refactor(RingTheory/Ideal/Heoght): minimize usages of Ideal.primeHeight

[Thmoas-Guan]

deprecate and private most of primeHeight APIs, providing height variant.

---

• depends on: feat(RingTheory): lemmas for minimal primes to lessen defeq abuse #38029

[github-actions]

PR summary 2ff88851d5

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff

+ Ideal.height_add_one_le_of_lt_of_isPrime
+ Ideal.height_eq_ringKrullDim_iff
+ Ideal.height_eq_zero_iff
+ Ideal.height_strict_mono_of_isPrime
+ Ideal.height_strict_mono_of_isPrime_of_isPrime
+ Ideal.isMaximal_of_height_eq_ringKrullDim
+ Ideal.le_minimalPrimes
+ Ideal.sup_height_isPrime_eq_ringKrullDim
+ Ideal.sup_height_of_maximal_eq_ringKrullDim
+ minimalPrimes_isPrime
- Ideal.height_strict_mono_of_is_prime

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.

---

No changes to technical debt.

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).

[erdOne]

If you are doing this separately maybe you should also go around and fix the current violations?

[Thmoas-Guan]

Sorry, what do you mean?

[erdOne]

Can you replace the hI.1.1's throughout mathlib with the new lemmas?
Can you go around the library and remove the mentions of primeHeight whenever inappropriate?
Thanks.

[Thmoas-Guan]

Except for strict monotonicity, I added all height analogus and if possible use it to imply the primeHeight verison, up to you to decide which to depreceate and which mark private.
I only know mem_minimalPrimes_of_primeHeight_eq_height is ready to deprecate?

We still have abuse of hp.1.2, I'll fix it tomorrow.

[Thmoas-Guan]

@erdOne I added Ideal.le_minimalPrimes for hp.1.2, what do you think now? I privated or deprecated all primeHeight related stuffs except for the two strict monotonicity lemmas.

[mathlib-dependent-issues]

This PR/issue depends on:

• feat(RingTheory): lemmas for minimal primes to lessen defeq abuse #38029
  By Dependent Issues (🤖). Happy coding!