Variants of the basic calculus of constructions

Abstract

In this paper, a number of different versions of the basic calculus of constructions that have appeared in the literature are compared and the exact relationships between them are determined. The biggest differences between versions are those between the original version of Coquand and the version in early papers on the subject by Seldin. None of these results is very deep, but it seems useful to collect them in one place.

Pedagogical lambda-cube: the 2 case

“…The properties 1, 2, 3, 4, 5, 6, 8, 9 are still valid for λ 2 e , modulo the addition of the corresponding explicit motivations.…”

“…Rules of CC are presented in fig. 1: close presentations can be found in [8], with well formed judgements; in [2] avoiding weakening rule; or [1] presenting usual properties of CC. Removing some rules of CC we obtain λ 2 of fig.…”

“…In λ 2 e examples has to be maintained during the whole proof: all premisses of rules use the same motivation. But we have seen that motivations can be exchanged (lem.…”

“…Outline of the article In section 2 we recall the usual notations, definitions and well-known results about the calculus of constructions (CC) and its subsystem of second order λ 2 . In sections 3, 4, 5 we define and study pedagogical subsystems of CC of second order: first with explicit and total examples (also called motivations) λ 2 e , then with total motivations λ 2 t and finally with partial motivations λ 2 p . Each is obtained from the previous by relaxing some constraints, the last one fully satisfying the definition of pedagogical subsystem of CC.…”

“…Let us assume that the type checking problem for λ 2 e can be decided by an algorithm D: D(Γ, σ, t, A) = true if and only if Γ ⊢ 2e σ t : A. We can then build an algorithm D ′ to decide the type checking problem for λ 2 t : But the type checking problem for λ 2 t is undecidable (thm. 85).…”