l3lab/ntp-mathlib huggingface dataset:
/-- This instance asserts that if `G` has a right-cancellative multiplication, a linear order, and
  multiplication is strictly monotone w.r.t. the second argument, then `G` has `TwoUniqueProds`. -/
@[to_additive
  "This instance asserts that if `G` has a right-cancellative addition, a linear order,
  and addition is strictly monotone w.r.t. the second argument, then `G` has `TwoUniqueSums`." ]
instance (priority := 100) of_covariant_right [IsRightCancelMul G]
    [LinearOrder G] [CovariantClass G G (· * ·) (· < ·)] :
    TwoUniqueProds G where
  uniqueMul_of_one_lt_card {A B} hc := by
    obtain ⟨hA, hB, -⟩ := Nat.one_lt_mul_iff.mp hc
    rw [card_pos] at hA hB
    rw [← card_product] at hc
    obtain ⟨a0, b0, ha0, hb0, he0⟩ := mem_mul.mp (max'_mem _ <| hA.mul hB)
    obtain ⟨a1, b1, ha1, hb1, he1⟩ := mem_mul.mp (min'_mem _ <| hA.mul hB)
    have : UniqueMul A B a0 b0
    · intro a b ha hb he
      obtain hl | rfl | hl := lt_trichotomy b b0
      · exact ((he0 ▸ he ▸ mul_lt_mul_left' hl a).not_le <| le_max' _ _ <| mul_mem_mul ha hb0).elim
      · exact ⟨mul_right_cancel he, rfl⟩
      · exact ((he0 ▸ mul_lt_mul_left' hl a0).not_le <| le_max' _ _ <| mul_mem_mul ha0 hb).elim
    refine ⟨_, mk_mem_product ha0 hb0, _, mk_mem_product ha1 hb1, fun he ↦ ?_, this, ?_⟩
    · rw [Prod.mk.inj_iff] at he; 

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
G : Type u
H : Type v
inst✝⁴ : Mul G
inst✝³ : Mul H
inst✝² : IsRightCancelMul G
inst✝¹ : LinearOrder G
inst✝ : CovariantClass G G (fun x x_1 => x * x_1) fun x x_1 => x < x_1
A B : Finset G
hc : 1 < card (A ×ˢ B)
hA : Finset.Nonempty A
hB : Finset.Nonempty B
a0 b0 : G
ha0 : a0 ∈ A
hb0 : b0 ∈ B
he0 : a0 * b0 = max' (A * B) (_ : Finset.Nonempty (A * B))
a1 b1 : G
ha1 : a1 ∈ A
hb1 : b1 ∈ B
he1 : a1 * b1 = min' (A * B) (_ : Finset.Nonempty (A * B))
this : UniqueMul A B a0 b0
he : a0 = a1 ∧ b0 = b1
⊢ False

Next tactic:
rw [he.1, he.2, he1] at he0
/-- This instance asserts that if `G` has a right-cancellative multiplication, a linear order, and
  multiplication is strictly monotone w.r.t. the second argument, then `G` has `TwoUniqueProds`. -/
@[to_additive
  "This instance asserts that if `G` has a right-cancellative addition, a linear order,
  and addition is strictly monotone w.r.t. the second argument, then `G` has `TwoUniqueSums`." ]
instance (priority := 100) of_covariant_right [IsRightCancelMul G]
    [LinearOrder G] [CovariantClass G G (· * ·) (· < ·)] :
    TwoUniqueProds G where
  uniqueMul_of_one_lt_card {A B} hc := by
    obtain ⟨hA, hB, -⟩ := Nat.one_lt_mul_iff.mp hc
    rw [card_pos] at hA hB
    rw [← card_product] at hc
    obtain ⟨a0, b0, ha0, hb0, he0⟩ := mem_mul.mp (max'_mem _ <| hA.mul hB)
    obtain ⟨a1, b1, ha1, hb1, he1⟩ := mem_mul.mp (min'_mem _ <| hA.mul hB)
    have : UniqueMul A B a0 b0
    · intro a b ha hb he
      obtain hl | rfl | hl := lt_trichotomy b b0
      · exact ((he0 ▸ he ▸ mul_lt_mul_left' hl a).not_le <| le_max' _ _ <| mul_mem_mul ha hb0).elim
      · exact ⟨mul_right_cancel he, rfl⟩
      · exact ((he0 ▸ mul_lt_mul_left' hl a0).not_le <| le_max' _ _ <| mul_mem_mul ha0 hb).elim
    refine ⟨_, mk_mem_product ha0 hb0, _, mk_mem_product ha1 hb1, fun he ↦ ?_, this, ?_⟩
    · rw [Prod.mk.inj_iff] at he; rw [he.1, he.2, he1] at he0
      

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1
G : Type u
H : Type v
inst✝⁴ : Mul G
inst✝³ : Mul H
inst✝² : IsRightCancelMul G
inst✝¹ : LinearOrder G
inst✝ : CovariantClass G G (fun x x_1 => x * x_1) fun x x_1 => x < x_1
A B : Finset G
hc : 1 < card (A ×ˢ B)
hA : Finset.Nonempty A
hB : Finset.Nonempty B
a0 b0 : G
ha0 : a0 ∈ A
hb0 : b0 ∈ B
a1 b1 : G
he0 : min' (A * B) (_ : Finset.Nonempty (A * B)) = max' (A * B) (_ : Finset.Nonempty (A * B))
ha1 : a1 ∈ A
hb1 : b1 ∈ B
he1 : a1 * b1 = min' (A * B) (_ : Finset.Nonempty (A * B))
this : UniqueMul A B a0 b0
he : a0 = a1 ∧ b0 = b1
⊢ False

Next tactic:
obtain ⟨⟨a2, b2⟩, h2, hne⟩ := exists_ne_of_one_lt_card hc (a0, b0)
/-- This instance asserts that if `G` has a right-cancellative multiplication, a linear order, and
  multiplication is strictly monotone w.r.t. the second argument, then `G` has `TwoUniqueProds`. -/
@[to_additive
  "This instance asserts that if `G` has a right-cancellative addition, a linear order,
  and addition is strictly monotone w.r.t. the second argument, then `G` has `TwoUniqueSums`." ]
instance (priority := 100) of_covariant_right [IsRightCancelMul G]
    [LinearOrder G] [CovariantClass G G (· * ·) (· < ·)] :
    TwoUniqueProds G where
  uniqueMul_of_one_lt_card {A B} hc := by
    obtain ⟨hA, hB, -⟩ := Nat.one_lt_mul_iff.mp hc
    rw [card_pos] at hA hB
    rw [← card_product] at hc
    obtain ⟨a0, b0, ha0, hb0, he0⟩ := mem_mul.mp (max'_mem _ <| hA.mul hB)
    obtain ⟨a1, b1, ha1, hb1, he1⟩ := mem_mul.mp (min'_mem _ <| hA.mul hB)
    have : UniqueMul A B a0 b0
    · intro a b ha hb he
      obtain hl | rfl | hl := lt_trichotomy b b0
      · exact ((he0 ▸ he ▸ mul_lt_mul_left' hl a).not_le <| le_max' _ _ <| mul_mem_mul ha hb0).elim
      · exact ⟨mul_right_cancel he, rfl⟩
      · exact ((he0 ▸ mul_lt_mul_left' hl a0).not_le <| le_max' _ _ <| mul_mem_mul ha0 hb).elim
    refine ⟨_, mk_mem_product ha0 hb0, _, mk_mem_product ha1 hb1, fun he ↦ ?_, this, ?_⟩
    · rw [Prod.mk.inj_iff] at he; rw [he.1, he.2, he1] at he0
      obtain ⟨⟨a2, b2⟩, h2, hne⟩ := exists_ne_of_one_lt_card hc (a0, b0)
      

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1.intro.mk.intro
G : Type u
H : Type v
inst✝⁴ : Mul G
inst✝³ : Mul H
inst✝² : IsRightCancelMul G
inst✝¹ : LinearOrder G
inst✝ : CovariantClass G G (fun x x_1 => x * x_1) fun x x_1 => x < x_1
A B : Finset G
hc : 1 < card (A ×ˢ B)
hA : Finset.Nonempty A
hB : Finset.Nonempty B
a0 b0 : G
ha0 : a0 ∈ A
hb0 : b0 ∈ B
a1 b1 : G
he0 : min' (A * B) (_ : Finset.Nonempty (A * B)) = max' (A * B) (_ : Finset.Nonempty (A * B))
ha1 : a1 ∈ A
hb1 : b1 ∈ B
he1 : a1 * b1 = min' (A * B) (_ : Finset.Nonempty (A * B))
this : UniqueMul A B a0 b0
he : a0 = a1 ∧ b0 = b1
a2 b2 : G
h2 : (a2, b2) ∈ A ×ˢ B
hne : (a2, b2) ≠ (a0, b0)
⊢ False

Next tactic:
rw [mem_product] at h2
/-- This instance asserts that if `G` has a right-cancellative multiplication, a linear order, and
  multiplication is strictly monotone w.r.t. the second argument, then `G` has `TwoUniqueProds`. -/
@[to_additive
  "This instance asserts that if `G` has a right-cancellative addition, a linear order,
  and addition is strictly monotone w.r.t. the second argument, then `G` has `TwoUniqueSums`." ]
instance (priority := 100) of_covariant_right [IsRightCancelMul G]
    [LinearOrder G] [CovariantClass G G (· * ·) (· < ·)] :
    TwoUniqueProds G where
  uniqueMul_of_one_lt_card {A B} hc := by
    obtain ⟨hA, hB, -⟩ := Nat.one_lt_mul_iff.mp hc
    rw [card_pos] at hA hB
    rw [← card_product] at hc
    obtain ⟨a0, b0, ha0, hb0, he0⟩ := mem_mul.mp (max'_mem _ <| hA.mul hB)
    obtain ⟨a1, b1, ha1, hb1, he1⟩ := mem_mul.mp (min'_mem _ <| hA.mul hB)
    have : UniqueMul A B a0 b0
    · intro a b ha hb he
      obtain hl | rfl | hl := lt_trichotomy b b0
      · exact ((he0 ▸ he ▸ mul_lt_mul_left' hl a).not_le <| le_max' _ _ <| mul_mem_mul ha hb0).elim
      · exact ⟨mul_right_cancel he, rfl⟩
      · exact ((he0 ▸ mul_lt_mul_left' hl a0).not_le <| le_max' _ _ <| mul_mem_mul ha0 hb).elim
    refine ⟨_, mk_mem_product ha0 hb0, _, mk_mem_product ha1 hb1, fun he ↦ ?_, this, ?_⟩
    · rw [Prod.mk.inj_iff] at he; rw [he.1, he.2, he1] at he0
      obtain ⟨⟨a2, b2⟩, h2, hne⟩ := exists_ne_of_one_lt_card hc (a0, b0)
      rw [mem_product] at h2
      

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1.intro.mk.intro
G : Type u
H : Type v
inst✝⁴ : Mul G
inst✝³ : Mul H
inst✝² : IsRightCancelMul G
inst✝¹ : LinearOrder G
inst✝ : CovariantClass G G (fun x x_1 => x * x_1) fun x x_1 => x < x_1
A B : Finset G
hc : 1 < card (A ×ˢ B)
hA : Finset.Nonempty A
hB : Finset.Nonempty B
a0 b0 : G
ha0 : a0 ∈ A
hb0 : b0 ∈ B
a1 b1 : G
he0 : min' (A * B) (_ : Finset.Nonempty (A * B)) = max' (A * B) (_ : Finset.Nonempty (A * B))
ha1 : a1 ∈ A
hb1 : b1 ∈ B
he1 : a1 * b1 = min' (A * B) (_ : Finset.Nonempty (A * B))
this : UniqueMul A B a0 b0
he : a0 = a1 ∧ b0 = b1
a2 b2 : G
h2 : (a2, b2).1 ∈ A ∧ (a2, b2).2 ∈ B
hne : (a2, b2) ≠ (a0, b0)
⊢ False

Next tactic:
refine (min'_lt_max' _ (mul_mem_mul ha0 hb0) (mul_mem_mul h2.1 h2.2) fun he ↦ hne ?_).ne he0
/-- This instance asserts that if `G` has a right-cancellative multiplication, a linear order, and
  multiplication is strictly monotone w.r.t. the second argument, then `G` has `TwoUniqueProds`. -/
@[to_additive
  "This instance asserts that if `G` has a right-cancellative addition, a linear order,
  and addition is strictly monotone w.r.t. the second argument, then `G` has `TwoUniqueSums`." ]
instance (priority := 100) of_covariant_right [IsRightCancelMul G]
    [LinearOrder G] [CovariantClass G G (· * ·) (· < ·)] :
    TwoUniqueProds G where
  uniqueMul_of_one_lt_card {A B} hc := by
    obtain ⟨hA, hB, -⟩ := Nat.one_lt_mul_iff.mp hc
    rw [card_pos] at hA hB
    rw [← card_product] at hc
    obtain ⟨a0, b0, ha0, hb0, he0⟩ := mem_mul.mp (max'_mem _ <| hA.mul hB)
    obtain ⟨a1, b1, ha1, hb1, he1⟩ := mem_mul.mp (min'_mem _ <| hA.mul hB)
    have : UniqueMul A B a0 b0
    · intro a b ha hb he
      obtain hl | rfl | hl := lt_trichotomy b b0
      · exact ((he0 ▸ he ▸ mul_lt_mul_left' hl a).not_le <| le_max' _ _ <| mul_mem_mul ha hb0).elim
      · exact ⟨mul_right_cancel he, rfl⟩
      · exact ((he0 ▸ mul_lt_mul_left' hl a0).not_le <| le_max' _ _ <| mul_mem_mul ha0 hb).elim
    refine ⟨_, mk_mem_product ha0 hb0, _, mk_mem_product ha1 hb1, fun he ↦ ?_, this, ?_⟩
    · rw [Prod.mk.inj_iff] at he; rw [he.1, he.2, he1] at he0
      obtain ⟨⟨a2, b2⟩, h2, hne⟩ := exists_ne_of_one_lt_card hc (a0, b0)
      rw [mem_product] at h2
      refine (min'_lt_max' _ (mul_mem_mul ha0 hb0) (mul_mem_mul h2.1 h2.2) fun he ↦ hne ?_).ne he0
      

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_1.intro.mk.intro
G : Type u
H : Type v
inst✝⁴ : Mul G
inst✝³ : Mul H
inst✝² : IsRightCancelMul G
inst✝¹ : LinearOrder G
inst✝ : CovariantClass G G (fun x x_1 => x * x_1) fun x x_1 => x < x_1
A B : Finset G
hc : 1 < card (A ×ˢ B)
hA : Finset.Nonempty A
hB : Finset.Nonempty B
a0 b0 : G
ha0 : a0 ∈ A
hb0 : b0 ∈ B
a1 b1 : G
he0 : min' (A * B) (_ : Finset.Nonempty (A * B)) = max' (A * B) (_ : Finset.Nonempty (A * B))
ha1 : a1 ∈ A
hb1 : b1 ∈ B
he1 : a1 * b1 = min' (A * B) (_ : Finset.Nonempty (A * B))
this : UniqueMul A B a0 b0
he✝ : a0 = a1 ∧ b0 = b1
a2 b2 : G
h2 : (a2, b2).1 ∈ A ∧ (a2, b2).2 ∈ B
hne : (a2, b2) ≠ (a0, b0)
he : a0 * b0 = (a2, b2).1 * (a2, b2).2
⊢ (a2, b2) = (a0, b0)

Next tactic:
exact Prod.ext_iff.mpr (this h2.1 h2.2 he.symm)
/-- This instance asserts that if `G` has a right-cancellative multiplication, a linear order, and
  multiplication is strictly monotone w.r.t. the second argument, then `G` has `TwoUniqueProds`. -/
@[to_additive
  "This instance asserts that if `G` has a right-cancellative addition, a linear order,
  and addition is strictly monotone w.r.t. the second argument, then `G` has `TwoUniqueSums`." ]
instance (priority := 100) of_covariant_right [IsRightCancelMul G]
    [LinearOrder G] [CovariantClass G G (· * ·) (· < ·)] :
    TwoUniqueProds G where
  uniqueMul_of_one_lt_card {A B} hc := by
    obtain ⟨hA, hB, -⟩ := Nat.one_lt_mul_iff.mp hc
    rw [card_pos] at hA hB
    rw [← card_product] at hc
    obtain ⟨a0, b0, ha0, hb0, he0⟩ := mem_mul.mp (max'_mem _ <| hA.mul hB)
    obtain ⟨a1, b1, ha1, hb1, he1⟩ := mem_mul.mp (min'_mem _ <| hA.mul hB)
    have : UniqueMul A B a0 b0
    · intro a b ha hb he
      obtain hl | rfl | hl := lt_trichotomy b b0
      · exact ((he0 ▸ he ▸ mul_lt_mul_left' hl a).not_le <| le_max' _ _ <| mul_mem_mul ha hb0).elim
      · exact ⟨mul_right_cancel he, rfl⟩
      · exact ((he0 ▸ mul_lt_mul_left' hl a0).not_le <| le_max' _ _ <| mul_mem_mul ha0 hb).elim
    refine ⟨_, mk_mem_product ha0 hb0, _, mk_mem_product ha1 hb1, fun he ↦ ?_, this, ?_⟩
    · rw [Prod.mk.inj_iff] at he; rw [he.1, he.2, he1] at he0
      obtain ⟨⟨a2, b2⟩, h2, hne⟩ := exists_ne_of_one_lt_card hc (a0, b0)
      rw [mem_product] at h2
      refine (min'_lt_max' _ (mul_mem_mul ha0 hb0) (mul_mem_mul h2.1 h2.2) fun he ↦ hne ?_).ne he0
      exact Prod.ext_iff.mpr (this h2.1 h2.2 he.symm)
    · 

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
G : Type u
H : Type v
inst✝⁴ : Mul G
inst✝³ : Mul H
inst✝² : IsRightCancelMul G
inst✝¹ : LinearOrder G
inst✝ : CovariantClass G G (fun x x_1 => x * x_1) fun x x_1 => x < x_1
A B : Finset G
hc : 1 < card (A ×ˢ B)
hA : Finset.Nonempty A
hB : Finset.Nonempty B
a0 b0 : G
ha0 : a0 ∈ A
hb0 : b0 ∈ B
he0 : a0 * b0 = max' (A * B) (_ : Finset.Nonempty (A * B))
a1 b1 : G
ha1 : a1 ∈ A
hb1 : b1 ∈ B
he1 : a1 * b1 = min' (A * B) (_ : Finset.Nonempty (A * B))
this : UniqueMul A B a0 b0
⊢ UniqueMul A B (a1, b1).1 (a1, b1).2

Next tactic:
intro a b ha hb he
/-- This instance asserts that if `G` has a right-cancellative multiplication, a linear order, and
  multiplication is strictly monotone w.r.t. the second argument, then `G` has `TwoUniqueProds`. -/
@[to_additive
  "This instance asserts that if `G` has a right-cancellative addition, a linear order,
  and addition is strictly monotone w.r.t. the second argument, then `G` has `TwoUniqueSums`." ]
instance (priority := 100) of_covariant_right [IsRightCancelMul G]
    [LinearOrder G] [CovariantClass G G (· * ·) (· < ·)] :
    TwoUniqueProds G where
  uniqueMul_of_one_lt_card {A B} hc := by
    obtain ⟨hA, hB, -⟩ := Nat.one_lt_mul_iff.mp hc
    rw [card_pos] at hA hB
    rw [← card_product] at hc
    obtain ⟨a0, b0, ha0, hb0, he0⟩ := mem_mul.mp (max'_mem _ <| hA.mul hB)
    obtain ⟨a1, b1, ha1, hb1, he1⟩ := mem_mul.mp (min'_mem _ <| hA.mul hB)
    have : UniqueMul A B a0 b0
    · intro a b ha hb he
      obtain hl | rfl | hl := lt_trichotomy b b0
      · exact ((he0 ▸ he ▸ mul_lt_mul_left' hl a).not_le <| le_max' _ _ <| mul_mem_mul ha hb0).elim
      · exact ⟨mul_right_cancel he, rfl⟩
      · exact ((he0 ▸ mul_lt_mul_left' hl a0).not_le <| le_max' _ _ <| mul_mem_mul ha0 hb).elim
    refine ⟨_, mk_mem_product ha0 hb0, _, mk_mem_product ha1 hb1, fun he ↦ ?_, this, ?_⟩
    · rw [Prod.mk.inj_iff] at he; rw [he.1, he.2, he1] at he0
      obtain ⟨⟨a2, b2⟩, h2, hne⟩ := exists_ne_of_one_lt_card hc (a0, b0)
      rw [mem_product] at h2
      refine (min'_lt_max' _ (mul_mem_mul ha0 hb0) (mul_mem_mul h2.1 h2.2) fun he ↦ hne ?_).ne he0
      exact Prod.ext_iff.mpr (this h2.1 h2.2 he.symm)
    · intro a b ha hb he
      

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2
G : Type u
H : Type v
inst✝⁴ : Mul G
inst✝³ : Mul H
inst✝² : IsRightCancelMul G
inst✝¹ : LinearOrder G
inst✝ : CovariantClass G G (fun x x_1 => x * x_1) fun x x_1 => x < x_1
A B : Finset G
hc : 1 < card (A ×ˢ B)
hA : Finset.Nonempty A
hB : Finset.Nonempty B
a0 b0 : G
ha0 : a0 ∈ A
hb0 : b0 ∈ B
he0 : a0 * b0 = max' (A * B) (_ : Finset.Nonempty (A * B))
a1 b1 : G
ha1 : a1 ∈ A
hb1 : b1 ∈ B
he1 : a1 * b1 = min' (A * B) (_ : Finset.Nonempty (A * B))
this : UniqueMul A B a0 b0
a b : G
ha : a ∈ A
hb : b ∈ B
he : a * b = (a1, b1).1 * (a1, b1).2
⊢ a = (a1, b1).1 ∧ b = (a1, b1).2

Next tactic:
obtain hl | rfl | hl := lt_trichotomy b b1
/-- This instance asserts that if `G` has a right-cancellative multiplication, a linear order, and
  multiplication is strictly monotone w.r.t. the second argument, then `G` has `TwoUniqueProds`. -/
@[to_additive
  "This instance asserts that if `G` has a right-cancellative addition, a linear order,
  and addition is strictly monotone w.r.t. the second argument, then `G` has `TwoUniqueSums`." ]
instance (priority := 100) of_covariant_right [IsRightCancelMul G]
    [LinearOrder G] [CovariantClass G G (· * ·) (· < ·)] :
    TwoUniqueProds G where
  uniqueMul_of_one_lt_card {A B} hc := by
    obtain ⟨hA, hB, -⟩ := Nat.one_lt_mul_iff.mp hc
    rw [card_pos] at hA hB
    rw [← card_product] at hc
    obtain ⟨a0, b0, ha0, hb0, he0⟩ := mem_mul.mp (max'_mem _ <| hA.mul hB)
    obtain ⟨a1, b1, ha1, hb1, he1⟩ := mem_mul.mp (min'_mem _ <| hA.mul hB)
    have : UniqueMul A B a0 b0
    · intro a b ha hb he
      obtain hl | rfl | hl := lt_trichotomy b b0
      · exact ((he0 ▸ he ▸ mul_lt_mul_left' hl a).not_le <| le_max' _ _ <| mul_mem_mul ha hb0).elim
      · exact ⟨mul_right_cancel he, rfl⟩
      · exact ((he0 ▸ mul_lt_mul_left' hl a0).not_le <| le_max' _ _ <| mul_mem_mul ha0 hb).elim
    refine ⟨_, mk_mem_product ha0 hb0, _, mk_mem_product ha1 hb1, fun he ↦ ?_, this, ?_⟩
    · rw [Prod.mk.inj_iff] at he; rw [he.1, he.2, he1] at he0
      obtain ⟨⟨a2, b2⟩, h2, hne⟩ := exists_ne_of_one_lt_card hc (a0, b0)
      rw [mem_product] at h2
      refine (min'_lt_max' _ (mul_mem_mul ha0 hb0) (mul_mem_mul h2.1 h2.2) fun he ↦ hne ?_).ne he0
      exact Prod.ext_iff.mpr (this h2.1 h2.2 he.symm)
    · intro a b ha hb he
      obtain hl | rfl | hl := lt_trichotomy b b1
      · 

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2.inl
G : Type u
H : Type v
inst✝⁴ : Mul G
inst✝³ : Mul H
inst✝² : IsRightCancelMul G
inst✝¹ : LinearOrder G
inst✝ : CovariantClass G G (fun x x_1 => x * x_1) fun x x_1 => x < x_1
A B : Finset G
hc : 1 < card (A ×ˢ B)
hA : Finset.Nonempty A
hB : Finset.Nonempty B
a0 b0 : G
ha0 : a0 ∈ A
hb0 : b0 ∈ B
he0 : a0 * b0 = max' (A * B) (_ : Finset.Nonempty (A * B))
a1 b1 : G
ha1 : a1 ∈ A
hb1 : b1 ∈ B
he1 : a1 * b1 = min' (A * B) (_ : Finset.Nonempty (A * B))
this : UniqueMul A B a0 b0
a b : G
ha : a ∈ A
hb : b ∈ B
he : a * b = (a1, b1).1 * (a1, b1).2
hl : b < b1
⊢ a = (a1, b1).1 ∧ b = (a1, b1).2

Next tactic:
exact ((he1 ▸ mul_lt_mul_left' hl a1).not_le <| min'_le _ _ <| mul_mem_mul ha1 hb).elim
/-- This instance asserts that if `G` has a right-cancellative multiplication, a linear order, and
  multiplication is strictly monotone w.r.t. the second argument, then `G` has `TwoUniqueProds`. -/
@[to_additive
  "This instance asserts that if `G` has a right-cancellative addition, a linear order,
  and addition is strictly monotone w.r.t. the second argument, then `G` has `TwoUniqueSums`." ]
instance (priority := 100) of_covariant_right [IsRightCancelMul G]
    [LinearOrder G] [CovariantClass G G (· * ·) (· < ·)] :
    TwoUniqueProds G where
  uniqueMul_of_one_lt_card {A B} hc := by
    obtain ⟨hA, hB, -⟩ := Nat.one_lt_mul_iff.mp hc
    rw [card_pos] at hA hB
    rw [← card_product] at hc
    obtain ⟨a0, b0, ha0, hb0, he0⟩ := mem_mul.mp (max'_mem _ <| hA.mul hB)
    obtain ⟨a1, b1, ha1, hb1, he1⟩ := mem_mul.mp (min'_mem _ <| hA.mul hB)
    have : UniqueMul A B a0 b0
    · intro a b ha hb he
      obtain hl | rfl | hl := lt_trichotomy b b0
      · exact ((he0 ▸ he ▸ mul_lt_mul_left' hl a).not_le <| le_max' _ _ <| mul_mem_mul ha hb0).elim
      · exact ⟨mul_right_cancel he, rfl⟩
      · exact ((he0 ▸ mul_lt_mul_left' hl a0).not_le <| le_max' _ _ <| mul_mem_mul ha0 hb).elim
    refine ⟨_, mk_mem_product ha0 hb0, _, mk_mem_product ha1 hb1, fun he ↦ ?_, this, ?_⟩
    · rw [Prod.mk.inj_iff] at he; rw [he.1, he.2, he1] at he0
      obtain ⟨⟨a2, b2⟩, h2, hne⟩ := exists_ne_of_one_lt_card hc (a0, b0)
      rw [mem_product] at h2
      refine (min'_lt_max' _ (mul_mem_mul ha0 hb0) (mul_mem_mul h2.1 h2.2) fun he ↦ hne ?_).ne he0
      exact Prod.ext_iff.mpr (this h2.1 h2.2 he.symm)
    · intro a b ha hb he
      obtain hl | rfl | hl := lt_trichotomy b b1
      · exact ((he1 ▸ mul_lt_mul_left' hl a1).not_le <| min'_le _ _ <| mul_mem_mul ha1 hb).elim
      · 

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2.inr.inl
G : Type u
H : Type v
inst✝⁴ : Mul G
inst✝³ : Mul H
inst✝² : IsRightCancelMul G
inst✝¹ : LinearOrder G
inst✝ : CovariantClass G G (fun x x_1 => x * x_1) fun x x_1 => x < x_1
A B : Finset G
hc : 1 < card (A ×ˢ B)
hA : Finset.Nonempty A
hB : Finset.Nonempty B
a0 b0 : G
ha0 : a0 ∈ A
hb0 : b0 ∈ B
he0 : a0 * b0 = max' (A * B) (_ : Finset.Nonempty (A * B))
a1 : G
ha1 : a1 ∈ A
this : UniqueMul A B a0 b0
a b : G
ha : a ∈ A
hb hb1 : b ∈ B
he1 : a1 * b = min' (A * B) (_ : Finset.Nonempty (A * B))
he : a * b = (a1, b).1 * (a1, b).2
⊢ a = (a1, b).1 ∧ b = (a1, b).2

Next tactic:
exact ⟨mul_right_cancel he, rfl⟩
/-- This instance asserts that if `G` has a right-cancellative multiplication, a linear order, and
  multiplication is strictly monotone w.r.t. the second argument, then `G` has `TwoUniqueProds`. -/
@[to_additive
  "This instance asserts that if `G` has a right-cancellative addition, a linear order,
  and addition is strictly monotone w.r.t. the second argument, then `G` has `TwoUniqueSums`." ]
instance (priority := 100) of_covariant_right [IsRightCancelMul G]
    [LinearOrder G] [CovariantClass G G (· * ·) (· < ·)] :
    TwoUniqueProds G where
  uniqueMul_of_one_lt_card {A B} hc := by
    obtain ⟨hA, hB, -⟩ := Nat.one_lt_mul_iff.mp hc
    rw [card_pos] at hA hB
    rw [← card_product] at hc
    obtain ⟨a0, b0, ha0, hb0, he0⟩ := mem_mul.mp (max'_mem _ <| hA.mul hB)
    obtain ⟨a1, b1, ha1, hb1, he1⟩ := mem_mul.mp (min'_mem _ <| hA.mul hB)
    have : UniqueMul A B a0 b0
    · intro a b ha hb he
      obtain hl | rfl | hl := lt_trichotomy b b0
      · exact ((he0 ▸ he ▸ mul_lt_mul_left' hl a).not_le <| le_max' _ _ <| mul_mem_mul ha hb0).elim
      · exact ⟨mul_right_cancel he, rfl⟩
      · exact ((he0 ▸ mul_lt_mul_left' hl a0).not_le <| le_max' _ _ <| mul_mem_mul ha0 hb).elim
    refine ⟨_, mk_mem_product ha0 hb0, _, mk_mem_product ha1 hb1, fun he ↦ ?_, this, ?_⟩
    · rw [Prod.mk.inj_iff] at he; rw [he.1, he.2, he1] at he0
      obtain ⟨⟨a2, b2⟩, h2, hne⟩ := exists_ne_of_one_lt_card hc (a0, b0)
      rw [mem_product] at h2
      refine (min'_lt_max' _ (mul_mem_mul ha0 hb0) (mul_mem_mul h2.1 h2.2) fun he ↦ hne ?_).ne he0
      exact Prod.ext_iff.mpr (this h2.1 h2.2 he.symm)
    · intro a b ha hb he
      obtain hl | rfl | hl := lt_trichotomy b b1
      · exact ((he1 ▸ mul_lt_mul_left' hl a1).not_le <| min'_le _ _ <| mul_mem_mul ha1 hb).elim
      · exact ⟨mul_right_cancel he, rfl⟩
      · 

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2.inr.inr
G : Type u
H : Type v
inst✝⁴ : Mul G
inst✝³ : Mul H
inst✝² : IsRightCancelMul G
inst✝¹ : LinearOrder G
inst✝ : CovariantClass G G (fun x x_1 => x * x_1) fun x x_1 => x < x_1
A B : Finset G
hc : 1 < card (A ×ˢ B)
hA : Finset.Nonempty A
hB : Finset.Nonempty B
a0 b0 : G
ha0 : a0 ∈ A
hb0 : b0 ∈ B
he0 : a0 * b0 = max' (A * B) (_ : Finset.Nonempty (A * B))
a1 b1 : G
ha1 : a1 ∈ A
hb1 : b1 ∈ B
he1 : a1 * b1 = min' (A * B) (_ : Finset.Nonempty (A * B))
this : UniqueMul A B a0 b0
a b : G
ha : a ∈ A
hb : b ∈ B
he : a * b = (a1, b1).1 * (a1, b1).2
hl : b1 < b
⊢ a = (a1, b1).1 ∧ b = (a1, b1).2

Next tactic:
exact ((he1 ▸ he ▸ mul_lt_mul_left' hl a).not_le <| min'_le _ _ <| mul_mem_mul ha hb1).elim
/-- auxiliary construction for `coconesIso` -/
@[simp]
def coconesIsoComponentHom {J : Type u} [Category.{v} J] {K : J ⥤ C} (Y : D)
    (t : ((cocones J D).obj (op (K ⋙ F))).obj Y) : (G ⋙ (cocones J C).obj (op K)).obj Y where
  app j := (adj.homEquiv (K.obj j) Y) (t.app j)
  naturality j j' f := by
    

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
D : Type u₂
inst✝¹ : Category.{v₂, u₂} D
F : C ⥤ D
G : D ⥤ C
adj : F ⊣ G
J : Type u
inst✝ : Category.{v, u} J
K : J ⥤ C
Y : D
t : ((cocones J D).obj (op (K ⋙ F))).obj Y
j j' : J
f : j ⟶ j'
⊢ (op (op K).unop).unop.map f ≫ (fun j => (homEquiv adj (K.obj j) Y) (t.app j)) j' =
    (fun j => (homEquiv adj (K.obj j) Y) (t.app j)) j ≫ ((const J).obj (G.obj Y)).map f

Next tactic:
erw [← adj.homEquiv_naturality_left, t.naturality]