fact stringlengths 8 1.54k | type stringclasses 19 values | library stringclasses 8 values | imports listlengths 1 10 | filename stringclasses 98 values | symbolic_name stringlengths 1 42 | docstring stringclasses 1 value |
|---|---|---|---|---|---|---|
pi_spec(x : T) : T -> Type :=
PiSpec y of x = y %[mod qT] : pi_spec x y. | Variant | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | pi_spec | |
piP(x : T) : pi_spec x (repr (\pi_qT x)).
Proof. by constructor; rewrite reprK. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | piP | |
mpiE: \mpi =1 \pi_qT.
Proof. by move=> x; rewrite !unlock. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | mpiE | |
quotWP : (forall y : T, P (\pi_qT y)) -> forall x : qT, P x.
Proof. by move=> Py x; rewrite -[x]reprK; apply: Py. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | quotW | |
quotPP : (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 | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | quotP | |
equal_toT (x : T) := EqualTo {
equal_val : T;
_ : x = equal_val
}. | Structure | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | equal_to | |
equal_toE(T : Type) (x : T) (m : equal_to x) : equal_val m = x.
Proof. by case: m. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | equal_toE | |
piE:= (@equal_toE _ _). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | piE | |
equal_to_piT (qT : quotType T) (x : T) :=
@EqualTo _ (\pi_qT x) (\pi x) (erefl _).
Arguments EqualTo {T x equal_val}. | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | equal_to_pi | |
pi_morph1: \pi (f a) = fq (equal_val x). Proof. by rewrite !piE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | pi_morph1 | |
pi_morph2: \pi (g a b) = gq (equal_val x) (equal_val y). Proof. by rewrite !piE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | pi_morph2 | |
pi_mono1: p a = pq (equal_val x). Proof. by rewrite !piE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | pi_mono1 | |
pi_mono2: r a b = rq (equal_val x) (equal_val y). Proof. by rewrite !piE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | pi_mono2 | |
pi_morph11: \pi (h a) = hq (equal_val x). Proof. by rewrite !piE. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | pi_morph11 | |
PiMorphpi_x := (EqualTo pi_x). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | PiMorph | |
PiMorph1pi_f :=
(fun a (x : {pi a}) => EqualTo (pi_morph1 pi_f a x)). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | PiMorph1 | |
PiMorph2pi_g :=
(fun a b (x : {pi a}) (y : {pi b}) => EqualTo (pi_morph2 pi_g a b x y)). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | PiMorph2 | |
PiMono1pi_p :=
(fun a (x : {pi a}) => EqualTo (pi_mono1 pi_p a x)). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | PiMono1 | |
PiMono2pi_r :=
(fun a b (x : {pi a}) (y : {pi b}) => EqualTo (pi_mono2 pi_r a b x y)). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | PiMono2 | |
PiMorph11pi_f :=
(fun a (x : {pi a}) => EqualTo (pi_morph11 pi_f a x)). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | PiMorph11 | |
lift_op1Q f := (locked (fun x : Q => \pi_Q (f (repr x)) : Q)). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | lift_op1 | |
lift_op2Q g :=
(locked (fun x y : Q => \pi_Q (g (repr x) (repr y)) : Q)). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | lift_op2 | |
lift_fun1Q f := (locked (fun x : Q => f (repr x))). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | lift_fun1 | |
lift_fun2Q g := (locked (fun x y : Q => g (repr x) (repr y))). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | lift_fun2 | |
lift_op11Q Q' f := (locked (fun x : Q => \pi_Q' (f (repr x)) : Q')). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | lift_op11 | |
lift_cstQ x := (locked (\pi_Q x : Q)). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | lift_cst | |
PiConsta := (@EqualTo _ _ a (lock _)). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | PiConst | |
lift_embedqT e := (locked (fun x => \pi_qT (e x) : qT)). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | lift_embed | |
eq_lockT T' e : e =1 (@locked (T -> T') (fun x : T => e x)).
Proof. by rewrite -lock. Qed.
Prenex Implicits eq_lock. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | eq_lock | |
PiEmbede :=
(fun x => @EqualTo _ _ (e x) (eq_lock (fun _ => \pi _) _)). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | PiEmbed | |
RecordisEqQuotient T (eq_quot_op : rel T) (Q : Type) of
isQuotient T Q & hasDecEq Q := {
pi_eq_quot : {mono \pi_Q : x y / eq_quot_op x y >-> x == y}
}.
#[short(type="eqQuotType")]
HB.structure Definition EqQuotient T eq_quot_op :=
{Q of isEqQuotient T eq_quot_op Q & Quotient T Q & hasDecEq Q}. | HB.mixin | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | Record | |
pi_eq_quot_monoT eq_quot_op eqT :=
PiMono2 (@pi_eq_quot T eq_quot_op eqT). | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | pi_eq_quot_mono | |
quot_type_ofT (qT : quotType T) : Type := qT.
Arguments quot_type_of T%_type qT%_type : clear implicits. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | quot_type_of | |
quot_typeQ := (quot_type_of _ Q).
HB.instance Definition _ T (qT : quotType T) := Quotient.on (quot_type qT). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | quot_type | |
Subx (px : repr (\pi_qT x) == x) := \pi_qT x. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | Sub | |
qreprKx Px : repr (@Sub x Px) = x.
Proof. by rewrite /Sub (eqP Px). Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | qreprK | |
sortPx(x : qT) : repr (\pi_qT (repr x)) == repr x.
Proof. by rewrite !reprK eqxx. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | sortPx | |
sort_Sub(x : qT) : x = Sub (sortPx x).
Proof. by rewrite /Sub reprK. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | sort_Sub | |
reprPK (PK : forall x Px, K (@Sub x Px)) u : K u.
Proof. by rewrite (sort_Sub u); apply: PK. Qed.
#[export]
HB.instance Definition _ := isSub.Build _ _ (quot_type qT) reprP qreprK.
#[export]
HB.instance Definition _ := [Equality of quot_type qT by <:]. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | reprP | |
Definition_ (T : choiceType) (qT : quotType T) :=
[Choice of quot_type qT by <:].
HB.instance Definition _ (T : countType) (qT : quotType T) :=
[Countable of quot_type qT by <:].
HB.instance Definition _ (T : finType) (qT : quotType T) :=
[Finite of quot_type qT by <:]. | HB.instance | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | Definition | |
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 | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | left_trans | |
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 | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | right_trans | |
equiv_class_of(equiv : rel T) :=
EquivClass of reflexive equiv & symmetric equiv & transitive equiv. | Variant | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | equiv_class_of | |
equiv_rel:= EquivRelPack {
equiv :> rel T;
_ : equiv_class_of equiv
}.
Variable e : equiv_rel. | Record | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | equiv_rel | |
equiv_class:=
let: EquivRelPack _ ce as e' := e return equiv_class_of e' in ce. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | equiv_class | |
equiv_pack(r : rel T) ce of phant_id ce equiv_class :=
@EquivRelPack r ce. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | equiv_pack | |
equiv_reflx : e x x. Proof. by case: e => [] ? []. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | equiv_refl | |
equiv_sym: symmetric e. Proof. by case: e => [] ? []. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | equiv_sym | |
equiv_trans: transitive e. Proof. by case: e => [] ? []. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | equiv_trans | |
eq_op_trans(T' : eqType) : transitive (@eq_op T').
Proof. by move=> x y z /eqP -> /eqP ->. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | eq_op_trans | |
equiv_ltrans: left_transitive e.
Proof. by apply: left_trans; [apply: equiv_sym|apply: equiv_trans]. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | equiv_ltrans | |
equiv_rtrans: right_transitive e.
Proof. by apply: right_trans; [apply: equiv_sym|apply: equiv_trans]. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | equiv_rtrans | |
EquivRelr er es et := (@EquivRelPack _ r (EquivClass er es et)). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | EquivRel | |
encModRel_class_of(r : rel D) :=
EncModRelClassPack of (forall x, r x x -> r (ED (DE x)) x) & (r =2 e). | Variant | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | encModRel_class_of | |
encModRel:= EncModRelPack {
enc_mod_rel :> rel D;
_ : encModRel_class_of enc_mod_rel
}.
Variable r : encModRel. | Record | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | encModRel | |
encModRelClass:=
let: EncModRelPack _ c as r' := r return encModRel_class_of r' in c. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | encModRelClass | |
encModRelP(x : D) : r x x -> r (ED (DE x)) x.
Proof. by case: r => [] ? [] /= he _ /he. Qed. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | encModRelP | |
encModRelE: r =2 e. Proof. by case: r => [] ? []. Qed. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | encModRelE | |
encoded_equiv: rel E := [rel x y | r (ED x) (ED y)]. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | encoded_equiv | |
EncModRelClassm :=
(EncModRelClassPack (fun x _ => m x) (fun _ _ => erefl _)). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | EncModRelClass | |
EncModRelr m := (@EncModRelPack _ _ _ _ _ r (EncModRelClass m)). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | EncModRel | |
enc_mod_rel_is_equiv: equiv_class_of (enc_mod_rel r).
Proof.
split => [x|x y|y x z]; rewrite !encModRelE //; first by rewrite equiv_sym.
by move=> exy /(equiv_trans exy).
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | enc_mod_rel_is_equiv | |
enc_mod_rel_equiv_rel:= EquivRelPack enc_mod_rel_is_equiv. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | enc_mod_rel_equiv_rel | |
encModEquivP(x : D) : r (ED (DE x)) x.
Proof. by rewrite encModRelP ?encModRelE. Qed.
Local Notation e' := (encoded_equiv r). | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | encModEquivP | |
encoded_equivE: e' =2 [rel x y | e (ED x) (ED y)].
Proof. by move=> x y; rewrite /encoded_equiv /= encModRelE. Qed.
Local Notation e'E := encoded_equivE. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | encoded_equivE | |
encoded_equiv_is_equiv: equiv_class_of e'.
Proof.
split => [x|x y|y x z]; rewrite !e'E //=; first by rewrite equiv_sym.
by move=> exy /(equiv_trans exy).
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | encoded_equiv_is_equiv | |
encoded_equiv_equiv_rel:= EquivRelPack encoded_equiv_is_equiv. | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | encoded_equiv_equiv_rel | |
encoded_equivPx : e' (DE (ED x)) x.
Proof. by rewrite /encoded_equiv /= encModEquivP. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | encoded_equivP | |
eC:= (encoded_equiv encD). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | eC | |
canonx := choose (eC x) (x). | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | canon | |
equivQuotient:= EquivQuotient {
erepr : C;
_ : (frel canon) erepr erepr
}. | Record | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | equivQuotient | |
type_ofof (phantom (rel _) encD) := equivQuotient. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | type_of | |
canon_id: forall x, (invariant canon canon) x.
Proof.
move=> x /=; rewrite /canon (@eq_choose _ _ (eC x)).
by rewrite (@choose_id _ (eC x) _ x) ?chooseP ?equiv_refl.
by move=> y; apply: equiv_ltrans; rewrite equiv_sym /= chooseP.
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | canon_id | |
pi:= locked (fun x => EquivQuotient (canon_id x)). | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | pi | |
ereprK: cancel erepr pi.
Proof.
pose T : subType _ := HB.pack equivQuotient [isSub for erepr].
by unlock pi; case=> x hx; apply/(@val_inj _ _ T)/eqP.
Qed.
Local Notation encDE := (encModRelE encD).
Local Notation encDP := (encModEquivP encD). | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | ereprK | |
encD_equiv_rel:= EquivRelPack (enc_mod_rel_is_equiv encD). | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | encD_equiv_rel | |
pi_CD(x y : C) : reflect (pi x = pi y) (eC x y).
Proof.
apply: (iffP idP) => hxy.
apply: (can_inj ereprK); unlock pi canon => /=.
rewrite -(@eq_choose _ (eC x) (eC y)); last first.
by move=> z; rewrite /eC /=; apply: equiv_ltrans.
by apply: choose_id; rewrite ?equiv_refl //.
rewrite (equiv_trans (chooseP (equiv_refl _ _))) //=.
move: hxy => /(f_equal erepr) /=; unlock pi canon => /= ->.
by rewrite equiv_sym /= chooseP.
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | pi_CD | |
pi_DC(x y : D) :
reflect (pi (DC x) = pi (DC y)) (eD x y).
Proof.
apply: (iffP idP)=> hxy.
apply/pi_CD; rewrite /eC /=.
by rewrite (equiv_ltrans (encDP _)) (equiv_rtrans (encDP _)) /= encDE.
rewrite -encDE -(equiv_ltrans (encDP _)) -(equiv_rtrans (encDP _)) /=.
exact/pi_CD.
Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | pi_DC | |
equivQTP: cancel (CD \o erepr) (pi \o DC).
Proof. by move=> x; rewrite /= (pi_CD _ (erepr x) _) ?ereprK /eC /= ?encDP. Qed.
Local Notation qT := (type_of (Phantom (rel D) encD)).
#[export]
HB.instance Definition _ := isQuotient.Build D qT equivQTP. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | equivQTP | |
eqmodPx y : reflect (x = y %[mod qT]) (eD x y).
Proof. by apply: (iffP (pi_DC _ _)); rewrite !unlock. Qed.
#[export]
HB.instance Definition _ := Choice.copy qT (can_type ereprK). | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | eqmodP | |
eqmodEx y : x == y %[mod qT] = eD x y.
Proof. exact: sameP eqP (@eqmodP _ _). Qed.
#[export]
HB.instance Definition _ := isEqQuotient.Build _ eD qT eqmodE. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | eqmodE | |
defaultEncModRelClass:=
@EncModRelClassPack D D id id r r (fun _ rxx => rxx) (fun _ _ => erefl _). | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | defaultEncModRelClass | |
defaultEncModRel:= EncModRelPack defaultEncModRelClass. | Canonical | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | defaultEncModRel | |
eC:= (encoded_equiv encD).
HB.instance Definition _ :=
Countable.copy {eq_quot encD} (can_type EquivQuot.ereprK). | Notation | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | eC | |
eqmodEx y : x == y %[mod_eq e] = e x y.
Proof. by rewrite pi_eq_quot. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | eqmodE | |
eqmodPx y : reflect (x = y %[mod_eq e]) (e x y).
Proof. by rewrite -eqmodE; apply/eqP. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | eqmodP | |
eqquotEx y : x == y %[mod Q] = e x y.
Proof. by rewrite pi_eq_quot. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | eqquotE | |
eqquotPx y : reflect (x = y %[mod Q]) (e x y).
Proof. by rewrite -eqquotE; apply/eqP. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype ssrnat choice",
"From mathcomp Require Import seq fintype"
] | boot/generic_quotient.v | eqquotP | |
RecordhasMul G := {
mul : G -> G -> G
}.
#[short(type="magmaType")]
HB.structure Definition Magma := {G of hasMul G}.
Bind Scope group_scope with Magma.sort.
HB.structure Definition ChoiceMagma := {G of Magma G & Choice G}.
Bind Scope group_scope with ChoiceMagma.sort.
Local Notation "*%g" := (@mul _) : function_scope.
Local Notation "x * y" := (mul x y) : group_scope. | HB.mixin | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | Record | |
commutex y := x * y = y * x. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | commute | |
commute_reflx : commute x x.
Proof. by []. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | commute_refl | |
commute_symx y : commute x y -> commute y x.
Proof. by []. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | commute_sym | |
mulg_closed:= {in S &, forall u v, u * v \in S}. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | mulg_closed | |
RecordMagma_isSemigroup G of Magma G := {
mulgA : associative (@mul G)
}.
#[short(type="semigroupType")]
HB.structure Definition Semigroup := {G of Magma_isSemigroup G & ChoiceMagma G}.
HB.factory Record isSemigroup G of Choice G := {
mul : G -> G -> G;
mulgA : associative mul
}.
HB.builders Context G of isSemigroup G.
HB.instance Definition _ := hasMul.Build G mul.
HB.instance Definition _ := Magma_isSemigroup.Build G mulgA.
HB.end.
Bind Scope group_scope with Semigroup.sort. | HB.mixin | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | Record | |
commuteMx y z : commute x y -> commute x z -> commute x (y * z).
Proof. by move=> cxy cxz; rewrite /commute -mulgA -cxz !mulgA cxy. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | commuteM | |
RecordhasOne G := {
one : G
}.
#[short(type="baseUMagmaType")]
HB.structure Definition BaseUMagma := {G of hasOne G & Magma G}.
Bind Scope group_scope with BaseUMagma.sort.
HB.structure Definition ChoiceBaseUMagma := {G of BaseUMagma G & Choice G}.
Bind Scope group_scope with ChoiceBaseUMagma.sort.
Local Notation "1" := (@one _) : group_scope.
Local Notation "s `_ i" := (nth 1 s i) : group_scope.
Local Notation "\prod_ ( i <- r | P ) F" := (\big[*%g/1]_(i <- r | P) F).
Local Notation "\prod_ ( i | P ) F" := (\big[*%g/1]_(i | P) F).
Local Notation "\prod_ ( i 'in' A ) F" := (\big[*%g/1]_(i in A) F).
Local Notation "\prod_ ( m <= i < n ) F" := (\big[*%g/1%g]_(m <= i < n) F%g). | HB.mixin | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | Record | |
natexp(G : baseUMagmaType) (x : G) n : G := iterop n *%g x 1.
Arguments natexp : simpl never.
Local Notation "x ^+ n" := (natexp x n) : group_scope. | Definition | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | natexp | |
expgnEx n : x ^+ n = iterop n mul x 1. Proof. by []. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | expgnE | |
expg0x : x ^+ 0 = 1. Proof. by []. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | expg0 | |
expg1x : x ^+ 1 = x. Proof. by []. Qed. | Lemma | boot | [
"From HB Require Import structures",
"From mathcomp Require Import ssreflect ssrfun ssrbool eqtype choice ssrnat seq",
"From mathcomp Require Import bigop fintype finfun"
] | boot/monoid.v | expg1 |
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