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import Mathlib

open symmDiff

-- open Set

-- #eval {1, 2, 3} ∆ {3, 6}

example [Monoid M] (N : Submonoid M) : Monoid N where
  mul := fun ⟨x, hx⟩ ⟨y, hy⟩ ↦ ⟨x*y, N.mul_mem hx hy⟩
  mul_assoc := by grind
  one := ⟨1, N.one_mem⟩
  one_mul := fun ⟨x, _⟩ ↦ SetCoe.ext (one_mul x)
  mul_one := fun ⟨x, _⟩ ↦ SetCoe.ext (mul_one x)

#check Set.symmDiff_def

#synth Add ℕ

-- #synth symmDiff (Set ℕ) (Set ℕ)

instance : Add (Set α) := ⟨fun a b ↦ a ∆ b⟩

@[simp]
lemma add_def {a b : Set α} : a + b = a ∆ b := rfl

instance : Zero (Set α) := ⟨{}⟩

@[simp]
lemma zero_def : (0 : Set α) = {} := rfl

instance : Neg (Set α) := ⟨id⟩

@[simp]
lemma neg_def {a : Set α} : -a = a := rfl

instance : Mul (Set α) := ⟨fun a b ↦ a ∩ b⟩

@[simp]
lemma mul_def {a b : Set α} : a * b = a ∩ b := rfl

instance : One (Set α) := ⟨.univ⟩

@[simp]
lemma one_def : (1 : Set α) = .univ := rfl

example : CommRing (Set α) where
  add_assoc := by
    simp only [add_def]
    grind
  zero_add := by simp
  add_zero := by simp
  nsmul := nsmulRec
  zsmul := zsmulRec
  neg_add_cancel := by simp
  add_comm := by
    simp only [add_def]
    grind
  left_distrib := by
    intro a b c
    ext x
    simp [symmDiff_def]
    grind
  right_distrib := by
    intro a b c
    ext x
    simp [symmDiff_def]
    grind
  zero_mul := by simp
  mul_zero := by simp
  mul_assoc := by
    simp only [mul_def]
    grind
  one_mul := by simp
  mul_one := by simp
  mul_comm := by
    simp only [mul_def]
    grind