eq_of_reflTransGen

formal source

  32/-- If a real-valued function agrees on all R-related pairs,
  33    it agrees on the reflexive-transitive closure of R.
  34    This is the graph-theoretic core: local agreement propagates globally. -/
  35theorem eq_of_reflTransGen {R : V → V → Prop} {f : V → ℝ}
  36    (hadj : ∀ a b, R a b → f a = f b) {u v : V}
  37    (huv : Relation.ReflTransGen R u v) : f u = f v := by
  38  induction huv with
  39  | refl => rfl
  40  | tail _ hbc ih => exact ih.trans (hadj _ _ hbc)
  41
  42/-- On a preconnected graph (every pair related by ReflTransGen),
  43    agreement on edges implies global constancy. -/
  44theorem constant_of_preconnected {R : V → V → Prop}
  45    (hconn : ∀ u v : V, Relation.ReflTransGen R u v) {f : V → ℝ}
  46    (hadj : ∀ a b, R a b → f a = f b) :
  47    ∀ v w : V, f v = f w := by
  48  intro v w
  49  exact eq_of_reflTransGen hadj (hconn v w)
  50
  51/-! ### Ratio rigidity (Result 1) -/
  52
  53/-- Edge cost is nonneg for positive fields. -/
  54theorem edge_cost_nonneg {x : V → ℝ} {v w : V}
  55    (hv : 0 < x v) (hw : 0 < x w) :
  56    0 ≤ Jcost (x v / x w) :=
  57  Jcost_nonneg (div_pos hv hw)
  58
  59/-- **RESULT 1 — Forward direction (Finite-Volume Rigidity).**
  60
  61On a connected graph, if the ratio cost J(x_v/x_w) vanishes on every edge,
  62then the positive field x is constant.
  63
  64Proof sketch:
  651. J(x_v/x_w) = 0 implies x_v/x_w = 1, i.e. x_v = x_w (unique zero of J).