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Definition abmonoidtomonoid : abmonoid β monoid := Ξ» X, make_monoid (pr1 X) (pr1 (pr2 X)). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoidtomonoid | 100 |
Definition commax (X : abmonoid) : iscomm (@op X) := pr2 (pr2 X). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | commax | 101 |
Definition abmonoidrer (X : abmonoid) (a b c d : X) : (a + b) + (c + d) = (a + c) + (b + d) := abmonoidoprer (pr2 X) a b c d. | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoidrer | 102 |
Definition abmonoid_of_monoid (X : monoid) (H : iscomm (@op X)) : abmonoid := make_abmonoid X (make_isabmonoidop (pr2 X) H). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoid_of_monoid | 103 |
Definition unitabmonoid_isabmonoid : isabmonoidop (@op unitmonoid). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | unitabmonoid_isabmonoid | 104 |
Definition unitabmonoid : abmonoid := make_abmonoid unitmonoid unitabmonoid_isabmonoid. | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | unitabmonoid | 105 |
Lemma abmonoidfuntounit_ismonoidfun (X : abmonoid) : ismonoidfun (Y := unitabmonoid) (Ξ» x : X, 0). Proof. use make_ismonoidfun. - use make_isbinopfun. intros x x'. use isProofIrrelevantUnit. - use isProofIrrelevantUnit. Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoidfuntounit_ismonoidfun | 106 |
Definition abmonoidfuntounit (X : abmonoid) : monoidfun X unitabmonoid := monoidfunconstr (abmonoidfuntounit_ismonoidfun X). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoidfuntounit | 107 |
Lemma abmonoidfunfromunit_ismonoidfun (X : abmonoid) : ismonoidfun (Y := X) (Ξ» x : unitabmonoid, 0). Proof. use make_ismonoidfun. - use make_isbinopfun. intros x x'. exact (!runax X _). - use idpath. Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoidfunfromunit_ismonoidfun | 108 |
Definition abmonoidfunfromunit (X : abmonoid) : monoidfun unitabmonoid X := monoidfunconstr (abmonoidfunfromunit_ismonoidfun X). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoidfunfromunit | 109 |
Lemma unelabmonoidfun_ismonoidfun (X Y : abmonoid) : ismonoidfun (Y := Y) (Ξ» x : X, 0). Proof. use make_ismonoidfun. - use make_isbinopfun. intros x x'. exact (!lunax _ _). - use idpath. Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | unelabmonoidfun_ismonoidfun | 110 |
Definition unelabmonoidfun (X Y : abmonoid) : monoidfun X Y := monoidfunconstr (unelabmonoidfun_ismonoidfun X Y). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | unelabmonoidfun | 111 |
Lemma abmonoidshombinop_ismonoidfun {X Y : abmonoid} (f g : monoidfun X Y) : @ismonoidfun X Y (Ξ» x : pr1 X, pr1 f x + pr1 g x). Proof. use make_ismonoidfun. - use make_isbinopfun. intros x x'. cbn. rewrite (pr1 (pr2 f)). rewrite (pr1 (pr2 g)). rewrite (assocax Y). rewrite (assocax Y). use maponpaths. rewrite <- (assocax Y). rewrite <- (assocax Y). use (maponpaths (Ξ» y : Y, y + pr1 g x')). use (commax Y). - refine (maponpaths (Ξ» h : Y, pr1 f 0 + h) (monoidfununel g) @ _). rewrite runax. exact (monoidfununel f). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoidshombinop_ismonoidfun | 112 |
Definition abmonoidshombinop {X Y : abmonoid} : binop (monoidfun X Y) := (Ξ» f g, monoidfunconstr (abmonoidshombinop_ismonoidfun f g)). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoidshombinop | 113 |
Lemma abmonoidsbinop_runax {X Y : abmonoid} (f : monoidfun X Y) : abmonoidshombinop f (unelmonoidfun X Y) = f. Proof. use monoidfun_paths. use funextfun. intros x. use (runax Y). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoidsbinop_runax | 114 |
Lemma abmonoidsbinop_lunax {X Y : abmonoid} (f : monoidfun X Y) : abmonoidshombinop (unelmonoidfun X Y) f = f. Proof. use monoidfun_paths. use funextfun. intros x. use (lunax Y). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoidsbinop_lunax | 115 |
Lemma abmonoidshombinop_assoc {X Y : abmonoid} (f g h : monoidfun X Y) : abmonoidshombinop (abmonoidshombinop f g) h = abmonoidshombinop f (abmonoidshombinop g h). Proof. use monoidfun_paths. use funextfun. intros x. use assocax. Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoidshombinop_assoc | 116 |
Lemma abmonoidshombinop_comm {X Y : abmonoid} (f g : monoidfun X Y) : abmonoidshombinop f g = abmonoidshombinop g f. Proof. use monoidfun_paths. use funextfun. intros x. use (commax Y). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoidshombinop_comm | 117 |
Lemma abmonoidshomabmonoid_ismonoidop (X Y : abmonoid) : @ismonoidop (make_hSet (monoidfun X Y) (isasetmonoidfun X Y)) (Ξ» f g : monoidfun X Y, abmonoidshombinop f g). Proof. use make_ismonoidop. - intros f g h. exact (abmonoidshombinop_assoc f g h). - use make_isunital. + exact (unelmonoidfun X Y). + use make_isunit. * intros f. exact (abmonoidsbinop_lunax f). * intros f. exact (abmonoidsbinop_runax f). Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoidshomabmonoid_ismonoidop | 118 |
Lemma abmonoidshomabmonoid_isabmonoid (X Y : abmonoid) : @isabmonoidop (make_hSet (monoidfun X Y) (isasetmonoidfun X Y)) (Ξ» f g : monoidfun X Y, abmonoidshombinop f g). Proof. use make_isabmonoidop. - exact (abmonoidshomabmonoid_ismonoidop X Y). - intros f g. exact (abmonoidshombinop_comm f g). Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoidshomabmonoid_isabmonoid | 119 |
Definition abmonoidshomabmonoid (X Y : abmonoid) : abmonoid. Proof. use make_abmonoid. - use make_setwithbinop. + use make_hSet. * exact (monoidfun X Y). * exact (isasetmonoidfun X Y). + intros f g. exact (abmonoidshombinop f g). - exact (abmonoidshomabmonoid_isabmonoid X Y). Defined. | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoidshomabmonoid | 120 |
Definition abmonoid_univalence_weq1 : abmonoid β abmonoid' := weqtotal2asstol (Ξ» X : setwithbinop, ismonoidop (@op X)) (Ξ» y : (β X : setwithbinop, ismonoidop op), iscomm (@op (pr1 y))). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoid_univalence_weq1 | 121 |
Definition abmonoid_univalence_weq1' (X Y : abmonoid) : (X = Y) β ((make_abmonoid' X) = (make_abmonoid' Y)) := make_weq _ (@isweqmaponpaths abmonoid abmonoid' abmonoid_univalence_weq1 X Y). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoid_univalence_weq1' | 122 |
Definition abmonoid_univalence_weq2 (X Y : abmonoid) : ((make_abmonoid' X) = (make_abmonoid' Y)) β ((pr1 (make_abmonoid' X)) = (pr1 (make_abmonoid' Y))). Proof. use subtypeInjectivity. intros w. use isapropiscomm. Defined. | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoid_univalence_weq2 | 123 |
Definition abmonoid_univalence_weq3 (X Y : abmonoid) : ((pr1 (make_abmonoid' X)) = (pr1 (make_abmonoid' Y))) β (monoidiso X Y) := monoid_univalence (pr1 (make_abmonoid' X)) (pr1 (make_abmonoid' Y)). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoid_univalence_weq3 | 124 |
Definition abmonoid_univalence_map (X Y : abmonoid) : (X = Y) β (monoidiso X Y). Proof. intro e. induction e. exact (idmonoidiso X). Defined. | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoid_univalence_map | 125 |
Lemma abmonoid_univalence_isweq (X Y : abmonoid) : isweq (abmonoid_univalence_map X Y). Proof. use isweqhomot. - exact (weqcomp (abmonoid_univalence_weq1' X Y) (weqcomp (abmonoid_univalence_weq2 X Y) (abmonoid_univalence_weq3 X Y))). - intros e. induction e. refine (weqcomp_to_funcomp_app @ _). use weqcomp_to_funcomp_app. - use weqproperty. Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoid_univalence_isweq | 126 |
Definition abmonoid_univalence (X Y : abmonoid) : (X = Y) β (monoidiso X Y) := make_weq (abmonoid_univalence_map X Y) (abmonoid_univalence_isweq X Y). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoid_univalence | 127 |
Definition subabmonoid (X : abmonoid) := submonoid X. | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | subabmonoid | 128 |
Lemma iscommcarrier {X : abmonoid} (A : submonoid X) : iscomm (@op A). Proof. intros a a'. apply (invmaponpathsincl _ (isinclpr1carrier A)). simpl. apply (pr2 (pr2 X)). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | iscommcarrier | 129 |
Definition isabmonoidcarrier {X : abmonoid} (A : submonoid X) : isabmonoidop (@op A) := ismonoidcarrier A ,, iscommcarrier A. | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | isabmonoidcarrier | 130 |
Definition carrierofsubabmonoid {X : abmonoid} (A : subabmonoid X) : abmonoid. Proof. unfold subabmonoid in A. exists A. apply isabmonoidcarrier. Defined. | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | carrierofsubabmonoid | 131 |
Definition subabmonoid_incl {X : abmonoid} (A : subabmonoid X) : monoidfun A X := submonoid_incl A. | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | subabmonoid_incl | 132 |
Lemma iscommquot {X : abmonoid} (R : binopeqrel X) : iscomm (@op (setwithbinopquot R)). Proof. intros. set (X0 := setwithbinopquot R). intros x x'. apply (setquotuniv2prop R (Ξ» x x' : X0, make_hProp _ (setproperty X0 (x + x') (x' + x)))). intros x0 x0'. apply (maponpaths (setquotpr R) ((commax X) x0 x0')). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | iscommquot | 133 |
Definition isabmonoidquot {X : abmonoid} (R : binopeqrel X) : isabmonoidop (@op (setwithbinopquot R)) := ismonoidquot R ,, iscommquot R. | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | isabmonoidquot | 134 |
Definition abmonoidquot {X : abmonoid} (R : binopeqrel X) : abmonoid. Proof. exists (setwithbinopquot R). apply isabmonoidquot. Defined. | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoidquot | 135 |
Lemma iscommdirprod (X Y : abmonoid) : iscomm (@op (setwithbinopdirprod X Y)). Proof. intros xy xy'. induction xy as [ x y ]. induction xy' as [ x' y' ]. simpl. apply pathsdirprod. - apply (commax X). - apply (commax Y). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | iscommdirprod | 136 |
Definition isabmonoiddirprod (X Y : abmonoid) : isabmonoidop (@op (setwithbinopdirprod X Y)) := ismonoiddirprod X Y ,, iscommdirprod X Y. | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | isabmonoiddirprod | 137 |
Definition abmonoiddirprod (X Y : abmonoid) : abmonoid. Proof. exists (setwithbinopdirprod X Y). apply isabmonoiddirprod. Defined. | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoiddirprod | 138 |
Definition abmonoidfracopint (X : abmonoid) (A : submonoid X) : binop (X Γ A) := @op (setwithbinopdirprod X A). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoidfracopint | 139 |
Definition hrelabmonoidfrac (X : abmonoid) (A : submonoid X) : hrel (setwithbinopdirprod X A) := Ξ» xa yb, β (a0 : A), (pr1 xa + pr1 (pr2 yb)) + pr1 a0 = (pr1 yb + pr1 (pr2 xa) + pr1 a0). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | hrelabmonoidfrac | 140 |
Lemma iseqrelabmonoidfrac (X : abmonoid) (A : submonoid X) : iseqrel (hrelabmonoidfrac X A). Proof. intros. set (assoc := assocax X). set (comm := commax X). set (R := hrelabmonoidfrac X A). apply iseqrelconstr. - unfold istrans. intros ab cd ef. simpl. apply hinhfun2. induction ab as [ a b ]. induction cd as [ c d ]. induction ef as [ e f ]. induction b as [ b isb ]. induction d as [ d isd ]. induction f as [ f isf ]. intros eq1 eq2. induction eq1 as [ x1 eq1 ]. induction eq2 as [ x2 eq2 ]. simpl in *. exists (@op A (d ,, isd) (@op A x1 x2)). induction x1 as [ x1 isx1 ]. induction x2 as [ x2 isx2 ]. induction A as [ A ax ]. simpl in *. rewrite (assoc a f (d + (x1 + x2))). rewrite (comm f (d + (x1 + x2))). induction (assoc a (d + (x1 + x2)) f). induction (assoc a d (x1 + x2)). induction (assoc (a + d) x1 x2). rewrite eq1. rewrite (comm x1 x2). rewrite (assoc e b (d + (x2 + x1))). rewrite (comm b (d + (x2 + x1))). induction (assoc e (d + (x2 + x1)) b). induction (assoc e d (x2 + x1)). induction (assoc (e + d) x2 x1). induction eq2. rewrite (assoc (c + b) x1 x2). rewrite (assoc (c + f) x2 x1). rewrite (comm x1 x2). rewrite (assoc (c + b) (x2 + x1) f). rewrite (assoc (c + f) (x2 + x1) b). rewrite (comm (x2 + x1) f). rewrite (comm (x2 + x1) b). induction (assoc (c + b) f (x2 + x1)). induction (assoc (c + f) b (x2 + x1)). rewrite (assoc c b f). rewrite (assoc c f b). rewrite (comm b f). apply idpath. - intro xa. simpl. apply hinhpr. exists (pr2 xa). apply idpath. - intros xa yb. unfold R. simpl. apply hinhfun. intro eq1. induction eq1 as [ x1 eq1 ]. exists x1. induction x1 as [ x1 isx1 ]. simpl. apply (!eq1). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | iseqrelabmonoidfrac | 141 |
Definition eqrelabmonoidfrac (X : abmonoid) (A : submonoid X) : eqrel (setwithbinopdirprod X A) := make_eqrel (hrelabmonoidfrac X A) (iseqrelabmonoidfrac X A). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | eqrelabmonoidfrac | 142 |
Lemma isbinophrelabmonoidfrac (X : abmonoid) (A : submonoid X) : @isbinophrel (setwithbinopdirprod X A) (eqrelabmonoidfrac X A). Proof. intros. apply (isbinopreflrel (eqrelabmonoidfrac X A) (eqrelrefl (eqrelabmonoidfrac X A))). set (rer := abmonoidoprer (pr2 X)). intros a b c d. simpl. apply hinhfun2. induction a as [ a a' ]. induction a' as [ a' isa' ]. induction b as [ b b' ]. induction b' as [ b' isb' ]. induction c as [ c c' ]. induction c' as [ c' isc' ]. induction d as [ d d' ]. induction d' as [ d' isd' ]. intros ax ay. induction ax as [ a1 eq1 ]. induction ay as [ a2 eq2 ]. exists (@op A a1 a2). induction a1 as [ a1 aa1 ]. induction a2 as [ a2 aa2 ]. simpl in *. rewrite (rer a c b' d'). rewrite (rer b d a' c'). rewrite (rer (a + b') (c + d') a1 a2). rewrite (rer (b + a') (d + c') a1 a2). induction eq1. induction eq2. apply idpath. Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | isbinophrelabmonoidfrac | 143 |
Definition abmonoidfracop (X : abmonoid) (A : submonoid X) : binop (setquot (hrelabmonoidfrac X A)) := setquotfun2 (hrelabmonoidfrac X A) (eqrelabmonoidfrac X A) (abmonoidfracopint X A) ((iscompbinoptransrel _ (eqreltrans _) (isbinophrelabmonoidfrac X A))). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoidfracop | 144 |
Definition binopeqrelabmonoidfrac (X : abmonoid) (A : subabmonoid X) : binopeqrel (abmonoiddirprod X A) := @make_binopeqrel (setwithbinopdirprod X A) (eqrelabmonoidfrac X A) (isbinophrelabmonoidfrac X A). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | binopeqrelabmonoidfrac | 145 |
Definition abmonoidfrac (X : abmonoid) (A : submonoid X) : abmonoid := abmonoidquot (binopeqrelabmonoidfrac X A). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoidfrac | 146 |
Definition prabmonoidfrac (X : abmonoid) (A : submonoid X) : X β A β abmonoidfrac X A := Ξ» (x : X) (a : A), setquotpr (eqrelabmonoidfrac X A) (x ,, a). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | prabmonoidfrac | 147 |
Lemma invertibilityinabmonoidfrac (X : abmonoid) (A : submonoid X) : β a a' : A, isinvertible (@op (abmonoidfrac X A)) (prabmonoidfrac X A (pr1 a) a'). Proof. intros a a'. set (R := eqrelabmonoidfrac X A). unfold isinvertible. assert (isl : islinvertible (@op (abmonoidfrac X A)) (prabmonoidfrac X A (pr1 a) a')). { unfold islinvertible. set (f := Ξ» x0 : abmonoidfrac X A, prabmonoidfrac X A (pr1 a) a' + x0). set (g := Ξ» x0 : abmonoidfrac X A, prabmonoidfrac X A (pr1 a') a + x0). assert (egf : β x0, g (f x0) = x0). { apply (setquotunivprop R (Ξ» x, _ = _)%logic). intro xb. simpl. apply (iscompsetquotpr R (pr1 a' + (pr1 a + pr1 xb) ,, (@op A) a ((@op A) a' (pr2 xb)))). simpl. apply hinhpr. exists (unel A). unfold pr1carrier. simpl. set (e := assocax X (pr1 a) (pr1 a') (pr1 (pr2 xb))). simpl in e. induction e. set (e := assocax X (pr1 xb) (pr1 a + pr1 a') (pr1 (pr2 xb))). simpl in e. induction e. set (e := assocax X (pr1 a') (pr1 a) (pr1 xb)). simpl in e. induction e. set (e := commax X (pr1 a) (pr1 a')). simpl in e. induction e. set (e := commax X (pr1 a + pr1 a') (pr1 xb)). simpl in e. induction e. apply idpath. } assert (efg : β x0, f (g x0) = x0). { apply (setquotunivprop R (Ξ» x0, _ = _)%logic). intro xb. simpl. apply (iscompsetquotpr R (pr1 a + (pr1 a' + pr1 xb) ,, (@op A) a' ((@op A) a (pr2 xb)))). simpl. apply hinhpr. exists (unel A). unfold pr1carrier. simpl. set (e := assocax X (pr1 a') (pr1 a) (pr1 (pr2 xb))). simpl in e. induction e. set (e := assocax X (pr1 xb) (pr1 a' + pr1 a) (pr1 (pr2 xb))). simpl in e. induction e. set (e := assocax X (pr1 a) (pr1 a') (pr1 xb)). simpl in e. induction e. set (e := commax X (pr1 a') (pr1 a)). simpl in e. induction e. set (e := commax X (pr1 a' + pr1 a) (pr1 xb)). simpl in e. induction e. apply idpath. } apply (isweq_iso _ _ egf efg). } exact ( isl ,, weqlinvertiblerinvertible op (commax (abmonoidfrac X A)) (prabmonoidfrac X A (pr1 a) a') isl ). Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | invertibilityinabmonoidfrac | 148 |
Definition toabmonoidfrac (X : abmonoid) (A : submonoid X) (x : X) : abmonoidfrac X A := setquotpr _ (x ,, unel A). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | toabmonoidfrac | 149 |
Lemma isbinopfuntoabmonoidfrac (X : abmonoid) (A : submonoid X) : isbinopfun (toabmonoidfrac X A). Proof. unfold isbinopfun. intros x1 x2. change (setquotpr _ (x1 + x2 ,, @unel A) = setquotpr (eqrelabmonoidfrac X A) (x1 + x2 ,, unel A + unel A)). apply (maponpaths (setquotpr _)). apply (@pathsdirprod X A). apply idpath. exact (!lunax A 0). Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | isbinopfuntoabmonoidfrac | 150 |
Lemma isunitalfuntoabmonoidfrac (X : abmonoid) (A : submonoid X) : toabmonoidfrac X A 0 = 0. Proof. apply idpath. Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | isunitalfuntoabmonoidfrac | 151 |
Definition ismonoidfuntoabmonoidfrac (X : abmonoid) (A : submonoid X) : ismonoidfun (toabmonoidfrac X A) := isbinopfuntoabmonoidfrac X A ,, isunitalfuntoabmonoidfrac X A. | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | ismonoidfuntoabmonoidfrac | 152 |
Definition hrel0abmonoidfrac (X : abmonoid) (A : submonoid X) : hrel (X Γ A) := Ξ» xa yb : setdirprod X A, (pr1 xa + pr1 (pr2 yb) = pr1 yb + pr1 (pr2 xa))%logic. | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | hrel0abmonoidfrac | 153 |
Lemma weqhrelhrel0abmonoidfrac (X : abmonoid) (A : submonoid X) (iscanc : β a : A, isrcancelable (@op X) (pr1carrier _ a)) (xa xa' : X Γ A) : (eqrelabmonoidfrac X A xa xa') β (hrel0abmonoidfrac X A xa xa'). Proof. unfold eqrelabmonoidfrac. unfold hrelabmonoidfrac. simpl. apply weqimplimpl. apply (@hinhuniv _ (pr1 xa + pr1 (pr2 xa') = pr1 xa' + pr1 (pr2 xa))%logic). intro ae. induction ae as [ a eq ]. apply (invmaponpathsincl _ (iscanc a) _ _ eq). intro eq. apply hinhpr. exists (unel A). rewrite (runax X). rewrite (runax X). apply eq. apply (isapropishinh _). apply (setproperty X). Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | weqhrelhrel0abmonoidfrac | 154 |
Lemma isinclprabmonoidfrac (X : abmonoid) (A : submonoid X) (iscanc : β a : A, isrcancelable (@op X) (pr1carrier _ a)) : β a' : A, isincl (Ξ» x, prabmonoidfrac X A x a'). Proof. intro a'. apply isinclbetweensets. - apply (setproperty X). - apply (setproperty (abmonoidfrac X A)). - intros x x'. intro e. set (e' := invweq (weqpathsinsetquot (eqrelabmonoidfrac X A) (x ,, a') (x' ,, a')) e). set (e'' := weqhrelhrel0abmonoidfrac X A iscanc (_ ,, _) (_ ,, _) e'). simpl in e''. apply (invmaponpathsincl _ (iscanc a')). apply e''. Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | isinclprabmonoidfrac | 155 |
Definition isincltoabmonoidfrac (X : abmonoid) (A : submonoid X) (iscanc : β a : A, isrcancelable (@op X) (pr1carrier _ a)) : isincl (toabmonoidfrac X A) := isinclprabmonoidfrac X A iscanc (unel A). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | isincltoabmonoidfrac | 156 |
Lemma isdeceqabmonoidfrac (X : abmonoid) (A : submonoid X) (iscanc : β a : A, isrcancelable (@op X) (pr1carrier _ a)) (is : isdeceq X) : isdeceq (abmonoidfrac X A). Proof. apply (isdeceqsetquot (eqrelabmonoidfrac X A)). intros xa xa'. apply (isdecpropweqb (weqhrelhrel0abmonoidfrac X A iscanc xa xa')). apply isdecpropif. unfold isaprop. simpl. set (int := setproperty X (pr1 xa + pr1 (pr2 xa')) (pr1 xa' + pr1 (pr2 xa))). simpl in int. apply int. unfold hrel0abmonoidfrac. simpl. apply (is _ _). Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | isdeceqabmonoidfrac | 157 |
Definition abmonoidfracrelint (X : abmonoid) (A : subabmonoid X) (L : hrel X) : hrel (setwithbinopdirprod X A) := Ξ» xa yb, β (c0 : A), L (((pr1 xa) + (pr1 (pr2 yb))) + (pr1 c0)) (((pr1 yb) + (pr1 (pr2 xa))) + (pr1 c0)). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoidfracrelint | 158 |
Lemma iscomprelabmonoidfracrelint (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) : iscomprelrel (eqrelabmonoidfrac X A) (abmonoidfracrelint X A L). Proof. set (assoc := (assocax X) : isassoc (@op X)). unfold isassoc in assoc. set (comm := commax X). unfold iscomm in comm. set (rer := abmonoidrer X). apply iscomprelrelif. apply (eqrelsymm (eqrelabmonoidfrac X A)). - intros xa xa' yb. unfold hrelabmonoidfrac. simpl. apply (@hinhfun2). intros t2e t2l. induction t2e as [ c1a e ]. induction t2l as [ c0a l ]. set (x := pr1 xa). set (a := pr1 (pr2 xa)). set (x' := pr1 xa'). set (a' := pr1 (pr2 xa')). set (y := pr1 yb). set (b := pr1 (pr2 yb)). set (c0 := pr1 c0a). set (c1 := pr1 c1a). exists ((pr2 xa) + c1a + c0a). change (L ((x' + b) + ((a + c1) + c0)) ((y + a') + ((a + c1) + c0))). change (x + a' + c1 = x' + a + c1) in e. rewrite (rer x' _ _ c0). induction (assoc x' a c1). induction e. rewrite (assoc x a' c1). rewrite (rer x _ _ c0). rewrite (assoc a c1 c0). rewrite (rer _ a' a _). rewrite (assoc a' c1 c0). rewrite (comm a' _). rewrite (comm c1 _). rewrite (assoc c0 c1 a'). induction (assoc (x + b) c0 (@op X c1 a')). induction (assoc (y + a) c0 (@op X c1 a')). apply ((pr2 is) _ _ _ (pr2 (@op A c1a (pr2 xa'))) l). - intros xa yb yb'. unfold hrelabmonoidfrac. simpl. apply (@hinhfun2). intros t2e t2l. induction t2e as [ c1a e ]. induction t2l as [ c0a l ]. set (x := pr1 xa). set (a := pr1 (pr2 xa)). set (y' := pr1 yb'). set (b' := pr1 (pr2 yb')). set (y := pr1 yb). set (b := pr1 (pr2 yb)). set (c0 := pr1 c0a). set (c1 := pr1 c1a). exists ((pr2 yb) + c1a + c0a). change (L ((x + b') + ((b + c1) + c0)) ((y' + a) + ((b + c1) + c0))). change (y + b' + c1 = y' + b + c1) in e. rewrite (rer y' _ _ c0). induction (assoc y' b c1). induction e. rewrite (assoc y b' c1). rewrite (rer y _ _ c0). rewrite (assoc b c1 c0). rewrite (rer _ b' b _). rewrite (assoc b' c1 c0). rewrite (comm b' _). rewrite (comm c1 _). rewrite (assoc c0 c1 b'). induction (assoc (x + b) c0 (@op X c1 b')). induction (assoc (y + a) c0 (@op X c1 b')). apply ((pr2 is) _ _ _ (pr2 (@op A c1a (pr2 yb'))) l). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | iscomprelabmonoidfracrelint | 159 |
Definition abmonoidfracrel (X : abmonoid) (A : submonoid X) {L : hrel X} (is : ispartbinophrel A L) := quotrel (iscomprelabmonoidfracrelint X A is). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoidfracrel | 160 |
Lemma istransabmonoidfracrelint (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (isl : istrans L) : istrans (abmonoidfracrelint X A L). Proof. intros. set (assoc := (assocax X) : isassoc (@op X)). unfold isassoc in assoc. set (comm := commax X). unfold iscomm in comm. set (rer := abmonoidrer X). intros xa1 xa2 xa3. unfold abmonoidfracrelint. simpl. apply hinhfun2. intros t2l1 t2l2. set (c1a := pr1 t2l1). set (l1 := pr2 t2l1). set (c2a := pr1 t2l2). set (l2 := pr2 t2l2). set (x1 := pr1 xa1). set (a1 := pr1 (pr2 xa1)). set (x2 := pr1 xa2). set (a2 := pr1 (pr2 xa2)). set (x3 := pr1 xa3). set (a3 := pr1 (pr2 xa3)). set (c1 := pr1 c1a). set (c2 := pr1 c2a). exists ((pr2 xa2) + (@op A c1a c2a)). change (L ((x1 + a3) + (a2 + (c1 + c2))) ((x3 + a1) + (a2 + (c1 + c2)))). assert (ll1 : L ((x1 + a3) + (a2 + (@op X c1 c2))) (((x2 + a1) + c1) + (c2 + a3))). { rewrite (rer _ a3 a2 _). rewrite (comm a3 (@op X c1 c2)). rewrite (assoc c1 c2 a3). induction (assoc (x1 + a2) c1 (@op X c2 a3)). apply ((pr2 is) _ _ _ (pr2 (@op A c2a (pr2 xa3))) l1). } assert (ll2 : L (((x2 + a3) + c2) + (@op X a1 c1)) ((x3 + a1) + (a2 + (@op X c1 c2)))). { rewrite (rer _ a1 a2 _). induction (assoc a1 c1 c2). rewrite (comm (a1 + c1) c2). induction (assoc (x3 + a2) c2 (@op X a1 c1)). apply ((pr2 is) _ _ _ (pr2 (@op A (pr2 xa1) c1a)) l2). } assert (e : x2 + a1 + c1 + (c2 + a3) = x2 + a3 + c2 + (a1 + c1)). { rewrite (assoc (x2 + a1) c1 _). rewrite (assoc (x2 + a3) c2 _). rewrite (assoc x2 a1 _). rewrite (assoc x2 a3 _). induction (assoc a1 c1 (c2 + a3)). induction (assoc a3 c2 (a1 + c1)). induction (comm (c2 + a3) (a1 + c1)). rewrite (comm a3 c2). apply idpath. } induction e. apply (isl _ _ _ ll1 ll2). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | istransabmonoidfracrelint | 161 |
Lemma istransabmonoidfracrel (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (isl : istrans L) : istrans (abmonoidfracrel X A is). Proof. apply istransquotrel. apply istransabmonoidfracrelint. - apply is. - apply isl. Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | istransabmonoidfracrel | 162 |
Lemma issymmabmonoidfracrelint (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (isl : issymm L) : issymm (abmonoidfracrelint X A L). Proof. intros xa1 xa2. unfold abmonoidfracrelint. simpl. apply hinhfun. intros t2l1. set (c1a := pr1 t2l1). set (l1 := pr2 t2l1). exists (c1a). apply (isl _ _ l1). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | issymmabmonoidfracrelint | 163 |
Lemma issymmabmonoidfracrel (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (isl : issymm L) : issymm (abmonoidfracrel X A is). Proof. apply issymmquotrel. apply issymmabmonoidfracrelint. - apply is. - apply isl. Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | issymmabmonoidfracrel | 164 |
Lemma isreflabmonoidfracrelint (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (isl : isrefl L) : isrefl (abmonoidfracrelint X A L). Proof. intro xa. unfold abmonoidfracrelint. simpl. apply hinhpr. exists (unel A). apply (isl _). Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | isreflabmonoidfracrelint | 165 |
Lemma isreflabmonoidfracrel (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (isl : isrefl L) : isrefl (abmonoidfracrel X A is). Proof. apply isreflquotrel. apply isreflabmonoidfracrelint. - apply is. - apply isl. Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | isreflabmonoidfracrel | 166 |
Lemma ispoabmonoidfracrelint (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (isl : ispreorder L) : ispreorder (abmonoidfracrelint X A L). Proof. exists (istransabmonoidfracrelint X A is (pr1 isl)). apply (isreflabmonoidfracrelint X A is (pr2 isl)). Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | ispoabmonoidfracrelint | 167 |
Lemma ispoabmonoidfracrel (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (isl : ispreorder L) : ispreorder (abmonoidfracrel X A is). Proof. apply ispoquotrel. apply ispoabmonoidfracrelint. apply is. apply isl. Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | ispoabmonoidfracrel | 168 |
Lemma iseqrelabmonoidfracrelint (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (isl : iseqrel L) : iseqrel (abmonoidfracrelint X A L). Proof. exists (ispoabmonoidfracrelint X A is (pr1 isl)). apply (issymmabmonoidfracrelint X A is (pr2 isl)). Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | iseqrelabmonoidfracrelint | 169 |
Lemma iseqrelabmonoidfracrel (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (isl : iseqrel L) : iseqrel (abmonoidfracrel X A is). Proof. apply iseqrelquotrel. apply iseqrelabmonoidfracrelint. - apply is. - apply isl. Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | iseqrelabmonoidfracrel | 170 |
Lemma isirreflabmonoidfracrelint (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (isl : isirrefl L) : isirrefl (abmonoidfracrelint X A L). Proof. unfold isirrefl. intro xa. unfold abmonoidfracrelint. simpl. unfold neg. apply (@hinhuniv _ (make_hProp _ isapropempty)). intro t2. apply (isl _ (pr2 t2)). Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | isirreflabmonoidfracrelint | 171 |
Lemma isirreflabmonoidfracrel (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (isl : isirrefl L) : isirrefl (abmonoidfracrel X A is). Proof. apply isirreflquotrel. apply isirreflabmonoidfracrelint. - apply is. - apply isl. Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | isirreflabmonoidfracrel | 172 |
Lemma isasymmabmonoidfracrelint (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (isl : isasymm L) : isasymm (abmonoidfracrelint X A L). Proof. intros. set (assoc := (assocax X) : isassoc (@op X)). unfold isassoc in assoc. set (comm := commax X). unfold iscomm in comm. unfold isasymm. intros xa1 xa2. unfold abmonoidfracrelint. simpl. apply (@hinhuniv2 _ _ (make_hProp _ isapropempty)). intros t2l1 t2l2. set (c1a := pr1 t2l1). set (l1 := pr2 t2l1). set (c2a := pr1 t2l2). set (l2 := pr2 t2l2). set (c1 := pr1 c1a). set (c2 := pr1 c2a). set (x1 := pr1 xa1). set (a1 := pr1 (pr2 xa1)). set (x2 := pr1 xa2). set (a2 := pr1 (pr2 xa2)). assert (ll1 : L ((x1 + a2) + (@op X c1 c2)) ((x2 + a1) + (@op X c1 c2))). { induction (assoc (x1 + a2) c1 c2). induction (assoc (x2 + a1) c1 c2). apply ((pr2 is) _ _ _ (pr2 c2a)). apply l1. } assert (ll2 : L ((x2 + a1) + (@op X c1 c2)) ((x1 + a2) + (@op X c1 c2))). { induction (comm c2 c1). induction (assoc (x1 + a2) c2 c1). induction (assoc (x2 + a1) c2 c1). apply ((pr2 is) _ _ _ (pr2 c1a)). apply l2. } apply (isl _ _ ll1 ll2). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | isasymmabmonoidfracrelint | 173 |
Lemma isasymmabmonoidfracrel (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (isl : isasymm L) : isasymm (abmonoidfracrel X A is). Proof. apply isasymmquotrel. apply isasymmabmonoidfracrelint. - apply is. - apply isl. Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | isasymmabmonoidfracrel | 174 |
Lemma iscoasymmabmonoidfracrelint (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (isl : iscoasymm L) : iscoasymm (abmonoidfracrelint X A L). Proof. intros. set (assoc := (assocax X) : isassoc (@op X)). unfold isassoc in assoc. set (comm := commax X). unfold iscomm in comm. unfold iscoasymm. intros xa1 xa2. intro nl0. set (nl := neghexisttoforallneg _ nl0 (unel A)). simpl in nl. set (l := isl _ _ nl). apply hinhpr. exists (unel A). apply l. Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | iscoasymmabmonoidfracrelint | 175 |
Lemma iscoasymmabmonoidfracrel (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (isl : iscoasymm L) : iscoasymm (abmonoidfracrel X A is). Proof. apply iscoasymmquotrel. apply iscoasymmabmonoidfracrelint. - apply is. - apply isl. Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | iscoasymmabmonoidfracrel | 176 |
Lemma istotalabmonoidfracrelint (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (isl : istotal L) : istotal (abmonoidfracrelint X A L). Proof. unfold istotal. intros x1 x2. unfold abmonoidfracrelint. set (int := isl (pr1 x1 + pr1 (pr2 x2)) (pr1 x2 + pr1 (pr2 x1))). generalize int. clear int. simpl. apply hinhfun. apply coprodf. intro l. apply hinhpr. exists (unel A). rewrite (runax X _). rewrite (runax X _). apply l. intro l. apply hinhpr. exists (unel A). rewrite (runax X _). rewrite (runax X _). apply l. Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | istotalabmonoidfracrelint | 177 |
Lemma istotalabmonoidfracrel (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (isl : istotal L) : istotal (abmonoidfracrel X A is). Proof. apply istotalquotrel. apply istotalabmonoidfracrelint. - apply is. - apply isl. Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | istotalabmonoidfracrel | 178 |
Lemma iscotransabmonoidfracrelint (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (isl : iscotrans L) : iscotrans (abmonoidfracrelint X A L). Proof. intros. set (assoc := (assocax X) : isassoc (@op X)). unfold isassoc in assoc. set (comm := (commax X) : iscomm (@op X)). unfold iscomm in comm. set (rer := abmonoidrer X). unfold iscotrans. intros xa1 xa2 xa3. unfold abmonoidfracrelint. simpl. apply (@hinhuniv _ (ishinh _)). intro t2. set (c0a := pr1 t2). set (l0 := pr2 t2). set (x1 := pr1 xa1). set (a1 := pr1 (pr2 xa1)). set (x2 := pr1 xa2). set (a2 := pr1 (pr2 xa2)). set (x3 := pr1 xa3). set (a3 := pr1 (pr2 xa3)). set (c0 := pr1 c0a). set (z1 := (x1 + a3 + (a2 + c0))). set (z2 := x2 + a1 + (a3 + c0)). set (z3 := x3 + a1 + (a2 + c0)). assert (int : L z1 z3). { unfold z1. unfold z3. rewrite (comm a2 c0). rewrite <- (assoc _ _ a2). rewrite <- (assoc _ _ a2). apply ((pr2 is) _ _ _ (pr2 (pr2 xa2)) l0). } set (int' := isl z1 z2 z3 int). generalize int'. clear int'. simpl. apply hinhfun. intro cc. induction cc as [ l12 | l23 ]. - apply ii1. apply hinhpr. exists ((pr2 xa3) + c0a). change (L (x1 + a2 + (a3 + c0)) (x2 + a1 + (a3 + c0))). rewrite (rer _ a2 a3 _). apply l12. - apply ii2. apply hinhpr. exists ((pr2 xa1) + c0a). change (L (x2 + a3 + (a1 + c0)) (x3 + a2 + (a1 + c0))). rewrite (rer _ a3 a1 _). rewrite (rer _ a2 a1 _). apply l23. Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | iscotransabmonoidfracrelint | 179 |
Lemma iscotransabmonoidfracrel (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (isl : iscotrans L) : iscotrans (abmonoidfracrel X A is). Proof. apply iscotransquotrel. apply iscotransabmonoidfracrelint. - apply is. - apply isl. Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | iscotransabmonoidfracrel | 180 |
Lemma isStrongOrder_abmonoidfrac {X : abmonoid} (Y : @submonoid X) (gt : hrel X) (Hgt : ispartbinophrel Y gt) : isStrongOrder gt β isStrongOrder (abmonoidfracrel X Y Hgt). Proof. intros H. split ; [ | split]. - apply istransabmonoidfracrel, (istrans_isStrongOrder H). - apply iscotransabmonoidfracrel, (iscotrans_isStrongOrder H). - apply isirreflabmonoidfracrel, (isirrefl_isStrongOrder H). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | isStrongOrder_abmonoidfrac | 181 |
Definition StrongOrder_abmonoidfrac {X : abmonoid} (Y : @submonoid X) (gt : StrongOrder X) (Hgt : ispartbinophrel Y gt) : StrongOrder (abmonoidfrac X Y) := abmonoidfracrel X Y Hgt,, isStrongOrder_abmonoidfrac Y gt Hgt (pr2 gt). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | StrongOrder_abmonoidfrac | 182 |
Lemma isantisymmnegabmonoidfracrel (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (isl : isantisymmneg L) : isantisymmneg (abmonoidfracrel X A is). Proof. intros. assert (int : β x1 x2, isaprop (neg (abmonoidfracrel X A is x1 x2) β neg (abmonoidfracrel X A is x2 x1) β x1 = x2)). { intros x1 x2. apply impred. intro. apply impred. intro. apply (isasetsetquot _ x1 x2). } unfold isantisymmneg. apply (setquotuniv2prop _ (Ξ» x1 x2, make_hProp _ (int x1 x2))). intros xa1 xa2. intros r r'. apply (weqpathsinsetquot _). generalize r r'. clear r r'. change (neg (abmonoidfracrelint X A L xa1 xa2) β neg (abmonoidfracrelint X A L xa2 xa1) β (eqrelabmonoidfrac X A xa1 xa2)). intros nr12 nr21. set (nr12' := neghexisttoforallneg _ nr12 (unel A)). set (nr21' := neghexisttoforallneg _ nr21 (unel A)). set (int' := isl _ _ nr12' nr21'). simpl. apply hinhpr. exists (unel A). apply int'. Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | isantisymmnegabmonoidfracrel | 183 |
Lemma isantisymmabmonoidfracrel (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (isl : isantisymm L) : isantisymm (abmonoidfracrel X A is). Proof. intros. set (assoc := (assocax X) : isassoc (@op X)). unfold isassoc in assoc. set (comm := commax X). unfold iscomm in comm. unfold isantisymm. assert (int : β x1 x2, isaprop ((abmonoidfracrel X A is x1 x2) β (abmonoidfracrel X A is x2 x1) β x1 = x2)). { intros x1 x2. apply impred. intro. apply impred. intro. apply (isasetsetquot _ x1 x2). } apply (setquotuniv2prop _ (Ξ» x1 x2, make_hProp _ (int x1 x2))). intros xa1 xa2. intros r r'. apply (weqpathsinsetquot _). generalize r r'. clear r r'. change ((abmonoidfracrelint X A L xa1 xa2) β (abmonoidfracrelint X A L xa2 xa1) β (eqrelabmonoidfrac X A xa1 xa2)). unfold abmonoidfracrelint. unfold eqrelabmonoidfrac. simpl. apply hinhfun2. intros t2l1 t2l2. set (c1a := pr1 t2l1). set (l1 := pr2 t2l1). set (c2a := pr1 t2l2). set (l2 := pr2 t2l2). set (c1 := pr1 c1a). set (c2 := pr1 c2a). exists (@op A c1a c2a). set (x1 := pr1 xa1). set (a1 := pr1 (pr2 xa1)). set (x2 := pr1 xa2). set (a2 := pr1 (pr2 xa2)). change (x1 + a2 + (c1 + c2) = x2 + a1 + (@op X c1 c2)). assert (ll1 : L ((x1 + a2) + (@op X c1 c2)) ((x2 + a1) + (@op X c1 c2))). { induction (assoc (x1 + a2) c1 c2). induction (assoc (x2 + a1) c1 c2). apply ((pr2 is) _ _ _ (pr2 c2a)). apply l1. } assert (ll2 : L ((x2 + a1) + (@op X c1 c2)) ((x1 + a2) + (@op X c1 c2))). { induction (comm c2 c1). induction (assoc (x1 + a2) c2 c1). induction (assoc (x2 + a1) c2 c1). apply ((pr2 is) _ _ _ (pr2 c1a)). apply l2. } apply (isl _ _ ll1 ll2). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | isantisymmabmonoidfracrel | 184 |
Lemma ispartbinopabmonoidfracrelint (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) : @ispartbinophrel (setwithbinopdirprod X A) (Ξ» xa, A (pr1 xa)) (abmonoidfracrelint X A L). Proof. intros. set (assoc := (assocax X) : isassoc (@op X)). unfold isassoc in assoc. set (comm := commax X). unfold iscomm in comm. set (rer := abmonoidrer X). apply ispartbinophrelif. apply (commax (abmonoiddirprod X A)). intros xa yb zc s. unfold abmonoidfracrelint. simpl. apply (@hinhfun). intro t2l. induction t2l as [ c0a l ]. set (x := pr1 xa). set (a := pr1 (pr2 xa)). set (y := pr1 yb). set (b := pr1 (pr2 yb)). set (z := pr1 zc). set (c := pr1 (pr2 zc)). set (c0 := pr1 c0a). exists c0a. change (L (((z + x) + (c + b)) + c0) (((z + y) + (c + a)) + c0)). change (pr1 (L ((x + b) + c0) ((y + a) + c0))) in l. rewrite (rer z _ _ b). rewrite (assoc (z + c) _ _). rewrite (rer z _ _ a). rewrite (assoc (z + c) _ _). apply ((pr1 is) _ _ _ (pr2 (@op A (make_carrier A z s) (pr2 zc)))). apply l. Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | ispartbinopabmonoidfracrelint | 185 |
Lemma ispartlbinopabmonoidfracrel (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (aa aa' : A) (z z' : abmonoidfrac X A) (l : abmonoidfracrel X A is z z') : abmonoidfracrel X A is ((prabmonoidfrac X A (pr1 aa) aa') + z) ((prabmonoidfrac X A (pr1 aa) aa') + z'). Proof. revert z z' l. set (assoc := (assocax X) : isassoc (@op X)). unfold isassoc in assoc. set (comm := commax X). unfold iscomm in comm. set (rer := abmonoidrer X). assert (int : β z z', isaprop (abmonoidfracrel X A is z z' β abmonoidfracrel X A is (prabmonoidfrac X A (pr1 aa) aa' + z) (prabmonoidfrac X A (pr1 aa) aa' + z'))). { intros z z'. apply impred. intro. apply (pr2 (abmonoidfracrel _ _ _ _ _)). } apply (setquotuniv2prop _ (Ξ» z z', make_hProp _ (int z z'))). intros xa1 xa2. change (abmonoidfracrelint X A L xa1 xa2 β abmonoidfracrelint X A L (@op (abmonoiddirprod X A) (pr1 aa ,, aa') xa1) (@op (abmonoiddirprod X A) (pr1 aa ,, aa') xa2)). unfold abmonoidfracrelint. simpl. apply hinhfun. intro t2l. set (a := pr1 aa). set (a' := pr1 aa'). set (c0a := pr1 t2l). set (l := pr2 t2l). set (c0 := pr1 c0a). set (x1 := pr1 xa1). set (a1 := pr1 (pr2 xa1)). set (x2 := pr1 xa2). set (a2 := pr1 (pr2 xa2)). exists c0a. change (L (a + x1 + (a' + a2) + c0) (a + x2 + (a' + a1) + c0)). rewrite (rer _ x1 a' _). rewrite (rer _ x2 a' _). rewrite (assoc _ (x1 + a2) c0). rewrite (assoc _ (x2 + a1) c0). apply ((pr1 is) _ _ _ (pr2 (@op A aa aa'))). apply l. Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | ispartlbinopabmonoidfracrel | 186 |
Lemma ispartrbinopabmonoidfracrel (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartbinophrel A L) (aa aa' : A) (z z' : abmonoidfrac X A) (l : abmonoidfracrel X A is z z') : abmonoidfracrel X A is (z + (prabmonoidfrac X A (pr1 aa) aa')) (z' + (prabmonoidfrac X A (pr1 aa) aa')). Proof. revert z z' l. set (assoc := (assocax X) : isassoc (@op X)). unfold isassoc in assoc. set (comm := commax X). unfold iscomm in comm. set (rer := abmonoidrer X). assert (int : β (z z' : abmonoidfrac X A), isaprop (abmonoidfracrel X A is z z' β abmonoidfracrel X A is (z + (prabmonoidfrac X A (pr1 aa) aa')) (z' + prabmonoidfrac X A (pr1 aa) aa'))). { intros z z'. apply impred. intro. apply (pr2 (abmonoidfracrel _ _ _ _ _)). } apply (setquotuniv2prop _ (Ξ» z z', make_hProp _ (int z z'))). intros xa1 xa2. change (abmonoidfracrelint X A L xa1 xa2 β abmonoidfracrelint X A L (@op (abmonoiddirprod X A) xa1 (pr1 aa ,, aa')) (@op (abmonoiddirprod X A) xa2 (pr1 aa ,, aa'))). unfold abmonoidfracrelint. simpl. apply hinhfun. intro t2l. set (a := pr1 aa). set (a' := pr1 aa'). set (c0a := pr1 t2l). set (l := pr2 t2l). set (c0 := pr1 c0a). set (x1 := pr1 xa1). set (a1 := pr1 (pr2 xa1)). set (x2 := pr1 xa2). set (a2 := pr1 (pr2 xa2)). exists c0a. change (L (x1 + a + (a2 + a') + c0) (x2 + a + (a1 + a') + c0)). rewrite (rer _ a a2 _). rewrite (rer _ a a1 _). rewrite (assoc (x1 + a2) _ c0). rewrite (assoc (x2 + a1) _ c0). rewrite (comm _ c0). induction (assoc (x1 + a2) c0 (a + a')). induction (assoc (x2 + a1) c0 (a + a')). apply ((pr2 is) _ _ _ (pr2 (@op A aa aa'))). apply l. Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | ispartrbinopabmonoidfracrel | 187 |
Lemma abmonoidfracrelimpl (X : abmonoid) (A : subabmonoid X) {L L' : hrel X} (is : ispartbinophrel A L) (is' : ispartbinophrel A L') (impl : β x x', L x x' β L' x x') (x x' : abmonoidfrac X A) (ql : abmonoidfracrel X A is x x') : abmonoidfracrel X A is' x x'. Proof. generalize ql. apply quotrelimpl. intros x0 x0'. unfold abmonoidfracrelint. simpl. apply hinhfun. intro t2. exists (pr1 t2). apply (impl _ _ (pr2 t2)). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoidfracrelimpl | 188 |
Lemma abmonoidfracrellogeq (X : abmonoid) (A : subabmonoid X) {L L' : hrel X} (is : ispartbinophrel A L) (is' : ispartbinophrel A L') (lg : β x x', L x x' <-> L' x x') (x x' : abmonoidfrac X A) : (abmonoidfracrel X A is x x') <-> (abmonoidfracrel X A is' x x'). Proof. apply quotrellogeq. intros x0 x0'. split. - unfold abmonoidfracrelint. simpl. apply hinhfun. intro t2. exists (pr1 t2). apply (pr1 (lg _ _) (pr2 t2)). - unfold abmonoidfracrelint. simpl. apply hinhfun. intro t2. exists (pr1 t2). apply (pr2 (lg _ _) (pr2 t2)). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoidfracrellogeq | 189 |
Definition isdecabmonoidfracrelint (X : abmonoid) (A : subabmonoid X) {L : hrel X} (is : ispartinvbinophrel A L) (isl : isdecrel L) : isdecrel (abmonoidfracrelint X A L). Proof. intros xa1 xa2. set (x1 := pr1 xa1). set (a1 := pr1 (pr2 xa1)). set (x2 := pr1 xa2). set (a2 := pr1 (pr2 xa2)). assert (int : coprod (L (x1 + a2) (x2 + a1)) (neg (L (x1 + a2) (x2 + a1)))) by apply (isl _ _). induction int as [ l | nl ]. - apply ii1. unfold abmonoidfracrelint. apply hinhpr. exists (unel A). rewrite (runax X _). rewrite (runax X _). apply l. - apply ii2. generalize nl. clear nl. apply negf. unfold abmonoidfracrelint. simpl. apply (@hinhuniv _ (make_hProp _ (pr2 (L _ _)))). intro t2l. induction t2l as [ c0a l ]. simpl. apply ((pr2 is) _ _ _ (pr2 c0a) l). Defined. | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | isdecabmonoidfracrelint | 190 |
Definition isdecabmonoidfracrel (X : abmonoid) (A : submonoid X) {L : hrel X} (is : ispartbinophrel A L) (isi : ispartinvbinophrel A L) (isl : isdecrel L) : isdecrel (abmonoidfracrel X A is). Proof. apply isdecquotrel. apply isdecabmonoidfracrelint. - apply isi. - apply isl. Defined. | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | isdecabmonoidfracrel | 191 |
Lemma iscomptoabmonoidfrac (X : abmonoid) (A : submonoid X) {L : hrel X} (is : ispartbinophrel A L) : iscomprelrelfun L (abmonoidfracrel X A is) (toabmonoidfrac X A). Proof. unfold iscomprelrelfun. intros x x' l. change (abmonoidfracrelint X A L (x ,, unel A) (x' ,, unel A)). simpl. apply (hinhpr). exists (unel A). apply ((pr2 is) _ _ 0). apply (pr2 (unel A)). apply ((pr2 is) _ _ 0). apply (pr2 (unel A)). apply l. Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | iscomptoabmonoidfrac | 192 |
Definition isaprel {X : UU} (ap : hrel X) := isirrefl ap Γ issymm ap Γ iscotrans ap. | Definition | Algebra | Require Import UniMath.Foundations.Propositions. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.DecidablePropositions. | Algebra\Apartness.v | isaprel | 193 |
Lemma isaprop_isaprel {X : UU} (ap : hrel X) : isaprop (isaprel ap). Proof. apply isapropdirprod. apply isaprop_isirrefl. apply isapropdirprod. apply isaprop_issymm. apply isaprop_iscotrans. Qed. | Lemma | Algebra | Require Import UniMath.Foundations.Propositions. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.DecidablePropositions. | Algebra\Apartness.v | isaprop_isaprel | 194 |
Definition aprel (X : UU) := β ap : hrel X, isaprel ap. | Definition | Algebra | Require Import UniMath.Foundations.Propositions. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.DecidablePropositions. | Algebra\Apartness.v | aprel | 195 |
Definition aprel_pr1 {X : UU} (ap : aprel X) : hrel X := pr1 ap. | Definition | Algebra | Require Import UniMath.Foundations.Propositions. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.DecidablePropositions. | Algebra\Apartness.v | aprel_pr1 | 196 |
Definition apSet := β X : hSet, aprel X. | Definition | Algebra | Require Import UniMath.Foundations.Propositions. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.DecidablePropositions. | Algebra\Apartness.v | apSet | 197 |
Definition apSet_pr1 (X : apSet) : hSet := pr1 X. | Definition | Algebra | Require Import UniMath.Foundations.Propositions. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.DecidablePropositions. | Algebra\Apartness.v | apSet_pr1 | 198 |
Definition apSet_pr2 (X : apSet) : aprel X := pr2 X. | Definition | Algebra | Require Import UniMath.Foundations.Propositions. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.DecidablePropositions. | Algebra\Apartness.v | apSet_pr2 | 199 |