Zulip Chat Archive

Stream: lean4

Topic: unification hint


view this post on Zulip Kenny Lau (Jan 06 2021 at 09:09):

What is unification hint and where can I learn more about it?

view this post on Zulip Eric Wieser (Jan 06 2021 at 09:23):

Also this question, but for lean3

view this post on Zulip Adam Topaz (Jan 08 2021 at 19:03):

Is there a way to make the following code work?

structure Magma where
  α : Type
  mul : α  α  α

instance : CoeSort Magma Type where
  coe M := M.α

infixl:70 (priority := high) "**" => Magma.mul

def Nat.Magma : Magma where
  α := Nat
  mul a b := Nat.mul a b

unif_hint (S : Magma) where
  S.α =?= Nat
  |-
  S =?= Nat.Magma

#check (3 : Nat) ** 3

view this post on Zulip Adam Topaz (Jan 08 2021 at 19:26):

https://leanprover.github.io/lean4/doc/unifhint.html
:sad:

view this post on Zulip Alexander Bentkamp (Jan 08 2021 at 19:42):

structure Magma where
  α : Type
  mul : α  α  α

-- instance : CoeSort Magma Type where
--   coe M := M.α

infixl:70 (priority := high) "**" => Magma.mul _ -- added underscore

def Nat.Magma : Magma where
  α := Nat
  mul a b := Nat.mul a b

unif_hint (S : Magma) where
  S =?= Nat.Magma
  |-
  S.α =?= Nat
  --swapped

#check (3 : Nat) ** (3 : Nat) -- added type hint

view this post on Zulip Adam Topaz (Jan 08 2021 at 19:43):

nice!

view this post on Zulip Adam Topaz (Jan 08 2021 at 19:45):

The underscore and type hint I completely understand, but what's going on with the swapped syntax in the unification hint itself?

view this post on Zulip Reid Barton (Jan 08 2021 at 19:47):

I guess because normally one would try to check a typing judgment by matching it against the conclusions of inference rules

view this post on Zulip Alexander Bentkamp (Jan 08 2021 at 19:48):

In the unification hint paper https://www.cs.unibo.it/~sacerdot/PAPERS/tphol09.pdf they also use this order.

view this post on Zulip Kenny Lau (Jan 08 2021 at 19:48):

Well this sure is exciting to see.

view this post on Zulip Kenny Lau (Jan 08 2021 at 19:48):

but how well can this scale?

view this post on Zulip Adam Topaz (Jan 08 2021 at 19:49):

Yeah that's something we should experiment with.

view this post on Zulip Kenny Lau (Jan 08 2021 at 19:50):

I mean, what if I want to reason with an arbitrary Magma? Do I need to keep writing S.\a all the time?

view this post on Zulip Adam Topaz (Jan 08 2021 at 19:51):

Yeah good question. Why can't we have the CoeSort as well?

view this post on Zulip Adam Topaz (Jan 08 2021 at 19:51):

(Looks like adding the CoeSort instance doesn't break anything.)

view this post on Zulip Adam Topaz (Jan 08 2021 at 19:52):

structure Magma where
  α : Type
  mul : α  α  α

instance : CoeSort Magma Type where
  coe M := M.α

infixl:70 (priority := high) "**" => Magma.mul _ -- added underscore

def Nat.Magma : Magma where
  α := Nat
  mul a b := Nat.mul a b

unif_hint (S : Magma) where
  S =?= Nat.Magma
  |-
  S.α =?= Nat

#check (3 : Nat) ** (3 : Nat) -- added type hint

variables (M : Magma) (m : M)

view this post on Zulip Kenny Lau (Jan 08 2021 at 19:57):

structure Magma where
  α : Type
  mul : α  α  α

instance : CoeSort Magma Type where
  coe M := M.α

infixl:70 (priority := high) "**" => Magma.mul _

def mul_assoc (S : Magma) : Prop :=
 x y z : S, (x ** y) ** z = x ** (y ** z)

view this post on Zulip Kenny Lau (Jan 08 2021 at 19:57):

looks like this works as-is

view this post on Zulip Kenny Lau (Jan 08 2021 at 20:00):

ok but can we build a bundled algberaic hierarchy without having to write every lemma 37 times?

view this post on Zulip Cyril Cohen (Jan 10 2021 at 03:51):

The answer is yes and the solution is in the literature: the "Packaging mathematical structures", "Competing inheritance paths in dependent type theory [...]" and "Hierarchy builder" papers are a good start to explain how to do so with Canonical structures which are a strict subset of unification hints. An extensive use case on a large hierarchy can be found in the Coq mathematical components library. I also wrote a short demo in lean3, for a lean meeting that someone found in the zulip archives and posted again in the topic of the same name but for lean3. Please have a look.

view this post on Zulip Alex Peattie (Jan 10 2021 at 15:48):

I think this is the post/demo in question: https://leanprover-community.github.io/archive/stream/113488-general/topic/unification.20hints.html#158613068


Last updated: May 18 2021 at 23:14 UTC