import Imports.AllImports

-- start_def problem_details
/--
function_signature: "def fibfib(n: int)"
docstring: |
    The FibFib number sequence is a sequence similar to the Fibbonacci sequnece that's defined as follows:
    fibfib(0) == 0
    fibfib(1) == 0
    fibfib(2) == 1
    fibfib(n) == fibfib(n-1) + fibfib(n-2) + fibfib(n-3).
    Please write a function to efficiently compute the n-th element of the fibfib number sequence.
Note(Meghana): While the specification asks for an efficient computation of fibfib, we cannot enforce this constraint currently.
test_cases:
  - input: 1
    output: 0
  - input: 5
    output: 4
  - input: 8
    output: 24
-/
-- end_def problem_details

-- start_def problem_spec
def problem_spec
-- function signature
(implementation: Nat → Nat)
-- inputs
(n: Nat) :=
-- spec
let spec (result: Nat) :=
fibonacci_non_computable_3 n result
-- program termination
∃ result, implementation n = result ∧
spec result
-- end_def problem_spec

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

-- start_def implementation_signature
def implementation (n: Nat) : Nat :=
-- 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 1 = 0
-- #test implementation 5 = 4
-- #test implementation 8 = 24
-- 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
unfold problem_spec
let result := implementation n
use result
simp [result]
repeat sorry
-- end_def correctness_proof
