refactor: SetSemiring as a 1-field structure

[Parcly-Taxel]

Ultimately inspired by #36934 (comment).

---

• depends on: refactor: review Kleene algebra axioms #35963
• depends on: feat: pushing Lattice and predecessors through Equiv #37605

[github-actions]

PR summary caa79c2e3c

Import changes exceeding 2%

%	File
+14.85%	Mathlib.Data.Set.Semiring

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.Data.Set.Semiring	485	557	+72 (+14.85%)

Import changes for all files

Files	Import difference
Mathlib.Data.Set.Semiring	72

---

Declarations diff

+ distribLattice
+ equiv
+ instIdemSemiring
+ instNonAssocSemiring
+ instNonUnitalSemiring
+ instance : Add (SetSemiring α) where add s t := ⟨s.down ∪ t.down⟩
+ instance : Mul (SetSemiring α) where mul s t := ⟨image2 (· * ·) s.down t.down⟩
+ instance : One (SetSemiring α) where one := ⟨1⟩
+ instance : OrderBot (SetSemiring α)
+ instance : PartialOrder (SetSemiring α)
+ instance : Zero (SetSemiring α) where zero := ⟨∅⟩
+ lattice
+ le
+ linearOrder
+ lt
+ lt_def
+ max
+ max_def
+ min
+ min_def
+ ord
+ ord_def
+ partialOrder
+ preorder
+ semilatticeInf
+ semilatticeSup
++ le_def
- down
- instance (priority := 100) IdemSemiring.toOrderBot [IdemSemiring α] : OrderBot α
- instance : Add (SetSemiring α) where add s t := (s.down ∪ t.down).up
- instance : One (SetSemiring α) where one := (1 : Set α).up
- instance : Zero (SetSemiring α) where zero := (∅ : Set α).up
- instance [Monoid α] : IdemSemiring (SetSemiring α)
- instance [MulOneClass α] : NonAssocSemiring (SetSemiring α)
- instance [Semigroup α] : NonUnitalSemiring (SetSemiring α)

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.

---

Decrease in tech debt: (relative, absolute) = (2.00, 0.00)

Current number	Change	Type
6899	-2	backward.isDefEq.respectTransparency

Current commit 7d62089e41
Reference commit caa79c2e3c

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).

[YaelDillies]

What's the motivation here? Can this be a separate PR?

[Parcly-Taxel]

It is already in a separate PR awaiting review, #35963.

[YaelDillies]

Ah but then this PR is blocked by that one! I have updated the PR description accordingly

[YaelDillies]

This too should be a separate PR, please 😄

[Parcly-Taxel]

#37605

[mathlib-dependent-issues]

This PR/issue depends on:

• refactor: review Kleene algebra axioms #35963
• feat: pushing Lattice and predecessors through Equiv #37605
  By Dependent Issues (🤖). Happy coding!