import Imports.AllImports

-- start_def problem_details
/--
function_signature: "def fruit_distribution(string: str, Nat n) -> Nat"
docstring: |
    In this task, you will be given a string that represents a number of apples and oranges
    that are distributed in a basket of fruit this basket contains
    apples, oranges, and mango fruits. Given the string that represents the total number of
    the oranges and apples and an integer that represent the total number of the fruits
    in the basket return the number of the mango fruits in the basket.
test_cases:
  - input: ["5 apples and 6 oranges", 19]
    expected_output: 8
  - input: ["0 apples and 1 oranges", 3]
    expected_output: 2
  - input: ["2 apples and 3 oranges", 100]
    expected_output: 95
  - input: ["100 apples and 1 oranges", 120]
    expected_output: 19
-/
-- end_def problem_details

-- start_def problem_spec
def problem_spec
-- function signature
(implementation: String → Nat → Nat)
-- inputs
(string: String)
(n : Nat) :=
-- spec
let spec (result: Nat) :=
∃ x y : Nat, x + y = n - result
∧ (String.join [x.repr, " apples and ", y.repr, " oranges"] = string)
-- program termination
∃ result, implementation string n = result ∧
spec result
-- end_def problem_spec

-- start_def generated_spec
def generated_spec
-- function signature
(impl: String → Nat → Nat)
-- inputs
(x: String) (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 (string: String) (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 "5 apples and 6 oranges" 19 = 8
-- #test implementation "0 apples and 1 oranges" 3 = 2
-- #test implementation "2 apples and 3 oranges" 100 = 95
-- #test implementation "100 apples and 1 oranges" 120 = 19
-- end_def test_cases

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