smooth
plain-language theorem explainer
The abbrev smooth sets the target regularity class to infinite differentiability for continuous solutions of the d'Alembert equation. Workers on the Recognition cost functional equation cite it when bridging continuous hypotheses to the analytic classification step. The definition is a direct one-line assignment of the top element in the extended naturals.
Claim. Let $smooth := ⊤ ∈ WithTop ℕ∞$ denote the class of $C^∞$ maps, i.e., the target for ContDiff ℝ ⊤ in statements about solutions of the d'Alembert equation.
background
The AczelProof module establishes that any continuous H : ℝ → ℝ satisfying H(t + u) + H(t - u) = 2 H(t) H(u) with H(0) = 1 is real analytic. The module doc states: 'Any continuous solution H : ℝ → ℝ of the d'Alembert functional equation with H(0) = 1 is real analytic (ContDiff ℝ ⊤).' The abbrev supplies the concrete value ⊤ used in all ContDiff statements inside the integration bootstrap and ODE derivation phases.
proof idea
One-line definition that directly assigns the top element of WithTop ℕ∞. No lemmas or tactics are applied; the identifier is then referenced by downstream ContDiff statements such as dAlembert_contDiff_smooth.
why it matters
The definition supplies the smoothness target inside AczelRegularityKernel, which is the regularity bridge for primitive_to_uniqueness_of_kernel. That theorem identifies the primitive cost function with Jcost and thereby closes the T5 J-uniqueness step. It also feeds standardLagrangian and alphaInv_linear_rate through the cost structures. The abbrev therefore anchors the analyticity claim that lets the phi-ladder and eight-tick octave operate on sufficiently regular solutions.
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