[Merged by Bors] - feat(Tactic/ComputeAsymptotics/Multiseries): non-primitive corecursion for Seq: FriendlyOperation API

[vasnesterov]

This is a continuation of #34311.

This PR adds more API about friendly operations:

• Basic friendly operations constructions
• FriendlyOperation.destruct: a "coinductive destructor" for FriendlyOperation

---

This is a part of the compute_asymptotics tactic (#28291).

[github-actions]

PR summary 507f3af613

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff

+ FriendlyOperation.cons
+ FriendlyOperation.cons_tail
+ FriendlyOperation.destruct_apply_eq_unfold
+ FriendlyOperation.of_dist_le_pow
+ FriendlyOperation.op_cons_head_eq
+ FriendlyOperation.op_head_eq
+ FriendlyOperation.unfold
+ dist_const_tail_cons_tail_le
+ dist_le_half_iff
+ friendlyOperation_iff_dist_le_dist

You can run this locally as follows

## summary with just the declaration names:
./scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh contains some details about this script.

---

Increase in tech debt: (relative, absolute) = (2.00, 0.00)

Current number	Change	Type
6901	2	backward.isDefEq.respectTransparency

Current commit 2ba6668de7
Reference commit 507f3af613

You can run this locally as

./scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[joneugster]

Thanks for the PR! I think the sequence of lemmas makes sense. Only a few comments

[joneugster]

Thank you for the PR and the applied suggestions.

maintainer merge

[github-actions]

🚀 Pull request has been placed on the maintainer queue by joneugster.

[Vierkantor]

I feel like the work starting here and going down is quite elementary and deserves to exist as own lemma, saying that two Seq α have distance less than 2⁻¹ iff they are either both nil or they are both cons with the same head. Something like dist_le_half_iff_nil_or_head_eq?

[vasnesterov]

I've added this lemma (dist_le_half_iff), but it's not clear for how to use it here.

[Vierkantor]

I see a lot of this working by cases in this file, where we specialize a lemma, do a case split, and then have to simplify the result. I wonder if we can't write a better cases / recursion principle.

[vasnesterov]

I'm not sure I understand. A better cases principle for Seq?

[joneugster]

@vasnesterov to get some movement on merging your work, could I suggest that we split this PR here at this point (again)? I think everything above could be approved quite directly, while the remainder still contains some heavy, long proofs.

I doubt that a maintainer will get around looking at the entire PR, but if we split it here, I think we might improve the changes of getting some progress.

[vasnesterov]

Done. The second part is in #37342

[Vierkantor]

Thanks for the split! I think this half is good to go now.

bors d+

[mathlib-bors]

✌️ vasnesterov can now approve this pull request. To approve and merge a pull request, simply reply with bors r+. More detailed instructions are available here.

[vasnesterov]

bors r+

[mathlib-bors]

Pull request successfully merged into master.

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