Alleged problems with equational reasoning caused by views

[Jacques Carette]

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)
> views.
>   

The problems are not unsolvable - see the Pattern Matching Calculus
http://www.cas.mcmaster.ca/~kahl/PMC/
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.

REFERENCES:
Moon, P. and D.E.Spencer. "Field Theory Handbook, 2nd Ed." Berlin: 
Springer-Verlag, 1971.
Spiegel, Murray R. "Mathematical Handbook of Formulas and Tables." New 
York: McGraw Hill Book Company, 1968. 126-130.

Jacques