Verifier for A_tm in polynomial time - how to formally prove it does not exist?
Suppose by contradiction $A_\mathrm{TM}$ has polytime verifier $\Longrightarrow$ $A_\mathrm{TM}$ is in NP $\Longrightarrow$ $A_\mathrm{TM}$ is decideable by some non-deterministic TM in polytime $\Longrightarrow$ $A_\mathrm{TM}$ is dcidiable by some deterministic $2^{\mathrm{polytime}}$ TM $\Longrightarrow$ $A_\mathrm{TM}$ is decidable $\Longrightarrow$ contradiction.