diff --git a/.gitignore b/.gitignore
new file mode 100644
index 0000000..a13427f
--- /dev/null
+++ b/.gitignore
@@ -0,0 +1 @@
+compiled
\ No newline at end of file
diff --git a/README.md b/README.md
index a7ceb45..20a9095 100644
--- a/README.md
+++ b/README.md
@@ -1,4 +1,4 @@
-# untyped NbE in racket
+# Untyped NbE in Typed Racket
 [![status-badge](https://woodpecker.electriclam.com/api/badges/4/status.svg)](https://woodpecker.electriclam.com/repos/4)
 
 An implementation of normalization by evaluation loosely based on [A
diff --git a/nbe-test.rkt b/nbe-test.rkt
index b7b4b81..300a90b 100644
--- a/nbe-test.rkt
+++ b/nbe-test.rkt
@@ -83,6 +83,7 @@
   `(succ ,a))
 
 (check-equal? (normalize `(app ,tm-id ,tm-id)) tm-id)
+(check βη-eq? `(app ,tm-id (U 0)) '(U 0))
 (check-equal? (normalize `(app (app (app ,tm-pair ,tm-id) ,tm-fst) ,tm-snd)) tm-fst)
 (check-equal? (normalize `(app (app (app ,tm-pair ,tm-id) ,tm-fst) ,tm-fst)) tm-id)
 (check-equal? (normalize (tm-app tm-snd (tm-app tm-pair tm-id tm-fst) tm-fst)) tm-fst)
@@ -91,5 +92,10 @@
 (check-equal? (normalize (tm-app tm-nat-to-pnat (tm-nat 10))) (tm-pnat 10))
 (check-equal? (normalize `(ind ,(tm-pnat 3) ,(tm-pnat 0) (var 1))) (tm-pnat 2))
 (check-equal? (normalize `(ind ,(tm-pnat 3) ,tm-loop (var 1))) (tm-pnat 2))
+(check-equal? (normalize (tm-padd (tm-pnat 1) '(succ (var 0)))) '(succ (succ (var 0))))
 (check-equal? (normalize (tm-padd (tm-pnat 10000) (tm-pnat 2000)))
               (tm-pnat 12000))
+(check βη-eq? (tm-padd (tm-pnat 10000) (tm-pnat 2000)) (tm-pnat 12000))
+(check βη-eq? (tm-abs (tm-app (tm-var 1) (tm-var 0))) (tm-app tm-id (tm-var 0)))
+(check βη-eq? `(Π (U 0) (Π (var 0) ,(tm-app tm-id '(var 1)))) '(Π (U 0) (Π (var 0) (var 1))))
+(check-false (βη-eq? '(U 0) '(var 0)))
diff --git a/nbe.rkt b/nbe.rkt
index 57db916..2c309dd 100644
--- a/nbe.rkt
+++ b/nbe.rkt
@@ -5,33 +5,36 @@
 (define-type denv (-> V (Promise D)))
 (define-type V Nonnegative-Integer)
 (define-type Term (∪ 'zero
+                     'nat
                      (List 'succ Term)
                      (List 'var V)
                      (List 'λ Term)
                      (List 'app Term Term)
-                     (List 'ind Term Term Term)))
+                     (List 'ind Term Term Term)
+                     (List 'Π Term Term)
+                     (List 'U V)))
 
 (define-type D (∪ 'zero
+                  (List 'Π (Promise D) (-> (Promise D) D))
                   (List 'succ (Promise D))
                   (List 'fun (-> (Promise D) D))
-                  (List 'neu D-ne)))
+                  (List 'neu D-ne)
+                  (List 'U V)))
 
 (define-type D-ne (∪ (List 'app D-ne D)
                      (List 'idx V)
                      (List 'ind D-ne D (-> (Promise D) (Promise D) D))))
 
-(: ext (-> denv (Promise D) denv))
+(: ext (All (A) (-> (-> V A) A (-> V A))))
 (define (ext ρ a)
     (lambda (i)
     (if (zero? i)
         a
         (ρ (- i 1)))))
 
-
-
-(: interp-fun (-> Term denv D))
+(: interp-fun (-> Term denv (-> (Promise D) D)))
 (define (interp-fun a ρ)
-  (list 'fun (λ (x) (interp a (ext ρ x)))))
+  (λ (x) (interp a (ext ρ x))))
 
 (: interp-fun2 (-> Term denv (-> (Promise D) (Promise D) D)))
 (define (interp-fun2 a ρ)
@@ -42,19 +45,21 @@
   (match a
     [`(neu ,u) `(neu (ind ,u ,(force b) ,c))]
     ['zero (force b)]
-    [`(succ ,a) (c a (delay (interp-ind (force a) b c)))]))
+    [`(succ ,a) (c a (delay (interp-ind (force a) b c)))]
+    [_ (error "type-error: ind")]))
 
-(: ap (-> D Term denv D))
+(: ap (-> Term Term denv D))
 (define (ap a b ρ)
-  (match a
-    ['zero (error "type-error: ap zero")]
-    [`(succ ,_) (error "type-error: ap succ")]
+  (match (interp a ρ)
     [`(fun ,f) (f (delay (interp b ρ)))]
-    [`(neu ,u) `(neu (app ,u ,(interp b ρ)))]))
+    [`(neu ,u) `(neu (app ,u ,(interp b ρ)))]
+    [_ (error "type-error: ap")]))
 
 (: interp (-> Term denv D))
 (define (interp a ρ)
   (match a
+    [`(U ,_) a]
+    [`(Π ,A ,B) `(Π ,(delay (interp A ρ)) ,(interp-fun B ρ))]
     [`(var ,i) (force (ρ i))]
     ['zero 'zero]
     [`(succ ,a) `(succ ,(delay (interp a ρ)))]
@@ -62,17 +67,26 @@
                       (interp a ρ)
                       (delay (interp b ρ))
                       (interp-fun2 c ρ))]
-    [`(λ ,a) (interp-fun a ρ)]
-    [`(app ,a ,b) (ap (interp a ρ) b ρ)]))
+    [`(λ ,a) (list 'fun (interp-fun a ρ))]
+    [`(app ,a ,b) (ap a b ρ)]))
 
 (: reify (-> V D Term))
 (define (reify n a)
   (match a
+    [`(Π ,A ,B) `(Π ,(reify n (force A)) ,(reify (+ n 1) (B (delay `(neu (idx ,n))))))]
     ['zero 'zero]
     [`(succ ,a) `(succ ,(reify n (force a)))]
+    [`(U ,_) a]
     [`(fun ,f) (list 'λ (reify (+ n 1) (f (delay `(neu (idx ,n))))))]
     [`(neu ,a) (reify-neu n a)]))
 
+(: var-to-idx (-> V V V))
+(define (var-to-idx s v)
+  (let ([ret (- s (+ v 1))])
+    (if (< ret 0)
+        (error "variable to index conversion failed")
+        ret)))
+
 (: reify-neu (-> V D-ne Term))
 (define (reify-neu n a)
   (match a
@@ -83,14 +97,16 @@
                                   (c (delay `(neu (idx ,n)))
                                      (delay `(neu (idx ,(+ 1 n)))))))]
     [`(app ,u ,v) (list 'app (reify-neu n u) (reify n v))]
-    [`(idx ,i) (list 'var (max 0 (- n (+ i 1))))]))
+    [`(idx ,i) (list 'var (var-to-idx n i))]))
 
 (: idsub (-> V V D))
-(define (idsub s i) `(neu (idx ,(max 0 (- s (+ i 1))))))
+(define (idsub s i) `(neu (idx ,(var-to-idx s i))))
 
 (: scope (-> Term V))
 (define (scope a)
   (match a
+    [`(Π ,A ,B) (max (scope A) (- (scope B) 2))]
+    [`(U ,_) 0]
     ['zero 0]
     [`(succ ,a) (scope a)]
     (`(ind ,a ,b ,c) (max (scope a) (scope b) (- (scope c) 2)))
@@ -104,56 +120,52 @@
   (let ([sa (scope a)])
     (reify sa (interp a (λ (x) (delay (idsub sa x)))))))
 
-;; (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))]))
+(: up (-> V (-> V Term) (-> V Term)))
+(define (up n ρ)
+  (ext (λ ([x : V]) (subst (λ (i) `(var ,(+ i n))) (ρ x))) '(var 0)))
 
-;; (define (idsub-tm i) `(var ,i))
-;; (define (subst1 b a)
-;;   (subst (ext idsub-tm b) a))
+(: subst (-> (-> V Term) Term Term))
+(define (subst ρ a)
+  (match a
+    [`(U ,_) a]
+    [`(Π ,A ,B) `(Π ,(subst ρ A) ,(subst (up 1 ρ) B))]
+    [`(var ,i) (ρ i)]
+    [`(app ,a ,b) `(app ,(subst ρ a) ,(subst ρ b))]
+    [`(λ ,a) `(λ ,(subst (up 1 ρ) a))]
+    ['zero 'zero]
+    [`(succ ,a) `(succ ,(subst ρ a))]
+    [`(ind ,a ,b ,c) `(ind ,(subst ρ a) ,(subst ρ b) ,(subst (up 2 ρ) c))]))
 
-;; (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))])]))
+(: idsub-tm (-> V Term))
+(define (idsub-tm i) `(var ,i))
+(: subst1 (-> Term Term Term))
+(define (subst1 b a)
+  (subst (ext idsub-tm b) a))
 
-;; (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
+(: η-eq? (-> Term Term Boolean))
+(define (η-eq? a b)
+  (match (list a b)
+    [`((U ,i) (U ,j)) (eqv? i j) ]
+    [`((Π ,A0 ,B0) (Π ,A1 ,B1))
+     (and (η-eq? A0 A1) (η-eq? B0 B1))]
+    ['(zero zero) true]
+    [`((succ ,a) (succ ,b)) (η-eq? a b)]
+    [`((ind ,a ,b ,c) (ind ,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]))
 
-;; ;; Coquand's algorithm but for β-normal forms
-;; (: η-eq? (-> Term Term Boolean))
-;; (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)))
+(: βη-eq? (-> Term Term Boolean))
+(define (βη-eq? a b)
+  (η-eq? (normalize a) (normalize b)))
 
 (: β-eq? (-> Term Term Boolean))
 (define (β-eq? a b)
   (equal? (normalize a) (normalize b)))
 
-(provide reify interp normalize β-eq? Term D V)
+(provide reify interp normalize β-eq? Term D V βη-eq?)