statement stringlengths 5 10.7k | proof stringlengths 0 9.25k | type stringclasses 4
values | symbolic_name stringlengths 1 80 | library stringclasses 103
values | filename stringclasses 746
values | imports listlengths 0 49 | deps listlengths 0 64 | docstring stringclasses 1
value | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
is-singleton : {𝓤 : Universe} → 𝓤 ̇ → 𝓤 ̇ | is-singleton X = Σ c ꞉ X , ((x : X) → Id c x) | function | is-singleton | Various | source/Various/UnivalenceFromScratch.lagda | [] | [
"Id"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
fiber : {𝓤 𝓥 : Universe} {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X → Y) → Y → 𝓤 ⊔ 𝓥 ̇ | fiber {𝓤} {𝓥} {X} {Y} f y = Σ x ꞉ X , Id (f x) y | function | fiber | Various | source/Various/UnivalenceFromScratch.lagda | [] | [
"Id"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
is-equiv : {𝓤 𝓥 : Universe} {X : 𝓤 ̇ } {Y : 𝓥 ̇ } → (X → Y) → 𝓤 ⊔ 𝓥 ̇ | is-equiv f = (y : _) → is-singleton(fiber f y) | function | is-equiv | Various | source/Various/UnivalenceFromScratch.lagda | [] | [
"fiber",
"is-singleton"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
Eq : {𝓤 𝓥 : Universe} → 𝓤 ̇ → 𝓥 ̇ → 𝓤 ⊔ 𝓥 ̇ | Eq X Y = Σ f ꞉ (X → Y) , is-equiv f | function | Eq | Various | source/Various/UnivalenceFromScratch.lagda | [] | [
"is-equiv"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
singletonType : {𝓤 : Universe} {X : 𝓤 ̇ } → X → 𝓤 ̇ | singletonType {𝓤} {X} x = Σ y ꞉ X , Id y x | function | singletonType | Various | source/Various/UnivalenceFromScratch.lagda | [] | [
"Id"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
η : {𝓤 : Universe} {X : 𝓤 ̇ } (x : X) → singletonType x | η x = (x , refl x) | function | η | Various | source/Various/UnivalenceFromScratch.lagda | [] | [
"refl",
"singletonType"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
singletonTypesAreSingletons : {𝓤 : Universe} {X : 𝓤 ̇ } (x : X) → is-singleton(singletonType x) | singletonTypesAreSingletons {𝓤} {X} = h
where
A : (y x : X) → Id y x → 𝓤 ̇
A y x p = Id (η x) (y , p)
f : (x : X) → A x x (refl x)
f x = refl (η x)
φ : (y x : X) (p : Id y x) → Id (η x) (y , p)
φ = J A f
g : (x : X) (σ : singletonType x) → Id (η x) σ
g x (y , p) = φ y x p
h : (x : X) → Σ c ꞉ single... | function | singletonTypesAreSingletons | Various | source/Various/UnivalenceFromScratch.lagda | [] | [
"Id",
"is-singleton",
"refl",
"singletonType"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
id : {𝓤 : Universe} (X : 𝓤 ̇ ) → X → X | id X x = x | function | id | Various | source/Various/UnivalenceFromScratch.lagda | [] | [] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
idIsEquiv : {𝓤 : Universe} (X : 𝓤 ̇ ) → is-equiv(id X) | idIsEquiv X = g
where
g : (x : X) → is-singleton (fiber (id X) x)
g = singletonTypesAreSingletons | function | idIsEquiv | Various | source/Various/UnivalenceFromScratch.lagda | [] | [
"fiber",
"id",
"is-equiv",
"is-singleton",
"singletonTypesAreSingletons"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
IdToEq : {𝓤 : Universe} (X Y : 𝓤 ̇ ) → Id X Y → Eq X Y | IdToEq {𝓤} = J A f
where
A : (X Y : 𝓤 ̇ ) → Id X Y → 𝓤 ̇
A X Y p = Eq X Y
f : (X : 𝓤 ̇ ) → A X X (refl X)
f X = (id X , idIsEquiv X) | function | IdToEq | Various | source/Various/UnivalenceFromScratch.lagda | [] | [
"Eq",
"Id",
"id",
"idIsEquiv",
"refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
is-univalent : (𝓤 : Universe) → 𝓤 ⁺ ̇ | is-univalent 𝓤 = (X Y : 𝓤 ̇ ) → is-equiv(IdToEq X Y) | function | is-univalent | Various | source/Various/UnivalenceFromScratch.lagda | [] | [
"IdToEq",
"is-equiv"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
𝓤 ̇ = Set 𝓤 -- This should be the only use of the Agda keyword 'Set' in this development. | function | 𝓤 | Various | source/Various/UnivalenceFromScratch.lagda | [] | [] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | ||
_＝ʷ_ : 𝕎 → 𝕎 → 𝓤 ⊔ 𝓥 ̇ | function | _＝ʷ_ | W | source/W/Properties.lagda | [
"MLTT.Spartan",
"UF.Base",
"UF.Equiv",
"UF.EquivalenceExamples",
"UF.FunExt",
"UF.Retracts",
"UF.Sets",
"UF.Subsingletons",
"UF.Subsingletons-FunExt",
"W.Type"
] | [] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | ||
＝ʷ-refl : (w : 𝕎) → w ＝ʷ w | ＝ʷ-refl (ssup x f) = refl , (λ a → ＝ʷ-refl (f a)) | function | ＝ʷ-refl | W | source/W/Properties.lagda | [
"MLTT.Spartan",
"UF.Base",
"UF.Equiv",
"UF.EquivalenceExamples",
"UF.FunExt",
"UF.Retracts",
"UF.Sets",
"UF.Subsingletons",
"UF.Subsingletons-FunExt",
"W.Type"
] | [
"refl",
"ssup"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
singleton-typeʷ : 𝕎 → 𝓤 ⊔ 𝓥 ̇ | singleton-typeʷ w = Σ t ꞉ 𝕎 , w ＝ʷ t | function | singleton-typeʷ | W | source/W/Properties.lagda | [
"MLTT.Spartan",
"UF.Base",
"UF.Equiv",
"UF.EquivalenceExamples",
"UF.FunExt",
"UF.Retracts",
"UF.Sets",
"UF.Subsingletons",
"UF.Subsingletons-FunExt",
"W.Type"
] | [] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
W-center : (w : 𝕎) → singleton-typeʷ w | W-center w = w , ＝ʷ-refl w | function | W-center | W | source/W/Properties.lagda | [
"MLTT.Spartan",
"UF.Base",
"UF.Equiv",
"UF.EquivalenceExamples",
"UF.FunExt",
"UF.Retracts",
"UF.Sets",
"UF.Subsingletons",
"UF.Subsingletons-FunExt",
"W.Type"
] | [
"singleton-typeʷ",
"＝ʷ-refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
W-centrality : Fun-Ext → (w : 𝕎) (σ : singleton-typeʷ w) → W-center w ＝ σ | W-centrality fe w@(ssup x f) σ@(ssup x g , refl , u) = IV
where
have-u : (a : A x) → f a ＝ʷ g a
have-u = u
IH : (a : A x) → W-center (f a) ＝ (g a , u a)
IH a = W-centrality fe (f a) (g a , u a)
I : (λ a → W-center (f a)) ＝ (λ a → g a , u a)
I = dfunext fe IH
π : (Σ h ꞉ (A x → 𝕎) , ((a : A x) → f a ＝ʷ... | function | W-centrality | W | source/W/Properties.lagda | [
"MLTT.Spartan",
"UF.Base",
"UF.Equiv",
"UF.EquivalenceExamples",
"UF.FunExt",
"UF.Retracts",
"UF.Sets",
"UF.Subsingletons",
"UF.Subsingletons-FunExt",
"W.Type"
] | [
"Fun-Ext",
"W-center",
"ap",
"dfunext",
"domain",
"fe",
"refl",
"singleton-typeʷ",
"ssup",
"ΠΣ-distr",
"＝ʷ-refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
singleton-typesʷ-are-singletons : Fun-Ext
→ (w : 𝕎) → is-singleton (singleton-typeʷ w) | singleton-typesʷ-are-singletons fe w = W-center w , W-centrality fe w | function | singleton-typesʷ-are-singletons | W | source/W/Properties.lagda | [
"MLTT.Spartan",
"UF.Base",
"UF.Equiv",
"UF.EquivalenceExamples",
"UF.FunExt",
"UF.Retracts",
"UF.Sets",
"UF.Subsingletons",
"UF.Subsingletons-FunExt",
"W.Type"
] | [
"Fun-Ext",
"W-center",
"W-centrality",
"fe",
"is-singleton",
"singleton-typeʷ"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
idtoidʷ : (w t : 𝕎) → w ＝ t → w ＝ʷ t | idtoidʷ w w refl = ＝ʷ-refl w | function | idtoidʷ | W | source/W/Properties.lagda | [
"MLTT.Spartan",
"UF.Base",
"UF.Equiv",
"UF.EquivalenceExamples",
"UF.FunExt",
"UF.Retracts",
"UF.Sets",
"UF.Subsingletons",
"UF.Subsingletons-FunExt",
"W.Type"
] | [
"refl",
"＝ʷ-refl"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
idtoidʷ-is-equiv : Fun-Ext → (w t : 𝕎) → is-equiv (idtoidʷ w t) | idtoidʷ-is-equiv fe w = I
where
f : singleton-type w → singleton-typeʷ w
f = NatΣ (idtoidʷ w)
f-is-equiv : is-equiv f
f-is-equiv = maps-of-singletons-are-equivs f
(singleton-types-are-singletons w)
(singleton-typesʷ-are-singletons fe w)
I : (t : 𝕎) → is-equiv (idtoidʷ w t)
... | function | idtoidʷ-is-equiv | W | source/W/Properties.lagda | [
"MLTT.Spartan",
"UF.Base",
"UF.Equiv",
"UF.EquivalenceExamples",
"UF.FunExt",
"UF.Retracts",
"UF.Sets",
"UF.Subsingletons",
"UF.Subsingletons-FunExt",
"W.Type"
] | [
"Fun-Ext",
"NatΣ",
"NatΣ-equiv-gives-fiberwise-equiv",
"fe",
"idtoidʷ",
"is-equiv",
"maps-of-singletons-are-equivs",
"singleton-type",
"singleton-types-are-singletons",
"singleton-typesʷ-are-singletons",
"singleton-typeʷ"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
idtoidʷ-≃ : Fun-Ext → (w t : 𝕎) → (w ＝ t) ≃ (w ＝ʷ t) | idtoidʷ-≃ fe w t = idtoidʷ w t , idtoidʷ-is-equiv fe w t | function | idtoidʷ-≃ | W | source/W/Properties.lagda | [
"MLTT.Spartan",
"UF.Base",
"UF.Equiv",
"UF.EquivalenceExamples",
"UF.FunExt",
"UF.Retracts",
"UF.Sets",
"UF.Subsingletons",
"UF.Subsingletons-FunExt",
"W.Type"
] | [
"Fun-Ext",
"fe",
"idtoidʷ",
"idtoidʷ-is-equiv"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
to-W-＝ : {x : X} {φ : A x → 𝕎}
{x' : X} {φ' : A x' → 𝕎}
→ (Σ p ꞉ x ＝ x' , (φ ＝ φ' ∘ transport A p))
→ ssup x φ ＝[ 𝕎 ] ssup x' φ' | to-W-＝ {x} {φ} {x} {φ'} (refl , f) = ap (ssup x) f | function | to-W-＝ | W | source/W/Properties.lagda | [
"MLTT.Spartan",
"UF.Base",
"UF.Equiv",
"UF.EquivalenceExamples",
"UF.FunExt",
"UF.Retracts",
"UF.Sets",
"UF.Subsingletons",
"UF.Subsingletons-FunExt",
"W.Type"
] | [
"ap",
"refl",
"ssup",
"transport"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
from-W-＝ : {x : X} {φ : A x → 𝕎}
{x' : X} {φ' : A x' → 𝕎}
→ ssup x φ ＝[ 𝕎 ] ssup x' φ'
→ (Σ p ꞉ x ＝ x' , (φ ＝ φ' ∘ transport A p)) | from-W-＝ refl = refl , refl | function | from-W-＝ | W | source/W/Properties.lagda | [
"MLTT.Spartan",
"UF.Base",
"UF.Equiv",
"UF.EquivalenceExamples",
"UF.FunExt",
"UF.Retracts",
"UF.Sets",
"UF.Subsingletons",
"UF.Subsingletons-FunExt",
"W.Type"
] | [
"refl",
"ssup",
"transport"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
to-from-W-＝ : {x : X} {φ : A x → 𝕎}
{x' : X} {φ' : A x' → 𝕎}
→ (q : ssup x φ ＝[ 𝕎 ] ssup x' φ')
→ to-W-＝ (from-W-＝ q) ＝ q | to-from-W-＝ refl = refl | function | to-from-W-＝ | W | source/W/Properties.lagda | [
"MLTT.Spartan",
"UF.Base",
"UF.Equiv",
"UF.EquivalenceExamples",
"UF.FunExt",
"UF.Retracts",
"UF.Sets",
"UF.Subsingletons",
"UF.Subsingletons-FunExt",
"W.Type"
] | [
"from-W-＝",
"refl",
"ssup",
"to-W-＝"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
from-to-W-＝ : {x : X} {φ : A x → 𝕎}
{x' : X} {φ' : A x' → 𝕎}
→ (σ : Σ p ꞉ x ＝ x' , (φ ＝ φ' ∘ transport A p))
→ from-W-＝ (to-W-＝ {x} {φ} {x'} {φ'} σ) ＝ σ | from-to-W-＝ (refl , refl) = refl | function | from-to-W-＝ | W | source/W/Properties.lagda | [
"MLTT.Spartan",
"UF.Base",
"UF.Equiv",
"UF.EquivalenceExamples",
"UF.FunExt",
"UF.Retracts",
"UF.Sets",
"UF.Subsingletons",
"UF.Subsingletons-FunExt",
"W.Type"
] | [
"from-W-＝",
"refl",
"to-W-＝",
"transport"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
W-＝ : {x : X} {φ : A x → 𝕎}
{x' : X} {φ' : A x' → 𝕎}
→ (ssup x φ ＝[ 𝕎 ] ssup x' φ')
≃ (Σ p ꞉ x ＝ x' , (φ ＝ φ' ∘ transport A p)) | W-＝ {x} {φ} {x'} {φ'} = qinveq
from-W-＝
(to-W-＝ ,
to-from-W-＝ ,
from-to-W-＝ {x} {φ} {x'} {φ'}) | function | W-＝ | W | source/W/Properties.lagda | [
"MLTT.Spartan",
"UF.Base",
"UF.Equiv",
"UF.EquivalenceExamples",
"UF.FunExt",
"UF.Retracts",
"UF.Sets",
"UF.Subsingletons",
"UF.Subsingletons-FunExt",
"W.Type"
] | [
"from-W-＝",
"from-to-W-＝",
"qinveq",
"ssup",
"to-W-＝",
"to-from-W-＝",
"transport"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
W-is-prop : funext 𝓥 (𝓤 ⊔ 𝓥)
→ is-prop X
→ is-prop 𝕎 | W-is-prop fe X-is-prop (ssup x φ) (ssup x' φ') = γ
where
p : x ＝ x'
p = X-is-prop x x'
IH : (a : A x) → φ a ＝ φ' (transport A p a)
IH a = W-is-prop fe X-is-prop (φ a) (φ' (transport A p a))
γ : ssup x φ ＝ ssup x' φ'
γ = to-W-＝ (p , dfunext fe IH) | function | W-is-prop | W | source/W/Properties.lagda | [
"MLTT.Spartan",
"UF.Base",
"UF.Equiv",
"UF.EquivalenceExamples",
"UF.FunExt",
"UF.Retracts",
"UF.Sets",
"UF.Subsingletons",
"UF.Subsingletons-FunExt",
"W.Type"
] | [
"dfunext",
"fe",
"funext",
"is-prop",
"ssup",
"to-W-＝",
"transport"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
W-is-set : funext 𝓥 (𝓤 ⊔ 𝓥)
→ is-set X
→ is-set 𝕎 | W-is-set fe X-is-set {ssup x φ} {ssup x' φ'} = γ
where
S = Σ p ꞉ x ＝ x' , (φ ∼ φ' ∘ transport A p)
IH : (p : x ＝ x') (a : A x) → is-prop (φ a ＝ φ' (transport A p a))
IH p a = W-is-set fe X-is-set {φ a} {φ' (transport A p a)}
α : is-prop S
α = Σ-is-prop X-is-set (λ p → Π-is-prop fe (IH p))
β : retract (s... | function | W-is-set | W | source/W/Properties.lagda | [
"MLTT.Spartan",
"UF.Base",
"UF.Equiv",
"UF.EquivalenceExamples",
"UF.FunExt",
"UF.Retracts",
"UF.Sets",
"UF.Subsingletons",
"UF.Subsingletons-FunExt",
"W.Type"
] | [
"ap",
"dfunext",
"fe",
"from-W-＝",
"funext",
"funext-happly",
"happly",
"id",
"is-prop",
"is-set",
"pr₁",
"pr₂",
"refl",
"retract",
"retract-of-prop",
"ssup",
"to-W-＝",
"transport",
"Π-is-prop",
"Σ-is-prop"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | |
𝕎 = W X A | function | 𝕎 | W | source/W/Properties.lagda | [
"MLTT.Spartan",
"UF.Base",
"UF.Equiv",
"UF.EquivalenceExamples",
"UF.FunExt",
"UF.Retracts",
"UF.Sets",
"UF.Subsingletons",
"UF.Subsingletons-FunExt",
"W.Type"
] | [] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | ||
ssup x f ＝ʷ ssup x' f' = Σ p ꞉ x ＝ x' , ((a : A x) → f a ＝ʷ f' (transport A p a)) | function | ssup | W | source/W/Properties.lagda | [
"MLTT.Spartan",
"UF.Base",
"UF.Equiv",
"UF.EquivalenceExamples",
"UF.FunExt",
"UF.Retracts",
"UF.Sets",
"UF.Subsingletons",
"UF.Subsingletons-FunExt",
"W.Type"
] | [
"f'",
"transport"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | ||
W {𝓤 𝓥 : Universe} (X : 𝓤 ̇ ) (A : X → 𝓥 ̇ ) : 𝓤 ⊔ 𝓥 ̇ where
ssup : (x : X) (φ : A x → W X A) → W X A | data | W | W | source/W/Type.lagda | [
"MLTT.Spartan"
] | [
"ssup"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | ||
Wᵢ {𝓤 𝓥 𝓦 : Universe}
(I : 𝓦 ̇ )
(A : 𝓤 ̇ )
(t : A → I)
(B : A → 𝓥 ̇ )
(s : (a : A) → B a → I)
: I → 𝓤 ⊔ 𝓥 ⊔ 𝓦 ̇
where
ssup : (a : A) → ((b : B a) → Wᵢ I A t B s (s a b)) → Wᵢ I A t B s (t a) | data | Wᵢ | W | source/W/Type.lagda | [
"MLTT.Spartan"
] | [
"ssup"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e | ||
WildCategory 𝓤 𝓥 : (𝓤 ⊔ 𝓥)⁺ ̇ where
field
ob : 𝓤 ̇
hom : ob → ob → 𝓥 ̇
idn : (x : ob) → hom x x
_<<_ : {x y z : ob} (g : hom y z) (f : hom x y) → hom x z
assoc
: {u v w x : ob} (f : hom u v) (g : hom v w) (h : hom w x)
→ h << (g << f) ＝ (h << g) << f
lunit
: {x y : ob} (f : hom x y)
→ f <... | record | WildCategory | WildCategories | source/WildCategories/Base.lagda | [
"MLTT.Spartan",
"UF.Subsingletons"
] | [
"assoc"
] | https://github.com/martinescardo/TypeTopology | 66c36add15b0583ae0e96523f29bb6b56e68459e |
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