feat(ModelTheory): add elementary chains

[NoneMore]

This PR adds the definition of elementary chains and construct their direct limits, shows that the canonical maps and compatible cocone lifts are elementary and derives theory preservation and cardinality bounds for the limit.

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• depends on: feat(SetTheory/Cardinal): add cardinality bounds for direct limits #42125

AI disclosure: Most declarations, comments, and documentation in this PR were generated with Codex. Some proofs were structurally simplified using the /golf command from Lean 4 Skills. I reviewed the resulting code and documentation and can vouch for all submitted content.

[github-actions]

PR summary 1e7695608f

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.SetTheory.Cardinal.DirectLimit (new file)	777
Mathlib.ModelTheory.ElementaryChain (new file)	809

---

Declarations diff (regex)

+ ElementaryChain
+ Limit
+ coe_natLERec
+ equivOfForallLE
+ exists_finite_common_stage
+ exists_toLimit
+ iSup_lift_mk_le_mk
+ liftElementary
+ limitEquiv
+ limit_models
+ mk_eq_iSup_lift_mk
+ mk_eq_of_forall_lift_mk_eq
+ mk_le_of_aleph0_le
+ mk_le_sum
+ mk_limit_eq_of_forall_lift_mk_eq
+ mk_limit_le_of_lift_mk_le
+ natLERec
+ natLERec_self
+ natLERec_succ
+ natLERec_trans
+ ofNatSucc
+ ofNatSucc_map_succ
+ realize_toLimit
+ toEmbeddingDirectedSystem
+ toLimit
+ toLimitElementary
+ toLimit_map

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 1e76956).

• +43 new declarations
• −0 removed declarations

+DirectLimit.equivOfForallLE
+DirectLimit.iSup_lift_mk_le_mk
+DirectLimit.mk_eq_iSup_lift_mk
+DirectLimit.mk_eq_of_forall_lift_mk_eq
+DirectLimit.mk_le_of_aleph0_le
+DirectLimit.mk_le_sum
+FirstOrder.Language.ElementaryChain
+FirstOrder.Language.ElementaryChain.Limit
+FirstOrder.Language.ElementaryChain.carrier
+FirstOrder.Language.ElementaryChain.casesOn
+FirstOrder.Language.ElementaryChain.ctorIdx
+FirstOrder.Language.ElementaryChain.exists_finite_common_stage
+FirstOrder.Language.ElementaryChain.exists_toLimit
+FirstOrder.Language.ElementaryChain.liftElementary
+FirstOrder.Language.ElementaryChain.limit_models
+FirstOrder.Language.ElementaryChain.map
+FirstOrder.Language.ElementaryChain.map.congr_simp
+FirstOrder.Language.ElementaryChain.map_map
+FirstOrder.Language.ElementaryChain.map_self
+FirstOrder.Language.ElementaryChain.mk
+FirstOrder.Language.ElementaryChain.mk.inj
+FirstOrder.Language.ElementaryChain.mk.injEq
+FirstOrder.Language.ElementaryChain.mk.noConfusion
+FirstOrder.Language.ElementaryChain.mk.sizeOf_spec
+FirstOrder.Language.ElementaryChain.mk_limit_eq_of_forall_lift_mk_eq
+FirstOrder.Language.ElementaryChain.mk_limit_le_of_lift_mk_le
+FirstOrder.Language.ElementaryChain.noConfusion
+FirstOrder.Language.ElementaryChain.noConfusionType
+FirstOrder.Language.ElementaryChain.ofNatSucc
+FirstOrder.Language.ElementaryChain.ofNatSucc_map_succ
+FirstOrder.Language.ElementaryChain.realize_toLimit
+FirstOrder.Language.ElementaryChain.rec
+FirstOrder.Language.ElementaryChain.recOn
+FirstOrder.Language.ElementaryChain.struc
+FirstOrder.Language.ElementaryChain.toEmbeddingDirectedSystem
+FirstOrder.Language.ElementaryChain.toLimit
+FirstOrder.Language.ElementaryChain.toLimitElementary
+FirstOrder.Language.ElementaryChain.toLimit_map
+FirstOrder.Language.ElementaryEmbedding.natLERec
+FirstOrder.Language.ElementaryEmbedding.natLERec.congr_simp
+FirstOrder.Language.ElementaryEmbedding.natLERec_self
+FirstOrder.Language.ElementaryEmbedding.natLERec_succ
+FirstOrder.Language.ElementaryEmbedding.natLERec_trans

---

No changes to strong technical debt.

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

Current number	Change	Type (weak)
5022	2	exposed public sections

Current commit 1e7695608f
Reference commit 6996953ff1

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:

• feat(SetTheory/Cardinal): add cardinality bounds for direct limits #42125
  By Dependent Issues (🤖). Happy coding!