Alleged problems with equational reasoning caused by views
carette at mcmaster.ca
Sat Feb 3 10:27:28 EST 2007
Arie Peterson wrote:
> J. Garrett Morris wrote (to Bulat Ziganshin):
>> Yes - you've reiterated Wadler's original design, with an automatic
>> creation of a type class. Erwig and Peyton-Jones, _Pattern Guards and
>> Transformational Patterns_
>> (http://research.microsoft.com/~simonpj/Papers/pat.htm) mentions
>> problems with equational reasoning raised by this approach.
> I just read this paper, in particular the part about the problems with
> equational reasoning that come up once you introduce (a certain form of)
The problems are not unsolvable - see the Pattern Matching Calculus
for one way to re-introduce equational reasoning in pattern-matching.
On another front, I am a big fan of the polar/cartesian 'view' of
Complex numbers as being a fundamental test case for "full" views. In
fact, that is quite restricted, one should instead be looking at the
following views for R^2: bipolar, cardioid, cassinian, cartesian,
elliptic, hyperbolic, invcassinian, invelliptic, logarithmic, logcosh,
maxwell, parabolic, polar, rose, and tangent.
In three dimensions, one then gets - bipolarcylindrical, bispherical,
cardioidal, cardioidcylindrical, casscylindrical, confocalellip,
confocalparab, conical, cylindrical, ellcylindrical, ellipsoidal,
hypercylindrical, invcasscylindrical, invellcylindrical,
invoblspheroidal, invprospheroidal, logcoshcylindrical, logcylindrical,
maxwellcylindrical, oblatespheroidal, paraboloidal, paraboloidal2,
paracylindrical, prolatespheroidal, rectangular, rosecylindrical,
sixsphere, spherical, tangentcylindrical, tangentsphere, and toroidal.
Moon, P. and D.E.Spencer. "Field Theory Handbook, 2nd Ed." Berlin:
Spiegel, Murray R. "Mathematical Handbook of Formulas and Tables." New
York: McGraw Hill Book Company, 1968. 126-130.
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