Problem: queue_enqueue

Your task is to implement a Queue and verify its First-In, First-Out correctness in Lean. The queue is modeled as a sequence where elements enter at the "back" and exit from the "front." You should implement and verify the enqueue (push) operation. Enqueue: Verify that the item is appended to the end of the model sequence. In a functional setting, this means returning a new queue with the item appended to the data list.


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 enqueue_precond {T} (q : VerifiableQueue T) (v : T) : Prop :=
  -- !benchmark @start precond
  q.is_valid
  -- !benchmark @end precond

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

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

-- Postcondition definitions
@[reducible, simp]
def enqueue_postcond {T} (q : VerifiableQueue T) (v : T) (result : VerifiableQueue T)
    (_ : enqueue_precond q v) : Prop :=
  -- !benchmark @start postcond
  result.is_valid ∧
  result.view = q.view ++ [v]
  -- !benchmark @end postcond

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

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