The paper presents DeLaM, a dependent layered modal type theory that extends the previous layered modal type theory developed by Hu and Pientka with contextual variables. This allows for recursion on the structure of code, which was not possible in the previous system.
The key aspects of DeLaM are:
Contextual variables: The type theory is "slightly" dependently typed, with both global and local contexts and types able to depend on contextual variables. This enables recursion on code structure.
Weakenings: The paper defines global and local weakenings, and proves various algebraic properties about their interactions with substitutions.
Syntax and typing: The paper defines the syntax and typing rules for the 2-layered modal type theory with contextual variables, including rules for terms, local substitutions, and global substitutions.
Syntactic properties: The paper establishes a set of important syntactic properties, such as composition of global/local substitutions, naturality, and global weakening lemmas. These properties are crucial for proving the main normalization result.
Consistency: By proving the decidability of conversion checking via a reducibility predicate argument, the paper shows the consistency of DeLaM, making it a suitable foundation for type-theory-based proof assistants.
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