Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck 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 ff and every ss-tt cut SS, ∣f∣+∑tail⁡(a)∉S,head⁡(a)∈Sf(a)=∑tail⁡(a)∈S,head⁡(a)∉Sf(a)≤c(S).|f|+\sum_{\operatorname{tail}(a)\notin S,\operatorname{head}(a)\in S}f(a)=\sum_{\operatorname{tail}(a)\in S,\operatorname{head}(a)\notin S}f(a)\le c(S). In particular ∣f∣≤c(S)|f|\le c(S).

Facts & Assumptions

Given: A finite integral network, a feasible flow ff, and an ss-tt cut SS.

[F1]

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

Proof

technique · direct
1.1

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

F1
1.2

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

F1
2.1

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

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 42 results over 16 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