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arxiv: 2604.19000 · v1 · submitted 2026-04-21 · 💻 cs.LG · cs.AI

Decompose, Structure, and Repair: A Neuro-Symbolic Framework for Autoformalization via Operator Trees

Xiaoyang Liu , Zineng Dong , Yifan Bai , Yantao Li , Yuntian Liu , Tao Luo This is my paper

Pith reviewed 2026-05-10 02:46 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords autoformalizationneuro-symbolic frameworkoperator treesformal mathematicsLean 4error repairmathematical statementsPRIME benchmark
0
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The pith

A neuro-symbolic framework improves autoformalization by mapping statements to operator trees for targeted error repair.

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

The paper presents Decompose, Structure, and Repair (DSR), which turns the task of translating natural language math into formal code into a three-step process. It first breaks statements into parts, then builds operator trees that capture their logical hierarchy, and finally uses those trees to find and fix mistakes in the formal output by adjusting sub-parts. This differs from previous methods that treat the formal code as a single sequence. The authors create a new test set of 156 theorems from textbooks formalized in Lean 4 and show their method performs better than other approaches while using similar amounts of computation.

Core claim

DSR restructures autoformalization into a modular pipeline that decomposes natural-language statements into logical components, maps them to structured operator trees as a topological blueprint, and leverages sub-tree refinement to precisely localize and repair errors in the generated formal code, achieving new state-of-the-art results on the PRIME benchmark of 156 undergraduate and graduate-level theorems.

What carries the argument

Operator trees, which represent the hierarchical logic of mathematical statements and serve as a blueprint for localizing errors and performing sub-tree refinement during repair.

If this is right

  • Treating formal code as hierarchical structures rather than flat sequences improves accuracy in translating natural language mathematics.
  • Error localization and repair become more precise when guided by the topological structure of operator trees.
  • Modular neuro-symbolic pipelines can outperform end-to-end large language models on formalization tasks under fixed resource limits.
  • The PRIME benchmark of annotated Lean 4 theorems provides a standardized way to measure progress on this task.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The tree-based decomposition may apply to other structured reasoning domains where logical hierarchy matters, such as proof generation.
  • Public release of the benchmark and models could let others test whether the same structure helps scale to more advanced theorems.
  • If operator trees capture essential mathematical topology, hybrid systems might combine this repair step with larger generative models for broader coverage.

Load-bearing premise

That mapping natural-language statements to structured operator trees enables precise localization and repair of errors via sub-tree refinement.

What would settle it

A head-to-head evaluation on the PRIME benchmark where DSR fails to produce higher formalization success rates than sequence-based baselines when both use the same computational budget.

Figures

Figures reproduced from arXiv: 2604.19000 by Tao Luo, Xiaoyang Liu, Yantao Li, Yifan Bai, Yuntian Liu, Zineng Dong.

Figure 1
Figure 1. Figure 1: The Decompose, Structure, and Repair (DSR) Framework. Given an NL statement, DSR first decomposes it into logical NL components. The framework then structures the translation by mapping each NL component to its corresponding FL component and its associated FL OPT. Finally, a tree-guided repair loop leverages the OPT to precisely localize and repair errors via sub-tree refinement. studies validate the effic… view at source ↗
Figure 2
Figure 2. Figure 2: Semantic Decomposition. The NL statement is de￾composed into NL components via Semantic Canonicalization to filter crossed-out noise and explicate green-highlighted implicit context, followed by Structural Role Alignment to categorize these components into CONDITIONS and CONCLUSION. 3.2. Structuring Translation with Operator Trees 3.2.1. MOTIVATION In our framework, the formalizer is designed to generate t… view at source ↗
Figure 3
Figure 3. Figure 3: Training Pipeline of the DSR Formalizer. (a) Data Construction: A structured corpus is built by aligning NL components, FL components, and FL OPTs. (b) Complexity Stratification: The corpus is stratified into three complexity levels, while basic NL and FL components constitute the atomic data. (c) Curriculum Learning: A four-stage strategy where the model first learns basic NL-to-FL component translation, … view at source ↗
Figure 4
Figure 4. Figure 4: A repair trajectory of the tree-guided repair process. More detailed repair examples can be found in Appendix D. 5 [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: , the benchmark spans diverse domains including Algebra, Analysis, and Number Theory. To ensure a high-fidelity ground truth, we adopted a metic￾ulous construction process. For each entry, we extracted the NL statement and its informal proof from the source text. Subsequently, the NL statements were manually for￾malized by Lean experts into corresponding Lean 4 theorem statements. This expert-in-the-loop a… view at source ↗
Figure 6
Figure 6. Figure 6: The repair starts by replacing the non-Lean type Z + with N. Then it repeatedly fixes the misuse of the informal order symbol o (treated as a numeral rather than a function) by switching to orderOf. A locally-compiling but semantically wrong patch o = m is later corrected at the global statement level into orderOf g = m. *Note: The declaration for type G is neglected during decomposition as this feature is… view at source ↗
Figure 7
Figure 7. Figure 7: The model first corrects the instance issue and the incorrect definition at the subcomponent level, and then tries to resolve the undefined T star. When this fails, it attempts to fix the issue in the parent scope, but still fails. Finally, it moves to the statement level and adds the missing namespace and the finite-dimensional instance. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The model first resolves a namespace error by correcting abs to Complex.abs at the subcomponent level. It subsequently addresses a semantic inconsistency regarding a domain error by shifting the summation index to n + 1 at the statement level. E. Prompt Templates Prompt Template for Autoformalization (Qwen3-Max) You are an expert in mathematics and Lean 4. Please autoformalize the following problem in Lean… view at source ↗
read the original abstract

Statement autoformalization acts as a critical bridge between human mathematics and formal mathematics by translating natural language problems into formal language. While prior works have focused on data synthesis and diverse training paradigms to optimize end-to-end Large Language Models (LLMs), they typically treat formal code as flat sequences, neglecting the hierarchical logic inherent in mathematical statements. In this work, we introduce Decompose, Structure, and Repair (DSR), a neuro-symbolic framework that restructures autoformalization into a modular pipeline. DSR decomposes statements into logical components and maps them to structured operator trees, leveraging this topological blueprint to precisely localize and repair errors via sub-tree refinement. Furthermore, we introduce PRIME, a benchmark of 156 undergraduate and graduate-level theorems selected from canonical textbooks and expertly annotated in Lean 4. Experimental results demonstrate that DSR establishes a new state-of-the-art, consistently outperforming baselines under equivalent computational budgets. The datasets, model, and code will be released to the public soon.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The paper introduces Decompose, Structure, and Repair (DSR), a neuro-symbolic framework for statement autoformalization. It decomposes natural-language mathematical statements into logical components, maps them to hierarchical operator trees, and performs error localization and repair via sub-tree refinement. The authors also present the PRIME benchmark (156 undergraduate/graduate theorems from canonical textbooks, annotated in Lean 4) and claim that DSR achieves new state-of-the-art performance, consistently outperforming baselines under equivalent computational budgets.

Significance. If the operator-tree repair mechanism can be shown to drive the reported gains, the work would meaningfully advance autoformalization by exploiting hierarchical structure instead of treating formal code as flat sequences. The introduction of the PRIME benchmark is a clear positive contribution that could serve as a reusable evaluation resource. However, the absence of ablations isolating the tree-based repair step substantially weakens the ability to assess the framework's true impact or novelty relative to decomposition and prompting alone.

major comments (3)
  1. [§4] §4 (Experimental Results) and abstract: the SOTA claim and outperformance under equivalent budgets are presented only as aggregate success rates on PRIME. No ablation removing the sub-tree refinement step, no quantitative localization accuracy, and no before/after examples of operator-tree edits are supplied, so it is impossible to attribute gains specifically to the topological repair mechanism rather than the initial decomposition or richer prompting.
  2. [§3.2] §3.2 (Operator Tree Construction and Repair): the central assertion that operator trees enable 'precise localization and repair of errors via sub-tree refinement' is load-bearing for the framework's novelty, yet the manuscript provides neither formal definition of the repair operator nor empirical verification (e.g., success rate of localization or edit distance before/after repair).
  3. [Table 1] Table 1 (or equivalent results table): without reported standard deviations, multiple random seeds, or controlled budget-matched comparisons that hold the LLM and prompt fixed while varying only the tree-repair component, the 'consistent outperformance' claim cannot be evaluated for statistical robustness.
minor comments (3)
  1. The abstract states that 'datasets, model, and code will be released to the public soon'; a concrete timeline or repository link would strengthen reproducibility claims.
  2. [§3] An illustrative example showing a natural-language statement, its operator-tree decomposition, the detected error subtree, and the repaired formal statement would greatly improve clarity of the pipeline in §3.
  3. Minor notation inconsistency: the term 'operator trees' is introduced without an explicit recursive definition or grammar; adding one would aid readers unfamiliar with the representation.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed review. We appreciate the recognition of the PRIME benchmark as a positive contribution and the potential significance of the operator-tree approach for advancing autoformalization. We respond point-by-point to the major comments below, acknowledging where additional evidence is needed and outlining specific revisions to address the concerns.

read point-by-point responses

  1. Referee: [§4] §4 (Experimental Results) and abstract: the SOTA claim and outperformance under equivalent budgets are presented only as aggregate success rates on PRIME. No ablation removing the sub-tree refinement step, no quantitative localization accuracy, and no before/after examples of operator-tree edits are supplied, so it is impossible to attribute gains specifically to the topological repair mechanism rather than the initial decomposition or richer prompting.

    Authors: We agree that the current presentation of aggregate results limits the ability to isolate the contribution of sub-tree refinement. The manuscript reports performance under equivalent computational budgets but does not include the requested ablation, localization metrics, or edit examples. In the revised version we will add an ablation study that removes the repair step while keeping decomposition and prompting fixed, report quantitative localization accuracy, and include concrete before/after operator-tree edit examples to directly attribute gains to the repair mechanism. revision: yes

  2. Referee: [§3.2] §3.2 (Operator Tree Construction and Repair): the central assertion that operator trees enable 'precise localization and repair of errors via sub-tree refinement' is load-bearing for the framework's novelty, yet the manuscript provides neither formal definition of the repair operator nor empirical verification (e.g., success rate of localization or edit distance before/after repair).

    Authors: We will add a formal definition of the repair operator, including its inputs, outputs, and sub-tree refinement procedure, to Section 3.2. We will also include empirical verification consisting of localization success rates and average edit distances before versus after repair, computed on the PRIME benchmark, to substantiate the claim that operator trees enable precise error localization and repair. revision: yes

  3. Referee: [Table 1] Table 1 (or equivalent results table): without reported standard deviations, multiple random seeds, or controlled budget-matched comparisons that hold the LLM and prompt fixed while varying only the tree-repair component, the 'consistent outperformance' claim cannot be evaluated for statistical robustness.

    Authors: We acknowledge that statistical robustness requires standard deviations and controlled comparisons. The revised manuscript will report mean success rates with standard deviations over at least three random seeds. We will further add a controlled experiment that holds the underlying LLM and prompt template fixed while varying only the presence of the tree-repair component, thereby providing direct evidence of its incremental contribution under matched budgets. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical framework with new benchmark and aggregate comparisons

full rationale

The paper introduces a modular neuro-symbolic pipeline (decompose to operator trees, then repair) and evaluates it on a newly constructed PRIME benchmark of 156 theorems. Claims of SOTA performance rest on reported success rates versus baselines under matched budgets, not on any internal derivation, fitted parameter renamed as prediction, or self-citation chain that reduces the result to its own inputs. No equations, uniqueness theorems, or ansatzes are presented that loop back by construction. The absence of ablations or localization metrics is a limitation of evidence strength, not a circularity in the claimed derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The framework rests on the domain assumption that mathematical statements possess decomposable logical structure representable by operator trees; no free parameters or invented entities with independent evidence are described in the abstract.

axioms (1)
  • domain assumption Mathematical statements can be decomposed into logical components that map to hierarchical operator trees.
    This is the core premise enabling the structure and repair stages of DSR.
invented entities (2)
  • Operator trees no independent evidence
    purpose: Represent hierarchical logic of statements for error localization and sub-tree repair.
    Introduced as the central data structure in the DSR pipeline.
  • PRIME benchmark no independent evidence
    purpose: Provide 156 annotated undergraduate and graduate theorems in Lean 4 for evaluation.
    Newly created dataset described in the abstract.

pith-pipeline@v0.9.0 · 5490 in / 1235 out tokens · 42794 ms · 2026-05-10T02:46:40.079196+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

50 extracted references · 2 canonical work pages

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  4. [4]

    ,→ (m n :N) ✓Fix: ReplaceZ+withN

    Component-Level Repair (m n :Z +) ×Error: Nonnegative integers should beN. ,→ (m n :N) ✓Fix: ReplaceZ+withN

  5. [5]

    ,→ o = m ✓(Partial) Fix: remove application to satisfy parser/typechecker locally

    Subcomponent-Level Repair o g = m ×Error: ‘Order’ is not defined asoin Mathlib. ,→ o = m ✓(Partial) Fix: remove application to satisfy parser/typechecker locally. (Semantically wrong; fixed globally later.)

  6. [6]

    ,→ orderOf h = n ✓Fix: useorderOffor element order in Mathlib

    Component-Level Repair o h = n ×Error: ‘Order’ is not defined asoin Mathlib. ,→ orderOf h = n ✓Fix: useorderOffor element order in Mathlib

  7. [7]

    ,→ orderOf (g ˆ n * h ˆ m) = Nat.lcm m n / Nat.gcd m n ✓Fix: replace the informal order notationo(·)byorderOf

    Component-Level Repair o (g ˆ n * h ˆ m) = Nat.lcm m n / Nat.gcd m n ×Error: ‘Order’ is not defined asoin Mathlib. ,→ orderOf (g ˆ n * h ˆ m) = Nat.lcm m n / Nat.gcd m n ✓Fix: replace the informal order notationo(·)byorderOf

  8. [8]

    ×Issue: hypothesis became ill-formed semantically after local patch; it no longer states the order ofg

    Statement-Level Repair ...(h2 : o = m) ... ×Issue: hypothesis became ill-formed semantically after local patch; it no longer states the order ofg. ,→ (h2 : orderOf g = m) ✓Fix: restore intended meaningo(g) =musingorderOf g. Consistency check passed. Final Verified Code: theorem test [Group G] (g h : G) (m n :N) (h1 : g * h = h * g) (h2 : orderOf g = m) (h...

  9. [9]

    ,→ [SeminormedAddCommGroup V] [InnerProductSpaceCV] ✓Fix: Add instance ‘SeminormedAddCommGroup’

    Subcomponent-Level Repair [InnerProductSpaceCV] ×Error: Failed to synthesize SeminormedAddCommGroup V . ,→ [SeminormedAddCommGroup V] [InnerProductSpaceCV] ✓Fix: Add instance ‘SeminormedAddCommGroup’

  10. [10]

    ,→ algebraMapR C ✓Fix:Cis an algebra overR, not the converse

    Subcomponent-Level Repair algebraMapC R ×Error: Failed to synthesize algebraMapC R. ,→ algebraMapR C ✓Fix:Cis an algebra overR, not the converse

  11. [11]

    ,→ T star : Module.EndCV ×Failed: Error is caused by the definition of adjoint map, not the type ofT star

    Subcomponent-Level Repair T star ×Error: T star is undefined. ,→ T star : Module.EndCV ×Failed: Error is caused by the definition of adjoint map, not the type ofT star

  12. [12]

    ,→ T = star T

    (Parent) Subcomponent-Level Repair T = T star↔ ∀(eigenvalue :C), (∃(v : V), v̸=0∧T v = eigenvalue·v)→ eigenvalue∈Set.range (algebraMapR C) ×Error: T star is undefined. ,→ T = star T ... ×Failed:star Tis not the adjoint map in Mathlib

  13. [13]

    (Parent) Component-Level Repair ∀(V : Type *) [SeminormedAddCommGroup V] [InnerProductSpaceCV] (T : Module.EndCV), T * T star = T star * T→(T = T star↔ ∀(eigenvalue :C), (∃(v : V), v̸=0∧T v = eigenvalue·v)→eigenvalue∈Set.range (algebraMapR C)) ×Error: T star is undefined. ,→ ... T = adjoint T ... ×Failed: The namespaceLinearMapis omitted

  14. [14]

    T * T star = T star * T→(T = T star↔...) ×Error: T star is undefined

    Statement-Level Repair ... T * T star = T star * T→(T = T star↔...) ×Error: T star is undefined. ,→ ...T * LinearMap.adjoint T = LinearMap.adjoint T * T→(T = LinearMap.adjoint T↔...) ✓Fix: Add the namespace and replace allT starwithLinearMap.adjoint T. Consistency check passed. Final Verified Code: theorem test :∀(V : Type *) [NormedAddCommGroup V] [Inner...

  15. [15]

    ,→ (Complex.abs k = 1∧z̸=1) ✓Fix: Add namespace ‘Complex’

    Subcomponent-Level Repair (abs k = 1∧z̸=1) ×Error: Function ‘abs’ not found in scope. ,→ (Complex.abs k = 1∧z̸=1) ✓Fix: Add namespace ‘Complex’

  16. [16]

    ,→ Summable fun n :N=> zˆ(n + 1) / (n + 1) ✓Fix: n + 1 can never be zero

    Statement-Level Repair Summable fun n :N=> zˆn / n ×Error: The seriesP zn n is undefined atn= 0, causing a division by zero. ,→ Summable fun n :N=> zˆ(n + 1) / (n + 1) ✓Fix: n + 1 can never be zero. Consistency check passed. Final Verified Code: theorem test:∀z :C, (Complex.abs z = 1∧z̸=1)→Summable fun n :N=> zˆ(n + 1) / (n + 1) := by sorry Figure 8.The m...

  17. [17]

    Compared to written expressions, give priority to mathematical expressions (LaTeX format)

  18. [18]

    Do not add extra information or interpret beyond the given content

    Translate each line as a whole, with each translation on a separate line ( −−>format). Do not add extra information or interpret beyond the given content

  19. [19]

    If the binder declares a variable (for example, ‘a‘) to be a type, simply write ’Let a be a type’

    Always translate the binders one−by−one, even if some of the binders can be merged in natural language. If the binder declares a variable (for example, ‘a‘) to be a type, simply write ’Let a be a type’

  20. [20]

    If the formal conclusion is a sequence of curried chain, you can start with ’If’, and then stating the binders one−by −one in the curried chain, finally give the theorem statement

  21. [21]

    If the input formal conditions are NULL, simply state the informal conditions as ’No conditions’. # Output Format Please present your response in the following structured format: **Informal Conditions:** **Informal Conclusion:** # Example # Input Data **Formal Conditions:** {a b c :R} (h : a * b * c = 1) (haux : 1 + a + a * b̸=0) **Formal Conclusion:** a ...

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    Only structure the statements

    Do Not Solve: Do not attempt to prove or solve the problem. Only structure the statements

  23. [23]

    Formalize Logic Only: Express each condition and conclusion using concise LaTeX−style mathematical notation

  24. [24]

    − Do NOT use phrases such as ”i.e.”, ”that is”, ”in other words”, ”namely”, or similar

    No Redundant Rephrasing: − Do NOT explain, restate, or paraphrase a condition in words after giving a formula. − Do NOT use phrases such as ”i.e.”, ”that is”, ”in other words”, ”namely”, or similar. − Each condition must be stated exactly once, in its most direct mathematical form

  25. [25]

    − Avoid prose descriptions like ”is an infinite sequence”; use quantifiers instead

    Explicit Quantifiers: − All variables must be explicitly quantified (e.g., $\forall n\in\mathbb{N}$). − Avoid prose descriptions like ”is an infinite sequence”; use quantifiers instead

  26. [26]

    − Each condition should represent a single logical fact

    Atomic Conditions: − Split compound statements into separate numbered conditions when possible. − Each condition should represent a single logical fact

  27. [27]

    − Do NOT add explanatory remarks about why they are needed

    Implicit Conditions: − Include standard domain constraints only if they are mathematically necessary. − Do NOT add explanatory remarks about why they are needed

  28. [28]

    − Avoid any natural language explanation beyond the formula itself

    Minimality: − Prefer the shortest mathematically complete formulation. − Avoid any natural language explanation beyond the formula itself

  29. [29]

    If the input problem statement contains only a conclusion and no conditions, state: **Conditions:** No conditions

  30. [30]

    Quantifier Classification Rule: − A quantified statement belongs to Conditions if and only if it restricts already−given objects and does not introduce any predicate whose truth is to be established. − If a universal quantifier governs a statement expressing divisibility, equality, inequality, or any nontrivial property, it MUST appear in the Conclusion a...

  31. [31]

    − Variables introduced by explicit quantifiers (∀,∃) MUST be declared only within their quantified statements and MUST NOT be separately declared as conditions

    Free Variable and Implicit Type Declaration Completion: − Every variable that appears free (i.e., not bound by∀or∃) in any condition or conclusion MUST be explicitly declared with a domain or type. − Variables introduced by explicit quantifiers (∀,∃) MUST be declared only within their quantified statements and MUST NOT be separately declared as conditions...

  32. [32]

    ”$A\subset X$” (type/membership declaration)

  33. [33]

    ”$A\subsetneq X$” (property condition) − All such free−variable and type declarations MUST be included as **Conditions** and MUST appear before all other non−declaration conditions

  34. [34]

    such that ...”, ”There exists

    Existential Quantifier Preservation: − If the input problem statement is of the form ”Find ... such that ...”, ”There exists ... such that ...”, or otherwise asserts existence, the Conclusion MUST be a single existentially quantified formula. − Do NOT place existentially quantified variables as separate domain declarations in Conditions.− Do NOT split an ...

  35. [35]

    − Do NOT omit or move universal quantifiers to Conditions; they must remain bound to their predicates in the Conclusion

    Universal Quantifier Preservation: − If the input problem statement asserts that a property holds for all elements of a certain domain (e.g., ”for all ...”, ” any ...”, ”every ...”), the Conclusion MUST include the universal quantifier explicitly. − Do NOT omit or move universal quantifiers to Conditions; they must remain bound to their predicates in the ...

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    − Do NOT merge multiple facts or properties into a single condition, even if they appear in the same sentence in natural language

    Atomic Condition Enforcement for Lean Formalization: − When extracting conditions from natural language statements, each condition MUST contain **only one atomic fact**. − Do NOT merge multiple facts or properties into a single condition, even if they appear in the same sentence in natural language. − Each atomic fact should be expressed as a separate num...

  37. [37]

    **Important Note:** The **Conclusion** section must always be a **single line**

    **Conclusion:** − ... **Important Note:** The **Conclusion** section must always be a **single line**. Do NOT split the conclusion into multiple numbered lines or separate statements. All predicates, quantifiers, and logical connectors related to the conclusion must be combined into this one line. −−− Now, perform the task for the following Input Data. **...

  38. [38]

    Your output will be programmatically used to strictly replace it

    Scope Consistency (CRITICAL): The ‘Broken Code‘ is a strict substring (an expression). Your output will be programmatically used to strictly replace it. − Output ONLY the expression. Do NOT add ‘def‘, ‘let‘, ‘have‘, ‘theorem‘, or assignment symbols (‘:=‘). − Do NOT output the surrounding code. If the input is ‘MulAction.orbitRel G H‘, do not return ‘(h1 :...

  39. [39]

    − Explicit/Implicit Arguments: Check if you need to make an argument explicit (using ‘@‘) or if you provided an explicit argument where an implicit one was expected

    Type−Driven Repair: Since there is no informal text, rely on Mathlib signatures and Type Theory: − Argument Order: Check if the function expects arguments in a different order. − Explicit/Implicit Arguments: Check if you need to make an argument explicit (using ‘@‘) or if you provided an explicit argument where an implicit one was expected. − Coercions: C...

  40. [40]

    Analyze the Current Error: Examine the ‘message‘ and ‘position‘ to identify if the issue is a Type Mismatch, Unknown Identifier, or Synthesis Failure

  41. [43]

    # Output Format Please present your response in the following structured format and do not include conversational filler

    Summarize: Write only a single sentence describing why the code failed (useful for classification). # Output Format Please present your response in the following structured format and do not include conversational filler. **Error Reason:** <One−sentence summary, keep it as simple as possible> **Corrected Code Snippet:** <The fixed expression ONLY.> −−− No...

  42. [44]

    Your output will be programmatically used to strictly replace the ‘Broken Code‘ string

    **Scope Consistency (CRITICAL):** The ‘Broken Code‘ is a strictly defined substring of a larger file. Your output will be programmatically used to strictly replace the ‘Broken Code‘ string. − Do NOT output the full theorem if the input was only a signature or a hypothesis. − Do NOT include surrounding keywords (like ‘theorem‘, ‘example‘, ‘:=‘, or ‘by‘) un...

  43. [45]

    Ensure the fix preserves the intended logic for that specific context

    Semantic Alignment: Compare the ‘Broken Code‘ against the ‘Informal Component‘. Ensure the fix preserves the intended logic for that specific context

  44. [46]

    − If there is a type mismatch, check the ‘Informal Component‘ to decide whether to cast/coerce variables or change the type definition

    Analyze the Current Error: Examine the ‘message‘ and ‘position‘ in the Error Message to pinpoint the exact failure (e.g., incorrect syntax, type mismatch, unknown identifier). − If there is a type mismatch, check the ‘Informal Component‘ to decide whether to cast/coerce variables or change the type definition

  45. [47]

    Check Context: Verify if the variables used are consistent with the ‘Previously Declared Variables‘ (If any)

  46. [49]

    # Output Format Please present your response in the following structured format and do not include conversational filler

    Summarize: Write only a single sentence describing why the code failed (useful for classification). # Output Format Please present your response in the following structured format and do not include conversational filler. **Error Reason:** <One−sentence summary, keep it as simple as possible> **Corrected Code Snippet:** <The fixed code snippet ONLY.> −−− ...

  47. [50]

    − Ensure the fix results in valid Lean 4 syntax

    **Analyze the Error Signal (CRITICAL BRANCHING):** **CASE A: ‘Error Message‘ is PRESENT:** − Focus primarily on fixing the reported syntax or type error (e.g., ”unknown identifier”, ”type mismatch”). − Ensure the fix results in valid Lean 4 syntax. **CASE B: ‘Error Message‘ is EMPTY/NULL:** − STOP DEBUGGING SYNTAX. The code already compiles. − Focus ONLY ...

  48. [51]

    Ensure the fixed code preserves the intended logic (quantifiers, implications, types) rather than just satisfying the compiler by changing the meaning

    Semantic Alignment: Compare the ‘Broken Statement‘ against the ‘Informal Statement‘. Ensure the fixed code preserves the intended logic (quantifiers, implications, types) rather than just satisfying the compiler by changing the meaning

  49. [52]

    Do not add any ‘import‘ statements ( assume Mathlib is present)

    Apply Minimal Fixes: Correct the code only at the source of the error. Do not add any ‘import‘ statements ( assume Mathlib is present)

  50. [53]

    # Output Format Please present your response in the following structured format and do not include conversational filler

    Summarize: Write only a single sentence describing why the code failed (useful for classification). # Output Format Please present your response in the following structured format and do not include conversational filler. **Error Reason:** <One−sentence summary, keep it as simple as possible> **Corrected Formal Statement:** <The fixed formal statement (th...