-- Mathlib/SetTheory/Game/Basic.lean
theorem quot_mul_assoc (x y z : PGame) : (⟦x * y * z⟧ : Game) = ⟦x * (y * z)⟧ :=
  match x, y, z with
  | mk xl xr xL xR, mk yl yr yL yR, mk zl zr zL zR => by
    let x := mk xl xr xL xR
    let y := mk yl yr yL yR
    let z := mk zl zr zL zR
    refine quot_eq_of_mk'_quot_eq ?_ ?_ ?_ ?_
    · fconstructor
      · rintro (⟨⟨_, _⟩ | ⟨_, _⟩, _⟩ | ⟨⟨_, _⟩ | ⟨_, _⟩, _⟩) <;>
          -- Porting note: as above, increased the `maxDepth` here by 1.
          solve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]
      · rintro (⟨_, ⟨_, _⟩ | ⟨_, _⟩⟩ | ⟨_, ⟨_, _⟩ | ⟨_, _⟩⟩) <;>
          solve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]
      · rintro (⟨⟨_, _⟩ | ⟨_, _⟩, _⟩ | ⟨⟨_, _⟩ | ⟨_, _⟩, _⟩) <;> rfl
      · rintro (⟨_, ⟨_, _⟩ | ⟨_, _⟩⟩ | ⟨_, ⟨_, _⟩ | ⟨_, _⟩⟩) <;> rfl
    · fconstructor
      · rintro (⟨⟨_, _⟩ | ⟨_, _⟩, _⟩ | ⟨⟨_, _⟩ | ⟨_, _⟩, _⟩) <;>
          solve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]
      · rintro (⟨_, ⟨_, _⟩ | ⟨_, _⟩⟩ | ⟨_, ⟨_, _⟩ | ⟨_, _⟩⟩) <;>
          solve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]
      · rintro (⟨⟨_, _⟩ | ⟨_, _⟩, _⟩ | ⟨⟨_, _⟩ | ⟨_, _⟩, _⟩) <;> rfl
      · rintro (⟨_, ⟨_, _⟩ | ⟨_, _⟩⟩ | ⟨_, ⟨_, _⟩ | ⟨_, _⟩⟩) <;> rfl
    -- Porting note: explicitly wrote out arguments to each recursive
    -- quot_mul_assoc reference below, because otherwise the decreasing_by block
    -- failed. Each branch previously ended with: `simp [quot_mul_assoc]; abel`
    -- See https://github.com/leanprover/lean4/issues/2288
    · rintro (⟨⟨i, j⟩ | ⟨i, j⟩, k⟩ | ⟨⟨i, j⟩ | ⟨i, j⟩, k⟩)
      · change
          ⟦(xL i * y + x * yL j - xL i * yL j) * z + x * y * zL k -
                (xL i * y + x * yL j - xL i * yL j) * zL k⟧ =
            ⟦xL i * (y * z) + x * (yL j * z + y * zL k - yL j * zL k) -
                xL i * (yL j * z + y * zL k - yL j * zL k)⟧
        simp only [quot_sub, quot_add, quot_right_distrib_sub, quot_right_distrib,
                   quot_left_distrib_sub, quot_left_distrib]
        rw [quot_mul_assoc (xL i) (mk yl yr yL yR) (mk zl zr zL zR)]
        rw [quot_mul_assoc (mk xl xr xL xR) (yL j) (mk zl zr zL zR)]
        rw [quot_mul_assoc (xL i) (yL j) (mk zl zr zL zR)]
        rw [quot_mul_assoc (mk xl xr xL xR) (mk yl yr yL yR) (zL k)]
        rw [quot_mul_assoc (xL i) (mk yl yr yL yR) (zL k)]
        rw [quot_mul_assoc (mk xl xr xL xR) (yL j) (zL k)]
        rw [quot_mul_assoc (xL i) (yL j) (zL k)]
        abel
      · change
          ⟦(xR i * y + x * yR j - xR i * yR j) * z + x * y * zL k -
                (xR i * y + x * yR j - xR i * yR j) * zL k⟧ =
            ⟦xR i * (y * z) + x * (yR j * z + y * zL k - yR j * zL k) -
                xR i * (yR j * z + y * zL k - yR j * zL k)⟧
        simp only [quot_sub, quot_add, quot_right_distrib_sub, quot_right_distrib,
                   quot_left_distrib_sub, quot_left_distrib]
        rw [quot_mul_assoc (xR i) (mk yl yr yL yR) (mk zl zr zL zR)]
        rw [quot_mul_assoc (mk xl xr xL xR) (yR j) (mk zl zr zL zR)]
        rw [quot_mul_assoc (xR i) (yR j) (mk zl zr zL zR)]
        rw [quot_mul_assoc (mk xl xr xL xR) (mk yl yr yL yR) (zL k)]
        rw [quot_mul_assoc (xR i) (mk yl yr yL yR) (zL k)]
        rw [quot_mul_assoc (mk xl xr xL xR) (yR j) (zL k)]
        rw [quot_mul_assoc (xR i) (yR j) (zL k)]
        abel
      · change
          ⟦(xL i * y + x * yR j - xL i * yR j) * z + x * y * zR k -
                (xL i * y + x * yR j - xL i * yR j) * zR k⟧ =
            ⟦xL i * (y * z) + x * (yR j * z + y * zR k - yR j * zR k) -
                xL i * (yR j * z + y * zR k - yR j * zR k)⟧
        simp only [quot_sub, quot_add, quot_right_distrib_sub, quot_right_distrib,
                   quot_left_distrib_sub, quot_left_distrib]
        rw [quot_mul_assoc (xL i) (mk yl yr yL yR) (mk zl zr zL zR)]
        rw [quot_mul_assoc (mk xl xr xL xR) (yR j) (mk zl zr zL zR)]
        rw [quot_mul_assoc (xL i) (yR j) (mk zl zr zL zR)

Proof state:
case refine_2
x✝ y✝ z✝ : PGame
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
zl zr : Type u_1
zL : zl → PGame
zR : zr → PGame
x : PGame := mk xl xr xL xR
y : PGame := mk yl yr yL yR
z : PGame := mk zl zr zL zR
⊢ (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR).RightMoves ≃
    (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR)).RightMoves

Tactic:
fconstructor

Result:
case refine_2.toFun
x✝ y✝ z✝ : PGame
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
zl zr : Type u_1
zL : zl → PGame
zR : zr → PGame
x : PGame := mk xl xr xL xR
y : PGame := mk yl yr yL yR
z : PGame := mk zl zr zL zR
⊢ (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR).RightMoves →
    (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR)).RightMoves

case refine_2.invFun
x✝ y✝ z✝ : PGame
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
zl zr : Type u_1
zL : zl → PGame
zR : zr → PGame
x : PGame := mk xl xr xL xR
y : PGame := mk yl yr yL yR
z : PGame := mk zl zr zL zR
⊢ (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR)).RightMoves →
    (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR).RightMoves

case refine_2.left_inv
x✝ y✝ z✝ : PGame
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
zl zr : Type u_1
zL : zl → PGame
zR : zr → PGame
x : PGame := mk xl xr xL xR
y : PGame := mk yl yr yL yR
z : PGame := mk zl zr zL zR
⊢ LeftInverse ?refine_2.invFun ?refine_2.toFun

case refine_2.right_inv
x✝ y✝ z✝ : PGame
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
zl zr : Type u_1
zL : zl → PGame
zR : zr → PGame
x : PGame := mk xl xr xL xR
y : PGame := mk yl yr yL yR
z : PGame := mk zl zr zL zR
⊢ RightInverse ?refine_2.invFun ?refine_2.toFun

Proof state:
case refine_4.inr.mk.inr.mk
x✝ y✝ z✝ : PGame
xl xr : Type u_1
xL : xl → PGame
xR : xr → PGame
yl yr : Type u_1
yL : yl → PGame
yR : yr → PGame
zl zr : Type u_1
zL : zl → PGame
zR : zr → PGame
x : PGame := mk xl xr xL xR
y : PGame := mk yl yr yL yR
z : PGame := mk zl zr zL zR
k : zl
i : xr
j : yl
⊢ ⟦xR i * (mk yl yr yL yR * mk zl zr zL zR)⟧ + ⟦mk xl xr xL xR * (yL j * mk zl zr zL zR)⟧ -
          ⟦xR i * (yL j * mk zl zr zL zR)⟧ +
        ⟦mk xl xr xL xR * (mk yl yr yL yR * zL k)⟧ -
      (⟦xR i * (mk yl yr yL yR * zL k)⟧ + ⟦mk xl xr xL xR * (yL j * zL k)⟧ - ⟦xR i * (yL j * zL k)⟧) =
    ⟦xR i * (y * z)⟧ + (⟦x * (yL j * z)⟧ + ⟦x * (y * zL k)⟧ - ⟦x * (yL j * zL k)⟧) -
      (⟦xR i * (yL j * z)⟧ + ⟦xR i * (y * zL k)⟧ - ⟦xR i * (yL j * zL k)⟧)

Tactic:
abel

Result:
no goals