# How to Build a Formal Verification AI Platform (Cajal Clone)<\/p>\n
A step-by-step technical guide for building a system that discovers and formally verifies mathematical proofs using multi-agent AI. Each step is scoped for a developer working with Claude Code and modern tooling.<\/p>\n
—<\/p>\n
## Step 1: Set Up the Proof Assistant Environment<\/p>\n
**Goal:** Get Lean 4 and mathlib running, expose them programmatically, and establish your baseline proof-checking infrastructure.<\/p>\n
Install Lean 4 via `elan` (the Lean version manager):<\/p>\n
“`bash
\ncurl https:\/\/raw.githubusercontent.com\/leanprover\/elan\/master\/elan-init.sh -sSf | sh
\nlake new proof_env
\ncd proof_env && lake add mathlib
\n“`<\/p>\n
Mathlib is the Lean community’s massive mathematics library \u00e2\u20ac\u201d over 150,000 theorems. This is your ground truth corpus and your starting vocabulary. You’ll need it.<\/p>\n
Build a thin Python wrapper around the Lean REPL (Read-Eval-Print Loop) using the `lean4-repl` project or by spawning Lean processes directly:<\/p>\n
“`python
\n# lean_env.py
\nimport subprocess, json<\/p>\n
class LeanEnvironment:
\n def __init__(self):
\n self.proc = subprocess.Popen(
\n [“lake”, “env”, “lean”, “–server”],
\n stdin=subprocess.PIPE, stdout=subprocess.PIPE
\n )<\/p>\n
def check_proof(self, tactic_block: str) -> dict:
\n payload = json.dumps({“cmd”: tactic_block, “env”: 0})
\n self.proc.stdin.write((payload + “\\n”).encode())
\n self.proc.stdin.flush()
\n return json.loads(self.proc.stdout.readline())
\n“`<\/p>\n
**Key metric:** Proof check latency should be under 50ms for simple tactics. Optimize this aggressively \u00e2\u20ac\u201d it’s your inner loop for everything that follows.<\/p>\n
Database schema for tracking proof states:<\/p>\n
“`sql
\nCREATE TABLE proof_attempts (
\n id UUID PRIMARY KEY DEFAULT gen_random_uuid(),
\n theorem_id UUID NOT NULL REFERENCES theorems(id),
\n tactic_sequence JSONB NOT NULL,
\n lean_output TEXT,
\n verified BOOLEAN DEFAULT FALSE,
\n error_msg TEXT,
\n check_latency_ms INTEGER,
\n created_at TIMESTAMPTZ DEFAULT NOW()
\n);<\/p>\n
CREATE TABLE theorems (
\n id UUID PRIMARY KEY DEFAULT gen_random_uuid(),
\n statement TEXT NOT NULL,
\n lean_statement TEXT NOT NULL,
\n domain TEXT, — ‘algebra’, ‘topology’, ‘number_theory’, etc.
\n difficulty_estimate FLOAT,
\n source TEXT,
\n verified_proof TEXT,
\n created_at TIMESTAMPTZ DEFAULT NOW()
\n);
\n“`<\/p>\n
—<\/p>\n
## Step 2: Build the Proof Search Engine<\/p>\n
**Goal:** Implement Monte Carlo Tree Search (MCTS) over the tactic space, using Lean as the state evaluator.<\/p>\n
Proof search is a tree problem. Each node is a proof state (a set of goals remaining), each edge is a tactic applied, and success is a leaf with zero remaining goals. MCTS is a strong baseline because it balances exploration (trying novel tactics) with exploitation (following paths that have worked before).<\/p>\n
“`python
\n# mcts.py
\nimport math
\nfrom dataclasses import dataclass, field
\nfrom typing import Optional<\/p>\n
@dataclass
\nclass ProofNode:
\n state: str # Lean tactic state as string
\n tactic: Optional[str] = None
\n parent: Optional[‘ProofNode’] = None
\n children: list = field(default_factory=list)
\n visits: int = 0
\n value: float = 0.0
\n is_terminal: bool = False
\n is_proved: bool = False<\/p>\n
def ucb_score(self, exploration_weight=1.4) -> float:
\n if self.visits == 0:
\n return float(‘inf’)
\n exploitation = self.value \/ self.visits
\n exploration = exploration_weight * math.sqrt(
\n math.log(self.parent.visits) \/ self.visits
\n )
\n return exploitation + exploration<\/p>\n
class MCTSProofSearch:
\n def __init__(self, lean_env, llm_client, num_simulations=500):
\n self.lean = lean_env
\n self.llm = llm_client
\n self.num_simulations = num_simulations<\/p>\n
def search(self, theorem: str) -> Optional[list[str]]:
\n root = ProofNode(state=theorem)
\n for _ in range(self.num_simulations):
\n node = self._select(root)
\n result = self._expand_and_simulate(node)
\n self._backpropagate(node, result)
\n if result[‘proved’]:
\n return self._extract_proof_path(node)
\n return None<\/p>\n
def _select(self, node: ProofNode) -> ProofNode:
\n while node.children and not node.is_terminal:
\n node = max(node.children, key=lambda n: n.ucb_score())
\n return node<\/p>\n
def _expand_and_simulate(self, node: ProofNode) -> dict:
\n # Ask LLM for candidate tactics given current proof state
\n tactics = self.llm.suggest_tactics(node.state, n=8)
\n for tactic in tactics:
\n result = self.lean.apply_tactic(node.state, tactic)
\n child = ProofNode(
\n state=result[‘new_state’],
\n tactic=tactic,
\n parent=node,
\n is_terminal=result[‘is_terminal’],
\n is_proved=result[‘is_proved’]
\n )
\n node.children.append(child)
\n return {‘proved’: any(c.is_proved for c in node.children)}
\n“`<\/p>\n
Add beam search as a complementary strategy for simpler theorems where MCTS overhead isn’t worth it. Switch between strategies based on estimated theorem difficulty.<\/p>\n
—<\/p>\n
## Step 3: Train a Proof-Generation Model<\/p>\n
**Goal:** Fine-tune a language model specifically on formal proof corpora so it generates valid Lean tactics rather than plausible-looking nonsense.<\/p>\n
Start with a strong base model (Qwen2.5-Math or DeepSeek-Prover are solid open-source options). Fine-tune on Lean 4 proof data using next-token prediction on tactic sequences.<\/p>\n
Data format for fine-tuning:<\/p>\n
“`jsonl
\n{“messages”: [
\n {“role”: “system”, “content”: “You are a Lean 4 proof assistant. Given a theorem statement and current proof state, suggest the next tactic.”},
\n {“role”: “user”, “content”: “Theorem: \u00e2\u02c6\u20ac n : \u00e2\u201e\u2022, n + 0 = n\\nCurrent state: \u00e2\u0160\u00a2 \u00e2\u02c6\u20ac n : \u00e2\u201e\u2022, n + 0 = n”},
\n {“role”: “assistant”, “content”: “intro n\\nsimp [Nat.add_zero]”}
\n]}
\n“`<\/p>\n
After supervised fine-tuning, apply GRPO (Group Relative Policy Optimization) with the Lean kernel as the reward function:<\/p>\n
“`python
\ndef compute_reward(proof_attempt: list[str], theorem: str, lean_env) -> float:
\n result = lean_env.check_full_proof(theorem, proof_attempt)
\n if result[‘verified’]:
\n return 1.0
\n # Partial credit for making progress (fewer goals remaining)
\n progress = result.get(‘goals_closed’, 0) \/ result.get(‘total_goals’, 1)
\n return progress * 0.3
\n“`<\/p>\n
The reward signal is clean and binary at the top level \u00e2\u20ac\u201d the proof either checks out or it doesn’t. This is what makes formal verification uniquely powerful for RL: no learned reward model, no reward hacking, no ambiguity.<\/p>\n
—<\/p>\n
## Step 4: Build the Tau Multi-Agent Orchestration System<\/p>\n
**Goal:** Coordinate multiple specialized agents to collaborate on proof discovery.<\/p>\n
Different agents handle different parts of the search. Implement a supervisor that routes tasks and aggregates results:<\/p>\n
“`python
\n# orchestrator.py
\nfrom enum import Enum
\nfrom typing import Protocol<\/p>\n
class AgentRole(Enum):
\n STRATEGIST = “strategist” # High-level proof plan
\n TACTICIAN = “tactician” # Low-level tactic generation
\n CRITIC = “critic” # Evaluates partial proofs
\n SPECIALIST = “specialist” # Domain expert (algebra, analysis, etc.)
\n VERIFIER = “verifier” # Calls Lean kernel<\/p>\n
class ProofAgent(Protocol):
\n role: AgentRole
\n async def act(self, state: dict) -> dict: …<\/p>\n
class TauOrchestrator:
\n def __init__(self, agents: list[ProofAgent], lean_env, max_rounds=50):
\n self.agents = {a.role: a for a in agents}
\n self.lean = lean_env
\n self.max_rounds = max_rounds<\/p>\n
async def prove(self, theorem: str) -> dict:
\n state = {
\n “theorem”: theorem,
\n “proof_steps”: [],
\n “current_goals”: [theorem],
\n “failed_tactics”: [],
\n “round”: 0
\n }<\/p>\n
while state[“round”] < self.max_rounds and state[“current_goals”]:
\n # Strategist sets the plan
\n strategy = await self.agents[AgentRole.STRATEGIST].act(state)
\n state[“strategy”] = strategy[“plan”]<\/p>\n
# Tactician generates concrete steps
\n tactics = await self.agents[AgentRole.TACTICIAN].act(state)<\/p>\n
# Critic filters bad moves before wasting Lean calls
\n filtered = await self.agents[AgentRole.CRITIC].act({
\n **state, “proposed_tactics”: tactics[“tactics”]
\n })<\/p>\n
# Apply surviving tactics, verify with Lean
\n for tactic in filtered[“approved_tactics”]:
\n result = self.lean.apply_tactic(
\n state[“current_goals”][0], tactic
\n )
\n if result[“success”]:
\n state[“proof_steps”].append(tactic)
\n state[“current_goals”] = result[“remaining_goals”]
\n break
\n else:
\n state[“failed_tactics”].append(tactic)<\/p>\n
state[“round”] += 1<\/p>\n
verified = self.lean.check_full_proof(theorem, state[“proof_steps”])
\n return {“proof”: state[“proof_steps”], “verified”: verified[“success”]}
\n“`<\/p>\n
Use a message queue (Redis Streams or RabbitMQ) for agent coordination in production. Each agent is a separate service; the orchestrator is the control plane.<\/p>\n
—<\/p>\n
## Step 5: Build the Dataset Pipeline<\/p>\n
**Goal:** Curate, formalize, and verify mathematical corpora at scale for sale to AI labs.<\/p>\n
This is your primary business asset. Build it like it matters \u00e2\u20ac\u201d because it does.<\/p>\n
Pipeline stages:<\/p>\n
1. **Ingest** \u00e2\u20ac\u201d Scrape arXiv math papers, ProofWiki, existing Lean\/Coq\/Isabelle libraries. Parse LaTeX with `latexml` or `plasTeX`.
\n2. **Formalize** \u00e2\u20ac\u201d Use your proof-generation model to translate informal math into Lean 4 statements.
\n3. **Verify** \u00e2\u20ac\u201d Every statement gets checked by the Lean kernel. Failed verifications go to a human review queue or back to the model.
\n4. **Grade** \u00e2\u20ac\u201d Assign difficulty scores, domain tags, and proof complexity metrics.
\n5. **Deduplicate** \u00e2\u20ac\u201d Embedding-based dedup to remove near-identical theorems.<\/p>\n
“`sql
\n— Dataset versioning schema
\nCREATE TABLE dataset_versions (
\n id UUID PRIMARY KEY DEFAULT gen_random_uuid(),
\n version_tag TEXT UNIQUE NOT NULL, — ‘v1.2.0’
\n proof_assistant TEXT NOT NULL, — ‘lean4’, ‘coq’, ‘isabelle’
\n theorem_count INTEGER,
\n verified_count INTEGER,
\n domain_breakdown JSONB,
\n created_at TIMESTAMPTZ DEFAULT NOW(),
\n s3_path TEXT NOT NULL
\n);<\/p>\n
CREATE TABLE theorem_provenance (
\n theorem_id UUID REFERENCES theorems(id),
\n dataset_version_id UUID REFERENCES dataset_versions(id),
\n source_url TEXT,
\n formalization_model TEXT,
\n human_reviewed BOOLEAN DEFAULT FALSE,
\n PRIMARY KEY (theorem_id, dataset_version_id)
\n);
\n“`<\/p>\n
Export in multiple formats: raw Lean files, JSONL for training, Parquet for analytics. Automate nightly builds and publish checksums.<\/p>\n
—<\/p>\n
## Step 6: Build the API Layer<\/p>\n
**Goal:** Expose your RL environments, datasets, and evaluation endpoints to paying customers.<\/p>\n
Three distinct API surfaces, each with different latency and throughput requirements.<\/p>\n
**RL Environment API** (latency-critical, sub-millisecond target):<\/p>\n
“`python
\n# FastAPI with async Lean pool
\n@app.post(“\/v1\/env\/step”)
\nasync def env_step(request: StepRequest, api_key: APIKey = Depends(verify_key)):
\n env = await lean_pool.acquire(request.env_id)
\n result = await env.apply_tactic_async(request.tactic)
\n return {
\n “observation”: result.new_state,
\n “reward”: 1.0 if result.proved else 0.0,
\n “done”: result.is_terminal,
\n “info”: {“goals_remaining”: result.goal_count}
\n }<\/p>\n
@app.post(“\/v1\/env\/reset”)
\nasync def env_reset(request: ResetRequest, api_key: APIKey = Depends(verify_key)):
\n env_id = await lean_pool.spawn(request.theorem)
\n return {“env_id”: env_id, “observation”: request.theorem}
\n“`<\/p>\n
Maintain a warm pool of pre-initialized Lean processes. Cold-starting Lean is slow (200\u00e2\u20ac\u201c500ms); warm instances check tactics in under 5ms.<\/p>\n
**Dataset API** (throughput-optimized):<\/p>\n
“`python
\n@app.get(“\/v1\/datasets\/{version}\/theorems”)
\nasync def get_theorems(
\n version: str,
\n domain: Optional[str] = None,
\n min_difficulty: float = 0.0,
\n limit: int = 1000,
\n offset: int = 0,
\n api_key: APIKey = Depends(verify_key)
\n):
\n # Stream from S3 or serve from read replica
\n …
\n“`<\/p>\n
**Eval API:**<\/p>\n
“`python
\n@app.post(“\/v1\/eval\/run”)
\nasync def run_evaluation(request: EvalRequest, api_key: APIKey = Depends(verify_key)):
\n job_id = await eval_queue.enqueue({
\n “model_endpoint”: request.model_endpoint,
\n “benchmark_id”: request.benchmark_id,
\n “pass_at_k”: request.k,
\n “timeout_per_problem”: request.timeout_s
\n })
\n return {“job_id”: job_id, “status”: “queued”}
\n“`<\/p>\n
—<\/p>\n
## Step 7: Deploy and Productize<\/p>\n
**Goal:** Ship to production, onboard customers, and build the billing\/usage infrastructure.<\/p>\n
**Infrastructure:**<\/p>\n
– API layer: Kubernetes on GKE or EKS, autoscaled on request latency
\n– Lean pool: Stateful pods, pre-warmed, drained gracefully before termination
\n– Database: Postgres (RDS or Cloud SQL) with read replicas for dataset queries
\n– Queue: Redis for RL environment session state, RabbitMQ for eval jobs
\n– Storage: S3 for dataset artifacts, versioned with lifecycle policies<\/p>\n
**Billing schema:**<\/p>\n
“`sql
\nCREATE TABLE usage_events (
\n id UUID PRIMARY KEY DEFAULT gen_random_uuid(),
\n org_id UUID REFERENCES organizations(id),
\n event_type TEXT NOT NULL, — ‘env_step’, ‘dataset_download’, ‘eval_run’
\n quantity INTEGER DEFAULT 1,
\n metadata JSONB,
\n billed_at TIMESTAMPTZ,
\n created_at TIMESTAMPTZ DEFAULT NOW()
\n);<\/p>\n
CREATE TABLE subscriptions (
\n org_id UUID REFERENCES organizations(id) PRIMARY KEY,
\n plan TEXT NOT NULL, — ‘research’, ‘enterprise’, ‘lab’
\n env_steps_quota BIGINT, — monthly RL environment steps
\n dataset_gb_quota INTEGER,
\n eval_runs_quota INTEGER,
\n overage_rate_usd NUMERIC(10,4),
\n stripe_subscription_id TEXT
\n);
\n“`<\/p>\n
**Customer onboarding checklist:**
\n– Provision org + API key via internal admin panel
\n– Send Lean environment quickstart (Python SDK + example RL training loop)
\n– Slack connect for enterprise customers
\n– Weekly usage report email<\/p>\n
**Monitoring:** Track `env_step_p99_latency`, `proof_verification_error_rate`, `dataset_download_throughput`. Page on p99 > 10ms for RL endpoints \u00e2\u20ac\u201d your customers are training on this in real time.<\/p>\n
The hardest part of this build is not the code. It’s accumulating enough verified theorems that your dataset is worth paying for, and getting your RL environment trusted enough that a lab plugs it into a live training run. Both of those are slow, trust-based processes. Start building both on day one.<\/p>\n","protected":false},"excerpt":{"rendered":"# How to Build a Formal Verification AI Platform (Cajal Clone) A step-by-step technical guide for building a…\n","protected":false},"author":2,"featured_media":31262,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[306,53,3154,25,182,3326,5070,186],"class_list":{"0":"post-31261","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-anthropic","8":"tag-ai-startups","9":"tag-anthropic","10":"tag-anthropic-claude","11":"tag-artificial-intelligence","12":"tag-claude","13":"tag-investors","14":"tag-startup-funding","15":"tag-tech-news"},"_links":{"self":[{"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/posts\/31261","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/comments?post=31261"}],"version-history":[{"count":0,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/posts\/31261\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/media\/31262"}],"wp:attachment":[{"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/media?parent=31261"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/categories?post=31261"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/tags?post=31261"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}