Legacy.Base.Structures.Terms

Interpretation of Terms in General Structures

This is the Legacy.Base.Structures.Terms module of the Agda Universal Algebra Library.

When we interpret a term in a structure we call the resulting function a term operation. Given a term p and a structure 𝑨, we denote by 𝑨 ⟦ p ⟧ the interpretation of p in 𝑨. This is defined inductively as follows.

1. If p is a variable symbol x : X and if a : X → ∣ 𝑨 ∣ is a tuple of elements of ∣ 𝑨 ∣, then define 𝑨 ⟦ p ⟧ a := a x.

2. If p = f t, where f : ∣ 𝑆 ∣ is an operation symbol, if t : (arity 𝐹) f → 𝑻 X is a tuple of terms, and if a : X → ∣ 𝑨 ∣ is a tuple from 𝑨, then define 𝑨 ⟦ p ⟧ a := (f ᵒ 𝑨) (λ i → 𝑨 ⟦ t i ⟧ a).

Thus interpretation of a term is defined by structural induction.

{-# OPTIONS --cubical-compatible --exact-split --safe #-}

module Legacy.Base.Structures.Terms where

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

open import Legacy.Base.Structures.Basic  using ( signature ; structure ; _ᵒ_ )
open import Legacy.Base.Terms.Basic

private variable
 𝓞₀ 𝓥₀ 𝓞₁ 𝓥₁ χ α ρ : Level
 𝐹 : signature 𝓞₀ 𝓥₀
 𝑅 : signature 𝓞₁ 𝓥₁
 X : Type χ

open signature
open structure

_⟦_⟧ : (𝑨 : structure 𝐹 𝑅 {α} {ρ}) → Term X → (X → carrier 𝑨) → carrier 𝑨
𝑨 ⟦ ℊ x ⟧ = λ a → a x
𝑨 ⟦ node f t ⟧ = λ a → (f ᵒ 𝑨) (λ i → (𝑨 ⟦ t i ⟧ ) a)