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Operations

This is the Overture.Operations module of the Agda Universal Algebra Library.

For consistency and readability, we reserve two universe variables for special purposes.

The first of these is π“ž which we used in the Overture.Signatures as the universe of the type of operation symbols of a signature.

The second is π“₯ which we reserve for types representing arities of relations or operations.

The type Op encodes the arity of an operation as an arbitrary type I : Type π“₯, which gives us a very general way to represent an operation as a function type with domain I β†’ A (the type of β€œtuples”) and codomain A.


{-# OPTIONS --without-K --exact-split --safe #-}

module Overture.Operations where

-- Imports from Agda and the Agda Standard Library -----------------------------
open import Agda.Primitive               using () renaming ( Set to Type )
open import Level                        using ( Level ; _βŠ”_ )

private variable a b ρ π“₯ : Level

-- The type of operations on A of arity I
Op : Type a β†’ Type π“₯ β†’ Type (a βŠ” π“₯)
Op A I = (I β†’ A) β†’ A

For example, the I-ary projection operations on A are represented as inhabitants of the type Op A I as follows.


-- Example (projections)
Ο€ : {I : Type π“₯} {A : Type a } β†’ I β†’ Op A I
Ο€ i = Ξ» x β†’ x i

Occasionally we want to extract the arity of a given operation symbol.


-- return the arity of a given operation symbol
arity[_] : {I : Type π“₯} {A : Type a } β†’ Op A I β†’ Type π“₯
arity[_] {I = I} f = I

← Overture.Signatures Base β†’