import Imports.AllImports

-- start_def problem_details
/--
function_signature: "def add(x: Int, y: Int) -> Int"
docstring: Add two numbers x and y
test_cases:
  - input: [2, 3]
    expected_output: 5
  - input: [5, 7]
    expected_output: 12
-/
-- end_def problem_details

-- start_def problem_spec
def problem_spec
-- function signature
(impl: Int → Int → Int)
-- inputs
(x y: Int) :=
-- spec
let spec (res: Int) :=
  res - x - y = 0
-- program terminates
∃ result, impl x y = result ∧
-- return value satisfies spec
spec result
-- if result then spec else ¬spec
-- end_def problem_spec

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

-- start_def implementation_signature
def implementation (x y: Int) : Int :=
-- 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 2 3 = 5
-- #test implementation 5 7 = 12
-- end_def test_cases


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