{-
   The Category Solver solveCat! solves equations in a category that
   hold up to associativity and unit laws

   This file shows some examples of how to use it.
-}

module Cubical.Tactics.CategorySolver.Examples where

open import Cubical.Foundations.Prelude

open import Cubical.Categories.Category
open import Cubical.Tactics.CategorySolver.Reflection

private
  variable
    ℓ ℓ' : Level

module Examples (𝓒 : Category ℓ ℓ') where
  open Category 𝓒

  _ : ∀ {A} → id {A} ≡ id {A}
  _ = solveCat! 𝓒

  _ : ∀ {A B}{f : 𝓒 [ A , B ]}
    → f ∘ id ≡ f
  _ = solveCat! 𝓒

  _ : ∀ {A B}{f : 𝓒 [ A , B ]}
    → id ∘ (id ∘ id ∘ f) ∘ id ≡ id ∘ id ∘ (id ∘ f)
  _ = solveCat! 𝓒

  _ : ∀ {A B C}{f : 𝓒 [ A , B ]}{g : 𝓒 [ B , C ]}
    → f ⋆ g ≡ g ∘ (id ∘ f) ∘⟨ 𝓒 ⟩ id
  _ = solveCat! 𝓒

  ex : ∀ {A B C}(f : 𝓒 [ A , B ])(g : 𝓒 [ B , C ])(h : 𝓒 [ A , C ])
    → (f ⋆ (id ⋆ g)) ≡ h
    → f ⋆ g ≡ h ⋆ id
  ex f g h p =
    f ⋆ g ≡⟨ solveCat! 𝓒 ⟩
    (f ⋆ (id ⋆ g)) ≡⟨ p ⟩
    h ≡⟨ solveCat! 𝓒 ⟩
    h ⋆ id ∎