---
license: apache-2.0
language:
- en
pretty_name: Mathlib Normalized S-Expressions
task_categories:
- text-generation
tags:
- lean4
- mathlib
- theorem-proving
- s-expressions
- formal-mathematics
size_categories:
- 100K<n<1M
configs:
- config_name: default
  data_files:
  - split: test
    path: test/test-*.parquet
  - split: train
    path: train/train-*.parquet
  - split: val
    path: val/val-*.parquet
dataset_info:
  splits:
  - name: test
    num_examples: 4392
  - name: train
    num_examples: 218727
  - name: val
    num_examples: 4157
---

# Mathlib Normalized S-Expressions

Lean 4 proof states from Mathlib, paired with the tactic applied at each
step, in three representations extracted directly from the Lean kernel:

1. **Source-faithful S-expressions** of the goal and every hypothesis, as
   Lean elaborated them.
2. **Normalized S-expressions** of the same state, with stable local-context
   indices suitable for model input.
3. **Annotated tactic syntax** -- the original tactic's syntax tree with
   identifier leaves resolved to the constants and hypotheses they refer to.

Row identity follows
[`cat-searcher/leandojo-benchmark-4-random`](https://huggingface.co/datasets/cat-searcher/leandojo-benchmark-4-random),
so `split` and `row_index` join directly against it and the `val`/`test`
splits remain comparable to published results on that benchmark.

Unlike pretty-printed proof states, S-expressions carry the elaborated term
structure: implicit arguments, instance resolution, and binder structure are
all explicit, which is what makes graph- and pointer-based models possible.

## Loading

```python
from datasets import load_dataset
import json

ds = load_dataset("jajostrains/Mathlib-Normalized-Sexpr", split="train")
row = ds[0]
print(row["model_goal_sexp"])

# Nested payloads are JSON strings; see the schema note below.
syntax_tree = json.loads(row["source_syntax"])
context = json.loads(row["local_context"])
```

## Coverage

**Read this before training on the `train` split.**

| split | rows published | upstream rows | coverage |
|---|---:|---:|---:|
| `test` | 4,392 | 4,506 | 97.47% |
| `train` | 218,727 | 250,814 | 87.21% |
| `val` | 4,157 | 4,260 | 97.58% |
| **total** | **227,276** | **259,580** | **87.56%** |

A row is published only when all three representations were extracted and
validated for it. Rows are missing for two distinct reasons, and the
difference matters:

- **Environment limits** -- a source file that did not elaborate within the
  timeout. These rows are absent for incidental reasons and are, in
  principle, recoverable. Their absence is close to random.
- **Extractor limits and row data** -- most importantly
  `goal_cardinality`: this extractor records states with exactly one pending
  goal, so every tactic invoked against multiple goals is excluded. Their
  absence is **systematic**.

### Known bias

The `train` split is not a uniform random subsample of its upstream split.
Because multi-goal invocations are excluded by construction, single-goal
proof states are over-represented relative to the full benchmark, as are the
tactics that tend to be applied to them. Any comparison against a model
trained on all upstream rows should account for this.

The `val` and `test` splits are near-complete, so evaluation numbers
computed on them remain comparable with published results.

### Per-split failure taxonomy

#### `test` -- 114 rows not published

| phase | rows | kind | reason |
|---|---:|---|---|
| `file_compile` | 59 | environment | Lean could not elaborate the source file within the timeout, or the REPL exited while compiling it. Retryable. |
| `goal_cardinality` | 25 | extractor limit | The tactic was invoked against multiple pending goals. This extractor records single-goal states only, so these rows are absent by design. |
| `invocation_alignment` | 24 | row data | The dataset's tactic text could not be matched to any invocation recorded by Lean in that file. |
| `ambiguous_invocation` | 6 | row data | The tactic text matched more than one invocation, so the proof state could not be attributed unambiguously. |

#### `train` -- per-phase breakdown unavailable

This split was extracted over several runs, and the manifest left on disk covers only 3,000 of its 250,814 rows. Rather than report phase counts measured on that fraction as if they described the split, they are omitted. The aggregate coverage above is measured from the published rows themselves and is exact; the causes listed under Coverage all apply here, `goal_cardinality` and `file_compile` foremost, but their relative sizes for this split are not established.

#### `val` -- 103 rows not published

| phase | rows | kind | reason |
|---|---:|---|---|
| `file_compile` | 53 | environment | Lean could not elaborate the source file within the timeout, or the REPL exited while compiling it. Retryable. |
| `goal_cardinality` | 24 | extractor limit | The tactic was invoked against multiple pending goals. This extractor records single-goal states only, so these rows are absent by design. |
| `invocation_alignment` | 18 | row data | The dataset's tactic text could not be matched to any invocation recorded by Lean in that file. |
| `ambiguous_invocation` | 4 | row data | The tactic text matched more than one invocation, so the proof state could not be attributed unambiguously. |
| `theorem_identity` | 4 | environment | The theorem's compilation unit could not be identified in the file. Retryable. |

## Schema

All three representations for a row ship in that same row, so nothing needs
to be joined across files. Recursive payloads (`source_syntax`,
`local_context`, the hypothesis lists) are stored as **JSON strings**: their
shape varies per tactic, and a typed nested column would force either schema
inference failures or a lowest-common-denominator union. Call `json.loads` on
them.

| column | description |
|---|---|
| `dataset` | Upstream dataset name the row came from. |
| `split` | Upstream split: `train`, `val`, or `test`. |
| `row_index` | Row index within the upstream split. Stable join key. |
| `theorem` | Fully qualified Mathlib theorem name. |
| `file_path` | Mathlib source path, relative to the repository root. |
| `repo_url` | Mathlib repository URL. |
| `repo_commit` | Mathlib commit the state was extracted at. |
| `tactic` | The tactic text applied at this step. |
| `text_state` | Pretty-printed goal before the tactic, as Lean printed it. |
| `text_target_state` | Pretty-printed goal after the tactic. |
| `raw_goal_sexp` | Source-faithful S-expression of the goal type. |
| `raw_hyp_sexps` | JSON string: list of `{name, sexp}` for each hypothesis, source-faithful. |
| `model_goal_sexp` | Normalized S-expression of the goal type. |
| `model_hyp_sexps` | JSON string: list of `{name, internal_name, context_index, binder_role, is_instance, is_let, sexp}`. |
| `source_syntax` | JSON string: the tactic's original syntax tree, with identifier leaves annotated by Lean-resolved references. This is the generation target. |
| `syntax_args` | JSON string: normalized argument slots of the tactic syntax. |
| `term_ranges` | JSON string: source ranges of elaborated tactic terms. |
| `local_context` | JSON string: the local context with stable indices, for pointer-style argument supervision. |
| `unit_index` | Index of the compilation unit within the file. |
| `invocation_index` | Index of the matched invocation within the unit. |
| `alignment_kind` | How the dataset tactic text was matched to the invocation. |
| `target_state_matches_invocation` | True when Lean's post-tactic goal matches the dataset's target state. A useful filter for strict agreement with upstream. |
| `pending_goal_count` | Pending goals at the invocation. Always 1; see coverage. |
| `hypothesis_count` | Number of hypotheses in the local context. |
| `hypothesis_names_match` | True when Lean's hypothesis names match those parsed from the upstream pretty-printed state. |
| `raw_schema_version` | Schema version of the source-faithful record. |
| `raw_extractor_version` | Extractor version that produced the raw record. |
| `model_schema_version` | Schema version of the normalized sidecar. |
| `model_normalization` | Normalization identifier applied to model S-expressions. |
| `trace_schema_version` | Schema version of the action trace. |
| `trace_extractor_version` | Extractor version that produced the action trace. |
| `pantograph_commit` | Pantograph commit used for raw extraction. |
| `model_pantograph_commit` | Pantograph commit used for normalization. |
| `raw_record_sha256` | Digest binding the sidecars to the raw record. Retained for audit; consumers need not check it, since all three targets ship in one row. |
| `state_sha256` | SHA-256 of the upstream state text. |
| `tactic_sha256` | SHA-256 of the upstream tactic text. |
| `target_state_sha256` | SHA-256 of the upstream target-state text. |

## Provenance

- Mathlib commit: `29dcec074de168ac2bf835a77ef68bbe069194c5`
- Upstream dataset: `cat-searcher/leandojo-benchmark-4-random`
- Extraction: each Mathlib source file is elaborated with a patched
  [Pantograph](https://github.com/lenianiva/Pantograph) REPL, and the proof
  state Lean records at each matched tactic invocation is captured. States
  are read from Lean, never parsed back from pretty-printed text.
- `test`: extractor `lean-action-trace-v3`, raw schema v3, Pantograph `unknown`
- `train`: extractor `lean-action-trace-v3`, raw schema v3, Pantograph `unknown`
- `val`: extractor `lean-action-trace-v3`, raw schema v3, Pantograph `unknown`

## Limitations

- Single-goal states only; see Coverage.
- `train` coverage is partial and biased; `val`/`test` are near-complete.
- S-expressions are large. Expect the goal and hypothesis columns to
  dominate memory; select only the representation you need.
- Normalized and source-faithful S-expressions are *different*
  representations of the same state, not alternatives of equal fidelity.
  Train on `model_*`; use `raw_*` when you need exactly what Lean emitted.

## License and attribution

This dataset is released under `apache-2.0`. It is derived work, and
both upstream licenses continue to apply to the material they cover:

- **Mathlib 4** is licensed Apache-2.0. Theorem names, tactic text, file
  paths, and every elaborated term in the S-expression columns derive from
  Mathlib at commit `29dcec074de168ac2bf835a77ef68bbe069194c5`. See
  <https://github.com/leanprover-community/mathlib4/blob/master/LICENSE>.
- **LeanDojo** is licensed MIT. The row selection and the `split` /
  `row_index` assignment are inherited unchanged from the LeanDojo
  benchmark, so its notice is reproduced below as MIT requires.

```
MIT License

Copyright (c) 2023 LeanDojo Team

Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
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The above copyright notice and this permission notice shall be included in all
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THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
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FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
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LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
SOFTWARE.
```

## Citation

Please cite both upstreams alongside this dataset: Mathlib for the
mathematical content, and LeanDojo for the benchmark this dataset's row
identity follows.