l3lab/ntp-mathlib huggingface dataset:
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 (⟨⟨_, _⟩ | ⟨_, _⟩, _⟩ | ⟨⟨_, _⟩ | ⟨_, _⟩, _⟩) <;>
          

Proof state:
case refine'_2.toFun.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
snd✝¹ : zl
fst✝ : xr
snd✝ : yl
⊢ RightMoves (mk xl xr xL xR * (mk yl yr yL yR * mk zl zr zL zR))

Next tactic:
solve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]
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]
      · 

Proof state:
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
⊢ RightMoves (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)

Next tactic:
rintro (⟨_, ⟨_, _⟩ | ⟨_, _⟩⟩ | ⟨_, ⟨_, _⟩ | ⟨_, _⟩⟩)
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 (⟨_, ⟨_, _⟩ | ⟨_, _⟩⟩ | ⟨_, ⟨_, _⟩ | ⟨_, _⟩⟩) <;>
          

Proof state:
case refine'_2.invFun.inl.mk.inl.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
fst✝¹ : xl
fst✝ : yl
snd✝ : zr
⊢ RightMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)

Next tactic:
solve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]
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 (⟨_, ⟨_, _⟩ | ⟨_, _⟩⟩ | ⟨_, ⟨_, _⟩ | ⟨_, _⟩⟩) <;>
          

Proof state:
case refine'_2.invFun.inl.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
fst✝¹ : xl
fst✝ : yr
snd✝ : zl
⊢ RightMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)

Next tactic:
solve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]
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 (⟨_, ⟨_, _⟩ | ⟨_, _⟩⟩ | ⟨_, ⟨_, _⟩ | ⟨_, _⟩⟩) <;>
          

Proof state:
case refine'_2.invFun.inr.mk.inl.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
fst✝¹ : xr
fst✝ : yl
snd✝ : zl
⊢ RightMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)

Next tactic:
solve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]
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 (⟨_, ⟨_, _⟩ | ⟨_, _⟩⟩ | ⟨_, ⟨_, _⟩ | ⟨_, _⟩⟩) <;>
          

Proof state:
case refine'_2.invFun.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
fst✝¹ : xr
fst✝ : yr
snd✝ : zr
⊢ RightMoves (mk xl xr xL xR * mk yl yr yL yR * mk zl zr zL zR)

Next tactic:
solve_by_elim (config := { maxDepth := 8 }) [Sum.inl, Sum.inr, Prod.mk]
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]
      · 

Proof state:
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
    (fun a =>
      Sum.casesOn a
        (fun val =>
          Prod.casesOn val fun fst snd =>
            Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))
              fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))
        fun val =>
        Prod.casesOn val fun fst snd =>
          Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd)) fun val =>
            Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))
    fun a =>
    Sum.casesOn a
      (fun val =>
        Prod.casesOn val fun fst snd =>
          Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd))) fun val =>
            Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))
      fun val =>
      Prod.casesOn val fun fst snd =>
        Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd))) fun val =>
          Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd))

Next tactic:
rintro (⟨⟨_, _⟩ | ⟨_, _⟩, _⟩ | ⟨⟨_, _⟩ | ⟨_, _⟩, _⟩)
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 (⟨⟨_, _⟩ | ⟨_, _⟩, _⟩ | ⟨⟨_, _⟩ | ⟨_, _⟩, _⟩) <;> 

Proof state:
case refine'_2.left_inv.inl.mk.inl.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
snd✝¹ : zr
fst✝ : xl
snd✝ : yl
⊢ (fun a =>
        Sum.casesOn a
          (fun val =>
            Prod.casesOn val fun fst snd =>
              Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))
                fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))
          fun val =>
          Prod.casesOn val fun fst snd =>
            Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))
              fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))
      ((fun a =>
          Sum.casesOn a
            (fun val =>
              Prod.casesOn val fun fst snd =>
                Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))
                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))
            fun val =>
            Prod.casesOn val fun fst snd =>
              Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))
                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))
        (Sum.inl (Sum.inl (fst✝, snd✝), snd✝¹))) =
    Sum.inl (Sum.inl (fst✝, snd✝), snd✝¹)

Next tactic:
rfl
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 (⟨⟨_, _⟩ | ⟨_, _⟩, _⟩ | ⟨⟨_, _⟩ | ⟨_, _⟩, _⟩) <;> 

Proof state:
case refine'_2.left_inv.inl.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
snd✝¹ : zr
fst✝ : xr
snd✝ : yr
⊢ (fun a =>
        Sum.casesOn a
          (fun val =>
            Prod.casesOn val fun fst snd =>
              Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))
                fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))
          fun val =>
          Prod.casesOn val fun fst snd =>
            Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))
              fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))
      ((fun a =>
          Sum.casesOn a
            (fun val =>
              Prod.casesOn val fun fst snd =>
                Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))
                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))
            fun val =>
            Prod.casesOn val fun fst snd =>
              Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))
                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))
        (Sum.inl (Sum.inr (fst✝, snd✝), snd✝¹))) =
    Sum.inl (Sum.inr (fst✝, snd✝), snd✝¹)

Next tactic:
rfl
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 (⟨⟨_, _⟩ | ⟨_, _⟩, _⟩ | ⟨⟨_, _⟩ | ⟨_, _⟩, _⟩) <;> 

Proof state:
case refine'_2.left_inv.inr.mk.inl.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
snd✝¹ : zl
fst✝ : xl
snd✝ : yr
⊢ (fun a =>
        Sum.casesOn a
          (fun val =>
            Prod.casesOn val fun fst snd =>
              Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))
                fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))
          fun val =>
          Prod.casesOn val fun fst snd =>
            Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))
              fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))
      ((fun a =>
          Sum.casesOn a
            (fun val =>
              Prod.casesOn val fun fst snd =>
                Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))
                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))
            fun val =>
            Prod.casesOn val fun fst snd =>
              Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))
                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))
        (Sum.inr (Sum.inl (fst✝, snd✝), snd✝¹))) =
    Sum.inr (Sum.inl (fst✝, snd✝), snd✝¹)

Next tactic:
rfl
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 (⟨⟨_, _⟩ | ⟨_, _⟩, _⟩ | ⟨⟨_, _⟩ | ⟨_, _⟩, _⟩) <;> 

Proof state:
case refine'_2.left_inv.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
snd✝¹ : zl
fst✝ : xr
snd✝ : yl
⊢ (fun a =>
        Sum.casesOn a
          (fun val =>
            Prod.casesOn val fun fst snd =>
              Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inl (fst, fst_1), snd))
                fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inl (fst, fst_1), snd))
          fun val =>
          Prod.casesOn val fun fst snd =>
            Sum.casesOn snd (fun val => Prod.casesOn val fun fst_1 snd => Sum.inr (Sum.inr (fst, fst_1), snd))
              fun val => Prod.casesOn val fun fst_1 snd => Sum.inl (Sum.inr (fst, fst_1), snd))
      ((fun a =>
          Sum.casesOn a
            (fun val =>
              Prod.casesOn val fun fst snd =>
                Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inl (snd_1, snd)))
                  fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inr (snd_1, snd)))
            fun val =>
            Prod.casesOn val fun fst snd =>
              Sum.casesOn fst (fun val => Prod.casesOn val fun fst snd_1 => Sum.inl (fst, Sum.inr (snd_1, snd)))
                fun val => Prod.casesOn val fun fst snd_1 => Sum.inr (fst, Sum.inl (snd_1, snd)))
        (Sum.inr (Sum.inr (fst✝, snd✝), snd✝¹))) =
    Sum.inr (Sum.inr (fst✝, snd✝), snd✝¹)

Next tactic:
rfl