# [racket] TR: Making recursive types as subtypes of non-recursive types

 From: Neil Toronto (neil.toronto at gmail.com) Date: Thu Feb 23 13:30:36 EST 2012 Previous message: [racket] TR: Making recursive types as subtypes of non-recursive types Next message: [racket] TR: Making recursive types as subtypes of non-recursive types Messages sorted by: [date] [thread] [subject] [author]

```On 02/22/2012 05:35 PM, Sam Tobin-Hochstadt wrote:
> You're very close -- you just need to give inference a little more
> help.  This definition works:
>
> (: card* (All (a) ((Set* a) ->  Hereditary-Set)))
> (define (card* A)
>    ((inst card (U a (Set* a))) A))

Turns out this wasn't enough to solve my problem in general, because
this function always returns the same type. Here's a simple one that
doesn't:

(: set-id (All (a) ((Set a) -> (Set a))))
(define (set-id A) A)

(: set-id* (All (a) ((Set* a) -> (Set* a))))
(define (set-id* A)
((inst set-id (U a (Set* a))) A))

Using `inst' makes the argument type pass, but I get an error on the
return type:

Expected  (Rec T (U Empty-Set
(Opaque-Set (U a T))))

but got          (U Empty-Set
(Opaque-Set (U a (Rec T (U Empty-Set
(Opaque-Set (U a T)))))))

I've formatted it so that it's easy to see that the body type is just an
unfold of the expected type. Is there a way to make TR realize that
they're isomorphic?

Neil ⊥

> On Wed, Feb 22, 2012 at 7:11 PM, Neil Toronto<neil.toronto at gmail.com>  wrote:
>> I'd like to use the same functions to operate on both "flat" container types
>> and arbitrarily nested container types. More precisely, I want a type for
>> `Set*' that allows this program otherwise unchanged:
>>
>>
>> #lang typed/racket
>>
>> (struct: Empty-Set ())
>> (struct: (a) Opaque-Set ([error-thunk : (->  a)]))   ; Phantom type
>>
>> (define-type (Set a) (U Empty-Set (Opaque-Set a)))  ; Flat sets
>> (define-type (Set* a) (Rec T (Set (U a T))))        ; Nested sets
>>
>> ;; Type of "pure" sets, currently doing double-duty for cardinals:
>> (define-type Hereditary-Set (Set* Nothing))
>>
>> ;; Cardinality operator
>> (: card (All (a) ((Set a) ->  Hereditary-Set)))
>> (define (card A) (error 'card "unimplementable"))
>>
>> (: card* (All (a) ((Set* a) ->  Hereditary-Set)))
>> (define (card* A)
>>   (card A))  ; checking fails here
>>
>>
>> I think the problem is that a (U a T) isn't a subtype of `a' - it's a
>> supertype. But I can't figure out how to make a recursive type that's a
>> subtype of its corresponding "flat" type.
>>
>> Neil ⊥
```

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