In this paper, we investigate the proof complexity of a wide range of substructural systems. For any proof system P at least as strong as Full Lambek calculus, FL, and polynomially simulated by the extended Frege system for some superintuitionistic logic of infinite branching, we present an exponential lower bound on the proof lengths. More precisely, we will provide a sequence of P-provable formulas ${\{A_n\}}_{n=1}^{\infty }$ such that the length of the shortest 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 superintuitionistic logic of infinite branching. As an example, Hilbert-style proof systems for any finitely axiomatizable extension of $\mathsf{FL}$ that are weaker than the intuitionistic logic, in particular the usual Hilbert-style proof systems for the logics $\mathsf{F}{\mathsf{L}}_{\mathsf{S}}$ for the set of structural rules $S\subseteq \{e,i,o,c\}$, fall in this category. We will also prove a similar result for the proof systems and logics extending Visser's basic propositional calculus 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 $\mathbf{CF}{\mathbf{L}}_{\mathbf{ew}}$.

在本文中，我们研究了范围广泛的子结构系统的证明复杂性。对于任何证明系统P至少强如全Lambek演算，FL，以及由扩展弗雷格系统无限分支的一些superintuitionistic逻辑多项式模拟，我们提出的指数下在样张长度约束。更准确地说，我们将提供一系列P可证明的公式${\{{\mathrm{一个}}_{ñ}\}}_{ñ=\mathrm{1个}}^{\infty }$使得最短P证明的长度为${\mathrm{一个}}_{ñ}$ 在长度上是指数的 ${\mathrm{一个}}_{ñ}$。下限也扩展到任何Frege系统（扩展的Frege系统）中的证明线（证明长度）的数量，以实现$\mathsf{佛罗里达州}$以及无限分支的任何超直觉逻辑。例如，对于任何有限公理可扩展的Hilbert样式的证明系统$\mathsf{佛罗里达州}$ 比直觉逻辑更弱的东西，特别是通常的逻辑希尔伯特式证明系统 $\mathsf{F}{\mathsf{大号}}_{\mathsf{小号}}$ 对于一组结构规则 $\mathrm{小号}\subseteq \{Ë，\mathrm{一世}，Ø，C\}$，属于此类别。对于扩展Visser基本命题演算BPC及其逻辑的证明系统和逻辑，我们还将证明相似的结果$\mathsf{BPC}$， 分别。最后，在经典的子结构设置中，我们将在由无割版本的多项式多项式模拟的任何证明系统中，在证明行数上建立指数下界。$\mathbf{碳纤维}{\mathbf{大号}}_{\mathbf{ew}}$。