import Imports.AllImports

-- start_def problem_details
/--
function_signature: "def special_factorial(n: int) -> int"
docstring: |
    The Brazilian factorial is defined as:
    brazilian_factorial(n) = n! * (n-1)! * (n-2)! * ... * 1!
    where n > 0. Please write a function that computes the Brazilian factorial.
test_cases:
  - input: 4
    expected_output: 288
-/
-- end_def problem_details

-- start_def problem_spec
def problem_spec
-- function signature
(impl: Nat → Nat)
-- inputs
(n: Nat) :=
-- spec
let spec (result: Nat) :=
let factorial := Nat.factorial n;
(0 < n → result / factorial = impl (n - 1)) ∧
(n = 0 → result = 1);
-- program termination
∃ result, impl n = result ∧
-- return value satisfies spec
spec result
-- end_def problem_spec

-- start_def generated_spec
def generated_spec
-- function signature
(impl: Nat → Nat)
-- inputs
(n: Int) : Prop :=
-- end_def generated_spec
--start_def generated_spec_body
sorry
--end_def generated_spec_body

-- start_def spec_isomorphism
theorem spec_isomorphism:
∀ impl,
(∀ n, problem_spec impl n) ↔
(∀ n, generated_spec impl n) :=
-- end_def spec_isomorphism
-- start_def spec_isomorphism_proof
sorry
-- end_def spec_isomorphism_proof

-- start_def implementation_signature
def implementation (n: Nat) : Nat :=
-- end_def implementation_signature
-- start_def implementation
sorry
-- end_def implementation

-- Uncomment the following test cases after implementing the function
-- start_def test_cases
-- #test implementation 4 = 288
-- end_def test_cases


-- start_def correctness_definition
theorem correctness
(n: Nat)
: problem_spec implementation n :=
-- end_def correctness_definition
-- start_def correctness_proof
sorry
-- end_def correctness_proof
