feat(ModelTheory): typeclass for theory to have nonempty models

[plp127]

We add a typeclass for a theory to have nonempty models. We provide instances for nonemptyTheory, infiniteTheory, and completeTheory.

---

• depends on: [Merged by Bors] - chore: make arguments explicit in nonempty_of_nonempty_constants #42508

[github-actions]

PR summary 4cc12ddf7c

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

---

Declarations diff (regex)

+ IsNonemptyTheory
+ IsNonemptyTheory.mono
+ IsNonemptyTheory.nonempty
+ instance : IsNonemptyTheory (infiniteTheory L)
+ instance : IsNonemptyTheory (nonemptyTheory L)
+ instance : Nonempty Language.ring.Constants := ⟨zeroFunc⟩
+ instance [Nonempty M] : IsNonemptyTheory (completeTheory L M)
+ isNonemptyTheory_of_cardGe_mem
+ isNonemptyTheory_of_nonempty_constants

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 4cc12dd).

• +14 new declarations
• −0 removed declarations

+FirstOrder.Language.IsNonemptyTheory
+FirstOrder.Language.IsNonemptyTheory.casesOn
+FirstOrder.Language.IsNonemptyTheory.mk
+FirstOrder.Language.IsNonemptyTheory.mono
+FirstOrder.Language.IsNonemptyTheory.nonempty
+FirstOrder.Language.IsNonemptyTheory.not_models'
+FirstOrder.Language.IsNonemptyTheory.rec
+FirstOrder.Language.IsNonemptyTheory.recOn
+FirstOrder.Language.instIsNonemptyTheoryCompleteTheoryOfNonempty
+FirstOrder.Language.instIsNonemptyTheoryInfiniteTheory
+FirstOrder.Language.instIsNonemptyTheoryNonemptyTheory
+FirstOrder.Language.isNonemptyTheory_of_cardGe_mem
+FirstOrder.Language.isNonemptyTheory_of_nonempty_constants
+FirstOrder.Ring.instNonemptyConstantsRing

---

No changes to strong technical debt.

No changes to weak technical debt.

Current commit 4cc12ddf7c
Reference commit 38c06cd7f3

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).

[mathlib-dependent-issues]

This PR/issue depends on:

• [Merged by Bors] - chore: make arguments explicit in nonempty_of_nonempty_constants #42508
  By Dependent Issues (🤖). Happy coding!