Common

This is a literate rzk file:

#lang rzk-1

Products of types

#def product
  ( A B : U)
  : U
  := Σ (x : A) , B

The following demonstrates the syntax for constructing terms in Sigma types:

#def diagonal
  ( A : U)
  ( a : A)
  : product A A
  := (a , a)

The type of logical equivalences between types

#def iff
  ( A B : U)
  : U
  := product (A → B) (B → A)

Basic function definitions

#section basic-functions

#variables A B C D : U

#def comp
  ( g : B → C)
  ( f : A → B)
  : A → C
  := \ z → g (f z)

#def triple-comp
  ( h : C → D)
  ( g : B → C)
  ( f : A → B)
  : A → D
  := \ z → h (g (f z))

#def identity
  : A → A
  := \ a → a

#def constant
  ( b : B)
  : A → B
  := \ a → b

#end basic-functions

Substitution

Reindexing a type family along a function into the base type

#def reindex
  ( A B : U)
  ( f : B → A)
  ( C : A → U)
  : B → U
  := \ b → C (f b)