiv_is_equiv : equiv_class_of equiv.	Proof. split=> [x\|x y\|y x z]; rewrite !equivE ?subrr ?rpred0 //. by rewrite -opprB rpredN. by move=> *; rewrite -[x](addrNK y) -addrA rpredD. Qed.	Lemma	equiv_is_equiv	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "addrA", "addrNK", "equiv", "equivE", "equiv_class_of", "opprB", "rpred0", "rpredD", "rpredN", "split", "subrr" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
equiv_equiv	:= EquivRelPack equiv_is_equiv.	Canonical	equiv_equiv	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "equiv_is_equiv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
equiv_encModRel	:= defaultEncModRel equiv.	Canonical	equiv_encModRel	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "defaultEncModRel", "equiv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quot	:= {eq_quot equiv}.	Definition	quot	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "equiv" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
idealrBE x y : ((x - y) \in I) = (x == y %[mod quot]).	Proof. by rewrite piE equivE. Qed.	Lemma	idealrBE	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "equivE", "piE", "quot" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
idealrDE x y : ((x + y) \in I) = (x == - y %[mod quot]).	Proof. by rewrite -idealrBE opprK. Qed.	Lemma	idealrDE	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "idealrBE", "opprK", "quot" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
zero : quot	:= lift_cst quot 0.	Definition	zero	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "lift_cst", "quot" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
add	:= lift_op2 quot +%R.	Definition	add	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "lift_op2", "quot" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
opp	:= lift_op1 quot -%R.	Definition	opp	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "lift_op1", "quot" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_zero_morph	:= PiConst zero.	Canonical	pi_zero_morph	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "PiConst", "zero" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_opp : {morph \pi : x / - x >-> opp x}.	Proof. move=> x; unlock opp; apply/eqP; rewrite piE equivE. by rewrite -opprD rpredN idealrDE opprK reprK. Qed.	Lemma	pi_opp	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "apply", "equivE", "idealrDE", "opp", "opprD", "opprK", "pi", "piE", "reprK", "rpredN" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_add : {morph \pi : x y / x + y >-> add x y}.	Proof. move=> x y /=; unlock add; apply/eqP; rewrite piE equivE. by rewrite opprD addrACA rpredD // (idealrBE, idealrDE) ?pi_opp ?reprK. Qed.	Lemma	pi_add	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "add", "addrACA", "apply", "equivE", "idealrBE", "idealrDE", "opprD", "pi", "piE", "pi_opp", "reprK", "rpredD" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addqA: associative add.	Proof. by move=> x y z; rewrite -[x]reprK -[y]reprK -[z]reprK !piE addrA. Qed.	Lemma	addqA	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "add", "addrA", "piE", "reprK" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addqC: commutative add.	Proof. by move=> x y; rewrite -[x]reprK -[y]reprK !piE addrC. Qed.	Lemma	addqC	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "add", "addrC", "piE", "reprK" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
add0q: left_id zero add.	Proof. by move=> x; rewrite -[x]reprK !piE add0r. Qed.	Lemma	add0q	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "add", "add0r", "piE", "reprK", "zero" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
addNq: left_inverse zero opp add.	Proof. by move=> x; rewrite -[x]reprK !piE addNr. Qed.	Lemma	addNq	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "add", "addNr", "opp", "piE", "reprK", "zero" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'quot' I }"	:= (quot I) : type_scope.	Notation	{ 'quot' I }	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "quot" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
I	:= (idealI : pred R).	Notation	I	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
one : {quot idealI}	:= lift_cst {quot idealI} 1.	Definition	one	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "lift_cst", "quot" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul	:= lift_op2 {quot idealI} *%R.	Definition	mul	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "lift_op2", "quot" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_one_morph	:= PiConst one.	Canonical	pi_one_morph	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "PiConst", "one" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_mul: {morph \pi : x y / x * y >-> mul x y}.	Proof. move=> x y; unlock mul; apply/eqP; rewrite piE equivE. rewrite -[_ * _](addrNK (x * repr (\pi_{quot idealI} y))) -mulrBr. rewrite -addrA -mulrBl rpredD //. by rewrite idealMr // idealrDE opprK reprK. by rewrite mulrC idealMr // idealrDE opprK reprK. Qed.	Lemma	pi_mul	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "addrA", "addrNK", "apply", "equivE", "idealMr", "idealrDE", "mul", "mulrBl", "mulrBr", "mulrC", "opprK", "pi", "piE", "quot", "repr", "reprK", "rpredD" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulqA: associative mul.	Proof. by move=> x y z; rewrite -[x]reprK -[y]reprK -[z]reprK !piE mulrA. Qed.	Lemma	mulqA	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "mul", "mulrA", "piE", "reprK" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulqC: commutative mul.	Proof. by move=> x y; rewrite -[x]reprK -[y]reprK !piE mulrC. Qed.	Lemma	mulqC	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "mul", "mulrC", "piE", "reprK" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mul1q: left_id one mul.	Proof. by move=> x; rewrite -[x]reprK !piE mul1r. Qed.	Lemma	mul1q	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "mul", "mul1r", "one", "piE", "reprK" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mulq_addl: left_distributive mul +%R.	Proof. move=> x y z; rewrite -[x]reprK -[y]reprK -[z]reprK. by apply/eqP; rewrite piE /= mulrDl equiv_refl. Qed.	Lemma	mulq_addl	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "apply", "equiv_refl", "mul", "mulrDl", "piE", "reprK" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
nonzero1q: one != 0.	Proof. by rewrite piE equivE subr0 idealr1. Qed.	Lemma	nonzero1q	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "equivE", "idealr1", "one", "piE", "subr0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
rquot_IdomainAxiom (x y : {quot I}): x * y = 0 -> (x == 0) \|\| (y == 0).	Proof. by move=> /eqP; rewrite -[x]reprK -[y]reprK !piE !equivE !subr0 prime_idealrM. Qed.	Lemma	rquot_IdomainAxiom	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "equivE", "piE", "prime_idealrM", "quot", "reprK", "subr0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{ 'ideal_quot' I }"	:= (@Quotient.quot _ I) : type_scope.	Notation	{ 'ideal_quot' I }	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[ "quot" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x == y %[ 'mod_ideal' I ]"	:= (x == y %[mod {ideal_quot I}]) : quotient_scope.	Notation	x == y %[ 'mod_ideal' I ]	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x = y %[ 'mod_ideal' I ]"	:= (x = y %[mod {ideal_quot I}]) : quotient_scope.	Notation	x = y %[ 'mod_ideal' I ]	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x != y %[ 'mod_ideal' I ]"	:= (x != y %[mod {ideal_quot I}]) : quotient_scope.	Notation	x != y %[ 'mod_ideal' I ]	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x <> y %[ 'mod_ideal' I ]"	:= (x <> y %[mod {ideal_quot I}]) : quotient_scope.	Notation	x <> y %[ 'mod_ideal' I ]	algebra	algebra/ring_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
env_nth (n : positive) l	:= nth rO l (Pos.to_nat n).-1.	Definition	env_nth	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "nth" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
env_jump (j : positive) l	:= drop (Pos.to_nat j) l.	Definition	env_jump	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "drop" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
env_jumpD l j j' : env_jump (Pos.add j j') l = env_jump j' (env_jump j l).	Proof. by rewrite /env_jump drop_drop Pos_to_natD addnC. Qed.	Lemma	env_jumpD	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "Pos_to_natD", "add", "addnC", "drop_drop", "env_jump" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
env_nth_jump l i j : env_nth i (env_jump j l) = env_nth (Pos.add i j) l.	Proof. by rewrite /env_nth nth_drop Pos_to_natD -!subn1 addnBA 1?addnC ?Pos_to_nat_gt0. Qed.	Lemma	env_nth_jump	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "Pos_to_natD", "Pos_to_nat_gt0", "add", "addnBA", "addnC", "env_jump", "env_nth", "nth_drop", "subn1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Peq	:= (Peq eq_op).	Notation	Peq	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
P0	:= (P0 0).	Notation	P0	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
P1	:= (P1 1).	Notation	P1	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mkX	:= (mkX 0 1).	Notation	mkX	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mkPX	:= (mkPX 0 eq_op).	Notation	mkPX	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PaddC	:= (PaddC +%R).	Notation	PaddC	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Padd	:= (Padd 0 +%R eq_op).	Notation	Padd	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PaddI	:= (PaddI +%R Padd).	Notation	PaddI	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "Padd" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PaddX	:= (PaddX 0 eq_op Padd).	Notation	PaddX	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "Padd" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Pmul	:= (Pmul 0 1 +%R *%R eq_op).	Notation	Pmul	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PmulC_aux	:= (PmulC_aux 0 *%R eq_op).	Notation	PmulC_aux	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PmulC	:= (PmulC 0 1 *%R eq_op).	Notation	PmulC	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PmulI	:= (PmulI 0 1 *%R eq_op Pmul).	Notation	PmulI	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "Pmul" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ppow_pos	:= (Ppow_pos 0 1 +%R *%R eq_op).	Notation	Ppow_pos	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Ppow_N	:= (Ppow_N 0 1 +%R *%R eq_op).	Notation	Ppow_N	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Peval_Peq l P P' : Peq P P' -> Peval l P = Peval l P'.	Proof. elim: P l P' => [c \| i P IH \| P IHP j Q IHQ] l [c' \| i' P' \| P' j' Q'] //=. - by move=> /eqP->. - by case: pos_nat_compareP => //; apply: IH. - by case: pos_nat_compareP => //; case: ifP => // /IHP-> /IHQ->. Qed.	Lemma	Peval_Peq	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "Peq", "Peval", "apply", "pos_nat_compareP" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Peval_mkPinj l j P : Peval l (mkPinj j P) = Peval (env_jump j l) P.	Proof. by case: P => [//\| j' Q /= \|//]; rewrite env_jumpD. Qed.	Lemma	Peval_mkPinj	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "Peval", "env_jump", "env_jumpD" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Peval_mkX l j : Peval l (mkX j) = env_nth j l.	Proof. case: j => [j\|j\|]/=; rewrite rmorph1 rmorph0 mul1r addr0 expr1 ?env_nth_jump//. congr (env_nth _ l); apply: Pos_to_natI; rewrite !Pos_to_natE. by rewrite [LHS]add1n prednK// -double0 ltn_double Pos_to_nat_gt0. Qed.	Lemma	Peval_mkX	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "Peval", "Pos_to_natE", "Pos_to_natI", "Pos_to_nat_gt0", "add1n", "addr0", "apply", "double0", "env_nth", "env_nth_jump", "expr1", "ltn_double", "mkX", "mul1r", "prednK", "rmorph0", "rmorph1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Peval_mkPX l P i Q : Peval l (mkPX P i Q) = Peval l P * (env_nth 1 l) ^+ Pos.to_nat i + Peval (env_jump 1 l) Q.	Proof. case: P => [c \|//\| P' i' [c \|//\|//]] /=. by case: eqP => [->\|//]; rewrite rmorph0 mul0r add0r Peval_mkPinj. by case: eqP => [->\|//]/=; rewrite rmorph0 addr0 Pos_to_natD exprD mulrA. Qed.	Lemma	Peval_mkPX	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "Peval", "Peval_mkPinj", "Pos_to_natD", "add0r", "addr0", "env_jump", "env_nth", "exprD", "mkPX", "mul0r", "mulrA", "rmorph0" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Popp_id P : Popp id P = P.	Proof. by elim: P => [\| j P /=->// \| P /=-> i Q ->]. Qed.	Lemma	Popp_id	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "Popp", "id" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PevalDC l P c : Peval l (PaddC P c) = Peval l P + R_of_C c.	Proof. by elim: P l => [c'\|i P IH\|P IHP j Q IHQ] l/=; rewrite ?rmorphD ?IH ?IHQ ?addrA. Qed.	Lemma	PevalDC	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "PaddC", "Peval", "addrA", "rmorphD" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Peval_addI l P j Q : (forall l P, Peval l (Padd P Q) = Peval l P + Peval l Q) -> Peval l (PaddI Q j P) = Peval l P + Peval l (Pinj j Q).	Proof. move=> PevalD; elim: P j l => [c \| j' Q' IH \| P IHP i Q' IHQ'] j l /=. - by rewrite Peval_mkPinj PevalDC addrC. - move: (Zint_pos_sub (pos_nat_Pos_to_nat j') (pos_nat_Pos_to_nat j)). case: ZintP => [//\| _ /eqP+ _ \| k nk _ /[swap] \| k nk _ /[swap] ]. + rewrite subr_eq0 => /eqP[]/Pos_to_natI->. by rewrite ...	Lemma	Peval_addI	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "NegzE", "Padd", "PaddI", "Peval", "PevalD", "PevalDC", "Peval_mkPinj", "Pos_to_natD", "Pos_to_natE", "Pos_to_natI", "Pos_to_nat_gt0", "ZintP", "Zint_pos_sub", "add1n", "addnC", "addrA", "addrC", "apply", "double0", "env_jump", "env_jumpD", "eqr_opp", "ltn_double", "opp...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Peval_addX l P P' i' : (forall l P, Peval l (Padd P P') = Peval l P + Peval l P') -> Peval l (PaddX P' i' P) = Peval l P + Peval l (PX P' i' P0).	Proof. rewrite /= rmorph0 addr0 addrC. move=> PevalD; elim: P i' l => [//\| j Q' IH \| P IHP i Q' IHQ'] i' l /=. - case: j => [j \| j \|//]/=; rewrite -env_jumpD//. congr (_ + Peval (env_jump _ l) _). apply: Pos_to_natI; rewrite !Pos_to_natE add1n prednK//. by rewrite -double0 ltn_double Pos_to_nat_gt0. - rewrite add...	Lemma	Peval_addX	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "NegzE", "P0", "Padd", "PaddX", "Peval", "PevalD", "Peval_mkPX", "Pos_to_natD", "Pos_to_natE", "Pos_to_natI", "Pos_to_nat_gt0", "ZintP", "Zint_pos_sub", "add1n", "addnC", "addr0", "addrA", "addrC", "apply", "double0", "env_jump", "env_jumpD", "eqr_opp", "exprD", "ltn_...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PevalD l P P' : Peval l (Padd P P') = Peval l P + Peval l P'.	Proof. elim: P' l P => [c \| j' Q' IH \| P' IHP' i' Q' IHQ'] l P /=. - by rewrite PevalDC. - by rewrite Peval_addI. case: P => [c \| j Q \| P i Q] /=. - by rewrite PevalDC addrA addrC. - rewrite addrCA; case: j => [j \| j \|] /=; rewrite IHQ'/= -?env_jumpD//. congr (_ + (Peval (env_jump _ l) _ + _)); apply: Pos_to_natI. ...	Lemma	PevalD	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "NegzE", "Padd", "Peval", "PevalDC", "Peval_addI", "Peval_addX", "Peval_mkPX", "Pos_to_natD", "Pos_to_natE", "Pos_to_natI", "Pos_to_nat_gt0", "ZintP", "Zint_pos_sub", "add1n", "addr0", "addrA", "addrACA", "addrC", "addrCA", "apply", "double0", "env_jump", "env_jumpD", "...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PsubI_addI P j Q : (forall P', Psub 0 +%R +%R id eq_op P' P = Padd P' P) -> PsubI +%R id (Psub 0 +%R +%R id eq_op) P j Q = PaddI P j Q.	Proof. elim: Q j => [c \| j' P' IH \| P' IHP i Q' IHQ ] j Psub_add /=. - by rewrite Popp_id. - by case: Z.pos_sub => [\| p \| p]; rewrite ?Psub_add ?IH. - by case: j => [p \| p \|]; rewrite ?Psub_add ?IHQ. Qed.	Lemma	PsubI_addI	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "Padd", "PaddI", "Popp_id", "Psub", "PsubI", "Psub_add", "id" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PsubX_addX P' i P : (forall P, Psub 0 +%R +%R id eq_op P P' = Padd P P') -> PsubX 0 id eq_op (Psub 0 +%R +%R id eq_op) P' i P = PaddX P' i P.	Proof. elim: P i => [c \| j P IH \| P IHP i' Q IHQ] i Psub_add /=. - by rewrite Popp_id. - by case: j => [p \| p \|] /=; rewrite Popp_id. - by case: Z.pos_sub => [\| p \| p]; rewrite ?Psub_add ?IHP. Qed.	Lemma	PsubX_addX	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "Padd", "PaddX", "Popp_id", "Psub", "PsubX", "Psub_add", "id" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Psub_add P P' : Psub 0 +%R +%R id eq_op P P' = Padd P P'.	Proof. elim: P' P => [//\| j' P' IH \| P' IHP' i' Q' IHQ' ] P /=. by rewrite PsubI_addI. case: P => [c \| j P \| P i Q] /=; first by rewrite !Popp_id. by case: j => [p \| p \|]; rewrite Popp_id IHQ'. by case: Z.pos_sub => [\| p \| p]; rewrite ?IHP' ?IHQ' ?PsubX_addX. Qed.	Lemma	Psub_add	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "Padd", "Popp_id", "Psub", "PsubI_addI", "PsubX_addX", "id" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Peval_mulC_aux l P c : Peval l (PmulC_aux P c) = Peval l P * R_of_C c.	Proof. elim: P l => [c' \| j Q IH \| P IHP i Q IHQ] l /=; first by rewrite rmorphM. by rewrite Peval_mkPinj IH. by rewrite Peval_mkPX IHP IHQ mulrAC -mulrDl. Qed.	Lemma	Peval_mulC_aux	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "Peval", "Peval_mkPX", "Peval_mkPinj", "PmulC_aux", "mulrAC", "mulrDl", "rmorphM" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PevalMC l P c : Peval l (PmulC P c) = Peval l P * R_of_C c.	Proof. rewrite /PmulC/=; case: eqP => [->\|_]; first by rewrite /= rmorph0 mulr0. by case: eqP => [->\|_]; rewrite ?rmorph1 ?mulr1 ?Peval_mulC_aux. Qed.	Lemma	PevalMC	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "Peval", "Peval_mulC_aux", "PmulC", "mulr0", "mulr1", "rmorph0", "rmorph1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Peval_mulI l P j Q : (forall l P, Peval l (Pmul P Q) = Peval l P * Peval l Q) -> Peval l (PmulI Q j P) = Peval l P * Peval l (Pinj j Q).	Proof. move=> PevalM; elim: P j l => [c \| j' Q' IH \| P IHP i Q' IHQ'] j l /=. - by rewrite Peval_mkPinj PevalMC mulrC. - move: (Zint_pos_sub (pos_nat_Pos_to_nat j') (pos_nat_Pos_to_nat j)). case: ZintP => [//\| _ /eqP+ _ \| k nk _ /[swap] \| k nk _ /[swap] ]. + rewrite subr_eq0 => /eqP[]/Pos_to_natI->. by rewrite ...	Lemma	Peval_mulI	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "NegzE", "Peval", "PevalM", "PevalMC", "Peval_mkPX", "Peval_mkPinj", "Pmul", "PmulI", "Pos_to_natD", "Pos_to_natE", "Pos_to_natI", "Pos_to_nat_gt0", "ZintP", "Zint_pos_sub", "add", "add1n", "addnC", "apply", "double0", "env_jumpD", "eqr_opp", "last", "ltn_double", "mulr...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PevalM l P P' : Peval l (Pmul P P') = Peval l P * Peval l P'.	Proof. elim: P' l P => [c \| j' Q' IH \| P' IHP' i' Q' IHQ'] l P /=. - by rewrite PevalMC. - by rewrite Peval_mulI. case: P => [c \| j Q \| P i Q] /=. - by rewrite PevalMC mulrC. - rewrite Peval_mkPX IHP' mulrDr mulrA. case: j => [j \| j \|] /=; rewrite IHQ'/= -?env_jumpD//. congr (_ + (Peval (env_jump _ l) _ * _)); appl...	Lemma	PevalM	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "Peval", "PevalD", "PevalMC", "Peval_mkPX", "Peval_mkPinj", "Peval_mulI", "Pmul", "Pos_to_natE", "Pos_to_natI", "Pos_to_nat_gt0", "add1n", "addr0", "addrACA", "apply", "double0", "env_jump", "env_jumpD", "ltn_double", "mulrA", "mulrAC", "mulrACA", "mulrC", "mulrDl", "mu...	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Peval_pow_pos l res P p : Peval l (Ppow_pos res P p) = Peval l res * Peval l P ^+ Pos.to_nat p.	Proof. elim: p l res P => [p IH \| p IH \|] l res P /=. - rewrite PevalM !IH -mulrAC -mulrA -exprS -mulrA -exprD. by rewrite addnS addnn Pos_to_natE. - by rewrite !IH -mulrA -exprD addnn Pos_to_natE. - by rewrite PevalM expr1. Qed.	Lemma	Peval_pow_pos	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "Peval", "PevalM", "Pos_to_natE", "Ppow_pos", "addnS", "addnn", "expr1", "exprD", "exprS", "mulrA", "mulrAC" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Peval_pow_N l P n : Peval l (Ppow_N P n) = Peval l P ^+ N.to_nat n.	Proof. by case: n => [\| p]/=; rewrite ?Peval_pow_pos /= rmorph1 ?mul1r. Qed.	Lemma	Peval_pow_N	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "Peval", "Peval_pow_pos", "Ppow_N", "mul1r", "rmorph1" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Popp	:= (Popp -%R).	Notation	Popp	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PsubC	:= (PsubC (fun x y => x - y)).	Notation	PsubC	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Psub	:= (Psub 0 +%R (fun x y => x - y) -%R eq_op).	Notation	Psub	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PsubI	:= (PsubI +%R -%R Psub).	Notation	PsubI	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "Psub" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PsubX	:= (PsubX 0 -%R eq_op Psub).	Notation	PsubX	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "Psub" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PevalN l P : Peval l (Popp P) = - Peval l P.	Proof. elim: P l => [c \| i P IH \| P IHP j Q IHQ] l /=; rewrite ?rmorphN ?IH//. by rewrite IHP IHQ opprD mulNr. Qed.	Lemma	PevalN	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "Peval", "Popp", "mulNr", "opprD", "rmorphN" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PevalBC l P c : Peval l (PsubC P c) = Peval l P - R_of_C c.	Proof. by elim: P l => [c'\|i P IH\|P IHP j Q IHQ] l/=; rewrite ?rmorphB ?IH ?IHQ ?addrA. Qed.	Lemma	PevalBC	algebra	algebra/ring_tactic.v	[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...	[ "Peval", "PsubC", "addrA", "rmorphB" ]	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_is_additive

:= pi_is_zmod_morphism.

Definition

pi_is_additive

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "pi_is_zmod_morphism" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Build T eqT zeroT oppT addT oneT mulT Q

:= (isNzRingQuotient.Build T eqT zeroT oppT addT oneT mulT Q) (only parsing).

Notation

Build

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "mulT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

isRingQuotient T eqT zeroT oppT addT oneT mulT Q

:= (isNzRingQuotient T eqT zeroT oppT addT oneT mulT Q) (only parsing).

Notation

isRingQuotient

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "mulT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

ringQuotType

:= (nzRingQuotType) (only parsing).

Notation

ringQuotType

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_one_quot_morph rqT

:= PiMorph (@pi_oner _ _ _ _ _ _ _ rqT).

Canonical

pi_one_quot_morph

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "PiMorph" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_mul_quot_morph rqT

:= PiMorph2 (@pi_mulr _ _ _ _ _ _ _ rqT).

Canonical

pi_mul_quot_morph

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "PiMorph2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_is_monoid_morphism : monoid_morphism \pi_Q.

Proof. by split; do ?move=> x y /=; rewrite !piE. Qed.

Lemma

pi_is_monoid_morphism

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "monoid_morphism", "piE", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_is_multiplicative

:= (fun g => (g.2,g.1)) pi_is_monoid_morphism.

Definition

pi_is_multiplicative

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "pi_is_monoid_morphism" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_unit_quot_morph urqT

:= PiMono1 (@pi_unitr _ _ _ _ _ _ _ _ _ urqT).

Canonical

pi_unit_quot_morph

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "PiMono1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_inv_quot_morph urqT

:= PiMorph1 (@pi_invr _ _ _ _ _ _ _ _ _ urqT).

Canonical

pi_inv_quot_morph

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "PiMorph1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

proper_ideal (R : nzRingType) (S : {pred R}) : Prop

:= 1 \notin S /\ forall a, {in S, forall u, a * u \in S}.

Definition

proper_ideal

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prime_idealr_closed (R : nzRingType) (S : {pred R}) : Prop

:= forall u v, u * v \in S -> (u \in S) || (v \in S).

Definition

prime_idealr_closed

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

idealr_closed (R : nzRingType) (S : {pred R})

:= [/\ 0 \in S, 1 \notin S & forall a, {in S &, forall u v, a * u + v \in S}].

Definition

idealr_closed

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

idealr_closed_nontrivial R S : @idealr_closed R S -> proper_ideal S.

Proof. by case=> S0 S1 hS; split => // a x xS; rewrite -[_ * _]addr0 hS. Qed.

Lemma

idealr_closed_nontrivial

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "S0", "S1", "addr0", "idealr_closed", "proper_ideal", "split" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

idealr_closedB R S : @idealr_closed R S -> zmod_closed S.

Proof. by case=> S0 _ hS; split=> // x y xS yS; rewrite -mulN1r addrC hS. Qed.

Lemma

idealr_closedB

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "S0", "addrC", "idealr_closed", "mulN1r", "split", "zmod_closed" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

I

:= (idealrI : pred R).

Notation

I

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

idealr1 : (1 \in I) = false.

Proof. apply: negPf; exact: proper_ideal_subproof.1. Qed.

Lemma

idealr1

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "apply" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

idealMr a u : u \in I -> a * u \in I.

Proof. exact: proper_ideal_subproof.2. Qed.

Lemma

idealMr

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

idealr0 : 0 \in I.

Proof. exact: rpred0. Qed.

Lemma

idealr0

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "rpred0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

I

:= (pidealI : pred R).

Notation

I

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prime_idealrM u v : (u * v \in I) = (u \in I) || (v \in I).

Proof. apply/idP/idP; last by case/orP => /idealMr hI; rewrite // mulrC. exact: prime_idealr_closed_subproof. Qed.

Lemma

prime_idealrM

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "apply", "hI", "idealMr", "last", "mulrC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

equiv (x y : R)

:= (x - y) \in I.

Definition

equiv

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

equivE x y : (equiv x y) = (x - y \in I).

Proof. by []. Qed.

Lemma

equivE

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "equiv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

equiv_is_equiv : equiv_class_of equiv.

Proof. split=> [x|x y|y x z]; rewrite !equivE ?subrr ?rpred0 //. by rewrite -opprB rpredN. by move=> *; rewrite -[x](addrNK y) -addrA rpredD. Qed.

Lemma

equiv_is_equiv

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "addrA", "addrNK", "equiv", "equivE", "equiv_class_of", "opprB", "rpred0", "rpredD", "rpredN", "split", "subrr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

equiv_equiv

:= EquivRelPack equiv_is_equiv.

Canonical

equiv_equiv

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "equiv_is_equiv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

equiv_encModRel

:= defaultEncModRel equiv.

Canonical

equiv_encModRel

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "defaultEncModRel", "equiv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quot

:= {eq_quot equiv}.

Definition

quot

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "equiv" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

idealrBE x y : ((x - y) \in I) = (x == y %[mod quot]).

Proof. by rewrite piE equivE. Qed.

Lemma

idealrBE

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "equivE", "piE", "quot" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

idealrDE x y : ((x + y) \in I) = (x == - y %[mod quot]).

Proof. by rewrite -idealrBE opprK. Qed.

Lemma

idealrDE

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "idealrBE", "opprK", "quot" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

zero : quot

:= lift_cst quot 0.

Definition

zero

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "lift_cst", "quot" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

add

:= lift_op2 quot +%R.

Definition

add

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "lift_op2", "quot" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

opp

:= lift_op1 quot -%R.

Definition

opp

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "lift_op1", "quot" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_zero_morph

:= PiConst zero.

Canonical

pi_zero_morph

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "PiConst", "zero" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_opp : {morph \pi : x / - x >-> opp x}.

Proof. move=> x; unlock opp; apply/eqP; rewrite piE equivE. by rewrite -opprD rpredN idealrDE opprK reprK. Qed.

Lemma

pi_opp

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "apply", "equivE", "idealrDE", "opp", "opprD", "opprK", "pi", "piE", "reprK", "rpredN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_add : {morph \pi : x y / x + y >-> add x y}.

Proof. move=> x y /=; unlock add; apply/eqP; rewrite piE equivE. by rewrite opprD addrACA rpredD // (idealrBE, idealrDE) ?pi_opp ?reprK. Qed.

Lemma

pi_add

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "add", "addrACA", "apply", "equivE", "idealrBE", "idealrDE", "opprD", "pi", "piE", "pi_opp", "reprK", "rpredD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addqA: associative add.

Proof. by move=> x y z; rewrite -[x]reprK -[y]reprK -[z]reprK !piE addrA. Qed.

Lemma

addqA

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "add", "addrA", "piE", "reprK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addqC: commutative add.

Proof. by move=> x y; rewrite -[x]reprK -[y]reprK !piE addrC. Qed.

Lemma

addqC

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "add", "addrC", "piE", "reprK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

add0q: left_id zero add.

Proof. by move=> x; rewrite -[x]reprK !piE add0r. Qed.

Lemma

add0q

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "add", "add0r", "piE", "reprK", "zero" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

addNq: left_inverse zero opp add.

Proof. by move=> x; rewrite -[x]reprK !piE addNr. Qed.

Lemma

addNq

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "add", "addNr", "opp", "piE", "reprK", "zero" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"{ 'quot' I }"

:= (quot I) : type_scope.

Notation

{ 'quot' I }

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "quot" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

I

:= (idealI : pred R).

Notation

I

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

one : {quot idealI}

:= lift_cst {quot idealI} 1.

Definition

one

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "lift_cst", "quot" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mul

:= lift_op2 {quot idealI} *%R.

Definition

mul

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "lift_op2", "quot" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_one_morph

:= PiConst one.

Canonical

pi_one_morph

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "PiConst", "one" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_mul: {morph \pi : x y / x * y >-> mul x y}.

Proof. move=> x y; unlock mul; apply/eqP; rewrite piE equivE. rewrite -[_ * _](addrNK (x * repr (\pi_{quot idealI} y))) -mulrBr. rewrite -addrA -mulrBl rpredD //. by rewrite idealMr // idealrDE opprK reprK. by rewrite mulrC idealMr // idealrDE opprK reprK. Qed.

Lemma

pi_mul

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "addrA", "addrNK", "apply", "equivE", "idealMr", "idealrDE", "mul", "mulrBl", "mulrBr", "mulrC", "opprK", "pi", "piE", "quot", "repr", "reprK", "rpredD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulqA: associative mul.

Proof. by move=> x y z; rewrite -[x]reprK -[y]reprK -[z]reprK !piE mulrA. Qed.

Lemma

mulqA

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "mul", "mulrA", "piE", "reprK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulqC: commutative mul.

Proof. by move=> x y; rewrite -[x]reprK -[y]reprK !piE mulrC. Qed.

Lemma

mulqC

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "mul", "mulrC", "piE", "reprK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mul1q: left_id one mul.

Proof. by move=> x; rewrite -[x]reprK !piE mul1r. Qed.

Lemma

mul1q

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "mul", "mul1r", "one", "piE", "reprK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mulq_addl: left_distributive mul +%R.

Proof. move=> x y z; rewrite -[x]reprK -[y]reprK -[z]reprK. by apply/eqP; rewrite piE /= mulrDl equiv_refl. Qed.

Lemma

mulq_addl

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "apply", "equiv_refl", "mul", "mulrDl", "piE", "reprK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

nonzero1q: one != 0.

Proof. by rewrite piE equivE subr0 idealr1. Qed.

Lemma

nonzero1q

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "equivE", "idealr1", "one", "piE", "subr0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

rquot_IdomainAxiom (x y : {quot I}): x * y = 0 -> (x == 0) || (y == 0).

Proof. by move=> /eqP; rewrite -[x]reprK -[y]reprK !piE !equivE !subr0 prime_idealrM. Qed.

Lemma

rquot_IdomainAxiom

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "equivE", "piE", "prime_idealrM", "quot", "reprK", "subr0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"{ 'ideal_quot' I }"

:= (@Quotient.quot _ I) : type_scope.

Notation

{ 'ideal_quot' I }

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[ "quot" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x == y %[ 'mod_ideal' I ]"

:= (x == y %[mod {ideal_quot I}]) : quotient_scope.

Notation

x == y %[ 'mod_ideal' I ]

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x = y %[ 'mod_ideal' I ]"

:= (x = y %[mod {ideal_quot I}]) : quotient_scope.

Notation

x = y %[ 'mod_ideal' I ]

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x != y %[ 'mod_ideal' I ]"

:= (x != y %[mod {ideal_quot I}]) : quotient_scope.

Notation

x != y %[ 'mod_ideal' I ]

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x <> y %[ 'mod_ideal' I ]"

:= (x <> y %[mod {ideal_quot I}]) : quotient_scope.

Notation

x <> y %[ 'mod_ideal' I ]

algebra

algebra/ring_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "choice", "ssrnat", "seq", "generic_quotient", "nmodule", "rings_modules_and_algebras", "divalg", "GRing.Theory", "Quotient.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

env_nth (n : positive) l

:= nth rO l (Pos.to_nat n).-1.

Definition

env_nth

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "nth" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

env_jump (j : positive) l

:= drop (Pos.to_nat j) l.

Definition

env_jump

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "drop" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

env_jumpD l j j' : env_jump (Pos.add j j') l = env_jump j' (env_jump j l).

Proof. by rewrite /env_jump drop_drop Pos_to_natD addnC. Qed.

Lemma

env_jumpD

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "Pos_to_natD", "add", "addnC", "drop_drop", "env_jump" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

env_nth_jump l i j : env_nth i (env_jump j l) = env_nth (Pos.add i j) l.

Proof. by rewrite /env_nth nth_drop Pos_to_natD -!subn1 addnBA 1?addnC ?Pos_to_nat_gt0. Qed.

Lemma

env_nth_jump

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "Pos_to_natD", "Pos_to_nat_gt0", "add", "addnBA", "addnC", "env_jump", "env_nth", "nth_drop", "subn1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Peq

:= (Peq eq_op).

Notation

Peq

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

P0

:= (P0 0).

Notation

P0

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

P1

:= (P1 1).

Notation

P1

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mkX

:= (mkX 0 1).

Notation

mkX

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mkPX

:= (mkPX 0 eq_op).

Notation

mkPX

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PaddC

:= (PaddC +%R).

Notation

PaddC

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Padd

:= (Padd 0 +%R eq_op).

Notation

Padd

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PaddI

:= (PaddI +%R Padd).

Notation

PaddI

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "Padd" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PaddX

:= (PaddX 0 eq_op Padd).

Notation

PaddX

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "Padd" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Pmul

:= (Pmul 0 1 +%R *%R eq_op).

Notation

Pmul

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PmulC_aux

:= (PmulC_aux 0 *%R eq_op).

Notation

PmulC_aux

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PmulC

:= (PmulC 0 1 *%R eq_op).

Notation

PmulC

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PmulI

:= (PmulI 0 1 *%R eq_op Pmul).

Notation

PmulI

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "Pmul" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ppow_pos

:= (Ppow_pos 0 1 +%R *%R eq_op).

Notation

Ppow_pos

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Ppow_N

:= (Ppow_N 0 1 +%R *%R eq_op).

Notation

Ppow_N

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Peval_Peq l P P' : Peq P P' -> Peval l P = Peval l P'.

Proof. elim: P l P' => [c | i P IH | P IHP j Q IHQ] l [c' | i' P' | P' j' Q'] //=. - by move=> /eqP->. - by case: pos_nat_compareP => //; apply: IH. - by case: pos_nat_compareP => //; case: ifP => // /IHP-> /IHQ->. Qed.

Lemma

Peval_Peq

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "Peq", "Peval", "apply", "pos_nat_compareP" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Peval_mkPinj l j P : Peval l (mkPinj j P) = Peval (env_jump j l) P.

Proof. by case: P => [//| j' Q /= |//]; rewrite env_jumpD. Qed.

Lemma

Peval_mkPinj

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "Peval", "env_jump", "env_jumpD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Peval_mkX l j : Peval l (mkX j) = env_nth j l.

Proof. case: j => [j|j|]/=; rewrite rmorph1 rmorph0 mul1r addr0 expr1 ?env_nth_jump//. congr (env_nth _ l); apply: Pos_to_natI; rewrite !Pos_to_natE. by rewrite [LHS]add1n prednK// -double0 ltn_double Pos_to_nat_gt0. Qed.

Lemma

Peval_mkX

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "Peval", "Pos_to_natE", "Pos_to_natI", "Pos_to_nat_gt0", "add1n", "addr0", "apply", "double0", "env_nth", "env_nth_jump", "expr1", "ltn_double", "mkX", "mul1r", "prednK", "rmorph0", "rmorph1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Peval_mkPX l P i Q : Peval l (mkPX P i Q) = Peval l P * (env_nth 1 l) ^+ Pos.to_nat i + Peval (env_jump 1 l) Q.

Proof. case: P => [c |//| P' i' [c |//|//]] /=. by case: eqP => [->|//]; rewrite rmorph0 mul0r add0r Peval_mkPinj. by case: eqP => [->|//]/=; rewrite rmorph0 addr0 Pos_to_natD exprD mulrA. Qed.

Lemma

Peval_mkPX

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "Peval", "Peval_mkPinj", "Pos_to_natD", "add0r", "addr0", "env_jump", "env_nth", "exprD", "mkPX", "mul0r", "mulrA", "rmorph0" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Popp_id P : Popp id P = P.

Proof. by elim: P => [| j P /=->// | P /=-> i Q ->]. Qed.

Lemma

Popp_id

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "Popp", "id" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PevalDC l P c : Peval l (PaddC P c) = Peval l P + R_of_C c.

Proof. by elim: P l => [c'|i P IH|P IHP j Q IHQ] l/=; rewrite ?rmorphD ?IH ?IHQ ?addrA. Qed.

Lemma

PevalDC

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "PaddC", "Peval", "addrA", "rmorphD" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Peval_addI l P j Q : (forall l P, Peval l (Padd P Q) = Peval l P + Peval l Q) -> Peval l (PaddI Q j P) = Peval l P + Peval l (Pinj j Q).

Proof. move=> PevalD; elim: P j l => [c | j' Q' IH | P IHP i Q' IHQ'] j l /=. - by rewrite Peval_mkPinj PevalDC addrC. - move: (Zint_pos_sub (pos_nat_Pos_to_nat j') (pos_nat_Pos_to_nat j)). case: ZintP => [//| _ /eqP+ _ | k nk _ /[swap] | k nk _ /[swap] ]. + rewrite subr_eq0 => /eqP[]/Pos_to_natI->. by rewrite ...

Lemma

Peval_addI

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "NegzE", "Padd", "PaddI", "Peval", "PevalD", "PevalDC", "Peval_mkPinj", "Pos_to_natD", "Pos_to_natE", "Pos_to_natI", "Pos_to_nat_gt0", "ZintP", "Zint_pos_sub", "add1n", "addnC", "addrA", "addrC", "apply", "double0", "env_jump", "env_jumpD", "eqr_opp", "ltn_double", "opp...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Peval_addX l P P' i' : (forall l P, Peval l (Padd P P') = Peval l P + Peval l P') -> Peval l (PaddX P' i' P) = Peval l P + Peval l (PX P' i' P0).

Proof. rewrite /= rmorph0 addr0 addrC. move=> PevalD; elim: P i' l => [//| j Q' IH | P IHP i Q' IHQ'] i' l /=. - case: j => [j | j |//]/=; rewrite -env_jumpD//. congr (_ + Peval (env_jump _ l) _). apply: Pos_to_natI; rewrite !Pos_to_natE add1n prednK//. by rewrite -double0 ltn_double Pos_to_nat_gt0. - rewrite add...

Lemma

Peval_addX

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "NegzE", "P0", "Padd", "PaddX", "Peval", "PevalD", "Peval_mkPX", "Pos_to_natD", "Pos_to_natE", "Pos_to_natI", "Pos_to_nat_gt0", "ZintP", "Zint_pos_sub", "add1n", "addnC", "addr0", "addrA", "addrC", "apply", "double0", "env_jump", "env_jumpD", "eqr_opp", "exprD", "ltn_...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PevalD l P P' : Peval l (Padd P P') = Peval l P + Peval l P'.

Proof. elim: P' l P => [c | j' Q' IH | P' IHP' i' Q' IHQ'] l P /=. - by rewrite PevalDC. - by rewrite Peval_addI. case: P => [c | j Q | P i Q] /=. - by rewrite PevalDC addrA addrC. - rewrite addrCA; case: j => [j | j |] /=; rewrite IHQ'/= -?env_jumpD//. congr (_ + (Peval (env_jump _ l) _ + _)); apply: Pos_to_natI. ...

Lemma

PevalD

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "NegzE", "Padd", "Peval", "PevalDC", "Peval_addI", "Peval_addX", "Peval_mkPX", "Pos_to_natD", "Pos_to_natE", "Pos_to_natI", "Pos_to_nat_gt0", "ZintP", "Zint_pos_sub", "add1n", "addr0", "addrA", "addrACA", "addrC", "addrCA", "apply", "double0", "env_jump", "env_jumpD", "...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PsubI_addI P j Q : (forall P', Psub 0 +%R +%R id eq_op P' P = Padd P' P) -> PsubI +%R id (Psub 0 +%R +%R id eq_op) P j Q = PaddI P j Q.

Lemma

PsubI_addI

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "Padd", "PaddI", "Popp_id", "Psub", "PsubI", "Psub_add", "id" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PsubX_addX P' i P : (forall P, Psub 0 +%R +%R id eq_op P P' = Padd P P') -> PsubX 0 id eq_op (Psub 0 +%R +%R id eq_op) P' i P = PaddX P' i P.

Lemma

PsubX_addX

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "Padd", "PaddX", "Popp_id", "Psub", "PsubX", "Psub_add", "id" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Psub_add P P' : Psub 0 +%R +%R id eq_op P P' = Padd P P'.

Lemma

Psub_add

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "Padd", "Popp_id", "Psub", "PsubI_addI", "PsubX_addX", "id" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Peval_mulC_aux l P c : Peval l (PmulC_aux P c) = Peval l P * R_of_C c.

Proof. elim: P l => [c' | j Q IH | P IHP i Q IHQ] l /=; first by rewrite rmorphM. by rewrite Peval_mkPinj IH. by rewrite Peval_mkPX IHP IHQ mulrAC -mulrDl. Qed.

Lemma

Peval_mulC_aux

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "Peval", "Peval_mkPX", "Peval_mkPinj", "PmulC_aux", "mulrAC", "mulrDl", "rmorphM" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PevalMC l P c : Peval l (PmulC P c) = Peval l P * R_of_C c.

Proof. rewrite /PmulC/=; case: eqP => [->|_]; first by rewrite /= rmorph0 mulr0. by case: eqP => [->|_]; rewrite ?rmorph1 ?mulr1 ?Peval_mulC_aux. Qed.

Lemma

PevalMC

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "Peval", "Peval_mulC_aux", "PmulC", "mulr0", "mulr1", "rmorph0", "rmorph1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Peval_mulI l P j Q : (forall l P, Peval l (Pmul P Q) = Peval l P * Peval l Q) -> Peval l (PmulI Q j P) = Peval l P * Peval l (Pinj j Q).

Proof. move=> PevalM; elim: P j l => [c | j' Q' IH | P IHP i Q' IHQ'] j l /=. - by rewrite Peval_mkPinj PevalMC mulrC. - move: (Zint_pos_sub (pos_nat_Pos_to_nat j') (pos_nat_Pos_to_nat j)). case: ZintP => [//| _ /eqP+ _ | k nk _ /[swap] | k nk _ /[swap] ]. + rewrite subr_eq0 => /eqP[]/Pos_to_natI->. by rewrite ...

Lemma

Peval_mulI

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "NegzE", "Peval", "PevalM", "PevalMC", "Peval_mkPX", "Peval_mkPinj", "Pmul", "PmulI", "Pos_to_natD", "Pos_to_natE", "Pos_to_natI", "Pos_to_nat_gt0", "ZintP", "Zint_pos_sub", "add", "add1n", "addnC", "apply", "double0", "env_jumpD", "eqr_opp", "last", "ltn_double", "mulr...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PevalM l P P' : Peval l (Pmul P P') = Peval l P * Peval l P'.

Proof. elim: P' l P => [c | j' Q' IH | P' IHP' i' Q' IHQ'] l P /=. - by rewrite PevalMC. - by rewrite Peval_mulI. case: P => [c | j Q | P i Q] /=. - by rewrite PevalMC mulrC. - rewrite Peval_mkPX IHP' mulrDr mulrA. case: j => [j | j |] /=; rewrite IHQ'/= -?env_jumpD//. congr (_ + (Peval (env_jump _ l) _ * _)); appl...

Lemma

PevalM

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "Peval", "PevalD", "PevalMC", "Peval_mkPX", "Peval_mkPinj", "Peval_mulI", "Pmul", "Pos_to_natE", "Pos_to_natI", "Pos_to_nat_gt0", "add1n", "addr0", "addrACA", "apply", "double0", "env_jump", "env_jumpD", "ltn_double", "mulrA", "mulrAC", "mulrACA", "mulrC", "mulrDl", "mu...

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Peval_pow_pos l res P p : Peval l (Ppow_pos res P p) = Peval l res * Peval l P ^+ Pos.to_nat p.

Proof. elim: p l res P => [p IH | p IH |] l res P /=. - rewrite PevalM !IH -mulrAC -mulrA -exprS -mulrA -exprD. by rewrite addnS addnn Pos_to_natE. - by rewrite !IH -mulrA -exprD addnn Pos_to_natE. - by rewrite PevalM expr1. Qed.

Lemma

Peval_pow_pos

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "Peval", "PevalM", "Pos_to_natE", "Ppow_pos", "addnS", "addnn", "expr1", "exprD", "exprS", "mulrA", "mulrAC" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Peval_pow_N l P n : Peval l (Ppow_N P n) = Peval l P ^+ N.to_nat n.

Proof. by case: n => [| p]/=; rewrite ?Peval_pow_pos /= rmorph1 ?mul1r. Qed.

Lemma

Peval_pow_N

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "Peval", "Peval_pow_pos", "Ppow_N", "mul1r", "rmorph1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Popp

:= (Popp -%R).

Notation

Popp

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PsubC

:= (PsubC (fun x y => x - y)).

Notation

PsubC

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Psub

:= (Psub 0 +%R (fun x y => x - y) -%R eq_op).

Notation

Psub

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PsubI

:= (PsubI +%R -%R Psub).

Notation

PsubI

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "Psub" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PsubX

:= (PsubX 0 -%R eq_op Psub).

Notation

PsubX

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "Psub" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PevalN l P : Peval l (Popp P) = - Peval l P.

Proof. elim: P l => [c | i P IH | P IHP j Q IHQ] l /=; rewrite ?rmorphN ?IH//. by rewrite IHP IHQ opprD mulNr. Qed.

Lemma

PevalN

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "Peval", "Popp", "mulNr", "opprD", "rmorphN" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PevalBC l P c : Peval l (PsubC P c) = Peval l P - R_of_C c.

Proof. by elim: P l => [c'|i P IH|P IHP j Q IHQ] l/=; rewrite ?rmorphB ?IH ?IHQ ?addrA. Qed.

Lemma

PevalBC

algebra

algebra/ring_tactic.v

[ "elpi", "derive.std", "param2", "micromega_plugin", "NatDef", "Corelib", "IntDef", "formula", "witness", "checker", "eval", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "seq", "preorder", "ssralg", "ssrint", "binnums", "mathcomp.algebra", "Extra", ...

[ "Peval", "PsubC", "addrA", "rmorphB" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d