## Stream: Coq devs & plugin devs

### Topic: Why induction is so slow

#### Jason Gross (May 10 2021 at 13:10):

@Pierre-Marie Pédrot Is asking about why induction is so slow (https://github.com/coq/coq/pull/14174#discussion_r627670736) a good topic to add to the Coq call this week?

#### Pierre-Marie Pédrot (May 10 2021 at 13:13):

Oh, sorry I missed that ping. I can have a look quickly though.

Thanks!

#### Pierre-Marie Pédrot (May 10 2021 at 13:52):

It's not super clear that there is a single source of slowdown. It looks spread around, although mainly inside unification.

#### Pierre-Marie Pédrot (May 10 2021 at 13:53):

Do you happen to have a slightly more extreme example?

#### Pierre-Marie Pédrot (May 10 2021 at 13:53):

I tried nesting more records but I didn't get slower.

#### Pierre-Marie Pédrot (May 10 2021 at 13:53):

It seems that the associativity of the records matter.

#### Jason Gross (May 10 2021 at 14:19):

@Pierre-Marie Pédrot Here's a slightly more extreme example (should work on Coq master too)

Set Implicit Arguments.
Definition ex_proj1 :=
fun {A : Prop} {P : A -> Prop} (x : exists y, P y) => match x with ex_intro _ a _ => a end.
Definition ex_proj2 :=
fun {A : Prop} {P : A -> Prop} (x : exists y, P y) => match x return P (ex_proj1 x) with ex_intro _ _ a => a end.
Axiom eq_sigT
: forall {A : Type} {P : A -> Type} (u v : {a : A & P a}) (p : projT1 u = projT1 v),
eq_rect (projT1 u) (fun a : A => P a) (projT2 u) (projT1 v) p = projT2 v -> u = v.
Axiom eq_sigT_rect_uncurried :
forall {A : Type} {P : A -> Type} {u v : {a : A & P a}} (Q : u = v -> Type),
(forall pq : exists p : projT1 u = projT1 v, eq_rect (projT1 u) (fun a : A => P a) (projT2 u) (projT1 v) p = projT2 v,
Q (eq_sigT u v (ex_proj1 pq) (ex_proj2 pq))) -> forall p : u = v, Q p.
Axiom ex_rect :
forall [A : Prop] [P : A -> Prop] (P0 : (exists y, P y) -> Type),
(forall (x : A) (p : P x), P0 (ex_intro P x p)) -> forall e : exists y, P y, P0 e.
Section inversion_sigma.
Unset Implicit Arguments.
Context A (B B' : A -> Prop) (C C' : forall a, B a -> Prop)
(D : forall a b, C a b -> Prop) (E : forall a b c, D a b c -> Prop)
(F : forall a b c d, E a b c d -> Prop) (G : forall a b c d e, F a b c d e -> Prop)
(H : forall a b c d e f, G a b c d e f -> Prop) (I : forall a b c d e f g, H a b c d e f g -> Prop).

Goal forall (x y : { a : A & { b : { b : B a & C a b } & { d : D a (projT1 b) (projT2 b) & { e : E _ _ _ d &
{ f : F _ _ _ _ e & { g : G _ _ _ _ _ f & { h : H _ _ _ _ _ _ g & I _ _ _ _ _ _ _ h } } } } } } })
(p : x = y), p = p.
Proof.
intros x y p; repeat match goal with H : sigT _ |- _ => destruct H end; cbn [projT1 projT2] in *.
Time induction p as [p] using (@eq_sigT_rect_uncurried _ _ _ _);
cbn beta iota
delta [proj1_sig proj2_sig proj3_sig projT1 projT2 projT3 ex_proj1 ex_proj2 sig_of_sig2 sigT_of_sigT2
sig_of_sigT sigT_of_sig sig2_of_sigT2 sigT2_of_sig2] in p.
Time induction p using ex_rect. (* Finished transaction in 2.99 secs (2.986u,0.003s) (successful) *)

#### Jason Gross (May 10 2021 at 14:20):

Indeed associativity seems to matter, but you should be able to keep adding sigmas on the right here.

#### Jason Gross (May 10 2021 at 14:26):

Strangely enough, if I do cbn beta match delta [projT1 projT2] before the second induction p, then the second induction gets much faster. Is the issue really that unification is slow because it spends a lot of time not realizing it needs to unfold projT1 and projT2?

#### Pierre-Marie Pédrot (May 10 2021 at 14:31):

So, there seems to be a problem with the conclusion.

#### Pierre-Marie Pédrot (May 10 2021 at 14:32):

The time is linear in the number of occurrences of p in the conclusion.

#### Pierre-Marie Pédrot (May 10 2021 at 14:32):

If you replace p = p with a dummy predicate Blah (p, p, p, p ...) you can observe it easily.

#### Jason Gross (May 10 2021 at 14:33):

I still think it's kind-of atrocious that the induction p as [p] using (@eq_sigT_rect_uncurried _ _ _ _) takes .2 seconds, especially since adding match type of p with @existT ?A ?P ?x ?y=@existT _ _ ?x' ?y' => set (Y := y) in *; set (Y' := y') in *; set (PV := P) in * end, which takes only 0.075s, drops the time of Time induction p as [p] using (@eq_sigT_rect_uncurried _ _ _ _). down to 0.005s, and then cbn [projT1 projT2] in *; induction p using ex_rect. takes only 0.006s rather than 0.157s

#### Pierre-Marie Pédrot (May 10 2021 at 14:34):

Yeah, I think that this is slow because of second-order induction.

#### Pierre-Marie Pédrot (May 10 2021 at 14:35):

Induction is trying to find your subterm in the conclusion to abstract it away.

#### Jason Gross (May 10 2021 at 14:35):

What do you mean by second-order induction?

^this

#### Pierre-Marie Pédrot (May 10 2021 at 14:35):

it tries to craft a predicate

#### Pierre-Marie Pédrot (May 10 2021 at 14:35):

so for this it needs to abstract the stuff you're inducting on

#### Pierre-Marie Pédrot (May 10 2021 at 14:36):

the pattern here is a quite big term

#### Pierre-Marie Pédrot (May 10 2021 at 14:37):

if you know upfront what this is going to be you should pass it manually

#### Pierre-Marie Pédrot (May 10 2021 at 14:37):

and here you do because you've just substituted a variable

#### Jason Gross (May 10 2021 at 14:38):

In general I don't, because I'm writing a general tactic to handle any equality of the form existT _ _ _ = existT _ _ _ which may be mentioned anywhere in the context and the goal

#### Jason Gross (May 10 2021 at 14:40):

But I'm a bit confused. It seems to me the unification problem should have a fast solution

#### Jason Gross (May 10 2021 at 14:41):

If instead of invoking induction I do revert p and then Time refine (@eq_sigT_rect_uncurried _ _ _ _ _ _)., this still takes 0.133s, but the unification problem is unifying something of the shape forall p : ?u = ?v, ?Q p with the goal, which has exactly the same shape modulo beta.

#### Jason Gross (May 10 2021 at 14:44):

Because p is not in the context of ?Q, this falls in the pattern fragment, and solving this should be as fast as just copying the goal. (So long as the goal syntactically has the form forall p : @eq (@sigT _ _) _ _, _, the instantiation is well-typed.) But Time match goal with |- ?G => pose G end. is instantaneous, and so is cbv beta, so this seems much slower than copying the goal.

#### Janno (May 10 2021 at 14:45):

I am not sure how useful this is but Set Debug Tactic Unification showed more failed unification attempts than my emacs buffer could handle. apply: (ex_rect _ _ p) (using Unicoq because evarconv cannot infer the predicate) doesn't show any failed unification whatsoever and takes a bit less than half the time that induction takes.

#### Janno (May 10 2021 at 14:49):

(It's probably not doing the same thing at all.. I was just curious about what unification was doing here.)

#### Jason Gross (May 10 2021 at 14:50):

@Janno That is interesting, but I'm not sure what to make of it. My claim is that both calls to induction should be instantaneous, and hence it sounds like even the Unicoq one is too slow. My reasoning is that, so long as the hypothesis I'm inducting on is not a section variable mentioned in the body of some definition used in the context or the goal (i.e., so long as revert dependent p succeeds), then both calls to induction are guaranteed to succeed and I can, at least in theory, craft the partial proof term and the resulting goal and context without ever invoking unification. So unification should not be doing anything hard.

#### Janno (May 10 2021 at 15:15):

I agree.. it is a bit baffling. And even the unicoq variant is definitely too slow. (I think unicoq spends a big chunk of its time instantiating evars but I am not sure about that.)

#### Jason Gross (May 11 2021 at 02:19):

I've made a parameterized version at https://github.com/coq/coq/issues/14301#issuecomment-837653213 where you can fill in the nat argument to make refine as slow as you'd like. The refine with holes is over 100x slower than the one where I solve the unification problem manually, despite the fact that the entire problem is in the pattern fragment

#### Matthieu Sozeau (May 17 2021 at 12:10):

I suspect we don't have a shortcut for the case what we're abstracting is actually a local variable...

#### Jason Gross (May 17 2021 at 20:26):

Hm, but it seems that the slowness is not pattern but apply (and to a lesser extent beta-iota normalization of intermediate and remaining goals)?

Last updated: Jun 04 2023 at 19:30 UTC