Proof Assistant Projects
Collection
Digesting proof assistant libraries for AI ingestion. • 84 items • Updated • 3
fact stringlengths 7 7.74k | type stringclasses 3
values | library stringclasses 4
values | imports listlengths 0 17 | filename stringlengths 14 69 | symbolic_name stringlengths 1 27 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
ua-in : ∀ {i} {A B : Set i} → A ≃ B → A == B | function | old | [
"open import Base public hiding (apd; Σ-eq)"
] | old/BaseOver.agda | ua-in | |
ua-out : ∀ {i} {A B : Set i} → A == B → A ≃ B | function | old | [
"open import Base public hiding (apd; Σ-eq)"
] | old/BaseOver.agda | ua-out | |
_ ∼_ {i j} (A : Set i) (B : Set j) : Set (max i j) where constructor _,_,_ field to : A → B from : B → A eq : ∞ ((a : A) (b : B) → ((to a ≡ b) ∼ (a ≡ from b))) -- Identity id-∼ : ∀ {i} (A : Set i) → A ∼ A id-∼ A = (id A) , (id A) , ♯ (λ a b → id-∼ (a ≡ b)) -- Transitivity trans-∼ : ∀ {i} {A B C : Set i} → A ∼ B → B ∼ C... | record | old | [
"open import Base",
"open import Coinduction"
] | old/CoindEquiv.agda | _ | |
hfiber : ∀ {i j} {A : Set i} {B : Set j} (f : A → B) (y : B) → Set (max i j) | function | old | [
"open import Types",
"open import Functions",
"open import Paths",
"open import HLevel"
] | old/Equivalences.agda | hfiber | |
is-equiv : ∀ {i j} {A : Set i} {B : Set j} (f : A → B) → Set (max i j) | function | old | [
"open import Types",
"open import Functions",
"open import Paths",
"open import HLevel"
] | old/Equivalences.agda | is-equiv | |
is-iso : ∀ {i j} {A : Set i} {B : Set j} (f : A → B) → Set (max i j) | function | old | [
"open import Types",
"open import Functions",
"open import Paths",
"open import HLevel"
] | old/Equivalences.agda | is-iso | |
is-hae : ∀ {i j} {A : Set i} {B : Set j} (f : A → B) → Set (max i j) | function | old | [
"open import Types",
"open import Functions",
"open import Paths",
"open import HLevel"
] | old/Equivalences.agda | is-hae | |
iso-is-hae : ∀ {i j} {A : Set i} {B : Set j} → (f : A → B) → is-iso f → is-hae f | function | old | [
"open import Types",
"open import Functions",
"open import Paths",
"open import HLevel"
] | old/Equivalences.agda | iso-is-hae | |
hae-is-eq : ∀ {i j} {A : Set i} {B : Set j} → (f : A → B) → is-hae f → is-equiv f | function | old | [
"open import Types",
"open import Functions",
"open import Paths",
"open import HLevel"
] | old/Equivalences.agda | hae-is-eq | |
id-is-equiv : ∀ {i} (A : Set i) → is-equiv (id A) | function | old | [
"open import Types",
"open import Functions",
"open import Paths",
"open import HLevel"
] | old/Equivalences.agda | id-is-equiv | |
id-equiv : ∀ {i} (A : Set i) → A ≃ A | function | old | [
"open import Types",
"open import Functions",
"open import Paths",
"open import HLevel"
] | old/Equivalences.agda | id-equiv | |
path-to-eq : ∀ {i} {A B : Set i} → (A ≡ B → A ≃ B) | function | old | [
"open import Types",
"open import Functions",
"open import Paths",
"open import HLevel"
] | old/Equivalences.agda | path-to-eq | |
contr-equiv-unit : ∀ {i j} {A : Set i} → (is-contr A → A ≃ unit {j}) | function | old | [
"open import Types",
"open import Functions",
"open import Paths",
"open import HLevel"
] | old/Equivalences.agda | contr-equiv-unit | |
total-map : Σ A P → Σ A Q | function | old | [
"open import Types",
"open import Functions",
"open import Paths",
"open import HLevel",
"open import Equivalences"
] | old/FiberEquivalences.agda | total-map | |
fiberwise-is-equiv : is-equiv total-map → ((x : A) → is-equiv (f x)) | function | old | [
"open import Types",
"open import Functions",
"open import Paths",
"open import HLevel",
"open import Equivalences"
] | old/FiberEquivalences.agda | fiberwise-is-equiv | |
id : ∀ {i} (A : Set i) → (A → A) | function | old | [
"open import Types"
] | old/Functions.agda | id | |
cst : ∀ {i j} {A : Set i} {B : Set j} (b : B) → (A → B) | function | old | [
"open import Types"
] | old/Functions.agda | cst | |
happly : ∀ {j} {P : A → Set j} {f g : Π A P} (p : f ≡ g) → ((x : A) → f x ≡ g x) | function | old | [
"open import Types",
"open import Paths",
"open import HLevel",
"open import Equivalences",
"open import Univalence"
] | old/Funext.agda | happly | |
is-contr : Set i → Set i | function | old | [
"open import Types",
"open import Paths"
] | old/HLevel.agda | is-contr | |
is-truncated : ℕ₋₂ → (Set i → Set i) | function | old | [
"open import Types",
"open import Paths"
] | old/HLevel.agda | is-truncated | |
is-hlevel : ℕ → (Set i → Set i) | function | old | [
"open import Types",
"open import Paths"
] | old/HLevel.agda | is-hlevel | |
has-all-paths : Set i → Set i | function | old | [
"open import Types",
"open import Paths"
] | old/HLevel.agda | has-all-paths | |
has-dec-eq : Set i → Set i | function | old | [
"open import Types",
"open import Paths"
] | old/HLevel.agda | has-dec-eq | |
hProp : (i : Level) → Set (suc i) | function | old | [
"open import Types",
"open import Functions",
"open import Paths",
"open import HLevel",
"open import Equivalences",
"open import Univalence",
"open import Funext"
] | old/HLevelBis.agda | hProp | |
hSet : (i : Level) → Set (suc i) | function | old | [
"open import Types",
"open import Functions",
"open import Paths",
"open import HLevel",
"open import Equivalences",
"open import Univalence",
"open import Funext"
] | old/HLevelBis.agda | hSet | |
_ <_ : ℕ → ℕ → Set where <n : {n : ℕ} → n < S n <S : {n m : ℕ} → (n < m) → (n < S m) _+_ : ℕ → ℕ → ℕ 0 + m = m S n + m = S (n + m) +S-is-S+ : (n m : ℕ) → n + S m ≡ S n + m +S-is-S+ O m = refl +S-is-S+ (S n) m = ap S (+S-is-S+ n m) +0-is-id : (n : ℕ) → n + 0 ≡ n +0-is-id O = refl +0-is-id (S n) = ap S (+0-is-id n) priva... | data | old | [
"open import Base"
] | old/Integers.agda | _ | |
succ : ℤ → ℤ | function | old | [
"open import Base"
] | old/Integers.agda | succ | |
pred : ℤ → ℤ | function | old | [
"open import Base"
] | old/Integers.agda | pred | |
succ-is-equiv : is-equiv succ | function | old | [
"open import Base"
] | old/Integers.agda | succ-is-equiv | |
succ-equiv : ℤ ≃ ℤ | function | old | [
"open import Base"
] | old/Integers.agda | succ-equiv | |
_ ≡_ {i} {A : Set i} (a : A) : A → Set i where refl : a ≡ a _==_ = _≡_ _≢_ : ∀ {i} {A : Set i} → (A → A → Set i) x ≢ y = ¬ (x ≡ y) -- -- This should not be provable -- K : {A : Set} → (x : A) → (p : x ≡ x) → p ≡ refl x -- K .x (refl x) = refl -- Composition and opposite of paths infixr 8 _∘_ -- \o _∘_ : ∀ {i} {A : Set ... | data | old | [
"open import Types",
"open import Functions"
] | old/Paths.agda | _ | |
bool {i} : Set i where true : bool false : bool -- Dependent sum | data | old | [] | old/Types.agda | bool | |
_ ⊔_ {i j} (A : Set i) (B : Set j) : Set (max i j) where -- \sqcup inl : A → A ⊔ B inr : B → A ⊔ B -- Product _×_ : ∀ {i j} (A : Set i) (B : Set j) → Set (max i j) -- \times A × B = Σ A (λ _ → B) -- Dependent product Π : ∀ {i j} (A : Set i) (P : A → Set j) → Set (max i j) Π A P = (x : A) → P x -- Natural numbers | data | old | [] | old/Types.agda | _ | |
ℕ : Set where -- \bn O : ℕ S : (n : ℕ) → ℕ {-# BUILTIN NATURAL ℕ #-} {-# BUILTIN ZERO O #-} {-# BUILTIN SUC S #-} -- Truncation index (isomorphic to the type of integers ≥ -2) | data | old | [] | old/Types.agda | ℕ | |
ℕ ₋₂ : Set where ⟨-2⟩ : ℕ₋₂ S : (n : ℕ₋₂) → ℕ₋₂ ⟨-1⟩ : ℕ₋₂ ⟨-1⟩ = S ⟨-2⟩ ⟨0⟩ : ℕ₋₂ ⟨0⟩ = S ⟨-1⟩ _-1 : ℕ → ℕ₋₂ O -1 = ⟨-1⟩ (S n) -1 = S (n -1) ⟨_⟩ : ℕ → ℕ₋₂ ⟨ n ⟩ = S (n -1) ⟨1⟩ = ⟨ 1 ⟩ ⟨2⟩ = ⟨ 2 ⟩ _+2+_ : ℕ₋₂ → ℕ₋₂ → ℕ₋₂ ⟨-2⟩ +2+ n = n S m +2+ n = S (m +2+ n) -- Integers | data | old | [] | old/Types.agda | ℕ | |
ℤ : Set where -- \bz O : ℤ pos : (n : ℕ) → ℤ neg : (n : ℕ) → ℤ -- Lifting | data | old | [] | old/Types.agda | ℤ | |
unit {i} : Set i where constructor tt ⊤ = unit -- \top -- Booleans | record | old | [] | old/Types.agda | unit | |
Σ {i j} (A : Set i) (P : A → Set j) : Set (max i j) where -- \Sigma constructor _,_ field π₁ : A -- \pi\_1 π₂ : P (π₁) -- \pi\_2 | record | old | [] | old/Types.agda | Σ | |
lift {i} (j : Level) (A : Set i) : Set (max i j) where constructor ↑ -- \u field ↓ : A -- \d | record | old | [] | old/Types.agda | lift | |
abort : ∀ {i j} {P : ⊥ {i} → Set j} → ((x : ⊥) → P x) | function | old | [] | old/Types.agda | abort | |
abort-nondep : ∀ {i j} {A : Set j} → (⊥ {i} → A) | function | old | [] | old/Types.agda | abort-nondep | |
Π : ∀ {i j} (A : Set i) (P : A → Set j) → Set (max i j) | function | old | [] | old/Types.agda | Π | |
_-1 : ℕ → ℕ₋₂ | function | old | [] | old/Types.agda | _-1 | |
path-to-eq-equiv : {A B : Set i} → ((A ≡ B) ≃ (A ≃ B)) | function | old | [
"open import Types",
"open import Functions",
"open import Paths",
"open import HLevel",
"open import Equivalences"
] | old/Univalence.agda | path-to-eq-equiv | |
eq-to-path-equiv : {A B : Set i} → ((A ≃ B) ≃ (A ≡ B)) | function | old | [
"open import Types",
"open import Functions",
"open import Paths",
"open import HLevel",
"open import Equivalences"
] | old/Univalence.agda | eq-to-path-equiv | |
eq-to-path : {A B : Set i} → (A ≃ B → A ≡ B) | function | old | [
"open import Types",
"open import Functions",
"open import Paths",
"open import HLevel",
"open import Equivalences"
] | old/Univalence.agda | eq-to-path | |
_ ==_ {i} {A : Type i} (a : A) : A → Type i where idp : a == a Path = _==_ {-# BUILTIN EQUALITY _==_ #-} {- Paulin-Mohring J rule At the time I’m writing this (July 2013), the identity type is somehow broken in Agda dev, it behaves more or less as the Martin-Löf identity type instead of behaving like the Paulin-Mohring... | data | core | [
"open import Agda.Primitive public using (lzero)"
] | core/lib/Base.agda | _ | |
ℕ : Type₀ where O : ℕ S : (n : ℕ) → ℕ Nat = ℕ {-# BUILTIN NATURAL ℕ #-} {- Lifting to a higher universe level The operation of lifting enjoys both β and η definitionally. It’s a bit annoying to use, but it’s not used much (for now). -} | data | core | [
"open import Agda.Primitive public using (lzero)"
] | core/lib/Base.agda | ℕ | |
TLevel : Type₀ where ⟨-2⟩ : TLevel S : (n : TLevel) → TLevel ℕ₋₂ = TLevel ⟨_⟩₋₂ : ℕ → ℕ₋₂ ⟨ O ⟩₋₂ = ⟨-2⟩ ⟨ S n ⟩₋₂ = S ⟨ n ⟩₋₂ {- Coproducts and case analysis -} | data | core | [
"open import Agda.Primitive public using (lzero)"
] | core/lib/Base.agda | TLevel | |
Coprod {i j} (A : Type i) (B : Type j) : Type (lmax i j) where inl : A → Coprod A B inr : B → Coprod A B infixr 80 _⊔_ _⊔_ = Coprod Dec : ∀ {i} (P : Type i) → Type i Dec P = P ⊔ ¬ P {- Pointed types and pointed maps. [A ⊙→ B] was pointed, but it was never used as a pointed type. -} infix 60 ⊙[_,_] | data | core | [
"open import Agda.Primitive public using (lzero)"
] | core/lib/Base.agda | Coprod | |
Phantom {i} {A : Type i} (a : A) : Type₀ where phantom : Phantom a {- Numeric literal overloading This enables writing numeric literals -} | data | core | [
"open import Agda.Primitive public using (lzero)"
] | core/lib/Base.agda | Phantom | |
Σ {i j} (A : Type i) (B : A → Type j) : Type (lmax i j) where constructor _,_ field fst : A snd : B fst | record | core | [
"open import Agda.Primitive public using (lzero)"
] | core/lib/Base.agda | Σ | |
Lift {i j} (A : Type i) : Type (lmax i j) where instance constructor lift field lower : A | record | core | [
"open import Agda.Primitive public using (lzero)"
] | core/lib/Base.agda | Lift | |
Ptd (i : ULevel) : Type (lsucc i) where constructor ⊙[_,_] field de⊙ : Type i pt : de⊙ | record | core | [
"open import Agda.Primitive public using (lzero)"
] | core/lib/Base.agda | Ptd | |
FromNat {i} (A : Type i) : Type (lsucc i) where field in-range : ℕ → Type i read : ∀ n → ⦃ _ : in-range n ⦄ → A | record | core | [
"open import Agda.Primitive public using (lzero)"
] | core/lib/Base.agda | FromNat | |
FromNeg {i} (A : Type i) : Type (lsucc i) where field in-range : ℕ → Type i read : ∀ n → ⦃ _ : in-range n ⦄ → A | record | core | [
"open import Agda.Primitive public using (lzero)"
] | core/lib/Base.agda | FromNeg | |
Type : (i : ULevel) → Set (lsucc i) | function | core | [
"open import Agda.Primitive public using (lzero)"
] | core/lib/Base.agda | Type | |
of-type : ∀ {i} (A : Type i) (u : A) → A | function | core | [
"open import Agda.Primitive public using (lzero)"
] | core/lib/Base.agda | of-type | |
coe : ∀ {i} {A B : Type i} (p : A == B) → A → B | function | core | [
"open import Agda.Primitive public using (lzero)"
] | core/lib/Base.agda | coe | |
Π : ∀ {i j} (A : Type i) (P : A → Type j) → Type (lmax i j) | function | core | [
"open import Agda.Primitive public using (lzero)"
] | core/lib/Base.agda | Π | |
idf : ∀ {i} (A : Type i) → (A → A) | function | core | [
"open import Agda.Primitive public using (lzero)"
] | core/lib/Base.agda | idf | |
cst : ∀ {i j} {A : Type i} {B : Type j} (b : B) → (A → B) | function | core | [
"open import Agda.Primitive public using (lzero)"
] | core/lib/Base.agda | cst | |
Dec : ∀ {i} (P : Type i) → Type i | function | core | [
"open import Agda.Primitive public using (lzero)"
] | core/lib/Base.agda | Dec | |
ptd : ∀ {i} (A : Type i) → A → Ptd i | function | core | [
"open import Agda.Primitive public using (lzero)"
] | core/lib/Base.agda | ptd | |
idf-is-equiv : ∀ {i} (A : Type i) → is-equiv (idf A) | function | core | [
"open import lib.Base",
"open import lib.NType",
"open import lib.PathFunctor",
"open import lib.PathGroupoid",
"open import lib.Function"
] | core/lib/Equivalence.agda | idf-is-equiv | |
ide : ∀ {i} (A : Type i) → A ≃ A | function | core | [
"open import lib.Base",
"open import lib.NType",
"open import lib.PathFunctor",
"open import lib.PathGroupoid",
"open import lib.Function"
] | core/lib/Equivalence.agda | ide | |
lower-equiv : ∀ {i j} {A : Type i} → Lift {j = j} A ≃ A | function | core | [
"open import lib.Base",
"open import lib.NType",
"open import lib.PathFunctor",
"open import lib.PathGroupoid",
"open import lib.Function"
] | core/lib/Equivalence.agda | lower-equiv | |
Σ₂-Empty : ∀ {i} {A : Type i} → Σ A (λ _ → Empty) ≃ Empty | function | core | [
"open import lib.Basics",
"open import lib.types.Sigma",
"open import lib.types.Pi",
"open import lib.types.Paths",
"open import lib.types.Unit",
"open import lib.types.Empty"
] | core/lib/Equivalence2.agda | Σ₂-Empty | |
CommSquare {i₀ i₁ j₀ j₁} {A₀ : Type i₀} {A₁ : Type i₁} {B₀ : Type j₀} {B₁ : Type j₁} (f₀ : A₀ → B₀) (f₁ : A₁ → B₁) (hA : A₀ → A₁) (hB : B₀ → B₁) : Type (lmax (lmax i₀ i₁) (lmax j₀ j₁)) where constructor comm-sqr field commutes : hB ∘ f₀ ∼ f₁ ∘ hA | record | core | [
"open import lib.Base",
"open import lib.PathGroupoid"
] | core/lib/Function.agda | CommSquare | |
is-surj : ∀ {i j} {A : Type i} {B : Type j} (f : A → B) → Type (lmax i j) | function | core | [
"open import lib.Basics",
"open import lib.types.Truncation"
] | core/lib/Function2.agda | is-surj | |
is-connected : ∀ {i} → ℕ₋₂ → Type i → Type i | function | core | [
"open import lib.Basics",
"open import lib.NType2",
"open import lib.Equivalence2",
"open import lib.path-seq.Rotations",
"open import lib.types.Unit",
"open import lib.types.Nat",
"open import lib.types.Pi",
"open import lib.types.Pointed",
"open import lib.types.Sigma",
"open import lib.types.Pa... | core/lib/NConnected.agda | is-connected | |
has-conn-fibers : ∀ {i j} {A : Type i} {B : Type j} → ℕ₋₂ → (A → B) → Type (lmax i j) | function | core | [
"open import lib.Basics",
"open import lib.NType2",
"open import lib.Equivalence2",
"open import lib.path-seq.Rotations",
"open import lib.types.Unit",
"open import lib.types.Nat",
"open import lib.types.Pi",
"open import lib.types.Pointed",
"open import lib.types.Sigma",
"open import lib.types.Pa... | core/lib/NConnected.agda | has-conn-fibers | |
-2-conn : ∀ {i} (A : Type i) → is-connected -2 A | function | core | [
"open import lib.Basics",
"open import lib.NType2",
"open import lib.Equivalence2",
"open import lib.path-seq.Rotations",
"open import lib.types.Unit",
"open import lib.types.Nat",
"open import lib.types.Pi",
"open import lib.types.Pointed",
"open import lib.types.Sigma",
"open import lib.types.Pa... | core/lib/NConnected.agda | -2-conn | |
inhab-conn : ∀ {i} {A : Type i} (a : A) → is-connected -1 A | function | core | [
"open import lib.Basics",
"open import lib.NType2",
"open import lib.Equivalence2",
"open import lib.path-seq.Rotations",
"open import lib.types.Unit",
"open import lib.types.Nat",
"open import lib.types.Pi",
"open import lib.types.Pointed",
"open import lib.types.Sigma",
"open import lib.types.Pa... | core/lib/NConnected.agda | inhab-conn | |
is-gpd : {i : ULevel} → Type i → Type i | function | core | [
"open import lib.Basics",
"open import lib.Equivalence2",
"open import lib.Relation2",
"open import lib.types.Paths",
"open import lib.types.Pi",
"open import lib.types.Sigma",
"open import lib.types.TLevel"
] | core/lib/NType2.agda | is-gpd | |
has-level-prop : ∀ {i} → ℕ₋₂ → SubtypeProp (Type i) i | function | core | [
"open import lib.Basics",
"open import lib.Equivalence2",
"open import lib.Relation2",
"open import lib.types.Paths",
"open import lib.types.Pi",
"open import lib.types.Sigma",
"open import lib.types.TLevel"
] | core/lib/NType2.agda | has-level-prop | |
_-Type_ : (n : ℕ₋₂) (i : ULevel) → Type (lsucc i) | function | core | [
"open import lib.Basics",
"open import lib.Equivalence2",
"open import lib.Relation2",
"open import lib.types.Paths",
"open import lib.types.Pi",
"open import lib.types.Sigma",
"open import lib.types.TLevel"
] | core/lib/NType2.agda | _-Type_ | |
hProp : (i : ULevel) → Type (lsucc i) | function | core | [
"open import lib.Basics",
"open import lib.Equivalence2",
"open import lib.Relation2",
"open import lib.types.Paths",
"open import lib.types.Pi",
"open import lib.types.Sigma",
"open import lib.types.TLevel"
] | core/lib/NType2.agda | hProp | |
hSet : (i : ULevel) → Type (lsucc i) | function | core | [
"open import lib.Basics",
"open import lib.Equivalence2",
"open import lib.Relation2",
"open import lib.types.Paths",
"open import lib.types.Pi",
"open import lib.types.Sigma",
"open import lib.types.TLevel"
] | core/lib/NType2.agda | hSet | |
_-Type₀ : (n : ℕ₋₂) → Type₁ | function | core | [
"open import lib.Basics",
"open import lib.Equivalence2",
"open import lib.Relation2",
"open import lib.types.Paths",
"open import lib.types.Pi",
"open import lib.types.Sigma",
"open import lib.types.TLevel"
] | core/lib/NType2.agda | _-Type₀ | |
hProp-is-set : (i : ULevel) → is-set (hProp i) | function | core | [
"open import lib.Basics",
"open import lib.Equivalence2",
"open import lib.Relation2",
"open import lib.types.Paths",
"open import lib.types.Pi",
"open import lib.types.Sigma",
"open import lib.types.TLevel"
] | core/lib/NType2.agda | hProp-is-set | |
hSet-level : (i : ULevel) → has-level 1 (hSet i) | function | core | [
"open import lib.Basics",
"open import lib.Equivalence2",
"open import lib.Relation2",
"open import lib.types.Paths",
"open import lib.types.Pi",
"open import lib.types.Sigma",
"open import lib.types.TLevel"
] | core/lib/NType2.agda | hSet-level | |
ap-idf : ∀ {i} {A : Type i} {u v : A} (p : u == v) → ap (idf A) p == p | function | core | [
"open import lib.Base",
"open import lib.PathGroupoid"
] | core/lib/PathFunctor.agda | ap-idf | |
Rel : ∀ (A : Type i) j → Type (lmax i (lsucc j)) | function | core | [
"open import lib.Base"
] | core/lib/Relation.agda | Rel | |
empty-rel : ∀ {A : Type i} → Rel A lzero | function | core | [
"open import lib.Base"
] | core/lib/Relation.agda | empty-rel | |
coe-equiv : ∀ {i} {A B : Type i} → A == B → A ≃ B | function | core | [
"open import lib.Base",
"open import lib.PathGroupoid",
"open import lib.PathFunctor",
"open import lib.Equivalence",
"open import lib.PathOver"
] | core/lib/Univalence.agda | coe-equiv | |
ua-equiv : ∀ {i} {A B : Type i} → (A ≃ B) ≃ (A == B) | function | core | [
"open import lib.Base",
"open import lib.PathGroupoid",
"open import lib.PathFunctor",
"open import lib.Equivalence",
"open import lib.PathOver"
] | core/lib/Univalence.agda | ua-equiv | |
Cube {i} {A : Type i} {a₀₀₀ : A} : {a₀₁₀ a₁₀₀ a₁₁₀ a₀₀₁ a₀₁₁ a₁₀₁ a₁₁₁ : A} {p₀₋₀ : a₀₀₀ == a₀₁₀} {p₋₀₀ : a₀₀₀ == a₁₀₀} {p₋₁₀ : a₀₁₀ == a₁₁₀} {p₁₋₀ : a₁₀₀ == a₁₁₀} (sq₋₋₀ : Square p₀₋₀ p₋₀₀ p₋₁₀ p₁₋₀) -- left {p₀₋₁ : a₀₀₁ == a₀₁₁} {p₋₀₁ : a₀₀₁ == a₁₀₁} {p₋₁₁ : a₀₁₁ == a₁₁₁} {p₁₋₁ : a₁₀₁ == a₁₁₁} (sq₋₋₁ : Square p₀₋₁ p₋... | data | core | [
"open import lib.Base",
"open import lib.PathGroupoid",
"open import lib.cubical.Square"
] | core/lib/cubical/Cube.agda | Cube | |
Square {i} {A : Type i} {a₀₀ : A} : {a₀₁ a₁₀ a₁₁ : A} → a₀₀ == a₀₁ → a₀₀ == a₁₀ → a₀₁ == a₁₁ → a₁₀ == a₁₁ → Type i where ids : Square idp idp idp idp hid-square : ∀ {i} {A : Type i} {a₀₀ a₀₁ : A} {p : a₀₀ == a₀₁} → Square p idp idp p hid-square {p = idp} = ids vid-square : ∀ {i} {A : Type i} {a₀₀ a₁₀ : A} {p : a₀₀ == a... | data | core | [
"open import lib.Base",
"open import lib.Equivalence",
"open import lib.NType",
"open import lib.PathFunctor",
"open import lib.PathGroupoid",
"open import lib.PathOver",
"open import lib.Univalence"
] | core/lib/cubical/Square.agda | Square | |
CommSquareᴳ {i₀ i₁ j₀ j₁} {G₀ : Group i₀} {G₁ : Group i₁} {H₀ : Group j₀} {H₁ : Group j₁} (φ₀ : G₀ →ᴳ H₀) (φ₁ : G₁ →ᴳ H₁) (ξG : G₀ →ᴳ G₁) (ξH : H₀ →ᴳ H₁) : Type (lmax (lmax i₀ i₁) (lmax j₀ j₁)) where constructor comm-sqrᴳ field commutesᴳ : ∀ g₀ → GroupHom.f (ξH ∘ᴳ φ₀) g₀ == GroupHom.f (φ₁ ∘ᴳ ξG) g₀ infixr 0 _□$ᴳ_ _□$ᴳ_... | record | core | [
"open import lib.Basics",
"open import lib.types.Sigma",
"open import lib.types.Group",
"open import lib.types.CommutingSquare",
"open import lib.groups.Homomorphism",
"open import lib.groups.Isomorphism"
] | core/lib/groups/CommutingSquare.agda | CommSquareᴳ | |
Πᴳ : ∀ {i j} (I : Type i) (F : I → Group j) → Group (lmax i j) | function | core | [
"open import lib.Basics",
"open import lib.types.Bool",
"open import lib.types.Group",
"open import lib.types.Nat",
"open import lib.types.Pi",
"open import lib.types.Sigma",
"open import lib.types.Coproduct",
"open import lib.types.Truncation",
"open import lib.groups.Homomorphism",
"open import ... | core/lib/groups/GroupProduct.agda | Πᴳ | |
GroupStructureHom {i j} {GEl : Type i} {HEl : Type j} (GS : GroupStructure GEl) (HS : GroupStructure HEl) : Type (lmax i j) where constructor group-structure-hom private module G = GroupStructure GS module H = GroupStructure HS field f : GEl → HEl pres-comp : preserves-comp G.comp H.comp f abstract pres-ident : f G.ide... | record | core | [
"open import lib.Basics",
"open import lib.Equivalence2",
"open import lib.Function2",
"open import lib.NType2",
"open import lib.types.Coproduct",
"open import lib.types.Fin",
"open import lib.types.Group",
"open import lib.types.Int",
"open import lib.types.Nat",
"open import lib.types.Pi",
"o... | core/lib/groups/Homomorphism.agda | GroupStructureHom | |
GroupHom {i j} (G : Group i) (H : Group j) : Type (lmax i j) where constructor group-hom private module G = Group G module H = Group H field f : G.El → H.El pres-comp : ∀ g₁ g₂ → f (G.comp g₁ g₂) == H.comp (f g₁) (f g₂) open GroupStructureHom {GS = G.group-struct} {HS = H.group-struct} record {f = f ; pres-comp = pres-... | record | core | [
"open import lib.Basics",
"open import lib.Equivalence2",
"open import lib.Function2",
"open import lib.NType2",
"open import lib.types.Coproduct",
"open import lib.types.Fin",
"open import lib.types.Group",
"open import lib.types.Int",
"open import lib.types.Nat",
"open import lib.types.Pi",
"o... | core/lib/groups/Homomorphism.agda | GroupHom | |
idhom : ∀ {i} (G : Group i) → (G →ᴳ G) | function | core | [
"open import lib.Basics",
"open import lib.Equivalence2",
"open import lib.Function2",
"open import lib.NType2",
"open import lib.types.Coproduct",
"open import lib.types.Fin",
"open import lib.types.Group",
"open import lib.types.Int",
"open import lib.types.Nat",
"open import lib.types.Pi",
"o... | core/lib/groups/Homomorphism.agda | idhom | |
idshom : ∀ {i} {GEl : Type i} (GS : GroupStructure GEl) → (GS →ᴳˢ GS) | function | core | [
"open import lib.Basics",
"open import lib.Equivalence2",
"open import lib.Function2",
"open import lib.NType2",
"open import lib.types.Coproduct",
"open import lib.types.Fin",
"open import lib.types.Group",
"open import lib.types.Int",
"open import lib.types.Nat",
"open import lib.types.Pi",
"o... | core/lib/groups/Homomorphism.agda | idshom | |
inv-hom : ∀ {i} (G : AbGroup i) → GroupHom (AbGroup.grp G) (AbGroup.grp G) | function | core | [
"open import lib.Basics",
"open import lib.Equivalence2",
"open import lib.Function2",
"open import lib.NType2",
"open import lib.types.Coproduct",
"open import lib.types.Fin",
"open import lib.types.Group",
"open import lib.types.Int",
"open import lib.types.Nat",
"open import lib.types.Pi",
"o... | core/lib/groups/Homomorphism.agda | inv-hom | |
ℤ-group-structure : GroupStructure ℤ | function | core | [
"open import lib.Basics",
"open import lib.NType2",
"open import lib.types.Int",
"open import lib.types.Group",
"open import lib.types.List",
"open import lib.types.Word",
"open import lib.groups.Homomorphism",
"open import lib.groups.Isomorphism",
"open import lib.groups.FreeAbelianGroup",
"open ... | core/lib/groups/Int.agda | ℤ-group-structure | |
ℤ-group : Group₀ | function | core | [
"open import lib.Basics",
"open import lib.NType2",
"open import lib.types.Int",
"open import lib.types.Group",
"open import lib.types.List",
"open import lib.types.Word",
"open import lib.groups.Homomorphism",
"open import lib.groups.Isomorphism",
"open import lib.groups.FreeAbelianGroup",
"open ... | core/lib/groups/Int.agda | ℤ-group | |
ℤ-group-is-abelian : is-abelian ℤ-group | function | core | [
"open import lib.Basics",
"open import lib.NType2",
"open import lib.types.Int",
"open import lib.types.Group",
"open import lib.types.List",
"open import lib.types.Word",
"open import lib.groups.Homomorphism",
"open import lib.groups.Isomorphism",
"open import lib.groups.FreeAbelianGroup",
"open ... | core/lib/groups/Int.agda | ℤ-group-is-abelian | |
ℤ-abgroup : AbGroup₀ | function | core | [
"open import lib.Basics",
"open import lib.NType2",
"open import lib.types.Int",
"open import lib.types.Group",
"open import lib.types.List",
"open import lib.types.Word",
"open import lib.groups.Homomorphism",
"open import lib.groups.Isomorphism",
"open import lib.groups.FreeAbelianGroup",
"open ... | core/lib/groups/Int.agda | ℤ-abgroup |
End of preview. Expand in Data Studio
Agda-HoTT
Structured dataset from HoTT-Agda — Homotopy Type Theory.
604 declarations extracted from Agda source files.
Applications
- Training language models on formal proofs
- Fine-tuning theorem provers
- Retrieval-augmented generation for proof assistants
- Learning proof embeddings and representations
Source
- Repository: https://github.com/HoTT/HoTT-Agda
Schema
| Column | Type | Description |
|---|---|---|
| fact | string | Declaration body |
| type | string | data, record, function |
| library | string | Source module |
| imports | list | Required imports |
| filename | string | Source file path |
| symbolic_name | string | Identifier |
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