Prove

/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