------------------------------------------------------------------------
-- A self-interpreter (without correctness proof)
------------------------------------------------------------------------

{-# OPTIONS --without-K --safe #-}

module Self-interpreter where

open import Prelude hiding (const)

-- To simplify the development, let's work with actual natural numbers
-- as variables and constants (see
-- Atom.one-can-restrict-attention-to-χ-ℕ-atoms).

open import Atom

open import Chi            χ-ℕ-atoms
open import Constants      χ-ℕ-atoms
open import Free-variables χ-ℕ-atoms

open χ-atoms χ-ℕ-atoms

-- Equality of natural numbers.

internal-equal : Exp
internal-equal =
  rec v-equal (lambda v-m (lambda v-n (case (var v-m) branches)))
  module Internal-equal where
  zero-branches =
    branch c-zero [] (const c-true []) 
    branch c-suc (v-n  []) (const c-false []) 
    []

  suc-branches =
    branch c-zero [] (const c-false []) 
    branch c-suc (v-n  []) (
      apply (apply (var v-equal) (var v-m)) (var v-n)) 
    []

  branches =
    branch c-zero [] (case (var v-n) zero-branches) 
    branch c-suc (v-m  []) (case (var v-n) suc-branches) 
    []

-- Membership of a list of natural numbers.

internal-member : Exp
internal-member =
  lambda v-x body
  module Internal-member where
  cons-branches =
    branch c-true [] (const c-true []) 
    branch c-false [] (apply (var v-member) (var v-xs)) 
    []

  branches =
    branch c-nil [] (const c-false []) 
    branch c-cons (v-y  v-xs  []) (
      case (apply (apply internal-equal (var v-x)) (var v-y))
        cons-branches) 
    []

  body = rec v-member (lambda v-xs (case (var v-xs) branches))

-- Substitution.

internal-subst : Exp
internal-subst = lambda v-x (lambda v-new body)
  module Internal-subst where
  private
    rec-or-lambda = λ c 
      branch c (v-y  v-e  []) (
        case (apply (apply internal-equal (var v-x)) (var v-y)) (
          branch c-true [] (const c (var v-y  var v-e  [])) 
          branch c-false [] (
            const c (var v-y  apply (var v-subst) (var v-e)  [])) 
          []))

  branches =
    branch c-apply (v-e₁  v-e₂  []) (
      const c-apply (apply (var v-subst) (var v-e₁) 
                     apply (var v-subst) (var v-e₂)  [])) 
    branch c-case (v-e  v-bs  []) (
      const c-case (
        apply (var v-subst) (var v-e) 
        apply (var v-subst) (var v-bs)  [])) 
    rec-or-lambda c-rec 
    rec-or-lambda c-lambda 
    branch c-const (v-c  v-es  []) (
      const c-const (var v-c  apply (var v-subst) (var v-es)  [])) 
    branch c-var (v-y  []) (
      case (apply (apply internal-equal (var v-x)) (var v-y)) (
        branch c-true [] (var v-new) 
        branch c-false [] (const c-var (var v-y  []))  [])) 
    branch c-branch (v-c  v-ys  v-e  []) (
      const c-branch (
        var v-c 
        var v-ys 
        case (apply (apply internal-member (var v-x)) (var v-ys)) (
          branch c-true [] (var v-e) 
          branch c-false [] (apply (var v-subst) (var v-e)) 
          [])  [])) 
    branch c-nil [] (const c-nil []) 
    branch c-cons (v-e  v-es  []) (
      const c-cons (apply (var v-subst) (var v-e) 
                    apply (var v-subst) (var v-es)  [])) 
    []

  body = rec v-subst (lambda v-e (case (var v-e) branches))

-- Searches for a branch matching a given natural number.

internal-lookup : Exp
internal-lookup =
  lambda v-c (rec v-lookup (lambda v-bs (case (var v-bs) (
    branch c-cons (v-b  v-bs  []) (case (var v-b) (
      branch c-branch (v-c′  v-underscore  v-underscore  []) (
        case (apply (apply internal-equal (var v-c)) (var v-c′)) (
          branch c-false [] (apply (var v-lookup) (var v-bs)) 
          branch c-true [] (var v-b)  []))  []))  []))))

-- Tries to apply multiple substitutions.

internal-substs : Exp
internal-substs =
  rec v-substs (lambda v-xs (lambda v-es (lambda v-e′ (
    case (var v-xs) (
      branch c-nil [] (case (var v-es) (
        branch c-nil [] (var v-e′)  [])) 
      branch c-cons (v-x  v-xs  []) (case (var v-es) (
        branch c-cons (v-e  v-es  []) (
          apply (apply (apply internal-subst (var v-x)) (var v-e))
            (apply (apply (apply (var v-substs) (var v-xs))
               (var v-es)) (var v-e′)))  []))  [])))))

-- A map function. The application of map to the function is done at
-- the meta-level in order to simplify proofs.

map : Exp  Exp
map f = rec v-map (lambda v-xs (case (var v-xs) branches))
  module Map where
  branches =
    branch c-nil [] (const c-nil []) 
    branch c-cons (v-x  v-xs  []) (
      const c-cons (apply f (var v-x) 
                    apply (var v-map) (var v-xs) 
                    [])) 
    []

-- The self-interpreter.

eval : Exp
eval =
  rec v-eval (lambda v-p (case (var v-p) branches))
  module Eval where
  apply-branch =
    branch c-lambda (v-x  v-e  []) (
      apply (var v-eval) (
        apply (apply (apply internal-subst (var v-x))
                 (apply (var v-eval) (var v-e₂)))
          (var v-e))) 
    []

  case-body₂ =
    case (apply (apply internal-lookup (var v-c)) (var v-bs)) (
      branch c-branch (v-underscore  v-xs  v-e  []) (
        apply (var v-eval) (
          apply (apply (apply internal-substs (var v-xs))
            (var v-es)) (var v-e))) 
      [])

  case-body₁ =
    case (apply (var v-eval) (var v-e)) (
      branch c-const (v-c  v-es  []) case-body₂ 
      [])

  branches =
    branch c-apply (v-e₁  v-e₂  []) (
      case (apply (var v-eval) (var v-e₁)) apply-branch) 
    branch c-case (v-e  v-bs  []) case-body₁ 
    branch c-rec (v-x  v-e  []) (
      apply (var v-eval) (
        apply (apply (apply internal-subst (var v-x))
                 (const c-rec (var v-x  var v-e  [])))
          (var v-e))) 
    branch c-lambda (v-x  v-e  []) (
      const c-lambda (var v-x  var v-e  [])) 
    branch c-const (v-c  v-es  []) (
      const c-const (var v-c 
                     apply (map (var v-eval)) (var v-es) 
                     [])) 
    []

eval-closed : Closed eval
eval-closed = from-⊎ (closed? eval)