Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-02
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.

A flow's value is its net flow across every source-sink cut and never exceeds the cut capacity

Statement

For a feasible flow f and every s-t cut S, ∣f∣+∑tail⁡(a)∉S,head⁡(a)∈Sf(a)=∑tail⁡(a)∈S,head⁡(a)∉Sf(a)≤c(S). In particular ∣f∣≤c(S).

Facts & Assumptions

Given: A finite integral network, a feasible flow f, and an s-t cut S.

[F1]

Flow is conserved at every vertex except s,t, its value is the outgoing flow at s, and a cut capacity sums outgoing capacities (Finite integral networks, feasible flows, values, cuts and residual networks).

Proof

technique · direct
1.1F1

Sum flow conservation over S∖{s} and cancel arcs whose two endpoints lie in S; because no original arc enters s, the resulting equality is the displayed equality of outgoing flow with ∣f∣ plus incoming flow.

1.2F1

Each outgoing flow term is at most its capacity, so the outgoing flow sum is at most c(S).

2.1step 1.1step 1.2∎

The incoming sum is nonnegative, hence the equality and inequality of steps 1.1--1.2 give ∣f∣≤c(S).

Depends on

Used by

Dependency tree · two levels

10 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