[Merged by Bors] - feat(Data/Nat): define Nat.nthRoot

[yury-harmonic]

Define Nat.nthRoot n a to be the nth root of a natural number a.

---

This is a dependency of a norm_num plugin that can handle (8 : Real) ^ (2 / 3 : Real).

[github-actions]

PR summary 4b0359cc77

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.Data.Nat.NthRoot.Defs (new file)	44
Mathlib.Analysis.SpecialFunctions.Pow.NthRootLemmas (new file)	1817

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Declarations diff

+ Nat.nthRoot
+ exists_pow_eq_iff
+ exists_pow_eq_iff'
+ instDecidableExistsPowEq
+ le_nthRoot_iff
+ lt_pow_nthRoot_add_one
+ nthRoot.lt_pow_go_succ_aux
+ nthRoot_eq_of_le_of_lt
+ nthRoot_lt_iff
+ nthRoot_one_left
+ nthRoot_one_right
+ nthRoot_pow
+ nthRoot_zero_left
+ nthRoot_zero_right
+ pow_nthRoot_le
+ pow_nthRoot_le_iff

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).

[eric-wieser]

I'll note there's a small overlap in concept here with #26935

[yury-harmonic]

AFAICT, there is no code overlap, but Nat.nthRoot can be used to write a norm_num extension for qpow too.

[yury-harmonic]

-awaiting-author

[pechersky]

golfs with grind (and making sure to replace omega). I was surprised that fun_induction didn't work

[yury-harmonic]

-awaiting-author

[pechersky]

Thanks! Looking forward to the norm_num extension.

maintainer merge

[github-actions]

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

[kbuzzard]

Thanks! Yes it's funny that we're using real numbers to make the API for this, but if it makes life easier then why not!

bors merge

[mathlib-bors]

Build failed (retrying...):

• Lint style

[bryangingechen]

bors r-

It turns out this PR is actually failing a style lint; not sure why it isn't throwing a red x here though: https://github.com/leanprover-community/mathlib4/actions/runs/18470107241/job/52621590367#step:2:275

@Vierkantor could you take a look when you have a chance?

edit: I think I know why it didn't fire. The lint-style-action action was not updated in the branch that this PR was based off, and pull_request workflows apparently run in that context rather than the version from master (!).

[mathlib-bors]

Canceled.

[bryangingechen]

bors d+

[mathlib-bors]

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

[yury-harmonic]

bors merge

[mathlib-bors]

Pull request successfully merged into master.

Build succeeded:

• Build
• Lint style
• Post-Build Step