"Nothing" as produced by intersection types does not always propagate outwards

@pnwamk You seem to be on a bugfix spree concerning intersection type, thanks a lot 😄 ! Here's another one (which fortunately already has a simple workaround).

What version of Racket are you using?

6.9.0.4

What program did you run?

#lang typed/racket
(define-type τ1 (U (Pairof (∩ Integer String) Any)
                  'other))
;; (:type τ1)
;; ⇒ (U (Pairof Nothing Any) 'other)
;; But should be simplified to:
;; ⇒ 'other

Currently, a trick to achieve the desired result is to re-intersect with a polymorphic type variable, instantiated in a way that does not change the final type:

#lang typed/racket
(define-type (τ2 A) (U (∩ (Pairof (∩ Integer String) Any) A)
                       'other))

(define-type τ2-cleaned (τ2 (Pairof Integer Any)))
;; (:type τ2-cleaned)
;; ⇒ 'other

Literally writing an empty intersection usually does not make much sense. The bug is more noticeable when using intersections to "select" parts of a type:

#lang typed/racket
(define-type (τ3 Select) (U (Pairof (∩ 'foo Select) Integer)
                            (Pairof (∩ 'bar Select) String)))

;; (:type (τ3 'foo))
;; ⇒ (U (Pairof 'foo Integer) (Pairof Nothing String))
;; But should be simplified to:
;; ⇒ (Pairof 'foo Integer)

;; (:type (τ3 'bar))
;; ⇒ (U (Pairof 'bar String) (Pairof Nothing Integer))
;; But should be simplified to:
;; ⇒ (Pairof 'bar String)

The same workaround/trick can be applied in this case too:

#lang typed/racket
(define-type (τ4 Select A) (U (∩ (Pairof (∩ 'foo Select) Integer) A)
                              (∩ (Pairof (∩ 'bar Select) String) A)))
;; (:type (τ4 'foo (Pairof Symbol Any)))
;; ⇒ (Pairof 'foo Integer)
;; (:type (τ4 'foo (Pairof Symbol Any)))
;; ⇒ (Pairof 'bar String)

The issue names no source files or tests; start by running its three Typed Racket reproductions and tracing how intersection types containing Nothing are simplified through pairs and unions. Done means the shown :type results reduce to the expected types, including the 'other and selected-pair cases.