Multi-level generating extensions, studied by Glück and Jørgensen, are generalization of (two-level) program generators, such as parser generators, to arbitrary many levels. By this generalization, the notion of persistent code-a quoted code fragment that can be used for different stages-naturally arises. In this paper we propose a typed lambda calculus λ , based on linear-time temporal logic, as a basis of programming languages for multi-level generating extensions with persistent code. The

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... idea of the type system is correspondence of (1) linearly ordered times in the logic to computation stages; (2) a formula A (next A) to a type of code that runs at the next stage; and (3) a formula A (always A) to a type of persistent code executable at and after the current stage. After formalizing λ , we prove its key property of time-ordered normalization that a well-typed program can never go back to a previous stage in a "time-ordered" execution, as well as basic properties such as subject reduction, confluence and strong normalization. Commuting conversion plays an important role for time-ordered normalization to hold.