l3lab/ntp-mathlib huggingface dataset:
@[to_additive]
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {M : Type*} [CommMonoid M]
    {k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
    (mulSingle k u : I → M) * mulSingle l v = mulSingle m u * mulSingle n v ↔
      k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n := by
  refine' ⟨fun h => _, _⟩
  · have hk := congr_fun h k
    have hl := congr_fun h l
    have hm := (congr_fun h m).symm
    have hn := (congr_fun h n).symm
    simp only [mul_apply, mulSingle_apply, if_pos rfl] at hk hl hm hn
    rcases eq_or_ne k m with (rfl | hkm)
    · refine' Or.inl ⟨rfl, not_ne_iff.mp fun hln => (hv _).elim⟩
      rcases eq_or_ne k l with (rfl | hkl)
      · rwa [if_neg hln.symm, if_neg hln.symm, one_mul, one_mul] at hn
      · rwa [if_neg hkl.symm, if_neg hln, one_mul, one_mul] at hl
    · 

Proof state:
case refine'_1.inr
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝¹ : DecidableEq I
M : Type u_3
inst✝ : CommMonoid M
k l m n : I
u v : M
hu : u ≠ 1
hv : v ≠ 1
h : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v
hk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = n then v else 1
hl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = n then v else 1
hm : ((if True then u else 1) * if m = n then v else 1) = (if m = k then u else 1) * if m = l then v else 1
hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1
hkm : k ≠ m
⊢ k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n

Next tactic:
rcases eq_or_ne m n with (rfl | hmn)
@[to_additive]
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {M : Type*} [CommMonoid M]
    {k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
    (mulSingle k u : I → M) * mulSingle l v = mulSingle m u * mulSingle n v ↔
      k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n := by
  refine' ⟨fun h => _, _⟩
  · have hk := congr_fun h k
    have hl := congr_fun h l
    have hm := (congr_fun h m).symm
    have hn := (congr_fun h n).symm
    simp only [mul_apply, mulSingle_apply, if_pos rfl] at hk hl hm hn
    rcases eq_or_ne k m with (rfl | hkm)
    · refine' Or.inl ⟨rfl, not_ne_iff.mp fun hln => (hv _).elim⟩
      rcases eq_or_ne k l with (rfl | hkl)
      · rwa [if_neg hln.symm, if_neg hln.symm, one_mul, one_mul] at hn
      · rwa [if_neg hkl.symm, if_neg hln, one_mul, one_mul] at hl
    · rcases eq_or_ne m n with (rfl | hmn)
      · 

Proof state:
case refine'_1.inr.inl
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝¹ : DecidableEq I
M : Type u_3
inst✝ : CommMonoid M
k l m : I
u v : M
hu : u ≠ 1
hv : v ≠ 1
hkm : k ≠ m
h : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle m v
hk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = m then v else 1
hl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = m then v else 1
hm : ((if True then u else 1) * if m = m then v else 1) = (if m = k then u else 1) * if m = l then v else 1
hn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = l then v else 1
⊢ k = m ∧ l = m ∨ u = v ∧ k = m ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = m

Next tactic:
rcases eq_or_ne k l with (rfl | hkl)
@[to_additive]
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {M : Type*} [CommMonoid M]
    {k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
    (mulSingle k u : I → M) * mulSingle l v = mulSingle m u * mulSingle n v ↔
      k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n := by
  refine' ⟨fun h => _, _⟩
  · have hk := congr_fun h k
    have hl := congr_fun h l
    have hm := (congr_fun h m).symm
    have hn := (congr_fun h n).symm
    simp only [mul_apply, mulSingle_apply, if_pos rfl] at hk hl hm hn
    rcases eq_or_ne k m with (rfl | hkm)
    · refine' Or.inl ⟨rfl, not_ne_iff.mp fun hln => (hv _).elim⟩
      rcases eq_or_ne k l with (rfl | hkl)
      · rwa [if_neg hln.symm, if_neg hln.symm, one_mul, one_mul] at hn
      · rwa [if_neg hkl.symm, if_neg hln, one_mul, one_mul] at hl
    · rcases eq_or_ne m n with (rfl | hmn)
      · rcases eq_or_ne k l with (rfl | hkl)
        · 

Proof state:
case refine'_1.inr.inl.inl
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝¹ : DecidableEq I
M : Type u_3
inst✝ : CommMonoid M
k m : I
u v : M
hu : u ≠ 1
hv : v ≠ 1
hkm : k ≠ m
h : mulSingle k u * mulSingle k v = mulSingle m u * mulSingle m v
hk : ((if True then u else 1) * if k = k then v else 1) = (if k = m then u else 1) * if k = m then v else 1
hl : ((if k = k then u else 1) * if True then v else 1) = (if k = m then u else 1) * if k = m then v else 1
hm : ((if True then u else 1) * if m = m then v else 1) = (if m = k then u else 1) * if m = k then v else 1
hn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = k then v else 1
⊢ k = m ∧ k = m ∨ u = v ∧ k = m ∧ k = m ∨ u * v = 1 ∧ k = k ∧ m = m

Next tactic:
rw [if_neg hkm.symm, if_neg hkm.symm, one_mul, if_pos rfl] at hm
@[to_additive]
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {M : Type*} [CommMonoid M]
    {k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
    (mulSingle k u : I → M) * mulSingle l v = mulSingle m u * mulSingle n v ↔
      k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n := by
  refine' ⟨fun h => _, _⟩
  · have hk := congr_fun h k
    have hl := congr_fun h l
    have hm := (congr_fun h m).symm
    have hn := (congr_fun h n).symm
    simp only [mul_apply, mulSingle_apply, if_pos rfl] at hk hl hm hn
    rcases eq_or_ne k m with (rfl | hkm)
    · refine' Or.inl ⟨rfl, not_ne_iff.mp fun hln => (hv _).elim⟩
      rcases eq_or_ne k l with (rfl | hkl)
      · rwa [if_neg hln.symm, if_neg hln.symm, one_mul, one_mul] at hn
      · rwa [if_neg hkl.symm, if_neg hln, one_mul, one_mul] at hl
    · rcases eq_or_ne m n with (rfl | hmn)
      · rcases eq_or_ne k l with (rfl | hkl)
        · rw [if_neg hkm.symm, if_neg hkm.symm, one_mul, if_pos rfl] at hm
          

Proof state:
case refine'_1.inr.inl.inl
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝¹ : DecidableEq I
M : Type u_3
inst✝ : CommMonoid M
k m : I
u v : M
hu : u ≠ 1
hv : v ≠ 1
hkm : k ≠ m
h : mulSingle k u * mulSingle k v = mulSingle m u * mulSingle m v
hk : ((if True then u else 1) * if k = k then v else 1) = (if k = m then u else 1) * if k = m then v else 1
hl : ((if k = k then u else 1) * if True then v else 1) = (if k = m then u else 1) * if k = m then v else 1
hm : (if True then u else 1) * v = 1
hn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = k then v else 1
⊢ k = m ∧ k = m ∨ u = v ∧ k = m ∧ k = m ∨ u * v = 1 ∧ k = k ∧ m = m

Next tactic:
exact Or.inr (Or.inr ⟨hm, rfl, rfl⟩)
@[to_additive]
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {M : Type*} [CommMonoid M]
    {k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
    (mulSingle k u : I → M) * mulSingle l v = mulSingle m u * mulSingle n v ↔
      k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n := by
  refine' ⟨fun h => _, _⟩
  · have hk := congr_fun h k
    have hl := congr_fun h l
    have hm := (congr_fun h m).symm
    have hn := (congr_fun h n).symm
    simp only [mul_apply, mulSingle_apply, if_pos rfl] at hk hl hm hn
    rcases eq_or_ne k m with (rfl | hkm)
    · refine' Or.inl ⟨rfl, not_ne_iff.mp fun hln => (hv _).elim⟩
      rcases eq_or_ne k l with (rfl | hkl)
      · rwa [if_neg hln.symm, if_neg hln.symm, one_mul, one_mul] at hn
      · rwa [if_neg hkl.symm, if_neg hln, one_mul, one_mul] at hl
    · rcases eq_or_ne m n with (rfl | hmn)
      · rcases eq_or_ne k l with (rfl | hkl)
        · rw [if_neg hkm.symm, if_neg hkm.symm, one_mul, if_pos rfl] at hm
          exact Or.inr (Or.inr ⟨hm, rfl, rfl⟩)
        · 

Proof state:
case refine'_1.inr.inl.inr
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝¹ : DecidableEq I
M : Type u_3
inst✝ : CommMonoid M
k l m : I
u v : M
hu : u ≠ 1
hv : v ≠ 1
hkm : k ≠ m
h : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle m v
hk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = m then v else 1
hl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = m then v else 1
hm : ((if True then u else 1) * if m = m then v else 1) = (if m = k then u else 1) * if m = l then v else 1
hn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = l then v else 1
hkl : k ≠ l
⊢ k = m ∧ l = m ∨ u = v ∧ k = m ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = m

Next tactic:
simp only [if_neg hkm, if_neg hkl, mul_one] at hk
@[to_additive]
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {M : Type*} [CommMonoid M]
    {k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
    (mulSingle k u : I → M) * mulSingle l v = mulSingle m u * mulSingle n v ↔
      k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n := by
  refine' ⟨fun h => _, _⟩
  · have hk := congr_fun h k
    have hl := congr_fun h l
    have hm := (congr_fun h m).symm
    have hn := (congr_fun h n).symm
    simp only [mul_apply, mulSingle_apply, if_pos rfl] at hk hl hm hn
    rcases eq_or_ne k m with (rfl | hkm)
    · refine' Or.inl ⟨rfl, not_ne_iff.mp fun hln => (hv _).elim⟩
      rcases eq_or_ne k l with (rfl | hkl)
      · rwa [if_neg hln.symm, if_neg hln.symm, one_mul, one_mul] at hn
      · rwa [if_neg hkl.symm, if_neg hln, one_mul, one_mul] at hl
    · rcases eq_or_ne m n with (rfl | hmn)
      · rcases eq_or_ne k l with (rfl | hkl)
        · rw [if_neg hkm.symm, if_neg hkm.symm, one_mul, if_pos rfl] at hm
          exact Or.inr (Or.inr ⟨hm, rfl, rfl⟩)
        · simp only [if_neg hkm, if_neg hkl, mul_one] at hk
          

Proof state:
case refine'_1.inr.inl.inr
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝¹ : DecidableEq I
M : Type u_3
inst✝ : CommMonoid M
k l m : I
u v : M
hu : u ≠ 1
hv : v ≠ 1
hkm : k ≠ m
h : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle m v
hl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = m then v else 1
hm : ((if True then u else 1) * if m = m then v else 1) = (if m = k then u else 1) * if m = l then v else 1
hn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = l then v else 1
hkl : k ≠ l
hk : (if True then u else 1) = 1
⊢ k = m ∧ l = m ∨ u = v ∧ k = m ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = m

Next tactic:
dsimp at hk
@[to_additive]
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {M : Type*} [CommMonoid M]
    {k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
    (mulSingle k u : I → M) * mulSingle l v = mulSingle m u * mulSingle n v ↔
      k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n := by
  refine' ⟨fun h => _, _⟩
  · have hk := congr_fun h k
    have hl := congr_fun h l
    have hm := (congr_fun h m).symm
    have hn := (congr_fun h n).symm
    simp only [mul_apply, mulSingle_apply, if_pos rfl] at hk hl hm hn
    rcases eq_or_ne k m with (rfl | hkm)
    · refine' Or.inl ⟨rfl, not_ne_iff.mp fun hln => (hv _).elim⟩
      rcases eq_or_ne k l with (rfl | hkl)
      · rwa [if_neg hln.symm, if_neg hln.symm, one_mul, one_mul] at hn
      · rwa [if_neg hkl.symm, if_neg hln, one_mul, one_mul] at hl
    · rcases eq_or_ne m n with (rfl | hmn)
      · rcases eq_or_ne k l with (rfl | hkl)
        · rw [if_neg hkm.symm, if_neg hkm.symm, one_mul, if_pos rfl] at hm
          exact Or.inr (Or.inr ⟨hm, rfl, rfl⟩)
        · simp only [if_neg hkm, if_neg hkl, mul_one] at hk
          dsimp at hk
          

Proof state:
case refine'_1.inr.inl.inr
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝¹ : DecidableEq I
M : Type u_3
inst✝ : CommMonoid M
k l m : I
u v : M
hu : u ≠ 1
hv : v ≠ 1
hkm : k ≠ m
h : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle m v
hl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = m then v else 1
hm : ((if True then u else 1) * if m = m then v else 1) = (if m = k then u else 1) * if m = l then v else 1
hn : ((if m = m then u else 1) * if True then v else 1) = (if m = k then u else 1) * if m = l then v else 1
hkl : k ≠ l
hk : u = 1
⊢ k = m ∧ l = m ∨ u = v ∧ k = m ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = m

Next tactic:
contradiction
@[to_additive]
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {M : Type*} [CommMonoid M]
    {k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
    (mulSingle k u : I → M) * mulSingle l v = mulSingle m u * mulSingle n v ↔
      k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n := by
  refine' ⟨fun h => _, _⟩
  · have hk := congr_fun h k
    have hl := congr_fun h l
    have hm := (congr_fun h m).symm
    have hn := (congr_fun h n).symm
    simp only [mul_apply, mulSingle_apply, if_pos rfl] at hk hl hm hn
    rcases eq_or_ne k m with (rfl | hkm)
    · refine' Or.inl ⟨rfl, not_ne_iff.mp fun hln => (hv _).elim⟩
      rcases eq_or_ne k l with (rfl | hkl)
      · rwa [if_neg hln.symm, if_neg hln.symm, one_mul, one_mul] at hn
      · rwa [if_neg hkl.symm, if_neg hln, one_mul, one_mul] at hl
    · rcases eq_or_ne m n with (rfl | hmn)
      · rcases eq_or_ne k l with (rfl | hkl)
        · rw [if_neg hkm.symm, if_neg hkm.symm, one_mul, if_pos rfl] at hm
          exact Or.inr (Or.inr ⟨hm, rfl, rfl⟩)
        · simp only [if_neg hkm, if_neg hkl, mul_one] at hk
          dsimp at hk
          contradiction
      · 

Proof state:
case refine'_1.inr.inr
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝¹ : DecidableEq I
M : Type u_3
inst✝ : CommMonoid M
k l m n : I
u v : M
hu : u ≠ 1
hv : v ≠ 1
h : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v
hk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = n then v else 1
hl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = n then v else 1
hm : ((if True then u else 1) * if m = n then v else 1) = (if m = k then u else 1) * if m = l then v else 1
hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1
hkm : k ≠ m
hmn : m ≠ n
⊢ k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n

Next tactic:
rw [if_neg hkm.symm, if_neg hmn, one_mul, mul_one] at hm
@[to_additive]
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {M : Type*} [CommMonoid M]
    {k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
    (mulSingle k u : I → M) * mulSingle l v = mulSingle m u * mulSingle n v ↔
      k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n := by
  refine' ⟨fun h => _, _⟩
  · have hk := congr_fun h k
    have hl := congr_fun h l
    have hm := (congr_fun h m).symm
    have hn := (congr_fun h n).symm
    simp only [mul_apply, mulSingle_apply, if_pos rfl] at hk hl hm hn
    rcases eq_or_ne k m with (rfl | hkm)
    · refine' Or.inl ⟨rfl, not_ne_iff.mp fun hln => (hv _).elim⟩
      rcases eq_or_ne k l with (rfl | hkl)
      · rwa [if_neg hln.symm, if_neg hln.symm, one_mul, one_mul] at hn
      · rwa [if_neg hkl.symm, if_neg hln, one_mul, one_mul] at hl
    · rcases eq_or_ne m n with (rfl | hmn)
      · rcases eq_or_ne k l with (rfl | hkl)
        · rw [if_neg hkm.symm, if_neg hkm.symm, one_mul, if_pos rfl] at hm
          exact Or.inr (Or.inr ⟨hm, rfl, rfl⟩)
        · simp only [if_neg hkm, if_neg hkl, mul_one] at hk
          dsimp at hk
          contradiction
      · rw [if_neg hkm.symm, if_neg hmn, one_mul, mul_one] at hm
        

Proof state:
case refine'_1.inr.inr
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝¹ : DecidableEq I
M : Type u_3
inst✝ : CommMonoid M
k l m n : I
u v : M
hu : u ≠ 1
hv : v ≠ 1
h : mulSingle k u * mulSingle l v = mulSingle m u * mulSingle n v
hk : ((if True then u else 1) * if k = l then v else 1) = (if k = m then u else 1) * if k = n then v else 1
hl : ((if l = k then u else 1) * if True then v else 1) = (if l = m then u else 1) * if l = n then v else 1
hm : (if True then u else 1) = if m = l then v else 1
hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = l then v else 1
hkm : k ≠ m
hmn : m ≠ n
⊢ k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n

Next tactic:
obtain rfl := (ite_ne_right_iff.mp (ne_of_eq_of_ne hm.symm hu)).1
@[to_additive]
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {M : Type*} [CommMonoid M]
    {k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
    (mulSingle k u : I → M) * mulSingle l v = mulSingle m u * mulSingle n v ↔
      k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n := by
  refine' ⟨fun h => _, _⟩
  · have hk := congr_fun h k
    have hl := congr_fun h l
    have hm := (congr_fun h m).symm
    have hn := (congr_fun h n).symm
    simp only [mul_apply, mulSingle_apply, if_pos rfl] at hk hl hm hn
    rcases eq_or_ne k m with (rfl | hkm)
    · refine' Or.inl ⟨rfl, not_ne_iff.mp fun hln => (hv _).elim⟩
      rcases eq_or_ne k l with (rfl | hkl)
      · rwa [if_neg hln.symm, if_neg hln.symm, one_mul, one_mul] at hn
      · rwa [if_neg hkl.symm, if_neg hln, one_mul, one_mul] at hl
    · rcases eq_or_ne m n with (rfl | hmn)
      · rcases eq_or_ne k l with (rfl | hkl)
        · rw [if_neg hkm.symm, if_neg hkm.symm, one_mul, if_pos rfl] at hm
          exact Or.inr (Or.inr ⟨hm, rfl, rfl⟩)
        · simp only [if_neg hkm, if_neg hkl, mul_one] at hk
          dsimp at hk
          contradiction
      · rw [if_neg hkm.symm, if_neg hmn, one_mul, mul_one] at hm
        obtain rfl := (ite_ne_right_iff.mp (ne_of_eq_of_ne hm.symm hu)).1
        

Proof state:
case refine'_1.inr.inr
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝¹ : DecidableEq I
M : Type u_3
inst✝ : CommMonoid M
k m n : I
u v : M
hu : u ≠ 1
hv : v ≠ 1
hkm : k ≠ m
hmn : m ≠ n
h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v
hk : ((if True then u else 1) * if k = m then v else 1) = (if k = m then u else 1) * if k = n then v else 1
hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1
hm : (if True then u else 1) = if m = m then v else 1
hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1
⊢ k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n

Next tactic:
rw [if_neg hkm, if_neg hkm, one_mul, mul_one] at hk
@[to_additive]
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {M : Type*} [CommMonoid M]
    {k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
    (mulSingle k u : I → M) * mulSingle l v = mulSingle m u * mulSingle n v ↔
      k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n := by
  refine' ⟨fun h => _, _⟩
  · have hk := congr_fun h k
    have hl := congr_fun h l
    have hm := (congr_fun h m).symm
    have hn := (congr_fun h n).symm
    simp only [mul_apply, mulSingle_apply, if_pos rfl] at hk hl hm hn
    rcases eq_or_ne k m with (rfl | hkm)
    · refine' Or.inl ⟨rfl, not_ne_iff.mp fun hln => (hv _).elim⟩
      rcases eq_or_ne k l with (rfl | hkl)
      · rwa [if_neg hln.symm, if_neg hln.symm, one_mul, one_mul] at hn
      · rwa [if_neg hkl.symm, if_neg hln, one_mul, one_mul] at hl
    · rcases eq_or_ne m n with (rfl | hmn)
      · rcases eq_or_ne k l with (rfl | hkl)
        · rw [if_neg hkm.symm, if_neg hkm.symm, one_mul, if_pos rfl] at hm
          exact Or.inr (Or.inr ⟨hm, rfl, rfl⟩)
        · simp only [if_neg hkm, if_neg hkl, mul_one] at hk
          dsimp at hk
          contradiction
      · rw [if_neg hkm.symm, if_neg hmn, one_mul, mul_one] at hm
        obtain rfl := (ite_ne_right_iff.mp (ne_of_eq_of_ne hm.symm hu)).1
        rw [if_neg hkm, if_neg hkm, one_mul, mul_one] at hk
        

Proof state:
case refine'_1.inr.inr
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝¹ : DecidableEq I
M : Type u_3
inst✝ : CommMonoid M
k m n : I
u v : M
hu : u ≠ 1
hv : v ≠ 1
hkm : k ≠ m
hmn : m ≠ n
h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle n v
hk : (if True then u else 1) = if k = n then v else 1
hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = n then v else 1
hm : (if True then u else 1) = if m = m then v else 1
hn : ((if n = m then u else 1) * if True then v else 1) = (if n = k then u else 1) * if n = m then v else 1
⊢ k = m ∧ m = n ∨ u = v ∧ k = n ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = n

Next tactic:
obtain rfl := (ite_ne_right_iff.mp (ne_of_eq_of_ne hk.symm hu)).1
@[to_additive]
theorem Pi.mulSingle_mul_mulSingle_eq_mulSingle_mul_mulSingle {M : Type*} [CommMonoid M]
    {k l m n : I} {u v : M} (hu : u ≠ 1) (hv : v ≠ 1) :
    (mulSingle k u : I → M) * mulSingle l v = mulSingle m u * mulSingle n v ↔
      k = m ∧ l = n ∨ u = v ∧ k = n ∧ l = m ∨ u * v = 1 ∧ k = l ∧ m = n := by
  refine' ⟨fun h => _, _⟩
  · have hk := congr_fun h k
    have hl := congr_fun h l
    have hm := (congr_fun h m).symm
    have hn := (congr_fun h n).symm
    simp only [mul_apply, mulSingle_apply, if_pos rfl] at hk hl hm hn
    rcases eq_or_ne k m with (rfl | hkm)
    · refine' Or.inl ⟨rfl, not_ne_iff.mp fun hln => (hv _).elim⟩
      rcases eq_or_ne k l with (rfl | hkl)
      · rwa [if_neg hln.symm, if_neg hln.symm, one_mul, one_mul] at hn
      · rwa [if_neg hkl.symm, if_neg hln, one_mul, one_mul] at hl
    · rcases eq_or_ne m n with (rfl | hmn)
      · rcases eq_or_ne k l with (rfl | hkl)
        · rw [if_neg hkm.symm, if_neg hkm.symm, one_mul, if_pos rfl] at hm
          exact Or.inr (Or.inr ⟨hm, rfl, rfl⟩)
        · simp only [if_neg hkm, if_neg hkl, mul_one] at hk
          dsimp at hk
          contradiction
      · rw [if_neg hkm.symm, if_neg hmn, one_mul, mul_one] at hm
        obtain rfl := (ite_ne_right_iff.mp (ne_of_eq_of_ne hm.symm hu)).1
        rw [if_neg hkm, if_neg hkm, one_mul, mul_one] at hk
        obtain rfl := (ite_ne_right_iff.mp (ne_of_eq_of_ne hk.symm hu)).1
        

Proof state:
case refine'_1.inr.inr
ι : Type u_1
α : Type u_2
I : Type u
f : I → Type v
x y : (i : I) → f i
i j : I
inst✝¹ : DecidableEq I
M : Type u_3
inst✝ : CommMonoid M
k m : I
u v : M
hu : u ≠ 1
hv : v ≠ 1
hkm : k ≠ m
hm : (if True then u else 1) = if m = m then v else 1
hmn : m ≠ k
h : mulSingle k u * mulSingle m v = mulSingle m u * mulSingle k v
hk : (if True then u else 1) = if k = k then v else 1
hl : ((if m = k then u else 1) * if True then v else 1) = (if m = m then u else 1) * if m = k then v else 1
hn : ((if k = m then u else 1) * if True then v else 1) = (if k = k then u else 1) * if k = m then v else 1
⊢ k = m ∧ m = k ∨ u = v ∧ k = k ∧ m = m ∨ u * v = 1 ∧ k = m ∧ m = k

Next tactic:
exact Or.inr (Or.inl ⟨hk.trans (if_pos rfl), rfl, rfl⟩)