feat(RingTheory/Idempotents): generalize to Semiring, add Corner and direct product decomposition

[alreadydone]

• Add CompleteOrthogonalIdempotents.iff_ortho_complete: if a family is complete orthogonal, it consists of idempotents.

• Add CompleteOrthogonalIdempotents.pair_iff'ₛ: x and y form a complete orthogonal family iff x * y = y * x = 0 and x + y = 1. Golf pair_iff.

• Given an element e in a semigroup R, define Subsemigroup.corner = eRe. If R is a non-unital semiring and e is an idempotent, then eRe is a semiring. Define IsIdempotentElem.Corner as the Type version of corner, and provide unital (commutative) (semi)ring instances on it.

• CompleteOrthogonalIdempotents.mulEquivOfIsMulCentral: A complete orthogonal family of central idempotents in a semiring give rise to a direct product decomposition. There exists CompleteOrthogonalIdempotents.bijective_pi but it uses subtraction and quotient so it's not suitable for semirings.

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[github-actions]

PR summary 56237b65cc

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

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Declarations diff

+ CompleteOrthogonalIdempotents.iff_ortho_complete
+ CompleteOrthogonalIdempotents.mulEquivOfComm
+ CompleteOrthogonalIdempotents.mulEquivOfIsMulCentral
+ CompleteOrthogonalIdempotents.pair_iff'ₛ
+ CompleteOrthogonalIdempotents.pair_iffₛ
+ NonUnitalRing.corner
+ NonUnitalSubsemiring.corner
+ _root_.IsIdempotentElem.Corner
+ corner
+ instance [NonUnitalCommRing R] (idem : IsIdempotentElem e) : CommRing idem.Corner
+ instance [NonUnitalCommSemiring R] (idem : IsIdempotentElem e) : CommSemiring idem.Corner
+ instance [NonUnitalRing R] (idem : IsIdempotentElem e) : Ring idem.Corner
+ instance [NonUnitalSemiring R] (idem : IsIdempotentElem e) : Semiring idem.Corner
+ mem_corner_iff
+ mem_corner_iff_mem_range_mul_left
+ mem_corner_iff_mem_range_mul_right
+ mem_corner_iff_mul_left
+ mem_corner_iff_mul_right

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.

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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).