import Imports.AllImports

-- start_def problem_details
/--
function_signature: "def find_fermat(n: int) -> int"
docstring: |
    Given an integer n, your task is to find the nth Fermat number.
    The nth Fermat number is defined as follows:
    - F_n = 2^(2^n) + 1
test_cases:
  - input:
      0
    expected_output: 3
  - input:
      1
    expected_output: 5
  - input:
      2
    expected_output: 17
  - input:
      3
    expected_output: 257
  - input:
      4
    expected_output: 65537
  - input:
      5
    expected_output: 4294967297
-/
-- end_def problem_details

-- start_def problem_spec
def problem_spec
-- function signature
(impl: Nat → Nat)
-- inputs
(n: Nat) :=
-- spec
let spec (result: Nat) :=
result = 2 ^ (2 ^ n) + 1
-- program terminates
∃ result, impl n = result ∧
-- return value satisfies spec
spec result
-- end_def problem_spec

-- start_def generated_spec
def generated_spec
-- function signature
(impl: Nat → Nat)
-- inputs
(n: Nat) : Prop :=
-- spec
sorry
-- end_def generated_spec

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

-- start_def implementation_signature
def implementation (n: Nat) : Nat :=
-- end_def implementation_signature
-- start_def implementation
Nat.fermatNumber n
-- end_def implementation

-- Uncomment the following test cases after implementing the function
-- start_def test_cases
#test implementation 0 = 3
#test implementation 1 = 5
#test implementation 2 = 17
#test implementation 3 = 257
#test implementation 4 = 65537
#test implementation 5 = 4294967297
-- end_def test_cases

-- start_def correctness_definition
theorem correctness
(n: Nat)
: problem_spec implementation n
:=
-- end_def correctness_definition
-- start_def correctness_proof
by sorry
-- end_def correctness_proof
