Foundation Generalized Dalembert Continuous Log Cost Of Continuous On Positive

Continuity and the cost function

The declaration continuous_log_cost_of_continuousOn_positive is a technical bridge. It starts with a function F that is continuous on the positive real numbers (numbers greater than zero). The theorem then states that the function G(t) = F(e^t), which is F evaluated at the exponential of t, is continuous everywhere on the real line. This is a standard result in analysis: the exponential map is a continuous bijection from the real line onto the positive reals, so continuity on the positive reals is exactly equivalent to continuity of the composition with the exponential on the whole line. The lemma is a formal, machine-checked proof of this equivalence.

This bridge matters because the framework's analysis of cost functions often works more naturally in the logarithmic coordinate t. The key functional equation, the d'Alembert equation, is stated for functions on the whole real line. By moving to log-coordinates, a cost function defined only on positive numbers can be studied with the full power of the classical theory. The declaration is a necessary preliminary step that allows the framework to apply the Aczél–Kannappan classification, which states that any continuous solution of the d'Alembert equation is either a constant, a hyperbolic cosine, or a trigonometric cosine. This classification is what ultimately forces the cost function to take one of a small set of specific forms.

The lemma itself does not claim anything about the specific form of the cost function. It does not say that F is a particular function, nor does it force any particular value for the cost. It only establishes a regularity property: if F is continuous on the positive reals, then its logarithmic counterpart is continuous everywhere. This is a hypothesis for the later classification theorems, not a conclusion about the nature of the cost itself. The classification of the cost function's form requires additional assumptions, such as the route-independence condition, which are not part of this single lemma.

In the broader context of the Recognition Science framework, this lemma is part of a move to relax an earlier, stricter assumption. Previously, the framework required the combiner function P to be a polynomial of degree at most two. This lemma, combined with the Aczél–Kannappan classification, shows that mere continuity of the combiner is sufficient to reach the same classification of cost functions. The polynomial case becomes a special instance of the continuous case. This is a step toward a more general and elegant foundation, though it is a step in a chain, not the final destination.

/-- Continuity of the derived cost on positive ratios lifts to continuity of
the log-coordinate cost `G(t) = F(exp t)`. -/
theorem continuous_log_cost_of_continuousOn_positive
    (F : ℝ → ℝ)
    (hF : ContinuousOn F (Set.Ioi (0 : ℝ))) :
    Continuous (fun t : ℝ => F (Real.exp t)) := by
  have hExpOn : ContinuousOn (fun t : ℝ => Real.exp t) (Set.univ : Set ℝ) :=
    Real.continuous_exp.continuousOn
  have hMaps : Set.MapsTo (fun t : ℝ => Real.exp t)
      (Set.univ : Set ℝ) (Set.Ioi (0 : ℝ)) := by
    intro t ht
    exact Real.exp_pos t
  have hComp : ContinuousOn (F ∘ fun t : ℝ => Real.exp t) (Set.univ : Set ℝ) :=
    hF.comp hExpOn hMaps
  have hCont := continuousOn_univ.mp hComp
  simpa [Function.comp_def] using hCont

What this page does not claim

The declaration does not determine the specific form of the cost function. The declaration does not prove the Aczél–Kannappan classification itself. The declaration does not establish that the cost function satisfies the route-independence condition.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/GeneralizedDAlembert.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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