Problem: ringbuffer_dequeue

Your task is to implement and verify the dequeue operation for a Ring Buffer data structure in Lean. Because the ring buffer uses a fixed-size underlying array, the "front" of the queue is tracked by a head pointer. Verification: Verify that the operation returns the element currently at the "front" of the logical sequence (index 0). Structural Update: You must prove that after the element is returned, the new model view is shortened by one and contains the remaining elements in their original relative order (tail of the view). Capacity remains unchanged.


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 dequeue_precond {T} (rb : RingBuffer T) : Prop :=
  -- !benchmark @start precond
  rb.is_valid ∧ rb.view.length > 0
  -- !benchmark @end precond

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

-- Main function definition
-- Returns (value, new_ringbuffer)
def dequeue {T} (rb : RingBuffer T)
    (h_precond : dequeue_precond rb) : T × RingBuffer T :=
  -- !benchmark @start code
  sorry
  -- !benchmark @end code

-- Postcondition definitions
@[reducible, simp]
def dequeue_postcond {T} (rb : RingBuffer T) (result : T × RingBuffer T)
    (_ : dequeue_precond rb) : Prop :=
  -- !benchmark @start postcond
  let (val, new_rb) := result
  new_rb.is_valid ∧
  new_rb.capacity = rb.capacity ∧
  -- Concise & Proof-Free: Old view is (val :: new_view)
  rb.view = val :: new_rb.view
  -- !benchmark @end postcond

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

-- Proof content
theorem dequeue_postcond_satisfied {T} (rb : RingBuffer T)
    (h_precond : dequeue_precond rb) :
    dequeue_postcond rb (dequeue rb h_precond) h_precond := by
  -- !benchmark @start proof
  sorry
  -- !benchmark @end proof