io.github.zengineco/gonzalgo

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║               where does a formal library spend its axioms?                ║
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#print axioms tells you whether one theorem depends on an axiom. It cannot tell you where an axiom is spent rather than inherited, how far that spending reaches, how much of it could be avoided, or — for a given theorem — which step introduced it. This does.

Works on Lean 4 / Mathlib and on Metamath databases (set.mm, iset.mm, nf.mm), by one program, so two foundations are compared under identical definitions rather than by analogy.

$ pip install gonzalgo
$ gonzalgo index

  THE KERNEL INDEX  (2026-08-05)   what formal libraries rest on
  library             system      theorems  unfinished  compiler   choice
  -----------------------------------------------------------------------
  Mathlib             Lean 4       437,429           0         0   66.62%
  Lean core (Init)    Lean 4        45,051           0         0   23.91%
  Std                 Lean 4        34,510           0         0   56.66%
  Batteries           Lean 4         5,249           0         0   32.63%
  set.mm              Metamath      47,621           0         -    1.22%
  ...
  14 libraries, 603,703 theorems, 0 resting on an unfinished proof.

That runs the moment it's installed — no Lean, no build, no files. Everything below needs a Lean project.

Pure Python. macOS, Windows, Linux. numpy is the only dependency.

Questions this answers

How do I know if my Lean proof depends on a sorry? Lean warns once, on the line you typed it. It does not warn you about the theorem three files later that uses that lemma and is therefore not proved either. Run gonzalgo trust and it reports every theorem that reaches a sorry anywhere upstream, however far.

How do I find a sorry I inherited from a dependency? Same command. The audit is over the whole environment, so a sorry in a library you import is reported exactly like one in your own file.

Does my project use native_decide anywhere? native_decide results are obtained by compiling and running code and believing the answer — the compiler and runtime are trusted, not the kernel, and soundness bugs have been found there. gonzalgo trust reports Lean.ofReduceBool and Lean.ofReduceNat, the axioms it emits, and how many theorems inherit them.

What axioms does this Lean theorem actually depend on, and why? #print axioms tells you whether. gonzalgo why <decl> -a <axiom> gives the shortest path from the theorem to the axiom, labelling each step as a statement dependency or a proof dependency — so you can see which step introduced it and whether it is reroutable.

Can I fail CI when a proof rests on something unfinished? Yes. --fail-on-trust, or the GitHub Action below.

If I change this definition, what breaks? gonzalgo impact splits dependents into those that name it in a statement — whose meaning changes with it — and those that only use it in a proof, which merely rebuild.

kernel > sorry

Related MCP server: Aristotle MCP Server

What it found

Pointed at Lean 4.32.1 with Mathlib — 790,171 declarations, 30 million dependency edges — the funnel from "the whole library" down to "provably removable" runs like this:

   532,605   theorems in Mathlib
  ─────────────────────────────────────────────────────────────────────
   324,808   ██████████████████████████████░░░░░░░░░░  depend on Classical.choice   61.0%
       144   ▏                                         actually SPEND it (entry points)
  ─────────────────────────────────────────────────────────────────────
    69,571   ██████░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░░  could be stated without it   13.1%
                                                       └─ a ceiling, not an estimate
  ─────────────────────────────────────────────────────────────────────
       805   substitutable sites — a choice-free instance existed, unused
       280   declarations whose ONLY route to the axiom runs through one
       276   ▏ attributable to a single tactic  ────────────────────┐
       275   ▏ kernel-verified choice-free after substitution       │
         4   ▏ kernel REJECTED — and they are exactly the 4 NOT ────┘
             ▏ attributable to that tactic. The partition was not designed.

That single tactic is omega, which supplies the Decidable arguments of six helper lemmas as a hardcoded Classical.propDecidable and never attempts instance synthesis — so proofs as elementary as a - b = 0 ↔ a ≤ b over Nat rest on the axiom of choice with no need. Filed upstream; the fix is one file.

Let a language model call it

$ pip install "gonzalgo[mcp]"

Add to your MCP client's configuration:

{
  "mcpServers": {
    "gonzalgo": { "command": "gonzalgo-mcp" }
  }
}

Ten tools: audit_trust, why, impact, axiom_reach, metamath_audit, kernel_index, plus the plumbing to produce a dump from a project.

There's also a scope tool that reports what gonzalgo can't do — read a paper, mark homework, judge whether text is any good. It's there so a model asked "is this proof correct?" about a page of prose doesn't grab the nearest proof-shaped tool and return something meaningless. Every other tool restates the precondition in its description.

The case it's built for: a generated Lean proof that fails to compile is easy to spot. One that compiles while resting on a sorry three lemmas upstream isn't, and Lean only mentions it once, in a warning, at the site.

kernel_index runs with no files and no network, so a model can call it cold for figures on known libraries.

Put it in CI, get a badge

Three lines in any Lean 4 project. Every commit is checked for theorems resting on an unfinished proof or on trusting the compiler rather than the kernel.

# .github/workflows/kernel-clean.yml
name: kernel-clean
on: [push, pull_request]
jobs:
  audit:
    runs-on: ubuntu-latest
    steps:
      - uses: actions/checkout@v4
      - uses: zengineco/gonzalgo@v1
        with:
          module: MyProject

Then the badge, which is just the workflow's own status — no extra service:

![kernel-clean](https://github.com/YOU/REPO/actions/workflows/kernel-clean.yml/badge.svg)

What the badge actually certifies. Not that the proofs are correct — Lean already checks that. That no theorem in the project is standing on a sorry somewhere upstream, and that none of them were decided by compiling and running code instead of by the kernel.

Lean warns about the sorry you just typed. It says nothing about the theorem three files later that quietly inherits it. In the worked example under examples/dirty, Lean reports one warning and the audit finds two contaminated theorems.

How it fits together

        your Lean project
               │
               │  gonzalgo lean-files ./scripts
               │  lake env lean scripts/Split.lean
               ▼
     ┌───────────────────────┐
     │   dependency graph    │   one row per declaration:
     │  statement │ proof    │   KIND · NAME · stmt-deps · proof-deps
     └───────────┬───────────┘
                 │
                 │  gonzalgo check      ← refuses a dump with no proof terms
                 ▼
     ┌───────────────────────────────────────────────────┐
     │                                                   │
     ▼                  ▼                ▼               ▼
  amplify           eligible            why            audit
  ───────           ────────            ───            ─────
  where is the      how much is         which step     which sites are
  axiom spent,      even eligible       introduced     substitutable, and
  and how far       for removal?        it?            which declarations
  does it reach?    (the ceiling)                      go clean if you fix
                                                       every one
                                                            │
                                                            ▼
                                                   lake env lean Rewrite.lean
                                                   ───────────────────────────
                                                   swap the instance in and ask
                                                   the KERNEL if the proof holds

Nothing above the kernel step is trusted on my say-so: Substitute.lean decides substitutability with collectAxioms, and Rewrite.lean submits the rewritten proof term to addDecl. A name-based screen was tried first and measured 41.5% precision, which is why none of this reads names.

Quickstart

Generate a dump from your own Lean project, then ask questions of it.

$ gonzalgo lean-files ./scripts        # writes the Lean extractors
$ cd my-lean-project
$ lake env lean scripts/Split.lean     # -> mathlib_split.tsv
$ gonzalgo check mathlib_split.tsv     # verify it actually contains proofs

Why does this theorem need choice?

$ gonzalgo why mathlib_split.tsv Int.mem_box

  Int.mem_box
    Int.mem_box
      --proof-->  Int.mem_box._proof_1_5
        --proof-->  Classical.propDecidable
          --proof-->  Classical.choice

Every hop is labelled stmt or proof, and that label is the point: a proof edge can often be rerouted by changing a tactic, a statement edge cannot be touched without changing what the theorem says. A path made only of proof edges is what makes a declaration worth patching at all.

If I change this, what breaks?

$ gonzalgo impact mathlib_split.tsv Nat.decLe

  Nat.decLe
    reached transitively    398,968   (295,411 of them theorems)
    ── direct ──
    in a STATEMENT              626   API surface: changing the
                                      type changes their meaning
    in a PROOF only           4,225   insulated: a type-preserving
                                      change costs a recompile

why run backwards. The statement/proof split is the value: a declaration whose type mentions the target has the target in its API, so its meaning moves when the target moves and its own users may need rewriting. One that merely calls it inside a proof needs nothing but a rebuild. A plain "who uses this" cannot tell them apart, which is why it can't tell you whether a change is safe.

How far does an axiom reach, and where is it spent?

$ gonzalgo amplify mathlib_split.tsv

  axiom            Classical.choice
  theorems              532,605
  dependents            324,808   reach 61.0%
  entry points              144   2.704e-04 per theorem
  amplification           2,256x

How much of that could even in principle be removed?

$ gonzalgo eligible mathlib_split.tsv

  statement CHOICE-FREE, proof dep    69,571   13.1%   <- eligible
  ...
  ceiling on removable classical dependence: 13.1%

A theorem whose statement mentions something choice-dependent cannot be made choice-free however it is proved. Only the rest are candidates, and that figure is a ceiling, not an estimate.

Metamath, same measurements:

$ gonzalgo mm set.mm iset.mm nf.mm

  set.mm
    theorems                     47,621
    logical axioms (|-)           1,561   used 1447
    median entries per axiom        2.0
    overall amplification         292.1x

Kernel Trust Profile

gonzalgo profile writes what a library rests on as JSON, meant to be committed and read by other programs:

$ lake exe gonzalgo MyProject myproject.tsv
$ gonzalgo profile myproject.tsv --name MyProject -o kernel-trust.json

{
  "ktp_version": "0.1",
  "subject": { "name": "Lean core + Std", "revision": "…", "system": "Lean 4" },
  "counts": { "theorems": 88842 },
  "unfinished":       { "theorems_reaching": 0, "axioms": ["sorryAx"] },
  "compiler_trusted": { "theorems_reaching": 0 },
  "assumptions": [
    { "name": "Classical.choice", "kind": "optional",
      "entry_points": 24, "reach": { "theorems": 31516, "fraction": 0.3548 },
      "via": "both" }
  ]
}

Two rules do most of the work. An unmeasured field is null, never 0 — Metamath has no native_decide, so its compiler_trusted count is not-applicable rather than zero, and writing zero would be a claim nobody made. And revision is refused on a dirty tree, because a profile whose commit does not describe what was measured cannot be reproduced by anyone else.

There is no summary score, deliberately. A theorem either reaches an unfinished proof or it does not; averaging that against anything else invents precision the measurement does not have.

Full specification: ktp/SPEC.md. A sample is in examples/lean-core-kernel-trust.json.

Reach versus amplification

Under inlining and factoring — operations that change how a library is written, not what it proves — the set of dependents is invariant while the set of entry points is not. Rerouting every use of an axiom through one gateway lemma, or inlining that lemma, moves amplification anywhere between 1 and the number of dependents without changing a single theorem.

So reach bears comparison between libraries; amplification describes one library's factorisation. The tool reports both and this README says which is which, because the distinction is easy to lose and expensive to lose.

One hazard worth knowing about

In Lean 4.32, ConstantInfo.value? returns none for theorems unless called as value? (allowOpaque := true), and this has changed across releases. An extractor written the obvious way records no proof terms at all: every theorem's value comes back empty, the analysis silently measures statements, and reports them as proofs. Nothing about the output looks wrong — the library just appears cleaner than it is.

gonzalgo check exists for this, and every subcommand runs it before trusting a dump:

$ gonzalgo check bad_dump.tsv
ERROR: bad_dump.tsv: 532,605 theorems, none carrying a proof term.
The extractor called `ConstantInfo.value?` without `(allowOpaque := true)` ...

It raises rather than warns. A dump with no proof terms does not produce slightly worse numbers; it produces confidently wrong ones.

Library use

from pathlib import Path
from gonzalgo import lean

dump = Path("mathlib_split.tsv")
lean.check_dump(dump)
g = lean.load(dump)

g.path_to("Int.mem_box", lean.AXIOM)      # why
g.entry_points(lean.AXIOM, among="T")     # where it is spent
g.dependents(lean.AXIOM)                  # boolean mask over all nodes
lean.eligibility(dump, g).ceiling         # what fraction could be removed

As a Lake package

This repository is also a Lean library, so the extractor can be a dependency rather than a file you copy and edit. Add to your lakefile.toml:

[[require]]
name = "gonzalgo"
git = "https://github.com/zengineco/gonzalgo"

Then no scratch file at all — it imports the target module at runtime:

$ lake exe gonzalgo MyProject myproject.tsv
importing MyProject ...
declarations written: 204543
now run:  gonzalgo check myproject.tsv

Everything the module imports is included, and the fifth output column records which module each declaration came from, which is what separates your project from its dependencies afterwards.

If you would rather call it from inside a file, the library is two lines:

import Gonzalgo
#eval Gonzalgo.dumpSplit "myproject.tsv"

The library requires Lean only — no Mathlib dependency, deliberately, since pinning a Mathlib revision here would force one on every project being measured.

Run against Lean core and Std alone, that gives:

axiom                  declared  direct  theorems
sorryAx                     yes       0         0
Lean.ofReduceBool           yes       0         0
Classical.choice            yes      32    31,516
propext                     yes  13,246    70,150

CLEAN: no theorem here rests on an unfinished proof or on the compiler.

88,842 theorems, and 32 declarations spend Classical.choice where 31,516 inherit it.

Shipped Lean sources

gonzalgo lean-files writes these into a directory of your choosing. Prefer the Lake package above for Split.lean's job; these are for the analyses that go beyond extraction, and for projects not using Lake.

Substitute.lean decides substitutability with the kernel's own bookkeeping rather than by name. A name-based screen measured 41.5% precision on set.mm; its characteristic failure is a lemma that relocates choice into an antecedent instead of discharging it, which looks like progress and is not.

Background

This package is the tooling behind Where Formal Libraries Spend Their Axioms: A Cross-Foundation Measurement, and an Avoidable Classical Dependency in Lean's omega — 10.5281/zenodo.21769846.

Applied to Lean 4.32.1 with Mathlib (790,171 declarations, 30M dependency edges), it finds 280 declarations whose only route to Classical.choice runs through a substitutable site, 276 of them attributable to a single cause in the omega decision procedure. Rewriting all 280 proof terms and submitting them to the kernel: 276 accepted, 4 rejected, 275 left free of Classical.choice.

Attribution and licence

Apache-2.0. See LICENSE and NOTICE.

OmegaFix.lean is a modified copy of Lean 4's src/Lean/Elab/Tactic/Omega/Frontend.lean, Copyright (c) 2023 Lean FRO, LLC, used under Apache-2.0. Its modifications are listed in a notice at the top of that file. It exists to demonstrate that a proposed fix compiles and produces choice-free proofs; it is not a replacement for omega and should not be used as one.

Not affiliated with or endorsed by the Lean FRO or the Mathlib community.