Problem: queue_dequeue

Your task is to implement a Queue and verify its First-In, First-Out correctness in Lean. The queue is modeled as a sequence. You should implement and verify the dequeue operation. Dequeue: Verify that the operation returns the first element (index 0) of the sequence and that the new model sequence contains all remaining elements in their original relative order (the tail of the list). In a functional setting, the function should accept a proof of non-emptiness and return both the dequeued value and the new queue state.


import Mathlib

structure VerifiableQueue (T : Type) where
  data : List T

def VerifiableQueue.view {T} (q : VerifiableQueue T) : List T :=
  q.data

def VerifiableQueue.is_valid {T} (q : VerifiableQueue T) : Prop :=
  True

-- Precondition definitions
@[reducible, simp]
def dequeue_precond {T} (q : VerifiableQueue T) : Prop :=
  -- !benchmark @start precond
  q.is_valid ∧ q.view.length > 0
  -- !benchmark @end precond

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

-- Main function definition
-- Returns the projected pair (val, new_queue)
def dequeue {T} (q : VerifiableQueue T)
    (h_precond : dequeue_precond q) : T × VerifiableQueue T :=
  -- !benchmark @start code
  sorry
  -- !benchmark @end code

-- Postcondition definitions
@[reducible, simp]
def dequeue_postcond {T} (q : VerifiableQueue T) (result : T × VerifiableQueue T)
    (_ : dequeue_precond q) : Prop :=
  -- !benchmark @start postcond
  let (val, new_queue) := result
  new_queue.is_valid ∧
  -- Concise & Proof-Free: The old queue consists of the return value followed by the new queue.
  q.view = val :: new_queue.view
  -- !benchmark @end postcond

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

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