feat: Make PNat.recOn induction eliminator

[EtienneC30]

Following this discussion on Zulip:
https://leanprover.zulipchat.com/#narrow/channel/113489-new-members/topic/Base.20Case.20for.20Induction.20on.20N.2B/near/492807806

---

[github-actions]

PR summary e823c5c32c

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff

No declarations were harmed in the making of this PR! 🐙

You can run this locally as follows

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

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

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

---

No changes to technical debt.

You can run this locally as

./scripts/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).

[riccardobrasca]

Can you have a quick look at where this is used in the library and replace it with the induction tactic? I guess it's not very common so it shouldn't be too much work.

[EtienneC30]

There is
refine (find_eq_iff _).2 ⟨PNat.find_spec h₂, fun n => PNat.recOn n ?_ ?_⟩
in Data.PNat.Find, should I change it too? It seems reasonable to me.

[riccardobrasca]

There is
refine (find_eq_iff _).2 ⟨PNat.find_spec h₂, fun n => PNat.recOn n ?_ ?_⟩
in Data.PNat.Find, should I change it too? It seems reasonable to me.

Yes, that's exactly what I mean.

[riccardobrasca]

There is
refine (find_eq_iff _).2 ⟨PNat.find_spec h₂, fun n => PNat.recOn n ?_ ?_⟩
in Data.PNat.Find, should I change it too? It seems reasonable to me.

Yes, that's exactly what I mean.

Well, only if it's not difficult to do it, don't spend too much time with it.

[EtienneC30]

Done!

[riccardobrasca]

There is a building error :)

Anyway I won't have time to have a look in the next few days, but it seems an improvements to me.

[edegeltje]

the build error is caused by these lines, where induction n using PNat.recOn is used with the old case names.

[mattrobball]

bors merge