  The aging law and the slip law are two representative evolution laws of the
rate- and state-dependent friction (RSF) law, based on canonical behaviors in
three types of laboratory experiments: slide-hold-slide (SHS), velocity-step
(VS), and steady-state (SS) tests. The aging law explains the SHS canon but
contradicts the VS canon, and vice versa for the slip law. The later proposed
composite law, which switches these two laws according to the slip rate $V$,
explains both canons but contradicts the SS canon. The present study constructs
evolution laws satisfying all three canons throughout the range of variables
where experiments have confirmed the canons. By recompiling the three canons,
we have derived constraints on the evolution law and find that the evolution
rates in the strengthening phases of the SHS and VS canons are so different
that complete reconciliation throughout the entire range of variables is
mathematically impossible. However, for the limited range of variables probed
by experiments so far, we have found that the SHS and VS canons can be
reconciled without violating the SS canon by switching the evolution function
according to $\Omega$, the ratio of the state $\theta$ to its steady-state
value $\theta_{\rm SS}$ for the instantaneous slip rate. We could generally
show that, as long as the state evolution rate $\dot \theta$ depends only on
the instantaneous values of $V$ and $\theta$, simultaneous reproduction of the
three canons, throughout the experimentally confirmed range, requires the
aging-law-like evolution for $\Omega$ sufficiently below a threshold $\beta$
and the slip-law-like evolution for $\Omega$ sufficiently above $\beta$. The
validity of the canons in existing experiments suggests $\beta \lesssim 0.01$.
