import Imports.AllImports

-- start_def problem_details
/--
function_signature: "def circular_shift(x: Int, shift: Int) -> String"
docstring: |
    Circular shift the digits of the integer x, shift the digits right by shift
    and return the result as a string.
    If shift > number of digits, return digits reversed.
test_cases:
  - input: [12, 1]
    expected_output: 21
  - input: [12, 2]
    expected_output: 12
-/
-- end_def problem_details

-- start_def problem_spec
def problem_spec
-- function signature
(implementation: Nat → Nat → String)
-- inputs
(x shift: Nat) :=
let isReverse (s: String) : Bool :=
  let n := s.length;
  ∀ i, i < n / 2 → s.get! ⟨i⟩ = s.get! ⟨n - 1 - i⟩;
-- spec
let spec (result: String) :=
let x_str := Nat.repr x;
result.length = x_str.length ∧
(x_str.length < shift → isReverse x_str) ∧
(shift ≤ x_str.length →
  x_str.take shift = result.drop (x_str.length - shift) ∧
  x_str.drop shift = result.take (x_str.length - shift));
-- program termination
∃ result, implementation x shift = result ∧
spec result
-- end_def problem_spec

-- start_def generated_spec
def generated_spec
-- function signature
(impl: Nat → Nat → String)
-- inputs
(x y: 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 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 shift: Nat) : String :=
-- 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 (12 : Int) (1 : Int) = "21"
-- #test implementation (12 : Int) (2 : Int) = "12"
-- end_def test_cases

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