1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
|
#lang racket
(require "lib.rkt")
;; The Simply-Typed Lambda Calculus with higher-order *impredicative* references
; Γ, x: τ₁ ⊢ e: τ₂ k > max-level(Γ, τ₁, τ₂)
; ---------------------------------------------
; Γ ⊢ λx:τ₁.e : τ₁ →ᵏ τ₂
; Γ ⊢ e₁: τ₁ →ᵏ τ₂ Γ ⊢ e₂: τ₁
; --------------------------------
; Γ ⊢ (e₁ e₂): τ₂
; --------------------------
; Nat::Type₀, Unit::Type₀
; τ::Type₀
; ---------------
; Ref τ :: Type₀
; τ::Typeᵢ, i ≥ 0
; ---------------
; Ref τ :: Typeᵢ₊₁
; τ₁::Typeᵢ, τ₂::Typeⱼ, k > max-level(τ₁, τ₂)
; -----------------------------------------
; τ₁ →ᵏ τ₂ :: Typeₖ
(require (only-in "stlc-ref.rkt" interpret))
;; (check Expr Type Table[Sym, Type]): Bool
(define (check expr with)
(check-core (desugar expr) with #hash()))
(define (check-core expr with Γ)
(match with
['sole (equal? 'Unit with)]
[n #:when (natural? n) (equal? 'Nat with)]
[x #:when (dict-has-key? Γ x)
(equal? (dict-ref Γ x) with)]
[`(new ,e)
(match with
[`(Ref ,t) (check-core e t Γ)]
[_ #f])]
[`(! ,e) (check-core e `(Ref ,with) Γ)]
[`(set ,e1 ,e2)
(match (infer-core e1 Γ)
[`(Ref ,t)
(and (equal? 'Unit with)
(check-core e2 t Γ))]
[_ #f])]
[`(λ (,x : ,t) ,e)
(match with
[`(,t1 → ,k ,t2)
(and (equal? t t1) (check-core e t2 (dict-set Γ x t))
(> k (max-level e t1 t2 (dict-set Γ x t1))))] ; KNOB
[_ #f])]
[`(,e1 ,e2)
(match (infer-core e1 Γ)
[`(,t1 → ,k ,t2)
(and (equal? t2 with)
(equal? t1 (infer-core e2 Γ)))]
[_ #f])]
[_ #f]))
;; (infer Expr Table[Sym, Type]): Type
(define (infer expr)
(infer-core (desugar expr) #hash()))
(define (infer-core expr Γ)
(match expr
['sole 'Unit]
[n #:when (natural? n) 'Nat]
[x #:when (dict-has-key? Γ x)
(dict-ref Γ x)]
[`(new ,e) `(Ref ,(infer-core e Γ))]
[`(! ,e)
(match (infer-core e Γ)
[`(Ref ,t) t]
[t (err "attempting to deref term not of Ref type!")])]
[`(set ,e1 ,e2)
(match (infer-core e1 Γ)
[`(Ref ,t)
(if (check-core e2 t Γ) 'Unit
(err (format "attempting to update ~a: ~a with term ~a: ~a of differing type"
e1 t e2 (infer-core e2 Γ))))]
[t (err (format "attempting to update non-reference ~a: ~a" e1 t))])]
[`(λ (,x : ,t1) ,e)
(let ([t2 (infer-core e (dict-set Γ x t1))])
(let ([k (+ 1 (max-level e (dict-set Γ x t1) t1 t2))]) ; KNOB
`(,t1 → ,k ,t2)))]
[`(,e1 ,e2)
(match (infer-core e1 Γ)
[`(,t1 → ,k ,t2)
(if (check-core e2 t1 Γ) t2
(err (format "inferred argument type ~a does not match arg ~a of type ~a" t1 e2 (infer-core e2 Γ))))]
[t (err (format "expected → type on application body, got ~a" t))])]
[e (err (format "attempting to infer an unknown expression ~a" e))]))
;; (max-level Table[Sym, Type] Expr Type Type): Natural
(define (max-level e Γ t1 t2)
(max
(level-type t1)
(level-type t2)
(level-body e Γ)))
;; (level-type Type): Natural
(define (level-type t)
(match t
['Unit 0]
['Nat 0]
[`(,t1 → ,k ,t2)
(if (or (< k (level-type t1)) (< k (level-type t2)))
(err (format "annotated level ~a is less than inferred levels of ~a and ~a!"
k t1 t2))
k)]
[`(Ref ,t)
(let ([k (level-type t)])
(if (zero? k) 0 (+ 1 k)))] ; KNOB
[t (err (format "attempting to infer the level of unknown type ~a" t))]))
;; (level-body Expr Table[Sym, Type]): Natural
(define (level-body e Γ)
(match e
['sole 0]
[n #:when (natural? n) 0]
[x #:when (dict-has-key? Γ x)
(level-type (dict-ref Γ x))]
[`(new ,e) (level-body e Γ)]
[`(! ,e) (level-body e Γ)]
[`(set ,e1 ,e2) (max (level-body e1 Γ) (level-body e2 Γ))]
[`(λ (,x : ,t) ,e) (level-body e (dict-set Γ x t))] ; todo: should be 0?
[`(,e1 ,e2) (max (level-body e1 Γ) (level-body e2 Γ))]))
|
