/opt/pydantic-deep-venv/lib/python3.12/site-packages/pydantic_ai/capabilities/builtin_or_local.py:74: UserWarning: WebSearch local fallback requires the `duckduckgo` optional group — `pip install "pydantic-ai-slim[duckduckgo]"`. Without it, WebSearch only works with models that support it natively.
  self.local = self._default_local()
/opt/pydantic-deep-venv/lib/python3.12/site-packages/pydantic_ai/capabilities/builtin_or_local.py:74: UserWarning: WebSearch local fallback requires the `duckduckgo` optional group — `pip install "pydantic-ai-slim[duckduckgo]"`. Without it, WebSearch only works with models that support it natively.
  self.local = self._default_local()
{
  "output": "Proof completed and compiled to `plus_comm.vo`. Here's what was fixed:\n\n- **Base case** (`n = 0`): After `simpl`, the goal is `m = m + 0`. Applied `Nat.add_0_r` to rewrite `m + 0` \u2192 `m`, then `reflexivity`.\n- **Inductive step** (`n = S n'`): After `simpl`, the goal is `S (n' + m) = m + S n'`. Applied `IHn'` to rewrite `n' + m` \u2192 `m + n'`, giving `S (m + n') = m + S n'`. Then `rewrite <- plus_n_Sm` (which states `S (n + m) = n + S m` \u2014 see [cs.ru.nl](http://www.cs.ru.nl/H.Geuvers/onderwijs/provingwithCA/coq_nat.html) and [rocq-prover.org](https://rocq-prover.org/doc/V8.10.1/stdlib/Coq.Arith.Plus.html)) to close the goal with `reflexivity`.",
  "usage": {
    "total_tokens": 171055,
    "request_tokens": 169874,
    "response_tokens": 1181,
    "requests": 9
  }
}
