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 |
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