-- A State maps an abstract set of edges/elements to a binary phase value (False = 0, True = 1)
def State (E : Type) := E → Bool

-- A Stabilizer is a functional that measures the total parity of a local geometric cycle
def Stabilizer (E : Type) := (E → Bool) → Bool

-- The predicate verifying that a state belongs to the null space of the parity checker
def Stabilizes {E : Type} (s : Stabilizer E) (state : State E) : Prop :=
  s state = false

-- The composite addition (XOR sum) representing the product of two stabilizer operators
def composite_stabilizer {E : Type} (s1 s2 : Stabilizer E) : Stabilizer E :=
  fun state => (s1 state) ≠ (s2 state)

/--
THEOREM: Closure of the Stabilizer Vacuum Code Space
Formally proves that if a pre-geometric vacuum state is stabilized by two 
discrete cycle operators, it is definitionally invariant under their binary composition.
-/
theorem stabilizer_group_closure {E : Type} (s1 s2 : Stabilizer E) (state : State E) :
    Stabilizes s1 state → Stabilizes s2 state → Stabilizes (composite_stabilizer s1 s2) state := by
  intro h1 h2
  unfold Stabilizes at *
  unfold composite_stabilizer
  -- Substitute the verified null-space values (false) into the target equation
  rw [h1, h2]
  -- Simplifies to: false ≠ false = false, which is definitionally true
  rfl