{-# OPTIONS --prop #-}
module Ahmed.Day1 where

open import Agda.Builtin.Sigma
open import Data.Bool
open import Data.Empty
open import Data.Fin
open import Data.Maybe
open import Data.Nat
open import Data.Product
open import Data.Sum
open import Relation.Nullary

id : {A : Set} → A → A
id {A} x = x

data type : Set where
  bool : type
  _-→_ : type → type → type

data term : Set where
  `_ : ℕ → term
  `true : term
  `false : term
  `if_then_else_ : term → term → term → term
  `λ[_]_ : (τ : type) → (e : term) → term
  _∙_ : term → term → term

data ctx : Set where
  nil : ctx
  cons : ctx → type → ctx

lookup : ctx → ℕ → Maybe type
lookup nil _ = nothing
lookup (cons ctx₁ x) zero = just x 
lookup (cons ctx₁ x) (suc n) = lookup ctx₁ n

data sub : Set where
  nil : sub
  cons : sub → term → sub

subst : term → term → term
subst (` zero) v = v
subst (` suc x) v = ` x
subst `true v = `true
subst `false v = `false
subst (`if x then x₁ else x₂) v = `if (subst x v) then (subst x₁ v) else (subst x₂ v)
subst (`λ[ τ ] x) v = `λ[ τ ] subst x v
subst (x ∙ x₁) v = (subst x v) ∙ (subst x₁ v)

data value-rel : type → term → Set where 
  v-`true : value-rel bool `true
  v-`false : value-rel bool `false
  v-`λ[_]_ : ∀ {τ e} → value-rel τ (`λ[ τ ] e)

data good-subst : ctx → sub → Set where
  nil : good-subst nil nil
  cons : ∀ {ctx τ γ v}
    → good-subst ctx γ
    → value-rel τ v
    → good-subst (cons ctx τ) γ

data step : term → term → Set where
  step-if-1 : ∀ {e₁ e₂} → step (`if `true then e₁ else e₂) e₁
  step-if-2 : ∀ {e₁ e₂} → step (`if `false then e₁ else e₂) e₂
  step-`λ : ∀ {τ e v} → step ((`λ[ τ ] e) ∙ v) (subst e v)

data steps : ℕ → term → term → Set where
  zero : ∀ {e} → steps zero e e
  suc : ∀ {e e' e''} → (n : ℕ) → step e e' → steps n e' e'' → steps (suc n) e e''

data well-typed : ctx → term → type → Set where
  type-`true : ∀ {ctx} → well-typed ctx `true bool
  type-`false : ∀ {ctx} → well-typed ctx `false bool
  type-`ifthenelse : ∀ {ctx e e₁ e₂ τ}
    → well-typed ctx e bool
    → well-typed ctx e₁ τ
    → well-typed ctx e₂ τ
    → well-typed ctx (`if e then e₁ else e₂) τ
  type-`x : ∀ {ctx x}
    → (p : Is-just (lookup ctx x))
    → well-typed ctx (` x) (to-witness p)
  type-`λ : ∀ {ctx τ τ₂ e}
    → well-typed (cons ctx τ) e τ₂
    → well-typed ctx (`λ[ τ ] e) (τ -→ τ₂)
  type-∙ : ∀ {ctx τ₁ τ₂ e₁ e₂}
    → well-typed ctx e₁ (τ₁ -→ τ₂)
    → well-typed ctx e₂ τ₂
    → well-typed ctx (e₁ ∙ e₂) τ₂

_⊢_∶_ : (Γ : ctx) → (e : term) → (τ : type) → Set
Γ ⊢ e ∶ τ = (well-typed Γ e τ) × (∀ {n} → (e' : term) → steps n e e' → value-rel τ e' ⊎ ∃ λ e'' → step e' e'')

irreducible : term → Set
irreducible e = ¬ (∃ λ e' → step e e')

data term-relation : type → term → Set where
  e-term : ∀ {τ e}
    → (∀ {n} → (e' : term) → steps n e e' → irreducible e' → value-rel τ e')
    → term-relation τ e

_⊨_∶_ : (Γ : ctx) → (e : term) → (τ : type) → Set
_⊨_∶_ Γ e τ = (γ : sub) → (good-subst Γ γ) → term-relation τ e

fundamental : ∀ {Γ e τ} → Γ ⊢ e ∶ τ → Γ ⊨ e ∶ τ
fundamental {Γ} {e} {τ} type-sound γ good-sub = e-term f
  where
    f : {n : ℕ} (e' : term) → steps n e e' → irreducible e' → value-rel τ e'
    f e' steps irred = [ id , (λ exists → ⊥-elim (irred exists)) ] (snd type-sound e' steps)