feat(ModelTheory): add syntax and semantics for infinitary logic

[cameronfreer]

Add basic syntax and semantics for infinitary model theory of L∞ω (and Lω₁ω).

The new BoundedFormulaInf L ι α n uses index type ι for all of a formula's infinitary conjunctions and disjunctions, and the definitional

abbrev BoundedFormulaω (α : Type u') (n : ℕ) := L.BoundedFormulaInf ℕ α n

lands in exactly the universe of the finitary BoundedFormula, as suggested by @plp127 in #mathlib4 > ModelTheory: API for infinitary formulas of L_{∞,ω} @ 💬.

This PR contains iSup/iInf constructors, derived connectives together with alls/exs, Realize with simp lemmas for every constructor and derived connective, and the embedding BoundedFormula.toInf with realize_toInf.

I developed this with extensive assistance from several Claude and GPT models along with lean4-skills and lean-lsp-mcp as part of the ongoing project cameronfreer/infinitary-logic.

[github-actions]

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[github-actions]

PR summary 8cfe4425f1

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.ModelTheory.Infinitary.Syntax (new file)	714
Mathlib.ModelTheory.Infinitary.Semantics (new file)	716

---

Declarations diff (regex)

+ BoundedFormulaInf
+ BoundedFormulaInf.realize_alls
+ BoundedFormulaInf.realize_exs
+ BoundedFormulaω
+ FormulaInf
+ FormulaInf.Realize
+ Formulaω
+ Realize
+ SentenceInf
+ SentenceInf.Realize
+ Sentenceω
+ alls
+ ex
+ exs
+ instance : Bot (L.BoundedFormulaInf ι α n)
+ instance : Inhabited (L.BoundedFormulaInf ι α n)
+ instance : Top (L.BoundedFormulaInf ι α n)
+ not
+ realize_all
+ realize_bot
+ realize_equal
+ realize_ex
+ realize_falsum
+ realize_iInf
+ realize_iSup
+ realize_imp
+ realize_not
+ realize_rel
+ realize_toInf
+ realize_top
+ toInf
+ toOmega
+ verum

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 8cfe442).

• +89 new declarations
• −0 removed declarations

+FirstOrder.Language.BoundedFormula.realize_toInf
+FirstOrder.Language.BoundedFormula.toInf
+FirstOrder.Language.BoundedFormula.toInf.eq_def
+FirstOrder.Language.BoundedFormula.toOmega
+FirstOrder.Language.BoundedFormulaInf
+FirstOrder.Language.BoundedFormulaInf.Realize
+FirstOrder.Language.BoundedFormulaInf.Realize.eq_def
+FirstOrder.Language.BoundedFormulaInf.all
+FirstOrder.Language.BoundedFormulaInf.all.elim
+FirstOrder.Language.BoundedFormulaInf.all.inj
+FirstOrder.Language.BoundedFormulaInf.all.injEq
+FirstOrder.Language.BoundedFormulaInf.all.noConfusion
+FirstOrder.Language.BoundedFormulaInf.all.sizeOf_spec
+FirstOrder.Language.BoundedFormulaInf.alls
+FirstOrder.Language.BoundedFormulaInf.alls.eq_def
+FirstOrder.Language.BoundedFormulaInf.below
+FirstOrder.Language.BoundedFormulaInf.brecOn
+FirstOrder.Language.BoundedFormulaInf.brecOn.eq
+FirstOrder.Language.BoundedFormulaInf.brecOn.go
+FirstOrder.Language.BoundedFormulaInf.casesOn
+FirstOrder.Language.BoundedFormulaInf.ctorElim
+FirstOrder.Language.BoundedFormulaInf.ctorElimType
+FirstOrder.Language.BoundedFormulaInf.ctorIdx
+FirstOrder.Language.BoundedFormulaInf.equal
+FirstOrder.Language.BoundedFormulaInf.equal.elim
+FirstOrder.Language.BoundedFormulaInf.equal.inj
+FirstOrder.Language.BoundedFormulaInf.equal.injEq
+FirstOrder.Language.BoundedFormulaInf.equal.noConfusion
+FirstOrder.Language.BoundedFormulaInf.equal.sizeOf_spec
+FirstOrder.Language.BoundedFormulaInf.ex
+FirstOrder.Language.BoundedFormulaInf.exs
+FirstOrder.Language.BoundedFormulaInf.exs.eq_def
+FirstOrder.Language.BoundedFormulaInf.falsum
+FirstOrder.Language.BoundedFormulaInf.falsum.elim
+FirstOrder.Language.BoundedFormulaInf.falsum.noConfusion
+FirstOrder.Language.BoundedFormulaInf.falsum.sizeOf_spec
+FirstOrder.Language.BoundedFormulaInf.iInf
+FirstOrder.Language.BoundedFormulaInf.iInf.elim
+FirstOrder.Language.BoundedFormulaInf.iInf.inj
+FirstOrder.Language.BoundedFormulaInf.iInf.injEq
+FirstOrder.Language.BoundedFormulaInf.iInf.noConfusion
+FirstOrder.Language.BoundedFormulaInf.iInf.sizeOf_spec
+FirstOrder.Language.BoundedFormulaInf.iSup
+FirstOrder.Language.BoundedFormulaInf.iSup.elim
+FirstOrder.Language.BoundedFormulaInf.iSup.inj
+FirstOrder.Language.BoundedFormulaInf.iSup.injEq
+FirstOrder.Language.BoundedFormulaInf.iSup.noConfusion
+FirstOrder.Language.BoundedFormulaInf.iSup.sizeOf_spec
+FirstOrder.Language.BoundedFormulaInf.imp
+FirstOrder.Language.BoundedFormulaInf.imp.elim
+FirstOrder.Language.BoundedFormulaInf.imp.inj
+FirstOrder.Language.BoundedFormulaInf.imp.injEq
+FirstOrder.Language.BoundedFormulaInf.imp.noConfusion
+FirstOrder.Language.BoundedFormulaInf.imp.sizeOf_spec
+FirstOrder.Language.BoundedFormulaInf.instBot
+FirstOrder.Language.BoundedFormulaInf.instInhabited
+FirstOrder.Language.BoundedFormulaInf.instTop
+FirstOrder.Language.BoundedFormulaInf.noConfusion
+FirstOrder.Language.BoundedFormulaInf.noConfusionType
+FirstOrder.Language.BoundedFormulaInf.not
+FirstOrder.Language.BoundedFormulaInf.realize_all
+FirstOrder.Language.BoundedFormulaInf.realize_alls
+FirstOrder.Language.BoundedFormulaInf.realize_bot
+FirstOrder.Language.BoundedFormulaInf.realize_equal
+FirstOrder.Language.BoundedFormulaInf.realize_ex
+FirstOrder.Language.BoundedFormulaInf.realize_exs
+FirstOrder.Language.BoundedFormulaInf.realize_falsum
+FirstOrder.Language.BoundedFormulaInf.realize_iInf
+FirstOrder.Language.BoundedFormulaInf.realize_iSup
+FirstOrder.Language.BoundedFormulaInf.realize_imp
+FirstOrder.Language.BoundedFormulaInf.realize_not
+FirstOrder.Language.BoundedFormulaInf.realize_rel
+FirstOrder.Language.BoundedFormulaInf.realize_top
+FirstOrder.Language.BoundedFormulaInf.rec
+FirstOrder.Language.BoundedFormulaInf.recOn
+FirstOrder.Language.BoundedFormulaInf.rel
+FirstOrder.Language.BoundedFormulaInf.rel.elim
+FirstOrder.Language.BoundedFormulaInf.rel.inj
+FirstOrder.Language.BoundedFormulaInf.rel.injEq
+FirstOrder.Language.BoundedFormulaInf.rel.noConfusion
+FirstOrder.Language.BoundedFormulaInf.rel.sizeOf_spec
+FirstOrder.Language.BoundedFormulaInf.verum
+FirstOrder.Language.BoundedFormulaω
+FirstOrder.Language.FormulaInf
+FirstOrder.Language.FormulaInf.Realize
+FirstOrder.Language.Formulaω
+FirstOrder.Language.SentenceInf
+FirstOrder.Language.SentenceInf.Realize
+FirstOrder.Language.Sentenceω

---

No changes to strong technical debt.

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

Current number	Change	Type (weak)
4932	2	exposed public sections

Current commit 8cfe4425f1
Reference commit caa4dbde35

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).

[cameronfreer]

LLM-generated