{-
This second-order equational theory was created from the following second-order syntax description:

syntax CommMonoid | CM

type
  * : 0-ary

term
  unit : * | ε
  add  : *  *  ->  * | _⊕_ l20

theory
  (εU⊕ᴸ) a |> add (unit, a) = a
  (εU⊕ᴿ) a |> add (a, unit) = a
  (⊕A) a b c |> add (add(a, b), c) = add (a, add(b, c))
  (⊕C) a b |> add(a, b) = add(b, a)
-}

module CommMonoid.Equality where

open import SOAS.Common
open import SOAS.Context
open import SOAS.Variable
open import SOAS.Families.Core
open import SOAS.Families.Build
open import SOAS.ContextMaps.Inductive

open import CommMonoid.Signature
open import CommMonoid.Syntax

open import SOAS.Metatheory.SecondOrder.Metasubstitution CM:Syn
open import SOAS.Metatheory.SecondOrder.Equality CM:Syn

private
  variable
    α β γ τ : *T
    Γ Δ Π : Ctx

infix 1 _▹_⊢_≋ₐ_

-- Axioms of equality
data _▹_⊢_≋ₐ_ : ∀ 𝔐 Γ {α} → (𝔐 ▷ CM) α Γ → (𝔐 ▷ CM) α Γ → Set where
  εU⊕ᴸ : ⁅ * ⁆̣             ▹ ∅ ⊢       ε ⊕ 𝔞 ≋ₐ 𝔞
  ⊕A   : ⁅ * ⁆ ⁅ * ⁆ ⁅ * ⁆̣ ▹ ∅ ⊢ (𝔞 ⊕ 𝔟) ⊕ 𝔠 ≋ₐ 𝔞 ⊕ (𝔟 ⊕ 𝔠)
  ⊕C   : ⁅ * ⁆ ⁅ * ⁆̣       ▹ ∅ ⊢       𝔞 ⊕ 𝔟 ≋ₐ 𝔟 ⊕ 𝔞

open EqLogic _▹_⊢_≋ₐ_
open ≋-Reasoning

-- Derived equations
εU⊕ᴿ : ⁅ * ⁆̣ ▹ ∅ ⊢ 𝔞 ⊕ ε ≋ 𝔞
εU⊕ᴿ = tr (ax ⊕C with《 𝔞 ◃ ε 》) (ax εU⊕ᴸ with《 𝔞 》)