Fascinating! But it looks like you still 'cheat' in your induction principles...
×-induction : ∀{A B} {P : A × B → Set} → ((x : A) → (y : B) → P (x , y)) → (p : A × B) → P p ×-induction {A} {B} {P} f p rewrite sym (×-η p) = f (fst p) (snd p) Can you somehow define x-induction {A} {B} {P} f p = p (P p) f On Tue, Sep 18, 2012 at 4:09 PM, Dan Doel <dan.d...@gmail.com> wrote: > This paper: > > http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.26.957 > > Induction is Not Derivable in Second Order Dependent Type Theory, > shows, well, that you can't encode naturals with a strong induction > principle in said theory. At all, no matter what tricks you try. > > However, A Logic for Parametric Polymorphism, > > http://www.era.lib.ed.ac.uk/bitstream/1842/205/1/Par_Poly.pdf > > Indicates that in a type theory incorporating relational parametricity > of its own types, the induction principle for the ordinary > Church-like encoding of natural numbers can be derived. I've done some > work here: > > http://code.haskell.org/~dolio/agda-share/html/ParamInduction.html > > for some simpler types (although, I've been informed that sigma was > novel, it not being a Simple Type), but haven't figured out natural > numbers yet (I haven't actually studied the second paper above, which > I was pointed to recently). > > -- Dan > > On Tue, Sep 18, 2012 at 5:41 PM, Ryan Ingram <ryani.s...@gmail.com> wrote: > > Oleg, do you have any references for the extension of lambda-encoding of > > data into dependently typed systems? > > > > In particular, consider Nat: > > > > nat_elim :: forall P:(Nat -> *). P 0 -> (forall n:Nat. P n -> P (succ > > n)) -> (n:Nat) -> P n > > > > The naive lambda-encoding of 'nat' in the untyped lambda-calculus has > > exactly the correct form for passing to nat_elim: > > > > nat_elim pZero pSucc n = n pZero pSucc > > > > with > > > > zero :: Nat > > zero pZero pSucc = pZero > > > > succ :: Nat -> Nat > > succ n pZero pSucc = pSucc (n pZero pSucc) > > > > But trying to encode the numerals this way leads to "Nat" referring to > its > > value in its type! > > > > type Nat = forall P:(Nat -> *). P 0 -> (forall n:Nat. P n -> P (succ > n)) > > -> P ??? > > > > Is there a way out of this quagmire? Or are we stuck defining actual > > datatypes if we want dependent types? > > > > -- ryan > > > > > > > > On Tue, Sep 18, 2012 at 1:27 AM, <o...@okmij.org> wrote: > >> > >> > >> There has been a recent discussion of ``Church encoding'' of lists and > >> the comparison with Scott encoding. > >> > >> I'd like to point out that what is often called Church encoding is > >> actually Boehm-Berarducci encoding. That is, often seen > >> > >> > newtype ChurchList a = > >> > CL { cataCL :: forall r. (a -> r -> r) -> r -> r } > >> > >> (in > http://community.haskell.org/%7Ewren/list-extras/Data/List/Church.hs ) > >> > >> is _not_ Church encoding. First of all, Church encoding is not typed > >> and it is not tight. The following article explains the other > >> difference between the encodings > >> > >> http://okmij.org/ftp/tagless-final/course/Boehm-Berarducci.html > >> > >> Boehm-Berarducci encoding is very insightful and influential. The > >> authors truly deserve credit. > >> > >> P.S. It is actually possible to write zip function using > Boehm-Berarducci > >> encoding: > >> http://okmij.org/ftp/ftp/Algorithms.html#zip-folds > >> > >> > >> > >> > >> _______________________________________________ > >> Haskell-Cafe mailing list > >> Haskell-Cafe@haskell.org > >> http://www.haskell.org/mailman/listinfo/haskell-cafe > > > > > > > > _______________________________________________ > > Haskell-Cafe mailing list > > Haskell-Cafe@haskell.org > > http://www.haskell.org/mailman/listinfo/haskell-cafe > > >
_______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe