Proof Assistant Projects
Collection
Digesting proof assistant libraries for AI ingestion. • 84 items • Updated
• 3
Datasets:
fact stringlengths 4 3.31k | type stringclasses 14 values | library stringclasses 23 values | imports listlengths 1 59 | filename stringlengths 20 105 | symbolic_name stringlengths 1 89 | docstring stringlengths 0 1.75k ⌀ |
|---|---|---|---|---|---|---|
abgr : UU := abelian_group_category. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgr | null |
make_abgr (X : setwithbinop) (is : isabgrop (@op X)) : abgr :=
X ,, is. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | make_abgr | null |
abgrconstr (X : abmonoid) (inv0 : X → X) (is : isinv (@op X) 0 inv0) : abgr. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrconstr | null |
abgrtogr : abgr → gr := λ X, make_gr (pr1 X) (pr1 (pr2 X)). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrtogr | null |
abgrtoabmonoid : abgr → abmonoid :=
λ X, make_abmonoid (pr1 X) (pr1 (pr1 (pr2 X)) ,, pr2 (pr2 X)). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrtoabmonoid | null |
abgr_of_gr (X : gr) (H : iscomm (@op X)) : abgr :=
make_abgr X (make_isabgrop (pr2 X) H). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgr_of_gr | null |
abelian_group_morphism
(X Y : abgr)
: UU
:= abelian_group_category⟦X, Y⟧%cat. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abelian_group_morphism | null |
abelian_group_morphism_to_group_morphism
{X Y : abgr}
(f : abelian_group_morphism X Y)
: group_morphism X Y
:= pr1 f ,, pr12 f. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abelian_group_morphism_to_group_morphism | null |
abelian_group_to_monoid_morphism
{X Y : abgr}
(f : abelian_group_morphism X Y)
: abelian_monoid_morphism X Y. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abelian_group_to_monoid_morphism | null |
make_abelian_group_morphism
{X Y : abgr}
(f : X → Y)
(H : isbinopfun f)
: abelian_group_morphism X Y
:= (f ,, H) ,, (((tt ,, binopfun_preserves_unit f H) ,, binopfun_preserves_inv f H) ,, tt). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | make_abelian_group_morphism | null |
binopfun_to_abelian_group_morphism
{X Y : abgr}
(f : binopfun X Y)
: abelian_group_morphism X Y
:= make_abelian_group_morphism f (binopfunisbinopfun f). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | binopfun_to_abelian_group_morphism | null |
abelian_group_morphism_paths
{X Y : abgr}
(f g : abelian_group_morphism X Y)
(H : (f : X → Y) = g)
: f = g. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abelian_group_morphism_paths | null |
abelian_group_morphism_eq
{X Y : abgr}
{f g : abelian_group_morphism X Y}
: (f = g) ≃ (∏ x, f x = g x). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abelian_group_morphism_eq | null |
identity_abelian_group_morphism
(X : abgr)
: abelian_group_morphism X X
:= identity X. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | identity_abelian_group_morphism | null |
composite_abelian_group_morphism
{X Y Z : abgr}
(f : abelian_group_morphism X Y)
(g : abelian_group_morphism Y Z)
: abelian_group_morphism X Z
:= (f · g)%cat. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | composite_abelian_group_morphism | null |
unitabgr_isabgrop : isabgrop (@op unitabmonoid). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | unitabgr_isabgrop | *** Construction of the trivial abgr consisting of one element given by unit. |
unitabgr : abgr := make_abgr unitabmonoid unitabgr_isabgrop. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | unitabgr | null |
unel_abelian_group_morphism (X Y : abgr) : abelian_group_morphism X Y :=
binopfun_to_abelian_group_morphism (unelmonoidfun X Y). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | unel_abelian_group_morphism | null |
abgrshombinop
{X Y : abgr} (f g : abelian_group_morphism X Y)
: abelian_group_morphism X Y. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrshombinop | *** Abelian group structure on morphism between abelian groups |
abgrshombinop_inv_isbinopfun {X Y : abgr} (f : abelian_group_morphism X Y) :
isbinopfun (λ x : X, grinv Y (f x)). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrshombinop_inv_isbinopfun | null |
abgrshombinop_inv {X Y : abgr} (f : abelian_group_morphism X Y) : abelian_group_morphism X Y :=
make_abelian_group_morphism _ (abgrshombinop_inv_isbinopfun f). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrshombinop_inv | null |
abgrshombinop_linvax {X Y : abgr} (f : abelian_group_morphism X Y) :
abgrshombinop (abgrshombinop_inv f) f = unel_abelian_group_morphism X Y. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrshombinop_linvax | null |
abgrshombinop_rinvax {X Y : abgr} (f : abelian_group_morphism X Y) :
abgrshombinop f (abgrshombinop_inv f) = unel_abelian_group_morphism X Y. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrshombinop_rinvax | null |
abgrshomabgr_isabgrop (X Y : abgr) :
isabgrop (abgrshombinop (X := X) (Y := Y)). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrshomabgr_isabgrop | null |
abgrshomabgr (X Y : abgr) : abgr. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrshomabgr | null |
subabgr (X : abgr) := subgr X. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | subabgr | * 2. Subobjects |
isabgrcarrier {X : abgr} (A : subgr X) : isabgrop (@op A). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | isabgrcarrier | null |
carrierofasubabgr {X : abgr} (A : subabgr X) : abgr. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | carrierofasubabgr | null |
subabgr_incl {X : abgr} (A : subabgr X) : abelian_group_morphism A X :=
binopfun_to_abelian_group_morphism (X := A) (submonoid_incl A). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | subabgr_incl | null |
abgr_kernel_hsubtype {A B : abgr} (f : abelian_group_morphism A B) : hsubtype A :=
monoid_kernel_hsubtype f. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgr_kernel_hsubtype | null |
abgr_image_hsubtype {A B : abgr} (f : abelian_group_morphism A B) : hsubtype B :=
(λ y : B, ∃ x : A, (f x) = y). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgr_image_hsubtype | null |
f : X → Y be a morphism of abelian groups. | Let | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | f | null |
abgr_Kernel_subabgr_issubgr {A B : abgr} (f : abelian_group_morphism A B) :
issubgr (abgr_kernel_hsubtype f). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgr_Kernel_subabgr_issubgr | ** Kernel as abelian group |
abgr_Kernel_subabgr {A B : abgr} (f : abelian_group_morphism A B) : @subabgr A :=
subgrconstr (@abgr_kernel_hsubtype A B f) (abgr_Kernel_subabgr_issubgr f). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgr_Kernel_subabgr | null |
abgr_Kernel_abelian_group_morphism_isbinopfun {A B : abgr} (f : abelian_group_morphism A B) :
isbinopfun (X := abgr_Kernel_subabgr f)
(make_incl (pr1carrier (abgr_kernel_hsubtype f))
(isinclpr1carrier (abgr_kernel_hsubtype f))). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgr_Kernel_abelian_group_morphism_isbinopfun | ** The inclusion Kernel f --> X is a morphism of abelian groups |
abgr_image_issubgr {A B : abgr} (f : abelian_group_morphism A B) : issubgr (abgr_image_hsubtype f). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgr_image_issubgr | ** Image of f is a subgroup |
abgr_image {A B : abgr} (f : abelian_group_morphism A B) : @subabgr B :=
@subgrconstr B (@abgr_image_hsubtype A B f) (abgr_image_issubgr f). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgr_image | null |
isabgrquot {X : abgr} (R : binopeqrel X) : isabgrop (@op (setwithbinopquot R)). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | isabgrquot | * 4. Quotient objects |
abgrquot {X : abgr} (R : binopeqrel X) : abgr. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrquot | null |
isabgrdirprod (X Y : abgr) : isabgrop (@op (setwithbinopdirprod X Y)). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | isabgrdirprod | * 5. Direct products |
abgrdirprod (X Y : abgr) : abgr. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrdirprod | null |
hrelabgrdiff (X : abmonoid) : hrel (X × X) :=
λ xa1 xa2, ∃ (x0 : X), (pr1 xa1 + pr2 xa2) + x0 = (pr1 xa2 + pr2 xa1) + x0. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | hrelabgrdiff | null |
abgrdiffphi (X : abmonoid) (xa : X × X) :
X × (totalsubtype X) := pr1 xa ,, make_carrier (λ x : X, htrue) (pr2 xa) tt. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrdiffphi | null |
hrelabgrdiff' (X : abmonoid) : hrel (X × X) :=
λ xa1 xa2, eqrelabmonoidfrac X (totalsubmonoid X) (abgrdiffphi X xa1) (abgrdiffphi X xa2). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | hrelabgrdiff | null |
logeqhrelsabgrdiff (X : abmonoid) : hrellogeq (hrelabgrdiff' X) (hrelabgrdiff X). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | logeqhrelsabgrdiff | null |
iseqrelabgrdiff (X : abmonoid) : iseqrel (hrelabgrdiff X). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | iseqrelabgrdiff | null |
eqrelabgrdiff (X : abmonoid) : @eqrel (abmonoiddirprod X X) :=
make_eqrel _ (iseqrelabgrdiff X). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | eqrelabgrdiff | null |
isbinophrelabgrdiff (X : abmonoid) : @isbinophrel (abmonoiddirprod X X) (hrelabgrdiff X). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | isbinophrelabgrdiff | null |
binopeqrelabgrdiff (X : abmonoid) : binopeqrel (abmonoiddirprod X X) :=
make_binopeqrel (eqrelabgrdiff X) (isbinophrelabgrdiff X). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | binopeqrelabgrdiff | null |
abgrdiffcarrier (X : abmonoid) : abmonoid := @abmonoidquot (abmonoiddirprod X X)
(binopeqrelabgrdiff X). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrdiffcarrier | null |
abgrdiffinvint (X : abmonoid) : X × X → X × X :=
λ xs, pr2 xs ,, pr1 xs. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrdiffinvint | null |
abgrdiffinvcomp (X : abmonoid) :
iscomprelrelfun (hrelabgrdiff X) (eqrelabgrdiff X) (abgrdiffinvint X). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrdiffinvcomp | null |
abgrdiffinv (X : abmonoid) : abgrdiffcarrier X → abgrdiffcarrier X :=
setquotfun (hrelabgrdiff X) (eqrelabgrdiff X) (abgrdiffinvint X) (abgrdiffinvcomp X). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrdiffinv | null |
abgrdiffisinv (X : abmonoid) :
isinv (@op (abgrdiffcarrier X)) 0 (abgrdiffinv X). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrdiffisinv | null |
abgrdiff (X : abmonoid) : abgr
:= abgrconstr (abgrdiffcarrier X) (abgrdiffinv X) (abgrdiffisinv X). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrdiff | null |
prabgrdiff (X : abmonoid) : X → X → abgrdiff X :=
λ x x' : X, setquotpr (eqrelabgrdiff X) (x ,, x'). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | prabgrdiff | null |
weqabgrdiffint (X : abmonoid) : weq (X × X) (X × totalsubtype X) :=
weqdirprodf (idweq X) (invweq (weqtotalsubtype X)). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | weqabgrdiffint | * 7. Abelian group of fractions and abelian monoid of fractions |
weqabgrdiff (X : abmonoid) : weq (abgrdiff X) (abmonoidfrac X (totalsubmonoid X)). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | weqabgrdiff | null |
toabgrdiff (X : abmonoid) (x : X) : abgrdiff X := setquotpr _ (x ,, 0). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | toabgrdiff | * 8. Canonical homomorphism to the abelian group of fractions |
isbinopfuntoabgrdiff (X : abmonoid) : isbinopfun (toabgrdiff X). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | isbinopfuntoabgrdiff | null |
isinclprabgrdiff (X : abmonoid) (iscanc : ∏ x : X, isrcancelable (@op X) x) :
∏ x' : X, isincl (λ x, prabgrdiff X x x'). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | isinclprabgrdiff | * 9. Abelian group of fractions in the case when all elements are cancelable |
isincltoabgrdiff (X : abmonoid) (iscanc : ∏ x : X, isrcancelable (@op X) x) :
isincl (toabgrdiff X) := isinclprabgrdiff X iscanc 0. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | isincltoabgrdiff | null |
isdeceqabgrdiff (X : abmonoid) (iscanc : ∏ x : X, isrcancelable (@op X) x) (is : isdeceq X) :
isdeceq (abgrdiff X). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | isdeceqabgrdiff | null |
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 | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrdiffrelint | * 10. Relations on the abelian group of fractions |
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 | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrdiffrelint | null |
logeqabgrdiffrelints (X : abmonoid) (L : hrel X) :
hrellogeq (abgrdiffrelint' X L) (abgrdiffrelint X L). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | logeqabgrdiffrelints | null |
iscomprelabgrdiffrelint (X : abmonoid) {L : hrel X} (is : isbinophrel L) :
iscomprelrel (eqrelabgrdiff X) (abgrdiffrelint X L). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | iscomprelabgrdiffrelint | null |
abgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) :=
quotrel (iscomprelabgrdiffrelint X is). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrdiffrel | null |
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 | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrdiffrel | null |
logeqabgrdiffrels (X : abmonoid) {L : hrel X} (is : isbinophrel L) :
hrellogeq (abgrdiffrel' X is) (abgrdiffrel X is). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | logeqabgrdiffrels | null |
istransabgrdiffrelint (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : istrans L) :
istrans (abgrdiffrelint X L). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | istransabgrdiffrelint | null |
istransabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : istrans L) :
istrans (abgrdiffrel X is). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | istransabgrdiffrel | null |
issymmabgrdiffrelint (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : issymm L) :
issymm (abgrdiffrelint X L). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | issymmabgrdiffrelint | null |
issymmabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : issymm L) :
issymm (abgrdiffrel X is). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | issymmabgrdiffrel | null |
isreflabgrdiffrelint (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : isrefl L) :
isrefl (abgrdiffrelint X L). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | isreflabgrdiffrelint | null |
isreflabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : isrefl L) :
isrefl (abgrdiffrel X is). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | isreflabgrdiffrel | null |
ispoabgrdiffrelint (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : ispreorder L) :
ispreorder (abgrdiffrelint X L). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | ispoabgrdiffrelint | null |
ispoabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : ispreorder L) :
ispreorder (abgrdiffrel X is). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | ispoabgrdiffrel | null |
iseqrelabgrdiffrelint (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : iseqrel L) :
iseqrel (abgrdiffrelint X L). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | iseqrelabgrdiffrelint | null |
iseqrelabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : iseqrel L) :
iseqrel (abgrdiffrel X is). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | iseqrelabgrdiffrel | null |
isantisymmnegabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L)
(isl : isantisymmneg L) : isantisymmneg (abgrdiffrel X is). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | isantisymmnegabgrdiffrel | null |
isantisymmabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : isantisymm L) :
isantisymm (abgrdiffrel X is). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | isantisymmabgrdiffrel | null |
isirreflabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : isirrefl L) :
isirrefl (abgrdiffrel X is). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | isirreflabgrdiffrel | null |
isasymmabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : isasymm L) :
isasymm (abgrdiffrel X is). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | isasymmabgrdiffrel | null |
iscoasymmabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : iscoasymm L) :
iscoasymm (abgrdiffrel X is). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | iscoasymmabgrdiffrel | null |
istotalabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : istotal L) :
istotal (abgrdiffrel X is). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | istotalabgrdiffrel | null |
iscotransabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) (isl : iscotrans L) :
iscotrans (abgrdiffrel X is). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | iscotransabgrdiffrel | null |
isStrongOrder_abgrdiff {X : abmonoid} (gt : hrel X)
(Hgt : isbinophrel gt) :
isStrongOrder gt → isStrongOrder (abgrdiffrel X Hgt). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | isStrongOrder_abgrdiff | null |
StrongOrder_abgrdiff {X : abmonoid} (gt : StrongOrder X)
(Hgt : isbinophrel gt) : StrongOrder (abgrdiff X) :=
abgrdiffrel X Hgt,, isStrongOrder_abgrdiff gt Hgt (pr2 gt). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | StrongOrder_abgrdiff | null |
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'. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrdiffrelimpl | null |
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'). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | abgrdiffrellogeq | null |
isbinopabgrdiffrelint (X : abmonoid) {L : hrel X} (is : isbinophrel L) :
@isbinophrel (setwithbinopdirprod X X) (abgrdiffrelint X L). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | isbinopabgrdiffrelint | null |
isbinopabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L) :
@isbinophrel (abgrdiff X) (abgrdiffrel X is). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | isbinopabgrdiffrel | null |
isdecabgrdiffrelint (X : abmonoid) {L : hrel X}
(is : isinvbinophrel L) (isl : isdecrel L) : isdecrel (abgrdiffrelint X L). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | isdecabgrdiffrelint | null |
isdecabgrdiffrel (X : abmonoid) {L : hrel X} (is : isbinophrel L)
(isi : isinvbinophrel L) (isl : isdecrel L) : isdecrel (abgrdiffrel X is). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | isdecabgrdiffrel | null |
iscomptoabgrdiff (X : abmonoid) {L : hrel X} (is : isbinophrel L) :
iscomprelrelfun L (abgrdiffrel X is) (toabgrdiff X). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianGroups.v | iscomptoabgrdiff | * 11. Relations and the canonical homomorphism to [abgrdiff] |
abmonoid : UU := abelian_monoid_category. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianMonoids.v | abmonoid | null |
make_abmonoid (t : setwithbinop) (H : isabmonoidop (@op t))
: abmonoid
:= t ,, H. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianMonoids.v | make_abmonoid | null |
abmonoidtomonoid : abmonoid → monoid :=
λ X, make_monoid (pr1 X) (pr1 (pr2 X)). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianMonoids.v | abmonoidtomonoid | null |
commax (X : abmonoid) : iscomm (@op X) := pr2 (pr2 X). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/AbelianMonoids.v | commax | null |
End of preview. Expand
in Data Studio
Coq-UniMath
Structured dataset of formalizations from the UniMath library (Univalent Mathematics in Coq).
Schema
| Column | Type | Description |
|---|---|---|
| fact | string | Full declaration (name, signature, body) |
| type | string | Definition, Lemma, Theorem, Proposition, etc. |
| library | string | Sub-library (Algebra, CategoryTheory, Foundations, etc.) |
| imports | list | Require Import statements |
| filename | string | Source file path |
| symbolic_name | string | Declaration identifier |
| docstring | string | Documentation comment (13% coverage) |
Statistics
By Type
| Type | Count |
|---|---|
| Definition | 30,272 |
| Lemma | 10,231 |
| Proposition | 3,807 |
| Let | 2,878 |
| Theorem | 359 |
| Notation | 233 |
| Ltac | 229 |
| Corollary | 181 |
| Other | 50 |
Docstring Coverage
- 6,648 entries (13%) have documentation
Example Entry
Use Cases
- Retrieval / RAG for Coq theorem proving
- Univalent mathematics research
- Training embeddings for formal mathematics
- Documentation generation
Citation
Changelog
- v2.0 (Jan 2025): Re-extraction with improved schema
- 38,661 -> 48,240 entries (+25%)
- Added docstring column (13% coverage)
- Changed imports from string to list
- Removed index_level column
- fact now includes full signature
- v1.0 (Dec 2024): Initial release
- Downloads last month
- 13
Source:
UniMath/UniMath
Entries:
48,240
Files processed:
1,557
License:
UniMath License
Size of downloaded dataset files:
2.92 MB
Size of the auto-converted Parquet files:
2.92 MB
Number of rows:
48,240