feat(Mathlib/Computability/Ackermann): Ackermann function is computable

[Komyyy]

It had been proved that Ackermann function is not primitive recursive, but not been proved that it's computable.
I proved that.

---

[github-actions]

PR summary b73698b1d3

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.Computability.Ackermann	710	716	+6 (+0.85%)

Import changes for all files

Files	Import difference
Mathlib.Computability.Ackermann	6

---

Declarations diff

+ _root_.computable₂_ack
+ eval_pappAck
+ eval_pappAck_step_succ
+ eval_pappAck_step_zero
+ pappAck
+ pappAck_prim
+ pappAck_step_prim

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).

[vihdzp]

Awesome! I'm not very familiar with the computability API, but I'll try to review this anyways.

[vihdzp]

These changes do make the code look nicer, though that's not quite what I meant. I was expecting something like

theorem primrec_ack_apply (n : ℕ) : Primrec (ack n) := sorry

This should then (hopefully) give you a simple proof of computability.

[Komyyy]

Good question!
Indeed, we can get a function of ℕ → Code from the primrec_ack_apply lemma by using choose tactic, but there are no warranty that the function is computable, this fact is used to the proof.
So, we should give an explicit code function which is computable.

[Komyyy]

By the way, there are uncountable noncomputable many code functions which evaluated to ack n.

[Komyyy]

@vihdzp If the review is done, please approve.

[vihdzp]

Is prim a standard abbreviation for Primrec in these files? I would've expected a name primrec_pappAck_step instead.

[Komyyy]

For the consistency with lemmas like Code.prec_prim.

[vihdzp]

This can also be used to prove the primrec_ack_apply theorem I mentioned earlier, right? Can you add that too?

[Komyyy]

This lemma can prove Computable (ack n) but not Primrec (ack n).
This is clearly weaker than Computable₂ ack so it's not worth to prove.