fact stringlengths 5 24.4k | type stringclasses 3
values | library stringclasses 2
values | imports listlengths 0 70 | filename stringlengths 18 57 | symbolic_name stringlengths 1 32 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
has-ski (_%_ : ↯ A → ↯ A → ↯ A) : Type (level-of A) where field K S : ↯ A K↓ : ⌞ K ⌟ S↓ : ⌞ S ⌟ K↓₁ : ∀ {x} → ⌞ x ⌟ → ⌞ K % x ⌟ K-β : ∀ {x y} → ⌞ x ⌟ → ⌞ y ⌟ → (K % x) % y ≡ x S↓₁ : ∀ {f} → ⌞ f ⌟ → ⌞ S % f ⌟ S↓₂ : ∀ {f g} → ⌞ f ⌟ → ⌞ g ⌟ → ⌞ (S % f) % g ⌟ S-β : ∀ {f g x} → ⌞ f ⌟ → ⌞ g ⌟ → ⌞ x ⌟ → ((S % f) % g) % x ≡ ((... | record | src | [
"open import 1Lab.Prelude",
"open import Data.Partial.Total",
"open import Data.Partial.Base",
"open import Data.Fin.Base hiding (_<_ ; _≤_)",
"open import Data.Vec.Base",
"open import Realisability.PCA"
] | src/Realisability/PCA/Combinatorial.lagda.md | has-ski | |
Termʰ (V : Type) : Type ℓ where var : V → Termʰ V const : ↯⁺ ⌞ 𝔸 ⌟ → Termʰ V app : Termʰ V → Termʰ V → Termʰ V lam : (V → Termʰ V) → Termʰ V ``` We will primarily use terms where the type of variables is taken to be the natural numbers, standing for de Bruijn *levels*. Since we can perform case analysis on natural num... | data | src | [
"open import 1Lab.Prelude",
"open import Data.Partial.Total",
"open import Data.Partial.Base",
"open import Data.Fin.Base hiding (_<_ ; _≤_)",
"open import Data.Nat.Base",
"open import Data.Vec.Base",
"open import Data.Irr",
"open import Realisability.PCA"
] | src/Realisability/PCA/Sugar.lagda.md | Termʰ | |
To-term {ℓ'} (V : Type) (X : Type ℓ') : Type (ℓ ⊔ ℓ') where field to : X → Termʰ V instance var-to-term : ∀ {V} → To-term V V var-to-term = record { to = var } const-to-term' : ∀ {V} → To-term V ⌞ 𝔸 ⌟ const-to-term' = record { to = λ x → const (pure x , tt) } const-to-term : ∀ {V} → To-term V (↯⁺ ⌞ 𝔸 ⌟) const-to-term... | record | src | [
"open import 1Lab.Prelude",
"open import Data.Partial.Total",
"open import Data.Partial.Base",
"open import Data.Fin.Base hiding (_<_ ; _≤_)",
"open import Data.Nat.Base",
"open import Data.Vec.Base",
"open import Data.Irr",
"open import Realisability.PCA"
] | src/Realisability/PCA/Sugar.lagda.md | To-term | |
expr_ : (t : ∀ {V} → Termʰ V) ⦃ _ : wf 0 t ⦄ → ↯ ⌞ 𝔸 ⌟ | function | src | [
"open import 1Lab.Prelude",
"open import Data.Partial.Total",
"open import Data.Partial.Base",
"open import Data.Fin.Base hiding (_<_ ; _≤_)",
"open import Data.Nat.Base",
"open import Data.Vec.Base",
"open import Data.Irr",
"open import Realisability.PCA"
] | src/Realisability/PCA/Sugar.lagda.md | expr_ | |
lam-syntax : ∀ {ℓ} {V : Type} {A : Type ℓ} ⦃ _ : To-term V A ⦄ → (V → A) → Termʰ V | function | src | [
"open import 1Lab.Prelude",
"open import Data.Partial.Total",
"open import Data.Partial.Base",
"open import Data.Fin.Base hiding (_<_ ; _≤_)",
"open import Data.Nat.Base",
"open import Data.Vec.Base",
"open import Data.Irr",
"open import Realisability.PCA"
] | src/Realisability/PCA/Sugar.lagda.md | lam-syntax | |
SK : Type where S K : SK _`·_ : SK → SK → SK ``` Note that this is *not* a pca, since the application $\tt{K}\ x\ y$ is an element of the inductive type distinct from $x$. We could fix this by making `SK`{.Agda} into a *higher* inductive type, with the two equations of SK combinator calculus as generating paths, but it... | data | src | [
"open import 1Lab.Prelude",
"open import Data.Set.Coequaliser",
"open import Data.Partial.Total",
"open import Data.Partial.Base",
"open import Realisability.PCA.Combinatorial",
"open import Realisability.PCA"
] | src/Realisability/PCA/Instances/Free.lagda.md | SK | |
_ ≻_ : SK → SK → Type where K-β : ∀ a b → K `· a `· b ≻ a S-β : ∀ f g x → S `· f `· g `· x ≻ f `· x `· (g `· x) _`·_ : ∀ {f f' x x'} → f ≻ f' → x ≻ x' → f `· x ≻ f' `· x' stop : ∀ {f} → f ≻ f ``` We then say that two terms $t$ and $t'$ are **convertible** if there exists a term $W$ and a sequence of reductions $t \succ... | data | src | [
"open import 1Lab.Prelude",
"open import Data.Set.Coequaliser",
"open import Data.Partial.Total",
"open import Data.Partial.Base",
"open import Realisability.PCA.Combinatorial",
"open import Realisability.PCA"
] | src/Realisability/PCA/Instances/Free.lagda.md | _ | |
_ ≻*_ : SK → SK → Type where step : ∀ {f f' f''} → f ≻ f' → f' ≻* f'' → f ≻* f'' stop : ∀ {f} → f ≻* f _∼_ : SK → SK → Type x ∼ y = ∃[ W ∈ SK ] (x ≻* W × y ≻* W) ``` <!-- ```agda | data | src | [
"open import 1Lab.Prelude",
"open import Data.Set.Coequaliser",
"open import Data.Partial.Total",
"open import Data.Partial.Base",
"open import Realisability.PCA.Combinatorial",
"open import Realisability.PCA"
] | src/Realisability/PCA/Instances/Free.lagda.md | _ | |
SK-conversion : Congruence SK _ | function | src | [
"open import 1Lab.Prelude",
"open import Data.Set.Coequaliser",
"open import Data.Partial.Total",
"open import Data.Partial.Base",
"open import Realisability.PCA.Combinatorial",
"open import Realisability.PCA"
] | src/Realisability/PCA/Instances/Free.lagda.md | SK-conversion | |
SK-is-pca : is-pca {A = conv.quotient} λ f x → ⦇ appl f x ⦈ | function | src | [
"open import 1Lab.Prelude",
"open import Data.Set.Coequaliser",
"open import Data.Partial.Total",
"open import Data.Partial.Base",
"open import Realisability.PCA.Combinatorial",
"open import Realisability.PCA"
] | src/Realisability/PCA/Instances/Free.lagda.md | SK-is-pca | |
SK-PCA : PCA lzero | function | src | [
"open import 1Lab.Prelude",
"open import Data.Set.Coequaliser",
"open import Data.Partial.Total",
"open import Data.Partial.Base",
"open import Realisability.PCA.Combinatorial",
"open import Realisability.PCA"
] | src/Realisability/PCA/Instances/Free.lagda.md | SK-PCA | |
S-K-no-common-reduct : ∀ x → K ≻* x → S ≻* x → ⊥ | function | src | [
"open import 1Lab.Prelude",
"open import Data.Set.Coequaliser",
"open import Data.Partial.Total",
"open import Data.Partial.Base",
"open import Realisability.PCA.Combinatorial",
"open import Realisability.PCA"
] | src/Realisability/PCA/Instances/Free.lagda.md | S-K-no-common-reduct | |
SK-nontriv : ¬ Path conv.quotient (inc K) (inc S) | function | src | [
"open import 1Lab.Prelude",
"open import Data.Set.Coequaliser",
"open import Data.Partial.Total",
"open import Data.Partial.Base",
"open import Realisability.PCA.Combinatorial",
"open import Realisability.PCA"
] | src/Realisability/PCA/Instances/Free.lagda.md | SK-nontriv | |
inspired : Most of those are conceived as being categories of _geometric_ | function | src | [
"open import Cat.Functor.Kan.Pointwise",
"open import Cat.Diagram.Colimit.Base",
"open import Cat.Diagram.Limit.Finite",
"open import Cat.Functor.Kan.Nerve",
"open import Cat.Instances.Functor",
"open import Cat.Diagram.Initial",
"open import Cat.Instances.Comma",
"open import Cat.Functor.Hom",
"ope... | src/Topoi/Classifying/Diaconescu.lagda.md | inspired |
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