Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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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.

A finite-step integrator gives a weighted sum over its jumps

Example

Let a<ba<b, let c1<⋯<cmc_1<\cdots<c_m be points of the open interval (a,b)(a,b), as in the one-jump example, and let α=C+∑j=1mwjHcj.\alpha=C+\sum_{j=1}^m w_jH_{c_j}. For every continuous ff, ∫abf dα=∑j=1mwjf(cj).\int_a^b f\,d\alpha=\sum_{j=1}^m w_jf(c_j).

Facts & Assumptions

Given: The displayed finite-step integrator and a continuous ff.

[L1]

For a<c<ba<c<b and HcH_c equal to 00 below cc and 11 from cc on, every continuous ff has ∫abf dHc=f(c)\int_a^bf\,dH_c=f(c): a single jump of weight one at an interior point evaluates ff there (A one-jump integrator evaluates a continuous integrand at the jump).

[L2]

The Stieltjes integral is linear in its integrator (Linearity and interval additivity of the Riemann–Stieltjes integral).

Verification

technique · computation
1.1

The constant term has every increment equal to zero. By [L1], each HcjH_{c_j} contributes f(cj)f(c_j), and finite linearity [L2] gives the displayed sum.

L1L2
2.1

Ordering the distinct jump points prevents double counting. The jump points are interior because [L1] places them strictly inside, and the restriction is not cosmetic: HaH_a takes the value 11 at every point of [a,b][a,b], so a jump placed at aa makes every increment zero and contributes nothing, while the weighted sum would still count wf(a)w f(a).

L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 69 results over 17 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