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theorem proved term proof

ratio_rigidity

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formal statement (Lean)

  67theorem ratio_rigidity {adj : V → V → Prop}
  68    (hconn : ∀ u v : V, Relation.ReflTransGen adj u v)
  69    {x : V → ℝ} (hpos : ∀ v, 0 < x v)
  70    (hzero : ∀ v w, adj v w → Jcost (x v / x w) = 0) :
  71    ∀ v w : V, x v = x w := by

proof body

Term-mode proof.

  72  apply constant_of_preconnected hconn
  73  intro v w hvw
  74  have hdiv_pos : 0 < x v / x w := div_pos (hpos v) (hpos w)
  75  have h1 : x v / x w = 1 := Jcost_zero_iff_one hdiv_pos (hzero v w hvw)
  76  rwa [div_eq_iff (ne_of_gt (hpos w)), one_mul] at h1
  77
  78/-- **Converse:** constant positive fields have zero ratio cost on every edge. -/

used by (14)

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depends on (11)

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