ContinuousRouteIndependence

formal source

 107/-- A *continuous-combiner* version of route-independence: there exists a
 108continuous, symmetric function `P : ℝ × ℝ → ℝ` such that
 109`F(xy) + F(x/y) = P(F(x), F(y))` on positive ratios. -/
 110def ContinuousRouteIndependence (C : ComparisonOperator) : Prop :=
 111  ∃ P : ℝ → ℝ → ℝ,
 112    Continuous (Function.uncurry P) ∧
 113    (∀ u v, P u v = P v u) ∧
 114    (∀ x y : ℝ, 0 < x → 0 < y →
 115       derivedCost C (x * y) + derivedCost C (x / y)
 116       = P (derivedCost C x) (derivedCost C y))
 117
 118/-- A *continuous Law of Logic* — the existing `SatisfiesLawsOfLogic`
 119with polynomial route-independence replaced by continuous
 120route-independence. -/
 121structure SatisfiesLawsOfLogicContinuous (C : ComparisonOperator) : Prop where
 122  identity            : Identity C
 123  non_contradiction   : NonContradiction C
 124  excluded_middle     : ExcludedMiddle C
 125  scale_invariant     : ScaleInvariant C
 126  route_independence  : ContinuousRouteIndependence C
 127  non_trivial         : NonTrivial C
 128
 129/-- Log-coordinate Aczél data extracted from a continuous-combiner Law of
 130Logic witness. -/
 131structure LogAczelData (G : ℝ → ℝ) (P : ℝ → ℝ → ℝ) : Prop where
 132  continuous_G : Continuous G
 133  zero_G       : G 0 = 0
 134  even_G       : Function.Even G
 135  continuous_P : Continuous (Function.uncurry P)
 136  symmetric_P  : ∀ u v, P u v = P v u
 137  aczel_eq     : ∀ t u : ℝ, G (t + u) + G (t - u) = P (G t) (G u)
 138
 139/-- Continuity of the derived cost on positive ratios lifts to continuity of
 140the log-coordinate cost `G(t) = F(exp t)`. -/