If function "natural-f" --<div> </div><div><div>;apply function f to x n times with a base case of TH where n=0</div> <div>;(check-expect (natural-f cons 'x empty 5) (list 'x 'x 'x 'x 'x))</div> <div>;(check-expect (natural-f + 1 5 9) 14)</div><div>(define (natural-f f x TH n)</div> <div>  (cond</div><div>    [(zero? n) TH]</div> <div>    [else (f x</div><div>             (natural-f f x TH (sub1 n) ))]))</div> <div> </div><div>-- properly abstracts over the the functions </div> <div> <table><tbody><tr><td><div align="left"><pre style="margin-left:2em;color:rgb(165, 42, 42)">;; <code style="color:rgb(165, 42, 42)">copy : N X  <tt>-></tt>  (listof X)</code> ;; to create a list that contains ;; <code style="color:rgb(165, 42, 42)">obj</code> <code style="color:rgb(165, 42, 42)">n</code> times (define (copy n obj) (cond [(zero? n) empty] [else (cons obj (copy (sub1 n) obj))])) </pre></div></td><td> </td> <td><div align="left"><pre style="margin-left:2em;color:rgb(165, 42, 42)">;; <code style="color:rgb(165, 42, 42)">n-adder : N number  <tt>-></tt>  number</code> ;; to add <code style="color:rgb(165, 42, 42)">n</code> to <code style="color:rgb(165, 42, 42)">x</code> using ;; <code style="color:rgb(165, 42, 42)">(+ 1 ...)</code> only (define (n-adder n x) (cond [(zero? n) x] [else (+ 1 (n-adder (sub1 n) x))]))</pre> </div></td></tr></tbody></table></div><div>--then what is the general contract for "natural-f"?</div> <div> </div><div>Considering "natural-f" as an abstraction of function "n-adder" above, the resulting contract would be:</div> <div>(number number -> number) number number number -> number  </div><div>(ex. (check-expect (natural-f + 1 5 9) 14)</div><div><div> </div><div>But what would the contract for "natural-f" be if it was viewed as an abstraction of function "copy?"</div> <div>(ex.(check-expect (natural-f cons 'x empty 5) (list 'x 'x 'x 'x 'x))</div> <div> </div><div>In that case, argument "f" would take the value of "cons," so the question (for me) comes down to how to represent the primitive "cons" as a function type within a contract. </div> <div> </div><div>Is "cons" represented as (X Y -> Y)</div> <div>where X is any Scheme ITEM, Y is the Scheme-value list, and the result is basic type Y (Scheme-value list)?</div> <div> </div><div>Or is "cons" represented as  (X list -> (listof X)</div> <div> where X is any Scheme ITEM, "list" is Scheme-value list, and the result is defined type (listof X)?</div> <div> </div><div>The difference between the two contracts appears to be in the output types.  The first output type is basic, which is consistent with the contract output for "n-adder"; the second output type is defined, which conflicts with "n-adder."  Avoiding conflict means choosing the first representation of "cons" in place of f in "natural-f"'s contract. S my guess is that the proper general contract for "natural-f" would be:</div> <div> </div><div>;natural-f: (X Y -> Y) X Y number -> Y </div><div> </div><div>Is this right? If not, where am I going wrong?</div> <div> </div><div>Thanks,</div> <div>Dave </div><div> </div><div> </div><div> </div><div> </div><div> </div><div> </div></div><div> </div><div> </div><div> </div><div> <div align="left"><pre style="margin-left:2em;color:rgb(165, 42, 42)"></pre></div></div></div>