This block takes in 4 bits representing a single digit of the 7-segment display and maps them to a 7-bit output representing digits 0-9. There are two approaches I could take to design this circuit (or atleast two that come to mind). Either I could (a) find the Boolean logic related to each element and create a network of gates that will decode the value on the fly, or I will (b) implement the logic as a MUX. After a bit of tinkering I found that the "Boolean" approach was far more efficient. 

Using some Karnough maps, I created the final design can be seen below. Note that I will be using a common anode 7-segment, so to turn on a segment, the output should be 0. 

Here are the maps I used for each element. I'm a little rusty on gate simplification so there may be some redundency hidden here. 

NOTE: there was incorrect logic for elelement B, this was discovered during validation (0010) is not a case where B is off