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

An integral flow decomposes into source-sink path flows and directed cycle flows; unit capacities make the paths arc-disjoint

Statement

Every feasible integral flow is a finite sum of unit s-t path flows and unit directed cycle flows. Here a labelled directed cycle flow also permits a two-arc antiparallel circuit, so that it records a circulation even when the underlying simple-digraph convention excludes two-cycles. If every capacity is at most one, the source-sink paths in such a decomposition are arc-disjoint.

Facts & Assumptions

Given: A feasible integral flow f in a finite integral network.

[F1]

Positive flow is conserved at nonterminals, the source has no incoming original arc, and the sink has no outgoing original arc (Finite integral networks, feasible flows, values, cuts and residual networks).

Proof

technique · induction on the finite total flow $\sum_{a\in A}f(a)$
1.1

The zero flow is the empty sum of path and cycle flows.

base
1.2

If f is nonzero, follow positive-flow arcs from a positive outgoing source arc; conservation continues the walk at every nonterminal until it reaches t or repeats a vertex.

F1
1.3

A repeated segment is a positive labelled circulation; subtract one on it and record a unit directed cycle flow, while an unrepeated walk to t yields a unit s-t path flow after subtracting one on its arcs.

F1
2.1

The remainder is feasible, integral, and has strictly smaller total flow, so the induction hypothesis decomposes it and adding the recorded term gives the desired decomposition.

step 1.3ih
3.1

If c(a)≤1, no arc can occur in two recorded path terms; together with the base case and step 2.1 this proves the claim for every finite total flow.

step 1.1step 2.1discharge-induction∎

Depends on

Used by

Dependency tree · two levels

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Sources