The weak form of the efficient market hypothesis (EMH) just says that the market is efficient to all prior information contained within price. By definition, the weak form of EMH obeys the Markov property such that the current state contains more information about the future state than all prior states combined. Thus, the weak form EHM is alone sufficient to show that market prices do indeed follow a random walk. Note that while the weak form potentially allows fundamental analysis and insider information as means of predicting future price, it rules out technical analysis as a possible predictor.
To your example, the arbitrage mechanism is the implied mechanism for how and why historically significant price signals should not persist into the future. Suppose there was a price signal which historically gave $(\textbf{past tense})$ a 65% chance of a steep price increases over some finite $(\textbf{future past})$ time horizon. Arbitrageurs' (i.e., traders, basically) knowledge of the existence of a predictable and significant pattern would then alter the nature of the signal causing it to become less predictable in the future. In this case, competition to be the first arbitrageur would result in buying in anticipation of the completion of the pattern. When you compound the actions of many arbitrageurs, the pricing anomaly is eventually eliminated. When you introduce machines to the mix, predictable price patterns are identified and arbitraged at speeds which defy human ability. In all cases, trading in anticipation of a predictable pattern causes price to evolve differently than in the past.
So, to your question: yes, price would've risen as predicted albeit more quickly and thereby differently, eventually to point where the pattern becomes unpredictable in the future.
In all of this, I think it's important to remember that the theory assumes that free lunches are consumed instantly, so that no free lunches exist across any finite time period. Reality, however, is different from the theory because it does take time and effort to identify and arbitrage opportunities for excess profit, but by that time the lunch is no longer free. So the theory is just an approximation, albeit a good one.
