Proof of Mining in Blockchain Systems

We propose a proof of mining system. Roughly speaking, in this system the mining stake ${\rm mstak}(A)$ with discrimination index $a\in[0,1]$ of an account $A$ is defined by the formula: $${\rm mstak}(A)=(1-a)\cdot\frac{1}{{\rm NOM}}+a\cdot\frac{{\rm NOBM}(A)}{L},$$ where $L$ is the length of the block-chain, ${\rm NOM}$ is the number of miners in the block-chain, and ${\rm NOBM}(A)$ is the number of blocks mined by $A$.