Postulates

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TOC

With postulates we can introduce elements in a type without actually giving the definition of the element itself.

The general form of a postulate declaration is as follow

  postulate
      c11 ... c1i : <Type>
      ...
      cn1 ... cnj : <Type>

Example

  postulate
    A B    : Set
    a      : A
    b      : B
    _=AB=_ : A -> B -> Set
    a==b   : a =AB= b

Introducing postulates is in general not something we recommend. Once postulates are introduced the consistency of the whole development is at risk, because there is nothing that prevents us from introducing an element in the empty set.

  data False : Set where

  postulate bottom : False

A preferable way to work is to define a module parametrised by the elements we need

  module Absurd (bt : False) where
    ...

  module M (A B : Set; a : A; b : B) 
           (_=AB=_ : A -> B -> Set) (a==b : a =AB= b) where
    ...