l3lab/ntp-mathlib huggingface dataset:
@[to_additive]
theorem card_pow_eq_card_pow_card_univ [∀ k : ℕ, DecidablePred (· ∈ S ^ k)] :
    ∀ k, Fintype.card G ≤ k → Fintype.card (↥(S ^ k)) = Fintype.card (↥(S ^ Fintype.card G)) := by
  have hG : 0 < Fintype.card G := Fintype.card_pos_iff.mpr ⟨1⟩
  by_cases hS : S = ∅
  · refine' fun k hk ↦ Fintype.card_congr _
    rw [hS, empty_pow (ne_of_gt (lt_of_lt_of_le hG hk)), empty_pow (ne_of_gt hG)]
  obtain ⟨a, ha⟩ := Set.nonempty_iff_ne_empty.2 hS
  have key : ∀ (a) (s t : Set G) [Fintype s] [Fintype t],
      (∀ b : G, b ∈ s → a * b ∈ t) → Fintype.card s ≤ Fintype.card t := by
    refine' fun a s t _ _ h ↦ Fintype.card_le_of_injective (fun ⟨b, hb⟩ ↦ ⟨a * b, h b hb⟩) _
    rintro ⟨b, hb⟩ ⟨c, hc⟩ hbc
    exact Subtype.ext (mul_left_cancel (Subtype.ext_iff.mp hbc))
  

Proof state:
case neg.intro
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
G : Type u_5
inst✝² : Group G
inst✝¹ : Fintype G
S : Set G
inst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k
hG : 0 < Fintype.card G
hS : ¬S = ∅
a : G
ha : a ∈ S
key :
  ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],
    (∀ b ∈ s, a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t
⊢ ∀ (k : ℕ), Fintype.card G ≤ k → Fintype.card ↑(S ^ k) = Fintype.card ↑(S ^ Fintype.card G)

Next tactic:
have mono : Monotone (fun n ↦ Fintype.card (↥(S ^ n)) : ℕ → ℕ) :=
    monotone_nat_of_le_succ fun n ↦ key a _ _ fun b hb ↦ Set.mul_mem_mul ha hb
@[to_additive]
theorem card_pow_eq_card_pow_card_univ [∀ k : ℕ, DecidablePred (· ∈ S ^ k)] :
    ∀ k, Fintype.card G ≤ k → Fintype.card (↥(S ^ k)) = Fintype.card (↥(S ^ Fintype.card G)) := by
  have hG : 0 < Fintype.card G := Fintype.card_pos_iff.mpr ⟨1⟩
  by_cases hS : S = ∅
  · refine' fun k hk ↦ Fintype.card_congr _
    rw [hS, empty_pow (ne_of_gt (lt_of_lt_of_le hG hk)), empty_pow (ne_of_gt hG)]
  obtain ⟨a, ha⟩ := Set.nonempty_iff_ne_empty.2 hS
  have key : ∀ (a) (s t : Set G) [Fintype s] [Fintype t],
      (∀ b : G, b ∈ s → a * b ∈ t) → Fintype.card s ≤ Fintype.card t := by
    refine' fun a s t _ _ h ↦ Fintype.card_le_of_injective (fun ⟨b, hb⟩ ↦ ⟨a * b, h b hb⟩) _
    rintro ⟨b, hb⟩ ⟨c, hc⟩ hbc
    exact Subtype.ext (mul_left_cancel (Subtype.ext_iff.mp hbc))
  have mono : Monotone (fun n ↦ Fintype.card (↥(S ^ n)) : ℕ → ℕ) :=
    monotone_nat_of_le_succ fun n ↦ key a _ _ fun b hb ↦ Set.mul_mem_mul ha hb
  

Proof state:
case neg.intro
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
G : Type u_5
inst✝² : Group G
inst✝¹ : Fintype G
S : Set G
inst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k
hG : 0 < Fintype.card G
hS : ¬S = ∅
a : G
ha : a ∈ S
key :
  ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],
    (∀ b ∈ s, a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t
mono : Monotone fun n => Fintype.card ↑(S ^ n)
⊢ ∀ (k : ℕ), Fintype.card G ≤ k → Fintype.card ↑(S ^ k) = Fintype.card ↑(S ^ Fintype.card G)

Next tactic:
refine' card_pow_eq_card_pow_card_univ_aux mono (fun n ↦ set_fintype_card_le_univ (S ^ n))
    fun n h ↦ le_antisymm (mono (n + 1).le_succ) (key a⁻¹ (S ^ (n + 2)) (S ^ (n + 1)) _)
@[to_additive]
theorem card_pow_eq_card_pow_card_univ [∀ k : ℕ, DecidablePred (· ∈ S ^ k)] :
    ∀ k, Fintype.card G ≤ k → Fintype.card (↥(S ^ k)) = Fintype.card (↥(S ^ Fintype.card G)) := by
  have hG : 0 < Fintype.card G := Fintype.card_pos_iff.mpr ⟨1⟩
  by_cases hS : S = ∅
  · refine' fun k hk ↦ Fintype.card_congr _
    rw [hS, empty_pow (ne_of_gt (lt_of_lt_of_le hG hk)), empty_pow (ne_of_gt hG)]
  obtain ⟨a, ha⟩ := Set.nonempty_iff_ne_empty.2 hS
  have key : ∀ (a) (s t : Set G) [Fintype s] [Fintype t],
      (∀ b : G, b ∈ s → a * b ∈ t) → Fintype.card s ≤ Fintype.card t := by
    refine' fun a s t _ _ h ↦ Fintype.card_le_of_injective (fun ⟨b, hb⟩ ↦ ⟨a * b, h b hb⟩) _
    rintro ⟨b, hb⟩ ⟨c, hc⟩ hbc
    exact Subtype.ext (mul_left_cancel (Subtype.ext_iff.mp hbc))
  have mono : Monotone (fun n ↦ Fintype.card (↥(S ^ n)) : ℕ → ℕ) :=
    monotone_nat_of_le_succ fun n ↦ key a _ _ fun b hb ↦ Set.mul_mem_mul ha hb
  refine' card_pow_eq_card_pow_card_univ_aux mono (fun n ↦ set_fintype_card_le_univ (S ^ n))
    fun n h ↦ le_antisymm (mono (n + 1).le_succ) (key a⁻¹ (S ^ (n + 2)) (S ^ (n + 1)) _)
  

Proof state:
case neg.intro
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
G : Type u_5
inst✝² : Group G
inst✝¹ : Fintype G
S : Set G
inst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k
hG : 0 < Fintype.card G
hS : ¬S = ∅
a : G
ha : a ∈ S
key :
  ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],
    (∀ b ∈ s, a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t
mono : Monotone fun n => Fintype.card ↑(S ^ n)
n : ℕ
h : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))
⊢ ∀ b ∈ S ^ (n + 2), a⁻¹ * b ∈ S ^ (n + 1)

Next tactic:
replace h₂ : {a} * S ^ n = S ^ (n + 1)
@[to_additive]
theorem card_pow_eq_card_pow_card_univ [∀ k : ℕ, DecidablePred (· ∈ S ^ k)] :
    ∀ k, Fintype.card G ≤ k → Fintype.card (↥(S ^ k)) = Fintype.card (↥(S ^ Fintype.card G)) := by
  have hG : 0 < Fintype.card G := Fintype.card_pos_iff.mpr ⟨1⟩
  by_cases hS : S = ∅
  · refine' fun k hk ↦ Fintype.card_congr _
    rw [hS, empty_pow (ne_of_gt (lt_of_lt_of_le hG hk)), empty_pow (ne_of_gt hG)]
  obtain ⟨a, ha⟩ := Set.nonempty_iff_ne_empty.2 hS
  have key : ∀ (a) (s t : Set G) [Fintype s] [Fintype t],
      (∀ b : G, b ∈ s → a * b ∈ t) → Fintype.card s ≤ Fintype.card t := by
    refine' fun a s t _ _ h ↦ Fintype.card_le_of_injective (fun ⟨b, hb⟩ ↦ ⟨a * b, h b hb⟩) _
    rintro ⟨b, hb⟩ ⟨c, hc⟩ hbc
    exact Subtype.ext (mul_left_cancel (Subtype.ext_iff.mp hbc))
  have mono : Monotone (fun n ↦ Fintype.card (↥(S ^ n)) : ℕ → ℕ) :=
    monotone_nat_of_le_succ fun n ↦ key a _ _ fun b hb ↦ Set.mul_mem_mul ha hb
  refine' card_pow_eq_card_pow_card_univ_aux mono (fun n ↦ set_fintype_card_le_univ (S ^ n))
    fun n h ↦ le_antisymm (mono (n + 1).le_succ) (key a⁻¹ (S ^ (n + 2)) (S ^ (n + 1)) _)
  replace h₂ : {a} * S ^ n = S ^ (n + 1)
  · 

Proof state:
case h₂
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
G : Type u_5
inst✝² : Group G
inst✝¹ : Fintype G
S : Set G
inst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k
hG : 0 < Fintype.card G
hS : ¬S = ∅
a : G
ha : a ∈ S
key :
  ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],
    (∀ b ∈ s, a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t
mono : Monotone fun n => Fintype.card ↑(S ^ n)
n : ℕ
h : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))
⊢ {a} * S ^ n = S ^ (n + 1)

Next tactic:
have : Fintype (Set.singleton a * S ^ n) := by
      classical!
      apply fintypeMul
@[to_additive]
theorem card_pow_eq_card_pow_card_univ [∀ k : ℕ, DecidablePred (· ∈ S ^ k)] :
    ∀ k, Fintype.card G ≤ k → Fintype.card (↥(S ^ k)) = Fintype.card (↥(S ^ Fintype.card G)) := by
  have hG : 0 < Fintype.card G := Fintype.card_pos_iff.mpr ⟨1⟩
  by_cases hS : S = ∅
  · refine' fun k hk ↦ Fintype.card_congr _
    rw [hS, empty_pow (ne_of_gt (lt_of_lt_of_le hG hk)), empty_pow (ne_of_gt hG)]
  obtain ⟨a, ha⟩ := Set.nonempty_iff_ne_empty.2 hS
  have key : ∀ (a) (s t : Set G) [Fintype s] [Fintype t],
      (∀ b : G, b ∈ s → a * b ∈ t) → Fintype.card s ≤ Fintype.card t := by
    refine' fun a s t _ _ h ↦ Fintype.card_le_of_injective (fun ⟨b, hb⟩ ↦ ⟨a * b, h b hb⟩) _
    rintro ⟨b, hb⟩ ⟨c, hc⟩ hbc
    exact Subtype.ext (mul_left_cancel (Subtype.ext_iff.mp hbc))
  have mono : Monotone (fun n ↦ Fintype.card (↥(S ^ n)) : ℕ → ℕ) :=
    monotone_nat_of_le_succ fun n ↦ key a _ _ fun b hb ↦ Set.mul_mem_mul ha hb
  refine' card_pow_eq_card_pow_card_univ_aux mono (fun n ↦ set_fintype_card_le_univ (S ^ n))
    fun n h ↦ le_antisymm (mono (n + 1).le_succ) (key a⁻¹ (S ^ (n + 2)) (S ^ (n + 1)) _)
  replace h₂ : {a} * S ^ n = S ^ (n + 1)
  · have : Fintype (Set.singleton a * S ^ n) := by
      

Proof state:
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
G : Type u_5
inst✝² : Group G
inst✝¹ : Fintype G
S : Set G
inst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k
hG : 0 < Fintype.card G
hS : ¬S = ∅
a : G
ha : a ∈ S
key :
  ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],
    (∀ b ∈ s, a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t
mono : Monotone fun n => Fintype.card ↑(S ^ n)
n : ℕ
h : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))
⊢ Fintype ↑(Set.singleton a * S ^ n)

Next tactic:
classical!
@[to_additive]
theorem card_pow_eq_card_pow_card_univ [∀ k : ℕ, DecidablePred (· ∈ S ^ k)] :
    ∀ k, Fintype.card G ≤ k → Fintype.card (↥(S ^ k)) = Fintype.card (↥(S ^ Fintype.card G)) := by
  have hG : 0 < Fintype.card G := Fintype.card_pos_iff.mpr ⟨1⟩
  by_cases hS : S = ∅
  · refine' fun k hk ↦ Fintype.card_congr _
    rw [hS, empty_pow (ne_of_gt (lt_of_lt_of_le hG hk)), empty_pow (ne_of_gt hG)]
  obtain ⟨a, ha⟩ := Set.nonempty_iff_ne_empty.2 hS
  have key : ∀ (a) (s t : Set G) [Fintype s] [Fintype t],
      (∀ b : G, b ∈ s → a * b ∈ t) → Fintype.card s ≤ Fintype.card t := by
    refine' fun a s t _ _ h ↦ Fintype.card_le_of_injective (fun ⟨b, hb⟩ ↦ ⟨a * b, h b hb⟩) _
    rintro ⟨b, hb⟩ ⟨c, hc⟩ hbc
    exact Subtype.ext (mul_left_cancel (Subtype.ext_iff.mp hbc))
  have mono : Monotone (fun n ↦ Fintype.card (↥(S ^ n)) : ℕ → ℕ) :=
    monotone_nat_of_le_succ fun n ↦ key a _ _ fun b hb ↦ Set.mul_mem_mul ha hb
  refine' card_pow_eq_card_pow_card_univ_aux mono (fun n ↦ set_fintype_card_le_univ (S ^ n))
    fun n h ↦ le_antisymm (mono (n + 1).le_succ) (key a⁻¹ (S ^ (n + 2)) (S ^ (n + 1)) _)
  replace h₂ : {a} * S ^ n = S ^ (n + 1)
  · have : Fintype (Set.singleton a * S ^ n) := by
      classical!
      

Proof state:
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
G : Type u_5
inst✝² : Group G
inst✝¹ : Fintype G
S : Set G
inst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k
hG : 0 < Fintype.card G
hS : ¬S = ∅
a : G
ha : a ∈ S
key :
  ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],
    (∀ b ∈ s, a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t
mono : Monotone fun n => Fintype.card ↑(S ^ n)
n : ℕ
h : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))
em✝ : (a : Prop) → Decidable a
⊢ Fintype ↑(Set.singleton a * S ^ n)

Next tactic:
apply fintypeMul
@[to_additive]
theorem card_pow_eq_card_pow_card_univ [∀ k : ℕ, DecidablePred (· ∈ S ^ k)] :
    ∀ k, Fintype.card G ≤ k → Fintype.card (↥(S ^ k)) = Fintype.card (↥(S ^ Fintype.card G)) := by
  have hG : 0 < Fintype.card G := Fintype.card_pos_iff.mpr ⟨1⟩
  by_cases hS : S = ∅
  · refine' fun k hk ↦ Fintype.card_congr _
    rw [hS, empty_pow (ne_of_gt (lt_of_lt_of_le hG hk)), empty_pow (ne_of_gt hG)]
  obtain ⟨a, ha⟩ := Set.nonempty_iff_ne_empty.2 hS
  have key : ∀ (a) (s t : Set G) [Fintype s] [Fintype t],
      (∀ b : G, b ∈ s → a * b ∈ t) → Fintype.card s ≤ Fintype.card t := by
    refine' fun a s t _ _ h ↦ Fintype.card_le_of_injective (fun ⟨b, hb⟩ ↦ ⟨a * b, h b hb⟩) _
    rintro ⟨b, hb⟩ ⟨c, hc⟩ hbc
    exact Subtype.ext (mul_left_cancel (Subtype.ext_iff.mp hbc))
  have mono : Monotone (fun n ↦ Fintype.card (↥(S ^ n)) : ℕ → ℕ) :=
    monotone_nat_of_le_succ fun n ↦ key a _ _ fun b hb ↦ Set.mul_mem_mul ha hb
  refine' card_pow_eq_card_pow_card_univ_aux mono (fun n ↦ set_fintype_card_le_univ (S ^ n))
    fun n h ↦ le_antisymm (mono (n + 1).le_succ) (key a⁻¹ (S ^ (n + 2)) (S ^ (n + 1)) _)
  replace h₂ : {a} * S ^ n = S ^ (n + 1)
  · have : Fintype (Set.singleton a * S ^ n) := by
      classical!
      apply fintypeMul
    

Proof state:
case h₂
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
G : Type u_5
inst✝² : Group G
inst✝¹ : Fintype G
S : Set G
inst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k
hG : 0 < Fintype.card G
hS : ¬S = ∅
a : G
ha : a ∈ S
key :
  ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],
    (∀ b ∈ s, a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t
mono : Monotone fun n => Fintype.card ↑(S ^ n)
n : ℕ
h : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))
this : Fintype ↑(Set.singleton a * S ^ n)
⊢ {a} * S ^ n = S ^ (n + 1)

Next tactic:
refine' Set.eq_of_subset_of_card_le _ (le_trans (ge_of_eq h) _)
@[to_additive]
theorem card_pow_eq_card_pow_card_univ [∀ k : ℕ, DecidablePred (· ∈ S ^ k)] :
    ∀ k, Fintype.card G ≤ k → Fintype.card (↥(S ^ k)) = Fintype.card (↥(S ^ Fintype.card G)) := by
  have hG : 0 < Fintype.card G := Fintype.card_pos_iff.mpr ⟨1⟩
  by_cases hS : S = ∅
  · refine' fun k hk ↦ Fintype.card_congr _
    rw [hS, empty_pow (ne_of_gt (lt_of_lt_of_le hG hk)), empty_pow (ne_of_gt hG)]
  obtain ⟨a, ha⟩ := Set.nonempty_iff_ne_empty.2 hS
  have key : ∀ (a) (s t : Set G) [Fintype s] [Fintype t],
      (∀ b : G, b ∈ s → a * b ∈ t) → Fintype.card s ≤ Fintype.card t := by
    refine' fun a s t _ _ h ↦ Fintype.card_le_of_injective (fun ⟨b, hb⟩ ↦ ⟨a * b, h b hb⟩) _
    rintro ⟨b, hb⟩ ⟨c, hc⟩ hbc
    exact Subtype.ext (mul_left_cancel (Subtype.ext_iff.mp hbc))
  have mono : Monotone (fun n ↦ Fintype.card (↥(S ^ n)) : ℕ → ℕ) :=
    monotone_nat_of_le_succ fun n ↦ key a _ _ fun b hb ↦ Set.mul_mem_mul ha hb
  refine' card_pow_eq_card_pow_card_univ_aux mono (fun n ↦ set_fintype_card_le_univ (S ^ n))
    fun n h ↦ le_antisymm (mono (n + 1).le_succ) (key a⁻¹ (S ^ (n + 2)) (S ^ (n + 1)) _)
  replace h₂ : {a} * S ^ n = S ^ (n + 1)
  · have : Fintype (Set.singleton a * S ^ n) := by
      classical!
      apply fintypeMul
    refine' Set.eq_of_subset_of_card_le _ (le_trans (ge_of_eq h) _)
    · 

Proof state:
case h₂.refine'_1
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
G : Type u_5
inst✝² : Group G
inst✝¹ : Fintype G
S : Set G
inst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k
hG : 0 < Fintype.card G
hS : ¬S = ∅
a : G
ha : a ∈ S
key :
  ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],
    (∀ b ∈ s, a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t
mono : Monotone fun n => Fintype.card ↑(S ^ n)
n : ℕ
h : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))
this : Fintype ↑(Set.singleton a * S ^ n)
⊢ {a} * S ^ n ⊆ S ^ (n + 1)

Next tactic:
exact mul_subset_mul (Set.singleton_subset_iff.mpr ha) Set.Subset.rfl
@[to_additive]
theorem card_pow_eq_card_pow_card_univ [∀ k : ℕ, DecidablePred (· ∈ S ^ k)] :
    ∀ k, Fintype.card G ≤ k → Fintype.card (↥(S ^ k)) = Fintype.card (↥(S ^ Fintype.card G)) := by
  have hG : 0 < Fintype.card G := Fintype.card_pos_iff.mpr ⟨1⟩
  by_cases hS : S = ∅
  · refine' fun k hk ↦ Fintype.card_congr _
    rw [hS, empty_pow (ne_of_gt (lt_of_lt_of_le hG hk)), empty_pow (ne_of_gt hG)]
  obtain ⟨a, ha⟩ := Set.nonempty_iff_ne_empty.2 hS
  have key : ∀ (a) (s t : Set G) [Fintype s] [Fintype t],
      (∀ b : G, b ∈ s → a * b ∈ t) → Fintype.card s ≤ Fintype.card t := by
    refine' fun a s t _ _ h ↦ Fintype.card_le_of_injective (fun ⟨b, hb⟩ ↦ ⟨a * b, h b hb⟩) _
    rintro ⟨b, hb⟩ ⟨c, hc⟩ hbc
    exact Subtype.ext (mul_left_cancel (Subtype.ext_iff.mp hbc))
  have mono : Monotone (fun n ↦ Fintype.card (↥(S ^ n)) : ℕ → ℕ) :=
    monotone_nat_of_le_succ fun n ↦ key a _ _ fun b hb ↦ Set.mul_mem_mul ha hb
  refine' card_pow_eq_card_pow_card_univ_aux mono (fun n ↦ set_fintype_card_le_univ (S ^ n))
    fun n h ↦ le_antisymm (mono (n + 1).le_succ) (key a⁻¹ (S ^ (n + 2)) (S ^ (n + 1)) _)
  replace h₂ : {a} * S ^ n = S ^ (n + 1)
  · have : Fintype (Set.singleton a * S ^ n) := by
      classical!
      apply fintypeMul
    refine' Set.eq_of_subset_of_card_le _ (le_trans (ge_of_eq h) _)
    · exact mul_subset_mul (Set.singleton_subset_iff.mpr ha) Set.Subset.rfl
    · 

Proof state:
case h₂.refine'_2
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
G : Type u_5
inst✝² : Group G
inst✝¹ : Fintype G
S : Set G
inst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k
hG : 0 < Fintype.card G
hS : ¬S = ∅
a : G
ha : a ∈ S
key :
  ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],
    (∀ b ∈ s, a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t
mono : Monotone fun n => Fintype.card ↑(S ^ n)
n : ℕ
h : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))
this : Fintype ↑(Set.singleton a * S ^ n)
⊢ Fintype.card ↑(S ^ n) ≤ Fintype.card ↑({a} * S ^ n)

Next tactic:
convert key a (S ^ n) ({a} * S ^ n) fun b hb ↦ Set.mul_mem_mul (Set.mem_singleton a) hb
@[to_additive]
theorem card_pow_eq_card_pow_card_univ [∀ k : ℕ, DecidablePred (· ∈ S ^ k)] :
    ∀ k, Fintype.card G ≤ k → Fintype.card (↥(S ^ k)) = Fintype.card (↥(S ^ Fintype.card G)) := by
  have hG : 0 < Fintype.card G := Fintype.card_pos_iff.mpr ⟨1⟩
  by_cases hS : S = ∅
  · refine' fun k hk ↦ Fintype.card_congr _
    rw [hS, empty_pow (ne_of_gt (lt_of_lt_of_le hG hk)), empty_pow (ne_of_gt hG)]
  obtain ⟨a, ha⟩ := Set.nonempty_iff_ne_empty.2 hS
  have key : ∀ (a) (s t : Set G) [Fintype s] [Fintype t],
      (∀ b : G, b ∈ s → a * b ∈ t) → Fintype.card s ≤ Fintype.card t := by
    refine' fun a s t _ _ h ↦ Fintype.card_le_of_injective (fun ⟨b, hb⟩ ↦ ⟨a * b, h b hb⟩) _
    rintro ⟨b, hb⟩ ⟨c, hc⟩ hbc
    exact Subtype.ext (mul_left_cancel (Subtype.ext_iff.mp hbc))
  have mono : Monotone (fun n ↦ Fintype.card (↥(S ^ n)) : ℕ → ℕ) :=
    monotone_nat_of_le_succ fun n ↦ key a _ _ fun b hb ↦ Set.mul_mem_mul ha hb
  refine' card_pow_eq_card_pow_card_univ_aux mono (fun n ↦ set_fintype_card_le_univ (S ^ n))
    fun n h ↦ le_antisymm (mono (n + 1).le_succ) (key a⁻¹ (S ^ (n + 2)) (S ^ (n + 1)) _)
  replace h₂ : {a} * S ^ n = S ^ (n + 1)
  · have : Fintype (Set.singleton a * S ^ n) := by
      classical!
      apply fintypeMul
    refine' Set.eq_of_subset_of_card_le _ (le_trans (ge_of_eq h) _)
    · exact mul_subset_mul (Set.singleton_subset_iff.mpr ha) Set.Subset.rfl
    · convert key a (S ^ n) ({a} * S ^ n) fun b hb ↦ Set.mul_mem_mul (Set.mem_singleton a) hb
  

Proof state:
case neg.intro
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
G : Type u_5
inst✝² : Group G
inst✝¹ : Fintype G
S : Set G
inst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k
hG : 0 < Fintype.card G
hS : ¬S = ∅
a : G
ha : a ∈ S
key :
  ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],
    (∀ b ∈ s, a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t
mono : Monotone fun n => Fintype.card ↑(S ^ n)
n : ℕ
h : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))
h₂ : {a} * S ^ n = S ^ (n + 1)
⊢ ∀ b ∈ S ^ (n + 2), a⁻¹ * b ∈ S ^ (n + 1)

Next tactic:
rw [pow_succ', ← h₂, mul_assoc, ← pow_succ', h₂]
@[to_additive]
theorem card_pow_eq_card_pow_card_univ [∀ k : ℕ, DecidablePred (· ∈ S ^ k)] :
    ∀ k, Fintype.card G ≤ k → Fintype.card (↥(S ^ k)) = Fintype.card (↥(S ^ Fintype.card G)) := by
  have hG : 0 < Fintype.card G := Fintype.card_pos_iff.mpr ⟨1⟩
  by_cases hS : S = ∅
  · refine' fun k hk ↦ Fintype.card_congr _
    rw [hS, empty_pow (ne_of_gt (lt_of_lt_of_le hG hk)), empty_pow (ne_of_gt hG)]
  obtain ⟨a, ha⟩ := Set.nonempty_iff_ne_empty.2 hS
  have key : ∀ (a) (s t : Set G) [Fintype s] [Fintype t],
      (∀ b : G, b ∈ s → a * b ∈ t) → Fintype.card s ≤ Fintype.card t := by
    refine' fun a s t _ _ h ↦ Fintype.card_le_of_injective (fun ⟨b, hb⟩ ↦ ⟨a * b, h b hb⟩) _
    rintro ⟨b, hb⟩ ⟨c, hc⟩ hbc
    exact Subtype.ext (mul_left_cancel (Subtype.ext_iff.mp hbc))
  have mono : Monotone (fun n ↦ Fintype.card (↥(S ^ n)) : ℕ → ℕ) :=
    monotone_nat_of_le_succ fun n ↦ key a _ _ fun b hb ↦ Set.mul_mem_mul ha hb
  refine' card_pow_eq_card_pow_card_univ_aux mono (fun n ↦ set_fintype_card_le_univ (S ^ n))
    fun n h ↦ le_antisymm (mono (n + 1).le_succ) (key a⁻¹ (S ^ (n + 2)) (S ^ (n + 1)) _)
  replace h₂ : {a} * S ^ n = S ^ (n + 1)
  · have : Fintype (Set.singleton a * S ^ n) := by
      classical!
      apply fintypeMul
    refine' Set.eq_of_subset_of_card_le _ (le_trans (ge_of_eq h) _)
    · exact mul_subset_mul (Set.singleton_subset_iff.mpr ha) Set.Subset.rfl
    · convert key a (S ^ n) ({a} * S ^ n) fun b hb ↦ Set.mul_mem_mul (Set.mem_singleton a) hb
  rw [pow_succ', ← h₂, mul_assoc, ← pow_succ', h₂]
  

Proof state:
case neg.intro
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
G : Type u_5
inst✝² : Group G
inst✝¹ : Fintype G
S : Set G
inst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k
hG : 0 < Fintype.card G
hS : ¬S = ∅
a : G
ha : a ∈ S
key :
  ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],
    (∀ b ∈ s, a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t
mono : Monotone fun n => Fintype.card ↑(S ^ n)
n : ℕ
h : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))
h₂ : {a} * S ^ n = S ^ (n + 1)
⊢ ∀ b ∈ {a} * S ^ (n + 1), a⁻¹ * b ∈ S ^ (n + 1)

Next tactic:
rintro _ ⟨b, c, hb, hc, rfl⟩
@[to_additive]
theorem card_pow_eq_card_pow_card_univ [∀ k : ℕ, DecidablePred (· ∈ S ^ k)] :
    ∀ k, Fintype.card G ≤ k → Fintype.card (↥(S ^ k)) = Fintype.card (↥(S ^ Fintype.card G)) := by
  have hG : 0 < Fintype.card G := Fintype.card_pos_iff.mpr ⟨1⟩
  by_cases hS : S = ∅
  · refine' fun k hk ↦ Fintype.card_congr _
    rw [hS, empty_pow (ne_of_gt (lt_of_lt_of_le hG hk)), empty_pow (ne_of_gt hG)]
  obtain ⟨a, ha⟩ := Set.nonempty_iff_ne_empty.2 hS
  have key : ∀ (a) (s t : Set G) [Fintype s] [Fintype t],
      (∀ b : G, b ∈ s → a * b ∈ t) → Fintype.card s ≤ Fintype.card t := by
    refine' fun a s t _ _ h ↦ Fintype.card_le_of_injective (fun ⟨b, hb⟩ ↦ ⟨a * b, h b hb⟩) _
    rintro ⟨b, hb⟩ ⟨c, hc⟩ hbc
    exact Subtype.ext (mul_left_cancel (Subtype.ext_iff.mp hbc))
  have mono : Monotone (fun n ↦ Fintype.card (↥(S ^ n)) : ℕ → ℕ) :=
    monotone_nat_of_le_succ fun n ↦ key a _ _ fun b hb ↦ Set.mul_mem_mul ha hb
  refine' card_pow_eq_card_pow_card_univ_aux mono (fun n ↦ set_fintype_card_le_univ (S ^ n))
    fun n h ↦ le_antisymm (mono (n + 1).le_succ) (key a⁻¹ (S ^ (n + 2)) (S ^ (n + 1)) _)
  replace h₂ : {a} * S ^ n = S ^ (n + 1)
  · have : Fintype (Set.singleton a * S ^ n) := by
      classical!
      apply fintypeMul
    refine' Set.eq_of_subset_of_card_le _ (le_trans (ge_of_eq h) _)
    · exact mul_subset_mul (Set.singleton_subset_iff.mpr ha) Set.Subset.rfl
    · convert key a (S ^ n) ({a} * S ^ n) fun b hb ↦ Set.mul_mem_mul (Set.mem_singleton a) hb
  rw [pow_succ', ← h₂, mul_assoc, ← pow_succ', h₂]
  rintro _ ⟨b, c, hb, hc, rfl⟩
  

Proof state:
case neg.intro.intro.intro.intro.intro
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
G : Type u_5
inst✝² : Group G
inst✝¹ : Fintype G
S : Set G
inst✝ : (k : ℕ) → DecidablePred fun x => x ∈ S ^ k
hG : 0 < Fintype.card G
hS : ¬S = ∅
a : G
ha : a ∈ S
key :
  ∀ (a : G) (s t : Set G) [inst : Fintype ↑s] [inst_1 : Fintype ↑t],
    (∀ b ∈ s, a * b ∈ t) → Fintype.card ↑s ≤ Fintype.card ↑t
mono : Monotone fun n => Fintype.card ↑(S ^ n)
n : ℕ
h : Fintype.card ↑(S ^ n) = Fintype.card ↑(S ^ (n + 1))
h₂ : {a} * S ^ n = S ^ (n + 1)
b c : G
hb : b ∈ {a}
hc : c ∈ S ^ (n + 1)
⊢ a⁻¹ * (fun x x_1 => x * x_1) b c ∈ S ^ (n + 1)

Next tactic:
rwa [Set.mem_singleton_iff.mp hb, inv_mul_cancel_left]
/-- The `R`-subalgebra of `S`-integers of `K`. -/
@[simps!]
def integer : Subalgebra R K :=
  {
    (⨅ (v) (_ : v ∉ S), (v : HeightOneSpectrum R).valuation.valuationSubring.toSubring).copy
        {x : K | ∀ (v) (_ : v ∉ S), (v : HeightOneSpectrum R).valuation x ≤ 1} <|
      Set.ext fun _ => by 

Proof state:
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsDedekindDomain R
S : Set (HeightOneSpectrum R)
K : Type v
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
x✝ : K
⊢ x✝ ∈ {x | ∀ v ∉ S, (HeightOneSpectrum.valuation v) x ≤ 1} ↔
    x✝ ∈ ↑(⨅ v, ⨅ (_ : v ∉ S), (Valuation.valuationSubring (HeightOneSpectrum.valuation v)).toSubring)

Next tactic:
simp [SetLike.mem_coe, Subring.mem_iInf]
theorem integer_eq :
    (S.integer K).toSubring =
      ⨅ (v) (_ : v ∉ S), (v : HeightOneSpectrum R).valuation.valuationSubring.toSubring :=
  SetLike.ext' <| by
    -- Porting note: was `simpa only [integer, Subring.copy_eq]`
    

Proof state:
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsDedekindDomain R
S : Set (HeightOneSpectrum R)
K : Type v
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
⊢ ↑(Subalgebra.toSubring (integer S K)) =
    ↑(⨅ v, ⨅ (_ : v ∉ S), (Valuation.valuationSubring (HeightOneSpectrum.valuation v)).toSubring)

Next tactic:
ext
theorem integer_eq :
    (S.integer K).toSubring =
      ⨅ (v) (_ : v ∉ S), (v : HeightOneSpectrum R).valuation.valuationSubring.toSubring :=
  SetLike.ext' <| by
    -- Porting note: was `simpa only [integer, Subring.copy_eq]`
    ext; 

Proof state:
case h
R : Type u
inst✝⁵ : CommRing R
inst✝⁴ : IsDomain R
inst✝³ : IsDedekindDomain R
S : Set (HeightOneSpectrum R)
K : Type v
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
x✝ : K
⊢ x✝ ∈ ↑(Subalgebra.toSubring (integer S K)) ↔
    x✝ ∈ ↑(⨅ v, ⨅ (_ : v ∉ S), (Valuation.valuationSubring (HeightOneSpectrum.valuation v)).toSubring)

Next tactic:
simp