[FOM] Simply Typed Lambda Calculus

Chad E Brown cebrown2323 at yahoo.com
Wed Jan 16 20:44:24 EST 2008


I suspect this is an example proving you cannot forget the types
without introducing new equations.

Let o be a base type, T and F be constants of type o,
Q be a constant of type ooo, and
N, C, and D be constants of type oo.
The following axioms are well typed, the left hand sides
are linear patterns, and the right hand sides only contain
free variables contained on the left:

(1) C x = Q x x
(2) D x = Q (N x) x
(3) C x = T
(4) D x = F

In the typed calculus, we cannot prove T = F.
(Consider the interpretation in which Q is equivalence and N is negation.)

Using untyped terms, we can prove T = F.  
Let Y = (W W) where W = (\z.(N (z z))) so that Y = (N Y).
T = C Y = Q Y Y = Q (N Y) Y = D Y = F.

Does this answer your question?


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