A recent strand of research in structural proof theory aims at exploring the notion of analytic calculi (i.e., those calculi that support general and modular proof-strategies for cut elimination) and at identifying classes of logics that can be captured in terms of these calculi. In this context, Wansing introduced the notion of proper display calculi as one possible design framework for proof calculi in which the analyticity desiderata are realized in a particularly transparent way. Recently, the theory of properly displayable logics (i.e., those logics that can be equivalently presented with some proper display calculus) has been developed in connection with generalized Sahlqvist theory (a.k.a. unified correspondence). Specifically, properly displayable logics have been syntactically characterized as those axiomatized by analytic inductive axioms, which can be equivalently and algorithmically transformed into analytic structural rules so the resulting proper display calculi enjoy a set of basic properties: soundness, completeness, conservativity, cut elimination, and the subformula property. In this context, the proof that the given calculus is complete w.r.t. the original logic is usually carried out syntactically, i.e., by showing that a (cut-free) derivation exists of each given axiom of the logic in the basic system to which the analytic structural rules algorithmically generated from the given axiom have been added. However, so far, this proof strategy for syntactic completeness has been implemented on a case-by-case base and not in general. In this article, we address this gap by proving syntactic completeness for properly displayable logics in any normal (distributive) lattice expansion signature. Specifically, we show that for every analytic inductive axiom a cut-free derivation can be effectively generated that has a specific shape, referred to as pre-normal form.

最近在结构证明理论方面的一系列研究旨在探索分析演算的概念（即，那些支持用于削减消除的通用和模块化证明策略的演算）并确定可以根据这些演算捕获的逻辑类别。在这种情况下，Wansing 引入了适当显示演算的概念，作为证明演算的一种可能设计框架，其中分析需求以特别透明的方式实现。最近，正确显示的理论逻辑（即，可以用一些适当的显示演算等效地表示的那些逻辑）是结合广义 Sahlqvist 理论（又名统一对应）开发的。具体来说，可正确显示的逻辑在句法上被表征为那些被分析归纳公理公理化的逻辑，可以等效地和算法转换为分析结构规则，因此得到的正确显示演算具有一组基本属性：健全性、完整性、保守性、削减消除、和子公式属性。在这种情况下，证明给定的演算是完整的原始逻辑通常是在句法上进行的，即，通过显示在基本系统中存在每个给定逻辑公理的（无截）推导，从给定公理通过算法生成的分析结构规则已添加到该基本系统中。然而，到目前为止，这种句法完整性证明策略是在逐个案例的基础上实施的，而不是普遍实施的。在本文中，我们通过证明任何正常（分布）格扩展签名中可正确显示的逻辑的句法完整性来解决这一差距。具体来说，我们表明，对于每个解析归纳公理，可以有效地生成具有特定形状的无截推导，称为准范式。