separatelyAdditive_iff_interactionDefect_zero

formal source

  90
  91/-- **Equivalence: separate additivity is identical vanishing of the
  92interaction defect.** -/
  93theorem separatelyAdditive_iff_interactionDefect_zero
  94    (P : ℝ → ℝ → ℝ) :
  95    SeparatelyAdditive P ↔ ∀ u v : ℝ, interactionDefect P u v = 0 :=
  96  ⟨interactionDefect_eq_zero_of_separatelyAdditive,
  97   separatelyAdditive_of_interactionDefect_zero⟩
  98
  99/-- **Equivalence: coupling is non-vanishing interaction defect.** -/
 100theorem isCouplingCombiner_iff_interactionDefect_nonzero
 101    (P : ℝ → ℝ → ℝ) :
 102    IsCouplingCombiner P ↔ ∃ u v : ℝ, interactionDefect P u v ≠ 0 := by
 103  unfold IsCouplingCombiner
 104  rw [separatelyAdditive_iff_interactionDefect_zero]
 105  constructor
 106  · intro h
 107    by_contra hno
 108    push_neg at hno
 109    exact h hno
 110  · rintro ⟨u, v, huv⟩ hall
 111    exact huv (hall u v)
 112
 113/-! ## The RCL Combiner -/
 114
 115/-- The polynomial combiner attached to the RCL family with parameter
 116`c`: `P(u, v) = 2u + 2v + c·u·v`. -/
 117def RCLCombiner (c : ℝ) : ℝ → ℝ → ℝ :=
 118  fun u v => 2 * u + 2 * v + c * u * v
 119
 120/-- The interaction defect of the RCL combiner at `(u, v)` is exactly
 121`c · u · v`. -/
 122theorem interactionDefect_RCLCombiner (c u v : ℝ) :
 123    interactionDefect (RCLCombiner c) u v = c * u * v := by