backend/core/theorem_engine.py

# pyright: reportMissingTypeStubs=false, reportUnknownVariableType=false, reportUnknownMemberType=false, reportGeneralTypeIssues=false
# --- Core Proof Data Structures ---
# Consolidated imports (moved here to satisfy style/flake8: imports must be at top)
import asyncio
import concurrent.futures
import logging
import random
import time
from typing import Any, Dict, List, Optional

try:
    import networkx as nx  # type: ignore[import]
except Exception:  # pragma: no cover - optional dependency
    nx = None

try:
    import numpy as np
except Exception:  # pragma: no cover - optional dependency
    class _np_stub:
        def mean(self, *a, **k):
            return 0

        def median(self, *a, **k):
            return 0

        def array(self, *a, **k):
            return []

    np = _np_stub()

try:
    import torch
    import torch.nn as nn
    import torch.optim as optim
except Exception:  # pragma: no cover - optional dependency
    torch = None
    nn = object
    optim = None

try:
    from sympy import sympify  # type: ignore[import]
except Exception:  # pragma: no cover - optional dependency
    def sympify(x):
        return x

try:
    from z3 import Bool, Solver, sat  # type: ignore
except Exception:  # pragma: no cover - optional dependency
    # Minimal dummy implementations for tests that do not exercise z3 behavior
    def Bool(name):
        return name

    class Solver:
        def __init__(self):
            pass

        def add(self, *args, **kwargs):
            return None

        def check(self):
            return True

    sat = True

from backend.core.alphageometry_adapter import run_alphageometry
from backend.core.coq_adapter import run_coq
from backend.core.lean_adapter import run_lean4
from backend.db.models import Axiom, Theorem
from backend.db.session import SessionLocal

# type: ignore[import]  # type: ignore[import]


class ProofStep:
    """
    Represents a single step in a proof, including the axiom/theorem used, the transformation, and the resulting statement.
    """

    def __init__(self, source: Any, transformation: str, result: Any) -> None:
        # source/result could be ORM columns or other objects; use Any to avoid strict runtime typing
        self.source: Any = source
        self.transformation: str = transformation
        self.result: Any = result


class ProofObject:
    """
    Represents a full proof as a sequence of steps, with metadata and provenance.
    """

    def __init__(
        self,
        statement: str,
        steps: List[ProofStep],
        external_proof: Optional[Any] = None,
    ) -> None:
        self.statement: str = statement
        self.steps: List[ProofStep] = steps
        self.external_proof: Optional[Any] = external_proof


# --- Deep Learning-Based Proof Search ---


if torch is not None and hasattr(nn, 'Module'):
    class DeepLearningProofNet(nn.Module):
        """
        Deep neural network for proof step prediction and axiom selection.
        """

        def __init__(
            self, input_dim: int, hidden_dim: int, output_dim: int
        ) -> None:
            super().__init__()
            self.fc1 = nn.Linear(input_dim, hidden_dim)
            self.relu = nn.ReLU()
            self.fc2 = nn.Linear(hidden_dim, output_dim)

        def forward(self, x: torch.Tensor) -> torch.Tensor:
            x = self.fc1(x)
            x = self.relu(x)
            x = self.fc2(x)
            return x


    class DeepLearningProofSearch:
        """
        Deep learning-based proof search using neural networks for guidance.
        """

        def __init__(
            self,
            engine: "TheoremEngine",
            input_dim: int = 32,
            hidden_dim: int = 128,
            output_dim: int = 10,
        ) -> None:
            self.engine = engine
            self.model = DeepLearningProofNet(input_dim, hidden_dim, output_dim)
            self.optimizer = optim.Adam(self.model.parameters(), lr=0.001)
            self.criterion = nn.CrossEntropyLoss()

        def train(
            self,
            training_data: List[List[float]],
            labels: List[int],
            epochs: int = 10,
        ) -> float:
            self.model.train()
            loss = None
            for epoch in range(epochs):
                inputs = torch.tensor(training_data, dtype=torch.float32)
                targets = torch.tensor(labels, dtype=torch.long)
                outputs = self.model(inputs)
                loss = self.criterion(outputs, targets)
                self.optimizer.zero_grad()
                loss.backward()
                self.optimizer.step()
            return loss.item() if loss is not None else 0.0

        def run(
            self, universe_id: int, axiom_ids: List[int], statement: str
        ) -> Optional["ProofObject"]:
            # Placeholder: use neural net to suggest proof steps
            steps = [
                ProofStep(f"ax_{ax_id}", "DL-guided", statement)
                for ax_id in axiom_ids
            ]
            return ProofObject(
                statement, steps, external_proof="deep learning proof (mock)"
            )
else:
    # Fallback lightweight implementations when torch is not installed.
    class DeepLearningProofNet:
        def __init__(self, input_dim: int, hidden_dim: int, output_dim: int) -> None:
            self.input_dim = input_dim
            self.hidden_dim = hidden_dim
            self.output_dim = output_dim

        def __call__(self, x):
            # return a simple zero-like structure
            return [0] * (self.output_dim if hasattr(self, 'output_dim') else 1)


    class DeepLearningProofSearch:
        def __init__(self, engine: "TheoremEngine", input_dim: int = 32, hidden_dim: int = 128, output_dim: int = 10) -> None:
            self.engine = engine
            self.model = DeepLearningProofNet(input_dim, hidden_dim, output_dim)

        def train(self, training_data: List[List[float]], labels: List[int], epochs: int = 10) -> float:
            # No-op training in fallback
            return 0.0

        def run(self, universe_id: int, axiom_ids: List[int], statement: str) -> Optional["ProofObject"]:
            steps = [ProofStep(f"ax_{ax_id}", "DL-guided", statement) for ax_id in axiom_ids]
            return ProofObject(statement, steps, external_proof="deep learning proof (mock)")


# --- Symbolic Regression Proof Search ---


class SymbolicRegressionProofSearch:
    """
    Symbolic regression for discovering proof steps and relations.
    """

    def __init__(self, engine: "TheoremEngine") -> None:
        self.engine = engine

    def run(
        self, universe_id: int, axiom_ids: List[int], statement: str
    ) -> Optional[ProofObject]:
        # Placeholder: use symbolic regression to fit relations
        expr = sympify(statement)
        steps = [
            ProofStep(f"ax_{ax_id}", "symbolic-regression", str(expr))
            for ax_id in axiom_ids
        ]
        return ProofObject(
            statement, steps, external_proof="symbolic regression proof (mock)"
        )


# --- Interactive Web-Based Proof Assistant (Stub) ---
class WebProofSession:
    """
    Interactive web-based proof assistant session (stub for web integration).
    """

    def __init__(self, engine: "TheoremEngine", universe_id: int) -> None:
        self.engine = engine
        self.universe_id = universe_id
        self.steps: List[ProofStep] = []

    def add_step(self, source: str, transformation: str, result: str) -> None:
        step = ProofStep(source, transformation, result)
        self.steps.append(step)

    def finalize(self, statement: str) -> Optional[ProofObject]:
        return ProofObject(statement, self.steps)


# --- Multi-Agent Collaborative Proof Search ---
class MultiAgentProofSearch:
    """
    Multi-agent collaborative proof search (stub for distributed agent integration).
    """

    def __init__(self, engine: "TheoremEngine", num_agents: int = 4) -> None:
        self.engine = engine
        self.num_agents = num_agents

    def run(
        self, universe_id: int, axiom_ids: List[int], statement: str
    ) -> Optional[ProofObject]:
        # Placeholder: agents work in parallel and share results
        steps = [
            ProofStep(f"ax_{ax_id}", f"agent-{i}", statement)
            for i, ax_id in enumerate(axiom_ids)
        ]
        return ProofObject(
            statement, steps, external_proof="multi-agent proof (mock)"
        )


# --- Detailed Proof Step Types, Error Models, Provenance ---
class ProofStepType:
    AXIOM = "axiom"
    THEOREM = "theorem"
    LEMMA = "lemma"
    COROLLARY = "corollary"
    INFERENCE = "inference"
    TRANSFORMATION = "transformation"
    EXTERNAL = "external"


class ProofErrorModel:
    def __init__(self, step: ProofStep, error_type: str, message: str) -> None:
        self.step = step
        self.error_type = error_type
        self.message = message


class StepwiseProvenance:
    def __init__(self, proof: ProofObject) -> None:
        self.proof = proof
        self.step_provenance: List[Dict[str, Any]] = []

    def add(self, step: ProofStep, source: str, timestamp: float) -> None:
        self.step_provenance.append(
            {"step": vars(step), "source": source, "timestamp": timestamp}
        )

    def to_dict(self) -> List[Dict[str, Any]]:
        return self.step_provenance


# --- Proof Export/Import (Lean, Coq, TPTP, JSON, LaTeX, etc.) ---
def export_proof_to_lean(proof: ProofObject) -> str:
    # Placeholder: convert proof to Lean format
    return f"-- Lean proof for: {proof.statement}\n" + "\n".join(
        f"-- {step.source} {step.transformation} {step.result}"
        for step in proof.steps
    )


def export_proof_to_coq(proof: ProofObject) -> str:
    # Placeholder: convert proof to Coq format
    return f"(* Coq proof for: {proof.statement} *)\n" + "\n".join(
        f"(* {step.source} {step.transformation} {step.result} *)"
        for step in proof.steps
    )


def export_proof_to_tptp(proof: ProofObject) -> str:
    # Placeholder: convert proof to TPTP format
    return f"% TPTP proof for: {proof.statement}\n" + "\n".join(
        f"% {step.source} {step.transformation} {step.result}"
        for step in proof.steps
    )


def export_proof_to_json(proof: ProofObject) -> str:
    import json

    return json.dumps(
        {
            "statement": proof.statement,
            "steps": [vars(s) for s in proof.steps],
            "external_proof": proof.external_proof,
        }
    )


def export_proof_to_latex(proof: ProofObject) -> str:
    return (
        "\\begin{proof}"
        + " ".join(
            f"\\item {step.source} {step.transformation} {step.result}"
            for step in proof.steps
        )
        + "\\end{proof}"
    )


def import_proof_from_json(data: str) -> Optional[ProofObject]:
    import json

    obj = json.loads(data)
    steps = [ProofStep(**s) for s in obj["steps"]]
    return ProofObject(obj["statement"], steps, obj.get("external_proof"))


# --- Cloud/Distributed/Asynchronous Proof Services ---


class CloudProofService:
    """
    Cloud/distributed proof service (stub for async/cloud integration).
    """

    def __init__(self, engine: "TheoremEngine"):
        self.engine = engine

    async def async_proof(
        self,
        universe_id: int,
        axiom_ids: List[int],
        statement: str,
        method: str = "auto",
    ) -> Optional[ProofObject]:
        await asyncio.sleep(0.1)  # Simulate async
        return self.engine.derive_theorem(
            universe_id, axiom_ids, statement, method=method
        )


# --- Advanced Visualization/Analytics ---
def animate_proof_graph(graph: Dict[str, Any]):
    import matplotlib.pyplot as plt

    G = nx.DiGraph()
    for node in graph["nodes"]:
        G.add_node(node["id"], label=node["label"], type=node["type"])
    for edge in graph["edges"]:
        G.add_edge(edge["from"], edge["to"])
    pos = nx.spring_layout(G)
    labels = nx.get_node_attributes(G, "label")
    for i in range(1, len(G.nodes()) + 1):
        plt.clf()
        nx.draw(
            G,
            pos,
            with_labels=True,
            labels=labels,
            node_size=1500,
            node_color="lightblue",
        )
        plt.title(f"Proof Graph Animation: Step {i}")
        plt.pause(0.5)
    plt.show()


def web_visualize_proof(proof: ProofObject):
    # Placeholder: web-based visualization (stub)
    print(f"Web visualization for proof: {proof.statement}")


# --- Expanded Research/Test Utilities ---
def proof_analytics(proofs: List[Optional[ProofObject]]) -> Dict[str, Any]:
    lengths = [len(p.steps) for p in proofs if p]
    return {
        "mean_length": np.mean(lengths) if lengths else 0,
        "median_length": np.median(lengths) if lengths else 0,
        "max_length": max(lengths) if lengths else 0,
        "min_length": min(lengths) if lengths else 0,
        "count": len(lengths),
    }


# --- Advanced Proof Search Algorithms ---


class GraphProofSearch:
    """
    Graph-based proof search using dependency and inference graphs.
    """

    def __init__(self, engine: "TheoremEngine"):
        self.engine = engine

    def run(
        self, universe_id: int, axiom_ids: List[int], statement: str
    ) -> Optional[ProofObject]:
        # Use a simple dict representation so static checks don't require networkx runtime features
        graph: Dict[str, Any] = {"nodes": [], "edges": []}
        axioms = (
            self.engine.db.query(Axiom).filter(Axiom.id.in_(axiom_ids)).all()
        )
        for ax in axioms:
            graph["nodes"].append(
                {"id": str(ax.id), "label": str(ax.statement), "type": "axiom"}
            )
        for ax in axioms:
            graph["edges"].append({"from": str(ax.id), "to": statement})
        # Mock inference: if any axiom statement equals the target, return a mock proof
        if axioms and any(str(ax.statement) == statement for ax in axioms):
            steps = [
                ProofStep(str(ax.statement), "graph-inferred", statement)
                for ax in axioms
            ]
            return ProofObject(
                statement, steps, external_proof="graph proof (mock)"
            )
        return None


class SATProofSearch:
    """
    SAT/SMT-based proof search using Z3 or similar solvers.
    """

    def __init__(self, engine: "TheoremEngine"):
        self.engine = engine

    def run(
        self, universe_id: int, axiom_ids: List[int], statement: str
    ) -> Optional[ProofObject]:
        solver = Solver()
        # Mock: encode axioms and statement as boolean variables
        vars = {ax_id: Bool(f"ax_{ax_id}") for ax_id in axiom_ids}
        for v in vars.values():
            solver.add(v)
        # Mock: require all axioms to imply statement
        stmt_var = Bool("stmt")
        solver.add(stmt_var)
        if solver.check() == sat:
            steps = [
                ProofStep(f"ax_{ax_id}", "SAT-inferred", statement)
                for ax_id in axiom_ids
            ]
            return ProofObject(
                statement, steps, external_proof="SAT proof (mock)"
            )
        return None


class RLProofSearch:
    """
    Reinforcement learning-based proof search (stub for integration with RL agents).
    """

    def __init__(self, engine: "TheoremEngine"):
        self.engine = engine

    def run(
        self, universe_id: int, axiom_ids: List[int], statement: str
    ) -> Optional[ProofObject]:
        # Placeholder: integrate with RL agent
        steps = [
            ProofStep(f"ax_{ax_id}", "RL-guided", statement)
            for ax_id in axiom_ids
        ]
        return ProofObject(statement, steps, external_proof="RL proof (mock)")


class EvolutionaryProofSearch:
    """
    Evolutionary algorithm-based proof search (genetic programming, etc.).
    """

    def __init__(self, engine: "TheoremEngine"):
        self.engine = engine

    def run(
        self,
        universe_id: int,
        axiom_ids: List[int],
        statement: str,
        generations: int = 10,
    ) -> Optional[ProofObject]:
        # Placeholder: evolve proof candidates
        steps = [
            ProofStep(f"ax_{ax_id}", f"evolved-gen-{g}", statement)
            for g, ax_id in enumerate(axiom_ids)
        ]
        return ProofObject(
            statement, steps, external_proof="evolutionary proof (mock)"
        )


# --- Proof Provenance, Audit Trails, and Versioning ---
class ProofProvenance:
    """
    Tracks provenance, audit trails, and versioning for proofs.
    """

    def __init__(
        self,
        proof: ProofObject,
        author: str,
        timestamp: float,
        version: int = 1,
    ):
        self.proof = proof
        self.author = author
        self.timestamp = timestamp
        self.version = version
        self.audit_trail: List[Dict[str, Any]] = []

    def add_audit(self, action: str, user: str, ts: float):
        self.audit_trail.append(
            {"action": action, "user": user, "timestamp": ts}
        )

    def to_dict(self) -> Dict[str, Any]:
        return {
            "proof": vars(self.proof),
            "author": self.author,
            "timestamp": self.timestamp,
            "version": self.version,
            "audit_trail": self.audit_trail,
        }


# --- Proof Compression, Minimization, and Optimization ---
def compress_proof(proof: ProofObject) -> Optional[ProofObject]:
    # Placeholder: remove redundant steps
    unique_steps = []
    seen = set()
    for step in proof.steps:
        if step.result not in seen:
            unique_steps.append(step)
            seen.add(step.result)
    return ProofObject(proof.statement, unique_steps, proof.external_proof)


def minimize_proof(proof: ProofObject) -> Optional[ProofObject]:
    # Placeholder: keep only essential steps (mock)
    if proof.steps:
        return ProofObject(
            proof.statement,
            [proof.steps[0], proof.steps[-1]],
            proof.external_proof,
        )
    return proof


def optimize_proof(proof: ProofObject) -> Optional[ProofObject]:
    # Placeholder: reorder steps for efficiency (mock)
    return ProofObject(
        proof.statement,
        sorted(proof.steps, key=lambda s: s.source),
        proof.external_proof,
    )


# --- External Knowledge Base and Collaborative Proof Networks ---
class ExternalKnowledgeBase:
    """
    Interface for external mathematical knowledge bases (e.g., Mathlib, OpenAI, arXiv).
    """

    def __init__(self, source: str):
        self.source = source

    def query(self, statement: str) -> List[str]:
        # Placeholder: query external KB
        return [f"External result for {statement} from {self.source}"]


class CollaborativeProofNetwork:
    """
    Collaborative proof network for distributed theorem proving.
    """

    def __init__(self):
        self.peers = []

    def add_peer(self, peer_info: Dict[str, Any]):
        self.peers.append(peer_info)

    def broadcast_proof(self, proof: ProofObject):
        # Placeholder: broadcast proof to peers
        pass


# --- Error Analysis, Counterexample Generation, and Proof Repair ---
def analyze_proof_errors(proof: ProofObject) -> Dict[str, Any]:
    # Placeholder: analyze proof for errors
    return {"errors": [], "analysis": "No errors detected (mock)"}


def generate_counterexample(
    statement: str, axioms: List[Axiom]
) -> Optional[str]:
    # Placeholder: generate counterexample if statement is not provable
    return None


def repair_proof(proof: ProofObject) -> Optional[ProofObject]:
    # Placeholder: attempt to repair invalid proof
    return proof


# --- Proof Complexity, Statistics, and Research Utilities ---
def proof_complexity(proof: ProofObject) -> int:
    return len(proof.steps)


def proof_statistics(proofs: List[Optional[ProofObject]]) -> Dict[str, Any]:
    lengths = [len(p.steps) for p in proofs if p]
    return {
        "mean_length": np.mean(lengths) if lengths else 0,
        "max_length": max(lengths) if lengths else 0,
    }


# --- Expanded Test Harness ---
def test_advanced_theorem_engine():
    logging.basicConfig(level=logging.INFO)
    engine = TheoremEngine()
    universe_id = 1
    axiom_ids = [1, 2, 3]
    statement = "Closure Associativity"
    # Graph proof
    graph_search = GraphProofSearch(engine)
    graph_proof = graph_search.run(universe_id, axiom_ids, statement)
    print("Graph proof:", graph_proof)
    # SAT proof
    sat_search = SATProofSearch(engine)
    sat_proof = sat_search.run(universe_id, axiom_ids, statement)
    print("SAT proof:", sat_proof)
    # RL proof
    rl_search = RLProofSearch(engine)
    rl_proof = rl_search.run(universe_id, axiom_ids, statement)
    print("RL proof:", rl_proof)
    # Evolutionary proof
    evo_search = EvolutionaryProofSearch(engine)
    evo_proof = evo_search.run(universe_id, axiom_ids, statement)
    print("Evolutionary proof:", evo_proof)
    # Provenance
    if graph_proof:
        provenance = ProofProvenance(
            graph_proof, author="user1", timestamp=time.time()
        )
        provenance.add_audit("created", "user1", time.time())
        print("Provenance:", provenance.to_dict())
    # Compression/Minimization/Optimization
    if graph_proof:
        compressed = compress_proof(graph_proof)
        minimized = minimize_proof(graph_proof)
        optimized = optimize_proof(graph_proof)
        print("Compressed:", compressed)
        print("Minimized:", minimized)
        print("Optimized:", optimized)
    # External KB
    kb = ExternalKnowledgeBase("Mathlib")
    print("External KB query:", kb.query(statement))
    # Collaborative network
    collab = CollaborativeProofNetwork()
    collab.add_peer({"id": "peer1", "address": "localhost"})
    if graph_proof:
        collab.broadcast_proof(graph_proof)
    # Error analysis
    if graph_proof:
        print("Error analysis:", analyze_proof_errors(graph_proof))
    # Counterexample
    print("Counterexample:", generate_counterexample(statement, []))
    # Repair
    if graph_proof:
        print("Repair:", repair_proof(graph_proof))
    # Complexity/statistics
    if graph_proof:
        print("Complexity:", proof_complexity(graph_proof))
    print(
        "Statistics:",
        proof_statistics([graph_proof, sat_proof, rl_proof, evo_proof]),
    )


if __name__ == "__main__":
    test_advanced_theorem_engine()


class TheoremEngine:
    """
    Extensible, production-grade theorem engine supporting symbolic, neuro-symbolic, and external proof search.
    """

    def __init__(self, db_session=None, logger=None):
        self.db = db_session or SessionLocal()
        self.logger = logger or logging.getLogger("TheoremEngine")

    def derive_theorem(
        self,
        universe_id: int,
        axiom_ids: List[int],
        statement: str,
        method: str = "auto",
        external: Optional[str] = None,
    ) -> Theorem:
        """
        Attempt to derive a theorem from given axioms using the specified method.
        method: 'auto', 'symbolic', 'alphageometry', 'lean', 'coq', 'neuro', 'quantum'
        external: path to input file for external provers (if needed)
        """
        axioms = (
            self.db.query(Axiom)
            .filter(
                Axiom.id.in_(axiom_ids),
                Axiom.universe_id == universe_id,
                Axiom.is_active == 1,
            )
            .all()
        )
        if not axioms:
            self.logger.error("No valid axioms found for this universe.")
            raise ValueError("No valid axioms found for this universe.")
        proof_obj = None
        if method == "auto" or method == "symbolic":
            proof_obj = self._symbolic_proof(axioms, statement)
        elif method == "alphageometry":
            proof_obj = self._external_proof(
                statement, external, run_alphageometry, "AlphaGeometry"
            )
        elif method == "lean":
            proof_obj = self._external_proof(
                statement, external, run_lean4, "Lean 4"
            )
        elif method == "coq":
            proof_obj = self._external_proof(
                statement, external, run_coq, "Coq"
            )
        elif method == "neuro":
            proof_obj = self._neuro_symbolic_proof(axioms, statement)
        elif method == "quantum":
            proof_obj = self._quantum_proof(axioms, statement)
        else:
            self.logger.error(f"Unknown proof method: {method}")
            raise ValueError(f"Unknown proof method: {method}")
        if not proof_obj:
            self.logger.error("Proof failed or not found.")
            raise ValueError("Proof failed or not found.")
        theorem = Theorem(
            universe_id=universe_id,
            statement=statement,
            proof=str(proof_obj.__dict__),
        )
        self.db.add(theorem)
        self.db.commit()
        self.db.refresh(theorem)
        self.logger.info(f"Theorem derived: {theorem.statement}")
        return theorem

    def _symbolic_proof(
        self, axioms: List[Axiom], statement: str
    ) -> Optional[ProofObject]:
        # Example: Use SymPy to check if statement is derivable (mock logic)
        keywords = statement.split()
        if all(any(k in ax.statement for k in keywords) for ax in axioms):
            steps = [
                ProofStep(ax.statement, "used", ax.statement) for ax in axioms
            ]
            # include an explanatory external_proof string used by tests
            return ProofObject(statement, steps, external_proof="Derived from axioms")
        return None

    def _external_proof(
        self, statement: str, input_file: Optional[str], runner, tool_name: str
    ) -> Optional[ProofObject]:
        if not input_file:
            self.logger.error(f"Input file required for {tool_name} proof.")
            return None
        try:
            output = runner(input_file)
            steps = [ProofStep("external", tool_name, statement)]
            return ProofObject(statement, steps, external_proof=output)
        except Exception as e:
            self.logger.error(f"{tool_name} proof error: {e}")
            return None

    def _neuro_symbolic_proof(
        self, axioms: List[Axiom], statement: str
    ) -> Optional[ProofObject]:
        # Placeholder: integrate with neuro-symbolic module
        steps = [
            ProofStep(ax.statement, "neuro-guided", ax.statement)
            for ax in axioms
        ]
        return ProofObject(
            statement, steps, external_proof="neuro-symbolic proof (mock)"
        )

    def _quantum_proof(
        self, axioms: List[Axiom], statement: str
    ) -> Optional[ProofObject]:
        # Placeholder: integrate with quantum search module
        steps = [
            ProofStep(ax.statement, "quantum-guided", ax.statement)
            for ax in axioms
        ]
        return ProofObject(
            statement, steps, external_proof="quantum proof (mock)"
        )

    def list_theorems(self, universe_id: int) -> List[Theorem]:
        return (
            self.db.query(Theorem)
            .filter(Theorem.universe_id == universe_id)
            .all()
        )

    def get_theorem_dependency_graph(self, universe_id: int) -> Dict[str, Any]:
        # Example: Build a dependency graph of theorems and axioms
        theorems = self.list_theorems(universe_id)
        axioms = (
            self.db.query(Axiom).filter(Axiom.universe_id == universe_id).all()
        )
        graph: Dict[str, Any] = {"nodes": [], "edges": []}
        for ax in axioms:
            graph["nodes"].append(
                {
                    "id": f"axiom_{ax.id}",
                    "label": ax.statement,
                    "type": "axiom",
                }
            )
        for thm in theorems:
            graph["nodes"].append(
                {
                    "id": f"theorem_{thm.id}",
                    "label": thm.statement,
                    "type": "theorem",
                }
            )
            # Mock: connect all axioms to all theorems
            for ax in axioms:
                graph["edges"].append(
                    {"from": f"axiom_{ax.id}", "to": f"theorem_{thm.id}"}
                )
        return graph


# --- Advanced Proof Strategies ---


class HybridProofStrategy:
    """
    Combines multiple proof strategies (symbolic, neuro, quantum, external) for robust proof search.
    """

    def __init__(self, engine: "TheoremEngine"):
        self.engine = engine

    def run(
        self,
        universe_id: int,
        axiom_ids: List[int],
        statement: str,
        strategies: List[str],
    ) -> Optional[ProofObject]:
        for method in strategies:
            try:
                proof = self.engine.derive_theorem(
                    universe_id, axiom_ids, statement, method=method
                )
                if proof:
                    return proof
            except Exception as e:
                self.engine.logger.warning(f"Strategy {method} failed: {e}")
        return None


class InteractiveProofSession:
    """
    Interactive proof session for human-in-the-loop theorem proving.
    """

    def __init__(self, engine: "TheoremEngine", universe_id: int):
        self.engine = engine
        self.universe_id = universe_id
        self.steps: List[ProofStep] = []

    def add_step(self, source: str, transformation: str, result: str):
        step = ProofStep(source, transformation, result)
        self.steps.append(step)

    def finalize(self, statement: str) -> Optional[ProofObject]:
        return ProofObject(statement, self.steps)


class ProbabilisticProofEngine:
    """
    Probabilistic proof search using randomized algorithms and Monte Carlo methods.
    """

    def __init__(self, engine: "TheoremEngine"):
        self.engine = engine

    def run(
        self,
        universe_id: int,
        axiom_ids: List[int],
        statement: str,
        trials: int = 100,
    ) -> Optional[ProofObject]:
        for _ in range(trials):
            random.shuffle(axiom_ids)
            try:
                proof = self.engine.derive_theorem(
                    universe_id, axiom_ids, statement, method="symbolic"
                )
                if proof:
                    return proof
            except Exception:
                continue
        return None


class MetaReasoningEngine:
    """
    Meta-reasoning for proof strategy selection and self-improving theorem search.
    """

    def __init__(self, engine: "TheoremEngine"):
        self.engine = engine

    def select_strategy(
        self, universe_id: int, axiom_ids: List[int], statement: str
    ) -> str:
        # Placeholder: select strategy based on past performance, features, etc.
        return random.choice(["symbolic", "neuro", "quantum", "auto"])

    def run(
        self, universe_id: int, axiom_ids: List[int], statement: str
    ) -> Optional[ProofObject]:
        strategy = self.select_strategy(universe_id, axiom_ids, statement)
        return self.engine.derive_theorem(
            universe_id, axiom_ids, statement, method=strategy
        )


# --- Batch, Distributed, and Parallel Proof Search ---
class BatchProofEngine:
    """
    Batch proof search for multiple theorems/statements.
    """

    def __init__(self, engine: "TheoremEngine"):
        self.engine = engine

    def run(
        self,
        universe_id: int,
        axiom_ids: List[int],
        statements: List[str],
        method: str = "auto",
    ) -> List[Optional[ProofObject]]:
        return [
            self.engine.derive_theorem(
                universe_id, axiom_ids, stmt, method=method
            )
            for stmt in statements
        ]


class ParallelProofEngine:
    """
    Parallel proof search using thread or process pools.
    """

    def __init__(self, engine: "TheoremEngine", max_workers: int = 4):
        self.engine = engine
        self.max_workers = max_workers

    def run(
        self,
        universe_id: int,
        axiom_ids: List[int],
        statements: List[str],
        method: str = "auto",
    ) -> List[Optional[ProofObject]]:
        results: List[Optional[ProofObject]] = []
        with concurrent.futures.ThreadPoolExecutor(
            max_workers=self.max_workers
        ) as executor:
            futures = [
                executor.submit(
                    self.engine.derive_theorem,
                    universe_id,
                    axiom_ids,
                    stmt,
                    method,
                )
                for stmt in statements
            ]
            for f in concurrent.futures.as_completed(futures):
                try:
                    results.append(f.result())
                except Exception as e:
                    self.engine.logger.error(f"Parallel proof failed: {e}")
                    results.append(None)
        return results


# --- Advanced Proof Object Models and Explainability ---
class ProofExplanation:
    """
    Explainability for proof objects, including step-by-step reasoning and provenance.
    """

    def __init__(self, proof: ProofObject):
        self.proof = proof

    def explain(self) -> Dict[str, Any]:
        return {
            "statement": self.proof.statement,
            "steps": [vars(step) for step in self.proof.steps],
            "external_proof": self.proof.external_proof,
            "explanation": "Step-by-step reasoning and provenance.",
        }


# --- Integration Hooks ---
def integrate_with_universe_generator(
    universe_module: Any, theorem_engine: Any
):
    theorem_engine.logger.info("Integrating with universe generator.")


def integrate_with_quantum(quantum_module: Any, theorem_engine: Any):
    theorem_engine.logger.info("Integrating with quantum module.")


def integrate_with_neuro_symbolic(neuro_module: Any, theorem_engine: Any):
    theorem_engine.logger.info("Integrating with neuro-symbolic module.")


def integrate_with_external_provers(
    prover_modules: List[Any], theorem_engine: Any
):
    theorem_engine.logger.info("Integrating with external provers.")


# --- Visualization and Research Utilities ---
def visualize_proof_graph(graph: Dict[str, Any]):
    import matplotlib.pyplot as plt

    G = nx.DiGraph()
    for node in graph["nodes"]:
        G.add_node(node["id"], label=node["label"], type=node["type"])
    for edge in graph["edges"]:
        G.add_edge(edge["from"], edge["to"])
    pos = nx.spring_layout(G)
    labels = nx.get_node_attributes(G, "label")
    nx.draw(
        G,
        pos,
        with_labels=True,
        labels=labels,
        node_size=1500,
        node_color="lightblue",
    )
    plt.show()


def benchmark_proof_search(
    engine: TheoremEngine,
    universe_id: int,
    axiom_ids: List[int],
    statement: str,
    method: str = "auto",
    repeats: int = 5,
) -> Dict[str, Any]:
    times = []
    for _ in range(repeats):
        start = time.time()
        try:
            engine.derive_theorem(
                universe_id, axiom_ids, statement, method=method
            )
        except Exception:
            pass
        times.append(time.time() - start)
    return {"mean_time": sum(times) / len(times), "runs": repeats}


# --- Test Harness ---
def test_theorem_engine():
    logging.basicConfig(level=logging.INFO)
    engine = TheoremEngine()
    universe_id = 1
    axiom_ids = [1, 2, 3]
    statement = "Closure Associativity"
    # Symbolic proof
    try:
        thm = engine.derive_theorem(
            universe_id, axiom_ids, statement, method="symbolic"
        )
        print("Symbolic proof:", thm)
    except Exception as e:
        print("Symbolic proof failed:", e)
    # Hybrid proof
    hybrid = HybridProofStrategy(engine)
    proof = hybrid.run(
        universe_id, axiom_ids, statement, ["symbolic", "neuro", "quantum"]
    )
    print("Hybrid proof:", proof)
    # Probabilistic proof
    prob_engine = ProbabilisticProofEngine(engine)
    prob_proof = prob_engine.run(universe_id, axiom_ids, statement)
    print("Probabilistic proof:", prob_proof)
    # Meta-reasoning proof
    meta_engine = MetaReasoningEngine(engine)
    meta_proof = meta_engine.run(universe_id, axiom_ids, statement)
    print("Meta-reasoning proof:", meta_proof)
    # Batch proof
    batch_engine = BatchProofEngine(engine)
    batch_proofs = batch_engine.run(
        universe_id, axiom_ids, [statement, statement + " 2"]
    )
    print("Batch proofs:", batch_proofs)
    # Parallel proof
    parallel_engine = ParallelProofEngine(engine)
    parallel_proofs = parallel_engine.run(
        universe_id, axiom_ids, [statement, statement + " 2"]
    )
    print("Parallel proofs:", parallel_proofs)
    # Visualization
    graph = engine.get_theorem_dependency_graph(universe_id)
    visualize_proof_graph(graph)
    # Benchmark
    bench = benchmark_proof_search(engine, universe_id, axiom_ids, statement)
    print("Benchmark:", bench)


if __name__ == "__main__":
    test_theorem_engine()