/prove

name: prove
description: Formal theorem proving with research, testing, and verification phases
triggers: ["prove", "verify", "show that", "is it true", "formalize"]
allowed-tools: [Bash, Read, Write, Edit, WebSearch, WebFetch, AskUserQuestion, Grep, Glob]
priority: high

/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)

   # 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

2. **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)?

3. **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)

4. **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`

2. Annotate each sorry:

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

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:

1. **Generate test cases**

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

2. **Run tests**

   lake env lean test_lemmas.lean

3. **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**

   lake build && grep "depends on axioms" output

• Standard: propext, Classical.choice, Quot.sound ✓
• Custom axioms: LIST EACH ONE

2. **Sorry Count**

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

• Must be 0 for "complete" proof

3. **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.m