import Axiom

namespace hmhhcnw7

structure HeadroomState where
  grossCapacity : Nat
  commitmentBurden : Nat

 def headroom (s : HeadroomState) : Nat :=
  s.grossCapacity - s.commitmentBurden

 theorem headroom_nonnegative (s : HeadroomState) :
    0 ≤ headroom s := by
  exact Nat.zero_le _

 theorem headroom_zero_of_burden_ge_capacity
    (s : HeadroomState)
    (h : s.grossCapacity ≤ s.commitmentBurden) :
    headroom s = 0 := by
  exact Nat.sub_eq_zero_of_le h

 theorem headroom_positive_iff_capacity_exceeds_burden
    (s : HeadroomState) :
    0 < headroom s ↔ s.commitmentBurden < s.grossCapacity := by
  simpa [headroom] using Nat.sub_pos_iff_lt

 def retirementHeadroomGain (releasedBurden capabilityLoss : Nat) : Nat :=
  releasedBurden - capabilityLoss

 theorem safe_retirement_gain_of_burden_release
    (releasedBurden capabilityLoss : Nat)
    (h : capabilityLoss < releasedBurden) :
    0 < retirementHeadroomGain releasedBurden capabilityLoss := by
  exact Nat.sub_pos_of_lt h

def headroom_theorem_boundaryPayload := "These theorems establish only arithmetic consequences of a natural-number headroom model. They do not prove that a real system's capacities are commensurable, that burden is fully observed, or that effective reachable futures are determined by one scalar."
axiom headroom_theorem_boundary_auth : Axiom.AuthenticatedTerm ("hitchhiker2thefuture", headroom_theorem_boundaryPayload, 1789059159)
def headroom_theorem_boundary : Axiom.Statement "hitchhiker2thefuture" headroom_theorem_boundaryPayload 1789059159 := ⟨headroom_theorem_boundary_auth⟩

end hmhhcnw7