A novel Nakamoto-like consensus was proposed by Tang et al. (ACISP 2019) to speed up the convergence (block finality) rate by determining a weight of a block in the blockchain by a tunable potential function of the block hash. However, the convergence of the scheme was evaluated only in an experimental way and a sudden utilization of another blockchain was not clearly explained. This article asymptotically analyses the convergence of Nakamoto-like consensus of Tang et al. by proposing a general framework for formalizing consensus schemes comprising both the classical Nakamoto consensus (bitcoin consensus) and the consensus of Tang et al. The framework contains two categories of schemes, namely, small-step consensus like the bitcoin consensus and giant-step consensus of Tang et al. Furthermore, the essence of the second chain, the even-trigger, is shown to be a necessity of realizing giant-step consensus.