[Agda] Standard WellFounded
Arseniy Alekseyev
arseniy.alekseyev at gmail.com
Fri Aug 10 15:05:51 CEST 2018
Two things:
1.
In your first attempt you did not re-define GCD (or you skipped that from
your e-mail, but that seems to be the important part). The original
definition can't work because [GCD x] isn't even a type so nothing can ever
return a value of this type.
You need to define something like (I'll stubbornly keep calling this P for
similarity with other examples but you're free to call it whatever):
P : Bin → Set
P a = (b : Bin) → GCD a b
2. Giving P explicitly can help improve error messages a lot. You can try
something like this:
gcd = <-rec _ P gc
I haven't look in much detail so maybe this is not actually helpful. Sorry
if it's not.
> WellFounded recursion thing is also available directly for the signature
of
> gcd : (a b : Bin) → GCD a b,
It's only available for signatures of form [(x : X) → P a] for well-founded
X, so yes, that's the case. By unifying the two types you get exactly the P
I wrote above.
On Fri, 10 Aug 2018 at 13:35, Sergei Meshveliani <mechvel at botik.ru> wrote:
> On Fri, 2018-08-10 at 14:36 +0300, Sergei Meshveliani wrote:
> > On Thu, 2018-08-09 at 23:15 +0100, Arseniy Alekseyev wrote:
> > > Something like this, I think:
> > >
> > >
> > > P : Bin → Set
> > >
> > > P _y = (x : Bin) → Bin
> > >
> > >
> > >
> > > gc : (y : Bin) → (∀ y' → y' < y → P y') → P y
> > >
> > >
> > > then after applying <-rec you get something of type [(y : Bin) → P y],
> > > which is just gcd with arguments swapped.
> > >
> > >
> > > (I wrote P in a general form so that it's more similar to "dependent"
> > > examples, but of course you don't need to)
> > >
> >
> >
> >
> > Thank you.
> > This works with the result of Bin.
> > I try to extend this to
> > gcd : (a b : Bin) -> GCD a b,
> > and fail.
> >
>
>
> Below r<x = rem< x y x≢0
> was a typo,
> it needs to be replaced with r<x = rem< y x x≢0.
>
> But this does not change the report of Agda.
>
>
> Another attempt
> ===============
>
> I change the signatures to
>
> ------------------------------------------
> BB = Bin × Bin
>
> _<p_ : Rel BB Level.zero
> _<p_ = _<_ on proj₁
>
> open import Induction.WellFounded using (WellFounded)
>
> postulate <p-wellFounded : WellFounded _<p_
>
> open All <p-wellFounded using () renaming (wfRec to <-rec)
>
> record GCD (pr : BB) : Set -- contrived
> where
> constructor gcd′
> a = proj₁ pr
> b = proj₂ pr
>
> field res : Bin
> divides-a : ∃ (\q → res * q ≡ a)
> divides-b : ∃ (\q → res * q ≡ b)
> -- and maximality axiom
>
> gcd : (pr : BB) → GCD pr
> ----------------------------------------------
>
> Now gcd is on Bin × Bin, and your approach works,
> the code is type-checked:
>
> gcd = <-rec _ _ gc
> where
> gc : (pr : BB) → (∀ pr' → pr' <p pr → GCD pr') → GCD pr
> gc (x , y) gcRec
> with x ≟ 0#
>
> ... | no x≢0 = f
> where
> f : GCD (x , y)
> f = liftGCD (gcRec (r , x) r<x)
> where
> r = rem y x x≢0
> r<x = rem< y x x≢0
>
> postulate liftGCD : GCD (r , x) → GCD (x , y)
> ... | ...
> -------------------------------------------------
>
>
> All right: I can convert from GCD-auxiliary (a , b)
> to GCD a b.
>
> But I have an impression that the WellFounded recursion thing is also
> available directly for the signature of
>
> gcd : (a b : Bin) → GCD a b,
>
> only am missing something.
> -- ?
>
> Regards,
>
> ------
> Sergei
>
>
>
>
>
>
> > ------------------------------------------------------------------
> > postulate
> > rem : Bin → (y : Bin) → y ≢ 0# → Bin -- remainder of x by y.
> >
> > rem< : (x y : Bin) → (y≢0 : y ≢ 0#) → rem x y y≢0 < y
> >
> >
> > record GCD (a b : Bin) : Set -- contrived
> > where
> > constructor gcd′
> >
> > field res : Bin
> > divides-a : ∃ (\q → res * q ≡ a)
> > divides-b : ∃ (\q → res * q ≡ b)
> > --
> > -- and maximality axiom
> >
> > -- Without using termination proof:
> > --
> > {-# TERMINATING #-}
> > gcd : (a b : Bin) → GCD a b
> > gcd x y
> > with x ≟ 0#
> > ... | yes x≡0 = gcd′ y (0# , y*0≡x) (1# , y*1≡y)
> > where
> > postulate y*0≡x : y * 0# ≡ x
> > y*1≡y : y * 1# ≡ y
> >
> > ... | no x≢0 = liftGCD (gcd r x)
> > where
> > r = rem y x x≢0
> >
> > postulate liftGCD : GCD r x → GCD x y
> > ---------------------------------------------------------------
> >
> > The second argument is divided by the first one in the loop -- this way
> > it is easier to use.
> >
> > This is type-checked.
> >
> > Now try WellFounded. As I understand, the approach is to reduce to a
> > function of a single argument:
> >
> > ---------------------------------------------------------------
> > gcd : (a : Bin) → GCD a
> > gcd = <-rec _ _ gc
> > where
> > gc : (x : Bin) → (∀ x' → x' < x → GCD x') → GCD x
> > gc x gcRec
> > with x ≟ 0#
> > ... | yes x≡0 = f
> > where
> > f : GCD x
> > f y = gcd′ y (0# , y*0≡x) (1# , y*1≡y)
> > where
> > postulate y*0≡x : y * 0# ≡ x
> > y*1≡y : y * 1# ≡ y
> >
> > ... | no x≢0 = f
> > where
> > f : GCD x
> > f y = liftGCD (gcRec r r<x x)
> > where
> > r = rem y x x≢0
> > r<x = rem< x y x≢0
> >
> > postulate liftGCD : GCD r x → GCD x y
> > ---------------------------------------------------------------
> >
> > Agda type-checks the function gc,
> > but it reports that (<-rec _ _ gc) does not return a value of the type
> > GCD a.
> >
> > Then I try
> >
> > gcd : Bin → (Bin → Set)
> > gcd = <-rec _ _ gc
> > where
> > postulate
> > gc : (x : Bin) → (∀ x' → x' < x → Bin → Set) → Bin → Set
> >
> > (which goal adequacy I do not understand).
> > It is type-checked,
> > but I fail to implement this version of gc.
> >
> > Can anybody advise, please?
> >
> > ------
> > Sergei
> >
> >
> >
> >
> > > On Thu, 9 Aug 2018 at 20:40, Sergei Meshveliani <mechvel at botik.ru>
> > > wrote:
> > >
> > > Thank you.
> > > After this sample of downFrom I was able to program divMod
> > > for Bin.
> > > But I am stuck with gcd for Bin.
> > > Consider a contrived simple version:
> > >
> > > ------------------------------------------------------
> > > postulate
> > > rem : Bin → (y : Bin) → y ≢ 0# → Bin -- remainder of x
> > > by y.
> > >
> > > rem< : (x y : Bin) → (y≢0 : y ≢ 0#) → rem x y y≢0 < y
> > >
> > > gcd : Bin → Bin → Bin
> > > gcd x y
> > > with y ≟ 0#
> > > ... | yes _ = x
> > > ... | no y≢0 = gcd y (rem x y y≢0)
> > >
> > > This lacks termination proof.
> > > The argument pair (x , y) is replaced in recursion with
> > > (y , r),
> > > where r < y. So, it is needed well-founded recursion:
> > >
> > > gcd' : Bin → Bin → Bin
> > > gcd' = <-rec _ _ gc
> > > where
> > > postulate
> > > gc : Bin → (b : Bin) → (∀ x y → y < b → Bin) → Bin
> > > -- ??
> > >
> > >
> > > I do not guess what signature to set for gc.
> > > I set a hole "?" for gc, and the type checker shows
> > >
> > > Induction.WellFounded.WfRec _<_ (λ _ → Bin → Bin)
> > > .Relation.Unary._.⊆′ (λ _ → Bin → Bin)
> > > -- ?
> > >
> > > Can anybody help, please?
> > >
> > > Thanks,
> > >
> > > ------
> > > Sergei
> > >
> > >
> > >
> > > On Wed, 2018-08-08 at 17:49 +0200, Sandro Stucki wrote:
> > > > > Can anybody demonstrate it on the following example?
> > > >
> > > > Here you go:
> > > >
> > > >
> > > --------------------------------------------------------------
> > > > open import Function using (_∘_; _on_)
> > > > open import Data.List using (List; []; _∷_)
> > > > open import Data.Bin using (Bin; toBits; pred; _<_; less;
> > > toℕ)
> > > > open import Data.Digit using (Bit)
> > > > import Data.Nat as Nat
> > > > import Induction.Nat as NatInd
> > > > open import Induction.WellFounded
> > > >
> > > > open Bin
> > > >
> > > > predBin : Bin → Bin
> > > > predBin = pred ∘ toBits
> > > >
> > > > postulate
> > > > predBin-< : (bs : List Bit) -> predBin (bs 1#) < (bs 1#)
> > > >
> > > > -- The strict order on binary naturals implies the strict
> > > order on the
> > > > -- corresponding unary naturals.
> > > > <⇒<ℕ : ∀ {b₁ b₂} → b₁ < b₂ → (Nat._<_ on toℕ) b₁ b₂
> > > > <⇒<ℕ (less lt) = lt
> > > >
> > > > -- We can derive well-foundedness of _<_ on binary naturals
> > > from
> > > > -- well-foundedness of _<_ on unary naturals.
> > > > <-wellFounded : WellFounded _<_
> > > > <-wellFounded =
> > > > Subrelation.wellFounded <⇒<ℕ (Inverse-image.wellFounded
> > > toℕ
> > > > NatInd.<-wellFounded)
> > > >
> > > > open All <-wellFounded using () renaming (wfRec to <-rec)
> > > >
> > > > downFrom : Bin → List Bin -- x ∷ x-1 ∷ x-2 ∷ ... ∷ 0# ∷
> > > []
> > > > downFrom = <-rec _ _ df
> > > > where
> > > > df : (b : Bin) → (∀ b′ → b′ < b → List Bin) → List Bin
> > > > df 0# dfRec = 0# ∷ []
> > > > df (bs 1#) dfRec = (bs 1#) ∷ (dfRec (predBin (bs 1#))
> > > (predBin-< bs))
> > > >
> > > --------------------------------------------------------------
> > > >
> > > > In order to use well-founded induction, we first have to
> > > prove that
> > > > the strict order < is indeed well-founded. Thankfully, the
> > > standard
> > > > library already contains such a proof for the strict order
> > > on (unary)
> > > > naturals as well as a collection of combinators for deriving
> > > > well-foundedness of relations from others (in this case the
> > > strict
> > > > order on unary naturals).
> > > >
> > > > The core of the implementation of `downFrom' via
> > > well-founded
> > > > recursion is the function `df', which has the same signature
> > > as
> > > > `downFrom' except for the additional argument `dfRec', which
> > > serves as
> > > > the 'induction hypothesis'. The argument `dfRec' is itself a
> > > function
> > > > with (almost) the same signature as `downFrom' allowing us
> > > to make
> > > > recursive calls (i.e. take a recursive step), provided we
> > > can prove
> > > > that the first argument of the recursive call (i.e. the
> > > argument to
> > > > the induction hypothesis) is smaller than the first argument
> > > of the
> > > > enclosing call to `df'. The proof that this is indeed the
> > > case is
> > > > passed to `dfRec' as an additional argument of type b′ < b.
> > > >
> > > > The following answer on Stackoverflow contains a nice
> > > explanation on
> > > > how all of this is implemented in Agda under the hood:
> > > > https://stackoverflow.com/a/19667260
> > > >
> > > > Cheers
> > > > /Sandro
> > > >
> > > >
> > > > On Wed, Aug 8, 2018 at 12:13 PM Sergei Meshveliani
> > > <mechvel at botik.ru> wrote:
> > > > >
> > > > > On Tue, 2018-08-07 at 20:51 +0300, Sergei Meshveliani
> > > wrote:
> > > > > > Dear all,
> > > > > >
> > > > > > I am trying to understand how to use WellFounded of
> > > Standard library.
> > > > > >
> > > > > > Can anybody demonstrate it on the following example?
> > > > > >
> > > > > >
> > > --------------------------------------------------------------
> > > > > > open import Function using (_∘_)
> > > > > > open import Data.List using (List; []; _∷_)
> > > > > > open import Data.Bin using (Bin; toBits; pred)
> > > > > >
> > > > > > open Bin
> > > > > >
> > > > > > predBin : Bin → Bin
> > > > > > predBin = pred ∘ toBits
> > > > > >
> > > > > > downFrom : Bin → List Bin -- x ∷ x-1 ∷ x-2 ∷ ... ∷
> > > 0# ∷ []
> > > > > > downFrom 0# = 0# ∷ []
> > > > > > downFrom (bs 1#) = (bs 1#) ∷ (downFrom (predBin (bs
> > > 1#)))
> > > > > >
> > > --------------------------------------------------------------
> > > > > >
> > > > > > downFrom is not recognized as terminating.
> > > > > > How to reorganize it with using items from
> > > > > > Induction/*, WellFounded.agda ?
> > > > >
> > > > >
> > > > >
> > > > > I presumed also that it is already given the property
> > > > >
> > > > > postulate
> > > > > predBin-< : (bs : List Bit) -> predBin (bs 1#) < (bs
> > > 1#)
> > > > >
> > > > > (I do not mean to deal here with its proof).
> > > > >
> > > > > --
> > > > > SM
> > > > >
> > > > >
> > > > > _______________________________________________
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