feat(RingTheory/DividedPowers/RatAlgebra): add definitions

[mariainesdff]

In this file we show that, for certain choices of a commutative (semi)ring A and an ideal I of
A, the family of maps ℕ → A → A given by fun n x ↦ x^n/n! is a divided power structure on I.

Co-authored-by: AntoineChambert-Loir [email protected]

---

• depends on: [Merged by Bors] - feat(Mathlib/Data/Nat/Factorial/NatCast): add IsUnit lemmas #22237
• depends on: [Merged by Bors] - feat(RingTheory/Ideal/Operations): add Ideal.pow_eq_zero_of_mem #22239
• depends on: [Merged by Bors] - feat(Algebra/GroupWithZero.Commute): add Ring.inverse_pow_mul_eq_iff_eq_mul #22240

[github-actions]

PR summary aefa7b2614

Import changes exceeding 2%

%	File
+100.86%	Mathlib.Data.Nat.Factorial.NatCast

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.Data.Nat.Factorial.NatCast	467	938	+471 (+100.86%)

Import changes for all files

Files	Import difference
Mathlib.Data.Nat.Factorial.NatCast	471
Mathlib.RingTheory.DividedPowers.RatAlgebra (new file)	1162

---

Declarations diff

+ Nat.castChoose_eq
+ dividedPowers_unique
+ dpow_add
+ dpow_add_of_lt
+ dpow_comp
+ dpow_comp_of_mul_lt
+ dpow_eq_inv_fact_smul
+ dpow_eq_of_not_mem
+ dpow_mem
+ dpow_mul
+ dpow_mul_of_add_lt
+ dpow_null
+ dpow_of_prime_le
+ dpow_of_two_le
+ dpow_one
+ dpow_zero
+ mul_dpow
++ dpow
++ dpow_apply
++ dpow_eq_of_mem
++++ dividedPowers

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).

[mathlib4-dependent-issues-bot]

This PR/issue depends on:

• [Merged by Bors] - feat(Mathlib/Data/Nat/Factorial/NatCast): add IsUnit lemmas #22237
• [Merged by Bors] - feat(RingTheory/Ideal/Operations): add Ideal.pow_eq_zero_of_mem #22239
• [Merged by Bors] - feat(Algebra/GroupWithZero.Commute): add Ring.inverse_pow_mul_eq_iff_eq_mul #22240
  By Dependent Issues (🤖). Happy coding!