import Imports.AllImports

-- start_def problem_details
/--
function_signature: "def sum_to_n(n: Nat) -> Nat"
docstring: |
    sum_to_n is a function that sums numbers from 1 to n.
test_cases:
  - input: 30
    expected_output: 465
  - input: 100
    expected_output: 4950
-/
-- end_def problem_details

-- start_def problem_spec
def problem_spec
-- function signature
(implementation: Nat → Nat)
-- inputs
(n : Nat) :=
-- spec
let spec (result: Nat) :=
  0 < n →
  (result = 1 ↔ n = 1) ∧
  (∀ i, implementation (i + 1) - (implementation i) = i + 1)
-- 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
(n : 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,
(∀ n, problem_spec impl n) ↔
(∀ n, generated_spec impl n) :=
-- 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 30 = 465
-- #test implementation 100 = 5050
-- #test implementation 5 = 15
-- #test implementation 10 = 55
-- #test implementation 1 = 1
-- 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]
sorry
-- end_def correctness_proof
