Type Inference for Pure Type Systems

Variants of the basic calculus of constructions

“…Severi [38] proved this as Theorem 4.9. Note that this theorem holds for TOC0A if ∆ is a well-formed environment.…”

“…Paula Severi [38] has studied variants of PTSs with this change; she calls them "PTSs without the Π -condition". If λ(S) is a regular PTS with specification S, then Severi calls the corresponding PTS without the Π -condition λ ω (S).…”

“…The addition of "P" will refer to a system whose abstraction and conversion rules are essentially those of a PTS; the addition of "A" will indicate a system with a modified abstraction rule, such as the systems λ ω (S) of [38], and the addition of "C" will denote a system with a modified rule of conversion between types. If both the abstraction and conversion rules are modified, "AC" will be added to the name.…”

“…The system TOC2A is obtained from the system TOC2P by replacing rule (Abstraction) of Definition 1 by the following rule: If S is the specification for the calculus of constructions, then this is the system λ ω (S) of [38], and the system λ(S) is TOC2P.…”

Type Isomorphisms and Proof Reuse in Dependent Type Theory

“…A key argument in the proof is that convertibility between legal types is decidable, see e.g. [6,31,37] for a recent survey of type-checking algorithms for dependent type theory.…”