import Imports.AllImports

-- start_def problem_details
/--
function_signature: "def largest_prime_factor(n: Nat) -> Nat"
docstring: |
    Return the largest prime factor of n. Assume n > 1 and is not a prime.
test_cases:
  - input: 13195
    expected_output: 29
  - input: 2048
    expected_output: 2
-/
-- end_def problem_details

-- start_def problem_spec
def problem_spec
-- function signature
(implementation: Nat → Nat)
-- inputs
(n: Nat) :=
-- spec
let spec (result: Nat) :=
  1 < n ∧ ¬ Nat.Prime n →
  (Nat.Prime result ∧ result ∣ n ∧
  ∀ i, i < n ∧ i ∣ n ∧ Nat.Prime i → i ≤ result);
-- program termination
∃ result, implementation n = result ∧
spec result
-- end_def problem_spec

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


-- start_def spec_isomorphism
theorem spec_isomorphism:
∀ impl,
(∀ x, problem_spec impl x) ↔
(∀ x, generated_spec impl x) :=
-- 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 13195 = 29
-- #test implementation 2048 = 2
-- end_def test_cases

-- start_def correctness_definition
theorem correctness
(n: Nat)
: problem_spec implementation n
:=
-- end_def correctness_definition
-- start_def correctness_proof
by
unfold problem_spec
let result := implementation n
use result
simp [result]
sorry
-- end_def correctness_proof
