-----Description-----
This task requires writing a Lean 4 function that calculates the minimum number of right shifts required to sort a given list of distinct positive integers.

A right shift operation on a list nums of length n moves the element at index i to index (i + 1) % n for all indices i. Effectively, the last element moves to the first position, and all other elements shift one position to the right.

The function should return the minimum number of right shifts needed to make the list sorted in ascending order. If the list is already sorted, the function should return 0. If it's impossible to sort the list using only right shifts, the function should return -1.

-----Input-----
The input consists of a single list of integers:
nums: A list (List Int) containing distinct positive integers.

-----Output-----
The output is a single integer (Int):
- If the list can be sorted using right shifts, return the minimum number of shifts required (an integer >= 0).
- If the list cannot be sorted using right shifts, return -1.


Difficulty: advanced

Lean 4 solution:
-- !benchmark @start import type=solution

-- !benchmark @end import

-- !benchmark @start solution_aux

-- !benchmark @end solution_aux

-- !benchmark @start precond_aux

-- !benchmark @end precond_aux
@[reducible, simp]
def minimumRightShifts_precond (nums : List Int) : Prop :=
  -- !benchmark @start precond
  List.Nodup nums
  -- !benchmark @end precond


-- !benchmark @start code_aux

-- !benchmark @end code_aux


def minimumRightShifts (nums : List Int) (h_precond : minimumRightShifts_precond (nums)) : Int :=
  -- !benchmark @start code
  let n := nums.length
  -- base cases: empty or single element list is already sorted
  if n <= 1 then 0 else

  -- local helper function to check if a list is sorted in ascending order
  let rec isSortedAux (l : List Int) : Bool :=
    match l with
    | [] => true       -- empty list is sorted
    | [_] => true      -- single element list is sorted
    | x :: y :: xs => if x <= y then isSortedAux (y :: xs) else false -- check pairwise

  -- check if the input list is already sorted
  if isSortedAux nums then 0 else

  -- local helper function to perform a single right shift
  -- assume the list `l` is non-empty based on the initial n > 1 check
  let rightShiftOnce (l : List Int) : List Int :=
     match l.reverse with
     | [] => [] -- should not happen for n > 1
     | last :: revInit => last :: revInit.reverse -- `last` is the original last element

  -- recursive function to check subsequent shifts
  -- `shifts_count` is the number of shifts already performed to get `current_list`
  -- we are checking if `current_list` is sorted
  let rec checkShifts (shifts_count : Nat) (current_list : List Int) : Int :=
    -- base case: stop recursion if we've checked n-1 shifts (count goes from 1 to n)
    -- the original list (0 shifts) has already been checked
    if shifts_count >= n then -1
    else
      -- check if the current state is sorted
      if isSortedAux current_list then
        (shifts_count : Int) -- found it after 'shifts_count' shifts
      else
        -- recursion: increment shift count, apply next shift
        checkShifts (shifts_count + 1) (rightShiftOnce current_list)
  -- specify the decreasing measure for the termination checker: n - shifts_count
  termination_by n - shifts_count

  -- start the checking process by performing the first shift and checking subsequent states
  -- the initial list (0 shifts) hass already been checked and is not sorted
  checkShifts 1 (rightShiftOnce nums) -- start checking from the state after 1 shift
  -- !benchmark @end code


-- !benchmark @start postcond_aux

-- !benchmark @end postcond_aux


@[reducible, simp]
def minimumRightShifts_postcond (nums : List Int) (result: Int) (h_precond : minimumRightShifts_precond (nums)) : Prop :=
  -- !benchmark @start postcond
  let n := nums.length

  let isSorted (l : List Int) := List.Pairwise (· ≤ ·) l
  let rightShift (k : Nat) (l : List Int) := l.rotateRight k

  -- specification logic based on the result value
  if n <= 1 then result = 0 else -- specification for base cases

  -- case 1: a non-negative result means a solution was found
  (result ≥ 0 ∧
   -- result corresponds to a valid shift count result < n
   result < n ∧
   -- applying result shifts results in a sorted list
   isSorted (rightShift result.toNat nums) ∧
   -- result is the minimum such non-negative shift count
   (List.range result.toNat |>.all (fun j => ¬ isSorted (rightShift j nums)))
  ) ∨

  -- case 2: result is -1 means no solution exists within n shifts
  (result = -1 ∧
   -- for all possible shift counts k from 0 to n-1, the resulting list is not sorted
   (List.range n |>.all (fun k => ¬ isSorted (rightShift k nums)))
  )
  -- !benchmark @end postcond


-- !benchmark @start proof_aux

-- !benchmark @end proof_aux


theorem minimumRightShifts_spec_satisfied (nums: List Int) (h_precond : minimumRightShifts_precond (nums)) :
    minimumRightShifts_postcond (nums) (minimumRightShifts (nums) h_precond) h_precond := by
  -- !benchmark @start proof
  sorry
  -- !benchmark @end proof




Tests:
input: nums: [3, 4, 5, 1, 2]
nums: [1, 3, 5]
nums: [2, 1, 4]
nums: [1]
nums: [2, 1]
nums: [1, 2, 3, 4, 5]
nums: [5, 1, 2, 3, 4]
nums: [1, 5, 2, 3, 4]
expected: 2
0
-1
0
1
0
4
-1
unexpected: -1
4
5
-1
3
0
2
3
-1
1
-1
0
2
-1
1
5
-1
0
5
0
1
5

Inputs rejected by the precondition:
nums: [1, 1]