l3lab/ntp-mathlib huggingface dataset:
/-- If a group has `UniqueProds`, then it actually has `TwoUniqueProds`.
  For an example of a semigroup `G` embeddable into a group that has `UniqueProds`
  but not `TwoUniqueProds`, see Example 10.13 in
  [J. Okniński, *Semigroup Algebras*][Okninski1991]. -/
@[to_additive] theorem toTwoUniqueProds_of_group {G}
    [Group G] [UniqueProds G] : TwoUniqueProds G where
  uniqueMul_of_one_lt_card {A B} hc := by
    simp_rw [Nat.one_lt_mul_iff, card_pos] at hc
    obtain ⟨a, ha, b, hb, hu⟩ := uniqueMul_of_nonempty hc.1 hc.2.1
    let C := A.map ⟨_, mul_right_injective a⁻¹⟩ -- C = a⁻¹A
    let D := B.map ⟨_, mul_left_injective b⁻¹⟩  -- D = Bb⁻¹
    have hcard : 1 < C.card ∨ 1 < D.card; · simp_rw [card_map]; exact hc.2.2
    have hC : 1 ∈ C := mem_map.mpr ⟨a, ha, inv_mul_self a⟩
    have hD : 1 ∈ D := mem_map.mpr ⟨b, hb, mul_inv_self b⟩
    suffices : ∃ c ∈ C, ∃ d ∈ D, (c ≠ 1 ∨ d ≠ 1) ∧ UniqueMul C D c d
    · simp_rw [mem_product]
      obtain ⟨c, hc, d, hd, hne, hu'⟩ := this
      obtain ⟨a0, ha0, rfl⟩ := mem_map.mp hc
      obtain ⟨b0, hb0, rfl⟩ := mem_map.mp hd
      refine ⟨(_, _), ⟨ha0, hb0⟩, (a, b), ⟨ha, hb⟩, ?_, fun a' b' ha' hb' he => ?_, hu⟩
      simp_rw [Function.Embedding.coeFn_mk, Ne, inv_mul_eq_one, mul_inv_eq_one] at hne
      · rwa [Ne, Prod.mk.inj_iff, not_and_or, eq_comm]
      specialize hu' (mem_map_of_mem _ ha') (mem_map_of_mem _ hb')
      simp_rw [Function.Embedding.coeFn_mk, mul_left_cancel_iff, mul_right_cancel_iff] at hu'
      rw [mul_assoc, ← mul_assoc a', he, mul_assoc, mul_assoc] at hu'
      exact hu' rfl
    classical
    let _ := Finset.mul (α := G)              -- E = D⁻¹C, F = DC⁻¹
    have := uniqueMul_of_nonempty (A := D.image (·⁻¹) * C) (B := D * C.image (·⁻¹)) ?_ ?_
    · obtain ⟨e, he, f, hf, hu⟩ := this
      clear_value C D
      simp only [UniqueMul, mem_mul, mem_image] at he hf hu
      obtain ⟨_, c1, ⟨d1, hd1, rfl⟩, hc1, rfl⟩ := he
      obtain ⟨d2, _, hd2, ⟨c2, hc2, rfl⟩, rfl⟩ := hf
      by_cases h12 : c1 ≠ 1 ∨ d2 ≠ 1
      · refine ⟨c1, hc1, d2, hd2, h12, fun c3 d3 hc3 hd3 he => ?_⟩
        specialize hu ⟨_, _, ⟨_, hd1, rfl⟩, hc3, rfl⟩ ⟨_, _, hd3, ⟨_, hc2, rfl⟩, rfl⟩
        rw [mul_left_cancel_iff, mul_right_cancel_iff,
            mul_assoc, ← mul_assoc c3, he, mul_assoc, mul_assoc] at hu; exact hu rfl
      push_neg at h12; obtain ⟨rfl, rfl⟩ := h12
      by_cases h21 : c2 ≠ 1 ∨ d1 ≠ 1
      · 

Proof state:
case pos
G✝ : Type u
H : Type v
inst✝³ : Mul G✝
inst✝² : Mul H
G : Type u_1
inst✝¹ : Group G
inst✝ : UniqueProds G
A B : Finset G
hc : Finset.Nonempty A ∧ Finset.Nonempty B ∧ (1 < card A ∨ 1 < card B)
a : G
ha : a ∈ A
b : G
hb : b ∈ B
hu✝ : UniqueMul A B a b
x✝ : Mul (Finset G) := Finset.mul
D : Finset G
hD : 1 ∈ D
C : Finset G
hC : 1 ∈ C
hcard : 1 < card C ∨ 1 < card D
d1 : G
hd1 : d1 ∈ D
c2 : G
hc2 : c2 ∈ C
hc1 : 1 ∈ C
hd2 : 1 ∈ D
hu :
  ∀ ⦃a b : G⦄,
    (∃ y z, (∃ a ∈ D, a⁻¹ = y) ∧ z ∈ C ∧ y * z = a) →
      (∃ y z, y ∈ D ∧ (∃ a ∈ C, a⁻¹ = z) ∧ y * z = b) → a * b = d1⁻¹ * 1 * (1 * c2⁻¹) → a = d1⁻¹ * 1 ∧ b = 1 * c2⁻¹
h21 : c2 ≠ 1 ∨ d1 ≠ 1
⊢ ∃ c ∈ C, ∃ d ∈ D, (c ≠ 1 ∨ d ≠ 1) ∧ UniqueMul C D c d

Next tactic:
refine ⟨c2, hc2, d1, hd1, h21, fun c4 d4 hc4 hd4 he => ?_⟩
open UniqueMul in
@[to_additive] instance {ι} (G : ι → Type*) [∀ i, Mul (G i)] [∀ i, UniqueProds (G i)] :
    UniqueProds (∀ i, G i) where
  uniqueMul_of_nonempty {A} := by
    

Proof state:
G✝ : Type u
H : Type v
inst✝³ : Mul G✝
inst✝² : Mul H
ι : Type ?u.379291
G : ι → Type u_1
inst✝¹ : (i : ι) → Mul (G i)
inst✝ : ∀ (i : ι), UniqueProds (G i)
A : Finset ((i : ι) → G i)
⊢ ∀ {B : Finset ((i : ι) → G i)}, Finset.Nonempty A → Finset.Nonempty B → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMul A B a0 b0

Next tactic:
classical
    let _ := isWellFounded_ssubset (α := ∀ i, G i) -- why need this?
    apply IsWellFounded.induction (· ⊂ ·) A; intro A ihA B hA
    apply IsWellFounded.induction (· ⊂ ·) B; intro B ihB hB
    by_cases hc : A.card ≤ 1 ∧ B.card ≤ 1
    · exact of_card_le_one hA hB hc.1 hc.2
    simp_rw [not_and_or, not_le] at hc
    obtain ⟨i, hc⟩ := exists_or.mpr (hc.imp exists_of_one_lt_card_pi exists_of_one_lt_card_pi)
    obtain ⟨ai, hA, bi, hB, hi⟩ := uniqueMul_of_nonempty (hA.image (· i)) (hB.image (· i))
    rw [mem_image, ← filter_nonempty_iff] at hA hB
    let A' := A.filter (· i = ai); let B' := B.filter (· i = bi)
    obtain ⟨a0, ha0, b0, hb0, hu⟩ : ∃ a0 ∈ A', ∃ b0 ∈ B', UniqueMul A' B' a0 b0
    · rcases hc with hc | hc; · exact ihA A' (hc.2 ai) hA hB
      by_cases hA' : A' = A; rw [hA']
      exacts [ihB B' (hc.2 bi) hB, ihA A' ((A.filter_subset _).ssubset_of_ne hA') hA hB]
    rw [mem_filter] at ha0 hb0
    exact ⟨a0, ha0.1, b0, hb0.1, of_image_filter (Pi.evalMulHom G i) ha0.2 hb0.2 hi hu⟩
open UniqueMul in
@[to_additive] instance {ι} (G : ι → Type*) [∀ i, Mul (G i)] [∀ i, UniqueProds (G i)] :
    UniqueProds (∀ i, G i) where
  uniqueMul_of_nonempty {A} := by
    classical
    

Proof state:
G✝ : Type u
H : Type v
inst✝³ : Mul G✝
inst✝² : Mul H
ι : Type ?u.379291
G : ι → Type u_1
inst✝¹ : (i : ι) → Mul (G i)
inst✝ : ∀ (i : ι), UniqueProds (G i)
A : Finset ((i : ι) → G i)
⊢ ∀ {B : Finset ((i : ι) → G i)}, Finset.Nonempty A → Finset.Nonempty B → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMul A B a0 b0

Next tactic:
let _ := isWellFounded_ssubset (α := ∀ i, G i)
open UniqueMul in
@[to_additive] instance {ι} (G : ι → Type*) [∀ i, Mul (G i)] [∀ i, UniqueProds (G i)] :
    UniqueProds (∀ i, G i) where
  uniqueMul_of_nonempty {A} := by
    classical
    let _ := isWellFounded_ssubset (α := ∀ i, G i) -- why need this?
    

Proof state:
G✝ : Type u
H : Type v
inst✝³ : Mul G✝
inst✝² : Mul H
ι : Type ?u.379291
G : ι → Type u_1
inst✝¹ : (i : ι) → Mul (G i)
inst✝ : ∀ (i : ι), UniqueProds (G i)
A : Finset ((i : ι) → G i)
x✝ : IsWellFounded (Finset ((i : ι) → G i)) fun x x_1 => x ⊂ x_1 := isWellFounded_ssubset
⊢ ∀ {B : Finset ((i : ι) → G i)}, Finset.Nonempty A → Finset.Nonempty B → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMul A B a0 b0

Next tactic:
apply IsWellFounded.induction (· ⊂ ·) A
open UniqueMul in
@[to_additive] instance {ι} (G : ι → Type*) [∀ i, Mul (G i)] [∀ i, UniqueProds (G i)] :
    UniqueProds (∀ i, G i) where
  uniqueMul_of_nonempty {A} := by
    classical
    let _ := isWellFounded_ssubset (α := ∀ i, G i) -- why need this?
    apply IsWellFounded.induction (· ⊂ ·) A; 

Proof state:
G✝ : Type u
H : Type v
inst✝³ : Mul G✝
inst✝² : Mul H
ι : Type ?u.379291
G : ι → Type u_1
inst✝¹ : (i : ι) → Mul (G i)
inst✝ : ∀ (i : ι), UniqueProds (G i)
A : Finset ((i : ι) → G i)
x✝ : IsWellFounded (Finset ((i : ι) → G i)) fun x x_1 => x ⊂ x_1 := isWellFounded_ssubset
⊢ ∀ (x : Finset ((i : ι) → G i)),
    (∀ y ⊂ x,
        ∀ {B : Finset ((i : ι) → G i)},
          Finset.Nonempty y → Finset.Nonempty B → ∃ a0 ∈ y, ∃ b0 ∈ B, UniqueMul y B a0 b0) →
      ∀ {B : Finset ((i : ι) → G i)}, Finset.Nonempty x → Finset.Nonempty B → ∃ a0 ∈ x, ∃ b0 ∈ B, UniqueMul x B a0 b0

Next tactic:
intro A ihA B hA
open UniqueMul in
@[to_additive] instance {ι} (G : ι → Type*) [∀ i, Mul (G i)] [∀ i, UniqueProds (G i)] :
    UniqueProds (∀ i, G i) where
  uniqueMul_of_nonempty {A} := by
    classical
    let _ := isWellFounded_ssubset (α := ∀ i, G i) -- why need this?
    apply IsWellFounded.induction (· ⊂ ·) A; intro A ihA B hA
    

Proof state:
G✝ : Type u
H : Type v
inst✝³ : Mul G✝
inst✝² : Mul H
ι : Type ?u.379291
G : ι → Type u_1
inst✝¹ : (i : ι) → Mul (G i)
inst✝ : ∀ (i : ι), UniqueProds (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)}, Finset.Nonempty y → Finset.Nonempty B → ∃ a0 ∈ y, ∃ b0 ∈ B, UniqueMul y B a0 b0
B : Finset ((i : ι) → G i)
hA : Finset.Nonempty A
⊢ Finset.Nonempty B → ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMul A B a0 b0

Next tactic:
apply IsWellFounded.induction (· ⊂ ·) B
open UniqueMul in
@[to_additive] instance {ι} (G : ι → Type*) [∀ i, Mul (G i)] [∀ i, UniqueProds (G i)] :
    UniqueProds (∀ i, G i) where
  uniqueMul_of_nonempty {A} := by
    classical
    let _ := isWellFounded_ssubset (α := ∀ i, G i) -- why need this?
    apply IsWellFounded.induction (· ⊂ ·) A; intro A ihA B hA
    apply IsWellFounded.induction (· ⊂ ·) B; 

Proof state:
G✝ : Type u
H : Type v
inst✝³ : Mul G✝
inst✝² : Mul H
ι : Type ?u.379291
G : ι → Type u_1
inst✝¹ : (i : ι) → Mul (G i)
inst✝ : ∀ (i : ι), UniqueProds (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)}, Finset.Nonempty y → Finset.Nonempty B → ∃ a0 ∈ y, ∃ b0 ∈ B, UniqueMul y B a0 b0
B : Finset ((i : ι) → G i)
hA : Finset.Nonempty A
⊢ ∀ (x : Finset ((i : ι) → G i)),
    (∀ y ⊂ x, Finset.Nonempty y → ∃ a0 ∈ A, ∃ b0 ∈ y, UniqueMul A y a0 b0) →
      Finset.Nonempty x → ∃ a0 ∈ A, ∃ b0 ∈ x, UniqueMul A x a0 b0

Next tactic:
intro B ihB hB
open UniqueMul in
@[to_additive] instance {ι} (G : ι → Type*) [∀ i, Mul (G i)] [∀ i, UniqueProds (G i)] :
    UniqueProds (∀ i, G i) where
  uniqueMul_of_nonempty {A} := by
    classical
    let _ := isWellFounded_ssubset (α := ∀ i, G i) -- why need this?
    apply IsWellFounded.induction (· ⊂ ·) A; intro A ihA B hA
    apply IsWellFounded.induction (· ⊂ ·) B; intro B ihB hB
    

Proof state:
G✝ : Type u
H : Type v
inst✝³ : Mul G✝
inst✝² : Mul H
ι : Type ?u.379291
G : ι → Type u_1
inst✝¹ : (i : ι) → Mul (G i)
inst✝ : ∀ (i : ι), UniqueProds (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)}, Finset.Nonempty y → Finset.Nonempty B → ∃ a0 ∈ y, ∃ b0 ∈ B, UniqueMul y B a0 b0
B✝ : Finset ((i : ι) → G i)
hA : Finset.Nonempty A
B : Finset ((i : ι) → G i)
ihB : ∀ y ⊂ B, Finset.Nonempty y → ∃ a0 ∈ A, ∃ b0 ∈ y, UniqueMul A y a0 b0
hB : Finset.Nonempty B
⊢ ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMul A B a0 b0

Next tactic:
by_cases hc : A.card ≤ 1 ∧ B.card ≤ 1
open UniqueMul in
@[to_additive] instance {ι} (G : ι → Type*) [∀ i, Mul (G i)] [∀ i, UniqueProds (G i)] :
    UniqueProds (∀ i, G i) where
  uniqueMul_of_nonempty {A} := by
    classical
    let _ := isWellFounded_ssubset (α := ∀ i, G i) -- why need this?
    apply IsWellFounded.induction (· ⊂ ·) A; intro A ihA B hA
    apply IsWellFounded.induction (· ⊂ ·) B; intro B ihB hB
    by_cases hc : A.card ≤ 1 ∧ B.card ≤ 1
    · 

Proof state:
case pos
G✝ : Type u
H : Type v
inst✝³ : Mul G✝
inst✝² : Mul H
ι : Type ?u.379291
G : ι → Type u_1
inst✝¹ : (i : ι) → Mul (G i)
inst✝ : ∀ (i : ι), UniqueProds (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)}, Finset.Nonempty y → Finset.Nonempty B → ∃ a0 ∈ y, ∃ b0 ∈ B, UniqueMul y B a0 b0
B✝ : Finset ((i : ι) → G i)
hA : Finset.Nonempty A
B : Finset ((i : ι) → G i)
ihB : ∀ y ⊂ B, Finset.Nonempty y → ∃ a0 ∈ A, ∃ b0 ∈ y, UniqueMul A y a0 b0
hB : Finset.Nonempty B
hc : card A ≤ 1 ∧ card B ≤ 1
⊢ ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMul A B a0 b0

Next tactic:
exact of_card_le_one hA hB hc.1 hc.2
open UniqueMul in
@[to_additive] instance {ι} (G : ι → Type*) [∀ i, Mul (G i)] [∀ i, UniqueProds (G i)] :
    UniqueProds (∀ i, G i) where
  uniqueMul_of_nonempty {A} := by
    classical
    let _ := isWellFounded_ssubset (α := ∀ i, G i) -- why need this?
    apply IsWellFounded.induction (· ⊂ ·) A; intro A ihA B hA
    apply IsWellFounded.induction (· ⊂ ·) B; intro B ihB hB
    by_cases hc : A.card ≤ 1 ∧ B.card ≤ 1
    · exact of_card_le_one hA hB hc.1 hc.2
    

Proof state:
case neg
G✝ : Type u
H : Type v
inst✝³ : Mul G✝
inst✝² : Mul H
ι : Type ?u.379291
G : ι → Type u_1
inst✝¹ : (i : ι) → Mul (G i)
inst✝ : ∀ (i : ι), UniqueProds (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)}, Finset.Nonempty y → Finset.Nonempty B → ∃ a0 ∈ y, ∃ b0 ∈ B, UniqueMul y B a0 b0
B✝ : Finset ((i : ι) → G i)
hA : Finset.Nonempty A
B : Finset ((i : ι) → G i)
ihB : ∀ y ⊂ B, Finset.Nonempty y → ∃ a0 ∈ A, ∃ b0 ∈ y, UniqueMul A y a0 b0
hB : Finset.Nonempty B
hc : ¬(card A ≤ 1 ∧ card B ≤ 1)
⊢ ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMul A B a0 b0

Next tactic:
simp_rw [not_and_or, not_le] at hc
open UniqueMul in
@[to_additive] instance {ι} (G : ι → Type*) [∀ i, Mul (G i)] [∀ i, UniqueProds (G i)] :
    UniqueProds (∀ i, G i) where
  uniqueMul_of_nonempty {A} := by
    classical
    let _ := isWellFounded_ssubset (α := ∀ i, G i) -- why need this?
    apply IsWellFounded.induction (· ⊂ ·) A; intro A ihA B hA
    apply IsWellFounded.induction (· ⊂ ·) B; intro B ihB hB
    by_cases hc : A.card ≤ 1 ∧ B.card ≤ 1
    · exact of_card_le_one hA hB hc.1 hc.2
    simp_rw [not_and_or, not_le] at hc
    

Proof state:
case neg
G✝ : Type u
H : Type v
inst✝³ : Mul G✝
inst✝² : Mul H
ι : Type ?u.379291
G : ι → Type u_1
inst✝¹ : (i : ι) → Mul (G i)
inst✝ : ∀ (i : ι), UniqueProds (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)}, Finset.Nonempty y → Finset.Nonempty B → ∃ a0 ∈ y, ∃ b0 ∈ B, UniqueMul y B a0 b0
B✝ : Finset ((i : ι) → G i)
hA : Finset.Nonempty A
B : Finset ((i : ι) → G i)
ihB : ∀ y ⊂ B, Finset.Nonempty y → ∃ a0 ∈ A, ∃ b0 ∈ y, UniqueMul A y a0 b0
hB : Finset.Nonempty B
hc : 1 < card A ∨ 1 < card B
⊢ ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMul A B a0 b0

Next tactic:
obtain ⟨i, hc⟩ := exists_or.mpr (hc.imp exists_of_one_lt_card_pi exists_of_one_lt_card_pi)
open UniqueMul in
@[to_additive] instance {ι} (G : ι → Type*) [∀ i, Mul (G i)] [∀ i, UniqueProds (G i)] :
    UniqueProds (∀ i, G i) where
  uniqueMul_of_nonempty {A} := by
    classical
    let _ := isWellFounded_ssubset (α := ∀ i, G i) -- why need this?
    apply IsWellFounded.induction (· ⊂ ·) A; intro A ihA B hA
    apply IsWellFounded.induction (· ⊂ ·) B; intro B ihB hB
    by_cases hc : A.card ≤ 1 ∧ B.card ≤ 1
    · exact of_card_le_one hA hB hc.1 hc.2
    simp_rw [not_and_or, not_le] at hc
    obtain ⟨i, hc⟩ := exists_or.mpr (hc.imp exists_of_one_lt_card_pi exists_of_one_lt_card_pi)
    

Proof state:
case neg.intro
G✝ : Type u
H : Type v
inst✝³ : Mul G✝
inst✝² : Mul H
ι : Type ?u.379291
G : ι → Type u_1
inst✝¹ : (i : ι) → Mul (G i)
inst✝ : ∀ (i : ι), UniqueProds (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)}, Finset.Nonempty y → Finset.Nonempty B → ∃ a0 ∈ y, ∃ b0 ∈ B, UniqueMul y B a0 b0
B✝ : Finset ((i : ι) → G i)
hA : Finset.Nonempty A
B : Finset ((i : ι) → G i)
ihB : ∀ y ⊂ B, Finset.Nonempty y → ∃ a0 ∈ A, ∃ b0 ∈ y, UniqueMul A y a0 b0
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
⊢ ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMul A B a0 b0

Next tactic:
obtain ⟨ai, hA, bi, hB, hi⟩ := uniqueMul_of_nonempty (hA.image (· i)) (hB.image (· i))
open UniqueMul in
@[to_additive] instance {ι} (G : ι → Type*) [∀ i, Mul (G i)] [∀ i, UniqueProds (G i)] :
    UniqueProds (∀ i, G i) where
  uniqueMul_of_nonempty {A} := by
    classical
    let _ := isWellFounded_ssubset (α := ∀ i, G i) -- why need this?
    apply IsWellFounded.induction (· ⊂ ·) A; intro A ihA B hA
    apply IsWellFounded.induction (· ⊂ ·) B; intro B ihB hB
    by_cases hc : A.card ≤ 1 ∧ B.card ≤ 1
    · exact of_card_le_one hA hB hc.1 hc.2
    simp_rw [not_and_or, not_le] at hc
    obtain ⟨i, hc⟩ := exists_or.mpr (hc.imp exists_of_one_lt_card_pi exists_of_one_lt_card_pi)
    obtain ⟨ai, hA, bi, hB, hi⟩ := uniqueMul_of_nonempty (hA.image (· i)) (hB.image (· i))
    

Proof state:
case neg.intro.intro.intro.intro.intro
G✝ : Type u
H : Type v
inst✝³ : Mul G✝
inst✝² : Mul H
ι : Type ?u.379291
G : ι → Type u_1
inst✝¹ : (i : ι) → Mul (G i)
inst✝ : ∀ (i : ι), UniqueProds (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)}, Finset.Nonempty y → Finset.Nonempty B → ∃ a0 ∈ y, ∃ b0 ∈ B, UniqueMul y B a0 b0
B✝ : Finset ((i : ι) → G i)
hA✝ : Finset.Nonempty A
B : Finset ((i : ι) → G i)
ihB : ∀ y ⊂ B, Finset.Nonempty y → ∃ a0 ∈ A, ∃ b0 ∈ y, UniqueMul A y a0 b0
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
ai : G i
hA : ai ∈ image (fun x => x i) A
bi : G i
hB : bi ∈ image (fun x => x i) B
hi : UniqueMul (image (fun x => x i) A) (image (fun x => x i) B) ai bi
⊢ ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMul A B a0 b0

Next tactic:
rw [mem_image, ← filter_nonempty_iff] at hA hB
open UniqueMul in
@[to_additive] instance {ι} (G : ι → Type*) [∀ i, Mul (G i)] [∀ i, UniqueProds (G i)] :
    UniqueProds (∀ i, G i) where
  uniqueMul_of_nonempty {A} := by
    classical
    let _ := isWellFounded_ssubset (α := ∀ i, G i) -- why need this?
    apply IsWellFounded.induction (· ⊂ ·) A; intro A ihA B hA
    apply IsWellFounded.induction (· ⊂ ·) B; intro B ihB hB
    by_cases hc : A.card ≤ 1 ∧ B.card ≤ 1
    · exact of_card_le_one hA hB hc.1 hc.2
    simp_rw [not_and_or, not_le] at hc
    obtain ⟨i, hc⟩ := exists_or.mpr (hc.imp exists_of_one_lt_card_pi exists_of_one_lt_card_pi)
    obtain ⟨ai, hA, bi, hB, hi⟩ := uniqueMul_of_nonempty (hA.image (· i)) (hB.image (· i))
    rw [mem_image, ← filter_nonempty_iff] at hA hB
    

Proof state:
case neg.intro.intro.intro.intro.intro
G✝ : Type u
H : Type v
inst✝³ : Mul G✝
inst✝² : Mul H
ι : Type ?u.379291
G : ι → Type u_1
inst✝¹ : (i : ι) → Mul (G i)
inst✝ : ∀ (i : ι), UniqueProds (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)}, Finset.Nonempty y → Finset.Nonempty B → ∃ a0 ∈ y, ∃ b0 ∈ B, UniqueMul y B a0 b0
B✝ : Finset ((i : ι) → G i)
hA✝ : Finset.Nonempty A
B : Finset ((i : ι) → G i)
ihB : ∀ y ⊂ B, Finset.Nonempty y → ∃ a0 ∈ A, ∃ b0 ∈ y, UniqueMul A y a0 b0
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
ai : G i
hA : Finset.Nonempty (filter (fun a => a i = ai) A)
bi : G i
hB : Finset.Nonempty (filter (fun a => a i = bi) B)
hi : UniqueMul (image (fun x => x i) A) (image (fun x => x i) B) ai bi
⊢ ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMul A B a0 b0

Next tactic:
let A' := A.filter (· i = ai)
open UniqueMul in
@[to_additive] instance {ι} (G : ι → Type*) [∀ i, Mul (G i)] [∀ i, UniqueProds (G i)] :
    UniqueProds (∀ i, G i) where
  uniqueMul_of_nonempty {A} := by
    classical
    let _ := isWellFounded_ssubset (α := ∀ i, G i) -- why need this?
    apply IsWellFounded.induction (· ⊂ ·) A; intro A ihA B hA
    apply IsWellFounded.induction (· ⊂ ·) B; intro B ihB hB
    by_cases hc : A.card ≤ 1 ∧ B.card ≤ 1
    · exact of_card_le_one hA hB hc.1 hc.2
    simp_rw [not_and_or, not_le] at hc
    obtain ⟨i, hc⟩ := exists_or.mpr (hc.imp exists_of_one_lt_card_pi exists_of_one_lt_card_pi)
    obtain ⟨ai, hA, bi, hB, hi⟩ := uniqueMul_of_nonempty (hA.image (· i)) (hB.image (· i))
    rw [mem_image, ← filter_nonempty_iff] at hA hB
    let A' := A.filter (· i = ai); 

Proof state:
case neg.intro.intro.intro.intro.intro
G✝ : Type u
H : Type v
inst✝³ : Mul G✝
inst✝² : Mul H
ι : Type ?u.379291
G : ι → Type u_1
inst✝¹ : (i : ι) → Mul (G i)
inst✝ : ∀ (i : ι), UniqueProds (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)}, Finset.Nonempty y → Finset.Nonempty B → ∃ a0 ∈ y, ∃ b0 ∈ B, UniqueMul y B a0 b0
B✝ : Finset ((i : ι) → G i)
hA✝ : Finset.Nonempty A
B : Finset ((i : ι) → G i)
ihB : ∀ y ⊂ B, Finset.Nonempty y → ∃ a0 ∈ A, ∃ b0 ∈ y, UniqueMul A y a0 b0
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
ai : G i
hA : Finset.Nonempty (filter (fun a => a i = ai) A)
bi : G i
hB : Finset.Nonempty (filter (fun a => a i = bi) B)
hi : UniqueMul (image (fun x => x i) A) (image (fun x => x i) B) ai bi
A' : Finset ((i : ι) → G i) := filter (fun x => x i = ai) A
⊢ ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMul A B a0 b0

Next tactic:
let B' := B.filter (· i = bi)
open UniqueMul in
@[to_additive] instance {ι} (G : ι → Type*) [∀ i, Mul (G i)] [∀ i, UniqueProds (G i)] :
    UniqueProds (∀ i, G i) where
  uniqueMul_of_nonempty {A} := by
    classical
    let _ := isWellFounded_ssubset (α := ∀ i, G i) -- why need this?
    apply IsWellFounded.induction (· ⊂ ·) A; intro A ihA B hA
    apply IsWellFounded.induction (· ⊂ ·) B; intro B ihB hB
    by_cases hc : A.card ≤ 1 ∧ B.card ≤ 1
    · exact of_card_le_one hA hB hc.1 hc.2
    simp_rw [not_and_or, not_le] at hc
    obtain ⟨i, hc⟩ := exists_or.mpr (hc.imp exists_of_one_lt_card_pi exists_of_one_lt_card_pi)
    obtain ⟨ai, hA, bi, hB, hi⟩ := uniqueMul_of_nonempty (hA.image (· i)) (hB.image (· i))
    rw [mem_image, ← filter_nonempty_iff] at hA hB
    let A' := A.filter (· i = ai); let B' := B.filter (· i = bi)
    

Proof state:
case neg.intro.intro.intro.intro.intro
G✝ : Type u
H : Type v
inst✝³ : Mul G✝
inst✝² : Mul H
ι : Type ?u.379291
G : ι → Type u_1
inst✝¹ : (i : ι) → Mul (G i)
inst✝ : ∀ (i : ι), UniqueProds (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)}, Finset.Nonempty y → Finset.Nonempty B → ∃ a0 ∈ y, ∃ b0 ∈ B, UniqueMul y B a0 b0
B✝ : Finset ((i : ι) → G i)
hA✝ : Finset.Nonempty A
B : Finset ((i : ι) → G i)
ihB : ∀ y ⊂ B, Finset.Nonempty y → ∃ a0 ∈ A, ∃ b0 ∈ y, UniqueMul A y a0 b0
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
ai : G i
hA : Finset.Nonempty (filter (fun a => a i = ai) A)
bi : G i
hB : Finset.Nonempty (filter (fun a => a i = bi) B)
hi : UniqueMul (image (fun x => x i) A) (image (fun x => x i) B) ai bi
A' : Finset ((i : ι) → G i) := filter (fun x => x i = ai) A
B' : Finset ((i : ι) → G i) := filter (fun x => x i = bi) B
⊢ ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMul A B a0 b0

Next tactic:
obtain ⟨a0, ha0, b0, hb0, hu⟩ : ∃ a0 ∈ A', ∃ b0 ∈ B', UniqueMul A' B' a0 b0