feat: partial sections for Filter.Product

[plp127]

Change the definition of Filter.Product to only use partial functions defined in some neighborhood of l. This is mathematically the correct definition, and differs from the previous definition in that with the previous definition IsEmpty (l.Product ε) ↔ ∃ x : α, IsEmpty (ε x) but with the new definition IsEmpty (l.Product ε) ↔ ∃ᶠ x in l, IsEmpty (ε x). See Zulip. See also #36763.

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[github-actions]

PR summary 4ccea1f1a1

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.ModelTheory.Ultraproducts	752	753	+1 (+0.13%)

Import changes for all files

Files	Import difference
10 files Mathlib.FieldTheory.AxGrothendieck Mathlib.ModelTheory.Algebra.Field.IsAlgClosed Mathlib.ModelTheory.Complexity Mathlib.ModelTheory.Equivalence Mathlib.ModelTheory.Graph Mathlib.ModelTheory.Order Mathlib.ModelTheory.Satisfiability Mathlib.ModelTheory.Topology.Types Mathlib.ModelTheory.Types Mathlib.ModelTheory.Ultraproducts	1

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Declarations diff (regex)

+ Germ.{u,
+ Product.{u,
+ boundedFormulaRealize_ofPartialFun
+ coe_inf
+ coe_sup
+ formulaRealize_ofPartialFun
+ funMap_ofPartialFun
+ inductionOnPartialFun
+ instance [(x : α) → Nonempty (ε x)] : Nonempty (l.Product ε)
+ instance [Subsingleton β] : Subsingleton (Germ l β)
+ instance [l.NeBot] [Nontrivial β] : Nontrivial (Germ l β)
+ instance [l.NeBot] [∀ x, Nontrivial (ε x)] : Nontrivial (l.Product ε)
+ instance [∀ x, Subsingleton (ε x)] : Subsingleton (l.Product ε)
+ isConstant_iff_exists
+ isEmpty_iff
+ liftOfPartialFun
+ liftOfPartialFun_ofPartialFun
+ liftOn_coe
+ liftOn_ofFun
+ liftPred_iff_map_eq_const_true
+ liftRel_eq_iff
+ liftRel_iff_map₂_eq_const_true
+ nonempty_iff
+ nontrivial_iff
+ ofFun_def
+ ofFun_eq_iff
+ ofFun_eq_ofPartialFun
+ ofPartialFun
+ ofPartialFun_eq_iff
+ ofPartialFun_eq_of_subset
+ relMap_ofPartialFun
+ subsingleton
+ subsingleton_iff
+ subsingleton_of_bot
+ termRealize_ofPartialFun
++- liftOn
+-+ ofFun
- Germ
- Product
- Product.instNonempty
- germSetoid
- instNontrivial
- mk'_eq_coe
- quot_mk_eq_coe
-++ inductionOn

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

## summary with just the declaration names:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
../mathlib-ci/scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit 4ccea1f).

• +39 new declarations
• −7 removed declarations

+Filter.Germ.coe_inf
+Filter.Germ.coe_sup
-Filter.Germ.instNontrivial
+Filter.Germ.instNontrivialOfNeBot
+Filter.Germ.instSubsingleton
+Filter.Germ.isConstant_iff_exists
+Filter.Germ.liftOn.congr_simp
+Filter.Germ.liftOn_coe
+Filter.Germ.liftPred_iff_map_eq_const_true
+Filter.Germ.liftRel_eq_iff
+Filter.Germ.liftRel_iff_map₂_eq_const_true
-Filter.Germ.mk'_eq_coe
-Filter.Germ.quot_mk_eq_coe
+Filter.Germ.subsingleton_of_bot
+Filter.Product.inductionOn
+Filter.Product.inductionOnPartialFun
+Filter.Product.instNonempty
+Filter.Product.instNontrivialOfNeBot
+Filter.Product.instSubsingleton
+Filter.Product.isEmpty_iff
+Filter.Product.liftOfPartialFun
+Filter.Product.liftOfPartialFun.congr_simp
+Filter.Product.liftOfPartialFun_ofPartialFun
+Filter.Product.liftOn
+Filter.Product.liftOn_ofFun
+Filter.Product.nonempty_iff
+Filter.Product.nontrivial_iff
+Filter.Product.ofFun
+Filter.Product.ofFun_def
+Filter.Product.ofFun_eq_iff
+Filter.Product.ofFun_eq_ofPartialFun
+Filter.Product.ofPartialFun
+Filter.Product.ofPartialFun.congr_simp
+Filter.Product.ofPartialFun_eq_iff
+Filter.Product.ofPartialFun_eq_of_subset
+Filter.Product.subsingleton
+Filter.Product.subsingleton_iff
-Filter.germSetoid
-Filter.productSetoid
-FirstOrder.Language.Ultraproduct.Product.instNonempty
+FirstOrder.Language.Ultraproduct.boundedFormulaRealize_ofPartialFun
+FirstOrder.Language.Ultraproduct.formulaRealize_ofPartialFun
+FirstOrder.Language.Ultraproduct.funMap_ofPartialFun
+FirstOrder.Language.Ultraproduct.relMap_ofPartialFun
-FirstOrder.Language.Ultraproduct.setoidPrestructure
+FirstOrder.Language.Ultraproduct.termRealize_ofPartialFun

---

Decrease in strong tech debt: (relative, absolute) = (1.00, 0.00)

Current number	Change	Type (strong)
498	-1	erw

Decrease in weak tech debt: (relative, absolute) = (1.00, 0.00)

Current number	Change	Type (weak)
5046	-1	exposed public sections

Current commit 4ccea1f1a1
Reference commit 9c0c555bde

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/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).