positive <evidence> / definite <object>

sure / certain

definite / positive

be certain / can depend on / be reliable / be dependable

urge on / hasten / hurry up / egg on / bustle

sponsor / promote

enhance / awaken / arouse / promote

increase the speed / accelerate / quicken / gather speed

facilitation / furtherance / speed up / acceleration / expediting / accelerate / help a movement forward / quicken / hasten / give an impetus / giving an impetus / hurry up / further / facilitate / promote / give a boost to / expedite / propel (the progress of *something) / giving a boost to / promotion

speed *something up / accelerate / quicken / hasten / advance / increase the speed of *something / promote