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

syntax Semiring | SR

type
  * : 0-ary

term
  zero : * | 𝟘
  add  : *  *  ->  * | _⊕_ l20
  one  : * | 𝟙
  mult : *  *  ->  * | _⊗_ l30

theory
  (𝟘U⊕ᴸ) a |> add (zero, a) = a
  (𝟘U⊕ᴿ) a |> add (a, zero) = a
  (⊕A) a b c |> add (add(a, b), c) = add (a, add(b, c))
  (⊕C) a b |> add(a, b) = add(b, a)
  (𝟙U⊗ᴸ) a |> mult (one, a) = a
  (𝟙U⊗ᴿ) a |> mult (a, one) = a
  (⊗A) a b c |> mult (mult(a, b), c) = mult (a, mult(b, c))
  (⊗D⊕ᴸ) a b c |> mult (a, add (b, c)) = add (mult(a, b), mult(a, c))
  (⊗D⊕ᴿ) a b c |> mult (add (a, b), c) = add (mult(a, c), mult(b, c))
  (𝟘X⊗ᴸ) a |> mult (zero, a) = zero
  (𝟘X⊗ᴿ) a |> mult (a, zero) = zero
-}

module Semiring.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 Semiring.Signature
open import Semiring.Syntax

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

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

infix  _▹_⊢_≋ₐ_

-- Axioms of equality
data _▹_⊢_≋ₐ_ : ∀ 𝔐 Γ {α} → (𝔐 ▷ SR) α Γ → (𝔐 ▷ SR) α Γ → Set where
  𝟘U⊕ᴸ : ⁅ * ⁆̣             ▹ ∅ ⊢       𝟘 ⊕ 𝔞 ≋ₐ 𝔞
  ⊕A   : ⁅ * ⁆ ⁅ * ⁆ ⁅ * ⁆̣ ▹ ∅ ⊢ (𝔞 ⊕ 𝔟) ⊕ 𝔠 ≋ₐ 𝔞 ⊕ (𝔟 ⊕ 𝔠)
  ⊕C   : ⁅ * ⁆ ⁅ * ⁆̣       ▹ ∅ ⊢       𝔞 ⊕ 𝔟 ≋ₐ 𝔟 ⊕ 𝔞
  𝟙U⊗ᴸ : ⁅ * ⁆̣             ▹ ∅ ⊢       𝟙 ⊗ 𝔞 ≋ₐ 𝔞
  𝟙U⊗ᴿ : ⁅ * ⁆̣             ▹ ∅ ⊢       𝔞 ⊗ 𝟙 ≋ₐ 𝔞
  ⊗A   : ⁅ * ⁆ ⁅ * ⁆ ⁅ * ⁆̣ ▹ ∅ ⊢ (𝔞 ⊗ 𝔟) ⊗ 𝔠 ≋ₐ 𝔞 ⊗ (𝔟 ⊗ 𝔠)
  ⊗D⊕ᴸ : ⁅ * ⁆ ⁅ * ⁆ ⁅ * ⁆̣ ▹ ∅ ⊢ 𝔞 ⊗ (𝔟 ⊕ 𝔠) ≋ₐ (𝔞 ⊗ 𝔟) ⊕ (𝔞 ⊗ 𝔠)
  ⊗D⊕ᴿ : ⁅ * ⁆ ⁅ * ⁆ ⁅ * ⁆̣ ▹ ∅ ⊢ (𝔞 ⊕ 𝔟) ⊗ 𝔠 ≋ₐ (𝔞 ⊗ 𝔠) ⊕ (𝔟 ⊗ 𝔠)
  𝟘X⊗ᴸ : ⁅ * ⁆̣             ▹ ∅ ⊢       𝟘 ⊗ 𝔞 ≋ₐ 𝟘
  𝟘X⊗ᴿ : ⁅ * ⁆̣             ▹ ∅ ⊢       𝔞 ⊗ 𝟘 ≋ₐ 𝟘

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

-- Derived equations
𝟘U⊕ᴿ : ⁅ * ⁆̣ ▹ ∅ ⊢ 𝔞 ⊕ 𝟘 ≋ 𝔞
𝟘U⊕ᴿ = tr (ax ⊕C with《 𝔞 ◃ 𝟘 》) (ax 𝟘U⊕ᴸ with《 𝔞 》)