Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-21
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.

The Henstock–Kurzweil integral has at most one value

Statement

A function on a compact interval has at most one Henstock–Kurzweil integral value.

Facts & Assumptions

Given: Alleged integral values I and J for the same function on [a,b].

[L1]

Every gauge on a compact interval admits a fine tagged partition (Cousin's lemma: every gauge on a compact interval admits a fine tagged partition).

Proof

technique · contradiction
1.1

Suppose, for contradiction, that I≠J; choose gauges controlling errors below ∣I−J∣/3, take their pointwise minimum, and use [L1] to obtain one tagged partition P fine for both.

givenL1assume-contra
2.1

The triangle inequality gives ∣I−J∣≤∣I−S(f,P)∣+∣S(f,P)−J∣<2∣I−J∣/3, a contradiction, so I=J.

step 1.1discharge-contradiction∎

Depends on

Used by

Cited to discharge well-definedness by The Henstock–Kurzweil integral on a compact interval.

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources