import Imports.AllImports

-- start_def problem_details
/--
function_signature: "def iscube(a: int) -> bool"
docstring: |
    Write a function that takes an integer a and returns True if this integer is a cube of some integer number.
    Note: you may assume the input is always valid.
test_cases:
  - input: 1
    expected_output: True
  - input: 2
    expected_output: False
  - input: -1
    expected_output: True
  - input: 64
    expected_output: True
  - input: 0
    expected_output: True
  - input: 180
    expected_output: False
-/
-- end_def problem_details


-- start_def problem_spec
def problem_spec
-- function signature
(implementation: Int → Bool)
-- inputs
(a: Int) :=
-- spec
let spec (result: Bool) :=
  result ↔ exists n: Int, a = n^3
-- program termination
∃ result, implementation a = result ∧
spec result
-- end_def problem_spec

-- start_def generated_spec
def generated_spec
-- function signature
(impl: Int → Bool)
-- inputs
(a: 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,
(∀ a, problem_spec impl a) ↔
(∀ a, generated_spec impl a) :=
-- end_def spec_isomorphism
-- start_def spec_isomorphism_proof
sorry
--end_def spec_isomorphism_proof

-- start_def implementation_signature
def implementation (a: Int) : Bool :=
-- 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 1 = True
-- #test implementation 2 = False
-- #test implementation -1 = True
-- #test implementation 64 = True
-- #test implementation 180 = False
-- #test implementation 1000 = True
-- #test implementation 0 = True
-- #test implementation 1729 = False
-- end_def test_cases

-- start_def correctness_definition
theorem correctness
(a: Int)
: problem_spec implementation a
:=
-- end_def correctness_definition
-- start_def correctness_proof
by
sorry
-- end_def correctness_proof
