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def definition def or abbrev

MultiplicativeL4Polynomial

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formal statement (Lean)

  98def MultiplicativeL4Polynomial (m : MultiplicativeRecognizer 𝒞) : Prop :=

proof body

Definition body.

  99  ∃ P : ℝ → ℝ → ℝ,
 100    (∃ a b c d e f : ℝ, ∀ u v, P u v = a + b*u + c*v + d*u*v + e*u^2 + f*v^2) ∧
 101    (∀ u v, P u v = P v u) ∧
 102    (∀ x y : ℝ, 0 < x → 0 < y →
 103      m.cost (x * y) + m.cost (x / y) = P (m.cost x) (m.cost y))
 104
 105/-! ## The Derivation Theorem -/
 106
 107/-- **L4 is automatic in the polynomial form for any multiplicative recognizer.**
 108
 109The route-independence field of `SatisfiesLawsOfLogic` already provides the
 110polynomial-degree-2 combiner satisfying the multiplicative L4. -/

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