{"steps":[{"command":{"type":"init_state","src_code":"\"\"\"Numerical derivative calculations - Level 4\"\"\"\n\nfrom core.arithmetic import add, divide, multiply, subtract\n\n\ndef numerical_derivative(func, x: float, h: float = 1e-8) -> float:\n    \"\"\"Calculate numerical derivative using central difference.\"\"\"\n    x_plus_h = add(x, h)\n    x_minus_h = subtract(x, h)\n\n    f_plus = func(x_plus_h)\n    f_minus = func(x_minus_h)\n\n    numerator = subtract(f_plus, f_minus)\n    denominator = multiply(2.0, h)\n\n    return divide(numerator, denominator)\n\n\ndef second_derivative(func, x: float, h: float = 1e-6) -> float:\n    \"\"\"Calculate second derivative.\"\"\"\n    x_plus_h = add(x, h)\n    x_minus_h = subtract(x, h)\n\n    f_center = func(x)\n    f_plus = func(x_plus_h)\n    f_minus = func(x_minus_h)\n\n    # f''(x) ≈ (f(x+h) - 2f(x) + f(x-h)) / h²\n    numerator = subtract(add(f_plus, f_minus), multiply(2.0, f_center))\n    denominator = multiply(h, h)\n\n    return divide(numerator, denominator)\n","src_lang":"Python"},"message":null,"task":null},{"command":{"type":"inject_formalization_context","context":"Do not use `[@@measure ...]` annotations on nested functions (closures). If a function needs a `[@@measure ...]` annotation, structure the code so that the nested function is moved to the top-level.\n\nWhen writing a recursive function where the recursion is controlled by an integer argument, put that integer argument as the first argument. Also try to arrange for the integer argument to count down to zero."},"message":null,"task":null},{"command":{"type":"edit_state_element","update":{"dependency":[{"src_module":{"name":"core/arithmetic.py","relative_path":"core/arithmetic.py","content":"\"\"\"Basic arithmetic operations - Level 1 (no dependencies)\"\"\"\n\n\ndef add(a: float, b: float) -> float:\n    \"\"\"Add two numbers.\"\"\"\n    return a + b\n\n\ndef subtract(a: float, b: float) -> float:\n    \"\"\"Subtract b from a.\"\"\"\n    return a - b\n\n\ndef multiply(a: float, b: float) -> float:\n    \"\"\"Multiply two numbers.\"\"\"\n    return a * b\n\n\ndef divide(a: float, b: float) -> float:\n    \"\"\"Divide a by b.\"\"\"\n    if b == 0:\n        raise ValueError(\"Division by zero\")\n    return a / b\n\n\ndef power(base: float, exponent: float) -> float:\n    \"\"\"Raise base to the power of exponent.\"\"\"\n    return base**exponent\n","src_lang":"Python"},"iml_module":{"name":"Core_arithmetic","relative_path":"core/arithmetic.py","content":"(* Basic arithmetic operations - Level 1 (no dependencies) *)\n\nlet add (a : real) (b : real) : real =\n  (* Add two numbers. *)\n  a +. b\n\nlet subtract (a : real) (b : real) : real =\n  (* Subtract b from a. *)\n  a -. b\n\nlet multiply (a : real) (b : real) : real =\n  (* Multiply two numbers. *)\n  a *. b\n\nlet divide (a : real) (b : real) : real option =\n  (* Divide a by b. *)\n  if b = 0.0 then\n    None\n  else\n    Some (a /. b)\n\nlet power : real -> real -> real = () [@@opaque]\n(* Raise base to the power of exponent. *)","src_lang":"IML"}}]}},"message":null,"task":null},{"command":{"type":"agent_formalizer","no_check_formalization_hitl":false,"no_refactor":false,"no_gen_model_hitl":true,"max_tries_wo_hitl":3,"max_tries":3},"message":null,"task":null},{"command":{"type":"gen_vgs","description":null},"message":null,"task":null},{"command":{"type":"gen_region_decomps","function_name":null},"message":null,"task":null}],"step_i":null,"info":{}}