Problem: ringbuffer_enqueue

Your task is to implement and verify the circular enqueue operation for a Ring Buffer in Lean. The ring buffer is modeled as a sequence with a fixed capacity. Enqueue: Verify that if the buffer is not full, the element is appended. If the buffer is full, the oldest element (head) is discarded, and the new element is appended, maintaining the fixed capacity (First-In, First-Out with Overwrite). In a functional setting, return a new Ring Buffer state.


import Mathlib

structure RingBuffer (T : Type) where
  capacity : Nat
  view : List T

def RingBuffer.is_valid {T} (rb : RingBuffer T) : Prop :=
  rb.capacity > 0 ∧ rb.view.length ≤ rb.capacity

-- Precondition definitions
@[reducible, simp]
def enqueue_precond {T} (rb : RingBuffer T) (v : T) : Prop :=
  -- !benchmark @start precond
  rb.is_valid
  -- !benchmark @end precond

-- !benchmark @start auxcode
-- !benchmark @end auxcode

-- Main function definition
def enqueue {T} (rb : RingBuffer T) (v : T)
    (h_precond : enqueue_precond rb v) : RingBuffer T :=
  -- !benchmark @start code
  sorry
  -- !benchmark @end code

-- Postcondition definitions
@[reducible, simp]
def enqueue_postcond {T} (rb : RingBuffer T) (v : T) (result : RingBuffer T)
    (_ : enqueue_precond rb v) : Prop :=
  -- !benchmark @start postcond
  result.is_valid ∧
  result.capacity = rb.capacity ∧
  (rb.view.length < rb.capacity → result.view = rb.view ++ [v]) ∧
  (rb.view.length = rb.capacity → result.view = rb.view.drop 1 ++ [v])
  -- !benchmark @end postcond

-- !benchmark @start lemma
-- !benchmark @end lemma

-- Proof content
theorem enqueue_postcond_satisfied {T} (rb : RingBuffer T) (v : T)
    (h_precond : enqueue_precond rb v) :
    enqueue_postcond rb v (enqueue rb v h_precond) h_precond := by
  -- !benchmark @start proof
  sorry
  -- !benchmark @end proof