It is not possible at the value level, because Haskell does not
support dependent types and thus cannot express the type of the
proposition "forall a . forall x:Tree a, mirror (mirror x) = x", and
therefore a proof term also cannot be constructed.
However, if you manage to express those trees at type level, probably
with typeclasses and type families, you might have some success.
2009/9/25 pat browne
Hi, Below is a function that returns a mirror of a tree, originally from:
http://www.nijoruj.org/~as/2009/04/20/A-little-fun.html
where it was used to demonstrate the use of Haskabelle(1) which converts Haskell programs to the Isabelle theorem prover. Isabelle was used to show that the Haskell implementation of mirror is a model for the axiom:
mirror (mirror x) = x
My question is this: Is there any way to achieve such a proof in Haskell itself? GHC appears to reject equations such has mirror (mirror x) = x mirror (mirror(Node x y z)) = Node x y z
Regards, Pat
=================CODE===================== module BTree where
data Tree a = Tip | Node (Tree a) a (Tree a)
mirror :: Tree a -> Tree a mirror (Node x y z) = Node (mirror z) y (mirror x) mirror Tip = Tip
(1)Thanks to John Ramsdell for the link to Haskabelle: http://www.cl.cam.ac.uk/research/hvg/Isabelle/haskabelle.html).
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