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LeanLTL: A unifying framework for linear temporal logics in Lean

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read LeanLTL is a Lean 4 framework that represents linear temporal formulas as sets of traces, allows arbitrary Lean expressions inside formulas, and formally proves that LTL and LTLf embed into it.

desk verdict A useful Lean framework for LTL with finite/infinite traces; the macro's next-push rule has a real LTLf soundness gap that should be addressed. read the letter →

arxiv 2507.01780 v1 pith:2JLF2FVZ submitted 2025-07-02 cs.LO cs.PL

Eric Vin , Kyle A. Miller , Daniel J. Fremont This is my paper
classification cs.LOcs.PL MSC 03B4403B7068Q60
keywords LinearTemporalLogicLean4InteractiveTheoremProvingLTLfFinitetracesTracesemanticsAutomatedreasoning
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

LeanLTL is a Lean 4 library that gives one representation for linear temporal logic over both infinite and finite traces. Its distinctive move is to let formulas contain arbitrary Lean expressions, not just proposition letters, so a property like "the queue length stays below 100" can be written and reasoned about directly. The paper provides formal embeddings of standard LTL and LTLf into the framework, with equisatisfaction proofs, plus automation that reduces temporal goals to arithmetic goals. If the framework works as claimed, users can verify systems where temporal behavior and numerical data interact, without needing a decidable theory.

What carries the argument

The load-bearing mechanism is the pair of types TraceSet σ and TraceFun σ α, together with the LLTL[...] macro that lowers conventional LTL notation to them. TraceSet σ is a structure wrapping the predicate Trace σ → Prop, so conjunction, disjunction, globally, until, and the rest are operations on sets of traces; this is what lets Lean's own tactics, after push_ltl expands satisfaction, work on ordinary first-order goals. TraceFun σ α evaluates a Lean expression on a trace and returns Option α; the sget/wget binders turn a none (the trace has ended) into False or True, which is the exact point where infinite-trace and finite-trace semantics differ. The macro first installs interpretations of Lean connectives as trace-set operations, then pushes X inside embedded trace functions, then lifts ← binders out, and finally coerces pure propositions to constant trace sets.

What would settle it

Find any Lean expression and any finite or infinite trace for which the satisfaction of the formula before a macro rewrite differs from the satisfaction after it, for instance comparing LLTL[X ((← a) < (← b))] with LLTL[(← X a) < (← X b)]; one such mismatch would refute semantic preservation.

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Extended reading notes

Core claim

The paper's central claim is that a single semantic representation, the trace set, can serve as a common target for linear temporal logics in Lean 4. A LeanLTL formula is literally a TraceSet σ, a predicate on traces of type σ, where a trace may be infinite or finite and is required to be nonempty. Arbitrary Lean expressions are admitted into formulas through TraceFun, an evaluator that returns an optional value at the current time; the strong-get and weak-get binders decide how a missing value (the end of a finite trace) is treated, and this is how the difference between LTL's next and LTLf's strong/weak next is captured. On top of these primitives the LLTL[...] macro reuses Lean's own connectives and notation, so G ((← queue) ≥ 0) looks like ordinary syntax but elaborates to trace-set operations. The paper formalizes embeddings of LTL and LTLf by defining their inductive syntax, translating it to TraceSet, and proving an equisatisfiability theorem between the original evaluation and the translation (abbreviated in Figure 2), and it demonstrates the library on a traffic-light example with arithmetic state.

Load-bearing premise

The whole construction rests on the assumption that the macro's rewrite rules, especially pushing the next operator into embedded trace functions, preserve the meaning of every formula a user can write, and this preservation is not proven in the paper.

Editorial extensions

If this is right

  • Users can state temporal specifications in which atomic parts are arbitrary Lean computations, for example inequalities over natural numbers, and prove them with Lean's existing arithmetic tactics.
  • The same framework covers infinite traces, finite traces, strong next, and weak next, so a single library substitutes for separate LTL and LTLf developments.
  • The formal equisatisfaction theorems mean any theorem proved for the LeanLTL translation transfers back to the original LTL or LTLf formula.
  • The push_ltl simp set reduces satisfaction of temporal operators to first-order goals that existing tactics such as linarith, omega, and nlinarith can often finish.
  • The traffic-light case study shows the library can be used to verify system-plus-environment assumptions against desired temporal specifications inside Lean.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: if semantic preservation of the macro is established, LeanLTL would cover the modulo-theories logics LTLMT and LTLfMT as a special case, since trace functions are more general than theory atoms.
  • Editorial inference: because a formula is just a predicate on traces, past-time and bounded-time operators could be added without changing the core representation; only new operators and rewrite rules would be needed.
  • Editorial inference: a likely testable extension is to prove embeddings of LTLMT and LTLfMT, which the paper itself lists as future work and which would connect the Lean development to the automated-solver literature.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper proposes LeanLTL, a Lean 4 framework for discrete-time linear temporal logics over both infinite and finite traces. Its core model is TraceSet (semantic sets of satisfying traces) and TraceFun (functions from traces to option values), with strong/weak get operators for embedding arbitrary Lean expressions into formulas. An LLTL[...] macro reinterprets Lean's connectives as LTL connectives and lifts out embedded trace functions. The paper claims formal embeddings of LTL and LTLf, presents a push_ltl simplification automation, and gives a traffic-light case study. Complete proofs are said to reside in an external repository.

Significance. The goal is worthwhile: a framework that supports both finite and infinite traces and arbitrary (including undecidable) theories in an interactive theorem prover would extend LTLMT-style reasoning beyond decidable fragments. The semantic presentation is clean, the use of Lean's existing tactics through a simp set is pragmatic, and the repository artifacts, if machine-checked, are valuable. The main risk is the soundness of the user-facing macro, which is not formally specified or proven.

major comments (2)
  1. [Section 3.2] The 'X notation is pushed into arrow notations' rule is not qualified with respect to LTLf's strong and weak next operators, and no preservation theorem is stated. Under the natural interpretation that `X` (like `Xs`) is a strong next on finite traces, the rule is unsound when a weak get is involved. Let `t` be a finite trace of length 1 and let `a,b : TraceFun σ Nat` be projections. The intended LTLf reading of `LLTL[Xs ((←w a) < (←w b))]` is false, because strong next at the last state is false. The push rule expands it to `LLTL[(←w Xs a) < (←w Xs b)]`; at `t` the shifted trace functions `Xs a` and `Xs b` have no value, and each weak get maps `none` to `True`, so the expanded formula is true. Thus a false formula elaborates to a true one. Even if the authors intend the push rule only for some next operator, the paper must state exactly which next operators are subject to the rule and prove that the macro expansion preserves the intended semantics for all embedded Lean expressions; the 'strong gets bound first' convention does not provide such a proof.
  2. [Section 4] The embeddings of LTL and LTLf are a central claimed contribution, but the paper gives only a code skeleton (Figure 2) and refers to a repository for the complete proofs. The exact statements of the equisatisfaction theorems and the main proof arguments are absent, so the claim cannot be checked from the manuscript. Please include the full theorem statements and at least a proof sketch, and make the repository artifact accessible for independent compilation.
minor comments (4)
  1. [Section 3.1] The type `N∞` is used without definition; please define the extended natural numbers or cite the Mathlib definition.
  2. [Figure 3] The symbol `X` appears both as a temporal next operator and as a trace-function shift (e.g., in `X (← TL1Queue)`); please disambiguate these two uses in the macro description.
  3. [Section 3.2] The phrase 'reliably causes the surrounding LTL formula to evaluate to false at the end of a trace' is misleading in general, since the cited convention applies only to strong gets in specific positions and does not account for weak gets or for next operators pushed into embedded expressions.
  4. [Section 3.2] The paper should clarify that `←s` and `←w` are not ordinary Lean values but macro-level binders; the current presentation may confuse readers about how `none` values interact with compound expressions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity found; LeanLTL's core claims are self-contained formalizations rather than derivations from their own outputs.

full rationale

LeanLTL is a formalization artifact: the TraceSet and TraceFun semantics are defined directly, and the claimed LTL and LTLf embeddings are proved against the standard semantics of those external logics (references [24] and [6]) with machine-checked Lean proofs. The equisat theorem in Figure 2 is a proved correspondence, not an input assumption, and no parameter is fitted to data and then relabeled as a prediction. The paper contains no load-bearing self-citation chain: the cited prior work is external, and the uniqueness of the framework is not imported from the authors' own theorems. The only notable weakness is that the Section 3.2 macro rule pushing X into arrow notations is stated without a proof of semantic preservation; that is a soundness/verification gap, not circularity, because the macro's behavior is an implementation choice rather than a claim whose conclusion is already contained in its premise. The framework is therefore self-contained against the external LTL/LTLf semantics it targets, and no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The framework's soundness rests on a few assumptions stated in the paper: the semantic definitions of traces and trace functions are taken as given, the macro's syntactic transformations are asserted to preserve meaning, and the LTL/LTLf embedding proofs are delegated to the external repository. No free parameters are fitted, and no new physical or logical entities are posited beyond the library's data structures.

assumptions (3)
  • domain assumption Traces are nonempty
    The Trace structure requires nempty : 0 < length, ensuring every proposition has a defined truth value, following LTLf conventions (Section 3.1).
  • ad hoc to paper The LLTL macro expansion preserves semantics for all embedded Lean expressions
    The macro transforms formulas by pushing X into trace functions and lifting arrow bindings, but the paper does not prove this transformation is semantics-preserving (Section 3.2).
  • standard math Standard LTL and LTLf semantics are correct as defined in the cited literature
    The embedding proofs rely on the classical semantics of LTL (Piterman and Pnueli 2018) and LTLf (De Giacomo and Vardi 2013).

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Cite this review

Pith. "Pith review of LeanLTL: A unifying framework for linear temporal logics in Lean." pith.science (2026). https://pith.science/paper/2JLF2FVZ

@misc{pith2026250701780,
  author       = {Pith},
  title        = {Pith review of: LeanLTL: A unifying framework for linear temporal logics in Lean},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2JLF2FVZ}},
  note         = {Machine review of arXiv:2507.01780}
}
read the original abstract

We propose LeanLTL, a unifying framework for linear temporal logics in Lean 4. LeanLTL supports reasoning about traces that represent either infinite or finite linear time. The library allows traditional LTL syntax to be combined with arbitrary Lean expressions, making it straightforward to define properties involving numerical or other types. We prove that standard flavors of LTL can be embedded in our framework. The library also provides automation for reasoning about LeanLTL formulas in a way that facilitates using Lean's existing tactics. Finally, we provide examples illustrating the utility of the library in reasoning about systems that come from applications.

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