Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-11
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Variation function and positive and negative variations

Definition

Let f:[a,b]→Rf:[a,b]\to\mathbb R have bounded variation (Bounded variation and total variation on an interval). Its variation function is

Vf(x):=Var⁡[a,x](f)(x∈[a,b]),V_f(x):=\operatorname{Var}_{[a,x]}(f)\qquad(x\in[a,b]),

so Vf(a)=0V_f(a)=0. This is defined on every subinterval by Total variation is additive over adjacent subintervals and decreases under restriction and the singleton convention. The positive variation and negative variation of ff are

Pf(x):=Vf(x)+f(x)−f(a)2,Nf(x):=Vf(x)−f(x)+f(a)2.P_f(x):=\frac{V_f(x)+f(x)-f(a)}2,\qquad N_f(x):=\frac{V_f(x)-f(x)+f(a)}2.

The names refer to upward and downward accumulated variation; their monotonicity and the resulting decomposition are proved separately.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 32 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources