Cost Aczel Classification H Continuous Of Positive Continuous

The continuity bridge

The declaration H_continuous_of_positive_continuous is a lemma in the machine-checked library of formal theorems that underpins Recognition Science. It establishes a bridge between two kinds of continuity for a function F that maps positive real numbers to real numbers. The first kind is continuity on the positive reals: as the input approaches any positive number, the output approaches the value at that number. The second kind is continuity everywhere: the same behavior holds for every real input, including zero and negative numbers. The lemma shows that if F has the first property, then a related function H, built from F by a logarithmic change of variables, has the second property.

The construction is concrete. Define H(t) = F(e^t) + 1. Since the exponential function e^t is continuous and always positive, composing it with F preserves continuity. Adding the constant 1 does not change that. The lemma packages this simple observation into a reusable step. Its role in the larger framework is to connect the primitive assumption of continuity on positive numbers, which is one of the five plain conditions for a cost function, to the smoothness that the classification theorem requires. The classification theorem, proved separately, says that any function satisfying the five conditions must equal the specific cost function J(x) = (x + 1/x)/2 - 1.

What the lemma does not claim is just as important. It does not assert that F itself is continuous everywhere, only that the transformed H is. It does not prove the classification theorem, nor does it establish the smoothness of F. The lemma is a single link in a chain: it supplies the continuity of H, which then feeds into a separate Aczél regularity kernel that derives smoothness and the differential equation H'' = H. The full uniqueness result requires all five primitive hypotheses, not just continuity. The lemma is a necessary but not sufficient ingredient.

In plain terms, this is the step that lets the framework move from a function that behaves well on positive inputs to one that behaves well everywhere, at least for the transformed variable. That move is what makes the later classification argument possible. Without it, the continuity assumption would not reach far enough to apply the classical theorems that force the unique cost function. The lemma is small, but it is the hinge that lets the continuity assumption do its work.

private theorem H_continuous_of_positive_continuous (F : ℝ → ℝ)
    (hCont : ContinuousOn F (Set.Ioi 0)) : Continuous (H F) := by
  have h := ContinuousOn.comp_continuous hCont Real.continuous_exp
  have h' : Continuous (fun t => F (Real.exp t)) :=
    h (by intro t; exact Set.mem_Ioi.mpr (Real.exp_pos t))
  have h_add : Continuous (fun t : ℝ => F (Real.exp t) + (1 : ℝ)) :=
    h'.add (continuous_const : Continuous fun _ : ℝ => (1 : ℝ))
  simpa [H, G] using h_add

private theorem H_continuous_of_positive_continuous (F : ℝ → ℝ)
    (hCont : ContinuousOn F (Set.Ioi 0)) : Continuous (H F) := by
  have h := ContinuousOn.comp_continuous hCont Real.continuous_exp
  have h' : Continuous (fun t => F (Real.exp t)) :=
    h (by intro t; exact Set.mem_Ioi.mpr (Real.exp_pos t))
  have h_add : Continuous (fun t : ℝ => F (Real.exp t) + (1 : ℝ)) :=
    h'.add (continuous_const : Continuous fun _ : ℝ => (1 : ℝ))
  simpa [H, G] using h_add

What this page does not claim

The lemma does not prove the full classification theorem that forces the unique cost function. The lemma does not establish smoothness of F, only continuity of the transformed H. The lemma does not apply to functions that are not continuous on the positive reals.

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/Cost/AczelClassification.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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