ExampleConstruction: AI-generatedVerification: AI-generatedprecheck 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 four-vertex integral network with an exhibited maximum flow and minimum cut of value five

Example

Let the vertices be $s, u, v, t$ , with labelled arcs and capacities $s ⁣ \to u : 3, s ⁣ \to v : 2, u ⁣ \to v : 1, u ⁣ \to t : 2, v ⁣ \to t : 3.$ The flow with values $3, 2, 1, 2, 3$ in that order has value five and is maximum.

Facts & Assumptions

Given: The displayed finite integral network and flow $f$ .

[L1]

Finite integral max-flow min-cut equates a maximum flow value and a minimum cut capacity (Ford-Fulkerson terminates for finite integer capacities and proves max-flow min-cut with an integral maximum flow).

Verification

Verification technique: direct.

1.1

At $u$ , incoming flow $3$ equals outgoing flow $1 + 2$ , and at $v$ , incoming flow $2 + 1$ equals outgoing flow $3$ ; every arc value is within its capacity.

given

1.2

The flow value is $3 + 2 = 5$ , while the cut $S = {s}$ has capacity $3 + 2 = 5$ .

1.3

By [L1], the exhibited feasible flow and cut of common value five are respectively maximum and minimum.

2.1

This network therefore has max-flow value and min-cut capacity both equal to five.

step 1.1step 1.2step 1.3∎

Depends on

Used by

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Dependency tree · two levels

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