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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 |