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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

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

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