proposition: Lemma connect_to_C1_id x y : x -[]-> y = x -[predC1 x]-> y.
proof: Proof. apply/idP/idP; last by apply: connect_toW. case/connectP => p /shortenP[p' Pxp' Uxp' Sxp' Lyxp']. apply/connectP; exists p' => //=. rewrite path_to Pxp'; apply/allP=> z zIp' /=. have /= /andP[H _] := Uxp'. by apply: contraNneq H => <-. Qed.
imports: mathcomp(all_ssreflect) extra