  A graph is word-representable if it can be represented in a certain way using
alternation of letters in words. Word-representable graphs generalise several
important and well-studied classes of graphs, and they can be characterised by
semi-transitive orientations. Recognising word-representability is an
NP-complete problem, and the bottleneck of the theory of word-representable
graphs is convincing someone that a graph is non-word-representable, keeping in
mind that references to (even publicly available and user-friendly) software
are not always welcome. (Word-representability can be justified by providing a
semi-transitive orientation as a certificate that can be checked in polynomial
time.)
  In the literature, a variety of (usually ad hoc) proofs of
non-word-representability for particular graphs, or families of graphs, appear,
but for a randomly selected graph, one should expect looking at O(2^{#{edges}})
orientations and justifying that none of them is semi-transitive.
  In this paper, we develop methods for an automatic search of human-verifiable
proofs of graph non-word-representability. As a proof-of-concept, we provide
``short'' proofs of non-word-representability, generated automatically by our
publicly available user-friendly software, of the Shrikhande graph on 16
vertices and 48 edges (6 ``lines'' of proof) and the Clebsch graph on 16
vertices and 40 edges (10 ``lines'' of proof). As a bi-product of our studies,
we correct two mistakes published multiple times (two graphs out of the 25
non-word-representable graphs on 7 vertices were actually word-representable,
while two non-word-representable graphs on 7 vertices were missing).
