The pool provides you with a block header to work on.  You double-hash the block header iteratively while incrementing the nonce, and return any results that are of higher difficulty than the pool difficulty to the pool.  If one of those results happens to also be higher difficulty than the network difficulty, you have solved the block for the pool.  The pool does not do any of the calculations associated with solving a block, so I don't see how you think the pool "solved" the block.