<div dir="ltr">s/(x : X) → P a/(x : X) → P x/</div><br><div class="gmail_quote"><div dir="ltr">On Fri, 10 Aug 2018 at 14:05, Arseniy Alekseyev <<a href="mailto:arseniy.alekseyev@gmail.com">arseniy.alekseyev@gmail.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><div dir="ltr">Two things:<div><br></div><div>1.</div><div>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.</div><div>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):</div><div>P : Bin → Set</div><div>P a = (b : Bin) → GCD a b</div><div><br></div><div>2. Giving P explicitly can help improve error messages a lot. You can try something like this:</div><div><span class="gmail-im" style="color:rgb(80,0,80)">gcd = <-rec _ P gc</span><br></div><div><span class="gmail-im" style="color:rgb(80,0,80)"><br></span></div><div><span class="gmail-im" style="color:rgb(80,0,80)">I haven't look in much detail so maybe this is not actually helpful. Sorry if it's not.</span></div><div><br></div><div><br></div><div><font color="#500050">> </font>WellFounded recursion thing is also available directly for the signature of </div><div>> gcd : (a b : Bin) → GCD a b,</div><div><br></div><div>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.</div></div><br><div class="gmail_quote"><div dir="ltr">On Fri, 10 Aug 2018 at 13:35, Sergei Meshveliani <<a href="mailto:mechvel@botik.ru">mechvel@botik.ru</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">On Fri, 2018-08-10 at 14:36 +0300, Sergei Meshveliani wrote:<br>
> On Thu, 2018-08-09 at 23:15 +0100, Arseniy Alekseyev wrote:<br>
> > Something like this, I think:<br>
> > <br>
> > <br>
> > P : Bin → Set <br>
> > <br>
> > P _y = (x : Bin) → Bin <br>
> > <br>
> > <br>
> > <br>
> > gc : (y : Bin) → (∀ y' → y' < y → P y') → P y<br>
> > <br>
> > <br>
> > then after applying <-rec you get something of type [(y : Bin) → P y],<br>
> > which is just gcd with arguments swapped.<br>
> > <br>
> > <br>
> > (I wrote P in a general form so that it's more similar to "dependent"<br>
> > examples, but of course you don't need to)<br>
> > <br>
> <br>
> <br>
> <br>
> Thank you. <br>
> This works with the result of Bin.<br>
> I try to extend this to <br>
> gcd : (a b : Bin) -> GCD a b,<br>
> and fail.<br>
> <br>
<br>
<br>
Below r<x = rem< x y x≢0<br>
was a typo,<br>
it needs to be replaced with r<x = rem< y x x≢0.<br>
<br>
But this does not change the report of Agda.<br>
<br>
<br>
Another attempt<br>
===============<br>
<br>
I change the signatures to <br>
<br>
------------------------------------------<br>
BB = Bin × Bin<br>
<br>
_<p_ : Rel BB Level.zero<br>
_<p_ = _<_ on proj₁<br>
<br>
open import Induction.WellFounded using (WellFounded)<br>
<br>
postulate <p-wellFounded : WellFounded _<p_<br>
<br>
open All <p-wellFounded using () renaming (wfRec to <-rec)<br>
<br>
record GCD (pr : BB) : Set -- contrived<br>
where<br>
constructor gcd′<br>
a = proj₁ pr<br>
b = proj₂ pr<br>
<br>
field res : Bin<br>
divides-a : ∃ (\q → res * q ≡ a)<br>
divides-b : ∃ (\q → res * q ≡ b)<br>
-- and maximality axiom<br>
<br>
gcd : (pr : BB) → GCD pr<br>
----------------------------------------------<br>
<br>
Now gcd is on Bin × Bin, and your approach works,<br>
the code is type-checked: <br>
<br>
gcd = <-rec _ _ gc<br>
where<br>
gc : (pr : BB) → (∀ pr' → pr' <p pr → GCD pr') → GCD pr<br>
gc (x , y) gcRec<br>
with x ≟ 0#<br>
<br>
... | no x≢0 = f<br>
where<br>
f : GCD (x , y)<br>
f = liftGCD (gcRec (r , x) r<x)<br>
where<br>
r = rem y x x≢0<br>
r<x = rem< y x x≢0<br>
<br>
postulate liftGCD : GCD (r , x) → GCD (x , y)<br>
... | ...<br>
-------------------------------------------------<br>
<br>
<br>
All right: I can convert from GCD-auxiliary (a , b) <br>
to GCD a b.<br>
<br>
But I have an impression that the WellFounded recursion thing is also<br>
available directly for the signature of<br>
<br>
gcd : (a b : Bin) → GCD a b,<br>
<br>
only am missing something.<br>
-- ?<br>
<br>
Regards,<br>
<br>
------<br>
Sergei<br>
<br>
<br>
<br>
<br>
<br>
<br>
> ------------------------------------------------------------------<br>
> postulate<br>
> rem : Bin → (y : Bin) → y ≢ 0# → Bin -- remainder of x by y.<br>
> <br>
> rem< : (x y : Bin) → (y≢0 : y ≢ 0#) → rem x y y≢0 < y<br>
> <br>
> <br>
> record GCD (a b : Bin) : Set -- contrived<br>
> where<br>
> constructor gcd′<br>
> <br>
> field res : Bin<br>
> divides-a : ∃ (\q → res * q ≡ a)<br>
> divides-b : ∃ (\q → res * q ≡ b)<br>
> --<br>
> -- and maximality axiom<br>
> <br>
> -- Without using termination proof:<br>
> --<br>
> {-# TERMINATING #-}<br>
> gcd : (a b : Bin) → GCD a b<br>
> gcd x y<br>
> with x ≟ 0#<br>
> ... | yes x≡0 = gcd′ y (0# , y*0≡x) (1# , y*1≡y)<br>
> where<br>
> postulate y*0≡x : y * 0# ≡ x<br>
> y*1≡y : y * 1# ≡ y<br>
> <br>
> ... | no x≢0 = liftGCD (gcd r x)<br>
> where<br>
> r = rem y x x≢0<br>
> <br>
> postulate liftGCD : GCD r x → GCD x y<br>
> ---------------------------------------------------------------<br>
> <br>
> The second argument is divided by the first one in the loop -- this way<br>
> it is easier to use.<br>
> <br>
> This is type-checked.<br>
> <br>
> Now try WellFounded. As I understand, the approach is to reduce to a<br>
> function of a single argument:<br>
> <br>
> --------------------------------------------------------------- <br>
> gcd : (a : Bin) → GCD a<br>
> gcd = <-rec _ _ gc<br>
> where<br>
> gc : (x : Bin) → (∀ x' → x' < x → GCD x') → GCD x<br>
> gc x gcRec<br>
> with x ≟ 0#<br>
> ... | yes x≡0 = f<br>
> where<br>
> f : GCD x<br>
> f y = gcd′ y (0# , y*0≡x) (1# , y*1≡y)<br>
> where<br>
> postulate y*0≡x : y * 0# ≡ x<br>
> y*1≡y : y * 1# ≡ y<br>
> <br>
> ... | no x≢0 = f<br>
> where<br>
> f : GCD x<br>
> f y = liftGCD (gcRec r r<x x)<br>
> where<br>
> r = rem y x x≢0<br>
> r<x = rem< x y x≢0<br>
> <br>
> postulate liftGCD : GCD r x → GCD x y<br>
> ---------------------------------------------------------------<br>
> <br>
> Agda type-checks the function gc, <br>
> but it reports that (<-rec _ _ gc) does not return a value of the type<br>
> GCD a.<br>
> <br>
> Then I try <br>
> <br>
> gcd : Bin → (Bin → Set)<br>
> gcd = <-rec _ _ gc<br>
> where<br>
> postulate <br>
> gc : (x : Bin) → (∀ x' → x' < x → Bin → Set) → Bin → Set<br>
> <br>
> (which goal adequacy I do not understand). <br>
> It is type-checked,<br>
> but I fail to implement this version of gc.<br>
> <br>
> Can anybody advise, please?<br>
> <br>
> ------<br>
> Sergei <br>
> <br>
> <br>
> <br>
> <br>
> > On Thu, 9 Aug 2018 at 20:40, Sergei Meshveliani <<a href="mailto:mechvel@botik.ru" target="_blank">mechvel@botik.ru</a>><br>
> > wrote:<br>
> > <br>
> > Thank you.<br>
> > After this sample of downFrom I was able to program divMod<br>
> > for Bin.<br>
> > But I am stuck with gcd for Bin.<br>
> > Consider a contrived simple version:<br>
> > <br>
> > ------------------------------------------------------<br>
> > postulate<br>
> > rem : Bin → (y : Bin) → y ≢ 0# → Bin -- remainder of x<br>
> > by y.<br>
> > <br>
> > rem< : (x y : Bin) → (y≢0 : y ≢ 0#) → rem x y y≢0 < y<br>
> > <br>
> > gcd : Bin → Bin → Bin<br>
> > gcd x y<br>
> > with y ≟ 0#<br>
> > ... | yes _ = x<br>
> > ... | no y≢0 = gcd y (rem x y y≢0)<br>
> > <br>
> > This lacks termination proof.<br>
> > The argument pair (x , y) is replaced in recursion with<br>
> > (y , r),<br>
> > where r < y. So, it is needed well-founded recursion: <br>
> > <br>
> > gcd' : Bin → Bin → Bin<br>
> > gcd' = <-rec _ _ gc<br>
> > where<br>
> > postulate<br>
> > gc : Bin → (b : Bin) → (∀ x y → y < b → Bin) → Bin<br>
> > -- ??<br>
> > <br>
> > <br>
> > I do not guess what signature to set for gc. <br>
> > I set a hole "?" for gc, and the type checker shows<br>
> > <br>
> > Induction.WellFounded.WfRec _<_ (λ _ → Bin → Bin)<br>
> > .Relation.Unary._.⊆′ (λ _ → Bin → Bin)<br>
> > -- ?<br>
> > <br>
> > Can anybody help, please?<br>
> > <br>
> > Thanks,<br>
> > <br>
> > ------<br>
> > Sergei <br>
> > <br>
> > <br>
> > <br>
> > On Wed, 2018-08-08 at 17:49 +0200, Sandro Stucki wrote:<br>
> > > > Can anybody demonstrate it on the following example?<br>
> > > <br>
> > > Here you go:<br>
> > > <br>
> > ><br>
> > --------------------------------------------------------------<br>
> > > open import Function using (_∘_; _on_)<br>
> > > open import Data.List using (List; []; _∷_)<br>
> > > open import Data.Bin using (Bin; toBits; pred; _<_; less;<br>
> > toℕ)<br>
> > > open import Data.Digit using (Bit)<br>
> > > import Data.Nat as Nat<br>
> > > import Induction.Nat as NatInd<br>
> > > open import Induction.WellFounded<br>
> > > <br>
> > > open Bin<br>
> > > <br>
> > > predBin : Bin → Bin<br>
> > > predBin = pred ∘ toBits<br>
> > > <br>
> > > postulate<br>
> > > predBin-< : (bs : List Bit) -> predBin (bs 1#) < (bs 1#)<br>
> > > <br>
> > > -- The strict order on binary naturals implies the strict<br>
> > order on the<br>
> > > -- corresponding unary naturals.<br>
> > > <⇒<ℕ : ∀ {b₁ b₂} → b₁ < b₂ → (Nat._<_ on toℕ) b₁ b₂<br>
> > > <⇒<ℕ (less lt) = lt<br>
> > > <br>
> > > -- We can derive well-foundedness of _<_ on binary naturals<br>
> > from<br>
> > > -- well-foundedness of _<_ on unary naturals.<br>
> > > <-wellFounded : WellFounded _<_<br>
> > > <-wellFounded =<br>
> > > Subrelation.wellFounded <⇒<ℕ (Inverse-image.wellFounded<br>
> > toℕ<br>
> > > NatInd.<-wellFounded)<br>
> > > <br>
> > > open All <-wellFounded using () renaming (wfRec to <-rec)<br>
> > > <br>
> > > downFrom : Bin → List Bin -- x ∷ x-1 ∷ x-2 ∷ ... ∷ 0# ∷<br>
> > []<br>
> > > downFrom = <-rec _ _ df<br>
> > > where<br>
> > > df : (b : Bin) → (∀ b′ → b′ < b → List Bin) → List Bin<br>
> > > df 0# dfRec = 0# ∷ []<br>
> > > df (bs 1#) dfRec = (bs 1#) ∷ (dfRec (predBin (bs 1#))<br>
> > (predBin-< bs))<br>
> > ><br>
> > --------------------------------------------------------------<br>
> > > <br>
> > > In order to use well-founded induction, we first have to<br>
> > prove that<br>
> > > the strict order < is indeed well-founded. Thankfully, the<br>
> > standard<br>
> > > library already contains such a proof for the strict order<br>
> > on (unary)<br>
> > > naturals as well as a collection of combinators for deriving<br>
> > > well-foundedness of relations from others (in this case the<br>
> > strict<br>
> > > order on unary naturals).<br>
> > > <br>
> > > The core of the implementation of `downFrom' via<br>
> > well-founded<br>
> > > recursion is the function `df', which has the same signature<br>
> > as<br>
> > > `downFrom' except for the additional argument `dfRec', which<br>
> > serves as<br>
> > > the 'induction hypothesis'. The argument `dfRec' is itself a<br>
> > function<br>
> > > with (almost) the same signature as `downFrom' allowing us<br>
> > to make<br>
> > > recursive calls (i.e. take a recursive step), provided we<br>
> > can prove<br>
> > > that the first argument of the recursive call (i.e. the<br>
> > argument to<br>
> > > the induction hypothesis) is smaller than the first argument<br>
> > of the<br>
> > > enclosing call to `df'. The proof that this is indeed the<br>
> > case is<br>
> > > passed to `dfRec' as an additional argument of type b′ < b.<br>
> > > <br>
> > > The following answer on Stackoverflow contains a nice<br>
> > explanation on<br>
> > > how all of this is implemented in Agda under the hood:<br>
> > > <a href="https://stackoverflow.com/a/19667260" rel="noreferrer" target="_blank">https://stackoverflow.com/a/19667260</a><br>
> > > <br>
> > > Cheers<br>
> > > /Sandro<br>
> > > <br>
> > > <br>
> > > On Wed, Aug 8, 2018 at 12:13 PM Sergei Meshveliani<br>
> > <<a href="mailto:mechvel@botik.ru" target="_blank">mechvel@botik.ru</a>> wrote:<br>
> > > ><br>
> > > > On Tue, 2018-08-07 at 20:51 +0300, Sergei Meshveliani<br>
> > wrote:<br>
> > > > > Dear all,<br>
> > > > ><br>
> > > > > I am trying to understand how to use WellFounded of<br>
> > Standard library.<br>
> > > > ><br>
> > > > > Can anybody demonstrate it on the following example?<br>
> > > > ><br>
> > > > ><br>
> > --------------------------------------------------------------<br>
> > > > > open import Function using (_∘_)<br>
> > > > > open import Data.List using (List; []; _∷_)<br>
> > > > > open import Data.Bin using (Bin; toBits; pred)<br>
> > > > ><br>
> > > > > open Bin<br>
> > > > ><br>
> > > > > predBin : Bin → Bin<br>
> > > > > predBin = pred ∘ toBits<br>
> > > > ><br>
> > > > > downFrom : Bin → List Bin -- x ∷ x-1 ∷ x-2 ∷ ... ∷<br>
> > 0# ∷ []<br>
> > > > > downFrom 0# = 0# ∷ []<br>
> > > > > downFrom (bs 1#) = (bs 1#) ∷ (downFrom (predBin (bs<br>
> > 1#)))<br>
> > > > ><br>
> > --------------------------------------------------------------<br>
> > > > ><br>
> > > > > downFrom is not recognized as terminating.<br>
> > > > > How to reorganize it with using items from<br>
> > > > > Induction/*, WellFounded.agda ?<br>
> > > ><br>
> > > ><br>
> > > ><br>
> > > > I presumed also that it is already given the property<br>
> > > ><br>
> > > > postulate<br>
> > > > predBin-< : (bs : List Bit) -> predBin (bs 1#) < (bs<br>
> > 1#)<br>
> > > ><br>
> > > > (I do not mean to deal here with its proof).<br>
> > > ><br>
> > > > --<br>
> > > > SM<br>
> > > ><br>
> > > ><br>
> > > > _______________________________________________<br>
> > > > Agda mailing list<br>
> > > > <a href="mailto:Agda@lists.chalmers.se" target="_blank">Agda@lists.chalmers.se</a><br>
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> > > <br>
> > <br>
> > <br>
> > _______________________________________________<br>
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<br>
</blockquote></div></blockquote></div>