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ℕ : Set where
data
latex
[]
latex/EffectfulForcing/dialogue.lagda
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
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.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
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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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