The inverse method is a saturation-based theorem-proving technique; it relies on a forward proof-search strategy and can be applied to cut-free calculi enjoying the subformula property. Here, we apply this method to derive the unprovability of a goal formula G in Intuitionistic Propositional Logic. To this aim we design a forward calculus FRJ ( G ) for Intuitionistic unprovability, which is appropriate for constructively ascertaining the unprovability of a formula G by providing a concise countermodel for it; in particular, we prove that the generated countermodels have minimal height. Moreover, we clarify the role of the saturated database obtained as result of a failed proof-search in FRJ ( G ) by showing how to extract from such a database a derivation witnessing the Intuitionistic validity of the goal.

中文翻译：

---

正向反驳中不可证明性与可证明性的对偶性——直觉命题逻辑搜索

逆方法是一种基于饱和度的定理证明技术；它依赖于前向证明搜索策略，可以应用于具有子公式性质的无切演算。在这里，我们应用这种方法来推导目标公式的不可证明性G在直觉命题逻辑中。为此，我们设计了一个前向演算FRJ(G) 用于直觉不可证明性，适用于建设性地确定公式的不可证明性G通过为其提供一个简洁的反模型；特别是，我们证明了生成的反模型具有最小的高度。此外，我们阐明了由于证明搜索失败而获得的饱和数据库的作用。FRJ(G) 通过展示如何从这样的数据库中提取证明目标的直觉有效性的推导。