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Add back beta-eta equality
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Yiyun Liu 2025-05-12 11:24:56 -04:00
parent aa233cc86b
commit 755119316b
2 changed files with 39 additions and 47 deletions
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3
nbe-test.rkt
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@ -91,5 +91,8 @@
(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)))

83
nbe.rkt
View file

@ -20,15 +20,13 @@
(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))
(define (interp-fun a ρ)
(list 'fun (λ (x) (interp a (ext ρ x)))))
@ -104,56 +102,47 @@
(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
[`(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)
['(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?)