We present techniques for automatically inferring formal properties of feed-forward neural networks. We observe that a significant part (if not all) of the logic of feed forward networks is captured in the activation status (on or off) of its neurons. We propose to extract patterns based on neuron decisions as preconditions that imply certain desirable output property e.g., the prediction being a certain class. Together, the inferred preconditions and the output property form a {\em contract} for the network. We present techniques to extract input properties} encoding convex predicates on the input space that imply given output properties and layer properties, representing network properties captured in the hidden layers that imply the desired output behavior. We apply our techniques on networks for the MNIST and ACASXU applications. Our experiments highlight the use of the inferred properties in a variety of tasks, such as explaining predictions, providing robustness guarantees, simplifying proofs, and network distillation.