## Stream: general

### Topic: Error due to several linear_order on N

#### Rémy Degenne (Jan 17 2021 at 10:36):

In the following code, I have an error at the last line of the lemma. It is apparently due to the two max involved (in h and in the goal) being defined through different linear_order on N. I am not exactly sure of what happens there. How do I go around that difficulty?

import data.finset.intervals

local attribute [instance] classical.prop_decidable

/-- A set s is sup-closed if for all t₁, t₂ ∈ s, t₁ ⊔ t₂ ∈ s. -/
def sup_closed {α} [has_sup α] (s : set α) : Prop := ∀ t1 t2, t1 ∈ s → t2 ∈ s → t1 ⊔ t2 ∈ s

lemma sup_closed_tail_finset_set (N : ℕ) :
sup_closed {s : finset ℕ | ∃ r : ℕ, s = finset.Ico N (N+r+1)} :=
begin
rintros s1 s2 ⟨r1, hs1⟩ ⟨r2, hs2⟩,
use (r1 ⊔ r2),
have hr : ∀ r : ℕ, N ≤ N + r + 1,
{ intro r,
nth_rewrite 0 ←add_zero N,
have h := finset.Ico.union' (hr r1) (hr r2),
rw min_self at h,
exact h, -- error caused by two different linear_order on ℕ
end


#### Shing Tak Lam (Jan 17 2021 at 10:38):

The example you posted works for me?

#### Mario Carneiro (Jan 17 2021 at 10:38):

Do you perhaps have open_locale classical in your file?

#### Rémy Degenne (Jan 17 2021 at 10:40):

I added the line that apparently causes the error : local attribute [instance] classical.prop_decidable

#### Rémy Degenne (Jan 17 2021 at 10:42):

but if I remove that from my original file, I get errors in many other theorems. Should I add decidable hypotheses to all lemma instead of having that line on top of the file?

#### Sebastien Gouezel (Jan 17 2021 at 10:44):

open_locale classical is better, because it adjusts more carefully the priorities.

#### Sebastien Gouezel (Jan 17 2021 at 10:44):

Your snippet works with open_locale classical, and hopefully the rest of your file should also work.

#### Rémy Degenne (Jan 17 2021 at 10:46):

I get errors at different places, of the same type, like

synthesized type class instance is not definitionally equal to expression inferred by typing rules, synthesized
semilattice_sup_bot.to_semilattice_sup (finset ℕ)
inferred
semilattice_sup_bot.to_semilattice_sup (finset ℕ)


#### Sebastien Gouezel (Jan 17 2021 at 10:48):

#mwe with open_locale classical?

#### Rémy Degenne (Jan 17 2021 at 10:49):

I'll try and make one, but it will take a while. That error is in the middle of a lemma near the end of a big new file with long proofs :)

#### Sebastien Gouezel (Jan 17 2021 at 10:52):

Otherwise, you can post a gist to your file

#### Rémy Degenne (Jan 17 2021 at 10:58):

but I'll try and extract something short

#### Kevin Buzzard (Jan 17 2021 at 11:02):

Are the errors like

invalid field notation, 'smul_finite' is not a valid "field" because environment does not contain 'measure_theory.measure.smul_finite'
μ
which has type
measure α


expected (in indep2_of_indep2_sets_of_pi_system_aux)? Is this related?

#### Rémy Degenne (Jan 17 2021 at 11:03):

That function should exist on that branch. There are two new lemma outside of independence.lean

#### Rémy Degenne (Jan 17 2021 at 11:04):

in measure_space.lean

#### Kevin Buzzard (Jan 17 2021 at 11:04):

Then maybe you should give a mathlib branch to look at and not a gist. It's independence, right?

yes it is

#### Rémy Degenne (Jan 17 2021 at 11:06):

In short, the situation is that all was working well until I merged master into the branch a few days ago. And now I am trying to understand what changed.

#### Kevin Buzzard (Jan 17 2021 at 11:14):

The issue seems to be that because finset is constructive (i.e. often wants a proof of decidable equality), and your finite sets of naturals sometimes seem to sometimes have the actual proof of decidable equality and sometimes the magic (classical) one. Her's a fix for aux_t1_inter_t2:

lemma aux_t1_inter_t2 {α} (N r : ℕ) (f1 f2 : ℕ → set α) (p1 p2 : finset ℕ)
(hp1 : p1 = finset.range N) (hp2 : p2 = finset.Ico N (N + r + 1)) :
((⋂ (i : ℕ) (hp : i ∈ p1), f1 i) ∩ ⋂ (i : ℕ) (hp : i ∈ p2), f2 i)
= ⋂ (i : ℕ) (h_le : i ∈ finset.range (N + r + 1)),
(ite (i ∈ p1) (f1 i) set.univ ∩ ite (i ∈ p2) (f2 i) set.univ) :=
begin
rw finset.Inter_inter_Inter_eq_Inter_ite,
have h_congr : p1 ∪ p2 = finset.range (N + r + 1),
{ rw [hp1, hp2, ←finset.Ico.zero_bot, ←finset.Ico.zero_bot],
have h_le : N ≤ N + r + 1,
nth_rewrite 0 ←add_zero N,
rw ←finset.Ico.union_consecutive (zero_le N) h_le },
congr,
ext1 i,
congr,
{ convert h_congr, },
ext1 x,
{ congr', convert h_congr, },
exact λ a a' haa', by congr',
end


but I am not offering an explanation as to what has changed, just a workaround.

#### Kevin Buzzard (Jan 17 2021 at 11:15):

I am changing refl to congr' and exact to convert because these tactics know that by proof irrelevance it often doesn't matter which algorithm we are using (the concrete one or the one that doesn't actually exist).

#### Kevin Buzzard (Jan 17 2021 at 11:16):

      { convert @is_measurable.ite α (s i) _ _ _ (hf1m i) (λ _, @is_measurable.univ α (s i)), },
{ convert @is_measurable.ite α (s i) _ _ _ (hf2m i) (λ _, @is_measurable.univ α (s i)), }, },


fixes head_n_indep_tail_n_pi_systems. This is a very impressive looking piece of work by the way! Many thanks!

#### Rémy Degenne (Jan 17 2021 at 11:17):

Thanks a lot for the help! I never used convert until now. I'll read its doc. There are so many tactics I am still not aware of.

#### Kevin Buzzard (Jan 17 2021 at 11:18):

and adding by convert fixes your last error:

  have h_pi_head : is_pi_system p_head,
from is_pi_system_pi_system_Union_Inter (λ n, (s n).is_measurable') h_pi {finset.range N}
(by convert sup_closed_singleton (finset.range N)),
...
have h_pi_tail : is_pi_system p_tail,
from is_pi_system_pi_system_Union_Inter (λ n, (s n).is_measurable') h_pi S_tail
(by convert sup_closed_tail_finset_set N),


#### Kevin Buzzard (Jan 17 2021 at 11:18):

Shall I just push to the branch? I am not answering your question, I am just solving the problem at hand.

#### Kevin Buzzard (Jan 17 2021 at 11:18):

This is just a trick I've seen before.

#### Rémy Degenne (Jan 17 2021 at 11:19):

Yes if you could push these, that would be great!

#### Kevin Buzzard (Jan 17 2021 at 11:19):

Are there errors in other files? I didn't check.

#### Rémy Degenne (Jan 17 2021 at 11:19):

no, the others are fine.

#### Kevin Buzzard (Jan 17 2021 at 11:20):

OK I pushed -- independence.lean is compiling again with open_locale classical.

Thanks!

#### Kevin Buzzard (Jan 17 2021 at 11:21):

Many thanks for this work! This looks like a very impressive addition to mathlib. I'm happy to help!

Last updated: May 14 2021 at 00:42 UTC