  In this paper, we investigate the proof complexity of a wide range of
substructural systems. For any proof system $\mathbf{P}$ at least as strong as
Full Lambek calculus, $\mathbf{FL}$, and polynomially simulated by the extended
Frege system for some infinite branching super-intuitionistic logic, we present
an exponential lower bound on the proof lengths. More precisely, we will
provide a sequence of $\mathbf{P}$-provable formulas $\{A_n\}_{n=1}^{\infty}$
such that the length of the shortest $\mathbf{P}$-proof for $A_n$ is
exponential in the length of $A_n$. The lower bound also extends to the number
of proof-lines (proof-lengths) in any Frege system (extended Frege system) for
a logic between $\mathsf{FL}$ and any infinite branching super-intuitionistic
logic. We will also prove a similar result for the proof systems and logics
extending Visser's basic propositional calculus $\mathbf{BPC}$ and its logic
$\mathsf{BPC}$, respectively. Finally, in the classical substructural setting,
we will establish an exponential lower bound on the number of proof-lines in
any proof system polynomially simulated by the cut-free version of
$\mathbf{CFL_{ew}}$.
