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Unit type

Since v0.5.1

#lang rzk-1

In the syntax, only Unit (the type) and unit (the only inhabitant) are provided. Everything else should be available from computation rules. More specifically, rzk takes the uniqueness property of the Unit type (see Section 1.5 of the HoTT book1) as the computation rule, meaning that any (well-typed) term of type Unit reduces to unit. This means in particular, that induction and uniqueness can be defined very easily:

#define ind-Unit
  (C : Unit -> U)
  (C-unit : C unit)
  (x : Unit)
  : C x
  := C-unit

#define uniq-Unit
  (x : Unit)
  : x = unit
  := refl

#define isProp-Unit
  (x y : Unit)
  : x = y
  := refl

As a non-trivial example, here is a proof that Unit is a Segal type:

#section isSegal-Unit

#variable extext : ExtExt

#define iscontr-Unit : isContr Unit
  := (unit, \_ -> refl)

#define isContr-Δ²→Unit uses (extext)
  : isContr (Δ² -> Unit)
  := (\_ -> unit, \k -> eq-ext-htpy extext
    (2 * 2) Δ² (\_ -> BOT)
    (\_ -> Unit) (\_ -> recBOT)
    (\_ -> unit) k
    (\_ -> refl)
    )

#define isSegal-Unit uses (extext)
  : isSegal Unit
  := \x y z f g -> isRetract-ofContr-isContr
    (∑ (h : hom Unit x z), hom2 Unit x y z f g h)
    (Δ² -> Unit)
    (\(_, k) -> k, (\k -> (\t -> k (t, t), k), \_ -> refl))
    isContr-Δ²→Unit

#end isSegal-Unit

  1. The Univalent Foundations Program (2013). Homotopy Type Theory: Univalent Foundations of Mathematics. https://homotopytypetheory.org/book ↩