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

import Mathlib

-- example (f : ℝ → ℝ → ℝ) (l r : ℝ) (h : ∀ x, f l x = f x r ∧ f l x = x) : l = r := by
--   have h₁ : f l r = l := by
--     specialize h l
--     grind
--   have h₂ : f l r = r := by
--     specialize h r
--     grind
--   grind

example {α} (f : α → α → α) (l r : α) (hl : ∀ x, f l x = x) (hr : ∀ x, f x r = x) : l = r := by
  have h₁ : f l r = l := by
    grind
  -- have h₂ : f l r = r := by
  --   grind
  grind

example {α} (f : α → α → α) (h : ∀ x y z, f x z = y ∧ ∀ z' ≠ z, f x z' ≠ y) : ∃ g : α → α → α, ∀ x y, f x (g x y) = g x (f x y) ∧ f x (g x y) = y := by
  sorry

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