# AFormal Proof of the Max-Flow Min-Cut Theorem for Countable Networks

 Title: A Formal Proof of the Max-Flow Min-Cut Theorem for Countable Networks Author: Andreas Lochbihler Submission date: 2016-05-09 Abstract: This article formalises a proof of the maximum-flow minimal-cut theorem for networks with countably many edges. A network is a directed graph with non-negative real-valued edge labels and two dedicated vertices, the source and the sink. A flow in a network assigns non-negative real numbers to the edges such that for all vertices except for the source and the sink, the sum of values on incoming edges equals the sum of values on outgoing edges. A cut is a subset of the vertices which contains the source, but not the sink. Our theorem states that in every network, there is a flow and a cut such that the flow saturates all the edges going out of the cut and is zero on all the incoming edges. The proof is based on the paper The Max-Flow Min-Cut theorem for countable networks by Aharoni et al. Additionally, we prove a characterisation of the lifting operation for relations on discrete probability distributions, which leads to a concise proof of its distributivity over relation composition. Change history: [2017-09-06]: derive characterisation for the lifting operations on discrete distributions from finite version of the max-flow min-cut theorem (revision a7a198f5bab0) [2020-12-19]: simpler proof of linkability for bounded unhindered bipartite webs, leading to a simpler proof for networks with bounded out-capacities (revision 93ca33f4d915) BibTeX: @article{MFMC_Countable-AFP, author = {Andreas Lochbihler}, title = {A Formal Proof of the Max-Flow Min-Cut Theorem for Countable Networks}, journal = {Archive of Formal Proofs}, month = may, year = 2016, note = {\url{https://isa-afp.org/entries/MFMC_Countable.html}, Formal proof development}, ISSN = {2150-914x}, } License: BSD License Depends on: EdmondsKarp_Maxflow Used by: Probabilistic_While