------------------------------------------------------------------------
-- Big-step semantics for the untyped λ-calculus
------------------------------------------------------------------------

module Lambda.Substitution.OneSemantics where

open import Coinduction
open import Data.Fin
open import Function
open import Data.Nat

open import Lambda.Syntax
open WHNF
open import Lambda.Substitution

-- Semantic domain. ⊥ represents non-termination.

data Sem (n : ) : Set where
     : Sem n
  val : (v : Value n)  Sem n

-- Big-step semantics.

mutual

  infix 4 _⇒_ _⇒?_

  data _⇒_ {n} : Tm n  Sem n  Set where
    val :  {v}   v   val v
    app :  {t₁ t₂ t v s}
          (t₁⇓   : t₁ ⇒? val (ƛ t))
          (t₂⇓   : t₂ ⇒? val v)
          (t₁t₂⇒ : t / sub  v  ⇒? s) 
          t₁ · t₂  s
    ·ˡ  :  {t₁ t₂}
          (t₁⇑ : t₁ ⇒? ) 
          t₁ · t₂  
    ·ʳ  :  {t₁ t₂ v}
          (t₁⇓ : t₁ ⇒? val v)
          (t₂⇑ : t₂ ⇒? ) 
          t₁ · t₂  

  -- Conditional coinduction: coinduction only for diverging
  -- computations.

  _⇒?_ :  {n}  Tm n  Sem n  Set
  t ⇒?      =  (t  )
  t ⇒? val v = t  val v

-- Example.

Ω-loops : Ω  
Ω-loops = app val val ( Ω-loops)