A Proof of theorems) such that H

Since c is the center point of the Poincaré hyperplane, the vector! The classification function f has the HEX property with respect to G if and only if for any constraint in G, the corresponding loss term is 0. Note that the loss term of the constraint being 0 implies that the corresponding constraint is respected. Our loss terms clearly connect the HEX property. According to the definition of HEX-property, f has the HEX property with respect to G if and only if the corresponding loss term of the corresponding constraint is 0. Corollary 1. Given a HEX graph G of labels and if the loss of the embeddings is 0, then the learned prediction function is logically consistent with respect to G. Hence, the loss being 0 implies that all losses are zeros (all constraints are satisfied).