l3lab/ntp-mathlib huggingface dataset:
/-- If all prime ideals in a commutative ring are principal, so are all other ideals. -/
theorem IsPrincipalIdealRing.of_prime (H : ∀ P : Ideal R, P.IsPrime → P.IsPrincipal) :
    IsPrincipalIdealRing R := by
  -- Suppose the set of `nonPrincipals` is not empty.
  rw [← nonPrincipals_eq_empty_iff, Set.eq_empty_iff_forall_not_mem]
  intro J hJ
  -- We will show a maximal element `I ∈ nonPrincipals R` (which exists by Zorn) is prime.
  obtain ⟨I, Ibad, -, Imax⟩ := zorn_nonempty_partialOrder₀ (nonPrincipals R) nonPrincipals_zorn _ hJ
  have Imax' : ∀ {J}, I < J → J.IsPrincipal := by
    intro J hJ
    by_contra He
    exact hJ.ne (Imax _ ((nonPrincipals_def R).2 He) hJ.le).symm
  by_cases hI1 : I = ⊤
  · subst hI1
    exact Ibad top_isPrincipal
  -- Let `x y : R` with `x * y ∈ I` and suppose WLOG `y ∉ I`.
  refine' Ibad (H I ⟨hI1, fun {x y} hxy => or_iff_not_imp_right.mpr fun hy => _⟩)
  obtain ⟨a, ha⟩ : (I ⊔ span {y}).IsPrincipal :=
    Imax' (left_lt_sup.mpr (mt I.span_singleton_le_iff_mem.mp hy))
  -- Then `x ∈ I.colon (span {y})`, which is equal to `I` if it's not principal.
  suffices He : ¬(I.colon (span {y})).IsPrincipal
  · rw [← Imax _ ((nonPrincipals_def R).2 He) fun a ha =>
        Ideal.mem_colon_singleton.2 (mul_mem_right _ _ ha)]
    exact Ideal.mem_colon_singleton.2 hxy
  -- So suppose for the sake of contradiction that both `I ⊔ span {y}` and `I.colon (span {y})`
  -- are principal.
  rintro ⟨b, hb⟩
  -- We will show `I` is generated by `a * b`.
  refine (nonPrincipals_def _).1 Ibad ⟨a * b, ?_⟩
  refine'
    le_antisymm (α := Ideal R) (fun i hi => _) <| (span_singleton_mul_span_singleton a b).ge.trans _
  · have hisup : i ∈ I ⊔ span {y} := Ideal.mem_sup_left hi
    have : y ∈ I ⊔ span {y} := Ideal.mem_sup_right (Ideal.mem_span_singleton_self y)
    erw [ha, mem_span_singleton'] at hisup this
    obtain ⟨v, rfl⟩ := this
    obtain ⟨u, rfl⟩ := hisup
    have hucolon : u ∈ I.colon (span {v * a}) := by
      rw [Ideal.mem_colon_singleton, mul_comm v, ← mul_assoc]
      exact mul_mem_right _ _ hi
    erw [hb, mem_span_singleton'] at hucolon
    obtain ⟨z, rfl⟩ := hucolon
    exact mem_span_singleton'.2 ⟨z, by ring⟩
  · rw [← Ideal.submodule_span_eq, ← ha, Ideal.sup_mul, sup_le_iff,
      span_singleton_mul_span_singleton, mul_comm y, Ideal.span_singleton_le_iff_mem]
    

Proof state:
case He.mk.intro.refine'_2
R : Type u
M : Type v
inst✝ : CommRing R
H : ∀ (P : Ideal R), IsPrime P → IsPrincipal P
J : Ideal R
hJ : J ∈ nonPrincipals R
I : Ideal R
Ibad : I ∈ nonPrincipals R
Imax : ∀ z ∈ nonPrincipals R, I ≤ z → z = I
Imax' : ∀ {J : Ideal R}, I < J → IsPrincipal J
hI1 : ¬I = ⊤
x y : R
hxy : x * y ∈ I
hy : y ∉ I
a : R
ha : I ⊔ Ideal.span {y} = Submodule.span R {a}
b : R
hb : colon I (Ideal.span {y}) = Submodule.span R {b}
⊢ I * Ideal.span {b} ≤ I ∧ b * y ∈ I

Next tactic:
exact ⟨mul_le_right, Ideal.mem_colon_singleton.1 <| hb.symm ▸ Ideal.mem_span_singleton_self b⟩
@[to_additive]
lemma orderOf_eq_two_iff (hG : Monoid.exponent G = 2) {x : G} :
    orderOf x = 2 ↔ x ≠ 1 :=
  ⟨by 

Proof state:
G : Type u_1
inst✝ : Group G
hG : Monoid.exponent G = 2
x : G
⊢ orderOf x = 2 → x ≠ 1

Next tactic:
rintro hx rfl
@[to_additive]
lemma orderOf_eq_two_iff (hG : Monoid.exponent G = 2) {x : G} :
    orderOf x = 2 ↔ x ≠ 1 :=
  ⟨by rintro hx rfl; 

Proof state:
G : Type u_1
inst✝ : Group G
hG : Monoid.exponent G = 2
hx : orderOf 1 = 2
⊢ False

Next tactic:
norm_num at hx
/-- In a group of exponent two, all elements commute. -/
@[to_additive]
lemma mul_comm_of_exponent_two (hG : Monoid.exponent G = 2) (x y : G) :
    x * y = y * x := by
  

Proof state:
G : Type u_1
inst✝ : Group G
hG : Monoid.exponent G = 2
x y : G
⊢ x * y = y * x

Next tactic:
simpa only [inv_eq_self_of_exponent_two hG] using mul_inv_rev x y
@[to_additive]
lemma mul_not_mem_of_orderOf_eq_two {G : Type*} [Group G] {x y : G} (hx : orderOf x = 2)
    (hy : orderOf y = 2) (hxy : x ≠ y) : x * y ∉ ({x, y, 1} : Set G) := by
  

Proof state:
G✝ : Type u_1
inst✝¹ : Group G✝
G : Type u_2
inst✝ : Group G
x y : G
hx : orderOf x = 2
hy : orderOf y = 2
hxy : x ≠ y
⊢ x * y ∉ {x, y, 1}

Next tactic:
simp only [Set.mem_singleton_iff, Set.mem_insert_iff, mul_right_eq_self, mul_left_eq_self,
    mul_eq_one_iff_eq_inv, inv_eq_self_of_orderOf_eq_two hy, not_or]
@[to_additive]
lemma mul_not_mem_of_orderOf_eq_two {G : Type*} [Group G] {x y : G} (hx : orderOf x = 2)
    (hy : orderOf y = 2) (hxy : x ≠ y) : x * y ∉ ({x, y, 1} : Set G) := by
  simp only [Set.mem_singleton_iff, Set.mem_insert_iff, mul_right_eq_self, mul_left_eq_self,
    mul_eq_one_iff_eq_inv, inv_eq_self_of_orderOf_eq_two hy, not_or]
  

Proof state:
G✝ : Type u_1
inst✝¹ : Group G✝
G : Type u_2
inst✝ : Group G
x y : G
hx : orderOf x = 2
hy : orderOf y = 2
hxy : x ≠ y
⊢ ¬y = 1 ∧ ¬x = 1 ∧ ¬x = y

Next tactic:
aesop
instance : IsAddKleinFour (ZMod 2 × ZMod 2) where
  card_four := by 

Proof state:
⊢ Nat.card (ZMod 2 × ZMod 2) = 4

Next tactic:
simp
instance : IsAddKleinFour (ZMod 2 × ZMod 2) where
  card_four := by simp
  exponent_two := by 

Proof state:
⊢ AddMonoid.exponent (ZMod 2 × ZMod 2) = 2

Next tactic:
simp [AddMonoid.exponent_prod]
instance : IsKleinFour (DihedralGroup 2) where
  card_four := by 

Proof state:
⊢ Nat.card (DihedralGroup 2) = 4

Next tactic:
simp only [Nat.card_eq_fintype_card]
instance : IsKleinFour (DihedralGroup 2) where
  card_four := by simp only [Nat.card_eq_fintype_card]; 

Proof state:
⊢ Fintype.card (DihedralGroup 2) = 4

Next tactic:
rfl
instance : IsKleinFour (DihedralGroup 2) where
  card_four := by simp only [Nat.card_eq_fintype_card]; rfl
  exponent_two := by 

Proof state:
⊢ Monoid.exponent (DihedralGroup 2) = 2

Next tactic:
simp [DihedralGroup.exponent]
instance {G : Type*} [Group G] [IsKleinFour G] :
    IsAddKleinFour (Additive G) where
  card_four := by 

Proof state:
G : Type u_1
inst✝¹ : Group G
inst✝ : IsKleinFour G
⊢ Nat.card (Additive G) = 4

Next tactic:
rw [← IsKleinFour.card_four (G := G)]
instance {G : Type*} [Group G] [IsKleinFour G] :
    IsAddKleinFour (Additive G) where
  card_four := by rw [← IsKleinFour.card_four (G := G)]; 

Proof state:
G : Type u_1
inst✝¹ : Group G
inst✝ : IsKleinFour G
⊢ Nat.card (Additive G) = Nat.card G

Next tactic:
congr!
instance {G : Type*} [Group G] [IsKleinFour G] :
    IsAddKleinFour (Additive G) where
  card_four := by rw [← IsKleinFour.card_four (G := G)]; congr!
  exponent_two := by 

Proof state:
G : Type u_1
inst✝¹ : Group G
inst✝ : IsKleinFour G
⊢ AddMonoid.exponent (Additive G) = 2

Next tactic:
simp
instance {G : Type*} [AddGroup G] [IsAddKleinFour G] :
    IsKleinFour (Multiplicative G) where
  card_four := by 

Proof state:
G : Type u_1
inst✝¹ : AddGroup G
inst✝ : IsAddKleinFour G
⊢ Nat.card (Multiplicative G) = 4

Next tactic:
rw [← IsAddKleinFour.card_four (G := G)]
instance {G : Type*} [AddGroup G] [IsAddKleinFour G] :
    IsKleinFour (Multiplicative G) where
  card_four := by rw [← IsAddKleinFour.card_four (G := G)]; 

Proof state:
G : Type u_1
inst✝¹ : AddGroup G
inst✝ : IsAddKleinFour G
⊢ Nat.card (Multiplicative G) = Nat.card G

Next tactic:
congr!
instance {G : Type*} [AddGroup G] [IsAddKleinFour G] :
    IsKleinFour (Multiplicative G) where
  card_four := by rw [← IsAddKleinFour.card_four (G := G)]; congr!
  exponent_two := by 

Proof state:
G : Type u_1
inst✝¹ : AddGroup G
inst✝ : IsAddKleinFour G
⊢ Monoid.exponent (Multiplicative G) = 2

Next tactic:
simp
@[to_additive]
instance instFinite {G : Type*} [Group G] [IsKleinFour G] : Finite G :=
  Nat.finite_of_card_ne_zero <| by 

Proof state:
G : Type u_1
inst✝¹ : Group G
inst✝ : IsKleinFour G
⊢ Nat.card G ≠ 0

Next tactic:
norm_num [IsKleinFour.card_four]
@[to_additive]
lemma not_isCyclic : ¬ IsCyclic G :=
  fun h ↦ by 

Proof state:
G : Type u_1
inst✝¹ : Group G
inst✝ : IsKleinFour G
h : IsCyclic G
⊢ False

Next tactic:
let _inst := Fintype.ofFinite G
@[to_additive]
lemma not_isCyclic : ¬ IsCyclic G :=
  fun h ↦ by let _inst := Fintype.ofFinite G; 

Proof state:
G : Type u_1
inst✝¹ : Group G
inst✝ : IsKleinFour G
h : IsCyclic G
_inst : Fintype G := Fintype.ofFinite G
⊢ False

Next tactic:
simpa using h.exponent_eq_card
@[to_additive]
lemma inv_eq_self (x : G) : x⁻¹ = x := inv_eq_self_of_exponent_two (by 

Proof state:
G : Type u_1
inst✝¹ : Group G
inst✝ : IsKleinFour G
x : G
⊢ Monoid.exponent G = 2

Next tactic:
simp
@[to_additive]
lemma mul_self (x : G) : x * x = 1 := by
  

Proof state:
G : Type u_1
inst✝¹ : Group G
inst✝ : IsKleinFour G
x : G
⊢ x * x = 1

Next tactic:
rw [mul_eq_one_iff_eq_inv, inv_eq_self]
@[to_additive]
lemma eq_finset_univ [Fintype G] [DecidableEq G]
    {x y : G} (hx : x ≠ 1) (hy : y ≠ 1) (hxy : x ≠ y) : {x * y, x, y, (1 : G)} = Finset.univ := by
  

Proof state:
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ {x * y, x, y, 1} = univ

Next tactic:
apply Finset.eq_univ_of_card
@[to_additive]
lemma eq_finset_univ [Fintype G] [DecidableEq G]
    {x y : G} (hx : x ≠ 1) (hy : y ≠ 1) (hxy : x ≠ y) : {x * y, x, y, (1 : G)} = Finset.univ := by
  apply Finset.eq_univ_of_card
  

Proof state:
case hs
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ card {x * y, x, y, 1} = Fintype.card G

Next tactic:
rw [card_four']
@[to_additive]
lemma eq_finset_univ [Fintype G] [DecidableEq G]
    {x y : G} (hx : x ≠ 1) (hy : y ≠ 1) (hxy : x ≠ y) : {x * y, x, y, (1 : G)} = Finset.univ := by
  apply Finset.eq_univ_of_card
  rw [card_four']
  

Proof state:
case hs
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ card {x * y, x, y, 1} = 4

Next tactic:
repeat rw [card_insert_of_not_mem]
@[to_additive]
lemma eq_finset_univ [Fintype G] [DecidableEq G]
    {x y : G} (hx : x ≠ 1) (hy : y ≠ 1) (hxy : x ≠ y) : {x * y, x, y, (1 : G)} = Finset.univ := by
  apply Finset.eq_univ_of_card
  rw [card_four']
  repeat 

Proof state:
case hs
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ card {x * y, x, y, 1} = 4

Next tactic:
rw [card_insert_of_not_mem]
@[to_additive]
lemma eq_finset_univ [Fintype G] [DecidableEq G]
    {x y : G} (hx : x ≠ 1) (hy : y ≠ 1) (hxy : x ≠ y) : {x * y, x, y, (1 : G)} = Finset.univ := by
  apply Finset.eq_univ_of_card
  rw [card_four']
  repeat 

Proof state:
case hs
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ card {x, y, 1} + 1 = 4
case hs
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ x * y ∉ {x, y, 1}

Next tactic:
rw [card_insert_of_not_mem]
@[to_additive]
lemma eq_finset_univ [Fintype G] [DecidableEq G]
    {x y : G} (hx : x ≠ 1) (hy : y ≠ 1) (hxy : x ≠ y) : {x * y, x, y, (1 : G)} = Finset.univ := by
  apply Finset.eq_univ_of_card
  rw [card_four']
  repeat 

Proof state:
case hs
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ card {y, 1} + 1 + 1 = 4
case hs
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ x ∉ {y, 1}
case hs
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ x * y ∉ {x, y, 1}

Next tactic:
rw [card_insert_of_not_mem]
@[to_additive]
lemma eq_finset_univ [Fintype G] [DecidableEq G]
    {x y : G} (hx : x ≠ 1) (hy : y ≠ 1) (hxy : x ≠ y) : {x * y, x, y, (1 : G)} = Finset.univ := by
  apply Finset.eq_univ_of_card
  rw [card_four']
  repeat 

Proof state:
case hs
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ card {1} + 1 + 1 + 1 = 4
case hs
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ y ∉ {1}
case hs
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ x ∉ {y, 1}
case hs
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ x * y ∉ {x, y, 1}

Next tactic:
rw [card_insert_of_not_mem]
@[to_additive]
lemma eq_finset_univ [Fintype G] [DecidableEq G]
    {x y : G} (hx : x ≠ 1) (hy : y ≠ 1) (hxy : x ≠ y) : {x * y, x, y, (1 : G)} = Finset.univ := by
  apply Finset.eq_univ_of_card
  rw [card_four']
  repeat rw [card_insert_of_not_mem]
  

Proof state:
case hs
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ card {1} + 1 + 1 + 1 = 4
case hs
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ y ∉ {1}
case hs
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ x ∉ {y, 1}
case hs
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ x * y ∉ {x, y, 1}

Next tactic:
on_goal 4 => simpa using mul_not_mem_of_exponent_two (by simp) hx hy hxy
@[to_additive]
lemma eq_finset_univ [Fintype G] [DecidableEq G]
    {x y : G} (hx : x ≠ 1) (hy : y ≠ 1) (hxy : x ≠ y) : {x * y, x, y, (1 : G)} = Finset.univ := by
  apply Finset.eq_univ_of_card
  rw [card_four']
  repeat rw [card_insert_of_not_mem]
  on_goal 4 => 

Proof state:
case hs
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ x * y ∉ {x, y, 1}

Next tactic:
simpa using mul_not_mem_of_exponent_two (by simp) hx hy hxy
@[to_additive]
lemma eq_finset_univ [Fintype G] [DecidableEq G]
    {x y : G} (hx : x ≠ 1) (hy : y ≠ 1) (hxy : x ≠ y) : {x * y, x, y, (1 : G)} = Finset.univ := by
  apply Finset.eq_univ_of_card
  rw [card_four']
  repeat rw [card_insert_of_not_mem]
  on_goal 4 => simpa using mul_not_mem_of_exponent_two (by 

Proof state:
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ Monoid.exponent G = 2

Next tactic:
simp
@[to_additive]
lemma eq_finset_univ [Fintype G] [DecidableEq G]
    {x y : G} (hx : x ≠ 1) (hy : y ≠ 1) (hxy : x ≠ y) : {x * y, x, y, (1 : G)} = Finset.univ := by
  apply Finset.eq_univ_of_card
  rw [card_four']
  repeat rw [card_insert_of_not_mem]
  on_goal 4 => simpa using mul_not_mem_of_exponent_two (by simp) hx hy hxy
  

Proof state:
case hs
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ card {1} + 1 + 1 + 1 = 4
case hs
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ y ∉ {1}
case hs
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ x ∉ {y, 1}

Next tactic:
all_goals aesop
@[to_additive]
lemma eq_finset_univ [Fintype G] [DecidableEq G]
    {x y : G} (hx : x ≠ 1) (hy : y ≠ 1) (hxy : x ≠ y) : {x * y, x, y, (1 : G)} = Finset.univ := by
  apply Finset.eq_univ_of_card
  rw [card_four']
  repeat rw [card_insert_of_not_mem]
  on_goal 4 => simpa using mul_not_mem_of_exponent_two (by simp) hx hy hxy
  all_goals 

Proof state:
case hs
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ card {1} + 1 + 1 + 1 = 4

Next tactic:
aesop
@[to_additive]
lemma eq_finset_univ [Fintype G] [DecidableEq G]
    {x y : G} (hx : x ≠ 1) (hy : y ≠ 1) (hxy : x ≠ y) : {x * y, x, y, (1 : G)} = Finset.univ := by
  apply Finset.eq_univ_of_card
  rw [card_four']
  repeat rw [card_insert_of_not_mem]
  on_goal 4 => simpa using mul_not_mem_of_exponent_two (by simp) hx hy hxy
  all_goals 

Proof state:
case hs
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ y ∉ {1}

Next tactic:
aesop
@[to_additive]
lemma eq_finset_univ [Fintype G] [DecidableEq G]
    {x y : G} (hx : x ≠ 1) (hy : y ≠ 1) (hxy : x ≠ y) : {x * y, x, y, (1 : G)} = Finset.univ := by
  apply Finset.eq_univ_of_card
  rw [card_four']
  repeat rw [card_insert_of_not_mem]
  on_goal 4 => simpa using mul_not_mem_of_exponent_two (by simp) hx hy hxy
  all_goals 

Proof state:
case hs
G : Type u_1
inst✝³ : Group G
inst✝² : IsKleinFour G
inst✝¹ : Fintype G
inst✝ : DecidableEq G
x y : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
⊢ x ∉ {y, 1}

Next tactic:
aesop
@[to_additive]
lemma eq_mul_of_ne_all {x y z : G} (hx : x ≠ 1)
    (hy : y ≠ 1) (hxy : x ≠ y) (hz : z ≠ 1) (hzx : z ≠ x) (hzy : z ≠ y) : z = x * y := by
  

Proof state:
G : Type u_1
inst✝¹ : Group G
inst✝ : IsKleinFour G
x y z : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
hz : z ≠ 1
hzx : z ≠ x
hzy : z ≠ y
⊢ z = x * y

Next tactic:
classical
  let _ := Fintype.ofFinite G
  apply eq_of_not_mem_of_mem_insert <| (eq_finset_univ hx hy hxy).symm ▸ mem_univ _
  simpa only [mem_singleton, mem_insert, not_or] using ⟨hzx, hzy, hz⟩
@[to_additive]
lemma eq_mul_of_ne_all {x y z : G} (hx : x ≠ 1)
    (hy : y ≠ 1) (hxy : x ≠ y) (hz : z ≠ 1) (hzx : z ≠ x) (hzy : z ≠ y) : z = x * y := by
  classical
  

Proof state:
G : Type u_1
inst✝¹ : Group G
inst✝ : IsKleinFour G
x y z : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
hz : z ≠ 1
hzx : z ≠ x
hzy : z ≠ y
⊢ z = x * y

Next tactic:
let _ := Fintype.ofFinite G
@[to_additive]
lemma eq_mul_of_ne_all {x y z : G} (hx : x ≠ 1)
    (hy : y ≠ 1) (hxy : x ≠ y) (hz : z ≠ 1) (hzx : z ≠ x) (hzy : z ≠ y) : z = x * y := by
  classical
  let _ := Fintype.ofFinite G
  

Proof state:
G : Type u_1
inst✝¹ : Group G
inst✝ : IsKleinFour G
x y z : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
hz : z ≠ 1
hzx : z ≠ x
hzy : z ≠ y
x✝ : Fintype G := Fintype.ofFinite G
⊢ z = x * y

Next tactic:
apply eq_of_not_mem_of_mem_insert <| (eq_finset_univ hx hy hxy).symm ▸ mem_univ _
@[to_additive]
lemma eq_mul_of_ne_all {x y z : G} (hx : x ≠ 1)
    (hy : y ≠ 1) (hxy : x ≠ y) (hz : z ≠ 1) (hzx : z ≠ x) (hzy : z ≠ y) : z = x * y := by
  classical
  let _ := Fintype.ofFinite G
  apply eq_of_not_mem_of_mem_insert <| (eq_finset_univ hx hy hxy).symm ▸ mem_univ _
  

Proof state:
G : Type u_1
inst✝¹ : Group G
inst✝ : IsKleinFour G
x y z : G
hx : x ≠ 1
hy : y ≠ 1
hxy : x ≠ y
hz : z ≠ 1
hzx : z ≠ x
hzy : z ≠ y
x✝ : Fintype G := Fintype.ofFinite G
⊢ z ∉ {x, y, 1}

Next tactic:
simpa only [mem_singleton, mem_insert, not_or] using ⟨hzx, hzy, hz⟩
/-- An equivalence between an `IsKleinFour` group `G₁` and a group `G₂` of exponent two which sends
`1 : G₁` to `1 : G₂` is in fact an isomorphism. -/
@[to_additive "An equivalence between an `IsAddKleinFour` group `G₁` and a group `G₂` of exponent
two which sends `0 : G₁` to `0 : G₂` is in fact an isomorphism."]
def mulEquiv' (e : G₁ ≃ G₂) (he : e 1 = 1) (h : Monoid.exponent G₂ = 2) : G₁ ≃* G₂ where
  toEquiv := e
  map_mul' := by
    

Proof state:
G : Type u_1
inst✝⁴ : Group G
inst✝³ : IsKleinFour G
G₁ : Type u_2
G₂ : Type u_3
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : IsKleinFour G₁
e : G₁ ≃ G₂
he : e 1 = 1
h : Monoid.exponent G₂ = 2
⊢ ∀ (x y : G₁), Equiv.toFun e (x * y) = Equiv.toFun e x * Equiv.toFun e y

Next tactic:
let _inst₁ := Fintype.ofFinite G₁
/-- An equivalence between an `IsKleinFour` group `G₁` and a group `G₂` of exponent two which sends
`1 : G₁` to `1 : G₂` is in fact an isomorphism. -/
@[to_additive "An equivalence between an `IsAddKleinFour` group `G₁` and a group `G₂` of exponent
two which sends `0 : G₁` to `0 : G₂` is in fact an isomorphism."]
def mulEquiv' (e : G₁ ≃ G₂) (he : e 1 = 1) (h : Monoid.exponent G₂ = 2) : G₁ ≃* G₂ where
  toEquiv := e
  map_mul' := by
    let _inst₁ := Fintype.ofFinite G₁
    

Proof state:
G : Type u_1
inst✝⁴ : Group G
inst✝³ : IsKleinFour G
G₁ : Type u_2
G₂ : Type u_3
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : IsKleinFour G₁
e : G₁ ≃ G₂
he : e 1 = 1
h : Monoid.exponent G₂ = 2
_inst₁ : Fintype G₁ := Fintype.ofFinite G₁
⊢ ∀ (x y : G₁), Equiv.toFun e (x * y) = Equiv.toFun e x * Equiv.toFun e y

Next tactic:
let _inst₂ := Fintype.ofEquiv G₁ e
/-- An equivalence between an `IsKleinFour` group `G₁` and a group `G₂` of exponent two which sends
`1 : G₁` to `1 : G₂` is in fact an isomorphism. -/
@[to_additive "An equivalence between an `IsAddKleinFour` group `G₁` and a group `G₂` of exponent
two which sends `0 : G₁` to `0 : G₂` is in fact an isomorphism."]
def mulEquiv' (e : G₁ ≃ G₂) (he : e 1 = 1) (h : Monoid.exponent G₂ = 2) : G₁ ≃* G₂ where
  toEquiv := e
  map_mul' := by
    let _inst₁ := Fintype.ofFinite G₁
    let _inst₂ := Fintype.ofEquiv G₁ e
    

Proof state:
G : Type u_1
inst✝⁴ : Group G
inst✝³ : IsKleinFour G
G₁ : Type u_2
G₂ : Type u_3
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : IsKleinFour G₁
e : G₁ ≃ G₂
he : e 1 = 1
h : Monoid.exponent G₂ = 2
_inst₁ : Fintype G₁ := Fintype.ofFinite G₁
_inst₂ : Fintype G₂ := Fintype.ofEquiv G₁ e
⊢ ∀ (x y : G₁), Equiv.toFun e (x * y) = Equiv.toFun e x * Equiv.toFun e y

Next tactic:
intro x y
/-- An equivalence between an `IsKleinFour` group `G₁` and a group `G₂` of exponent two which sends
`1 : G₁` to `1 : G₂` is in fact an isomorphism. -/
@[to_additive "An equivalence between an `IsAddKleinFour` group `G₁` and a group `G₂` of exponent
two which sends `0 : G₁` to `0 : G₂` is in fact an isomorphism."]
def mulEquiv' (e : G₁ ≃ G₂) (he : e 1 = 1) (h : Monoid.exponent G₂ = 2) : G₁ ≃* G₂ where
  toEquiv := e
  map_mul' := by
    let _inst₁ := Fintype.ofFinite G₁
    let _inst₂ := Fintype.ofEquiv G₁ e
    intro x y
    

Proof state:
G : Type u_1
inst✝⁴ : Group G
inst✝³ : IsKleinFour G
G₁ : Type u_2
G₂ : Type u_3
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : IsKleinFour G₁
e : G₁ ≃ G₂
he : e 1 = 1
h : Monoid.exponent G₂ = 2
_inst₁ : Fintype G₁ := Fintype.ofFinite G₁
_inst₂ : Fintype G₂ := Fintype.ofEquiv G₁ e
x y : G₁
⊢ Equiv.toFun e (x * y) = Equiv.toFun e x * Equiv.toFun e y

Next tactic:
by_cases hx : x = 1
/-- An equivalence between an `IsKleinFour` group `G₁` and a group `G₂` of exponent two which sends
`1 : G₁` to `1 : G₂` is in fact an isomorphism. -/
@[to_additive "An equivalence between an `IsAddKleinFour` group `G₁` and a group `G₂` of exponent
two which sends `0 : G₁` to `0 : G₂` is in fact an isomorphism."]
def mulEquiv' (e : G₁ ≃ G₂) (he : e 1 = 1) (h : Monoid.exponent G₂ = 2) : G₁ ≃* G₂ where
  toEquiv := e
  map_mul' := by
    let _inst₁ := Fintype.ofFinite G₁
    let _inst₂ := Fintype.ofEquiv G₁ e
    intro x y
    by_cases hx : x = 1 <;> 

Proof state:
case pos
G : Type u_1
inst✝⁴ : Group G
inst✝³ : IsKleinFour G
G₁ : Type u_2
G₂ : Type u_3
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : IsKleinFour G₁
e : G₁ ≃ G₂
he : e 1 = 1
h : Monoid.exponent G₂ = 2
_inst₁ : Fintype G₁ := Fintype.ofFinite G₁
_inst₂ : Fintype G₂ := Fintype.ofEquiv G₁ e
x y : G₁
hx : x = 1
⊢ Equiv.toFun e (x * y) = Equiv.toFun e x * Equiv.toFun e y

Next tactic:
by_cases hy : y = 1
/-- An equivalence between an `IsKleinFour` group `G₁` and a group `G₂` of exponent two which sends
`1 : G₁` to `1 : G₂` is in fact an isomorphism. -/
@[to_additive "An equivalence between an `IsAddKleinFour` group `G₁` and a group `G₂` of exponent
two which sends `0 : G₁` to `0 : G₂` is in fact an isomorphism."]
def mulEquiv' (e : G₁ ≃ G₂) (he : e 1 = 1) (h : Monoid.exponent G₂ = 2) : G₁ ≃* G₂ where
  toEquiv := e
  map_mul' := by
    let _inst₁ := Fintype.ofFinite G₁
    let _inst₂ := Fintype.ofEquiv G₁ e
    intro x y
    by_cases hx : x = 1 <;> 

Proof state:
case neg
G : Type u_1
inst✝⁴ : Group G
inst✝³ : IsKleinFour G
G₁ : Type u_2
G₂ : Type u_3
inst✝² : Group G₁
inst✝¹ : Group G₂
inst✝ : IsKleinFour G₁
e : G₁ ≃ G₂
he : e 1 = 1
h : Monoid.exponent G₂ = 2
_inst₁ : Fintype G₁ := Fintype.ofFinite G₁
_inst₂ : Fintype G₂ := Fintype.ofEquiv G₁ e
x y : G₁
hx : ¬x = 1
⊢ Equiv.toFun e (x * y) = Equiv.toFun e x * Equiv.toFun e y

Next tactic:
by_cases hy : y = 1