class MathematicalNativeAGI:
    def __init__(self):
        self.symbolic_reasoner = SymbolicReasoningEngine()
        self.arithmetic_engine = ArithmeticEngine()
        self.lemmas_store = LemmasStore()
        self.proof_system = ProofSystem(self.symbolic_reasoner, self.arithmetic_engine)
        
    def define_lemma(self, lemma_name, expression):
        """Define a new lemma."""
        self.lemmas_store.add_lemma(lemma_name, expression)

    def prove_expression(self, target_expression):
        """Prove the given expression using existing lemmas and rules."""
        return self.proof_system.prove(target_expression)
    
class SymbolicReasoningEngine:
    def __init__(self):
        self.rules_of_inference = {"modus_ponens": lambda p, q: (p and q)}
        
    def infer(self, premises, conclusion):
        """Apply inference rules to the given premises."""
        for rule_name, rule_func in self.rules_of_inference.items():
            if all(rule_func(premise) for premise in premises):
                return conclusion
        return None

class ArithmeticEngine:
    def __init__(self):
        self.arithmetic_ops = {"add": lambda a, b: a + b,
                               "subtract": lambda a, b: a - b,
                               "multiply": lambda a, b: a * b}
        
    def evaluate_expression(self, expr):
        """Evaluate an arithmetic expression."""
        if expr in self.arithmetic_ops:
            return self.arithmetic_ops[expr]
        else:
            raise ValueError("Expression not recognized")

class LemmasStore:
    def __init__(self):
        self.lemmas = {}
        
    def add_lemma(self, lemma_name, expression):
        """Add a new lemma to the store."""
        if lemma_name in self.lemmas:
            print(f"Lemma '{lemma_name}' already exists.")
        else:
            self.lemmas[lemma_name] = expression

class ProofSystem:
    def __init__(self, symbolic_reasoner, arithmetic_engine):
        self.symbolic_reasoner = symbolic_reasoner
        self.arithmetic_engine = arithmetic_engine
        
    def prove(self, target_expression):
        """Prove the given expression using existing lemmas and rules."""
        premises = []
        for lemma_name in self.lemmas_store.lemmas.keys():
            if lemma_name != "target":
                premises.append(lemma_name)
        conclusion = self.symbolic_reasoner.infer(premises, target_expression)
        return conclusion

# Example usage:
math_agi = MathematicalNativeAGI()
math_agi.define_lemma("identity", "x + 0 = x")
result = math_agi.prove_expression("x + 0 = x")
print(result)  # Should print: "x + 0 = x"