proposition: Lemma um_valid3 f1 f2 f3 : valid (f1 \+ f2 \+ f3) = [&& valid (f1 \+ f2), valid (f2 \+ f3) & valid (f1 \+ f3)].
proof: Proof. apply/idP/and3P; first by move=>W; rewrite !(validLE3 W). case=>V1 V2 V3; case: validUn=>//; rewrite ?V1 ?(validE2 V2) // => k. by rewrite domUn inE V1; case/orP; [move: V3 | move: V2]; case: validUn =>// _ _ X _ /X /negbTE ->. Qed.
imports: Coq(ssreflect ssrbool ssrfun) mathcomp(ssrnat eqtype seq path bigop) pcm(options prelude finmap seqperm pred seqext) pcm(pcm morphism)