We present a logic named L_{LF} whose intended use is to formalize properties of specifications developed in the dependently typed lambda calculus LF. The logic is parameterized by the LF signature that constitutes the specification. Atomic formulas correspond to typing derivations relative to this signature. The logic includes a collection of propositional connectives and quantifiers. Quantification ranges over expressions that denote LF terms and LF contexts. Quantifiers of the first variety are qualified by simple types that describe the functional structure associated with the variables they bind; deeper, dependency related properties are expressed by the body of the formula. Context-level quantifiers are qualified by context schemas that identify patterns of declarations out of which actual contexts may be constructed. The semantics of variable-free atomic formulas is articulated via the derivability in LF of the judgements they encode. Propositional constants and connectives are understood in the usual manner and the meaning of quantifiers is explicated through substitutions of expressions that adhere to the type qualifications. The logic is complemented by a proof system that enables reasoning that is sound with respect to the described semantics. The main novelties of the proof system are the provision for case-analysis style reasoning about LF judgements, support for inductive reasoning over the heights of LF derivations and the encoding of LF meta-theorems. The logic is motivated by the paradigmatic example of type assignment in the simply-typed lambda calculus and the proof system is illustrated through the formalization of a proof of type uniqueness for this calculus.
翻译:我们提出了一个名为 L ⁇ LLUD} 的逻辑, 它的用意是正式确定在依附型的 lambda 计算器中开发的规格的特性。 逻辑由构成该规格的LF 签名参数参数参数化。 原子公式对应与该符号相对的输入导出。 逻辑包括一组配方连接和量化符。 用于表示LF 术语和LF 背景的表达式的量化范围。 第一类的量化符由简单类型的描述与它们所绑定的变量相关的功能结构的功能结构的简单类型来限定; 较深的、 与依赖相关的属性由公式体积表示。 上标码型类型化的量化符由背景图案图案来限定, 用以确定可能构建实际环境的宣布模式。 变量自由原子公式的语义性通过它们编码的LFLF 的衍生法的衍生法来表述。 预设常数常数常数常数常数常数和连接值常数以通常的方式被理解, 量化的证明通过替代符合该类型资格的表达的表达的表达式特性; 逻辑级的逻辑的逻辑的逻辑由验证系统加以补充, 推算的逻辑的推理的逻辑的推理的逻辑的逻辑的推理的推理的推理, 的推理的推理的推理的推理法的推理的推理的推理法的推理的推理的推理, 的推理的推理的推理的推理的推理, 的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理, 的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理, 的推理, 的推理的系统是对的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理的推理