/prove - Machine-Verified Proofs (5-Phase Workflow)

For mathematicians who want verified proofs without learning Lean syntax.

Prerequisites

Before using this skill, check Lean4 is installed:

Check if lake is available

command -v lake &>/dev/null && echo "Lean4 installed" || echo "Lean4 NOT installed"

If not installed:

Install elan (Lean version manager)

curl https://raw.githubusercontent.com/leanprover/elan/master/elan-init.sh -sSf | sh

Restart shell, then verify

lake --version

First run of /prove will download Mathlib (~2GB) via lake build .

Usage

/prove every group homomorphism preserves identity /prove Monsky's theorem /prove continuous functions on compact sets are uniformly continuous

The 5-Phase Workflow

┌─────────────────────────────────────────────────────────────┐ │ 📚 RESEARCH → 🏗️ DESIGN → 🧪 TEST → ⚙️ IMPLEMENT → ✅ VERIFY │ └─────────────────────────────────────────────────────────────┘

Phase 1: RESEARCH (before any Lean)

Goal: Understand if/how this can be formalized.

Search Mathlib with Loogle (PRIMARY - type-aware search)

Use loogle for type signature search - finds lemmas by shape

loogle-search "pattern_here"

Examples:

loogle-search "Nontrivial _ ↔ _" # Find Nontrivial lemmas loogle-search "(?a → ?b) → List ?a → List ?b" # Map-like functions loogle-search "IsCyclic, center" # Multiple concepts

Query syntax:

• _ = any single type

• ?a , ?b = type variables (same var = same type)

• Foo, Bar = must mention both

Search External - What's the known proof strategy?

• Use Nia MCP if available: mcp__nia__search

• Use Perplexity MCP if available: mcp__perplexity__search

• Fall back to WebSearch for papers/references

• Check: Is there an existing formalization elsewhere (Coq, Isabelle)?

Identify Obstacles

• What lemmas are NOT in Mathlib?

• Does proof require axioms beyond ZFC? (Choice, LEM, etc.)

• Is the statement even true? (search for counterexamples)

Output: Brief summary of proof strategy and obstacles

CHECKPOINT: If obstacles found, use AskUserQuestion:

• "This requires [X]. Options: (a) restricted version, (b) accept axiom, (c) abort"

Phase 2: DESIGN (skeleton with sorries)

Goal: Build proof structure before filling details.

Create Lean file with:

• Imports

• Definitions needed

• Main theorem statement

• Helper lemmas as sorry

Annotate each sorry:

-- SORRY: needs proof (straightforward) -- SORRY: needs proof (complex - ~50 lines) -- AXIOM CANDIDATE: v₂ constraint - will test in Phase 3

Verify skeleton compiles (with sorries)

Output: proofs/<theorem_name>.lean with annotated structure

Phase 3: TEST (counterexample search)

Goal: Catch false lemmas BEFORE trying to prove them.

For each AXIOM CANDIDATE sorry:

Generate test cases

-- Create #eval or example statements #eval testLemma (randomInput1) -- should return true #eval testLemma (randomInput2) -- should return true

Run tests

lake env lean test_lemmas.lean

If counterexample found:

• Report the counterexample

• Use AskUserQuestion: "Lemma is FALSE. Options: (a) restrict domain, (b) reformulate, (c) abort"

CHECKPOINT: Only proceed if all axiom candidates pass testing.

Phase 4: IMPLEMENT (fill sorries)

Goal: Complete the proofs.

Standard iteration loop:

• Pick a sorry

• Write proof attempt

• Compiler-in-the-loop checks (hook fires automatically)

• If error, Godel-Prover suggests fixes

• Iterate until sorry is filled

• Repeat for all sorries

Tools active:

• compiler-in-the-loop hook (on every Write)

• Godel-Prover suggestions (on errors)

Phase 5: VERIFY (audit)

Goal: Confirm proof quality.

Axiom Audit

lake build && grep "depends on axioms" output

• Standard: propext, Classical.choice, Quot.sound ✓

• Custom axioms: LIST EACH ONE

Sorry Count

grep -c "sorry" proofs/<file>.lean

• Must be 0 for "complete" proof

Generate Summary

✓ MACHINE VERIFIED (or ⚠️ PARTIAL - N axioms)

Theorem: <statement> Proof Strategy: <brief description>

Proved:

• <lemma 1>
• <lemma 2>

Axiomatized (if any):

• <axiom>: <why it's needed>

File: proofs/<name>.lean

Research Tool Priority

Use whatever's available, in order:

Tool Best For Command

Loogle Type signature search (PRIMARY) loogle-search "pattern"

Nia MCP Library documentation mcp__nia__search

Perplexity MCP Proof strategies, papers mcp__perplexity__search

WebSearch General references WebSearch tool

WebFetch Specific paper/page content WebFetch tool

Loogle setup: Requires ~/tools/loogle with Mathlib index. Run loogle-server & for fast queries.

If no search tools available, proceed with caution and note "research phase skipped".

Checkpoints (automatic)

The workflow pauses for user input when:

• ⚠️ Research finds obstacles

• ❌ Testing finds counterexamples

• 🔄 Implementation hits unfillable sorry after N attempts

Output Format

┌─────────────────────────────────────────────────────┐ │ ✓ MACHINE VERIFIED │ │ │ │ Theorem: ∀ φ : G →* H, φ(1_G) = 1_H │ │ │ │ Proof Strategy: Direct application of │ │ MonoidHom.map_one from Mathlib. │ │ │ │ Phases: │ │ 📚 Research: Found in Mathlib.Algebra.Group.Hom │ │ 🏗️ Design: Single lemma, no sorries needed │ │ 🧪 Test: N/A (trivial) │ │ ⚙️ Implement: 3 lines │ │ ✅ Verify: 0 custom axioms, 0 sorries │ │ │ │ File: proofs/group_hom_identity.lean │ └─────────────────────────────────────────────────────┘

What I Can Prove

Domain Examples

Category Theory Functors, natural transformations, Yoneda

Abstract Algebra Groups, rings, homomorphisms

Topology Continuity, compactness, connectedness

Analysis Limits, derivatives, integrals

Logic Propositional, first-order

Limitations

• Complex proofs may take multiple iterations

• Novel research-level proofs may exceed capabilities

• Some statements are unprovable over ℚ (need ℝ extension)

Behind The Scenes

• Lean 4.26.0 - Theorem prover

• Mathlib - 100K+ formalized theorems

• Godel-Prover - AI tactic suggestions (via LMStudio)

• Compiler-in-the-loop - Automatic verification on every write

• Research tools - Nia, Perplexity, WebSearch (graceful degradation)

See Also

• /loogle-search

• Search Mathlib by type signature (used in Phase 1 RESEARCH)

• /math-router

• For computation (integrals, equations)

• /lean4

• Direct Lean syntax access