awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
⊢ Nat.card.{u}
      (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
        (comap (Subgroup.subtype (normalizer (↑H : Set G))) H)) ≡
    Nat.card.{u} (G ⧸ H) [MOD p]
have rw : Nat.card.{u} (↑(MulAction.fixedPoints (↥H) (G ⧸ H)) : Type u) ≡ Nat.card.{u} (G ⧸ H) [MOD p]
Tactic: rw [← Nat.card_congr (fixedPointsMulLeftCosetsEquivQuotient H)]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
⊢ Nat.card.{u} ↥(normalizer (↑H : Set G)) ≡ Nat.card G [MOD p ^ (n + 1)]
have have : ∀ (this : ↥(H.subgroupOf (normalizer (↑H : Set G))) ≃ ↥H),
  Nat.card.{u} ↥(normalizer (↑H : Set G)) ≡ Nat.card G [MOD p ^ (n + 1)]
Tactic: have : H.subgroupOf (normalizer H) ≃ H := (subgroupOfEquivOfLe le_normalizer).toEquiv
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
this : ↥(H.subgroupOf (normalizer (↑H : Set G))) ≃ ↥H
⊢ Nat.card.{u} ↥(normalizer (↑H : Set G)) ≡ Nat.card G [MOD p ^ (n + 1)]
have rw : Nat.card.{u}
      (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
        (H.subgroupOf (normalizer (↑H : Set G)))) *
    p ^ n ≡
  Nat.card.{u} (G ⧸ H) * p ^ n [MOD p * p ^ n]
Tactic: rw [card_eq_card_quotient_mul_card_subgroup H,
    card_eq_card_quotient_mul_card_subgroup (H.subgroupOf (normalizer H)), Nat.card_congr this,
    hH, pow_succ']
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
inst_2 : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
⊢ Nat.card.{u} (G ⧸ H) * Nat.card.{u} ↥H = s * p * Nat.card.{u} ↥H
have rw : Nat.card G = s * p * Nat.card.{u} ↥H
Tactic: rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
inst_2 : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
rw : Nat.card G = s * p * Nat.card.{u} ↥H
⊢ Nat.card.{u} (G ⧸ H) * Nat.card.{u} ↥H = s * p * Nat.card.{u} ↥H
have rw₁ : Nat.card G = s * p * p ^ n
Tactic: rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
inst_2 : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
rw : Nat.card G = s * p * Nat.card.{u} ↥H
rw₁ : Nat.card G = s * p * p ^ n
⊢ Nat.card.{u} (G ⧸ H) * Nat.card.{u} ↥H = s * p * Nat.card.{u} ↥H
have rw₂ : s * p ^ (n + 1) = s * p * p ^ n
Tactic: rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
inst_2 : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
rw : Nat.card G = s * p * Nat.card.{u} ↥H
rw₁ : Nat.card G = s * p * p ^ n
rw₂ : s * p ^ (n + 1) = s * p * p ^ n
⊢ Nat.card.{u} (G ⧸ H) * Nat.card.{u} ↥H = s * p * Nat.card.{u} ↥H
have rw₃ : s * (p * p ^ n) = s * p * p ^ n
Tactic: rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
inst_2 : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
rw : Nat.card G = s * p * Nat.card.{u} ↥H
rw₁ : Nat.card G = s * p * p ^ n
rw₂ : s * p ^ (n + 1) = s * p * p ^ n
rw₃ : s * (p * p ^ n) = s * p * p ^ n
⊢ Nat.card.{u} (G ⧸ H) * Nat.card.{u} ↥H = s * p * Nat.card.{u} ↥H
have rw₄ : s * (p * p ^ n) = s * (p * p ^ n)
Tactic: rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
inst_2 : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
rw : Nat.card G = s * p * Nat.card.{u} ↥H
rw₁ : Nat.card G = s * p * p ^ n
rw₂ : s * p ^ (n + 1) = s * p * p ^ n
rw₃ : s * (p * p ^ n) = s * p * p ^ n
rw₄ : s * (p * p ^ n) = s * (p * p ^ n)
⊢ Nat.card.{u} (G ⧸ H) * Nat.card.{u} ↥H = s * p * Nat.card.{u} ↥H
have rw₅ : s * (p ^ n * p) = s * (p ^ n * p)
Tactic: rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
inst_2 : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
hcard : Nat.card.{u} (G ⧸ H) = s * p
hm :
  s * p % p =
    Nat.card.{u}
        (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
          (comap (Subgroup.subtype (normalizer (↑H : Set G))) H)) %
      p
⊢ Nat.card.{u}
        (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
          (comap (Subgroup.subtype (normalizer (↑H : Set G))) H)) %
      p =
    0
have rwa : ∀
  (hm :
    0 =
      Nat.card.{u}
          (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
            (comap (Subgroup.subtype (normalizer (↑H : Set G))) H)) %
        p),
  Nat.card.{u}
        (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
          (comap (Subgroup.subtype (normalizer (↑H : Set G))) H)) %
      p =
    0
Tactic: rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm
Cleared hypotheses: hm
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
inst_2 : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
hcard : Nat.card.{u} (G ⧸ H) = s * p
hm :
  s * p % p =
    Nat.card.{u}
        (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
          (comap (Subgroup.subtype (normalizer (↑H : Set G))) H)) %
      p
rwa :
  ∀
    (hm :
      0 =
        Nat.card.{u}
            (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
              (comap (Subgroup.subtype (normalizer (↑H : Set G))) H)) %
          p),
    Nat.card.{u}
          (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
            (comap (Subgroup.subtype (normalizer (↑H : Set G))) H)) %
        p =
      0
⊢ Nat.card.{u}
        (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
          (comap (Subgroup.subtype (normalizer (↑H : Set G))) H)) %
      p =
    0
have rwa₁ : ∀
  (hm :
    Nat.card.{u}
          (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
            (comap (Subgroup.subtype (normalizer (↑H : Set G))) H)) %
        p =
      0),
  Nat.card.{u}
        (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
          (comap (Subgroup.subtype (normalizer (↑H : Set G))) H)) %
      p =
    0
Tactic: rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm
Cleared hypotheses: hm
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
⊢ Nat.card.{u} (G ⧸ H) * Nat.card.{u} ↥H = s * p * Nat.card.{u} ↥H
have rw : Nat.card G = s * p * Nat.card.{u} ↥H
Tactic: rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
rw : Nat.card G = s * p * Nat.card.{u} ↥H
⊢ Nat.card.{u} (G ⧸ H) * Nat.card.{u} ↥H = s * p * Nat.card.{u} ↥H
have rw₁ : Nat.card G = s * p * p ^ n
Tactic: rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
rw : Nat.card G = s * p * Nat.card.{u} ↥H
rw₁ : Nat.card G = s * p * p ^ n
⊢ Nat.card.{u} (G ⧸ H) * Nat.card.{u} ↥H = s * p * Nat.card.{u} ↥H
have rw₂ : s * p ^ (n + 1) = s * p * p ^ n
Tactic: rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
rw : Nat.card G = s * p * Nat.card.{u} ↥H
rw₁ : Nat.card G = s * p * p ^ n
rw₂ : s * p ^ (n + 1) = s * p * p ^ n
⊢ Nat.card.{u} (G ⧸ H) * Nat.card.{u} ↥H = s * p * Nat.card.{u} ↥H
have rw₃ : s * (p * p ^ n) = s * p * p ^ n
Tactic: rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
rw : Nat.card G = s * p * Nat.card.{u} ↥H
rw₁ : Nat.card G = s * p * p ^ n
rw₂ : s * p ^ (n + 1) = s * p * p ^ n
rw₃ : s * (p * p ^ n) = s * p * p ^ n
⊢ Nat.card.{u} (G ⧸ H) * Nat.card.{u} ↥H = s * p * Nat.card.{u} ↥H
have rw₄ : s * (p * p ^ n) = s * (p * p ^ n)
Tactic: rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
rw : Nat.card G = s * p * Nat.card.{u} ↥H
rw₁ : Nat.card G = s * p * p ^ n
rw₂ : s * p ^ (n + 1) = s * p * p ^ n
rw₃ : s * (p * p ^ n) = s * p * p ^ n
rw₄ : s * (p * p ^ n) = s * (p * p ^ n)
⊢ Nat.card.{u} (G ⧸ H) * Nat.card.{u} ↥H = s * p * Nat.card.{u} ↥H
have rw₅ : s * (p ^ n * p) = s * (p ^ n * p)
Tactic: rw [← card_eq_card_quotient_mul_card_subgroup H, hH, hs, pow_succ', mul_assoc, mul_comm p]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
hcard : Nat.card.{u} (G ⧸ H) = s * p
hm :
  s * p % p =
    Nat.card.{u}
        (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
          (H.subgroupOf (normalizer (↑H : Set G)))) %
      p
⊢ Nat.card.{u}
        (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
          (H.subgroupOf (normalizer (↑H : Set G)))) %
      p =
    0
have rwa : ∀
  (hm :
    0 =
      Nat.card.{u}
          (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
            (H.subgroupOf (normalizer (↑H : Set G)))) %
        p),
  Nat.card.{u}
        (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
          (H.subgroupOf (normalizer (↑H : Set G)))) %
      p =
    0
Tactic: rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm
Cleared hypotheses: hm
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
hcard : Nat.card.{u} (G ⧸ H) = s * p
hm :
  s * p % p =
    Nat.card.{u}
        (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
          (H.subgroupOf (normalizer (↑H : Set G)))) %
      p
rwa :
  ∀
    (hm :
      0 =
        Nat.card.{u}
            (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
              (H.subgroupOf (normalizer (↑H : Set G)))) %
          p),
    Nat.card.{u}
          (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
            (H.subgroupOf (normalizer (↑H : Set G)))) %
        p =
      0
⊢ Nat.card.{u}
        (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
          (H.subgroupOf (normalizer (↑H : Set G)))) %
      p =
    0
have rwa₁ : ∀
  (hm :
    Nat.card.{u}
          (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
            (H.subgroupOf (normalizer (↑H : Set G)))) %
        p =
      0),
  Nat.card.{u}
        (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
          (H.subgroupOf (normalizer (↑H : Set G)))) %
      p =
    0
Tactic: rwa [Nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm
Cleared hypotheses: hm
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
hcard : Nat.card.{u} (G ⧸ H) = s * p
hm :
  s * p % p =
    Nat.card.{u}
        (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
          (H.subgroupOf (normalizer (↑H : Set G)))) %
      p
hm' :
  p ∣
    Nat.card.{u}
      (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
        (H.subgroupOf (normalizer (↑H : Set G))))
x :
  HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
    (H.subgroupOf (normalizer (↑H : Set G)))
hx : orderOf x = p
hequiv : ↥H ≃ ↥(H.subgroupOf (normalizer (↑H : Set G)))
⊢ Nat.card.{u}
      ↥(Subgroup.map (Subgroup.subtype (normalizer (↑H : Set G)))
          (comap (mk' (H.subgroupOf (normalizer (↑H : Set G)))) (zpowers x))) =
    p ^ (n + 1)
have suffices : Nat.card.{u}
    (↑(Subtype.val (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (normalizer (↑H : Set G)) x) ''
          (↑(comap (mk' (H.subgroupOf (normalizer (↑H : Set G)))) (zpowers x)) : Set ↥(normalizer (↑H : Set G)))) :
      Type u) =
  p ^ (n + 1)
Tactic: suffices Nat.card (Subtype.val ''
      ((zpowers x).comap (mk' (H.subgroupOf (normalizer H))) : Set (normalizer H))) = p ^ (n + 1)
      by convert! this using 2
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
hcard : Nat.card.{u} (G ⧸ H) = s * p
hm :
  s * p % p =
    Nat.card.{u}
        (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
          (H.subgroupOf (normalizer (↑H : Set G)))) %
      p
hm' :
  p ∣
    Nat.card.{u}
      (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
        (H.subgroupOf (normalizer (↑H : Set G))))
x :
  HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
    (H.subgroupOf (normalizer (↑H : Set G)))
hx : orderOf x = p
hequiv : ↥H ≃ ↥(H.subgroupOf (normalizer (↑H : Set G)))
⊢ Nat.card.{u}
      (↑(Subtype.val (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (normalizer (↑H : Set G)) x) ''
            (↑(comap (mk' (H.subgroupOf (normalizer (↑H : Set G)))) (zpowers x)) : Set ↥(normalizer (↑H : Set G)))) :
        Type u) =
    p ^ (n + 1)
have rw : Nat.card.{u}
    (↑(↑(comap (mk' (H.subgroupOf (normalizer (↑H : Set G)))) (zpowers x)) : Set ↥(normalizer (↑H : Set G))) : Type u) =
  Nat.card.{u} (↥(H.subgroupOf (normalizer (↑H : Set G))) × ↥(zpowers x))
Tactic: rw [Nat.card_image_of_injective Subtype.val_injective
        ((zpowers x).comap (mk' (H.subgroupOf (normalizer H))) : Set (normalizer (H : Set G))),
      pow_succ, ← hH, Nat.card_congr hequiv, ← hx, ← Nat.card_zpowers, ← Nat.card_prod]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
hcard : Nat.card.{u} (G ⧸ H) = s * p
hm :
  s * p % p =
    Nat.card.{u}
        (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
          (H.subgroupOf (normalizer (↑H : Set G)))) %
      p
hm' :
  p ∣
    Nat.card.{u}
      (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
        (H.subgroupOf (normalizer (↑H : Set G))))
x :
  HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
    (H.subgroupOf (normalizer (↑H : Set G)))
hx : orderOf x = p
hequiv : ↥H ≃ ↥(H.subgroupOf (normalizer (↑H : Set G)))
y : G
hy : y ∈ H
⊢ y ∈
    Subgroup.map (Subgroup.subtype (normalizer (↑H : Set G)))
      (comap (mk' (H.subgroupOf (normalizer (↑H : Set G)))) (zpowers x))
have simp : ∃ (x_1 : ↥(normalizer (↑H : Set G))),
  Membership.mem (γ :=
      Subgroup
        (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
          (H.subgroupOf (normalizer (↑H : Set G)))))
      (zpowers x) (↑x_1 : ↥(normalizer (↑H : Set G)) ⧸ H.subgroupOf (normalizer (↑H : Set G))) ∧
    (↑x_1 : G) = y
Tactic: simp only [Subgroup.coe_subtype, mk'_apply, Subgroup.mem_map, Subgroup.mem_comap]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
hcard : Nat.card.{u} (G ⧸ H) = s * p
hm :
  s * p % p =
    Nat.card.{u}
        (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
          (H.subgroupOf (normalizer (↑H : Set G)))) %
      p
hm' :
  p ∣
    Nat.card.{u}
      (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
        (H.subgroupOf (normalizer (↑H : Set G))))
x :
  HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
    (H.subgroupOf (normalizer (↑H : Set G)))
hx : orderOf x = p
hequiv : ↥H ≃ ↥(H.subgroupOf (normalizer (↑H : Set G)))
y : G
hy : y ∈ H
⊢ ∃ (x_1 : ↥(normalizer (↑H : Set G))),
    Membership.mem (γ :=
        Subgroup
          (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
            (H.subgroupOf (normalizer (↑H : Set G)))))
        (zpowers x) (↑x_1 : ↥(normalizer (↑H : Set G)) ⧸ H.subgroupOf (normalizer (↑H : Set G))) ∧
      (↑x_1 : G) = y
have refine : Eq (α :=
  HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
    (H.subgroupOf (normalizer (↑H : Set G))))
  ((fun (x_1 : ℤ) ↦ x ^ x_1) 0)
  (↑(Subtype.mk (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (normalizer (↑H : Set G)) x) y sorry) :
    ↥(normalizer (↑H : Set G)) ⧸ H.subgroupOf (normalizer (↑H : Set G)))
Tactic: refine ⟨⟨y, le_normalizer hy⟩, ⟨0, ?_⟩, rfl⟩
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
hp : Fact (Nat.Prime p)
hdvd : p ^ (n + 1) ∣ Nat.card G
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
s : ℕ
hs : Nat.card G = s * p ^ (n + 1)
hcard : Nat.card.{u} (G ⧸ H) = s * p
hm :
  s * p % p =
    Nat.card.{u}
        (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
          (H.subgroupOf (normalizer (↑H : Set G)))) %
      p
hm' :
  p ∣
    Nat.card.{u}
      (HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
        (H.subgroupOf (normalizer (↑H : Set G))))
x :
  HasQuotient.Quotient ↥(normalizer (↑H : Set G)) (B := Subgroup ↥(normalizer (↑H : Set G)))
    (H.subgroupOf (normalizer (↑H : Set G)))
hx : orderOf x = p
hequiv : ↥H ≃ ↥(H.subgroupOf (normalizer (↑H : Set G)))
y : G
hy : y ∈ H
⊢ HPow.hPow (β := ℤ) x 0 =
    (↑(Subtype.mk (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (normalizer (↑H : Set G)) x) y ⋯) :
      ↥(normalizer (↑H : Set G)) ⧸ H.subgroupOf (normalizer (↑H : Set G)))
have rw : Membership.mem (γ := Subgroup ↥(normalizer (↑H : Set G))) (H.subgroupOf (normalizer (↑H : Set G)))
  (Subtype.mk (p := fun (x : G) ↦ Membership.mem (γ := Subgroup G) (normalizer (↑H : Set G)) x) y sorry)
Tactic: rw [zpow_zero, eq_comm, QuotientGroup.eq_one_iff]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p : ℕ
_hp : Fact (Nat.Prime p)
n m : ℕ
hdvd : p ^ m ∣ Nat.card G
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
hnm_1 : n ≤ m
hnm : n < m
h0m : 0 < m
hnm1 : n ≤ m - 1
K : Subgroup G
hK : Nat.card.{u} ↥K = p ^ (m - 1) ∧ H ≤ K
hdvd' : p ^ (m - 1 + 1) ∣ Nat.card G
K' : Subgroup G
hK' : Nat.card.{u} ↥K' = p ^ (m - 1 + 1) ∧ K ≤ K'
⊢ Nat.card.{u} ↥K' = p ^ m
have rw : p ^ (m - 1 + 1) = p ^ m
Tactic: rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p : ℕ
_hp : Fact (Nat.Prime p)
n m : ℕ
hdvd : p ^ m ∣ Nat.card G
H : Subgroup G
hH : Nat.card.{u} ↥H = p ^ n
hnm_1 : n ≤ m
hnm : n < m
h0m : 0 < m
hnm1 : n ≤ m - 1
K : Subgroup G
hK : Nat.card.{u} ↥K = p ^ (m - 1) ∧ H ≤ K
hdvd' : p ^ (m - 1 + 1) ∣ Nat.card G
K' : Subgroup G
hK' : Nat.card.{u} ↥K' = p ^ (m - 1 + 1) ∧ K ≤ K'
rw : p ^ (m - 1 + 1) = p ^ m
⊢ Nat.card.{u} ↥K' = p ^ m
have rw₁ : p ^ m = p ^ m
Tactic: rw [hK'.1, tsub_add_cancel_of_le h0m.nat_succ_le]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
inst_2 : Fact (Nat.Prime p)
hdvd : p ^ n ∣ Nat.card G
⊢ Nat.card.{u} ↥⊥ = HPow.hPow (β := ℕ) p 0
have rw : 1 = HPow.hPow (β := ℕ) p 0
Tactic: rw [card_bot, pow_zero]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
inst_1 : Finite G
p n : ℕ
inst_2 : Fact (Nat.Prime p)
hdvd : p ^ n ∣ Nat.card G
rw : 1 = HPow.hPow (β := ℕ) p 0
⊢ Nat.card.{u} ↥⊥ = HPow.hPow (β := ℕ) p 0
have rw₁ : Eq (α := ℕ) 1 1
Tactic: rw [card_bot, pow_zero]
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (hypothesis)
Goal:
G : Type u
inst : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
hn : p ^ n ≤ Nat.card G
⊢ ∃ (H : Subgroup G), Nat.card.{u} ↥H = p ^ n
have this : Fact (Nat.Prime p)
Tactic: have : Fact p.Prime := ⟨hp⟩
-- Lean 4 proof step from Mathlib.GroupTheory.Sylow (goal)
Goal:
G : Type u
inst : Group G
n p : ℕ
hp : Nat.Prime p
h : IsPGroup p G
hn : p ^ n ≤ Nat.card G
this : Fact (Nat.Prime p)
⊢ Nat.card G ≠ 0
have linarith : Nat.card G = 0 →
  -1 + (HPow.hPow (α := ℤ) (↑p : ℤ) n - (↑(Nat.card G) : ℤ)) + HSub.hSub (α := ℤ) (↑(Nat.card G) : ℤ) 0 +
      (1 - HPow.hPow (α := ℤ) (↑p : ℤ) n) =
    0
Tactic: linarith [Nat.one_le_pow n p hp.pos]