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 ⌀ |
|---|---|---|---|---|---|---|
tightap (X : UU) := ∑ ap : hrel X, istightap ap. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | tightap | null |
tightap_aprel {X : UU} (ap : tightap X) : aprel X := pr1 ap ,, (pr1 (pr2 ap)). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | tightap_aprel | null |
tightapSet := ∑ X : hSet, tightap X. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | tightapSet | null |
tightapSet_apSet (X : tightapSet) : apSet := pr1 X ,, (tightap_aprel (pr2 X)). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | tightapSet_apSet | null |
tightapSet_rel (X : tightapSet) : hrel X := (pr1 (pr2 X)). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | tightapSet_rel | null |
isirrefltightapSet {X : tightapSet} :
∏ x : X, ¬ (x ≠ x). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | isirrefltightapSet | Some lemmas |
issymmtightapSet {X : tightapSet} :
∏ x y : X, x ≠ y → y ≠ x. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | issymmtightapSet | null |
iscotranstightapSet {X : tightapSet} :
∏ x y z : X, x ≠ z → x ≠ y ∨ y ≠ z. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | iscotranstightapSet | null |
istighttightapSet {X : tightapSet} :
∏ x y : X, ¬ (x ≠ y) → x = y. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | istighttightapSet | null |
istighttightapSet_rev {X : tightapSet} :
∏ x y : X, x = y → ¬ (x ≠ y). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | istighttightapSet_rev | null |
tightapSet_dec {X : tightapSet} :
LEM → ∏ x y : X, (x != y <-> x ≠ y). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | tightapSet_dec | null |
isapunop {X : tightapSet} (op :unop X) :=
∏ x y : X, op x ≠ op y → x ≠ y. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | isapunop | ** Operations and apartness |
isaprop_isapunop {X : tightapSet} (op :unop X) :
isaprop (isapunop op). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | isaprop_isapunop | null |
islapbinop {X : tightapSet} (op : binop X) :=
∏ x, isapunop (λ y, op y x). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | islapbinop | null |
israpbinop {X : tightapSet} (op : binop X) :=
∏ x, isapunop (λ y, op x y). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | israpbinop | null |
isapbinop {X : tightapSet} (op : binop X) :=
(islapbinop op) × (israpbinop op). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | isapbinop | null |
isaprop_islapbinop {X : tightapSet} (op : binop X) :
isaprop (islapbinop op). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | isaprop_islapbinop | null |
isaprop_israpbinop {X : tightapSet} (op : binop X) :
isaprop (israpbinop op). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | isaprop_israpbinop | null |
isaprop_isapbinop {X : tightapSet} (op :binop X) :
isaprop (isapbinop op). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | isaprop_isapbinop | null |
apbinop (X : tightapSet) := ∑ op : binop X, isapbinop op. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | apbinop | null |
apbinop_pr1 {X : tightapSet} (op : apbinop X) : binop X := pr1 op. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | apbinop_pr1 | null |
apsetwithbinop := ∑ X : tightapSet, apbinop X. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | apsetwithbinop | null |
apsetwithbinop_pr1 (X : apsetwithbinop) : tightapSet := pr1 X. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | apsetwithbinop_pr1 | null |
apsetwithbinop_setwithbinop : apsetwithbinop → setwithbinop :=
λ X : apsetwithbinop, (apSet_pr1 (apsetwithbinop_pr1 X)),, (pr1 (pr2 X)). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | apsetwithbinop_setwithbinop | null |
op {X : apsetwithbinop} : binop X := op (X := apsetwithbinop_setwithbinop X). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | op | null |
apsetwith2binop := ∑ X : tightapSet, apbinop X × apbinop X. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | apsetwith2binop | null |
apsetwith2binop_pr1 (X : apsetwith2binop) : tightapSet := pr1 X. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | apsetwith2binop_pr1 | null |
apsetwith2binop_setwith2binop : apsetwith2binop → setwith2binop :=
λ X : apsetwith2binop,
apSet_pr1 (apsetwith2binop_pr1 X),, pr1 (pr1 (pr2 X)),, pr1 (pr2 (pr2 X)). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | apsetwith2binop_setwith2binop | null |
op1 {X : apsetwith2binop} : binop X := op1 (X := apsetwith2binop_setwith2binop X). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | op1 | null |
op2 {X : apsetwith2binop} : binop X := op2 (X := apsetwith2binop_setwith2binop X). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | op2 | null |
islapbinop_op :
∏ x x' y : X, op x y ≠ op x' y → x ≠ x'. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | islapbinop_op | null |
israpbinop_op :
∏ x y y' : X, op x y ≠ op x y' → y ≠ y'. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | israpbinop_op | null |
isapbinop_op :
∏ x x' y y' : X, op x y ≠ op x' y' → x ≠ x' ∨ y ≠ y'. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | isapbinop_op | null |
apsetwith2binop_apsetwithbinop1 : apsetwithbinop :=
(pr1 X) ,, (pr1 (pr2 X)). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | apsetwith2binop_apsetwithbinop1 | null |
apsetwith2binop_apsetwithbinop2 : apsetwithbinop :=
(pr1 X) ,, (pr2 (pr2 X)). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | apsetwith2binop_apsetwithbinop2 | null |
islapbinop_op1 :
∏ x x' y : X, op1 x y ≠ op1 x' y → x ≠ x'. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | islapbinop_op1 | null |
israpbinop_op1 :
∏ x y y' : X, op1 x y ≠ op1 x y' → y ≠ y'. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | israpbinop_op1 | null |
isapbinop_op1 :
∏ x x' y y' : X, op1 x y ≠ op1 x' y' → x ≠ x' ∨ y ≠ y'. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | isapbinop_op1 | null |
islapbinop_op2 :
∏ x x' y : X, op2 x y ≠ op2 x' y → x ≠ x'. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | islapbinop_op2 | null |
israpbinop_op2 :
∏ x y y' : X, op2 x y ≠ op2 x y' → y ≠ y'. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | israpbinop_op2 | null |
isapbinop_op2 :
∏ x x' y y' : X, op2 x y ≠ op2 x' y' → x ≠ x' ∨ y ≠ y'. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Apartness.v | isapbinop_op2 | null |
natmult {X : monoid} (n : nat) (x : X) : X :=
match n with
| O => 0%addmonoid
| S O => x
| S m => (x + natmult m x)%addmonoid
end. | Fixpoint | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | natmult | ** The standard function from the natural numbers to a monoid |
nattorig {X : rig} (n : nat) : X :=
natmult (X := rigaddabmonoid X) n 1%rig. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | nattorig | null |
nattoring {X : ring} (n : nat) : X :=
nattorig (X := ringtorig X) n. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | nattoring | null |
natmultS :
∏ {X : monoid} (n : nat) (x : X),
natmult (S n) x = (x + natmult n x)%addmonoid. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | natmultS | null |
nattorigS {X : rig} :
∏ (n : nat), nattorig (X := X) (S n) = (1 + nattorig n)%rig. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | nattorigS | null |
nattorig_natmult :
∏ {X : rig} (n : nat) (x : X),
(nattorig n * x)%rig = natmult (X := rigaddabmonoid X) n x. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | nattorig_natmult | null |
natmult_plus :
∏ {X : monoid} (n m : nat) (x : X),
natmult (n + m) x = (natmult n x + natmult m x)%addmonoid. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | natmult_plus | null |
nattorig_plus :
∏ {X : rig} (n m : nat),
(nattorig (n + m) : X) = (nattorig n + nattorig m)%rig. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | nattorig_plus | null |
natmult_mult :
∏ {X : monoid} (n m : nat) (x : X),
natmult (n * m) x = (natmult n (natmult m x))%addmonoid. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | natmult_mult | null |
nattorig_mult :
∏ {X : rig} (n m : nat),
(nattorig (n * m) : X) = (nattorig n * nattorig m)%rig. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | nattorig_mult | null |
natmult_op {X : monoid} :
∏ (n : nat) (x y : X),
(x + y = y + x)%addmonoid
→ natmult n (x + y)%addmonoid = (natmult n x + natmult n y)%addmonoid. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | natmult_op | null |
natmult_binophrel {X : monoid} (R : hrel X) :
istrans R → isbinophrel R →
∏ (n : nat) (x y : X), R x y → R (natmult (S n) x) (natmult (S n) y). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | natmult_binophrel | null |
setquot_aux {X : monoid} (R : hrel X) : hrel X :=
λ x y : X, ∃ c : X, R (x + c)%addmonoid (y + c)%addmonoid. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | setquot_aux | ** relation |
istrans_setquot_aux {X : abmonoid} (R : hrel X) :
istrans R → isbinophrel R → istrans (setquot_aux R). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | istrans_setquot_aux | null |
isbinophrel_setquot_aux {X : abmonoid} (R : hrel X) :
isbinophrel R → isbinophrel (setquot_aux R). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isbinophrel_setquot_aux | null |
isequiv_setquot_aux {X : abmonoid} (R : hrel X) :
isinvbinophrel R →
∏ x y : X, (setquot_aux R) x y <-> R x y. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isequiv_setquot_aux | null |
isarchmonoid {X : abmonoid} (R : hrel X) :=
∏ x y1 y2 : X,
R y1 y2 →
(∃ n : nat, R (natmult n y1 + x)%addmonoid (natmult n y2))
× (∃ n : nat, R (natmult n y1) (natmult n y2 + x)%addmonoid). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchmonoid | ** Archimedean property in a monoid |
isarchmonoid_1 {X : abmonoid} (R : hrel X) :
isarchmonoid R →
∏ x y1 y2 : X,
R y1 y2 →
∃ n : nat, R (natmult n y1 + x)%addmonoid (natmult n y2) :=
λ H x y1 y2 Hy, (pr1 (H x y1 y2 Hy)). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchmonoid_1 | null |
isarchmonoid_2 {X : abmonoid} (R : hrel X) :
isarchmonoid R →
∏ x y1 y2 : X,
R y1 y2 →
∃ n : nat, R (natmult n y1) (natmult n y2 + x)%addmonoid :=
λ H x y1 y2 Hy, (pr2 (H x y1 y2 Hy)). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchmonoid_2 | null |
isarchgr {X : abgr} (R : hrel X) :=
∏ x y1 y2 : X,
R y1 y2 →
∃ n : nat, R (natmult n y1 + x)%addmonoid (natmult n y2). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchgr | ** Archimedean property in a group |
isarchgr_isarchmonoid_aux {X : abgr} (R : hrel X) :
isbinophrel R →
∏ (n : nat) (x y1 y2 : X),
R (natmult n y1 * grinv X x)%multmonoid (natmult n y2) → R (natmult n y1) (natmult n y2 * x)%multmonoid. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchgr_isarchmonoid_aux | null |
isarchgr_isarchmonoid {X : abgr} (R : hrel X) :
isbinophrel R →
isarchgr R →
isarchmonoid (X := abgrtoabmonoid X) R. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchgr_isarchmonoid | null |
isarchmonoid_isarchgr {X : abgr} (R : hrel X) :
isarchmonoid (X := abgrtoabmonoid X) R →
isarchgr R. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchmonoid_isarchgr | null |
isarchabgrdiff_aux {X : abmonoid} (R : hrel X)
(Hr : isbinophrel R)
(Hr' : istrans R)
(y1 y2 x : abmonoiddirprod X X)
(n1 : nat)
(Hn1 : setquot_aux R (natmult n1 (pr1 y1 * pr2 y2) * pr1 x)%multmonoid
(natmult n1 (pr1 y2 * pr2 y1)%multmonoid))
(n2 : nat)
... | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchabgrdiff_aux | null |
isarchabgrdiff {X : abmonoid} (R : hrel X) (Hr : isbinophrel R) :
istrans R →
isarchmonoid (setquot_aux R) →
isarchgr (X := abgrdiff X) (abgrdiffrel X (L := R) Hr). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchabgrdiff | null |
isarchrig {X : rig} (R : hrel X) :=
(∏ y1 y2 : X, R y1 y2 → ∃ n : nat, R (nattorig n * y1)%rig (1 + nattorig n * y2)%rig)
× (∏ x : X, ∃ n : nat, R (nattorig n) x)
× (∏ x : X, ∃ n : nat, R (nattorig n + x)%rig 0%rig). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchrig | ** Archimedean property in a rig |
isarchrig_diff {X : rig} (R : hrel X) :
isarchrig R →
∏ y1 y2 : X, R y1 y2 → ∃ n : nat, R (nattorig n * y1)%rig (1 + nattorig n * y2)%rig :=
pr1. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchrig_diff | null |
isarchrig_gt {X : rig} (R : hrel X) :
isarchrig R →
∏ x : X, ∃ n : nat, R (nattorig n) x :=
λ H, (pr1 (pr2 H)). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchrig_gt | null |
isarchrig_pos {X : rig} (R : hrel X) :
isarchrig R →
∏ x : X, ∃ n : nat, R (nattorig n + x)%rig 0%rig :=
λ H, (pr2 (pr2 H)). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchrig_pos | null |
isarchrig_setquot_aux {X : rig} (R : hrel X) :
isinvbinophrel (X := rigaddabmonoid X) R
→ isarchrig R
→ isarchrig (setquot_aux (X := rigaddabmonoid X) R). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchrig_setquot_aux | null |
isarchrig_isarchmonoid_1_aux {X : rig} (R : hrel X)
(Hr1 : R 1%rig 0%rig)
(Hr : istrans R)
(Hop1 : isbinophrel (X := rigaddabmonoid X) R)
(x y1 y2 : rigaddabmonoid X)
(m : nat)
(Hm : R (nattorig m * y1)%ring (1%rig + nattorig m * y2)%ring)
(n : nat)
(Hn : R (nattorig n + ... | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchrig_isarchmonoid_1_aux | null |
isarchrig_isarchmonoid_2_aux {X : rig} (R : hrel X)
(Hr1 : R 1%rig 0%rig)
(Hr : istrans R)
(Hop1 : isbinophrel (X := rigaddabmonoid X) R)
(x y1 y2 : rigaddabmonoid X)
(m : nat)
(Hm : R (nattorig m * y1)%ring (1%rig + nattorig m * y2)%ring)
(n : nat)
(Hn : R (nattorig n) x... | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchrig_isarchmonoid_2_aux | null |
isarchrig_isarchmonoid {X : rig} (R : hrel X) :
R 1%rig 0%rig →
istrans R → isbinophrel (X := rigaddabmonoid X) R →
isarchrig R → isarchmonoid (X := rigaddabmonoid X) R. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchrig_isarchmonoid | null |
isarchmonoid_isarchrig {X : rig} (R : hrel X) :
(R 1%rig 0%rig)
→ isarchmonoid (X := rigaddabmonoid X) R
→ isarchrig R. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchmonoid_isarchrig | null |
isarchring {X : ring} (R : hrel X) :=
(∏ x : X, R x 0%ring → ∃ n : nat, R (nattoring n * x)%ring 1%ring)
× (∏ x : X, ∃ n : nat, R (nattoring n) x). | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchring | ** Archimedean property in a ring |
isarchring_1 {X : ring} (R : hrel X) :
isarchring R →
∏ x : X, R x 0%ring → ∃ n : nat, R (nattoring n * x)%ring 1%ring := pr1. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchring_1 | null |
isarchring_2 {X : ring} (R : hrel X) :
isarchring R →
∏ x : X, ∃ n : nat, R (nattoring n) x := pr2. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchring_2 | null |
isarchring_isarchrig {X : ring} (R : hrel X) :
isbinophrel (X := rigaddabmonoid X) R →
isarchring R → isarchrig (X := ringtorig X) R. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchring_isarchrig | null |
isarchrig_isarchring {X : ring} (R : hrel X) :
isbinophrel (X := rigaddabmonoid X) R →
isarchrig (X := ringtorig X) R → isarchring R. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchrig_isarchring | null |
isarchring_isarchgr {X : ring} (R : hrel X) :
R 1%ring 0%ring →
istrans R →
isbinophrel (X := X) R →
isarchring R → isarchgr (X := X) R. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchring_isarchgr | null |
isarchgr_isarchring {X : ring} (R : hrel X) :
R 1%ring 0%ring →
istrans R →
isbinophrel (X := X) R →
isarchgr (X := X) R → isarchring R. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchgr_isarchring | null |
isarchrigtoring :
∏ (X : rig) (R : hrel X) (Hr : R 1%rig 0%rig)
(Hadd : isbinophrel (X := rigaddabmonoid X) R)
(Htra : istrans R)
(Harch : isarchrig (setquot_aux (X := rigaddabmonoid X) R)), isarchring (X := rigtoring X) (rigtoringrel X Hadd). | Theorem | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchrigtoring | null |
natmult_commringfrac {X : commring} {S : subabmonoid _} :
∏ n (x : X × S), natmult (X := commringfrac X S) n (setquotpr (eqrelcommringfrac X S) x) = setquotpr (eqrelcommringfrac X S) (natmult (X := X) n (pr1 x) ,, (pr2 x)). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | natmult_commringfrac | null |
isarchcommringfrac {X : commring} {S : subabmonoid _} (R : hrel X) Hop1 Hop2 Hs:
R 1%ring 0%ring →
istrans R →
isarchring R → isarchring (X := commringfrac X S) (commringfracgt X S (R := R) Hop1 Hop2 Hs). | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchcommringfrac | null |
isarchfld {X : fld} (R : hrel X) :=
∏ x : X, ∃ n : nat, R (nattoring n) x. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchfld | ** Archimedean property in a field |
isarchfld_isarchring {X : fld} (R : hrel X) :
∏ (Hadd : isbinophrel (X := rigaddabmonoid X) R) ( Hmult : isringmultgt X R)
(Hirr : isirrefl R),
isarchfld R → isarchring R. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchfld_isarchring | null |
isarchring_isarchfld {X : fld} (R : hrel X) :
isarchring R → isarchfld R. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchring_isarchfld | null |
isarchfldfrac ( X : intdom ) ( is : isdeceq X ) { R : hrel X } ( is0 : @isbinophrel X R ) ( is1 : isringmultgt X R ) ( is2 : R 1%ring 0%ring ) ( nc : neqchoice R ) ( irr : isirrefl R ) ( tra : istrans R ) :
isarchring R → isarchfld (X := fldfrac X is ) (fldfracgt _ is is0 is1 is2 nc). | Theorem | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchfldfrac | null |
isarchCF {X : ConstructiveField} (R : hrel X) :=
∏ x : X, ∃ n : nat, R (nattoring n) x. | Definition | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchCF | ** Archimedean property in a constructive field |
isarchCF_isarchring {X : ConstructiveField} (R : hrel X) :
∏ (Hadd : isbinophrel (X := rigaddabmonoid X) R) ( Hmult : isringmultgt X R)
(Hirr : isirrefl R),
(∏ x : X, R x 0%CF → (x ≠ 0)%CF) →
isarchCF R → isarchring R. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchCF_isarchring | null |
isarchring_isarchCF {X : ConstructiveField} (R : hrel X) :
isarchring R → isarchCF R. | Lemma | Algebra | [
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath",
"UniMath"
] | UniMath/Algebra/Archimedean.v | isarchring_isarchCF | null |
unop (X : UU) : UU := X → X. | Definition | Algebra | [
"UniMath",
"UniMath"
] | UniMath/Algebra/BinaryOperations.v | unop | * 1. Unary operations |
islcancelable {X : UU} (opp : binop X) (x : X) : UU := isincl (λ x0 : X, opp x x0). | Definition | Algebra | [
"UniMath",
"UniMath"
] | UniMath/Algebra/BinaryOperations.v | islcancelable | * 2.1. General definitions |
lcancel {X : UU} {opp : binop X} {x : X} (H_x : islcancelable opp x) (y z : X) :
opp x y = opp x z → y = z. | Definition | Algebra | [
"UniMath",
"UniMath"
] | UniMath/Algebra/BinaryOperations.v | lcancel | null |
isrcancelable {X : UU} (opp : binop X) (x : X) : UU := isincl (λ x0 : X, opp x0 x). | Definition | Algebra | [
"UniMath",
"UniMath"
] | UniMath/Algebra/BinaryOperations.v | isrcancelable | null |
rcancel {X : UU} {opp : binop X} {x : X} (H_x : isrcancelable opp x) (y z : X) :
opp y x = opp z x → y = z. | Definition | Algebra | [
"UniMath",
"UniMath"
] | UniMath/Algebra/BinaryOperations.v | rcancel | null |
iscancelable {X : UU} (opp : binop X) (x : X) : UU :=
(islcancelable opp x) × (isrcancelable opp x). | Definition | Algebra | [
"UniMath",
"UniMath"
] | UniMath/Algebra/BinaryOperations.v | iscancelable | null |
islinvertible {X : UU} (opp : binop X) (x : X) : UU := isweq (λ x0 : X, opp x x0). | Definition | Algebra | [
"UniMath",
"UniMath"
] | UniMath/Algebra/BinaryOperations.v | islinvertible | null |
isrinvertible {X : UU} (opp : binop X) (x : X) : UU := isweq (λ x0 : X, opp x0 x). | Definition | Algebra | [
"UniMath",
"UniMath"
] | UniMath/Algebra/BinaryOperations.v | isrinvertible | null |
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