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{-# OPTIONS --without-K --safe #-}
open import Relation.Binary using (Setoid)
module Cfe.Judgement.Base
{c ℓ} (over : Setoid c ℓ)
where
open import Cfe.Expression over as E
open import Cfe.Type over renaming (_∙_ to _∙ₜ_; _∨_ to _∨ₜ_)
open import Cfe.Type.Construct.Lift over
open import Data.Fin as F
open import Data.Nat as ℕ hiding (_⊔_)
open import Data.Vec hiding (_⊛_)
open import Level hiding (Lift) renaming (suc to lsuc)
open import Relation.Binary.PropositionalEquality
infix 2 _,_⊢_∶_
infix 4 _≅_
data _,_⊢_∶_ : {m : ℕ} → {n : ℕ} → Vec (Type ℓ ℓ) m → Vec (Type ℓ ℓ) n → Expression (n ℕ.+ m) → Type ℓ ℓ → Set (c ⊔ lsuc ℓ) where
Eps : ∀ {m n} {Γ : Vec _ m} {Δ : Vec _ n} → Γ , Δ ⊢ ε ∶ Lift ℓ ℓ τε
Char : ∀ {m n} {Γ : Vec _ m} {Δ : Vec _ n} c → Γ , Δ ⊢ Char c ∶ Lift ℓ ℓ τ[ c ]
Bot : ∀ {m n} {Γ : Vec _ m} {Δ : Vec _ n} → Γ , Δ ⊢ ⊥ ∶ Lift ℓ ℓ τ⊥
Var : ∀ {m n : ℕ} {Γ : Vec _ m} {Δ : Vec _ n} {i : Fin (n ℕ.+ m)} (i≥n : toℕ i ≥ n) → Γ , Δ ⊢ Var i ∶ lookup Γ (reduce≥ i i≥n)
Fix : ∀ {m n} {Γ : Vec _ m} {Δ : Vec _ n} {e τ} → Γ , τ ∷ Δ ⊢ e ∶ τ → Γ , Δ ⊢ μ e ∶ τ
Cat : ∀ {m n} {Γ : Vec _ m} {Δ : Vec _ n} {e₁ e₂ τ₁ τ₂} → Γ , Δ ⊢ e₁ ∶ τ₁ → Δ ++ Γ , [] ⊢ e₂ ∶ τ₂ → (τ₁⊛τ₂ : τ₁ ⊛ τ₂) → Γ , Δ ⊢ e₁ ∙ e₂ ∶ τ₁ ∙ₜ τ₂
Vee : ∀ {m n} {Γ : Vec _ m} {Δ : Vec _ n} {e₁ e₂ τ₁ τ₂} → Γ , Δ ⊢ e₁ ∶ τ₁ → Γ , Δ ⊢ e₂ ∶ τ₂ → (τ₁#τ₂ : τ₁ # τ₂) → Γ , Δ ⊢ e₁ ∨ e₂ ∶ τ₁ ∨ₜ τ₂
vcast : ∀ {a A m n} → .(m ≡ n) → Vec {a} A m → Vec A n
vcast {n = suc n} eq (x ∷ xs) = x ∷ vcast (suc-injective eq) xs
where
open import Data.Nat.Properties using (suc-injective)
vcast {n = ℕ.zero} eq [] = []
data _≅_ {a A} : {m n : ℕ} → Vec {a} A m → Vec A n → Set a where
[]≅[] : [] ≅ []
_∷_ : ∀ {m n x y} {xs : Vec _ m} {ys : Vec _ n} → (x≡y : x ≡ y) → xs ≅ ys → x ∷ xs ≅ y ∷ ys
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