import Imports.AllImports

-- start_def problem_details
/--
function_signature: "def encode_cyclic(s: str) -> str"
docstring: |
  Returns an encoded string by cycling each group of three consecutive characters.
  Specifically, each group of exactly three characters 'abc' is transformed to 'bca'.
  Groups of fewer than three characters at the end of the string remain unchanged.
test_cases:
  - input: "abcdef"
    expected_output: "bcaefd"
  - input: "abcde"
    expected_output: "bcade"
  - input: "ab"
    expected_output: "ab"
-/
-- end_def problem_details

-- start_def problem_spec
def problem_spec
(impl: String → String)
(s: String) :=
let n := s.length;
let extract (chars: List Char) (start_index: ℕ) (end_index: ℕ) :=
  (chars.drop start_index).take (end_index - start_index + 1);
let spec (result: String) :=
  let encoded_chars := result.toList;
  let original_chars := s.toList;
  encoded_chars.length = n ∧
  (∀ i : ℕ, i * 3 + 3 ≤ n →
    extract encoded_chars (i * 3) (i * 3 + 2) =
      [original_chars.get! (i * 3 + 1), original_chars.get! (i * 3 + 2), original_chars.get! (i * 3)]) ∧
  (n % 3 ≠ 0 → extract encoded_chars (n - n % 3) (n - 1) =
    extract original_chars (n - n % 3) (n - 1));
-- program termination
∃ result,
  impl s = result ∧
  spec result
-- end_def problem_spec

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

-- start_def implementation_signature
def implementation (s: String) : String :=
-- end_def implementation_signature
-- start_def implementation
sorry
-- end_def implementation

-- start_def test_cases
-- #test implementation "abcdef" = "bcaefd"
-- #test implementation "abcde" = "bcade"
-- #test implementation "ab" = "ab"
-- end_def test_cases

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