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
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
|
#lang racket
(require "lib.rkt")
;; The Simply-Typed Lambda Calculus, with simple extensions
;; Unit/String/Natural/Boolean, pairs, sums, lists, ascryption
;; (interpret Expr Table[Sym, Expr]): Value
(define (interpret expr [ctx #hash()])
(interpret- (strip (desugar expr)) ctx))
(define (interpret- expr ctx heap)
(match expr
['sole 'sole]
[s #:when (string? s) s]
[n #:when (natural? n) n]
[b #:when (boolean? b) b]
[x #:when (dict-has-key? ctx x) (dict-ref ctx x)]
[`(inc ,e)
(match (interpret- e ctx heap)
[n #:when (natural? n) (+ n 1)]
[e (format "incrementing an unknown value ~a" e)])]
[`(if ,c ,e1 ,e2)
(match (interpret- c ctx heap)
['#t (interpret- e1 ctx heap)]
['#f (interpret- e2 ctx heap)]
[e (err (format "calling if on unknown expression ~a" e))])]
[`(pair ,e1 ,e2)
`(pair ,(interpret- e1 ctx) ,(interpret- e2 ctx))]
[`(car ,e)
(match (interpret- e ctx)
[`(pair ,e1 ,e2) e1]
[e (err (format "calling car on unknown expression ~a" e))])]
[`(cdr ,e)
(match (interpret- e ctx)
[`(pair ,e1 ,e2) e2]
[e (err (format "calling cdr on unknown expression ~a" e))])]
[`(inl ,e) `(inl ,(interpret- e ctx))]
[`(inr ,e) `(inr ,(interpret- e ctx))]
[`(case ,e ,f1 ,f2)
(match (interpret- e ctx)
[`(inl ,e) (interpret- `(,f1 ,e) ctx)]
[`(inr ,e) (interpret- `(,f2 ,e) ctx)]
[e (err (format "calling case on unknown expression ~a" e))])]
['nil 'nil]
[`(nil? ,e)
(match (interpret- e ctx)
['nil '#t]
[`(cons ,e1 ,e2) '#f]
[e (err (format "calling isnil on unknown expression ~a" e))])]
[`(cons ,e1 ,e2)
`(cons ,(interpret- e1 ctx) ,(interpret- e2 ctx))]
[`(head ,e)
(match (interpret- e ctx)
[`(cons ,e1 ,e2) (interpret- e1 ctx)]
[e (err (format "calling head on unknown expression ~a" e))])]
[`(tail ,e)
(match (interpret- e ctx)
[`(cons ,e1 ,e2) (interpret- e2 ctx)]
[e (err (format "calling tail on unknown expression ~a" e))])]
[`(λ ,id ,body) `(λ ,id ,body ,ctx)]
[`(,body ,arg)
(match (interpret- body ctx)
[`(λ ,id ,body ,env)
(interpret- body (dict-set env id (interpret- arg ctx)))]
[e (err (format "applying arg ~a to unknown expression ~a" arg e))])]
[e (err (format "interpreting an unknown expression ~a" e))]))
;; (check Expr Type Table[Sym, Type]): Bool
(define (check expr with [Γ #hash()])
(check- (desugar expr) with Γ))
(define (check- expr with Γ)
(let ([with (if (dict-has-key? Γ with) (dict-ref Γ with) with)])
(match* (expr with)
[('sole 'Unit) #t]
[(s 'Str) #:when (string? s) #t]
[(n 'Nat) #:when (natural? n) #t]
[(b 'Bool) #:when (boolean? b) #t]
[(e `(+ ,t1 ,t2))
(or (check- e t1 Γ) (check- e t2 Γ))]
[(x _) #:when (dict-has-key? Γ x)
(equal? (dict-ref Γ x) with)]
[(`(type ,t1 ,t2 ,in) with)
(check- in with (dict-set Γ t1 t2))]
[(`(pair ,e1 ,e2) `(× ,t1 ,t2))
(and (check- e1 t1 Γ) (check- e2 t2 Γ))]
[(`(car ,e) with)
(match (infer- e Γ)
[`(× ,t1 ,t2) (equal? t1 with)]
[t #f])]
[(`(cdr ,e) with)
(match (infer- e Γ)
[`(× ,t1 ,t2) (equal? t2 with)]
[t #f])]
[(`(inl ,e) with)
(match (infer- e Γ)
[`(+ ,t1 ,t2) (equal? t1 with)]
[t #f])]
[(`(inr ,e) with)
(match (infer- e Γ)
[`(+ ,t1 ,t2) (equal? t2 with)]
[t #f])]
[(`(case ,e ,f1 ,f2) with)
(match* ((infer- f1 Γ) (infer- f2 Γ))
[(`(→ ,a1 ,t1) `(→ ,a2 ,t2))
(and (check- e `(+ ,a1 ,a2))
(check- f1 `(→ ,a1 ,with) Γ) (check- f2 `(→ ,a2 ,with) Γ))]
[(t1 t2) #f])]
[(`(,e (: ,t)) with)
(and (equal? t with) (check- e t Γ))]
[('nil `(List ,t)) #t]
[(`(cons ,f1 ,f2) `(List ,t))
(and (check- f1 t Γ) (check- f2 `(List ,t) Γ))]
[(`(head ,e) with)
(match (infer- e)
[`(List ,t) (equal? t with)]
[t #f])]
[(`(tail ,e) with)
(equal? (infer- e Γ) with)]
[(`(λ ,x (: ,t) ,e) `(→ ,t1 ,t2))
(and (equal? t t1) (check- e t2 (dict-set Γ x t1)))]
[(`(,e1 ,e2) t)
(match (infer- e1 Γ)
[`(→ ,t1 ,t2)
(and (equal? t2 t) (equal? t1 (infer- e2 Γ)))]
[t #f])]
[(e t) #f])))
;; (infer Expr Table[Sym, Type]): Type
(define (infer expr [Γ #hash()])
(infer- (desugar expr) Γ))
(define (infer- expr Γ)
(match expr
['sole 'Unit]
[s #:when (string? s) 'Str]
[n #:when (natural? n) 'Nat]
[b #:when (boolean? b) 'Bool]
[x #:when (dict-has-key? Γ x)
(dict-ref Γ x)]
[`(type ,t1 ,t2 ,in)
(infer in (dict-set Γ t1 t2))]
[`(pair ,e1 ,e2)
`(× ,(infer- e1 Γ) ,(infer- e2 Γ))]
[`(car ,e)
(match (infer- e Γ)
[`(× ,t1 ,t2) t1]
[t (err (format "calling car on incorrect type ~a" t))])]
[`(cdr ,e)
(match (infer- e Γ)
[`(× ,t1 ,t2) t2]
[t (err (format "calling cdr on incorrect type ~a" t))])]
[`(inl ,e)
(match (infer- e Γ)
[`(+ ,t1 ,t2) t1]
[t (err (format "calling inl on incorrect type ~a" t))])]
[`(inr ,e)
(match (infer- e Γ)
[`(+ ,t1 ,t2) t2]
[t (err (format "calling inr on incorrect type ~a" t))])]
[`(case ,e ,f1 ,f2)
(match* ((infer- f1 Γ) (infer- f2 Γ))
[(`(→ ,a1 ,t1) `(→ ,a2 ,t2))
(if (and (check- e `(+ ,a1 ,a2)) (equal? t1 t2)) t1
(err (format "case ~a is not of consistent type!" `(case ,e ,f1 ,f2))))]
[(t1 t2) (err (format "case ~a is malformed!" `(case ,e ,f1 ,f2)))])]
[`(,e (: ,t))
(if (check- e t Γ) t
(err (format "annotated expression ~a is not of annotated type ~a" e t)))]
['nil (err (format "unable to infer type of empty list!"))]
[`(cons ,e1 ,e2)
(let ([t (infer- e1 Γ)])
(if (check- e2 `(List ,t) Γ) `(List ,t)
(err (format "list ~a is not of consistent type!" `(cons ,e1 ,e2)))))]
[`(head ,e)
(match (infer- e Γ)
[`(List ,t) t]
[t (err (format "calling head on incorrect type ~a" t))])]
[`(tail ,e)
(match (infer- e Γ)
[`(List ,t) `(List ,t)]
[t (err (format "calling tail on incorrect type ~a" t))])]
[`(λ ,x (: ,t) ,e)
`(→ ,t ,(infer- e (dict-set Γ x t)))]
[`(,e1 ,e2)
(match (infer- e1 Γ)
[`(→ ,t1 ,t2)
(if (check- e2 t1 Γ) t2
(err (format "inferred argument type ~a does not match arg ~a" t1 e2)))]
[t (err (format "expected → type on application body, got ~a" t))])]
[e (err (format "inferring an unknown expression ~a" e))]))
|
