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-- https://cjquines.com/files/binaryoperations.pdf

import Mathlib

example {α} (f : α → α → α) l r (hl : ∀ x, f l x = x) (hr : ∀ x, f x r = x) : l = r := by
  have h₁ : f l r = l := by
    grind
  grind

example {α} [Nonempty α] (f : α → α → α) (h : ∀ x y, ∃ z, f x z = y ∧ ∀ z', f x z = f x z' → z = z') : ∃ g : α → α → α, ∀ x y, f x (g x y) = y ∧ g x (f x y) = y := by
  let g x y := (f x).invFun y
  use g
  intro x y
  constructor
  have h₁ x : (f x).Surjective := by
    intro y
    specialize h x y
    grind
  apply Function.rightInverse_invFun (h₁ x)
  have h₁ x : (f x).Injective := by
    intro a b
    specialize h x (f x a)
    obtain ⟨a', ⟨h₂, h₃⟩⟩ := h
    rw [← h₃ a h₂]
    exact h₃ b
  apply Function.leftInverse_invFun (h₁ x)

example {α} (f g : α → α → α) i j (hid : ∀ x, f i x = x ∧ f x i = x ∧ g j x = x ∧ g x j = x) (h : ∀ x y z w, f (g x y) (g z w) = g (f x z) (f y w)) : f = g := by
  have h₁ x : f x j = x := by
    specialize h x j j i
    grind
  have h₂ : i = j := by
    specialize hid j
    grind
  have h₃ x y : f x y = g x y := by
    specialize h x i i y
    grind
  grind