Is Bitcoin future-proof? As I understand things, the security and integrity of Bitcoin rely on the assumption that its hashing is "hard" in some sense.  However, in the past, processing power has increased exponentially.  This has made some problems solvable in a reasonable amount of time that were not solvable before.  How does Bitcoin account for the possibility that processing power will continue to increase in the future?
BitCoin is futureproof, because in the event that processing power (or more relevantly, mathematical advances) make its algorithms weak or obsolete, the protocol can simply be upgraded.

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Upgrades to the Bitcoin protocol that break compatibility are implementable by social mechanisms but not by any technical process, which is by design.  In order to change the hashing or signing algorithms in the event that new techniques weaken them, the following process could be followed:


As soon as the existing algorithms are discovered to be weakened, have a trusted entity sign a copy of the existing blockchain, marking it as authoritative via a non-weakened algorithm.
Create a new protocol that 'respects' the existing blockchain (which contains all records of bitcoin ownership), and allows for an upgrade path to exist indefinitely.
Implement the new protocol dormantly in one or more clients, with a designated block number to switch over, and the trusted version of the blockchain hardcoded in.
Convince the vast majority of Bitcoin users that your trusted entity is trustworthy, and to switch to the new client(s) before the designated block is mined.  There really is no equilibrium allowing both versions to coexist, since people who own bitcoins will be able to spend them in both communities.  Unless the modified client achieves near-universal adoption before the designated block is mined, users will switch back before the change goes live.