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Definition abgr : UU := ∑ (X : setwithbinop), isabgrop (@op X). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgr | 0 |
Definition make_abgr (X : setwithbinop) (is : isabgrop (@op X)) : abgr := X ,, is. | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | make_abgr | 1 |
Definition abgrconstr (X : abmonoid) (inv0 : X → X) (is : isinv (@op X) 0 inv0) : abgr := make_abgr X (make_isgrop (pr2 X) (inv0 ,, is) ,, commax X). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrconstr | 2 |
Definition abgrtogr : abgr → gr := λ X, make_gr (pr1 X) (pr1 (pr2 X)). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrtogr | 3 |
Definition abgrtoabmonoid : abgr → abmonoid := λ X, make_abmonoid (pr1 X) (pr1 (pr1 (pr2 X)) ,, pr2 (pr2 X)). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrtoabmonoid | 4 |
Definition abgr_of_gr (X : gr) (H : iscomm (@op X)) : abgr := make_abgr X (make_isabgrop (pr2 X) H). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgr_of_gr | 5 |
Definition unitabgr_isabgrop : isabgrop (@op unitabmonoid). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | unitabgr_isabgrop | 6 |
Definition unitabgr : abgr := make_abgr unitabmonoid unitabgr_isabgrop. | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | unitabgr | 7 |
Lemma abgrfuntounit_ismonoidfun (X : abgr) : ismonoidfun (Y := unitabgr) (λ x : X, 0). Proof. use make_ismonoidfun. - use make_isbinopfun. intros x x'. use isProofIrrelevantUnit. - use isProofIrrelevantUnit. Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrfuntounit_ismonoidfun | 8 |
Definition abgrfuntounit (X : abgr) : monoidfun X unitabgr := monoidfunconstr (abgrfuntounit_ismonoidfun X). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrfuntounit | 9 |
Lemma abgrfunfromunit_ismonoidfun (X : abgr) : ismonoidfun (Y := X) (λ x : unitabgr, 0). Proof. use make_ismonoidfun. - use make_isbinopfun. intros x x'. exact (!runax X _). - use idpath. Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrfunfromunit_ismonoidfun | 10 |
Definition abgrfunfromunit (X : abgr) : monoidfun unitabgr X := monoidfunconstr (abgrfunfromunit_ismonoidfun X). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrfunfromunit | 11 |
Lemma unelabgrfun_ismonoidfun (X Y : abgr) : ismonoidfun (Y := Y) (λ x : X, 0). Proof. use make_ismonoidfun. - use make_isbinopfun. intros x x'. exact (!lunax _ _). - use idpath. Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | unelabgrfun_ismonoidfun | 12 |
Definition unelabgrfun (X Y : abgr) : monoidfun X Y := monoidfunconstr (unelgrfun_ismonoidfun X Y). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | unelabgrfun | 13 |
Definition abgrshombinop_inv_ismonoidfun {X Y : abgr} (f : monoidfun X Y) : ismonoidfun (λ x : X, grinv Y (pr1 f x)). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrshombinop_inv_ismonoidfun | 14 |
Definition abgrshombinop_inv {X Y : abgr} (f : monoidfun X Y) : monoidfun X Y := monoidfunconstr (abgrshombinop_inv_ismonoidfun f). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrshombinop_inv | 15 |
Definition abgrshombinop_linvax {X Y : abgr} (f : monoidfun X Y) : @abmonoidshombinop X Y (abgrshombinop_inv f) f = unelmonoidfun X Y. | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrshombinop_linvax | 16 |
Definition abgrshombinop_rinvax {X Y : abgr} (f : monoidfun X Y) : @abmonoidshombinop X Y f (abgrshombinop_inv f) = unelmonoidfun X Y. | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrshombinop_rinvax | 17 |
Lemma abgrshomabgr_isabgrop (X Y : abgr) : @isabgrop (abmonoidshomabmonoid X Y) (λ f g : monoidfun X Y, @abmonoidshombinop X Y f g). Proof. use make_isabgrop. - use make_isgrop. + exact (abmonoidshomabmonoid_ismonoidop X Y). + use make_invstruct. * intros f. exact (abgrshombinop_inv f). * use make_isinv. intros f. exact (abgrshombinop_linvax f). intros f. exact (abgrshombinop_rinvax f). - intros f g. exact (abmonoidshombinop_comm f g). Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrshomabgr_isabgrop | 18 |
Definition abgrshomabgr (X Y : abgr) : abgr. Proof. use make_abgr. - exact (abmonoidshomabmonoid X Y). - exact (abgrshomabgr_isabgrop X Y). Defined. | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrshomabgr | 19 |
Definition abgr_univalence_weq1' (X Y : abgr) : (X = Y) ≃ (make_abgr' X = make_abgr' Y) := make_weq _ (@isweqmaponpaths abgr abgr' abgr_univalence_weq1 X Y). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgr_univalence_weq1' | 20 |
Definition abgr_univalence_weq2 (X Y : abgr) : (make_abgr' X = make_abgr' Y) ≃ (pr1 (make_abgr' X) = pr1 (make_abgr' Y)). Proof. use subtypeInjectivity. intros w. use isapropiscomm. Defined. | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgr_univalence_weq2 | 21 |
Definition abgr_univalence_weq3 (X Y : abgr) : (pr1 (make_abgr' X) = pr1 (make_abgr' Y)) ≃ (monoidiso X Y) := gr_univalence (pr1 (make_abgr' X)) (pr1 (make_abgr' Y)). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgr_univalence_weq3 | 22 |
Definition abgr_univalence_map (X Y : abgr) : (X = Y) → (monoidiso X Y). Proof. intro e. induction e. exact (idmonoidiso X). Defined. | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgr_univalence_map | 23 |
Lemma abgr_univalence_isweq (X Y : abgr) : isweq (abgr_univalence_map X Y). Proof. use isweqhomot. - exact (weqcomp (abgr_univalence_weq1' X Y) (weqcomp (abgr_univalence_weq2 X Y) (abgr_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.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgr_univalence_isweq | 24 |
Definition abgr_univalence (X Y : abgr) : (X = Y) ≃ (monoidiso X Y) := make_weq (abgr_univalence_map X Y) (abgr_univalence_isweq X Y). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgr_univalence | 25 |
Definition subabgr (X : abgr) := subgr X. | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | subabgr | 26 |
Lemma isabgrcarrier {X : abgr} (A : subgr X) : isabgrop (@op A). Proof. exists (isgrcarrier A). apply (pr2 (@isabmonoidcarrier X A)). Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | isabgrcarrier | 27 |
Definition carrierofasubabgr {X : abgr} (A : subabgr X) : abgr. Proof. exists A. apply isabgrcarrier. Defined. | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | carrierofasubabgr | 28 |
Definition subabgr_incl {X : abgr} (A : subabgr X) : monoidfun A X := submonoid_incl A. | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | subabgr_incl | 29 |
Definition abgr_kernel_hsubtype {A B : abgr} (f : monoidfun A B) : hsubtype A := monoid_kernel_hsubtype f. | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgr_kernel_hsubtype | 30 |
Definition abgr_image_hsubtype {A B : abgr} (f : monoidfun A B) : hsubtype B := (λ y : B, ∃ x : A, (f x) = y). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgr_image_hsubtype | 31 |
Definition abgr_Kernel_subabgr_issubgr {A B : abgr} (f : monoidfun A B) : issubgr (abgr_kernel_hsubtype f). Proof. use make_issubgr. - apply kernel_issubmonoid. - intros x a. apply (grrcan B (f x)). refine (! (binopfunisbinopfun f (grinv A x) x) @ _). refine (maponpaths (λ a : A, f a) (grlinvax A x) @ _). refine (monoidfununel f @ !_). refine (lunax B (f x) @ _). exact a. Defined. | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgr_Kernel_subabgr_issubgr | 32 |
Definition abgr_Kernel_subabgr {A B : abgr} (f : monoidfun A B) : @subabgr A := subgrconstr (@abgr_kernel_hsubtype A B f) (abgr_Kernel_subabgr_issubgr f). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgr_Kernel_subabgr | 33 |
Definition abgr_Kernel_monoidfun_ismonoidfun {A B : abgr} (f : monoidfun A B) : @ismonoidfun (abgr_Kernel_subabgr f) A (make_incl (pr1carrier (abgr_kernel_hsubtype f)) (isinclpr1carrier (abgr_kernel_hsubtype f))). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgr_Kernel_monoidfun_ismonoidfun | 34 |
Definition abgr_image_issubgr {A B : abgr} (f : monoidfun A B) : issubgr (abgr_image_hsubtype f). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgr_image_issubgr | 35 |
Definition abgr_image {A B : abgr} (f : monoidfun A B) : @subabgr B := @subgrconstr B (@abgr_image_hsubtype A B f) (abgr_image_issubgr f). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgr_image | 36 |
Lemma isabgrquot {X : abgr} (R : binopeqrel X) : isabgrop (@op (setwithbinopquot R)). Proof. exists (isgrquot R). apply (pr2 (@isabmonoidquot X R)). Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | isabgrquot | 37 |
Definition abgrquot {X : abgr} (R : binopeqrel X) : abgr. Proof. exists (setwithbinopquot R). apply isabgrquot. Defined. | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrquot | 38 |
Lemma isabgrdirprod (X Y : abgr) : isabgrop (@op (setwithbinopdirprod X Y)). Proof. exists (isgrdirprod X Y). apply (pr2 (isabmonoiddirprod X Y)). Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | isabgrdirprod | 39 |
Definition abgrdirprod (X Y : abgr) : abgr. Proof. exists (setwithbinopdirprod X Y). apply isabgrdirprod. Defined. | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrdirprod | 40 |
Definition hrelabgrdiff (X : abmonoid) : hrel (X × X) := λ xa1 xa2, ∃ (x0 : X), (pr1 xa1 + pr2 xa2) + x0 = (pr1 xa2 + pr2 xa1) + x0. | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | hrelabgrdiff | 41 |
Definition abgrdiffphi (X : abmonoid) (xa : X × X) : X × (totalsubtype X) := pr1 xa ,, make_carrier (λ x : X, htrue) (pr2 xa) tt. | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrdiffphi | 42 |
Definition hrelabgrdiff' (X : abmonoid) : hrel (X × X) := λ xa1 xa2, eqrelabmonoidfrac X (totalsubmonoid X) (abgrdiffphi X xa1) (abgrdiffphi X xa2). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | hrelabgrdiff' | 43 |
Lemma logeqhrelsabgrdiff (X : abmonoid) : hrellogeq (hrelabgrdiff' X) (hrelabgrdiff X). Proof. split. simpl. apply hinhfun. intro t2. set (a0 := pr1 (pr1 t2)). exists a0. apply (pr2 t2). simpl. apply hinhfun. intro t2. set (x0 := pr1 t2). exists (x0 ,, tt). apply (pr2 t2). Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | logeqhrelsabgrdiff | 44 |
Lemma iseqrelabgrdiff (X : abmonoid) : iseqrel (hrelabgrdiff X). Proof. apply (iseqrellogeqf (logeqhrelsabgrdiff X)). apply (iseqrelconstr). intros xx' xx'' xx'''. intros r1 r2. apply (eqreltrans (eqrelabmonoidfrac X (totalsubmonoid X)) _ _ _ r1 r2). intro xx. apply (eqrelrefl (eqrelabmonoidfrac X (totalsubmonoid X)) _). intros xx xx'. intro r. apply (eqrelsymm (eqrelabmonoidfrac X (totalsubmonoid X)) _ _ r). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | iseqrelabgrdiff | 45 |
Definition eqrelabgrdiff (X : abmonoid) : @eqrel (abmonoiddirprod X X) := make_eqrel _ (iseqrelabgrdiff X). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | eqrelabgrdiff | 46 |
Lemma isbinophrelabgrdiff (X : abmonoid) : @isbinophrel (abmonoiddirprod X X) (hrelabgrdiff X). Proof. apply (@isbinophrellogeqf (abmonoiddirprod X X) _ _ (logeqhrelsabgrdiff X)). split. intros a b c r. apply (pr1 (isbinophrelabmonoidfrac X (totalsubmonoid X)) _ _ (pr1 c ,, make_carrier (λ x : X, htrue) (pr2 c) tt) r). intros a b c r. apply (pr2 (isbinophrelabmonoidfrac X (totalsubmonoid X)) _ _ (pr1 c ,, make_carrier (λ x : X, htrue) (pr2 c) tt) r). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | isbinophrelabgrdiff | 47 |
Definition binopeqrelabgrdiff (X : abmonoid) : binopeqrel (abmonoiddirprod X X) := make_binopeqrel (eqrelabgrdiff X) (isbinophrelabgrdiff X). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | binopeqrelabgrdiff | 48 |
Definition abgrdiffcarrier (X : abmonoid) : abmonoid := @abmonoidquot (abmonoiddirprod X X) (binopeqrelabgrdiff X). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrdiffcarrier | 49 |
Definition abgrdiffinvint (X : abmonoid) : X × X → X × X := λ xs, pr2 xs ,, pr1 xs. | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrdiffinvint | 50 |
Lemma abgrdiffinvcomp (X : abmonoid) : iscomprelrelfun (hrelabgrdiff X) (eqrelabgrdiff X) (abgrdiffinvint X). Proof. unfold iscomprelrelfun. unfold eqrelabgrdiff. unfold hrelabgrdiff. unfold eqrelabmonoidfrac. unfold hrelabmonoidfrac. simpl. intros xs xs'. apply (hinhfun). intro tt0. set (x := pr1 xs). set (s := pr2 xs). set (x' := pr1 xs'). set (s' := pr2 xs'). exists (pr1 tt0). induction tt0 as [ a eq ]. change (s + x' + a = s' + x + a). set(e := commax X s' x). simpl in e. rewrite e. clear e. set (e := commax X s x'). simpl in e. rewrite e. clear e. exact (!eq). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrdiffinvcomp | 51 |
Definition abgrdiffinv (X : abmonoid) : abgrdiffcarrier X → abgrdiffcarrier X := setquotfun (hrelabgrdiff X) (eqrelabgrdiff X) (abgrdiffinvint X) (abgrdiffinvcomp X). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrdiffinv | 52 |
Lemma abgrdiffisinv (X : abmonoid) : isinv (@op (abgrdiffcarrier X)) 0 (abgrdiffinv X). Proof. set (R := eqrelabgrdiff X). assert (isl : islinv (@op (abgrdiffcarrier X)) 0 (abgrdiffinv X)). { unfold islinv. apply (setquotunivprop R (λ x, _ = _)%logic). intro xs. set (x := pr1 xs). set (s := pr2 xs). apply (iscompsetquotpr R (@op (abmonoiddirprod X X) (abgrdiffinvint X xs) xs) 0). simpl. apply hinhpr. exists (unel X). change (s + x + 0 + 0 = 0 + (x + s) + 0). induction (commax X x s). induction (commax X 0 (x + s)). apply idpath. } exact (isl ,, weqlinvrinv (@op (abgrdiffcarrier X)) (commax (abgrdiffcarrier X)) 0 (abgrdiffinv X) isl). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrdiffisinv | 53 |
Definition abgrdiff (X : abmonoid) : abgr := abgrconstr (abgrdiffcarrier X) (abgrdiffinv X) (abgrdiffisinv X). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrdiff | 54 |
Definition prabgrdiff (X : abmonoid) : X → X → abgrdiff X := λ x x' : X, setquotpr (eqrelabgrdiff X) (x ,, x'). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | prabgrdiff | 55 |
Definition weqabgrdiffint (X : abmonoid) : weq (X × X) (X × totalsubtype X) := weqdirprodf (idweq X) (invweq (weqtotalsubtype X)). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | weqabgrdiffint | 56 |
Definition weqabgrdiff (X : abmonoid) : weq (abgrdiff X) (abmonoidfrac X (totalsubmonoid X)). Proof. intros. apply (weqsetquotweq (eqrelabgrdiff X) (eqrelabmonoidfrac X (totalsubmonoid X)) (weqabgrdiffint X)). - simpl. intros x x'. induction x as [ x1 x2 ]. induction x' as [ x1' x2' ]. simpl in *. apply hinhfun. intro tt0. induction tt0 as [ xx0 is0 ]. exists (make_carrier (λ x : X, htrue) xx0 tt). apply is0. - simpl. intros x x'. induction x as [ x1 x2 ]. induction x' as [ x1' x2' ]. simpl in *. apply hinhfun. intro tt0. induction tt0 as [ xx0 is0 ]. exists (pr1 xx0). apply is0. Defined. | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | weqabgrdiff | 57 |
Definition toabgrdiff (X : abmonoid) (x : X) : abgrdiff X := setquotpr _ (x ,, 0). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | toabgrdiff | 58 |
Lemma isbinopfuntoabgrdiff (X : abmonoid) : isbinopfun (toabgrdiff X). Proof. unfold isbinopfun. intros x1 x2. change (setquotpr _ (x1 + x2 ,, 0) = setquotpr (eqrelabgrdiff X) (x1 + x2 ,, 0 + 0)). apply (maponpaths (setquotpr _)). apply (@pathsdirprod X X). - apply idpath. - exact (!lunax X 0). Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | isbinopfuntoabgrdiff | 59 |
Lemma isunitalfuntoabgrdiff (X : abmonoid) : toabgrdiff X 0 = 0. Proof. apply idpath. Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | isunitalfuntoabgrdiff | 60 |
Definition ismonoidfuntoabgrdiff (X : abmonoid) : ismonoidfun (toabgrdiff X) := isbinopfuntoabgrdiff X ,, isunitalfuntoabgrdiff X. | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | ismonoidfuntoabgrdiff | 61 |
Lemma isinclprabgrdiff (X : abmonoid) (iscanc : ∏ x : X, isrcancelable (@op X) x) : ∏ x' : X, isincl (λ x, prabgrdiff X x x'). Proof. intros. set (int := isinclprabmonoidfrac X (totalsubmonoid X) (λ a : totalsubmonoid X, iscanc (pr1 a)) (make_carrier (λ x : X, htrue) x' tt)). set (int1 := isinclcomp (make_incl _ int) (weqtoincl (invweq (weqabgrdiff X)))). apply int1. Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | isinclprabgrdiff | 62 |
Definition isincltoabgrdiff (X : abmonoid) (iscanc : ∏ x : X, isrcancelable (@op X) x) : isincl (toabgrdiff X) := isinclprabgrdiff X iscanc 0. | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | isincltoabgrdiff | 63 |
Lemma isdeceqabgrdiff (X : abmonoid) (iscanc : ∏ x : X, isrcancelable (@op X) x) (is : isdeceq X) : isdeceq (abgrdiff X). Proof. intros. apply (isdeceqweqf (invweq (weqabgrdiff X))). apply (isdeceqabmonoidfrac X (totalsubmonoid X) (λ a : totalsubmonoid X, iscanc (pr1 a)) is). Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | isdeceqabgrdiff | 64 |
Definition abgrdiffrelint (X : abmonoid) (L : hrel X) : hrel (setwithbinopdirprod X X) := λ xa yb, ∃ (c0 : X), L ((pr1 xa + pr2 yb) + c0) ((pr1 yb + pr2 xa) + c0). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrdiffrelint | 65 |
Definition abgrdiffrelint' (X : abmonoid) (L : hrel X) : hrel (setwithbinopdirprod X X) := λ xa1 xa2, abmonoidfracrelint _ (totalsubmonoid X) L (abgrdiffphi X xa1) (abgrdiffphi X xa2). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrdiffrelint' | 66 |
Lemma logeqabgrdiffrelints (X : abmonoid) (L : hrel X) : hrellogeq (abgrdiffrelint' X L) (abgrdiffrelint X L). Proof. split. unfold abgrdiffrelint. unfold abgrdiffrelint'. simpl. apply hinhfun. intro t2. set (a0 := pr1 (pr1 t2)). exists a0. apply (pr2 t2). simpl. apply hinhfun. intro t2. set (x0 := pr1 t2). exists (x0 ,, tt). apply (pr2 t2). Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | logeqabgrdiffrelints | 67 |
Lemma iscomprelabgrdiffrelint (X : abmonoid) {L : hrel X} (is : isbinophrel L) : iscomprelrel (eqrelabgrdiff X) (abgrdiffrelint X L). Proof. apply (iscomprelrellogeqf1 _ (logeqhrelsabgrdiff X)). apply (iscomprelrellogeqf2 _ (logeqabgrdiffrelints X L)). intros x x' x0 x0' r r0. apply (iscomprelabmonoidfracrelint _ (totalsubmonoid X) (isbinoptoispartbinop _ _ is) _ _ _ _ r r0). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | iscomprelabgrdiffrelint | 68 |
Definition abgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) := quotrel (iscomprelabgrdiffrelint X is). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrdiffrel | 69 |
Definition abgrdiffrel' (X : abmonoid) {L : hrel X} (is : isbinophrel L) : hrel (abgrdiff X) := λ x x', abmonoidfracrel X (totalsubmonoid X) (isbinoptoispartbinop _ _ is) (weqabgrdiff X x) (weqabgrdiff X x'). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrdiffrel' | 70 |
Definition logeqabgrdiffrels (X : abmonoid) {L : hrel X} (is : isbinophrel L) : hrellogeq (abgrdiffrel' X is) (abgrdiffrel X is). Proof. intros x1 x2. split. - assert (int : ∏ x x', isaprop (abgrdiffrel' X is x x' → abgrdiffrel X is x x')). { intros x x'. apply impred. intro. apply (pr2 _). } generalize x1 x2. clear x1 x2. apply (setquotuniv2prop _ (λ x x', make_hProp _ (int x x'))). intros x x'. change ((abgrdiffrelint' X L x x') → (abgrdiffrelint _ L x x')). apply (pr1 (logeqabgrdiffrelints X L x x')). - assert (int : ∏ x x', isaprop (abgrdiffrel X is x x' → abgrdiffrel' X is x x')). intros x x'. apply impred. intro. apply (pr2 _). generalize x1 x2. clear x1 x2. apply (setquotuniv2prop _ (λ x x', make_hProp _ (int x x'))). intros x x'. change ((abgrdiffrelint X L x x') → (abgrdiffrelint' _ L x x')). apply (pr2 (logeqabgrdiffrelints X L x x')). Defined. | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | logeqabgrdiffrels | 71 |
Lemma istransabgrdiffrelint (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : istrans L) : istrans (abgrdiffrelint X L). Proof. apply (istranslogeqf (logeqabgrdiffrelints X L)). intros a b c rab rbc. apply (istransabmonoidfracrelint _ (totalsubmonoid X) (isbinoptoispartbinop _ _ is) isl _ _ _ rab rbc). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | istransabgrdiffrelint | 72 |
Lemma istransabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : istrans L) : istrans (abgrdiffrel X is). Proof. refine (istransquotrel _ _). apply istransabgrdiffrelint. - apply is. - apply isl. Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | istransabgrdiffrel | 73 |
Lemma issymmabgrdiffrelint (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : issymm L) : issymm (abgrdiffrelint X L). Proof. apply (issymmlogeqf (logeqabgrdiffrelints X L)). intros a b rab. apply (issymmabmonoidfracrelint _ (totalsubmonoid X) (isbinoptoispartbinop _ _ is) isl _ _ rab). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | issymmabgrdiffrelint | 74 |
Lemma issymmabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : issymm L) : issymm (abgrdiffrel X is). Proof. refine (issymmquotrel _ _). apply issymmabgrdiffrelint. - apply is. - apply isl. Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | issymmabgrdiffrel | 75 |
Lemma isreflabgrdiffrelint (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : isrefl L) : isrefl (abgrdiffrelint X L). Proof. intro xa. unfold abgrdiffrelint. simpl. apply hinhpr. exists (unel X). apply (isl _). Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | isreflabgrdiffrelint | 76 |
Lemma isreflabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : isrefl L) : isrefl (abgrdiffrel X is). Proof. refine (isreflquotrel _ _). apply isreflabgrdiffrelint. - apply is. - apply isl. Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | isreflabgrdiffrel | 77 |
Lemma ispoabgrdiffrelint (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : ispreorder L) : ispreorder (abgrdiffrelint X L). Proof. exists (istransabgrdiffrelint X is (pr1 isl)). apply (isreflabgrdiffrelint X is (pr2 isl)). Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | ispoabgrdiffrelint | 78 |
Lemma ispoabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : ispreorder L) : ispreorder (abgrdiffrel X is). Proof. refine (ispoquotrel _ _). apply ispoabgrdiffrelint. - apply is. - apply isl. Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | ispoabgrdiffrel | 79 |
Lemma iseqrelabgrdiffrelint (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : iseqrel L) : iseqrel (abgrdiffrelint X L). Proof. exists (ispoabgrdiffrelint X is (pr1 isl)). apply (issymmabgrdiffrelint X is (pr2 isl)). Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | iseqrelabgrdiffrelint | 80 |
Lemma iseqrelabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : iseqrel L) : iseqrel (abgrdiffrel X is). Proof. refine (iseqrelquotrel _ _). apply iseqrelabgrdiffrelint. - apply is. - apply isl. Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | iseqrelabgrdiffrel | 81 |
Lemma isantisymmnegabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : isantisymmneg L) : isantisymmneg (abgrdiffrel X is). Proof. apply (isantisymmneglogeqf (logeqabgrdiffrels X is)). intros a b rab rba. set (int := isantisymmnegabmonoidfracrel _ (totalsubmonoid X) (isbinoptoispartbinop _ _ is) isl (weqabgrdiff X a) (weqabgrdiff X b) rab rba). apply (invmaponpathsweq _ _ _ int). Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | isantisymmnegabgrdiffrel | 82 |
Lemma isantisymmabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : isantisymm L) : isantisymm (abgrdiffrel X is). Proof. apply (isantisymmlogeqf (logeqabgrdiffrels X is)). intros a b rab rba. set (int := isantisymmabmonoidfracrel _ (totalsubmonoid X) (isbinoptoispartbinop _ _ is) isl (weqabgrdiff X a) (weqabgrdiff X b) rab rba). apply (invmaponpathsweq _ _ _ int). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | isantisymmabgrdiffrel | 83 |
Lemma isirreflabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : isirrefl L) : isirrefl (abgrdiffrel X is). Proof. apply (isirrefllogeqf (logeqabgrdiffrels X is)). intros a raa. apply (isirreflabmonoidfracrel _ (totalsubmonoid X) (isbinoptoispartbinop _ _ is) isl (weqabgrdiff X a) raa). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | isirreflabgrdiffrel | 84 |
Lemma isasymmabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : isasymm L) : isasymm (abgrdiffrel X is). Proof. apply (isasymmlogeqf (logeqabgrdiffrels X is)). intros a b rab rba. apply (isasymmabmonoidfracrel _ (totalsubmonoid X) (isbinoptoispartbinop _ _ is) isl (weqabgrdiff X a) (weqabgrdiff X b) rab rba). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | isasymmabgrdiffrel | 85 |
Lemma iscoasymmabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : iscoasymm L) : iscoasymm (abgrdiffrel X is). Proof. apply (iscoasymmlogeqf (logeqabgrdiffrels X is)). intros a b rab. apply (iscoasymmabmonoidfracrel _ (totalsubmonoid X) (isbinoptoispartbinop _ _ is) isl (weqabgrdiff X a) (weqabgrdiff X b) rab). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | iscoasymmabgrdiffrel | 86 |
Lemma istotalabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : istotal L) : istotal (abgrdiffrel X is). Proof. apply (istotallogeqf (logeqabgrdiffrels X is)). intros a b. apply (istotalabmonoidfracrel _ (totalsubmonoid X) (isbinoptoispartbinop _ _ is) isl (weqabgrdiff X a) (weqabgrdiff X b)). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | istotalabgrdiffrel | 87 |
Lemma iscotransabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : iscotrans L) : iscotrans (abgrdiffrel X is). Proof. apply (iscotranslogeqf (logeqabgrdiffrels X is)). intros a b c. apply (iscotransabmonoidfracrel _ (totalsubmonoid X) (isbinoptoispartbinop _ _ is) isl (weqabgrdiff X a) (weqabgrdiff X b) (weqabgrdiff X c)). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | iscotransabgrdiffrel | 88 |
Lemma isStrongOrder_abgrdiff {X : abmonoid} (gt : hrel X) (Hgt : isbinophrel gt) : isStrongOrder gt → isStrongOrder (abgrdiffrel X Hgt). Proof. intros H. repeat split. - apply istransabgrdiffrel, (istrans_isStrongOrder H). - apply iscotransabgrdiffrel, (iscotrans_isStrongOrder H). - apply isirreflabgrdiffrel, (isirrefl_isStrongOrder H). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | isStrongOrder_abgrdiff | 89 |
Definition StrongOrder_abgrdiff {X : abmonoid} (gt : StrongOrder X) (Hgt : isbinophrel gt) : StrongOrder (abgrdiff X) := abgrdiffrel X Hgt,, isStrongOrder_abgrdiff gt Hgt (pr2 gt). | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | StrongOrder_abgrdiff | 90 |
Lemma abgrdiffrelimpl (X : abmonoid) {L L' : hrel X} (is : isbinophrel L) (is' : isbinophrel L') (impl : ∏ x x', L x x' → L' x x') (x x' : abgrdiff X) (ql : abgrdiffrel X is x x') : abgrdiffrel X is' x x'. Proof. generalize ql. refine (quotrelimpl _ _ _ _ _). intros x0 x0'. simpl. apply hinhfun. intro t2. exists (pr1 t2). apply (impl _ _ (pr2 t2)). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrdiffrelimpl | 91 |
Lemma abgrdiffrellogeq (X : abmonoid) {L L' : hrel X} (is : isbinophrel L) (is' : isbinophrel L') (lg : ∏ x x', L x x' <-> L' x x') (x x' : abgrdiff X) : (abgrdiffrel X is x x') <-> (abgrdiffrel X is' x x'). Proof. refine (quotrellogeq _ _ _ _ _). intros x0 x0'. split. - simpl. apply hinhfun. intro t2. exists (pr1 t2). apply (pr1 (lg _ _) (pr2 t2)). - simpl. apply hinhfun. intro t2. exists (pr1 t2). apply (pr2 (lg _ _) (pr2 t2)). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | abgrdiffrellogeq | 92 |
Lemma isbinopabgrdiffrelint (X : abmonoid) {L : hrel X} (is : isbinophrel L) : @isbinophrel (setwithbinopdirprod X X) (abgrdiffrelint X L). Proof. apply (isbinophrellogeqf (logeqabgrdiffrelints X L)). split. - intros a b c lab. apply (pr1 (ispartbinopabmonoidfracrelint _ (totalsubmonoid X) (isbinoptoispartbinop _ _ is)) (abgrdiffphi X a) (abgrdiffphi X b) (abgrdiffphi X c) tt lab). - intros a b c lab. apply (pr2 (ispartbinopabmonoidfracrelint _ (totalsubmonoid X) (isbinoptoispartbinop _ _ is)) (abgrdiffphi X a) (abgrdiffphi X b) (abgrdiffphi X c) tt lab). Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | isbinopabgrdiffrelint | 93 |
Lemma isbinopabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) : @isbinophrel (abgrdiff X) (abgrdiffrel X is). Proof. intros. apply (isbinopquotrel (binopeqrelabgrdiff X) (iscomprelabgrdiffrelint X is)). apply (isbinopabgrdiffrelint X is). Defined. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | isbinopabgrdiffrel | 94 |
Definition isdecabgrdiffrelint (X : abmonoid) {L : hrel X} (is : isinvbinophrel L) (isl : isdecrel L) : isdecrel (abgrdiffrelint X L). Proof. intros xa1 xa2. set (x1 := pr1 xa1). set (a1 := pr2 xa1). set (x2 := pr1 xa2). set (a2 := 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 abgrdiffrelint. apply hinhpr. exists 0. rewrite (runax X _). rewrite (runax X _). apply l. - apply ii2. generalize nl. clear nl. apply negf. unfold abgrdiffrelint. simpl. apply (@hinhuniv _ (make_hProp _ (pr2 (L _ _)))). intro t2l. induction t2l as [ c0a l ]. simpl. apply ((pr2 is) _ _ c0a l). Defined. | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | isdecabgrdiffrelint | 95 |
Definition isdecabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isi : isinvbinophrel L) (isl : isdecrel L) : isdecrel (abgrdiffrel X is). Proof. refine (isdecquotrel _ _). apply isdecabgrdiffrelint. - apply isi. - apply isl. Defined. | Definition | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | isdecabgrdiffrel | 96 |
Lemma iscomptoabgrdiff (X : abmonoid) {L : hrel X} (is : isbinophrel L) : iscomprelrelfun L (abgrdiffrel X is) (toabgrdiff X). Proof. unfold iscomprelrelfun. intros x x' l. change (abgrdiffrelint X L (x ,, 0) (x' ,, 0)). simpl. apply (hinhpr). exists (unel X). apply ((pr2 is) _ _ 0). apply ((pr2 is) _ _ 0). apply l. Qed. | Lemma | Algebra | Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes. | Algebra\AbelianGroups.v | iscomptoabgrdiff | 97 |
Definition abmonoid : UU := ∑ (X : setwithbinop), isabmonoidop (@op X). | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | abmonoid | 98 |
Definition make_abmonoid (t : setwithbinop) (H : isabmonoidop (@op t)) : abmonoid := t ,, H. | Definition | Algebra | Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.Algebra.Monoids2. | Algebra\AbelianMonoids.v | make_abmonoid | 99 |
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UniMath Dataset
Dataset Description
The UniMath Dataset is derived from the UniMath repository, focusing on the formalization of Univalent Mathematics in the Coq proof assistant. This dataset processes .v files from the core mathematical libraries to extract mathematical content in a structured format. This work builds upon the format established by Andreas Florath (@florath) in his Coq Facts, Propositions and Proofs dataset, providing a more focused and structured view of the UniMath library specifically.
Dataset Structure
The dataset includes the following fields:
- fact: The mathematical statement body
- type: The statement type (Definition/Lemma/Theorem/etc.)
- library: The originating library (Algebra/CategoryTheory/etc.)
- imports: The Require Import statements from the source file
- filename: The source file path within UniMath
- symbolic_name: The identifier of the mathematical object
- index_level_0: Sequential index for the dataset
Example Row
fact: "(X : abmonoid) {L : hrel X} (is : isbinophrel L) (isi : isinvbinophrel L) (isl : isdecrel L) : isdecrel (abgrdiffrel X is)" type: "Definition" library: "Algebra" imports: "Require Import UniMath.MoreFoundations.Orders. Require Import UniMath.MoreFoundations.Subtypes." filename: "Algebra/AbelianGroups.v" symbolic_name: "isdecabgrdiffrel" index_level_0: 55
Source Code
The dataset was generated using a custom Python script that processes core UniMath libraries including Algebra, CategoryTheory, Foundations, and others. The extraction focuses on mathematical content while preserving the structure and relationships between definitions, lemmas, and their source files.
Coverage
The dataset includes content from the following UniMath libraries:
- Algebra
- Bicategories
- CategoryTheory
- Combinatorics
- Foundations
- HomologicalAlgebra
- Ktheory
- MoreFoundations
- NumberSystems
- OrderTheory
- PAdics
- RealNumbers
- SubstitutionSystems
- Topology
Usage
This dataset is designed for:
- Formal Methods Research: Analyzing formal proofs and definitions in Univalent Mathematics
- Machine Learning Applications: Training models on formal verification, code completion, and theorem proving tasks
- Educational Purposes: Providing structured examples of UniMath formalizations
- Mathematical Analysis: Studying the structure and patterns in formalized mathematical content
License
This dataset is distributed under the BSD 2-clause license, aligning with the license of the original UniMath repository.
Acknowledgments
- Original repository: UniMath (https://github.com/UniMath/UniMath)
- Inspiration: Hugging Face user Andreas Florath (@florath) and his comprehensive Coq dataset
- Downloads last month
- 26
Size of downloaded dataset files:
4.43 MB
Size of the auto-converted Parquet files:
4.43 MB
Number of rows:
38,661