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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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