[Agda] congruence on dependent types
Sergei Meshveliani
mechvel at botik.ru
Fri Jun 21 12:28:46 CEST 2013
On Fri, 2013-06-21 at 11:30 +0200, Andreas Abel wrote:
> Probably this does not answer your question, but if you wonder how to
> treat ideals <a> in Agda, I would model them as predicates, not as types.
>
> < a > : C -> Set
> < a > b = exists \ x -> b == x * a
>
> Basically < a > b is just "a divides b".
>
> If you want to prove that associated a, b give rise to identical ideals,
> you would prove logical equivalence of predicates.
But this <a> also needs to be an algebraic domain, with various
operations on it: of Setoid, of Semigroup, and such.
Because a subsemigroup <a> also needs to be of a Semigroup, and so
on.
So, if you represent it as a relation, still you will need to relate
some type to it, and this type will depend on a.
As to the language extension which I asked of, may be, this relates to
conditional pattern matching (maybe, conditional unification) on the
type expressions.
May be, this is a matter of conditional rewriting rules provided by
the programmer for the type expressions.
I do not know.
In principle, Agda expresses algebra (mathematics) much more adequately
than Haskell.
But let us point it out:
there remains a certain disagreement between the Agda type system
and domain membership proofs in algebra.
Namely: in an algebra proof, <a> can be replaced with any
equivalent <b> in the statement f ∈ <a>,
while the Agda checker does not allow this for
f : P-Subsemigroup a.
Regards,
------
Sergei
>
> On 20.06.13 8:49 PM, Sergei Meshveliani wrote:
> > We wrote
> >
> >>> I knew the meaning of dotted patterns, but decided to avoid them --
> >>> while possible, and see further. I have forgotten of them, and now
> >>> was
> >>> not able to use them in this concrete example.
> >>>
> >>> Further, I have a feeling that something is strange in the Language.
> >>> Suppose that a type
> >>> Foo a
> >>>
> >>> depends on a : C, A : Setoid, C = Carrier A,
> >>> the equivalence p : a \~~ b is proved,
> >>> f : Foo a is built,
> >>> and we need to return f : Foo b,
> >>> and also somehow to express that f : Foo a can be replaced with
> >>> f : Foo b if a \~~ b.
> >>>
> >>> Can this be expressed in Agda ?
> >
> >> No, you can't prove this for any dependent type `Foo`. If it is true
> >> in your particular case, you will need to prove it specifically for
> >> that.
> >>
> >> To see why it's not true in general, pick any `a : C`, and define
> >>
> >> Foo : C → Set
> >> Foo x = a ≡ x
> >>
> >> Now, if you had:
> >>
> >> cong≈ : (a ≈ b) → Foo a → Foo b
> >>
> >> you could prove:
> >>
> >> discr : (b : C) → a ≈ b → a ≡ b
> >> discr b p = cong≈ p refl
> >>
> >> which, abstracted on `a`, would imply that `A` is discrete.
> >
> >
> > There remains a certain deviation of Agda from mathematics.
> > Consider the example.
> > For a CommutativeMonoid M, an element a ∈ M,
> > call a p-Subsemigroup <a> in M a semigroup of the set
> > {x ∙ a | x ∈ M}. Hence <a> and
> > Hence <a> and <b> is the same p-subsemigroup in M iff
> > associated a b
> > where
> > associated a b = ∃₂ λ (x y : C) → (x ∙ a) ≈ b × (y ∙ b) ≈ a
> >
> > -- that is a and b divide each other.
> >
> > Hence, if assoc-a-b : associated a b, then f ∈ <a>
> > can be replaced with f ∈ <b>
> > in a mathematical proof.
> >
> > Now, it is desirable for an algebraic domain to be expressed in Agda as a
> > _dependent type_. Respectively, f ∈ <a> need to correspond to f : PSSmg a,
> > where PSSmg a is the type expressing <a>.
> >
> > I can express in Agda the Set of all p-subsemigroups in M, and the equality
> > _=pss_ on this set, with _=pss_ defined via associated a b.
> > But for p : associated a b the type checker will not allow to replace
> > f : PSSmg a with f : PSSmg b.
> >
> > Is it possible to add some construct foo to the language so that the checker
> > would allow to replace f : PSSmg a with f : PSSmg b, if p : associated a b
> > and foo is set for PSSmg
> > ?
> >
> > May be, this is rather naive.
> > All right, if this is not possible, still dependent types make a great deal.
> >
> > Here is the implementation for the Set of all p-subsemigroups which I keep in mind:
> >
> > ------------------------------------------------------------------------
> > module SubsemigroupPack (c l : Level) (M : CommutativeMonoid c l)
> > where
> > C = CommutativeMonoid.Carrier M
> > _≈_ = CommutativeMonoid._≈_ M
> > _∙_ = CommutativeMonoid._∙_ M
> >
> > associated : C → C → Set _ -- a and b divide each other
> > associated a b = ∃₂ λ (x y : C) → (x ∙ a) ≈ b × (y ∙ b) ≈ a
> > --
> > -- <a> == <b> iff associated a b
> >
> > record PSSmg (a : C) : Set (c ⊔L l) -- p-Subsemigroup generated by a
> > where
> > constructor pSSmg
> >
> > field factor : C
> >
> > element : C
> > element = factor ∙ a
> >
> > -- Example. pSSmg 3 : PSSmg 5
> > -- expresses 15 as an element of the subsemigroup of <5>.
> >
> > _=s_ : Rel C _
> > x =s y = (x ∙ a) ≈ (y ∙ a)
> >
> > data PSSmgs : Set (c ⊔L l) where -- the set of p-subsemigroups in M
> > pssmg : (a : C) → PSSmg a → PSSmgs
> >
> > _=pss_ : Rel PSSmgs _
> > (pssmg a smg) =pss (pssmg b smg') = associated a b
> > ---------------------------------------------------------------------------
> >
> > So a p-subsemigroup <a> is a type, and it can be made a Semigroup, and
> > a program can compute over <a>.
> > And also a program can compare <a> and <b>, and to have the Setoid of
> > p-subsemigroups in M.
> > And there remains this difference:
> >
> > What to add to the language in order to make this possible:
> > postulate p0 : extSmg@(pssmg a smg) =pss extSmg'@(pssmg b smg')
> > postulate f : smg
> > g : smg'
> > g = f
> > ?
> >
> >
> >
> >
> >
> >
> >
> >
> >
> >
> >
> >
> >
> > _______________________________________________
> > Agda mailing list
> > Agda at lists.chalmers.se
> > https://lists.chalmers.se/mailman/listinfo/agda
> >
>
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