Proof Assistant Projects
Collection
Digesting proof assistant libraries for AI ingestion. • 84 items • Updated • 3
fact stringlengths 5 169 | type stringclasses 3
values | library stringclasses 59
values | imports listlengths 0 31 | filename stringclasses 625
values | symbolic_name stringlengths 1 80 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
ℕ : Set where | data | latex | [] | latex/EffectfulForcing/dialogue.lagda | ℕ | |
ℕ₂ : Set where | data | latex | [] | latex/EffectfulForcing/dialogue.lagda | ℕ₂ | |
List (X : Set) : Set where | data | latex | [] | latex/EffectfulForcing/dialogue.lagda | List | |
Tree (X : Set) : Set where | data | latex | [] | latex/EffectfulForcing/dialogue.lagda | Tree | |
Σ {X : Set} (Y : X → Set) : Set where | data | latex | [] | latex/EffectfulForcing/dialogue.lagda | Σ | |
_ ≡_ {X : Set} : X → X → Set where | data | latex | [] | latex/EffectfulForcing/dialogue.lagda | _ | |
D (X Y Z : Set) : Set where | data | latex | [] | latex/EffectfulForcing/dialogue.lagda | D | |
type : Set where | data | latex | [] | latex/EffectfulForcing/dialogue.lagda | type | |
T : (σ : type) → Set where | data | latex | [] | latex/EffectfulForcing/dialogue.lagda | T | |
TΩ : (σ : type) → Set where | data | latex | [] | latex/EffectfulForcing/dialogue.lagda | TΩ | |
Ķ : ∀{X Y : Set} → X → Y → X | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | Ķ | |
Ş : ∀{X Y Z : Set} → (X → Y → Z) → (X → Y) → X → Z | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | Ş | |
rec : ∀{X : Set} → (X → X) → X → ℕ → X | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | rec | |
π₀ : ∀{X : Set} {Y : X → Set} → (Σ \(x : X) → Y x) → X | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | π₀ | |
π₁ : ∀{X : Set} {Y : X → Set} → ∀(t : Σ \(x : X) → Y x) → Y(π₀ t) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | π₁ | |
sym : ∀{X : Set} → ∀{x y : X} → x ≡ y → y ≡ x | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | sym | |
trans : ∀{X : Set} → ∀{x y z : X} → x ≡ y → y ≡ z → x ≡ z | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | trans | |
cong : ∀{X Y : Set} → ∀(f : X → Y) → ∀{x₀ x₁ : X} → x₀ ≡ x₁ → f x₀ ≡ f x₁ | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | cong | |
cong₂ : ∀{X Y Z : Set} → ∀(f : X → Y → Z) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | cong₂ | |
dialogue : ∀{X Y Z : Set} → D X Y Z → (X → Y) → Z | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | dialogue | |
eloquent : ∀{X Y Z : Set} → ((X → Y) → Z) → Set | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | eloquent | |
Baire : Set | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | Baire | |
B : Set → Set | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | B | |
continuous : (Baire → ℕ) → Set | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | continuous | |
dialogue-continuity : ∀(d : B ℕ) → continuous(dialogue d) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | dialogue-continuity | |
continuity-extensional : ∀(f g : Baire → ℕ) → (∀ α → f α ≡ g α) → continuous f → continuous g | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | continuity-extensional | |
eloquent-is-continuous : ∀(f : Baire → ℕ) → eloquent f → continuous f | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | eloquent-is-continuous | |
Cantor : Set | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | Cantor | |
C : Set → Set | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | C | |
uniformly-continuous : (Cantor → ℕ) → Set | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | uniformly-continuous | |
dialogue-UC : ∀(d : C ℕ) → uniformly-continuous(dialogue d) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | dialogue-UC | |
UC-extensional : ∀(f g : Cantor → ℕ) → (∀(α : Cantor) → f α ≡ g α) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | UC-extensional | |
eloquent-is-UC : ∀(f : Cantor → ℕ) → eloquent f → uniformly-continuous f | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | eloquent-is-UC | |
embed-ℕ₂-ℕ : ℕ₂ → ℕ | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | embed-ℕ₂-ℕ | |
embed-C-B : Cantor → Baire | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | embed-C-B | |
C-restriction : (Baire → ℕ) → (Cantor → ℕ) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | C-restriction | |
prune : B ℕ → C ℕ | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | prune | |
prune-behaviour : ∀(d : B ℕ)(α : Cantor) → dialogue (prune d) α ≡ C-restriction(dialogue d) α | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | prune-behaviour | |
eloquent-restriction : ∀(f : Baire → ℕ) → eloquent f → eloquent(C-restriction f) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | eloquent-restriction | |
T-definable : ∀{σ : type} → Set⟦ σ ⟧ → Set | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | T-definable | |
embed : ∀{σ : type} → T σ → TΩ σ | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | embed | |
kleisli-extension : ∀{X Y : Set} → (X → B Y) → B X → B Y | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | kleisli-extension | |
B-functor : ∀{X Y : Set} → (X → Y) → B X → B Y | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | B-functor | |
decode : ∀{X : Set} → Baire → B X → X | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | decode | |
decode-α-is-natural : ∀{X Y : Set}(g : X → Y)(d : B X)(α : Baire) → g(decode α d) ≡ decode α (B-functor g d) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | decode-α-is-natural | |
decode-kleisli-extension : ∀{X Y : Set}(f : X → B Y)(d : B X)(α : Baire) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | decode-kleisli-extension | |
Kleisli-extension : ∀{X : Set} {σ : type} → (X → B-Set⟦ σ ⟧) → B X → B-Set⟦ σ ⟧ | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | Kleisli-extension | |
generic : B ℕ → B ℕ | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | generic | |
generic-diagram : ∀(α : Baire)(d : B ℕ) → α(decode α d) ≡ decode α (generic d) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | generic-diagram | |
zero' : B ℕ | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | zero' | |
succ' : B ℕ → B ℕ | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | succ' | |
rec' : ∀{σ : type} → (B-Set⟦ σ ⟧ → B-Set⟦ σ ⟧) → B-Set⟦ σ ⟧ → B ℕ → B-Set⟦ σ ⟧ | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | rec' | |
dialogue-tree : T((ι ⇒ ι) ⇒ ι) → B ℕ | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | dialogue-tree | |
preservation : ∀{σ : type} → ∀(t : T σ) → ∀(α : Baire) → ⟦ t ⟧ ≡ ⟦ embed t ⟧' α | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | preservation | |
R : ∀{σ : type} → (Baire → Set⟦ σ ⟧) → B-Set⟦ σ ⟧ → Set | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | R | |
R-kleisli-lemma : ∀(σ : type)(g : ℕ → Baire → Set⟦ σ ⟧)(g' : ℕ → B-Set⟦ σ ⟧) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | R-kleisli-lemma | |
main-lemma : ∀{σ : type}(t : TΩ σ) → R ⟦ t ⟧' (B⟦ t ⟧) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | main-lemma | |
dialogue-tree-correct : ∀(t : T((ι ⇒ ι) ⇒ ι))(α : Baire) → ⟦ t ⟧ α ≡ decode α (dialogue-tree t) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | dialogue-tree-correct | |
eloquence-theorem : ∀(f : Baire → ℕ) → T-definable f → eloquent f | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | eloquence-theorem | |
corollary₀ : ∀(f : Baire → ℕ) → T-definable f → continuous f | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | corollary₀ | |
corollary₁ : ∀(f : Baire → ℕ) → T-definable f → uniformly-continuous(C-restriction f) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | corollary₁ | |
mod-cont : T((ι ⇒ ι) ⇒ ι) → Baire → List ℕ | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | mod-cont | |
mod-cont-obs : ∀(t : T((ι ⇒ ι) ⇒ ι))(α : Baire) → mod-cont t α ≡ π₀(dialogue-continuity (dialogue-tree t) α) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | mod-cont-obs | |
flatten : {X : Set} → Tree X → List X | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | flatten | |
mod-unif : T((ι ⇒ ι) ⇒ ι) → List ℕ | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | mod-unif | |
I : ∀{σ : type} → T(σ ⇒ σ) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | I | |
I-behaviour : ∀{σ : type}{x : Set⟦ σ ⟧} → ⟦ I ⟧ x ≡ x | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | I-behaviour | |
number : ℕ → T ι | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | number | |
t₀ : T((ι ⇒ ι) ⇒ ι) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | t₀ | |
t₀-interpretation : ⟦ t₀ ⟧ ≡ λ α → 17 | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | t₀-interpretation | |
v : ∀{γ : type} → T(γ ⇒ γ) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | v | |
Number : ∀{γ} → ℕ → T(γ ⇒ ι) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | Number | |
t₁ : T((ι ⇒ ι) ⇒ ι) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | t₁ | |
t₁-interpretation : ⟦ t₁ ⟧ ≡ λ α → α 17 | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | t₁-interpretation | |
example₁ : List ℕ | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | example₁ | |
t₂ : T((ι ⇒ ι) ⇒ ι) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | t₂ | |
t₂-interpretation : ⟦ t₂ ⟧ ≡ λ α → rec α (α 17) (α 17) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | t₂-interpretation | |
Add : T(ι ⇒ ι ⇒ ι) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | Add | |
t₃ : T((ι ⇒ ι) ⇒ ι) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | t₃ | |
t₃-interpretation : ⟦ t₃ ⟧ ≡ λ α → rec α (α 1) (rec succ (α 2) (α 3)) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | t₃-interpretation | |
length : {X : Set} → List X → ℕ | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | length | |
max : ℕ → ℕ → ℕ | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | max | |
Max : List ℕ → ℕ | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | Max | |
t₄ : T((ι ⇒ ι) ⇒ ι) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | t₄ | |
t₄-interpretation : ⟦ t₄ ⟧ ≡ λ α → rec α (rec succ (α (α 2)) (α 3)) (rec α (α 1) (rec succ (α 2) (α 3))) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | t₄-interpretation | |
t₅ : T((ι ⇒ ι) ⇒ ι) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | t₅ | |
t₅-explicitly : t₅ ≡ (S · (S · Rec · (S · I · (S · (S · (K · (Rec · Succ)) · (S · I · (S · (S · Rec · | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | t₅-explicitly | |
t₅-interpretation : ⟦ t₅ ⟧ ≡ λ α → rec α (α(rec succ (α(rec α (α 17) (α 17))) (rec α (rec succ (α (α 2)) (α 3)) | function | latex | [] | latex/EffectfulForcing/dialogue.lagda | t₅-interpretation | |
Strong-Apartness : 𝓤 ̇ → (𝓥 : Universe) → 𝓥 ⁺ ⊔ 𝓤 ̇ | function | Apartness | [
"open import MLTT.Spartan\nopen import UF.DiscreteAndSeparated hiding (tight)",
"open import UF.FunExt\nopen import UF.Lower-FunExt",
"open import UF.NotNotStablePropositions\nopen import UF.PropTrunc",
"open import UF.Sets\nopen import UF.Sets-Properties",
"open import UF.Subsingletons\nopen import UF.Subs... | source/Apartness/Definition.lagda | Strong-Apartness | |
double-negation-of-equality-gives-negation-of-apartness : {X : 𝓤 ̇ } (x y : X) (_♯_ : X → X → 𝓥 ̇ ) | function | Apartness | [
"open import MLTT.Spartan\nopen import UF.DiscreteAndSeparated hiding (tight)",
"open import UF.FunExt\nopen import UF.Lower-FunExt",
"open import UF.NotNotStablePropositions\nopen import UF.PropTrunc",
"open import UF.Sets\nopen import UF.Sets-Properties",
"open import UF.Subsingletons\nopen import UF.Subs... | source/Apartness/Definition.lagda | double-negation-of-equality-gives-negation-of-apartness | |
tight-types-are-sets' : {X : 𝓤 ̇ } (_♯_ : X → X → 𝓥 ̇ ) | function | Apartness | [
"open import MLTT.Spartan\nopen import UF.DiscreteAndSeparated hiding (tight)",
"open import UF.FunExt\nopen import UF.Lower-FunExt",
"open import UF.NotNotStablePropositions\nopen import UF.PropTrunc",
"open import UF.Sets\nopen import UF.Sets-Properties",
"open import UF.Subsingletons\nopen import UF.Subs... | source/Apartness/Definition.lagda | tight-types-are-sets' | |
is-strongly-extensional : ∀ {𝓣} {X : 𝓤 ̇ } {Y : 𝓥 ̇ } | function | Apartness | [
"open import Apartness.Definition\nopen import MLTT.Spartan",
"open import UF.FunExt\nopen import UF.Subsingletons"
] | source/Apartness/Morphisms.lagda | is-strongly-extensional | |
being-strongly-extensional-is-prop : Fun-Ext | function | Apartness | [
"open import Apartness.Definition\nopen import MLTT.Spartan",
"open import UF.FunExt\nopen import UF.Subsingletons"
] | source/Apartness/Morphisms.lagda | being-strongly-extensional-is-prop | |
preserves : ∀ {𝓣} {X : 𝓤 ̇ } {Y : 𝓥 ̇ } | function | Apartness | [
"open import Apartness.Definition\nopen import MLTT.Spartan",
"open import UF.FunExt\nopen import UF.Subsingletons"
] | source/Apartness/Morphisms.lagda | preserves | |
elements-that-are-not-apart-have-the-same-apartness-class : {X : 𝓤 ̇ } (x y : X) (_♯_ : X → X → 𝓥 ̇ ) | function | Apartness | [
"open import UF.PropTrunc\n\nmodule Apartness.Negation",
"open import Apartness.Definition\nopen import MLTT.Spartan"
] | source/Apartness/Negation.lagda | elements-that-are-not-apart-have-the-same-apartness-class | |
elements-with-the-same-apartness-class-are-not-apart : {X : 𝓤 ̇ } (x y : X) (_♯_ : X → X → 𝓥 ̇ ) | function | Apartness | [
"open import UF.PropTrunc\n\nmodule Apartness.Negation",
"open import Apartness.Definition\nopen import MLTT.Spartan"
] | source/Apartness/Negation.lagda | elements-with-the-same-apartness-class-are-not-apart | |
has-two-points-apart : {X : 𝓤 ̇ } → Apartness X 𝓥 → 𝓥 ⊔ 𝓤 ̇ | function | Apartness | [
"open import UF.PropTrunc\n\nmodule Apartness.Properties",
"open import Apartness.Definition\nopen import MLTT.Spartan",
"open import Naturals.Properties\nopen import NotionsOfDecidability.DecidableClassifier",
"open import Taboos.LPO\nopen import Taboos.WLPO",
"open import TypeTopology.Cantor renaming (_♯_... | source/Apartness/Properties.lagda | has-two-points-apart | |
Nontrivial-Apartness : 𝓤 ̇ → (𝓥 : Universe) → 𝓥 ⁺ ⊔ 𝓤 ̇ | function | Apartness | [
"open import UF.PropTrunc\n\nmodule Apartness.Properties",
"open import Apartness.Definition\nopen import MLTT.Spartan",
"open import Naturals.Properties\nopen import NotionsOfDecidability.DecidableClassifier",
"open import Taboos.LPO\nopen import Taboos.WLPO",
"open import TypeTopology.Cantor renaming (_♯_... | source/Apartness/Properties.lagda | Nontrivial-Apartness | |
WEM-gives-that-type-with-two-distinct-points-has-nontrivial-apartness : funext 𝓤 𝓤₀ | function | Apartness | [
"open import UF.PropTrunc\n\nmodule Apartness.Properties",
"open import Apartness.Definition\nopen import MLTT.Spartan",
"open import Naturals.Properties\nopen import NotionsOfDecidability.DecidableClassifier",
"open import Taboos.LPO\nopen import Taboos.WLPO",
"open import TypeTopology.Cantor renaming (_♯_... | source/Apartness/Properties.lagda | WEM-gives-that-type-with-two-distinct-points-has-nontrivial-apartness | |
WEM-gives-non-trivial-apartness-on-universe : funext (𝓤 ⁺) 𝓤₀ | function | Apartness | [
"open import UF.PropTrunc\n\nmodule Apartness.Properties",
"open import Apartness.Definition\nopen import MLTT.Spartan",
"open import Naturals.Properties\nopen import NotionsOfDecidability.DecidableClassifier",
"open import Taboos.LPO\nopen import Taboos.WLPO",
"open import TypeTopology.Cantor renaming (_♯_... | source/Apartness/Properties.lagda | WEM-gives-non-trivial-apartness-on-universe |
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Agda-TypeTopology
Structured declarations from TypeTopology - Martín Escardó's Agda development exploring logical manifestations of topological concepts via the univalent point of view. Source: github.com/martinescardo/TypeTopology
Schema
| Column | Type | Description |
|---|---|---|
fact |
string | Declaration body (without type keyword) |
type |
string | Declaration type (Lemma, Definition, etc.) |
library |
string | Source module |
imports |
list | Import statements |
filename |
string | Source file path |
symbolic_name |
string | Declaration identifier |
Statistics
- Total entries: 7781
- Unique files: 625
- Declaration types: data, function, record
Usage
from datasets import load_dataset
ds = load_dataset("phanerozoic/Agda-TypeTopology")
License
gpl-3.0
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