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

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: vince-gonzalez/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