Proof that every circuit with diodes has exactly one solution
I assume this is for a contrived problem where there is a circuit with known passives and some I's and V's given and spots marked for diodes of unknown direction. My answer is:

Hopefully the creators of the problems have constrained themselves to instances where their assumptions lead to their conclusions.

It could be theoretically unsolvable by having a diode be extraneous; consider grounding both sides of a diode. There could be non-trivial cases using virtual grounds or other equal voltages that could be hard to spot. 

There surely could exist valid circuits that only differ by the direction of a diode for any value of "valid circuit" that includes diodes. Consider modeling switches using those ideal diode rules, how can you decide if a switch was meant to be on or off? 
Hopefully the given currents and voltages give enough hints. And hopefully they haven't given you conflicting hints.

This shifts the question to "How can you tell if an instance has enough information to be unique?" I remember the answer being something like you need one independent given for each independent unknown, but I'm sure I couldn't prove that or come up with a general test for the independence of either.