Proof Assistant Projects
Collection
Digesting proof assistant libraries for AI ingestion.
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84 items
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Updated
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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
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data
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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ℕ
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ℕ₂ : Set where
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data
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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ℕ₂
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List (X : Set) : Set where
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data
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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List
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Tree (X : Set) : Set where
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data
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
|
Tree
| |
Σ {X : Set} (Y : X → Set) : Set where
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data
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
|
Σ
| |
_ ≡_ {X : Set} : X → X → Set where
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data
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
|
_
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D (X Y Z : Set) : Set where
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data
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
|
D
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type : Set where
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data
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
|
type
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T : (σ : type) → Set where
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data
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
|
T
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TΩ : (σ : type) → Set where
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data
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
|
TΩ
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Ķ : ∀{X Y : Set} → X → Y → X
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function
|
latex
|
[] |
latex/EffectfulForcing/dialogue.lagda
|
Ķ
| |
Ş : ∀{X Y Z : Set} → (X → Y → Z) → (X → Y) → X → Z
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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Ş
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rec : ∀{X : Set} → (X → X) → X → ℕ → X
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
|
rec
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π₀ : ∀{X : Set} {Y : X → Set} → (Σ \(x : X) → Y x) → X
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
|
π₀
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π₁ : ∀{X : Set} {Y : X → Set} → ∀(t : Σ \(x : X) → Y x) → Y(π₀ t)
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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π₁
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sym : ∀{X : Set} → ∀{x y : X} → x ≡ y → y ≡ x
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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sym
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trans : ∀{X : Set} → ∀{x y z : X} → x ≡ y → y ≡ z → x ≡ z
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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trans
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cong : ∀{X Y : Set} → ∀(f : X → Y) → ∀{x₀ x₁ : X} → x₀ ≡ x₁ → f x₀ ≡ f x₁
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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cong
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cong₂ : ∀{X Y Z : Set} → ∀(f : X → Y → Z)
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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cong₂
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dialogue : ∀{X Y Z : Set} → D X Y Z → (X → Y) → Z
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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dialogue
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eloquent : ∀{X Y Z : Set} → ((X → Y) → Z) → Set
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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eloquent
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Baire : Set
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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Baire
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B : Set → Set
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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B
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continuous : (Baire → ℕ) → Set
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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continuous
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dialogue-continuity : ∀(d : B ℕ) → continuous(dialogue d)
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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dialogue-continuity
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continuity-extensional : ∀(f g : Baire → ℕ) → (∀ α → f α ≡ g α) → continuous f → continuous g
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
|
continuity-extensional
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eloquent-is-continuous : ∀(f : Baire → ℕ) → eloquent f → continuous f
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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eloquent-is-continuous
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Cantor : Set
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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Cantor
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C : Set → Set
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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C
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uniformly-continuous : (Cantor → ℕ) → Set
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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uniformly-continuous
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dialogue-UC : ∀(d : C ℕ) → uniformly-continuous(dialogue d)
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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dialogue-UC
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UC-extensional : ∀(f g : Cantor → ℕ) → (∀(α : Cantor) → f α ≡ g α)
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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UC-extensional
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eloquent-is-UC : ∀(f : Cantor → ℕ) → eloquent f → uniformly-continuous f
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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eloquent-is-UC
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embed-ℕ₂-ℕ : ℕ₂ → ℕ
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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embed-ℕ₂-ℕ
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embed-C-B : Cantor → Baire
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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embed-C-B
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C-restriction : (Baire → ℕ) → (Cantor → ℕ)
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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C-restriction
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prune : B ℕ → C ℕ
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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prune
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prune-behaviour : ∀(d : B ℕ)(α : Cantor) → dialogue (prune d) α ≡ C-restriction(dialogue d) α
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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prune-behaviour
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eloquent-restriction : ∀(f : Baire → ℕ) → eloquent f → eloquent(C-restriction f)
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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eloquent-restriction
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T-definable : ∀{σ : type} → Set⟦ σ ⟧ → Set
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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T-definable
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embed : ∀{σ : type} → T σ → TΩ σ
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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embed
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kleisli-extension : ∀{X Y : Set} → (X → B Y) → B X → B Y
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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kleisli-extension
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B-functor : ∀{X Y : Set} → (X → Y) → B X → B Y
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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B-functor
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decode : ∀{X : Set} → Baire → B X → X
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
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decode
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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)
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
|
decode-α-is-natural
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decode-kleisli-extension : ∀{X Y : Set}(f : X → B Y)(d : B X)(α : Baire)
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
|
decode-kleisli-extension
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Kleisli-extension : ∀{X : Set} {σ : type} → (X → B-Set⟦ σ ⟧) → B X → B-Set⟦ σ ⟧
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
|
Kleisli-extension
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generic : B ℕ → B ℕ
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function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
|
generic
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generic-diagram : ∀(α : Baire)(d : B ℕ) → α(decode α d) ≡ decode α (generic d)
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function
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latex
|
[] |
latex/EffectfulForcing/dialogue.lagda
|
generic-diagram
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zero' : B ℕ
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function
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latex
|
[] |
latex/EffectfulForcing/dialogue.lagda
|
zero'
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succ' : B ℕ → B ℕ
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function
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latex
|
[] |
latex/EffectfulForcing/dialogue.lagda
|
succ'
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rec' : ∀{σ : type} → (B-Set⟦ σ ⟧ → B-Set⟦ σ ⟧) → B-Set⟦ σ ⟧ → B ℕ → B-Set⟦ σ ⟧
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function
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latex
|
[] |
latex/EffectfulForcing/dialogue.lagda
|
rec'
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dialogue-tree : T((ι ⇒ ι) ⇒ ι) → B ℕ
|
function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
|
dialogue-tree
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preservation : ∀{σ : type} → ∀(t : T σ) → ∀(α : Baire) → ⟦ t ⟧ ≡ ⟦ embed t ⟧' α
|
function
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latex
|
[] |
latex/EffectfulForcing/dialogue.lagda
|
preservation
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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
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latex
|
[] |
latex/EffectfulForcing/dialogue.lagda
|
R-kleisli-lemma
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main-lemma : ∀{σ : type}(t : TΩ σ) → R ⟦ t ⟧' (B⟦ t ⟧)
|
function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
|
main-lemma
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dialogue-tree-correct : ∀(t : T((ι ⇒ ι) ⇒ ι))(α : Baire) → ⟦ t ⟧ α ≡ decode α (dialogue-tree t)
|
function
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latex
|
[] |
latex/EffectfulForcing/dialogue.lagda
|
dialogue-tree-correct
| |
eloquence-theorem : ∀(f : Baire → ℕ) → T-definable f → eloquent f
|
function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
|
eloquence-theorem
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corollary₀ : ∀(f : Baire → ℕ) → T-definable f → continuous f
|
function
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latex
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[] |
latex/EffectfulForcing/dialogue.lagda
|
corollary₀
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corollary₁ : ∀(f : Baire → ℕ) → T-definable f → uniformly-continuous(C-restriction f)
|
function
|
latex
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[] |
latex/EffectfulForcing/dialogue.lagda
|
corollary₁
| |
mod-cont : T((ι ⇒ ι) ⇒ ι) → Baire → List ℕ
|
function
|
latex
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[] |
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.Subsingletons-FunExt"
] |
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.Subsingletons-FunExt"
] |
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.Subsingletons-FunExt"
] |
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 (_♯_ to _♯[𝟚ᴺ]_) hiding (_=⟦_⟧_)",
"open import TypeTopology.TotallySeparated\nopen import UF.Base",
"open import UF.ClassicalLogic\nopen import UF.DiscreteAndSeparated renaming (_♯_ to ♯[Π])",
"open import UF.FunExt\nopen import UF.Size",
"open import UF.Subsingletons\nopen import UF.Subsingletons-FunExt"
] |
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 (_♯_ to _♯[𝟚ᴺ]_) hiding (_=⟦_⟧_)",
"open import TypeTopology.TotallySeparated\nopen import UF.Base",
"open import UF.ClassicalLogic\nopen import UF.DiscreteAndSeparated renaming (_♯_ to ♯[Π])",
"open import UF.FunExt\nopen import UF.Size",
"open import UF.Subsingletons\nopen import UF.Subsingletons-FunExt"
] |
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 (_♯_ to _♯[𝟚ᴺ]_) hiding (_=⟦_⟧_)",
"open import TypeTopology.TotallySeparated\nopen import UF.Base",
"open import UF.ClassicalLogic\nopen import UF.DiscreteAndSeparated renaming (_♯_ to ♯[Π])",
"open import UF.FunExt\nopen import UF.Size",
"open import UF.Subsingletons\nopen import UF.Subsingletons-FunExt"
] |
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 (_♯_ to _♯[𝟚ᴺ]_) hiding (_=⟦_⟧_)",
"open import TypeTopology.TotallySeparated\nopen import UF.Base",
"open import UF.ClassicalLogic\nopen import UF.DiscreteAndSeparated renaming (_♯_ to ♯[Π])",
"open import UF.FunExt\nopen import UF.Size",
"open import UF.Subsingletons\nopen import UF.Subsingletons-FunExt"
] |
source/Apartness/Properties.lagda
|
WEM-gives-non-trivial-apartness-on-universe
|
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
| 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 |
from datasets import load_dataset
ds = load_dataset("phanerozoic/Agda-TypeTopology")
gpl-3.0