define-type is just a type alias...

silly.rkt

#lang typed/racket

(define-type A (U String Integer))
(define-type B (U Integer))
(define-type D (U String Real Integer))

(: foo (A -> Integer))
(define (foo a)
  (cond [(string? a) (string-length a)]
        [(integer? a) a]))

(: bar (B -> Integer))
(define (bar a)
  (cond [(integer? a) a]))

(: baz (D -> Integer))
(define (baz a)
  (cond [(string? a) (string-length a)]
        [(integer? a) (numerator (inexact->exact (truncate a)))]
        [(real? a) (numerator (inexact->exact (truncate a)))]))

(: bad (A -> Integer))
(define (bad a) (baz a))

(foo "hello") ;; -> 5
(foo 10) ;; -> 10
(bar 10) ;; -> 10
(bad "444") ;; -> 3