import Imports.AllImports

-- start_def problem_details
/--
function_signature: "def correct_bracketing(brackets: str) -> Bool"
docstring: |
    brackets is a string of "<" and ">".
    return True if every opening bracket has a corresponding closing bracket, i.e., (each open bracket is properly closed)
test_cases:
  - input: "<"
    expected_output: False
  - input: "<>"
    expected_output: True
  - input: "<<><>>"
    expected_output: True
  - input: "><<>"
    expected_output: False
-/
-- end_def problem_details

-- start_def problem_spec
def problem_spec
-- function signature
(impl: String → Bool)
-- inputs
(brackets: String) :=
-- spec
let spec (result: Bool) :=
  brackets.data.all (fun c => c = '<' ∨ c = '>') →
  (result ↔ balanced_paren_non_computable brackets '<' '>')
-- program terminates
∃ result, impl brackets = result ∧
-- return value satisfies spec
spec result
-- end_def problem_spec

-- start_def generated_spec
def generated_spec
-- function signature
(impl: String → Bool)
-- inputs
(brackets : String) : Prop :=
--end_def generated_spec
--start_def generated_spec_body
sorry
--end_def generated_spec_body


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

-- start_def implementation_signature
def implementation (brackets: String) : 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 "<" = false
-- #test implementation "<>" = true
-- #test implementation "<<><>>" = true
-- #test implementation "><<>" = false
-- end_def test_cases

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