l3lab/ntp-mathlib huggingface dataset:
@[to_additive] instance {ι} (G : ι → Type*) [∀ i, Mul (G i)] [∀ i, TwoUniqueProds (G i)] :
    TwoUniqueProds (∀ i, G i) where
  uniqueMul_of_one_lt_card {A} := by
    classical
    let _ := isWellFounded_ssubset (α := ∀ i, G i) -- why need this?
    apply IsWellFounded.induction (· ⊂ ·) A; intro A ihA B
    apply IsWellFounded.induction (· ⊂ ·) B; intro B ihB hc
    obtain ⟨hA, hB, hc⟩ := Nat.one_lt_mul_iff.mp hc
    rw [card_pos] at hA hB
    obtain ⟨i, hc⟩ := exists_or.mpr (hc.imp exists_of_one_lt_card_pi exists_of_one_lt_card_pi)
    obtain ⟨p1, h1, p2, h2, hne, hi1, hi2⟩ := uniqueMul_of_one_lt_card (Nat.one_lt_mul_iff.mpr
      ⟨card_pos.2 (hA.image _), card_pos.2 (hB.image _), hc.imp And.left And.left⟩)
    simp_rw [mem_product, mem_image, ← filter_nonempty_iff] at h1 h2
    replace h1 := uniqueMul_of_twoUniqueMul ?_ h1.1 h1.2
    replace h2 := uniqueMul_of_twoUniqueMul ?_ h2.1 h2.2
    · obtain ⟨a1, ha1, b1, hb1, hu1⟩ := h1
      obtain ⟨a2, ha2, b2, hb2, hu2⟩ := h2
      rw [mem_filter] at ha1 hb1 ha2 hb2
      simp_rw [mem_product]
      refine ⟨(a1, b1), ⟨ha1.1, hb1.1⟩, (a2, b2), ⟨ha2.1, hb2.1⟩, ?_,
        UniqueMul.of_image_filter (Pi.evalMulHom G i) ha1.2 hb1.2 hi1 hu1,
        UniqueMul.of_image_filter (Pi.evalMulHom G i) ha2.2 hb2.2 hi2 hu2⟩
      

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2.refine_2.intro.intro.intro.intro.intro.intro.intro.intro
G✝ : Type u
H : Type v
inst✝³ : Mul G✝
inst✝² : Mul H
ι : Type ?u.450063
G : ι → Type u_1
inst✝¹ : (i : ι) → Mul (G i)
inst✝ : ∀ (i : ι), TwoUniqueProds (G i)
A✝ : Finset ((i : ι) → G i)
x✝ : IsWellFounded (Finset ((i : ι) → G i)) fun x x_1 => x ⊂ x_1 := isWellFounded_ssubset
A : Finset ((i : ι) → G i)
ihA :
  ∀ y ⊂ A,
    ∀ {B : Finset ((i : ι) → G i)},
      1 < card y * card B → ∃ p1 ∈ y ×ˢ B, ∃ p2 ∈ y ×ˢ B, p1 ≠ p2 ∧ UniqueMul y B p1.1 p1.2 ∧ UniqueMul y B p2.1 p2.2
B✝ B : Finset ((i : ι) → G i)
ihB :
  ∀ y ⊂ B,
    1 < card A * card y → ∃ p1 ∈ A ×ˢ y, ∃ p2 ∈ A ×ˢ y, p1 ≠ p2 ∧ UniqueMul A y p1.1 p1.2 ∧ UniqueMul A y p2.1 p2.2
hc✝¹ : 1 < card A * card B
hA : Finset.Nonempty A
hB : Finset.Nonempty B
hc✝ : 1 < card A ∨ 1 < card B
i : ι
hc :
  (1 < card (image (fun x => x i) A) ∧ ∀ (ai : G i), filter (fun x => x i = ai) A ⊂ A) ∨
    1 < card (image (fun x => x i) B) ∧ ∀ (ai : G i), filter (fun x => x i = ai) B ⊂ B
p1 p2 : G i × G i
hne : p1 ≠ p2
hi1 : UniqueMul (image (fun x => x i) A) (image (fun x => x i) B) p1.1 p1.2
hi2 : UniqueMul (image (fun x => x i) A) (image (fun x => x i) B) p2.1 p2.2
a1 : (i : ι) → G i
ha1 : a1 ∈ A ∧ a1 i = p1.1
b1 : (i : ι) → G i
hb1 : b1 ∈ B ∧ b1 i = p1.2
hu1 : UniqueMul (filter (fun a => a i = p1.1) A) (filter (fun a => a i = p1.2) B) a1 b1
a2 : (i : ι) → G i
ha2 : a2 ∈ A ∧ a2 i = p2.1
b2 : (i : ι) → G i
hb2 : b2 ∈ B ∧ b2 i = p2.2
hu2 : UniqueMul (filter (fun a => a i = p2.1) A) (filter (fun a => a i = p2.2) B) a2 b2
⊢ (a1, b1) ≠ (a2, b2)

Next tactic:
contrapose! hne
@[to_additive] instance {ι} (G : ι → Type*) [∀ i, Mul (G i)] [∀ i, TwoUniqueProds (G i)] :
    TwoUniqueProds (∀ i, G i) where
  uniqueMul_of_one_lt_card {A} := by
    classical
    let _ := isWellFounded_ssubset (α := ∀ i, G i) -- why need this?
    apply IsWellFounded.induction (· ⊂ ·) A; intro A ihA B
    apply IsWellFounded.induction (· ⊂ ·) B; intro B ihB hc
    obtain ⟨hA, hB, hc⟩ := Nat.one_lt_mul_iff.mp hc
    rw [card_pos] at hA hB
    obtain ⟨i, hc⟩ := exists_or.mpr (hc.imp exists_of_one_lt_card_pi exists_of_one_lt_card_pi)
    obtain ⟨p1, h1, p2, h2, hne, hi1, hi2⟩ := uniqueMul_of_one_lt_card (Nat.one_lt_mul_iff.mpr
      ⟨card_pos.2 (hA.image _), card_pos.2 (hB.image _), hc.imp And.left And.left⟩)
    simp_rw [mem_product, mem_image, ← filter_nonempty_iff] at h1 h2
    replace h1 := uniqueMul_of_twoUniqueMul ?_ h1.1 h1.2
    replace h2 := uniqueMul_of_twoUniqueMul ?_ h2.1 h2.2
    · obtain ⟨a1, ha1, b1, hb1, hu1⟩ := h1
      obtain ⟨a2, ha2, b2, hb2, hu2⟩ := h2
      rw [mem_filter] at ha1 hb1 ha2 hb2
      simp_rw [mem_product]
      refine ⟨(a1, b1), ⟨ha1.1, hb1.1⟩, (a2, b2), ⟨ha2.1, hb2.1⟩, ?_,
        UniqueMul.of_image_filter (Pi.evalMulHom G i) ha1.2 hb1.2 hi1 hu1,
        UniqueMul.of_image_filter (Pi.evalMulHom G i) ha2.2 hb2.2 hi2 hu2⟩
      contrapose! hne; 

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2.refine_2.intro.intro.intro.intro.intro.intro.intro.intro
G✝ : Type u
H : Type v
inst✝³ : Mul G✝
inst✝² : Mul H
ι : Type ?u.450063
G : ι → Type u_1
inst✝¹ : (i : ι) → Mul (G i)
inst✝ : ∀ (i : ι), TwoUniqueProds (G i)
A✝ : Finset ((i : ι) → G i)
x✝ : IsWellFounded (Finset ((i : ι) → G i)) fun x x_1 => x ⊂ x_1 := isWellFounded_ssubset
A : Finset ((i : ι) → G i)
ihA :
  ∀ y ⊂ A,
    ∀ {B : Finset ((i : ι) → G i)},
      1 < card y * card B → ∃ p1 ∈ y ×ˢ B, ∃ p2 ∈ y ×ˢ B, p1 ≠ p2 ∧ UniqueMul y B p1.1 p1.2 ∧ UniqueMul y B p2.1 p2.2
B✝ B : Finset ((i : ι) → G i)
ihB :
  ∀ y ⊂ B,
    1 < card A * card y → ∃ p1 ∈ A ×ˢ y, ∃ p2 ∈ A ×ˢ y, p1 ≠ p2 ∧ UniqueMul A y p1.1 p1.2 ∧ UniqueMul A y p2.1 p2.2
hc✝¹ : 1 < card A * card B
hA : Finset.Nonempty A
hB : Finset.Nonempty B
hc✝ : 1 < card A ∨ 1 < card B
i : ι
hc :
  (1 < card (image (fun x => x i) A) ∧ ∀ (ai : G i), filter (fun x => x i = ai) A ⊂ A) ∨
    1 < card (image (fun x => x i) B) ∧ ∀ (ai : G i), filter (fun x => x i = ai) B ⊂ B
p1 p2 : G i × G i
hi1 : UniqueMul (image (fun x => x i) A) (image (fun x => x i) B) p1.1 p1.2
hi2 : UniqueMul (image (fun x => x i) A) (image (fun x => x i) B) p2.1 p2.2
a1 : (i : ι) → G i
ha1 : a1 ∈ A ∧ a1 i = p1.1
b1 : (i : ι) → G i
hb1 : b1 ∈ B ∧ b1 i = p1.2
hu1 : UniqueMul (filter (fun a => a i = p1.1) A) (filter (fun a => a i = p1.2) B) a1 b1
a2 : (i : ι) → G i
ha2 : a2 ∈ A ∧ a2 i = p2.1
b2 : (i : ι) → G i
hb2 : b2 ∈ B ∧ b2 i = p2.2
hu2 : UniqueMul (filter (fun a => a i = p2.1) A) (filter (fun a => a i = p2.2) B) a2 b2
hne : (a1, b1) = (a2, b2)
⊢ p1 = p2

Next tactic:
rw [Prod.mk.inj_iff] at hne ⊢
@[to_additive] instance {ι} (G : ι → Type*) [∀ i, Mul (G i)] [∀ i, TwoUniqueProds (G i)] :
    TwoUniqueProds (∀ i, G i) where
  uniqueMul_of_one_lt_card {A} := by
    classical
    let _ := isWellFounded_ssubset (α := ∀ i, G i) -- why need this?
    apply IsWellFounded.induction (· ⊂ ·) A; intro A ihA B
    apply IsWellFounded.induction (· ⊂ ·) B; intro B ihB hc
    obtain ⟨hA, hB, hc⟩ := Nat.one_lt_mul_iff.mp hc
    rw [card_pos] at hA hB
    obtain ⟨i, hc⟩ := exists_or.mpr (hc.imp exists_of_one_lt_card_pi exists_of_one_lt_card_pi)
    obtain ⟨p1, h1, p2, h2, hne, hi1, hi2⟩ := uniqueMul_of_one_lt_card (Nat.one_lt_mul_iff.mpr
      ⟨card_pos.2 (hA.image _), card_pos.2 (hB.image _), hc.imp And.left And.left⟩)
    simp_rw [mem_product, mem_image, ← filter_nonempty_iff] at h1 h2
    replace h1 := uniqueMul_of_twoUniqueMul ?_ h1.1 h1.2
    replace h2 := uniqueMul_of_twoUniqueMul ?_ h2.1 h2.2
    · obtain ⟨a1, ha1, b1, hb1, hu1⟩ := h1
      obtain ⟨a2, ha2, b2, hb2, hu2⟩ := h2
      rw [mem_filter] at ha1 hb1 ha2 hb2
      simp_rw [mem_product]
      refine ⟨(a1, b1), ⟨ha1.1, hb1.1⟩, (a2, b2), ⟨ha2.1, hb2.1⟩, ?_,
        UniqueMul.of_image_filter (Pi.evalMulHom G i) ha1.2 hb1.2 hi1 hu1,
        UniqueMul.of_image_filter (Pi.evalMulHom G i) ha2.2 hb2.2 hi2 hu2⟩
      contrapose! hne; rw [Prod.mk.inj_iff] at hne ⊢
      

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2.refine_2.intro.intro.intro.intro.intro.intro.intro.intro
G✝ : Type u
H : Type v
inst✝³ : Mul G✝
inst✝² : Mul H
ι : Type ?u.450063
G : ι → Type u_1
inst✝¹ : (i : ι) → Mul (G i)
inst✝ : ∀ (i : ι), TwoUniqueProds (G i)
A✝ : Finset ((i : ι) → G i)
x✝ : IsWellFounded (Finset ((i : ι) → G i)) fun x x_1 => x ⊂ x_1 := isWellFounded_ssubset
A : Finset ((i : ι) → G i)
ihA :
  ∀ y ⊂ A,
    ∀ {B : Finset ((i : ι) → G i)},
      1 < card y * card B → ∃ p1 ∈ y ×ˢ B, ∃ p2 ∈ y ×ˢ B, p1 ≠ p2 ∧ UniqueMul y B p1.1 p1.2 ∧ UniqueMul y B p2.1 p2.2
B✝ B : Finset ((i : ι) → G i)
ihB :
  ∀ y ⊂ B,
    1 < card A * card y → ∃ p1 ∈ A ×ˢ y, ∃ p2 ∈ A ×ˢ y, p1 ≠ p2 ∧ UniqueMul A y p1.1 p1.2 ∧ UniqueMul A y p2.1 p2.2
hc✝¹ : 1 < card A * card B
hA : Finset.Nonempty A
hB : Finset.Nonempty B
hc✝ : 1 < card A ∨ 1 < card B
i : ι
hc :
  (1 < card (image (fun x => x i) A) ∧ ∀ (ai : G i), filter (fun x => x i = ai) A ⊂ A) ∨
    1 < card (image (fun x => x i) B) ∧ ∀ (ai : G i), filter (fun x => x i = ai) B ⊂ B
p1 p2 : G i × G i
hi1 : UniqueMul (image (fun x => x i) A) (image (fun x => x i) B) p1.1 p1.2
hi2 : UniqueMul (image (fun x => x i) A) (image (fun x => x i) B) p2.1 p2.2
a1 : (i : ι) → G i
ha1 : a1 ∈ A ∧ a1 i = p1.1
b1 : (i : ι) → G i
hb1 : b1 ∈ B ∧ b1 i = p1.2
hu1 : UniqueMul (filter (fun a => a i = p1.1) A) (filter (fun a => a i = p1.2) B) a1 b1
a2 : (i : ι) → G i
ha2 : a2 ∈ A ∧ a2 i = p2.1
b2 : (i : ι) → G i
hb2 : b2 ∈ B ∧ b2 i = p2.2
hu2 : UniqueMul (filter (fun a => a i = p2.1) A) (filter (fun a => a i = p2.2) B) a2 b2
hne : a1 = a2 ∧ b1 = b2
⊢ p1.1 = p2.1 ∧ p1.2 = p2.2

Next tactic:
rw [← ha1.2, ← hb1.2, ← ha2.2, ← hb2.2, hne.1, hne.2]
@[to_additive] instance {ι} (G : ι → Type*) [∀ i, Mul (G i)] [∀ i, TwoUniqueProds (G i)] :
    TwoUniqueProds (∀ i, G i) where
  uniqueMul_of_one_lt_card {A} := by
    classical
    let _ := isWellFounded_ssubset (α := ∀ i, G i) -- why need this?
    apply IsWellFounded.induction (· ⊂ ·) A; intro A ihA B
    apply IsWellFounded.induction (· ⊂ ·) B; intro B ihB hc
    obtain ⟨hA, hB, hc⟩ := Nat.one_lt_mul_iff.mp hc
    rw [card_pos] at hA hB
    obtain ⟨i, hc⟩ := exists_or.mpr (hc.imp exists_of_one_lt_card_pi exists_of_one_lt_card_pi)
    obtain ⟨p1, h1, p2, h2, hne, hi1, hi2⟩ := uniqueMul_of_one_lt_card (Nat.one_lt_mul_iff.mpr
      ⟨card_pos.2 (hA.image _), card_pos.2 (hB.image _), hc.imp And.left And.left⟩)
    simp_rw [mem_product, mem_image, ← filter_nonempty_iff] at h1 h2
    replace h1 := uniqueMul_of_twoUniqueMul ?_ h1.1 h1.2
    replace h2 := uniqueMul_of_twoUniqueMul ?_ h2.1 h2.2
    · obtain ⟨a1, ha1, b1, hb1, hu1⟩ := h1
      obtain ⟨a2, ha2, b2, hb2, hu2⟩ := h2
      rw [mem_filter] at ha1 hb1 ha2 hb2
      simp_rw [mem_product]
      refine ⟨(a1, b1), ⟨ha1.1, hb1.1⟩, (a2, b2), ⟨ha2.1, hb2.1⟩, ?_,
        UniqueMul.of_image_filter (Pi.evalMulHom G i) ha1.2 hb1.2 hi1 hu1,
        UniqueMul.of_image_filter (Pi.evalMulHom G i) ha2.2 hb2.2 hi2 hu2⟩
      contrapose! hne; rw [Prod.mk.inj_iff] at hne ⊢
      rw [← ha1.2, ← hb1.2, ← ha2.2, ← hb2.2, hne.1, hne.2]; 

Proof state:
case intro.intro.intro.intro.intro.intro.intro.intro.intro.refine_2.refine_2.intro.intro.intro.intro.intro.intro.intro.intro
G✝ : Type u
H : Type v
inst✝³ : Mul G✝
inst✝² : Mul H
ι : Type ?u.450063
G : ι → Type u_1
inst✝¹ : (i : ι) → Mul (G i)
inst✝ : ∀ (i : ι), TwoUniqueProds (G i)
A✝ : Finset ((i : ι) → G i)
x✝ : IsWellFounded (Finset ((i : ι) → G i)) fun x x_1 => x ⊂ x_1 := isWellFounded_ssubset
A : Finset ((i : ι) → G i)
ihA :
  ∀ y ⊂ A,
    ∀ {B : Finset ((i : ι) → G i)},
      1 < card y * card B → ∃ p1 ∈ y ×ˢ B, ∃ p2 ∈ y ×ˢ B, p1 ≠ p2 ∧ UniqueMul y B p1.1 p1.2 ∧ UniqueMul y B p2.1 p2.2
B✝ B : Finset ((i : ι) → G i)
ihB :
  ∀ y ⊂ B,
    1 < card A * card y → ∃ p1 ∈ A ×ˢ y, ∃ p2 ∈ A ×ˢ y, p1 ≠ p2 ∧ UniqueMul A y p1.1 p1.2 ∧ UniqueMul A y p2.1 p2.2
hc✝¹ : 1 < card A * card B
hA : Finset.Nonempty A
hB : Finset.Nonempty B
hc✝ : 1 < card A ∨ 1 < card B
i : ι
hc :
  (1 < card (image (fun x => x i) A) ∧ ∀ (ai : G i), filter (fun x => x i = ai) A ⊂ A) ∨
    1 < card (image (fun x => x i) B) ∧ ∀ (ai : G i), filter (fun x => x i = ai) B ⊂ B
p1 p2 : G i × G i
hi1 : UniqueMul (image (fun x => x i) A) (image (fun x => x i) B) p1.1 p1.2
hi2 : UniqueMul (image (fun x => x i) A) (image (fun x => x i) B) p2.1 p2.2
a1 : (i : ι) → G i
ha1 : a1 ∈ A ∧ a1 i = p1.1
b1 : (i : ι) → G i
hb1 : b1 ∈ B ∧ b1 i = p1.2
hu1 : UniqueMul (filter (fun a => a i = p1.1) A) (filter (fun a => a i = p1.2) B) a1 b1
a2 : (i : ι) → G i
ha2 : a2 ∈ A ∧ a2 i = p2.1
b2 : (i : ι) → G i
hb2 : b2 ∈ B ∧ b2 i = p2.2
hu2 : UniqueMul (filter (fun a => a i = p2.1) A) (filter (fun a => a i = p2.2) B) a2 b2
hne : a1 = a2 ∧ b1 = b2
⊢ a2 i = a2 i ∧ b2 i = b2 i

Next tactic:
exact ⟨rfl, rfl⟩
open ULift in
@[to_additive] instance [TwoUniqueProds G] [TwoUniqueProds H] : TwoUniqueProds (G × H) := by
  

Proof state:
G : Type u
H : Type v
inst✝³ : Mul G
inst✝² : Mul H
inst✝¹ : TwoUniqueProds G
inst✝ : TwoUniqueProds H
⊢ TwoUniqueProds (G × H)

Next tactic:
have : ∀ b, TwoUniqueProds (I G H b) := Bool.rec ?_ ?_
open ULift in
@[to_additive] instance [TwoUniqueProds G] [TwoUniqueProds H] : TwoUniqueProds (G × H) := by
  have : ∀ b, TwoUniqueProds (I G H b) := Bool.rec ?_ ?_
  · 

Proof state:
case refine_3
G : Type u
H : Type v
inst✝³ : Mul G
inst✝² : Mul H
inst✝¹ : TwoUniqueProds G
inst✝ : TwoUniqueProds H
⊢ TwoUniqueProds (I G H true)

Next tactic:
exact mulHom_image_of_injective (downMulHom H) down_injective ‹_›
open ULift in
@[to_additive] instance [TwoUniqueProds G] [TwoUniqueProds H] : TwoUniqueProds (G × H) := by
  have : ∀ b, TwoUniqueProds (I G H b) := Bool.rec ?_ ?_
  · exact mulHom_image_of_injective (downMulHom H) down_injective ‹_›
  · 

Proof state:
case refine_1
G : Type u
H : Type v
inst✝³ : Mul G
inst✝² : Mul H
inst✝¹ : TwoUniqueProds G
inst✝ : TwoUniqueProds H
this : ∀ (b : Bool), TwoUniqueProds (I G H b)
⊢ TwoUniqueProds (G × H)

Next tactic:
refine mulHom_image_of_injective (Prod.upMulHom G H) (fun x y he ↦ Prod.ext ?_ ?_)
      (instTwoUniqueProdsForAllInstMul <| I G H)
open ULift in
@[to_additive] instance [TwoUniqueProds G] [TwoUniqueProds H] : TwoUniqueProds (G × H) := by
  have : ∀ b, TwoUniqueProds (I G H b) := Bool.rec ?_ ?_
  · exact mulHom_image_of_injective (downMulHom H) down_injective ‹_›
  · refine mulHom_image_of_injective (Prod.upMulHom G H) (fun x y he ↦ Prod.ext ?_ ?_)
      (instTwoUniqueProdsForAllInstMul <| I G H) <;> 

Proof state:
case refine_1.refine_1
G : Type u
H : Type v
inst✝³ : Mul G
inst✝² : Mul H
inst✝¹ : TwoUniqueProds G
inst✝ : TwoUniqueProds H
this : ∀ (b : Bool), TwoUniqueProds (I G H b)
x y : G × H
he : (Prod.upMulHom G H) x = (Prod.upMulHom G H) y
⊢ x.1 = y.1

Next tactic:
apply up_injective
open ULift in
@[to_additive] instance [TwoUniqueProds G] [TwoUniqueProds H] : TwoUniqueProds (G × H) := by
  have : ∀ b, TwoUniqueProds (I G H b) := Bool.rec ?_ ?_
  · exact mulHom_image_of_injective (downMulHom H) down_injective ‹_›
  · refine mulHom_image_of_injective (Prod.upMulHom G H) (fun x y he ↦ Prod.ext ?_ ?_)
      (instTwoUniqueProdsForAllInstMul <| I G H) <;> 

Proof state:
case refine_1.refine_2
G : Type u
H : Type v
inst✝³ : Mul G
inst✝² : Mul H
inst✝¹ : TwoUniqueProds G
inst✝ : TwoUniqueProds H
this : ∀ (b : Bool), TwoUniqueProds (I G H b)
x y : G × H
he : (Prod.upMulHom G H) x = (Prod.upMulHom G H) y
⊢ x.2 = y.2

Next tactic:
apply up_injective
open ULift in
@[to_additive] instance [TwoUniqueProds G] [TwoUniqueProds H] : TwoUniqueProds (G × H) := by
  have : ∀ b, TwoUniqueProds (I G H b) := Bool.rec ?_ ?_
  · exact mulHom_image_of_injective (downMulHom H) down_injective ‹_›
  · refine mulHom_image_of_injective (Prod.upMulHom G H) (fun x y he ↦ Prod.ext ?_ ?_)
      (instTwoUniqueProdsForAllInstMul <| I G H) <;> apply up_injective
    

Proof state:
case refine_1.refine_1.a
G : Type u
H : Type v
inst✝³ : Mul G
inst✝² : Mul H
inst✝¹ : TwoUniqueProds G
inst✝ : TwoUniqueProds H
this : ∀ (b : Bool), TwoUniqueProds (I G H b)
x y : G × H
he : (Prod.upMulHom G H) x = (Prod.upMulHom G H) y
⊢ { down := x.1 } = { down := y.1 }
case refine_1.refine_2.a
G : Type u
H : Type v
inst✝³ : Mul G
inst✝² : Mul H
inst✝¹ : TwoUniqueProds G
inst✝ : TwoUniqueProds H
this : ∀ (b : Bool), TwoUniqueProds (I G H b)
x y : G × H
he : (Prod.upMulHom G H) x = (Prod.upMulHom G H) y
⊢ { down := x.2 } = { down := y.2 }

Next tactic:
exacts [congr_fun he false, congr_fun he true]
open ULift in
@[to_additive] instance [TwoUniqueProds G] [TwoUniqueProds H] : TwoUniqueProds (G × H) := by
  have : ∀ b, TwoUniqueProds (I G H b) := Bool.rec ?_ ?_
  · exact mulHom_image_of_injective (downMulHom H) down_injective ‹_›
  · refine mulHom_image_of_injective (Prod.upMulHom G H) (fun x y he ↦ Prod.ext ?_ ?_)
      (instTwoUniqueProdsForAllInstMul <| I G H) <;> apply up_injective
    exacts [congr_fun he false, congr_fun he true]
  · 

Proof state:
case refine_2
G : Type u
H : Type v
inst✝³ : Mul G
inst✝² : Mul H
inst✝¹ : TwoUniqueProds G
inst✝ : TwoUniqueProds H
⊢ TwoUniqueProds (I G H false)

Next tactic:
exact mulHom_image_of_injective (downMulHom G) down_injective ‹_›
open MulOpposite in
@[to_additive]
theorem of_mulOpposite (h : TwoUniqueProds Gᵐᵒᵖ) : TwoUniqueProds G where
  uniqueMul_of_one_lt_card hc := by
    

Proof state:
G : Type u
H : Type v
inst✝¹ : Mul G
inst✝ : Mul H
h : TwoUniqueProds Gᵐᵒᵖ
A✝ B✝ : Finset G
hc : 1 < card A✝ * card B✝
⊢ ∃ p1 ∈ A✝ ×ˢ B✝, ∃ p2 ∈ A✝ ×ˢ B✝, p1 ≠ p2 ∧ UniqueMul A✝ B✝ p1.1 p1.2 ∧ UniqueMul A✝ B✝ p2.1 p2.2

Next tactic:
let f : G ↪ Gᵐᵒᵖ := ⟨op, op_injective⟩
open MulOpposite in
@[to_additive]
theorem of_mulOpposite (h : TwoUniqueProds Gᵐᵒᵖ) : TwoUniqueProds G where
  uniqueMul_of_one_lt_card hc := by
    let f : G ↪ Gᵐᵒᵖ := ⟨op, op_injective⟩
    

Proof state:
G : Type u
H : Type v
inst✝¹ : Mul G
inst✝ : Mul H
h : TwoUniqueProds Gᵐᵒᵖ
A✝ B✝ : Finset G
hc : 1 < card A✝ * card B✝
f : G ↪ Gᵐᵒᵖ := { toFun := op, inj' := (_ : Function.Injective op) }
⊢ ∃ p1 ∈ A✝ ×ˢ B✝, ∃ p2 ∈ A✝ ×ˢ B✝, p1 ≠ p2 ∧ UniqueMul A✝ B✝ p1.1 p1.2 ∧ UniqueMul A✝ B✝ p2.1 p2.2

Next tactic:
rw [← card_map f, ← card_map f, mul_comm] at hc
open MulOpposite in
@[to_additive]
theorem of_mulOpposite (h : TwoUniqueProds Gᵐᵒᵖ) : TwoUniqueProds G where
  uniqueMul_of_one_lt_card hc := by
    let f : G ↪ Gᵐᵒᵖ := ⟨op, op_injective⟩
    rw [← card_map f, ← card_map f, mul_comm] at hc
    

Proof state:
G : Type u
H : Type v
inst✝¹ : Mul G
inst✝ : Mul H
h : TwoUniqueProds Gᵐᵒᵖ
A✝ B✝ : Finset G
f : G ↪ Gᵐᵒᵖ := { toFun := op, inj' := (_ : Function.Injective op) }
hc : 1 < card (map f B✝) * card (map f A✝)
⊢ ∃ p1 ∈ A✝ ×ˢ B✝, ∃ p2 ∈ A✝ ×ˢ B✝, p1 ≠ p2 ∧ UniqueMul A✝ B✝ p1.1 p1.2 ∧ UniqueMul A✝ B✝ p2.1 p2.2

Next tactic:
obtain ⟨p1, h1, p2, h2, hne, hu1, hu2⟩ := h.uniqueMul_of_one_lt_card hc
open MulOpposite in
@[to_additive]
theorem of_mulOpposite (h : TwoUniqueProds Gᵐᵒᵖ) : TwoUniqueProds G where
  uniqueMul_of_one_lt_card hc := by
    let f : G ↪ Gᵐᵒᵖ := ⟨op, op_injective⟩
    rw [← card_map f, ← card_map f, mul_comm] at hc
    obtain ⟨p1, h1, p2, h2, hne, hu1, hu2⟩ := h.uniqueMul_of_one_lt_card hc
    

Proof state:
case intro.intro.intro.intro.intro.intro
G : Type u
H : Type v
inst✝¹ : Mul G
inst✝ : Mul H
h : TwoUniqueProds Gᵐᵒᵖ
A✝ B✝ : Finset G
f : G ↪ Gᵐᵒᵖ := { toFun := op, inj' := (_ : Function.Injective op) }
hc : 1 < card (map f B✝) * card (map f A✝)
p1 : Gᵐᵒᵖ × Gᵐᵒᵖ
h1 : p1 ∈ map f B✝ ×ˢ map f A✝
p2 : Gᵐᵒᵖ × Gᵐᵒᵖ
h2 : p2 ∈ map f B✝ ×ˢ map f A✝
hne : p1 ≠ p2
hu1 : UniqueMul (map f B✝) (map f A✝) p1.1 p1.2
hu2 : UniqueMul (map f B✝) (map f A✝) p2.1 p2.2
⊢ ∃ p1 ∈ A✝ ×ˢ B✝, ∃ p2 ∈ A✝ ×ˢ B✝, p1 ≠ p2 ∧ UniqueMul A✝ B✝ p1.1 p1.2 ∧ UniqueMul A✝ B✝ p2.1 p2.2

Next tactic:
simp_rw [mem_product] at h1 h2 ⊢
open MulOpposite in
@[to_additive]
theorem of_mulOpposite (h : TwoUniqueProds Gᵐᵒᵖ) : TwoUniqueProds G where
  uniqueMul_of_one_lt_card hc := by
    let f : G ↪ Gᵐᵒᵖ := ⟨op, op_injective⟩
    rw [← card_map f, ← card_map f, mul_comm] at hc
    obtain ⟨p1, h1, p2, h2, hne, hu1, hu2⟩ := h.uniqueMul_of_one_lt_card hc
    simp_rw [mem_product] at h1 h2 ⊢
    

Proof state:
case intro.intro.intro.intro.intro.intro
G : Type u
H : Type v
inst✝¹ : Mul G
inst✝ : Mul H
h : TwoUniqueProds Gᵐᵒᵖ
A✝ B✝ : Finset G
f : G ↪ Gᵐᵒᵖ := { toFun := op, inj' := (_ : Function.Injective op) }
hc : 1 < card (map f B✝) * card (map f A✝)
p1 p2 : Gᵐᵒᵖ × Gᵐᵒᵖ
hne : p1 ≠ p2
hu1 : UniqueMul (map f B✝) (map f A✝) p1.1 p1.2
hu2 : UniqueMul (map f B✝) (map f A✝) p2.1 p2.2
h1 :
  p1.1 ∈ map { toFun := op, inj' := (_ : Function.Injective op) } B✝ ∧
    p1.2 ∈ map { toFun := op, inj' := (_ : Function.Injective op) } A✝
h2 :
  p2.1 ∈ map { toFun := op, inj' := (_ : Function.Injective op) } B✝ ∧
    p2.2 ∈ map { toFun := op, inj' := (_ : Function.Injective op) } A✝
⊢ ∃ p1,
    (p1.1 ∈ A✝ ∧ p1.2 ∈ B✝) ∧
      ∃ p2, (p2.1 ∈ A✝ ∧ p2.2 ∈ B✝) ∧ p1 ≠ p2 ∧ UniqueMul A✝ B✝ p1.1 p1.2 ∧ UniqueMul A✝ B✝ p2.1 p2.2

Next tactic:
refine ⟨(_, _), ⟨?_, ?_⟩, (_, _), ⟨?_, ?_⟩, ?_, hu1.of_mulOpposite, hu2.of_mulOpposite⟩
open MulOpposite in
@[to_additive]
theorem of_mulOpposite (h : TwoUniqueProds Gᵐᵒᵖ) : TwoUniqueProds G where
  uniqueMul_of_one_lt_card hc := by
    let f : G ↪ Gᵐᵒᵖ := ⟨op, op_injective⟩
    rw [← card_map f, ← card_map f, mul_comm] at hc
    obtain ⟨p1, h1, p2, h2, hne, hu1, hu2⟩ := h.uniqueMul_of_one_lt_card hc
    simp_rw [mem_product] at h1 h2 ⊢
    refine ⟨(_, _), ⟨?_, ?_⟩, (_, _), ⟨?_, ?_⟩, ?_, hu1.of_mulOpposite, hu2.of_mulOpposite⟩
    pick_goal 5
    · 

Proof state:
case intro.intro.intro.intro.intro.intro.refine_5
G : Type u
H : Type v
inst✝¹ : Mul G
inst✝ : Mul H
h : TwoUniqueProds Gᵐᵒᵖ
A✝ B✝ : Finset G
f : G ↪ Gᵐᵒᵖ := { toFun := op, inj' := (_ : Function.Injective op) }
hc : 1 < card (map f B✝) * card (map f A✝)
p1 p2 : Gᵐᵒᵖ × Gᵐᵒᵖ
hne : p1 ≠ p2
hu1 : UniqueMul (map f B✝) (map f A✝) p1.1 p1.2
hu2 : UniqueMul (map f B✝) (map f A✝) p2.1 p2.2
h1 :
  p1.1 ∈ map { toFun := op, inj' := (_ : Function.Injective op) } B✝ ∧
    p1.2 ∈ map { toFun := op, inj' := (_ : Function.Injective op) } A✝
h2 :
  p2.1 ∈ map { toFun := op, inj' := (_ : Function.Injective op) } B✝ ∧
    p2.2 ∈ map { toFun := op, inj' := (_ : Function.Injective op) } A✝
⊢ (p1.2.1, p1.1.1) ≠ (p2.2.1, p2.1.1)

Next tactic:
contrapose! hne
open MulOpposite in
@[to_additive]
theorem of_mulOpposite (h : TwoUniqueProds Gᵐᵒᵖ) : TwoUniqueProds G where
  uniqueMul_of_one_lt_card hc := by
    let f : G ↪ Gᵐᵒᵖ := ⟨op, op_injective⟩
    rw [← card_map f, ← card_map f, mul_comm] at hc
    obtain ⟨p1, h1, p2, h2, hne, hu1, hu2⟩ := h.uniqueMul_of_one_lt_card hc
    simp_rw [mem_product] at h1 h2 ⊢
    refine ⟨(_, _), ⟨?_, ?_⟩, (_, _), ⟨?_, ?_⟩, ?_, hu1.of_mulOpposite, hu2.of_mulOpposite⟩
    pick_goal 5
    · contrapose! hne; 

Proof state:
case intro.intro.intro.intro.intro.intro.refine_5
G : Type u
H : Type v
inst✝¹ : Mul G
inst✝ : Mul H
h : TwoUniqueProds Gᵐᵒᵖ
A✝ B✝ : Finset G
f : G ↪ Gᵐᵒᵖ := { toFun := op, inj' := (_ : Function.Injective op) }
hc : 1 < card (map f B✝) * card (map f A✝)
p1 p2 : Gᵐᵒᵖ × Gᵐᵒᵖ
hu1 : UniqueMul (map f B✝) (map f A✝) p1.1 p1.2
hu2 : UniqueMul (map f B✝) (map f A✝) p2.1 p2.2
h1 :
  p1.1 ∈ map { toFun := op, inj' := (_ : Function.Injective op) } B✝ ∧
    p1.2 ∈ map { toFun := op, inj' := (_ : Function.Injective op) } A✝
h2 :
  p2.1 ∈ map { toFun := op, inj' := (_ : Function.Injective op) } B✝ ∧
    p2.2 ∈ map { toFun := op, inj' := (_ : Function.Injective op) } A✝
hne : (p1.2.1, p1.1.1) = (p2.2.1, p2.1.1)
⊢ p1 = p2

Next tactic:
rw [Prod.ext_iff] at hne ⊢
open MulOpposite in
@[to_additive]
theorem of_mulOpposite (h : TwoUniqueProds Gᵐᵒᵖ) : TwoUniqueProds G where
  uniqueMul_of_one_lt_card hc := by
    let f : G ↪ Gᵐᵒᵖ := ⟨op, op_injective⟩
    rw [← card_map f, ← card_map f, mul_comm] at hc
    obtain ⟨p1, h1, p2, h2, hne, hu1, hu2⟩ := h.uniqueMul_of_one_lt_card hc
    simp_rw [mem_product] at h1 h2 ⊢
    refine ⟨(_, _), ⟨?_, ?_⟩, (_, _), ⟨?_, ?_⟩, ?_, hu1.of_mulOpposite, hu2.of_mulOpposite⟩
    pick_goal 5
    · contrapose! hne; rw [Prod.ext_iff] at hne ⊢
      

Proof state:
case intro.intro.intro.intro.intro.intro.refine_5
G : Type u
H : Type v
inst✝¹ : Mul G
inst✝ : Mul H
h : TwoUniqueProds Gᵐᵒᵖ
A✝ B✝ : Finset G
f : G ↪ Gᵐᵒᵖ := { toFun := op, inj' := (_ : Function.Injective op) }
hc : 1 < card (map f B✝) * card (map f A✝)
p1 p2 : Gᵐᵒᵖ × Gᵐᵒᵖ
hu1 : UniqueMul (map f B✝) (map f A✝) p1.1 p1.2
hu2 : UniqueMul (map f B✝) (map f A✝) p2.1 p2.2
h1 :
  p1.1 ∈ map { toFun := op, inj' := (_ : Function.Injective op) } B✝ ∧
    p1.2 ∈ map { toFun := op, inj' := (_ : Function.Injective op) } A✝
h2 :
  p2.1 ∈ map { toFun := op, inj' := (_ : Function.Injective op) } B✝ ∧
    p2.2 ∈ map { toFun := op, inj' := (_ : Function.Injective op) } A✝
hne : (p1.2.1, p1.1.1).1 = (p2.2.1, p2.1.1).1 ∧ (p1.2.1, p1.1.1).2 = (p2.2.1, p2.1.1).2
⊢ p1.1 = p2.1 ∧ p1.2 = p2.2

Next tactic:
exact ⟨unop_injective hne.2, unop_injective hne.1⟩