phanerozoic/Coq-MathComp · Datasets at Hugging Face

statement stringlengths 1 4.33k	proof stringlengths 0 37.9k	type stringclasses 25 values	symbolic_name stringlengths 1 67	library stringclasses 10 values	filename stringclasses 112 values	imports listlengths 2 138	deps listlengths 0 64	docstring stringclasses 798 values	source_url stringclasses 1 value	commit stringclasses 1 value
Zp_mulA : associative Zp_mul.	Proof. by move=> x y z; apply: val_inj; rewrite /= modnMml modnMmr mulnA. Qed.	Lemma	Zp_mulA	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "Zp_mul", "apply", "modnMml", "modnMmr", "mulnA", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Zp_mul_addr : right_distributive Zp_mul Zp_add.	Proof. by move=> x y z; apply: val_inj; rewrite /= modnMmr modnDm mulnDr. Qed.	Lemma	Zp_mul_addr	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "Zp_add", "Zp_mul", "apply", "modnDm", "modnMmr", "mulnDr", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Zp_mul_addl : left_distributive Zp_mul Zp_add.	Proof. by move=> x y z; rewrite -!(Zp_mulC z) Zp_mul_addr. Qed.	Lemma	Zp_mul_addl	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "Zp_add", "Zp_mul", "Zp_mulC", "Zp_mul_addr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Zp_inv_out i : ~~ coprime p i -> Zp_inv i = i.	Proof. by rewrite /Zp_inv => /negPf->. Qed.	Lemma	Zp_inv_out	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "Zp_inv", "coprime" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
inZp (i : nat) : 'I_p	:= Ordinal (ltn_pmod i (ltn0Sn p')).	Definition	inZp	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "ltn0Sn", "ltn_pmod", "nat" ]	Standard injection; val (inZp i) = i %% p	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
valZpK x : inZp x = x.	Proof. by apply: val_inj; rewrite /= modZp. Qed.	Lemma	valZpK	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "inZp", "modZp", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Zp0 : 'I_p	:= ord0.	Definition	Zp0	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "ord0" ]	Operations	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Zp1	:= inZp 1.	Definition	Zp1	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "inZp" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Zp_add0z : left_id Zp0 Zp_add.	Proof. by move=> x; apply: val_inj; rewrite /= modZp. Qed.	Lemma	Zp_add0z	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "Zp0", "Zp_add", "apply", "modZp", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Zp_addNz : left_inverse Zp0 Zp_opp Zp_add.	Proof. by move=> x; apply: val_inj; rewrite /= modnDml subnK ?modnn // ltnW. Qed.	Lemma	Zp_addNz	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "Zp0", "Zp_add", "Zp_opp", "apply", "ltnW", "modnDml", "modnn", "subnK", "val_inj" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prod_enum	:= [seq (x1, x2) \| x1 <- enum T1, x2 <- enum T2].	Definition	prod_enum	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "enum", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
predX_prod_enum (A1 : {pred T1}) (A2 : {pred T2}) : count [predX A1 & A2] prod_enum = #\|A1\| * #\|A2\|.	Proof. rewrite !cardE !size_filter -!enumT /prod_enum. elim: (enum T1) => //= x1 s1 IHs; rewrite count_cat {}IHs count_map /preim /=. by case: (x1 \in A1); rewrite ?count_pred0. Qed.	Lemma	predX_prod_enum	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "cardE", "count", "count_cat", "count_map", "count_pred0", "enum", "enumT", "predX", "prod_enum", "s1", "size_filter" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
prod_enumP : Finite.axiom prod_enum.	Proof. by case=> x1 x2; rewrite (predX_prod_enum (pred1 x1) (pred1 x2)) !card1. Qed.	Lemma	prod_enumP	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "axiom", "card1", "pred1", "predX_prod_enum", "prod_enum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
cardX (A1 : {pred T1}) (A2 : {pred T2}) : #\|[predX A1 & A2]\| = #\|A1\| * #\|A2\|.	Proof. by rewrite -predX_prod_enum unlock size_filter unlock. Qed.	Lemma	cardX	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "predX", "predX_prod_enum", "size_filter" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_prod : #\|{: T1 * T2}\| = #\|T1\| * #\|T2\|.	Proof. by rewrite -cardX; apply: eq_card; case. Qed.	Lemma	card_prod	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "cardX", "eq_card" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_card_prod (A : {pred (T1 * T2)}) : A =i predT -> #\|A\| = #\|T1\| * #\|T2\|.	Proof. exact: eq_card_trans card_prod. Qed.	Lemma	eq_card_prod	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "card_prod", "eq_card_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tag_enum	:= flatten [seq [seq Tagged T_ x \| x <- enumF (T_ i)] \| i <- enumF I].	Definition	tag_enum	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "enumF", "flatten", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
tag_enumP : Finite.axiom tag_enum.	Proof. case=> i x; rewrite -(enumP i) /tag_enum -enumT. elim: (enum I) => //= j e IHe. rewrite count_cat count_map {}IHe; congr (_ + _). rewrite -size_filter -cardE /=; case: eqP => [-> \| ne_j_i]. by apply: (@eq_card1 _ x) => y; rewrite -topredE /= tagged_asE ?eqxx. by apply: eq_card0 => y. Qed.	Lemma	tag_enumP	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "axiom", "cardE", "count_cat", "count_map", "enum", "enumP", "enumT", "eq_card0", "eq_card1", "eqxx", "size_filter", "tag_enum", "tagged_asE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_tagged : #\|{: {i : I & T_ i}}\| = sumn (map (fun i => #\|T_ i\|) (enum I)).	Proof. rewrite cardE !enumT [in LHS]unlock size_flatten /shape -map_comp. by congr (sumn _); apply: eq_map => i; rewrite /= size_map -enumT -cardE. Qed.	Lemma	card_tagged	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "cardE", "enum", "enumT", "eq_map", "map", "map_comp", "shape", "size_flatten", "size_map", "sumn" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sum_enum	:= [seq inl _ x \| x <- enumF T1] ++ [seq inr _ y \| y <- enumF T2].	Definition	sum_enum	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "enumF", "seq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sum_enum_uniq : uniq sum_enum.	Proof. rewrite cat_uniq -!enumT !(enum_uniq, map_inj_uniq); try by move=> ? ? []. by rewrite andbT; apply/hasP=> [[_ /mapP[x _ ->] /mapP[]]]. Qed.	Lemma	sum_enum_uniq	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "apply", "cat_uniq", "enumT", "enum_uniq", "hasP", "mapP", "map_inj_uniq", "sum_enum", "uniq" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mem_sum_enum u : u \in sum_enum.	Proof. by case: u => x; rewrite mem_cat -!enumT map_f ?mem_enum ?orbT. Qed.	Lemma	mem_sum_enum	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "enumT", "map_f", "mem_cat", "mem_enum", "sum_enum" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
card_sum : #\|{: T1 + T2}\| = #\|T1\| + #\|T2\|.	Proof. by rewrite !cardT !enumT [in LHS]unlock size_cat !size_map. Qed.	Lemma	card_sum	boot	boot/fintype.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]	[ "cardT", "enumT", "size_cat", "size_map" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_subdef	:= @quot_pi_subdef _ qT.	Definition	pi_subdef	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "qT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\pi"	:= pi_subdef.	Notation	\pi	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "pi_subdef" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
repr_ofK : cancel (@repr_of _ _) \pi.	Proof. exact: repr_ofK_subproof. Qed.	Lemma	repr_ofK	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\pi_ Q"	:= (@pi _ Q) : quotient_scope.	Notation	\pi_ Q	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\pi"	:= (@pi _ _) (only parsing) : quotient_scope.	Notation	\pi	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x == y %[mod Q ]"	:= (\pi_Q x == \pi_Q y) : quotient_scope.	Notation	x == y %[mod Q ]	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x = y %[mod Q ]"	:= (\pi_Q x = \pi_Q y) : quotient_scope.	Notation	x = y %[mod Q ]	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x != y %[mod Q ]"	:= (\pi_Q x != \pi_Q y) : quotient_scope.	Notation	x != y %[mod Q ]	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"x <> y %[mod Q ]"	:= (\pi_Q x <> \pi_Q y) : quotient_scope.	Notation	x <> y %[mod Q ]	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"\mpi"	:= (@mpi _ _).	Notation	\mpi	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mpi_unlock	:= Unlockable mpi.unlock.	Canonical	mpi_unlock	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_unlock	:= Unlockable pi.unlock.	Canonical	pi_unlock	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
repr_unlock	:= Unlockable repr.unlock.	Canonical	repr_unlock	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "repr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
reprK : cancel repr \pi_qT.	Proof. by move=> x; rewrite !unlock repr_ofK. Qed.	Lemma	reprK	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "repr", "repr_ofK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_spec (x : T) : T -> Type	:= PiSpec y of x = y %[mod qT] : pi_spec x y.	Variant	pi_spec	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "qT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
piP (x : T) : pi_spec x (repr (\pi_qT x)).	Proof. by constructor; rewrite reprK. Qed.	Lemma	piP	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "pi_spec", "repr", "reprK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
mpiE : \mpi =1 \pi_qT.	Proof. by move=> x; rewrite !unlock. Qed.	Lemma	mpiE	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotW P : (forall y : T, P (\pi_qT y)) -> forall x : qT, P x.	Proof. by move=> Py x; rewrite -[x]reprK; apply: Py. Qed.	Lemma	quotW	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "apply", "qT", "reprK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quotP P : (forall y : T, repr (\pi_qT y) = y -> P (\pi_qT y)) -> forall x : qT, P x.	Proof. by move=> Py x; rewrite -[x]reprK; apply: Py; rewrite reprK. Qed.	Lemma	quotP	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "apply", "qT", "repr", "reprK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
equal_to T (x : T)	:= EqualTo { equal_val : T; _ : x = equal_val }.	Structure	equal_to	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[]	This was pi_morph T (x : T) := PiMorph { pi_op : T; _ : x = pi_op }.	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
equal_toE (T : Type) (x : T) (m : equal_to x) : equal_val m = x.	Proof. by case: m. Qed.	Lemma	equal_toE	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "equal_to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
piE	:= (@equal_toE _ _).	Notation	piE	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "equal_toE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
equal_to_pi T (qT : quotType T) (x : T)	:= @EqualTo _ (\pi_qT x) (\pi x) (erefl _).	Canonical	equal_to_pi	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "pi", "qT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_f : {morph \pi : x / f x >-> fq x}.		Hypothesis	pi_f	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_g : {morph \pi : x y / g x y >-> gq x y}.		Hypothesis	pi_g	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_p : {mono \pi : x / p x >-> pq x}.		Hypothesis	pi_p	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_r : {mono \pi : x y / r x y >-> rq x y}.		Hypothesis	pi_r	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_h : forall (x : T), \pi_qU (h x) = hq (\pi_qT x).		Hypothesis	pi_h	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_morph1 : \pi (f a) = fq (equal_val x).	Proof. by rewrite !piE. Qed.	Lemma	pi_morph1	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "pi", "piE" ]	Internal Lemmas : do not use directly	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_morph2 : \pi (g a b) = gq (equal_val x) (equal_val y).	Proof. by rewrite !piE. Qed.	Lemma	pi_morph2	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "pi", "piE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_mono1 : p a = pq (equal_val x).	Proof. by rewrite !piE. Qed.	Lemma	pi_mono1	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "piE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_mono2 : r a b = rq (equal_val x) (equal_val y).	Proof. by rewrite !piE. Qed.	Lemma	pi_mono2	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "piE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_morph11 : \pi (h a) = hq (equal_val x).	Proof. by rewrite !piE. Qed.	Lemma	pi_morph11	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "pi", "piE" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{pi_ Q a }"	:= (equal_to (\pi_Q a)) : quotient_scope.	Notation	{pi_ Q a }	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "equal_to" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"{pi a }"	:= (equal_to (\pi a)) : quotient_scope.	Notation	{pi a }	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "equal_to", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PiMorph pi_x	:= (EqualTo pi_x).	Notation	PiMorph	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[]	Declaration of morphisms	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PiMorph1 pi_f	:= (fun a (x : {pi a}) => EqualTo (pi_morph1 pi_f a x)).	Notation	PiMorph1	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "pi", "pi_f", "pi_morph1" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PiMorph2 pi_g	:= (fun a b (x : {pi a}) (y : {pi b}) => EqualTo (pi_morph2 pi_g a b x y)).	Notation	PiMorph2	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "pi", "pi_g", "pi_morph2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PiMono1 pi_p	:= (fun a (x : {pi a}) => EqualTo (pi_mono1 pi_p a x)).	Notation	PiMono1	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "pi", "pi_mono1", "pi_p" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PiMono2 pi_r	:= (fun a b (x : {pi a}) (y : {pi b}) => EqualTo (pi_mono2 pi_r a b x y)).	Notation	PiMono2	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "pi", "pi_mono2", "pi_r" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PiMorph11 pi_f	:= (fun a (x : {pi a}) => EqualTo (pi_morph11 pi_f a x)).	Notation	PiMorph11	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "pi", "pi_f", "pi_morph11" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lift_op1 Q f	:= (locked (fun x : Q => \pi_Q (f (repr x)) : Q)).	Notation	lift_op1	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "repr" ]	lifting helpers	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lift_op2 Q g	:= (locked (fun x y : Q => \pi_Q (g (repr x) (repr y)) : Q)).	Notation	lift_op2	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "repr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lift_fun1 Q f	:= (locked (fun x : Q => f (repr x))).	Notation	lift_fun1	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "repr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lift_fun2 Q g	:= (locked (fun x y : Q => g (repr x) (repr y))).	Notation	lift_fun2	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "repr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lift_op11 Q Q' f	:= (locked (fun x : Q => \pi_Q' (f (repr x)) : Q')).	Notation	lift_op11	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "repr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lift_cst Q x	:= (locked (\pi_Q x : Q)).	Notation	lift_cst	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[]	constant declaration	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PiConst a	:= (@EqualTo _ _ a (lock _)).	Notation	PiConst	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
lift_embed qT e	:= (locked (fun x => \pi_qT (e x) : qT)).	Notation	lift_embed	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "qT" ]	embedding declaration, please don't redefine \pi	https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_lock T T' e : e =1 (@locked (T -> T') (fun x : T => e x)).	Proof. by rewrite -lock. Qed.	Lemma	eq_lock	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "T'" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
PiEmbed e	:= (fun x => @EqualTo _ _ (e x) (eq_lock (fun _ => \pi _) _)).	Notation	PiEmbed	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "eq_lock", "pi" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
pi_eq_quot_mono T eq_quot_op eqT	:= PiMono2 (@pi_eq_quot T eq_quot_op eqT).	Canonical	pi_eq_quot_mono	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "PiMono2" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quot_type_of T (qT : quotType T) : Type	:= qT.	Definition	quot_type_of	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "qT" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
quot_type Q	:= (quot_type_of _ Q).	Notation	quot_type	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "quot_type_of" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
Sub x (px : repr (\pi_qT x) == x)	:= \pi_qT x.	Definition	Sub	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "repr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
qreprK x Px : repr (@Sub x Px) = x.	Proof. by rewrite /Sub (eqP Px). Qed.	Lemma	qreprK	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "Px", "Sub", "repr" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sortPx (x : qT) : repr (\pi_qT (repr x)) == repr x.	Proof. by rewrite !reprK eqxx. Qed.	Lemma	sortPx	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "eqxx", "qT", "repr", "reprK" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
sort_Sub (x : qT) : x = Sub (sortPx x).	Proof. by rewrite /Sub reprK. Qed.	Lemma	sort_Sub	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "Sub", "qT", "reprK", "sortPx" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
reprP K (PK : forall x Px, K (@Sub x Px)) u : K u.	Proof. by rewrite (sort_Sub u); apply: PK. Qed.	Lemma	reprP	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "Px", "Sub", "apply", "sort_Sub" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'Sub' Q 'of' T 'by' %/ ]"	:= (SubType.copy Q%type (quot_type_of T Q%type)) (format "[ 'Sub' Q 'of' T 'by' %/ ]") : form_scope.	Notation	[ 'Sub' Q 'of' T 'by' %/ ]	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "copy", "quot_type_of", "type" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'Sub' Q 'by' %/ ]"	:= (SubType.copy Q%type (quot_type Q)) (format "[ 'Sub' Q 'by' %/ ]") : form_scope.	Notation	[ 'Sub' Q 'by' %/ ]	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "copy", "quot_type", "type" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'Equality' 'of' Q 'by' <:%/ ]"	:= (Equality.copy Q%type (quot_type Q)) (format "[ 'Equality' 'of' Q 'by' <:%/ ]") : form_scope.	Notation	[ 'Equality' 'of' Q 'by' <:%/ ]	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "copy", "quot_type", "type" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'Choice' 'of' Q 'by' <:%/ ]"	:= (Choice.copy Q%type (quot_type Q)) (format "[ 'Choice' 'of' Q 'by' <:%/ ]") : form_scope.	Notation	[ 'Choice' 'of' Q 'by' <:%/ ]	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "copy", "quot_type", "type" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'Countable' 'of' Q 'by' <:%/ ]"	:= (Countable.copy Q%type (quot_type Q)) (format "[ 'Countable' 'of' Q 'by' <:%/ ]") : form_scope.	Notation	[ 'Countable' 'of' Q 'by' <:%/ ]	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "copy", "quot_type", "type" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
"[ 'Finite' 'of' Q 'by' <:%/ ]"	:= (Finite.copy Q%type (quot_type Q)) (format "[ 'Finite' 'of' Q 'by' <:%/ ]") : form_scope.	Notation	[ 'Finite' 'of' Q 'by' <:%/ ]	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "copy", "quot_type", "type" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
left_trans (e : rel T) : symmetric e -> transitive e -> left_transitive e.	Proof. by move=> s t ? * ?; apply/idP/idP; apply: t; rewrite // s. Qed.	Lemma	left_trans	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "apply", "rel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
right_trans (e : rel T) : symmetric e -> transitive e -> right_transitive e.	Proof. by move=> s t ? * x; rewrite ![e x _]s; apply: left_trans. Qed.	Lemma	right_trans	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "apply", "left_trans", "rel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
equiv_class_of (equiv : rel T)	:= EquivClass of reflexive equiv & symmetric equiv & transitive equiv.	Variant	equiv_class_of	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "equiv", "rel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
equiv_rel	:= EquivRelPack { equiv :> rel T; _ : equiv_class_of equiv }.	Record	equiv_rel	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "equiv", "equiv_class_of", "rel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
equiv_class	:= let: EquivRelPack _ ce as e' := e return equiv_class_of e' in ce.	Definition	equiv_class	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "e'", "equiv_class_of" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
equiv_pack (r : rel T) ce & phant_id ce equiv_class	:= @EquivRelPack r ce.	Definition	equiv_pack	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "equiv_class", "rel" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
equiv_refl x : e x x.	Proof. by case: e => [] ? []. Qed.	Lemma	equiv_refl	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
equiv_sym : symmetric e.	Proof. by case: e => [] ? []. Qed.	Lemma	equiv_sym	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
equiv_trans : transitive e.	Proof. by case: e => [] ? []. Qed.	Lemma	equiv_trans	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
eq_op_trans (T' : eqType) : transitive (@eq_op T').	Proof. by move=> x y z /eqP -> /eqP ->. Qed.	Lemma	eq_op_trans	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "T'" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
equiv_ltrans: left_transitive e.	Proof. by apply: left_trans; [apply: equiv_sym\|apply: equiv_trans]. Qed.	Lemma	equiv_ltrans	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "apply", "equiv_sym", "equiv_trans", "left_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d
equiv_rtrans: right_transitive e.	Proof. by apply: right_trans; [apply: equiv_sym\|apply: equiv_trans]. Qed.	Lemma	equiv_rtrans	boot	boot/generic_quotient.v	[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]	[ "apply", "equiv_sym", "equiv_trans", "right_trans" ]		https://github.com/math-comp/math-comp	91d97df9cf3204b4dab84f4e24bc633e84b6473d

Zp_mulA : associative Zp_mul.

Proof. by move=> x y z; apply: val_inj; rewrite /= modnMml modnMmr mulnA. Qed.

Lemma

Zp_mulA

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "Zp_mul", "apply", "modnMml", "modnMmr", "mulnA", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Zp_mul_addr : right_distributive Zp_mul Zp_add.

Proof. by move=> x y z; apply: val_inj; rewrite /= modnMmr modnDm mulnDr. Qed.

Lemma

Zp_mul_addr

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "Zp_add", "Zp_mul", "apply", "modnDm", "modnMmr", "mulnDr", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Zp_mul_addl : left_distributive Zp_mul Zp_add.

Proof. by move=> x y z; rewrite -!(Zp_mulC z) Zp_mul_addr. Qed.

Lemma

Zp_mul_addl

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "Zp_add", "Zp_mul", "Zp_mulC", "Zp_mul_addr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Zp_inv_out i : ~~ coprime p i -> Zp_inv i = i.

Proof. by rewrite /Zp_inv => /negPf->. Qed.

Lemma

Zp_inv_out

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "Zp_inv", "coprime" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

inZp (i : nat) : 'I_p

:= Ordinal (ltn_pmod i (ltn0Sn p')).

Definition

inZp

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "ltn0Sn", "ltn_pmod", "nat" ]

Standard injection; val (inZp i) = i %% p

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

valZpK x : inZp x = x.

Proof. by apply: val_inj; rewrite /= modZp. Qed.

Lemma

valZpK

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "inZp", "modZp", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Zp0 : 'I_p

:= ord0.

Definition

Zp0

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "ord0" ]

Operations

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Zp1

:= inZp 1.

Definition

Zp1

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "inZp" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Zp_add0z : left_id Zp0 Zp_add.

Proof. by move=> x; apply: val_inj; rewrite /= modZp. Qed.

Lemma

Zp_add0z

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "Zp0", "Zp_add", "apply", "modZp", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Zp_addNz : left_inverse Zp0 Zp_opp Zp_add.

Proof. by move=> x; apply: val_inj; rewrite /= modnDml subnK ?modnn // ltnW. Qed.

Lemma

Zp_addNz

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "Zp0", "Zp_add", "Zp_opp", "apply", "ltnW", "modnDml", "modnn", "subnK", "val_inj" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prod_enum

:= [seq (x1, x2) | x1 <- enum T1, x2 <- enum T2].

Definition

prod_enum

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "enum", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

predX_prod_enum (A1 : {pred T1}) (A2 : {pred T2}) : count [predX A1 & A2] prod_enum = #|A1| * #|A2|.

Proof. rewrite !cardE !size_filter -!enumT /prod_enum. elim: (enum T1) => //= x1 s1 IHs; rewrite count_cat {}IHs count_map /preim /=. by case: (x1 \in A1); rewrite ?count_pred0. Qed.

Lemma

predX_prod_enum

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "cardE", "count", "count_cat", "count_map", "count_pred0", "enum", "enumT", "predX", "prod_enum", "s1", "size_filter" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

prod_enumP : Finite.axiom prod_enum.

Proof. by case=> x1 x2; rewrite (predX_prod_enum (pred1 x1) (pred1 x2)) !card1. Qed.

Lemma

prod_enumP

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "axiom", "card1", "pred1", "predX_prod_enum", "prod_enum" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

cardX (A1 : {pred T1}) (A2 : {pred T2}) : #|[predX A1 & A2]| = #|A1| * #|A2|.

Proof. by rewrite -predX_prod_enum unlock size_filter unlock. Qed.

Lemma

cardX

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "predX", "predX_prod_enum", "size_filter" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_prod : #|{: T1 * T2}| = #|T1| * #|T2|.

Proof. by rewrite -cardX; apply: eq_card; case. Qed.

Lemma

card_prod

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "cardX", "eq_card" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_card_prod (A : {pred (T1 * T2)}) : A =i predT -> #|A| = #|T1| * #|T2|.

Proof. exact: eq_card_trans card_prod. Qed.

Lemma

eq_card_prod

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "card_prod", "eq_card_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tag_enum

:= flatten [seq [seq Tagged T_ x | x <- enumF (T_ i)] | i <- enumF I].

Definition

tag_enum

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "enumF", "flatten", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

tag_enumP : Finite.axiom tag_enum.

Proof. case=> i x; rewrite -(enumP i) /tag_enum -enumT. elim: (enum I) => //= j e IHe. rewrite count_cat count_map {}IHe; congr (_ + _). rewrite -size_filter -cardE /=; case: eqP => [-> | ne_j_i]. by apply: (@eq_card1 _ x) => y; rewrite -topredE /= tagged_asE ?eqxx. by apply: eq_card0 => y. Qed.

Lemma

tag_enumP

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "axiom", "cardE", "count_cat", "count_map", "enum", "enumP", "enumT", "eq_card0", "eq_card1", "eqxx", "size_filter", "tag_enum", "tagged_asE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_tagged : #|{: {i : I & T_ i}}| = sumn (map (fun i => #|T_ i|) (enum I)).

Proof. rewrite cardE !enumT [in LHS]unlock size_flatten /shape -map_comp. by congr (sumn _); apply: eq_map => i; rewrite /= size_map -enumT -cardE. Qed.

Lemma

card_tagged

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "cardE", "enum", "enumT", "eq_map", "map", "map_comp", "shape", "size_flatten", "size_map", "sumn" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sum_enum

:= [seq inl _ x | x <- enumF T1] ++ [seq inr _ y | y <- enumF T2].

Definition

sum_enum

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "enumF", "seq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sum_enum_uniq : uniq sum_enum.

Proof. rewrite cat_uniq -!enumT !(enum_uniq, map_inj_uniq); try by move=> ? ? []. by rewrite andbT; apply/hasP=> [[_ /mapP[x _ ->] /mapP[]]]. Qed.

Lemma

sum_enum_uniq

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "apply", "cat_uniq", "enumT", "enum_uniq", "hasP", "mapP", "map_inj_uniq", "sum_enum", "uniq" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mem_sum_enum u : u \in sum_enum.

Proof. by case: u => x; rewrite mem_cat -!enumT map_f ?mem_enum ?orbT. Qed.

Lemma

mem_sum_enum

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "enumT", "map_f", "mem_cat", "mem_enum", "sum_enum" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

card_sum : #|{: T1 + T2}| = #|T1| + #|T2|.

Proof. by rewrite !cardT !enumT [in LHS]unlock size_cat !size_map. Qed.

Lemma

card_sum

boot

boot/fintype.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "ssrnotations", "eqtype", "ssrnat", "seq", "choice", "path", "div", "FiniteQuant.Exports", "Finite" ]

[ "cardT", "enumT", "size_cat", "size_map" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_subdef

:= @quot_pi_subdef _ qT.

Definition

pi_subdef

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "qT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\pi"

:= pi_subdef.

Notation

\pi

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "pi_subdef" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

repr_ofK : cancel (@repr_of _ _) \pi.

Proof. exact: repr_ofK_subproof. Qed.

Lemma

repr_ofK

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\pi_ Q"

:= (@pi _ Q) : quotient_scope.

Notation

\pi_ Q

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\pi"

:= (@pi _ _) (only parsing) : quotient_scope.

Notation

\pi

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x == y %[mod Q ]"

:= (\pi_Q x == \pi_Q y) : quotient_scope.

Notation

x == y %[mod Q ]

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x = y %[mod Q ]"

:= (\pi_Q x = \pi_Q y) : quotient_scope.

Notation

x = y %[mod Q ]

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x != y %[mod Q ]"

:= (\pi_Q x != \pi_Q y) : quotient_scope.

Notation

x != y %[mod Q ]

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"x <> y %[mod Q ]"

:= (\pi_Q x <> \pi_Q y) : quotient_scope.

Notation

x <> y %[mod Q ]

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"\mpi"

:= (@mpi _ _).

Notation

\mpi

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mpi_unlock

:= Unlockable mpi.unlock.

Canonical

mpi_unlock

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_unlock

:= Unlockable pi.unlock.

Canonical

pi_unlock

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

repr_unlock

:= Unlockable repr.unlock.

Canonical

repr_unlock

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "repr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

reprK : cancel repr \pi_qT.

Proof. by move=> x; rewrite !unlock repr_ofK. Qed.

Lemma

reprK

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "repr", "repr_ofK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_spec (x : T) : T -> Type

:= PiSpec y of x = y %[mod qT] : pi_spec x y.

Variant

pi_spec

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "qT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

piP (x : T) : pi_spec x (repr (\pi_qT x)).

Proof. by constructor; rewrite reprK. Qed.

Lemma

piP

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "pi_spec", "repr", "reprK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

mpiE : \mpi =1 \pi_qT.

Proof. by move=> x; rewrite !unlock. Qed.

Lemma

mpiE

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotW P : (forall y : T, P (\pi_qT y)) -> forall x : qT, P x.

Proof. by move=> Py x; rewrite -[x]reprK; apply: Py. Qed.

Lemma

quotW

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "apply", "qT", "reprK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quotP P : (forall y : T, repr (\pi_qT y) = y -> P (\pi_qT y)) -> forall x : qT, P x.

Proof. by move=> Py x; rewrite -[x]reprK; apply: Py; rewrite reprK. Qed.

Lemma

quotP

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "apply", "qT", "repr", "reprK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

equal_to T (x : T)

:= EqualTo { equal_val : T; _ : x = equal_val }.

Structure

equal_to

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[]

This was pi_morph T (x : T) := PiMorph { pi_op : T; _ : x = pi_op }.

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

equal_toE (T : Type) (x : T) (m : equal_to x) : equal_val m = x.

Proof. by case: m. Qed.

Lemma

equal_toE

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "equal_to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

piE

:= (@equal_toE _ _).

Notation

piE

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "equal_toE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

equal_to_pi T (qT : quotType T) (x : T)

:= @EqualTo _ (\pi_qT x) (\pi x) (erefl _).

Canonical

equal_to_pi

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "pi", "qT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_f : {morph \pi : x / f x >-> fq x}.

Hypothesis

pi_f

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_g : {morph \pi : x y / g x y >-> gq x y}.

Hypothesis

pi_g

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_p : {mono \pi : x / p x >-> pq x}.

Hypothesis

pi_p

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_r : {mono \pi : x y / r x y >-> rq x y}.

Hypothesis

pi_r

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_h : forall (x : T), \pi_qU (h x) = hq (\pi_qT x).

Hypothesis

pi_h

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_morph1 : \pi (f a) = fq (equal_val x).

Proof. by rewrite !piE. Qed.

Lemma

pi_morph1

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "pi", "piE" ]

Internal Lemmas : do not use directly

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_morph2 : \pi (g a b) = gq (equal_val x) (equal_val y).

Proof. by rewrite !piE. Qed.

Lemma

pi_morph2

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "pi", "piE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_mono1 : p a = pq (equal_val x).

Proof. by rewrite !piE. Qed.

Lemma

pi_mono1

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "piE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_mono2 : r a b = rq (equal_val x) (equal_val y).

Proof. by rewrite !piE. Qed.

Lemma

pi_mono2

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "piE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_morph11 : \pi (h a) = hq (equal_val x).

Proof. by rewrite !piE. Qed.

Lemma

pi_morph11

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "pi", "piE" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"{pi_ Q a }"

:= (equal_to (\pi_Q a)) : quotient_scope.

Notation

{pi_ Q a }

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "equal_to" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"{pi a }"

:= (equal_to (\pi a)) : quotient_scope.

Notation

{pi a }

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "equal_to", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PiMorph pi_x

:= (EqualTo pi_x).

Notation

PiMorph

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[]

Declaration of morphisms

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PiMorph1 pi_f

:= (fun a (x : {pi a}) => EqualTo (pi_morph1 pi_f a x)).

Notation

PiMorph1

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "pi", "pi_f", "pi_morph1" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PiMorph2 pi_g

:= (fun a b (x : {pi a}) (y : {pi b}) => EqualTo (pi_morph2 pi_g a b x y)).

Notation

PiMorph2

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "pi", "pi_g", "pi_morph2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PiMono1 pi_p

:= (fun a (x : {pi a}) => EqualTo (pi_mono1 pi_p a x)).

Notation

PiMono1

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "pi", "pi_mono1", "pi_p" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PiMono2 pi_r

:= (fun a b (x : {pi a}) (y : {pi b}) => EqualTo (pi_mono2 pi_r a b x y)).

Notation

PiMono2

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "pi", "pi_mono2", "pi_r" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PiMorph11 pi_f

:= (fun a (x : {pi a}) => EqualTo (pi_morph11 pi_f a x)).

Notation

PiMorph11

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "pi", "pi_f", "pi_morph11" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lift_op1 Q f

:= (locked (fun x : Q => \pi_Q (f (repr x)) : Q)).

Notation

lift_op1

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "repr" ]

lifting helpers

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lift_op2 Q g

:= (locked (fun x y : Q => \pi_Q (g (repr x) (repr y)) : Q)).

Notation

lift_op2

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "repr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lift_fun1 Q f

:= (locked (fun x : Q => f (repr x))).

Notation

lift_fun1

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "repr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lift_fun2 Q g

:= (locked (fun x y : Q => g (repr x) (repr y))).

Notation

lift_fun2

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "repr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lift_op11 Q Q' f

:= (locked (fun x : Q => \pi_Q' (f (repr x)) : Q')).

Notation

lift_op11

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "repr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lift_cst Q x

:= (locked (\pi_Q x : Q)).

Notation

lift_cst

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[]

constant declaration

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PiConst a

:= (@EqualTo _ _ a (lock _)).

Notation

PiConst

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

lift_embed qT e

:= (locked (fun x => \pi_qT (e x) : qT)).

Notation

lift_embed

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "qT" ]

embedding declaration, please don't redefine \pi

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_lock T T' e : e =1 (@locked (T -> T') (fun x : T => e x)).

Proof. by rewrite -lock. Qed.

Lemma

eq_lock

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "T'" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

PiEmbed e

:= (fun x => @EqualTo _ _ (e x) (eq_lock (fun _ => \pi _) _)).

Notation

PiEmbed

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "eq_lock", "pi" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

pi_eq_quot_mono T eq_quot_op eqT

:= PiMono2 (@pi_eq_quot T eq_quot_op eqT).

Canonical

pi_eq_quot_mono

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "PiMono2" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quot_type_of T (qT : quotType T) : Type

:= qT.

Definition

quot_type_of

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "qT" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

quot_type Q

:= (quot_type_of _ Q).

Notation

quot_type

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "quot_type_of" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

Sub x (px : repr (\pi_qT x) == x)

:= \pi_qT x.

Definition

Sub

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "repr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

qreprK x Px : repr (@Sub x Px) = x.

Proof. by rewrite /Sub (eqP Px). Qed.

Lemma

qreprK

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "Px", "Sub", "repr" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sortPx (x : qT) : repr (\pi_qT (repr x)) == repr x.

Proof. by rewrite !reprK eqxx. Qed.

Lemma

sortPx

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "eqxx", "qT", "repr", "reprK" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

sort_Sub (x : qT) : x = Sub (sortPx x).

Proof. by rewrite /Sub reprK. Qed.

Lemma

sort_Sub

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "Sub", "qT", "reprK", "sortPx" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

reprP K (PK : forall x Px, K (@Sub x Px)) u : K u.

Proof. by rewrite (sort_Sub u); apply: PK. Qed.

Lemma

reprP

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "Px", "Sub", "apply", "sort_Sub" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'Sub' Q 'of' T 'by' %/ ]"

:= (SubType.copy Q%type (quot_type_of T Q%type)) (format "[ 'Sub' Q 'of' T 'by' %/ ]") : form_scope.

Notation

[ 'Sub' Q 'of' T 'by' %/ ]

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "copy", "quot_type_of", "type" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'Sub' Q 'by' %/ ]"

:= (SubType.copy Q%type (quot_type Q)) (format "[ 'Sub' Q 'by' %/ ]") : form_scope.

Notation

[ 'Sub' Q 'by' %/ ]

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "copy", "quot_type", "type" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'Equality' 'of' Q 'by' <:%/ ]"

:= (Equality.copy Q%type (quot_type Q)) (format "[ 'Equality' 'of' Q 'by' <:%/ ]") : form_scope.

Notation

[ 'Equality' 'of' Q 'by' <:%/ ]

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "copy", "quot_type", "type" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'Choice' 'of' Q 'by' <:%/ ]"

:= (Choice.copy Q%type (quot_type Q)) (format "[ 'Choice' 'of' Q 'by' <:%/ ]") : form_scope.

Notation

[ 'Choice' 'of' Q 'by' <:%/ ]

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "copy", "quot_type", "type" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'Countable' 'of' Q 'by' <:%/ ]"

:= (Countable.copy Q%type (quot_type Q)) (format "[ 'Countable' 'of' Q 'by' <:%/ ]") : form_scope.

Notation

[ 'Countable' 'of' Q 'by' <:%/ ]

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "copy", "quot_type", "type" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

"[ 'Finite' 'of' Q 'by' <:%/ ]"

:= (Finite.copy Q%type (quot_type Q)) (format "[ 'Finite' 'of' Q 'by' <:%/ ]") : form_scope.

Notation

[ 'Finite' 'of' Q 'by' <:%/ ]

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "copy", "quot_type", "type" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

left_trans (e : rel T) : symmetric e -> transitive e -> left_transitive e.

Proof. by move=> s t ? * ?; apply/idP/idP; apply: t; rewrite // s. Qed.

Lemma

left_trans

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "apply", "rel" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

right_trans (e : rel T) : symmetric e -> transitive e -> right_transitive e.

Proof. by move=> s t ? * x; rewrite ![e x _]s; apply: left_trans. Qed.

Lemma

right_trans

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "apply", "left_trans", "rel" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

equiv_class_of (equiv : rel T)

:= EquivClass of reflexive equiv & symmetric equiv & transitive equiv.

Variant

equiv_class_of

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "equiv", "rel" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

equiv_rel

:= EquivRelPack { equiv :> rel T; _ : equiv_class_of equiv }.

Record

equiv_rel

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "equiv", "equiv_class_of", "rel" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

equiv_class

:= let: EquivRelPack _ ce as e' := e return equiv_class_of e' in ce.

Definition

equiv_class

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "e'", "equiv_class_of" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

equiv_pack (r : rel T) ce & phant_id ce equiv_class

:= @EquivRelPack r ce.

Definition

equiv_pack

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "equiv_class", "rel" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

equiv_refl x : e x x.

Proof. by case: e => [] ? []. Qed.

Lemma

equiv_refl

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

equiv_sym : symmetric e.

Proof. by case: e => [] ? []. Qed.

Lemma

equiv_sym

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

equiv_trans : transitive e.

Proof. by case: e => [] ? []. Qed.

Lemma

equiv_trans

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

eq_op_trans (T' : eqType) : transitive (@eq_op T').

Proof. by move=> x y z /eqP -> /eqP ->. Qed.

Lemma

eq_op_trans

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "T'" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

equiv_ltrans: left_transitive e.

Proof. by apply: left_trans; [apply: equiv_sym|apply: equiv_trans]. Qed.

Lemma

equiv_ltrans

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "apply", "equiv_sym", "equiv_trans", "left_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d

equiv_rtrans: right_transitive e.

Proof. by apply: right_trans; [apply: equiv_sym|apply: equiv_trans]. Qed.

Lemma

equiv_rtrans

boot

boot/generic_quotient.v

[ "HB", "structures", "mathcomp", "ssreflect", "ssrfun", "ssrbool", "eqtype", "ssrnat", "choice", "seq", "fintype", "QuotSubType.Exports", "EquivQuot.Exports" ]

[ "apply", "equiv_sym", "equiv_trans", "right_trans" ]

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

91d97df9cf3204b4dab84f4e24bc633e84b6473d