module ahmed.day1 where

open import Agda.Primitive
open import Data.Nat
open import Data.Fin
open import Data.Product

data Type : Set where
  Bool : Type
  _-→_ : Type → Type → Type

data Term (n : ℕ) : Set where
  Var : (m : Fin n) → Term n
  True : Term n
  False : Term n
  If_Then_Else_ : Term n → Term n → Term n → Term n
  λ[_::_]_ : Term n → Type → Term n → Term n
  _∙_ : Term n → Term n → Term n

data isValue (n : ℕ) : Term n → Set where
  TrueValue : isValue n True
  FalseValue : isValue n False
  LambdaValue : (x : Term n) → (τ : Type) → (e : Term n) → isValue n (λ[ x :: τ ] e)

Value = (n : ℕ) → Σ (Term n) (isValue n)

data evaluationContext : (n : ℕ) → Set where
  dot : evaluationContext 0
  EIf_Then_Else_ : {n : ℕ} → evaluationContext n → Term n → Term n → evaluationContext n
  EAppLeft : {n : ℕ} → evaluationContext n → Term n → evaluationContext n
  EAppRight : {n : ℕ} → Value → evaluationContext n → evaluationContext n

data isValidValueForType (n : ℕ) (t : Type) : (e : Term n) → Set where
  TrueV : isValidValueForType n Bool True