Proof of Mining in Blockchain Systems

We propose a proof of mining system. Roughly speaking, in this system the mining stake $\mathrm{m}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{k}(A)$ with discrimination index $a\in [0,1]$ of an account $A$ is defined by the formula: $$\mathrm{m}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{k}(A)=(1-a)\cdot \frac{1}{\mathrm{N}\mathrm{O}\mathrm{M}}+a\cdot \frac{\mathrm{N}\mathrm{O}\mathrm{B}\mathrm{M}(A)}{L},$$ where $L$ is the length of the block-chain, $\mathrm{N}\mathrm{O}\mathrm{M}$ is the number of miners in the block-chain, and ${\rm NOBM}(A)$isthenumberofblocksminedby$A$.