Numerical Comparison: "required coercions"

[Michael Greenwald <Greenwald@STONY-BROOK.SCRC.Symbolics.COM>]

    Date: Thu, 26 Mar 87 13:34 EST
    From: Michael Greenwald <Greenwald@STONY-BROOK.SCRC.Symbolics.COM>

	Date: Thu, 26 Mar 87 10:47 EST
	From: David C. Plummer <DCP@QUABBIN.SCRC.Symbolics.COM>

	    Date: Thu, 26 Mar 87 10:29 EST
	    From: Guy Steele <gls@Think.COM>

	    This issue makes me want to restate an earlier proposed model
	    for the comparison functions:

		    (? x1 x2 x3 x4 ... xn-1 xn) <=>
			    (AND (? x1 x2) (? x1 x3) (? x1 x4) ... (? x1 xn-1) (? x1 xn)
					   (? x2 x3) (? x2 x4) ... (? x2 xn-1) (? x2 xn)
						     (? x3 x4) ... (? x3 xn-1) (? x3 xn)
									.          .
									.          .
										   .
									       (? xn-1 xn))
	    where "?" may be =, /=, <, >, <=, or >=.
	    That is, in principle every pair of arguments is compared.
	    This is the model that justifies the "all different" interpretation for /=,
	    and it is recognized that in practice the other five operations may
	    be implemented cleverly through exploitation of transitivity.  Well,
	    maybe such reliance on transitivity is wrong.

    Why "maybe"?  (<= 1234d10 12340000000001 1234e10 12340000000000) => T
    when you depend on transitivity, but should => NIL according to the
    above model.

    On the other hand, SORT clearly depends on transitivity, and one might
    reasonably expect (APPLY '<= (SORT SEQUENCE '<=)) => T.

    (Or even more reasonably (APPLY '< (SORT SEQUENCE '<)) => T, which
    unfortunately isn't true since  
      (SORT '(1234d10 12340000000001 1234e10 12340000000000) '<) returns the
    original sequence.)

I was wrong.  The < case isn't a reasonable expectation.  Mike McMahon
points out:  (APPLY '< (SORT '(1 1 1 1 1 1 1) '<)).  I guess I wasn't
thinking too clearly.