/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.

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

```bash # 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

1. Search External - What's the known proof strategy?

- Use Nia MCP if available: mcpniasearch - Use Perplexity MCP if available: mcpperplexitysearch - Fall back to WebSearch for papers/references - Check: Is there an existing formalization elsewhere (Coq, Isabelle)?

1. 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)

1. 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.

1. Create Lean file with:

- Imports - Definitions needed - Main theorem statement - Helper lemmas as sorry

1. Annotate each sorry:

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

1. 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:

1. Generate test cases

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

1. Run tests

``bash lake env lean test_lemmas.lean ``

1. 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:

1. Pick a sorry
2. Write proof attempt
3. Compiler-in-the-loop checks (hook fires automatically)
4. If error, Godel-Prover suggests fixes
5. Iterate until sorry is filled
6. 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.

1. Axiom Audit

``bash lake build && grep "depends on axioms" output `` - Standard: propext, Classical.choice, Quot.sound ✓ - Custom axioms: LIST EACH ONE

1. Sorry Count

``bash grep -c "sorry" proofs/<file>.lean `` - Must be 0 for "complete" proof

1. 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:

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

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