import Axiom

namespace j1b75r6z

structure CapacityLedger where
  grossCapacity : Int
  commitmentBurden : Int

def ledgerHeadroom (s : CapacityLedger) : Int :=
  s.grossCapacity - s.commitmentBurden

def transitionHeadroom (s : CapacityLedger) (deltaCapacity deltaBurden : Int) : Int :=
  (s.grossCapacity + deltaCapacity) - (s.commitmentBurden + deltaBurden)

theorem headroom_transition_accounting
    (s : CapacityLedger) (deltaCapacity deltaBurden : Int) :
    transitionHeadroom s deltaCapacity deltaBurden =
      ledgerHeadroom s + deltaCapacity - deltaBurden := by
  dsimp [transitionHeadroom, ledgerHeadroom]
  omega

def backlogAfter (backlog arrivals service : Int) : Int :=
  backlog + arrivals - service

theorem backlog_grows_when_arrivals_exceed_service
    (backlog arrivals service : Int)
    (h : service < arrivals) :
    backlog < backlogAfter backlog arrivals service := by
  dsimp [backlogAfter]
  omega

def structuralUpdate (state gain : Int) : Int :=
  state + gain

theorem zero_gain_preserves_structure (state : Int) :
    structuralUpdate state 0 = state := by
  simp [structuralUpdate]

def compressedPolicy {History State Action : Type}
    (representation : History → State)
    (policy : State → Action)
    (history : History) : Action :=
  policy (representation history)

theorem compressed_policy_equal_on_equal_representation
    {History State Action : Type}
    (representation : History → State)
    (policy : State → Action)
    (h₁ h₂ : History)
    (h : representation h₁ = representation h₂) :
    compressedPolicy representation policy h₁ =
      compressedPolicy representation policy h₂ := by
  exact congrArg policy h

def theorem_scope_boundaryPayload := "The arithmetic theorems establish consequences of explicit Int models. The zero-gain theorem is a limiting case, not a stability result. The compressed-policy theorem establishes action equality when representations are equal; it does not prove that a representation is sufficient for a real continuation contract. Persistent backlog growth and update-rate thrashing require additional dynamical assumptions and observations."
axiom theorem_scope_boundary_auth : Axiom.AuthenticatedTerm ("hitchhiker2thefuture", theorem_scope_boundaryPayload, 1789061575)
def theorem_scope_boundary : Axiom.Statement "hitchhiker2thefuture" theorem_scope_boundaryPayload 1789061575 := ⟨theorem_scope_boundary_auth⟩

end j1b75r6z