#lang typed/racket
;; Grammar (Λ)
;; t := λ t | app t t | i

(define-type V Nonnegative-Integer)

(define-type Term (∪ Var Abs App))
(struct Var ([get : V]))
(struct Abs ([body : Term]))
(struct App ([fun : Term] [arg : Term]))

(define-type D (∪ D-ne Clos))
(struct Idx ([get : V]))
(struct D-ne ([get : (∪ Idx DApp)]))
(struct Clos ([get : (-> (Promise D) (Promise D))] ))
(struct DApp ([fun : D-ne] [arg : (Promise D)]))



(: ext (-> (-> V (Promise D)) (Promise D) (-> V (Promise D))))
(define (ext ρ a)
    (lambda (i)
    (if (zero? i)
        a
        (ρ (- i 1)))))

(: ap (-> (Promise D) (Promise D) D))
(define (ap a b)
  (match (force a)
    [(Clos f) (force (f b))]
    [(D-ne u) (D-ne (DApp (D-ne u) b))]))

;; (define-syntax-rule (ap a b)
;;   (match (force a)
;;     [`(fun ,f) (force (f b))]
;;     [`(neu ,u) `(neu (app ,u ,b))]
;;     [_ (error "ap: type error")]))


;; Domain
;; D := neu D_ne | fun [(var -> var) -> D → D]
;; D_ne := var i | app D_ne D

;; (define (tm? a)
;;   (match a
;;     ['zero true]
;;     [`(succ ,a) (tm? a)]
;;     [`(if-zero ,a ,b ,c) (and (tm? a) (tm? b) (tm? c))]
;;     [`(λ ,a) (tm? a)]
;;     [`(app ,a ,b) (and (tm? a) (tm? b))]
;;     [`(var ,i) (exact-nonnegative-integer? i)]
;;     [_ false]))

;; (define-syntax-rule (ap a b)
;;   (match (force a)
;;     [`(fun ,f) (force (f b))]
;;     [`(neu ,u) `(neu (app ,u ,b))]
;;     [_ (error "ap: type error")]))

;; (define-syntax-rule (ifz a b c)
;;   (match (force a)
;;     ['zero (force b)]
;;     [`(succ ,u) (ap c u)]
;;     [`(neu ,u) `(neu (if-zero ,u ,b ,c))]))

;; (define-syntax-rule (ext ρ a)
;;   (lambda (i)
;;     (if (zero? i)
;;         a
;;         (ρ (- i 1)))))

;; (define-syntax-rule (interp-fun a ρ)
;;   (list 'fun (λ (x) (interp a (ext ρ x)))))

;; (: interp (-> Term (-> Term)))

;; (define (interp a ρ)
;;   (delay (match a
;;     [`(var ,i) (force (ρ i))]
;;     ['zero 'zero]
;;     [`(succ ,a) `(succ ,(interp a ρ))]
;;     [`(if-zero ,a ,b ,c) (ifz (interp a ρ) (interp b ρ) (interp-fun c ρ))]
;;     [`(λ ,a) (interp-fun a ρ)]
;;     [`(app ,a ,b) (ap (interp a ρ) (interp b ρ))])))

;; (define (reify n a)
;;   (match (force a)
;;     ['zero 'zero]
;;     [`(succ ,a) `(succ ,(reify n a))]
;;     [`(fun ,f) (list 'λ (reify (+ n 1) (f `(neu (var ,n)))))]
;;     [`(neu ,a) (reify-neu n a)]))

;; (define (extract-body a)
;;   (match a
;;     [`(λ ,a) a]
;;     [_ (error "reify-neu: not reifiable")]))

;; (define (reify-neu n a)
;;   (match a
;;     [`(if-zero ,a ,b ,c) (list 'if (reify-neu n a) (reify n b) (extract-body (reify n c)))]
;;     [`(app ,u ,v) (list 'app (reify-neu n u) (reify n v))]
;;     [`(var ,i) (list 'var (- n (+ i 1)))]))

;; (define (idsub s i) `(neu (var ,(- s (+ i 1)))))

;; (define (scope a)
;;   (match a
;;     ['zero 0]
;;     [`(succ ,a) (scope a)]
;;     [`(if-zero ,a ,b ,c) (max (scope a) (scope b) (scope c))]
;;     [`(λ ,a) (max 0 (- (scope a) 1))]
;;     [`(app ,a ,b) (max (scope a) (scope b))]
;;     [`(var ,i) (+ i 1)]))

;; (define (normalize a)
;;   (let ([sa (scope a)])
;;     (reify sa (interp a (curry idsub sa)))))

;; (define (subst ρ a)
;;   (match a
;;     [`(var ,i) (ρ i)]
;;     [`(app ,a ,b) `(app ,(subst ρ a) ,(subst ρ b))]
;;     [`(λ ,a) `(λ ,(subst (ext (compose (curry subst (λ (i) `(var ,(+ i 1)))) ρ)
;;                               '(var 0)) a))]))

;; (define (idsub-tm i) `(var ,i))
;; (define (subst1 b a)
;;   (subst (ext idsub-tm b) a))

;; (define (eval-tm a)
;;   (match a
;;     [(list 'var _) a]
;;     [(list 'λ a) `(λ ,(eval-tm a))]
;;     [(list 'app a b)
;;      (match (eval-tm a)
;;        [(list 'λ a) (eval-tm (subst1 b a))]
;;        [v `(app ,v ,(eval-tm b))])]))

;; (define (eval-tm-strict a)
;;   (match a
;;     [(list 'var _) a]
;;     [(list 'λ a) `(λ ,(eval-tm-strict a))]
;;     [(list 'app a b)
;;      (match (eval-tm-strict a)
;;        [(list 'λ a) (eval-tm-strict (subst1 (eval-tm-strict b) a))]
;;        [v `(app ,v ,(eval-tm-strict b))])]))

;; ;; Coquand's algorithm but for β-normal forms
;; (define (η-eq? a b)
;;   (match (list a b)
;;     ['(zero zero) true]
;;     [`((succ ,a) (succ ,b)) (η-eq? a b)]
;;     [`((if-zero ,a ,b ,c) (if-zero ,a0 ,b0 ,c0))
;;      (and (η-eq? a a0) (η-eq? b b0) (η-eq? c c0))]
;;     [`((λ ,a) (λ ,b)) (η-eq? a b)]
;;     [`((λ ,a) ,u) (η-eq? a `(app ,(subst (λ (i) `(var ,(+ i 1))) u) (var 0)))]
;;     [`(,u (λ ,a)) (η-eq? `(app ,(subst (λ (i) `(var ,(+ i 1))) u) (var 0)) a)]
;;     [`((app ,u0 ,v0) (app ,u1 ,v1)) (and (η-eq? u0 u1) (η-eq? v0 v1))]
;;     [`((var ,i) (var ,j)) (eqv? i j)]
;;     [_ false]))


;; (define (βη-eq? a b)
;;   (η-eq? (normalize a) (normalize b)))

;; (define (β-eq? a b)
;;   (equal? (normalize a) (normalize b)))

;; (provide eval-tm eval-tm-strict reify interp normalize tm? η-eq? βη-eq? β-eq?)