internlm/Lean-Github huggingface dataset:
Theorem: Aux.log_add_one_le_sum_inv
Proof state before:
case calc_2.e_a
n : ℕ
⊢ 1 = ↑1
Tactic:
norm_num
Proof state after:
no goals
Theorem: Aux.log_add_one_le_sum_inv
Proof state before:
case calc_3
n : ℕ
⊢ ∫ (x : ℝ) in ↑1 ..↑(n + 1), x⁻¹ ≤ ∑ d ∈ Icc 1 n, (↑d)⁻¹
Tactic:
apply AntitoneOn.integral_le_sum_Ico (by norm_num)
Proof state after:
case calc_3
n : ℕ
⊢ AntitoneOn (fun x => x⁻¹) (Set.Icc ↑1 ↑(n + 1))
Theorem: Aux.log_add_one_le_sum_inv
Proof state before:
case calc_3
n : ℕ
⊢ AntitoneOn (fun x => x⁻¹) (Set.Icc ↑1 ↑(n + 1))
Tactic:
apply inv_antitoneOn_Icc
Proof state after:
case calc_3.ha
n : ℕ
⊢ 0 < ↑1
Theorem: Aux.log_add_one_le_sum_inv
Proof state before:
case calc_3.ha
n : ℕ
⊢ 0 < ↑1
Tactic:
norm_num
Proof state after:
no goals
Theorem: Aux.log_add_one_le_sum_inv
Proof state before:
n : ℕ
⊢ 1 ≤ n + 1
Tactic:
norm_num
Proof state after:
no goals
Theorem: Aux.log_le_sum_inv
Proof state before:
y : ℝ
hy : 1 ≤ y
⊢ y.log ≤ ∑ d ∈ Icc 1 ⌊y⌋₊, (↑d)⁻¹
Tactic:
calc _ ≤ Real.log ↑(Nat.floor y + 1) := ?_
     _ ≤ _ := ?_
Proof state after:
case calc_1
y : ℝ
hy : 1 ≤ y
⊢ y.log ≤ (↑(⌊y⌋₊ + 1)).log

case calc_2
y : ℝ
hy : 1 ≤ y
⊢ (↑(⌊y⌋₊ + 1)).log ≤ ∑ d ∈ Icc 1 ⌊y⌋₊, (↑d)⁻¹
Theorem: Aux.log_le_sum_inv
Proof state before:
case calc_1
y : ℝ
hy : 1 ≤ y
⊢ y.log ≤ (↑(⌊y⌋₊ + 1)).log
Tactic:
gcongr
Proof state after:
case calc_1.hxy
y : ℝ
hy : 1 ≤ y
⊢ y ≤ ↑(⌊y⌋₊ + 1)
Theorem: Aux.log_le_sum_inv
Proof state before:
case calc_1.hxy
y : ℝ
hy : 1 ≤ y
⊢ y ≤ ↑(⌊y⌋₊ + 1)
Tactic:
apply (le_ceil y).trans
Proof state after:
case calc_1.hxy
y : ℝ
hy : 1 ≤ y
⊢ ↑⌈y⌉₊ ≤ ↑(⌊y⌋₊ + 1)
Theorem: Aux.log_le_sum_inv
Proof state before:
case calc_1.hxy
y : ℝ
hy : 1 ≤ y
⊢ ↑⌈y⌉₊ ≤ ↑(⌊y⌋₊ + 1)
Tactic:
norm_cast
Proof state after:
case calc_1.hxy
y : ℝ
hy : 1 ≤ y
⊢ ⌈y⌉₊ ≤ ⌊y⌋₊ + 1
Theorem: Aux.log_le_sum_inv
Proof state before:
case calc_1.hxy
y : ℝ
hy : 1 ≤ y
⊢ ⌈y⌉₊ ≤ ⌊y⌋₊ + 1
Tactic:
exact ceil_le_floor_add_one y
Proof state after:
no goals
Theorem: Aux.log_le_sum_inv
Proof state before:
case calc_2
y : ℝ
hy : 1 ≤ y
⊢ (↑(⌊y⌋₊ + 1)).log ≤ ∑ d ∈ Icc 1 ⌊y⌋₊, (↑d)⁻¹
Tactic:
apply log_add_one_le_sum_inv
Proof state after:
no goals
Theorem: Aux.sum_inv_le_log
Proof state before:
n : ℕ
hn : 1 ≤ n
⊢ ∑ d ∈ Icc 1 n, (↑d)⁻¹ ≤ 1 + (↑n).log
Tactic:
rw [← Finset.sum_erase_add (Icc 1 n) _ (by simp [hn] : 1 ∈ Icc 1 n), add_comm]
Proof state after:
n : ℕ
hn : 1 ≤ n
⊢ (↑1)⁻¹ + ∑ x ∈ (Icc 1 n).erase 1, (↑x)⁻¹ ≤ 1 + (↑n).log
Theorem: Aux.sum_inv_le_log
Proof state before:
n : ℕ
hn : 1 ≤ n
⊢ (↑1)⁻¹ + ∑ x ∈ (Icc 1 n).erase 1, (↑x)⁻¹ ≤ 1 + (↑n).log
Tactic:
gcongr
Proof state after:
case h₁
n : ℕ
hn : 1 ≤ n
⊢ (↑1)⁻¹ ≤ 1

case h₂
n : ℕ
hn : 1 ≤ n
⊢ ∑ x ∈ (Icc 1 n).erase 1, (↑x)⁻¹ ≤ (↑n).log
Theorem: Aux.sum_inv_le_log
Proof state before:
case h₂
n : ℕ
hn : 1 ≤ n
⊢ ∑ x ∈ (Icc 1 n).erase 1, (↑x)⁻¹ ≤ (↑n).log
Tactic:
simp only [gt_iff_lt, lt_one_iff, mem_Icc, true_and, not_le, Icc_erase_left]
Proof state after:
case h₂
n : ℕ
hn : 1 ≤ n
⊢ ∑ x ∈ Ioc 1 n, (↑x)⁻¹ ≤ (↑n).log
Theorem: Aux.sum_inv_le_log
Proof state before:
case h₂
n : ℕ
hn : 1 ≤ n
⊢ ∑ x ∈ Ioc 1 n, (↑x)⁻¹ ≤ (↑n).log
Tactic:
calc
  ∑ d : ℕ in Ico 2 (n + 1), (d : ℝ)⁻¹ = ∑ d in Ico 2 (n + 1), (↑(d + 1) - 1)⁻¹ := ?_
  _ ≤ ∫ x in (2).. ↑(n + 1), (x - 1)⁻¹  := ?_
  _ = Real.log ↑n := ?_
Proof state after:
case h₂.calc_1
n : ℕ
hn : 1 ≤ n
⊢ ∑ d ∈ Ico 2 (n + 1), (↑d)⁻¹ = ∑ d ∈ Ico 2 (n + 1), (↑(d + 1) - 1)⁻¹

case h₂.calc_2
n : ℕ
hn : 1 ≤ n
⊢ ∑ d ∈ Ico 2 (n + 1), (↑(d + 1) - 1)⁻¹ ≤ ∫ (x : ℝ) in 2 ..↑(n + 1), (x - 1)⁻¹

case h₂.calc_3
n : ℕ
hn : 1 ≤ n
⊢ ∫ (x : ℝ) in 2 ..↑(n + 1), (x - 1)⁻¹ = (↑n).log
Theorem: Aux.sum_inv_le_log
Proof state before:
case h₂.calc_3
n : ℕ
hn : 1 ≤ n
⊢ ∫ (x : ℝ) in 2 ..↑(n + 1), (x - 1)⁻¹ = (↑n).log
Tactic:
rw [intervalIntegral.integral_comp_sub_right _ 1, integral_inv]
Proof state after:
case h₂.calc_3
n : ℕ
hn : 1 ≤ n
⊢ ((↑(n + 1) - 1) / (2 - 1)).log = (↑n).log

case h₂.calc_3
n : ℕ
hn : 1 ≤ n
⊢ 0 ∉ Set.uIcc (2 - 1) (↑(n + 1) - 1)
Theorem: Aux.sum_inv_le_log
Proof state before:
case h₂.calc_3
n : ℕ
hn : 1 ≤ n
⊢ 0 ∉ Set.uIcc (2 - 1) (↑(n + 1) - 1)
Tactic:
norm_num
Proof state after:
case h₂.calc_3
n : ℕ
hn : 1 ≤ n
⊢ 0 ∉ Set.uIcc 1 ↑n
Theorem: Aux.sum_inv_le_log
Proof state before:
case h₂.calc_3
n : ℕ
hn : 1 ≤ n
⊢ 0 ∉ Set.uIcc 1 ↑n
Tactic:
simp[hn, show (0:ℝ) < 1 by norm_num]
Proof state after:
no goals
Theorem: Aux.sum_inv_le_log
Proof state before:
n : ℕ
hn : 1 ≤ n
⊢ 1 ∈ Icc 1 n
Tactic:
simp [hn]
Proof state after:
no goals
Theorem: Aux.sum_inv_le_log
Proof state before:
case h₁
n : ℕ
hn : 1 ≤ n
⊢ (↑1)⁻¹ ≤ 1
Tactic:
norm_num
Proof state after:
no goals
Theorem: Aux.sum_inv_le_log
Proof state before:
case h₂.calc_1
n : ℕ
hn : 1 ≤ n
⊢ ∑ d ∈ Ico 2 (n + 1), (↑d)⁻¹ = ∑ d ∈ Ico 2 (n + 1), (↑(d + 1) - 1)⁻¹
Tactic:
congr
Proof state after:
case h₂.calc_1.e_f
n : ℕ
hn : 1 ≤ n
⊢ (fun d => (↑d)⁻¹) = fun d => (↑(d + 1) - 1)⁻¹
Theorem: Aux.sum_inv_le_log
Proof state before:
case h₂.calc_1.e_f
n : ℕ
hn : 1 ≤ n
⊢ (fun d => (↑d)⁻¹) = fun d => (↑(d + 1) - 1)⁻¹
Tactic:
norm_num
Proof state after:
no goals
Theorem: Aux.sum_inv_le_log
Proof state before:
case h₂.calc_2
n : ℕ
hn : 1 ≤ n
⊢ ∑ d ∈ Ico 2 (n + 1), (↑(d + 1) - 1)⁻¹ ≤ ∫ (x : ℝ) in 2 ..↑(n + 1), (x - 1)⁻¹
Tactic:
apply @AntitoneOn.sum_le_integral_Ico 2 (n + 1) fun x : ℝ => (x - 1)⁻¹
Proof state after:
case h₂.calc_2.hab
n : ℕ
hn : 1 ≤ n
⊢ 2 ≤ n + 1

case h₂.calc_2.hf
n : ℕ
hn : 1 ≤ n
⊢ AntitoneOn (fun x => (x - 1)⁻¹) (Set.Icc ↑2 ↑(n + 1))
Theorem: Aux.sum_inv_le_log
Proof state before:
case h₂.calc_2.hf
n : ℕ
hn : 1 ≤ n
⊢ AntitoneOn (fun x => (x - 1)⁻¹) (Set.Icc ↑2 ↑(n + 1))
Tactic:
apply inv_sub_antitoneOn_Icc
Proof state after:
case h₂.calc_2.hf.ha
n : ℕ
hn : 1 ≤ n
⊢ 1 < ↑2
Theorem: Aux.sum_inv_le_log
Proof state before:
case h₂.calc_2.hf.ha
n : ℕ
hn : 1 ≤ n
⊢ 1 < ↑2
Tactic:
norm_num
Proof state after:
no goals
Theorem: Aux.sum_inv_le_log
Proof state before:
case h₂.calc_2.hab
n : ℕ
hn : 1 ≤ n
⊢ 2 ≤ n + 1
Tactic:
linarith [hn]
Proof state after:
no goals
Theorem: Aux.sum_inv_le_log
Proof state before:
case h₂.calc_3
n : ℕ
hn : 1 ≤ n
⊢ ((↑(n + 1) - 1) / (2 - 1)).log = (↑n).log
Tactic:
norm_num
Proof state after:
no goals
Theorem: Aux.sum_inv_le_log
Proof state before:
n : ℕ
hn : 1 ≤ n
⊢ 0 < 1
Tactic:
norm_num
Proof state after:
no goals
Theorem: Aux.sum_inv_le_log_real
Proof state before:
y : ℝ
hy : 1 ≤ y
⊢ ∑ d ∈ Icc 1 ⌊y⌋₊, (↑d)⁻¹ ≤ 1 + y.log
Tactic:
trans (1 + Real.log (⌊y⌋₊))
Proof state after:
y : ℝ
hy : 1 ≤ y
⊢ ∑ d ∈ Icc 1 ⌊y⌋₊, (↑d)⁻¹ ≤ 1 + (↑⌊y⌋₊).log

y : ℝ
hy : 1 ≤ y
⊢ 1 + (↑⌊y⌋₊).log ≤ 1 + y.log
Theorem: Aux.sum_inv_le_log_real
Proof state before:
y : ℝ
hy : 1 ≤ y
⊢ 1 + (↑⌊y⌋₊).log ≤ 1 + y.log
Tactic:
gcongr
Proof state after:
case bc.hx
y : ℝ
hy : 1 ≤ y
⊢ 0 < ↑⌊y⌋₊

case bc.hxy
y : ℝ
hy : 1 ≤ y
⊢ ↑⌊y⌋₊ ≤ y
Theorem: Aux.sum_inv_le_log_real
Proof state before:
y : ℝ
hy : 1 ≤ y
⊢ ∑ d ∈ Icc 1 ⌊y⌋₊, (↑d)⁻¹ ≤ 1 + (↑⌊y⌋₊).log
Tactic:
apply sum_inv_le_log (⌊y⌋₊)
Proof state after:
y : ℝ
hy : 1 ≤ y
⊢ 1 ≤ ⌊y⌋₊
Theorem: Aux.sum_inv_le_log_real
Proof state before:
y : ℝ
hy : 1 ≤ y
⊢ 1 ≤ ⌊y⌋₊
Tactic:
apply le_floor
Proof state after:
case h
y : ℝ
hy : 1 ≤ y
⊢ ↑1 ≤ y
Theorem: Aux.sum_inv_le_log_real
Proof state before:
case h
y : ℝ
hy : 1 ≤ y
⊢ ↑1 ≤ y
Tactic:
norm_cast
Proof state after:
no goals
Theorem: Aux.sum_inv_le_log_real
Proof state before:
case bc.hx
y : ℝ
hy : 1 ≤ y
⊢ 0 < ↑⌊y⌋₊
Tactic:
norm_cast
Proof state after:
case bc.hx
y : ℝ
hy : 1 ≤ y
⊢ 0 < ⌊y⌋₊
Theorem: Aux.sum_inv_le_log_real
Proof state before:
case bc.hx
y : ℝ
hy : 1 ≤ y
⊢ 0 < ⌊y⌋₊
Tactic:
apply Nat.lt_of_succ_le
Proof state after:
case bc.hx.h
y : ℝ
hy : 1 ≤ y
⊢ succ 0 ≤ ⌊y⌋₊
Theorem: Aux.sum_inv_le_log_real
Proof state before:
case bc.hx.h
y : ℝ
hy : 1 ≤ y
⊢ succ 0 ≤ ⌊y⌋₊
Tactic:
apply le_floor
Proof state after:
case bc.hx.h.h
y : ℝ
hy : 1 ≤ y
⊢ ↑(succ 0) ≤ y
Theorem: Aux.sum_inv_le_log_real
Proof state before:
case bc.hx.h.h
y : ℝ
hy : 1 ≤ y
⊢ ↑(succ 0) ≤ y
Tactic:
norm_cast
Proof state after:
no goals
Theorem: Aux.sum_inv_le_log_real
Proof state before:
case bc.hxy
y : ℝ
hy : 1 ≤ y
⊢ ↑⌊y⌋₊ ≤ y
Tactic:
apply floor_le
Proof state after:
case bc.hxy.ha
y : ℝ
hy : 1 ≤ y
⊢ 0 ≤ y
Theorem: Aux.sum_inv_le_log_real
Proof state before:
case bc.hxy.ha
y : ℝ
hy : 1 ≤ y
⊢ 0 ≤ y
Tactic:
linarith
Proof state after:
no goals
Theorem: Aux.Nat.le_prod
Proof state before:
ι : Type u_1
inst✝ : DecidableEq ι
f : ι → ℕ
s : Finset ι
i : ι
hi : i ∈ s
hf : ∀ i ∈ s, f i ≠ 0
⊢ f i ≤ ∏ j ∈ s, f j
Tactic:
rw [←prod_erase_mul (a:=i) (h:= hi)]
Proof state after:
ι : Type u_1
inst✝ : DecidableEq ι
f : ι → ℕ
s : Finset ι
i : ι
hi : i ∈ s
hf : ∀ i ∈ s, f i ≠ 0
⊢ f i ≤ (∏ x ∈ s.erase i, f x) * f i
Theorem: Aux.Nat.le_prod
Proof state before:
ι : Type u_1
inst✝ : DecidableEq ι
f : ι → ℕ
s : Finset ι
i : ι
hi : i ∈ s
hf : ∀ i ∈ s, f i ≠ 0
⊢ f i ≤ (∏ x ∈ s.erase i, f x) * f i
Tactic:
exact Nat.le_mul_of_pos_left _ (prod_pos fun j hj => Nat.pos_of_ne_zero (hf j (mem_of_mem_erase hj)))
Proof state after:
no goals
Theorem: Aux.sum_pow_cardDistinctFactors_div_self_le_log_pow
Proof state before:
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
⊢ (∑ d ∈ P.divisors, if ↑d ≤ x then ↑k ^ ω d / ↑d else 0) ≤ (1 + x.log) ^ k
Tactic:
have hx_pos : 0 < x := by
  linarith
Proof state after:
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
⊢ (∑ d ∈ P.divisors, if ↑d ≤ x then ↑k ^ ω d / ↑d else 0) ≤ (1 + x.log) ^ k
Theorem: Aux.sum_pow_cardDistinctFactors_div_self_le_log_pow
Proof state before:
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
⊢ (∑ d ∈ P.divisors, if ↑d ≤ x then ↑k ^ ω d / ↑d else 0) ≤ (1 + x.log) ^ k
Tactic:
calc
  _ = ∑ d in P.divisors,
        ∑ a in Fintype.piFinset fun _i : Fin k => P.divisors,
          if ∏ i, a i = d ∧ d ∣ P then if ↑d ≤ x then (d : ℝ)⁻¹ else 0 else 0 := ?_
  _ = ∑ a in Fintype.piFinset fun _i : Fin k => P.divisors,
        if ∏ i, a i ∣ P then if ↑(∏ i, a i) ≤ x then ∏ i, (a i : ℝ)⁻¹ else 0 else 0 := ?_
  _ ≤ ∑ a in Fintype.piFinset fun _i : Fin k => P.divisors,
        if ↑(∏ i, a i) ≤ x then ∏ i, (a i : ℝ)⁻¹ else 0 := ?_ _ ≤ ∑ a in Fintype.piFinset fun _i : Fin k => P.divisors,
        ∏ i, if ↑(a i) ≤ x then (a i : ℝ)⁻¹ else 0 := ?_
  _ = ∏ _i : Fin k, ∑ d in P.divisors, if ↑d ≤ x then (d : ℝ)⁻¹ else 0 := by rw [prod_univ_sum]
  _ = (∑ d in P.divisors, if ↑d ≤ x then (d : ℝ)⁻¹ else 0) ^ k := by rw [prod_const, card_fin]
  _ ≤ (1 + Real.log x) ^ k := ?_
Proof state after:
case calc_1
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
⊢ (∑ d ∈ P.divisors, if ↑d ≤ x then ↑k ^ ω d / ↑d else 0) =
    ∑ d ∈ P.divisors,
      ∑ a ∈ Fintype.piFinset fun _i => P.divisors,
        if ∏ i : Fin k, a i = d ∧ d ∣ P then if ↑d ≤ x then (↑d)⁻¹ else 0 else 0

case calc_2
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
⊢ (∑ d ∈ P.divisors,
      ∑ a ∈ Fintype.piFinset fun _i => P.divisors,
        if ∏ i : Fin k, a i = d ∧ d ∣ P then if ↑d ≤ x then (↑d)⁻¹ else 0 else 0) =
    ∑ a ∈ Fintype.piFinset fun _i => P.divisors,
      if ∏ i : Fin k, a i ∣ P then if ↑(∏ i : Fin k, a i) ≤ x then ∏ i : Fin k, (↑(a i))⁻¹ else 0 else 0

case calc_3
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
⊢ (∑ a ∈ Fintype.piFinset fun _i => P.divisors,
      if ∏ i : Fin k, a i ∣ P then if ↑(∏ i : Fin k, a i) ≤ x then ∏ i : Fin k, (↑(a i))⁻¹ else 0 else 0) ≤
    ∑ a ∈ Fintype.piFinset fun _i => P.divisors, if ↑(∏ i : Fin k, a i) ≤ x then ∏ i : Fin k, (↑(a i))⁻¹ else 0

case calc_4
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
⊢ (∑ a ∈ Fintype.piFinset fun _i => P.divisors, if ↑(∏ i : Fin k, a i) ≤ x then ∏ i : Fin k, (↑(a i))⁻¹ else 0) ≤
    ∑ a ∈ Fintype.piFinset fun _i => P.divisors, ∏ i : Fin k, if ↑(a i) ≤ x then (↑(a i))⁻¹ else 0

case calc_5
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
⊢ (∑ d ∈ P.divisors, if ↑d ≤ x then (↑d)⁻¹ else 0) ^ k ≤ (1 + x.log) ^ k
Theorem: Aux.sum_pow_cardDistinctFactors_div_self_le_log_pow
Proof state before:
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
⊢ 0 < x
Tactic:
linarith
Proof state after:
no goals
Theorem: Aux.sum_pow_cardDistinctFactors_div_self_le_log_pow
Proof state before:
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
⊢ (∑ a ∈ Fintype.piFinset fun _i => P.divisors, ∏ i : Fin k, if ↑(a i) ≤ x then (↑(a i))⁻¹ else 0) =
    ∏ _i : Fin k, ∑ d ∈ P.divisors, if ↑d ≤ x then (↑d)⁻¹ else 0
Tactic:
rw [prod_univ_sum]
Proof state after:
no goals
Theorem: Aux.sum_pow_cardDistinctFactors_div_self_le_log_pow
Proof state before:
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
⊢ (∏ _i : Fin k, ∑ d ∈ P.divisors, if ↑d ≤ x then (↑d)⁻¹ else 0) = (∑ d ∈ P.divisors, if ↑d ≤ x then (↑d)⁻¹ else 0) ^ k
Tactic:
rw [prod_const, card_fin]
Proof state after:
no goals
Theorem: Aux.sum_pow_cardDistinctFactors_div_self_le_log_pow
Proof state before:
case calc_1
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
⊢ (∑ d ∈ P.divisors, if ↑d ≤ x then ↑k ^ ω d / ↑d else 0) =
    ∑ d ∈ P.divisors,
      ∑ a ∈ Fintype.piFinset fun _i => P.divisors,
        if ∏ i : Fin k, a i = d ∧ d ∣ P then if ↑d ≤ x then (↑d)⁻¹ else 0 else 0
Tactic:
apply sum_congr rfl
Proof state after:
case calc_1
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
⊢ ∀ x_1 ∈ P.divisors,
    (if ↑x_1 ≤ x then ↑k ^ ω x_1 / ↑x_1 else 0) =
      ∑ a ∈ Fintype.piFinset fun _i => P.divisors,
        if ∏ i : Fin k, a i = x_1 ∧ x_1 ∣ P then if ↑x_1 ≤ x then (↑x_1)⁻¹ else 0 else 0
Theorem: Aux.sum_pow_cardDistinctFactors_div_self_le_log_pow
Proof state before:
case calc_1
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
⊢ ∀ x_1 ∈ P.divisors,
    (if ↑x_1 ≤ x then ↑k ^ ω x_1 / ↑x_1 else 0) =
      ∑ a ∈ Fintype.piFinset fun _i => P.divisors,
        if ∏ i : Fin k, a i = x_1 ∧ x_1 ∣ P then if ↑x_1 ≤ x then (↑x_1)⁻¹ else 0 else 0
Tactic:
intro d hd
Proof state after:
case calc_1
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
d : ℕ
hd : d ∈ P.divisors
⊢ (if ↑d ≤ x then ↑k ^ ω d / ↑d else 0) =
    ∑ a ∈ Fintype.piFinset fun _i => P.divisors,
      if ∏ i : Fin k, a i = d ∧ d ∣ P then if ↑d ≤ x then (↑d)⁻¹ else 0 else 0
Theorem: Aux.sum_pow_cardDistinctFactors_div_self_le_log_pow
Proof state before:
case calc_1
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
d : ℕ
hd : d ∈ P.divisors
⊢ (if ↑d ≤ x then ↑k ^ ω d / ↑d else 0) =
    ∑ a ∈ Fintype.piFinset fun _i => P.divisors,
      if ∏ i : Fin k, a i = d ∧ d ∣ P then if ↑d ≤ x then (↑d)⁻¹ else 0 else 0
Tactic:
rw [mem_divisors] at hd
Proof state after:
case calc_1
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
d : ℕ
hd : d ∣ P ∧ P ≠ 0
⊢ (if ↑d ≤ x then ↑k ^ ω d / ↑d else 0) =
    ∑ a ∈ Fintype.piFinset fun _i => P.divisors,
      if ∏ i : Fin k, a i = d ∧ d ∣ P then if ↑d ≤ x then (↑d)⁻¹ else 0 else 0
Theorem: Aux.sum_pow_cardDistinctFactors_div_self_le_log_pow
Proof state before:
case calc_1
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
d : ℕ
hd : d ∣ P ∧ P ≠ 0
⊢ (if ↑d ≤ x then ↑k ^ ω d / ↑d else 0) =
    ∑ a ∈ Fintype.piFinset fun _i => P.divisors,
      if ∏ i : Fin k, a i = d ∧ d ∣ P then if ↑d ≤ x then (↑d)⁻¹ else 0 else 0
Tactic:
simp_rw [ite_and]
Proof state after:
case calc_1
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
d : ℕ
hd : d ∣ P ∧ P ≠ 0
⊢ (if ↑d ≤ x then ↑k ^ ω d / ↑d else 0) =
    ∑ x_1 ∈ Fintype.piFinset fun _i => P.divisors,
      if ∏ i : Fin k, x_1 i = d then if d ∣ P then if ↑d ≤ x then (↑d)⁻¹ else 0 else 0 else 0
Theorem: Aux.sum_pow_cardDistinctFactors_div_self_le_log_pow
Proof state before:
case calc_1
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
d : ℕ
hd : d ∣ P ∧ P ≠ 0
⊢ (if ↑d ≤ x then ↑k ^ ω d / ↑d else 0) =
    ∑ x_1 ∈ Fintype.piFinset fun _i => P.divisors,
      if ∏ i : Fin k, x_1 i = d then if d ∣ P then if ↑d ≤ x then (↑d)⁻¹ else 0 else 0 else 0
Tactic:
rw [←sum_filter, Finset.sum_const, ←finMulAntidiagonal_univ_eq hd.1 hd.2, card_finMulAntidiagonal
  <| hP.squarefree_of_dvd hd.1, if_pos hd.1]
Proof state after:
case calc_1
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
d : ℕ
hd : d ∣ P ∧ P ≠ 0
⊢ (if ↑d ≤ x then ↑k ^ ω d / ↑d else 0) = k ^ ω d • if ↑d ≤ x then (↑d)⁻¹ else 0
Theorem: Aux.sum_pow_cardDistinctFactors_div_self_le_log_pow
Proof state before:
case calc_1
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
d : ℕ
hd : d ∣ P ∧ P ≠ 0
⊢ (if ↑d ≤ x then ↑k ^ ω d / ↑d else 0) = k ^ ω d • if ↑d ≤ x then (↑d)⁻¹ else 0
Tactic:
simp only [div_eq_mul_inv, one_mul, nsmul_eq_mul, cast_pow, mul_ite, mul_zero]
Proof state after:
no goals
Theorem: Aux.sum_pow_cardDistinctFactors_div_self_le_log_pow
Proof state before:
case calc_2
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
⊢ (∑ d ∈ P.divisors,
      ∑ a ∈ Fintype.piFinset fun _i => P.divisors,
        if ∏ i : Fin k, a i = d ∧ d ∣ P then if ↑d ≤ x then (↑d)⁻¹ else 0 else 0) =
    ∑ a ∈ Fintype.piFinset fun _i => P.divisors,
      if ∏ i : Fin k, a i ∣ P then if ↑(∏ i : Fin k, a i) ≤ x then ∏ i : Fin k, (↑(a i))⁻¹ else 0 else 0
Tactic:
rw [sum_comm]
Proof state after:
case calc_2
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
⊢ (∑ y ∈ Fintype.piFinset fun _i => P.divisors,
      ∑ x_1 ∈ P.divisors, if ∏ i : Fin k, y i = x_1 ∧ x_1 ∣ P then if ↑x_1 ≤ x then (↑x_1)⁻¹ else 0 else 0) =
    ∑ a ∈ Fintype.piFinset fun _i => P.divisors,
      if ∏ i : Fin k, a i ∣ P then if ↑(∏ i : Fin k, a i) ≤ x then ∏ i : Fin k, (↑(a i))⁻¹ else 0 else 0
Theorem: Aux.sum_pow_cardDistinctFactors_div_self_le_log_pow
Proof state before:
case calc_2
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
⊢ (∑ y ∈ Fintype.piFinset fun _i => P.divisors,
      ∑ x_1 ∈ P.divisors, if ∏ i : Fin k, y i = x_1 ∧ x_1 ∣ P then if ↑x_1 ≤ x then (↑x_1)⁻¹ else 0 else 0) =
    ∑ a ∈ Fintype.piFinset fun _i => P.divisors,
      if ∏ i : Fin k, a i ∣ P then if ↑(∏ i : Fin k, a i) ≤ x then ∏ i : Fin k, (↑(a i))⁻¹ else 0 else 0
Tactic:
apply sum_congr rfl
Proof state after:
case calc_2
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
⊢ ∀ x_1 ∈ Fintype.piFinset fun _i => P.divisors,
    (∑ x_2 ∈ P.divisors, if ∏ i : Fin k, x_1 i = x_2 ∧ x_2 ∣ P then if ↑x_2 ≤ x then (↑x_2)⁻¹ else 0 else 0) =
      if ∏ i : Fin k, x_1 i ∣ P then if ↑(∏ i : Fin k, x_1 i) ≤ x then ∏ i : Fin k, (↑(x_1 i))⁻¹ else 0 else 0
Theorem: Aux.sum_pow_cardDistinctFactors_div_self_le_log_pow
Proof state before:
case calc_2
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
⊢ ∀ x_1 ∈ Fintype.piFinset fun _i => P.divisors,
    (∑ x_2 ∈ P.divisors, if ∏ i : Fin k, x_1 i = x_2 ∧ x_2 ∣ P then if ↑x_2 ≤ x then (↑x_2)⁻¹ else 0 else 0) =
      if ∏ i : Fin k, x_1 i ∣ P then if ↑(∏ i : Fin k, x_1 i) ≤ x then ∏ i : Fin k, (↑(x_1 i))⁻¹ else 0 else 0
Tactic:
intro a _
Proof state after:
case calc_2
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
a : Fin k → ℕ
a✝ : a ∈ Fintype.piFinset fun _i => P.divisors
⊢ (∑ x_1 ∈ P.divisors, if ∏ i : Fin k, a i = x_1 ∧ x_1 ∣ P then if ↑x_1 ≤ x then (↑x_1)⁻¹ else 0 else 0) =
    if ∏ i : Fin k, a i ∣ P then if ↑(∏ i : Fin k, a i) ≤ x then ∏ i : Fin k, (↑(a i))⁻¹ else 0 else 0
Theorem: Aux.sum_pow_cardDistinctFactors_div_self_le_log_pow
Proof state before:
case calc_2
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
a : Fin k → ℕ
a✝ : a ∈ Fintype.piFinset fun _i => P.divisors
⊢ (∑ x_1 ∈ P.divisors, if ∏ i : Fin k, a i = x_1 ∧ x_1 ∣ P then if ↑x_1 ≤ x then (↑x_1)⁻¹ else 0 else 0) =
    if ∏ i : Fin k, a i ∣ P then if ↑(∏ i : Fin k, a i) ≤ x then ∏ i : Fin k, (↑(a i))⁻¹ else 0 else 0
Tactic:
rw [sum_eq_single (∏ i, a i)]
Proof state after:
case calc_2
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
a : Fin k → ℕ
a✝ : a ∈ Fintype.piFinset fun _i => P.divisors
⊢ (if ∏ i : Fin k, a i = ∏ i : Fin k, a i ∧ ∏ i : Fin k, a i ∣ P then
      if ↑(∏ i : Fin k, a i) ≤ x then (↑(∏ i : Fin k, a i))⁻¹ else 0
    else 0) =
    if ∏ i : Fin k, a i ∣ P then if ↑(∏ i : Fin k, a i) ≤ x then ∏ i : Fin k, (↑(a i))⁻¹ else 0 else 0

case calc_2.h₀
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
a : Fin k → ℕ
a✝ : a ∈ Fintype.piFinset fun _i => P.divisors
⊢ ∀ b ∈ P.divisors,
    b ≠ ∏ i : Fin k, a i → (if ∏ i : Fin k, a i = b ∧ b ∣ P then if ↑b ≤ x then (↑b)⁻¹ else 0 else 0) = 0

case calc_2.h₁
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
a : Fin k → ℕ
a✝ : a ∈ Fintype.piFinset fun _i => P.divisors
⊢ ∏ i : Fin k, a i ∉ P.divisors →
    (if ∏ i : Fin k, a i = ∏ i : Fin k, a i ∧ ∏ i : Fin k, a i ∣ P then
        if ↑(∏ i : Fin k, a i) ≤ x then (↑(∏ i : Fin k, a i))⁻¹ else 0
      else 0) =
      0
Theorem: Aux.sum_pow_cardDistinctFactors_div_self_le_log_pow
Proof state before:
case calc_2
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
a : Fin k → ℕ
a✝ : a ∈ Fintype.piFinset fun _i => P.divisors
⊢ (if ∏ i : Fin k, a i = ∏ i : Fin k, a i ∧ ∏ i : Fin k, a i ∣ P then
      if ↑(∏ i : Fin k, a i) ≤ x then (↑(∏ i : Fin k, a i))⁻¹ else 0
    else 0) =
    if ∏ i : Fin k, a i ∣ P then if ↑(∏ i : Fin k, a i) ≤ x then ∏ i : Fin k, (↑(a i))⁻¹ else 0 else 0
Tactic:
apply if_ctx_congr _ _ (fun _ => rfl)
Proof state after:
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
a : Fin k → ℕ
a✝ : a ∈ Fintype.piFinset fun _i => P.divisors
⊢ ∏ i : Fin k, a i = ∏ i : Fin k, a i ∧ ∏ i : Fin k, a i ∣ P ↔ ∏ i : Fin k, a i ∣ P

P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
a : Fin k → ℕ
a✝ : a ∈ Fintype.piFinset fun _i => P.divisors
⊢ ∏ i : Fin k, a i ∣ P →
    (if ↑(∏ i : Fin k, a i) ≤ x then (↑(∏ i : Fin k, a i))⁻¹ else 0) =
      if ↑(∏ i : Fin k, a i) ≤ x then ∏ i : Fin k, (↑(a i))⁻¹ else 0
Theorem: Aux.sum_pow_cardDistinctFactors_div_self_le_log_pow
Proof state before:
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
a : Fin k → ℕ
a✝ : a ∈ Fintype.piFinset fun _i => P.divisors
⊢ ∏ i : Fin k, a i = ∏ i : Fin k, a i ∧ ∏ i : Fin k, a i ∣ P ↔ ∏ i : Fin k, a i ∣ P

P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
a : Fin k → ℕ
a✝ : a ∈ Fintype.piFinset fun _i => P.divisors
⊢ ∏ i : Fin k, a i ∣ P →
    (if ↑(∏ i : Fin k, a i) ≤ x then (↑(∏ i : Fin k, a i))⁻¹ else 0) =
      if ↑(∏ i : Fin k, a i) ≤ x then ∏ i : Fin k, (↑(a i))⁻¹ else 0
Tactic:
rw [Iff.comm, iff_and_self]
Proof state after:
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
a : Fin k → ℕ
a✝ : a ∈ Fintype.piFinset fun _i => P.divisors
⊢ ∏ i : Fin k, a i ∣ P → ∏ i : Fin k, a i = ∏ i : Fin k, a i

P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
a : Fin k → ℕ
a✝ : a ∈ Fintype.piFinset fun _i => P.divisors
⊢ ∏ i : Fin k, a i ∣ P →
    (if ↑(∏ i : Fin k, a i) ≤ x then (↑(∏ i : Fin k, a i))⁻¹ else 0) =
      if ↑(∏ i : Fin k, a i) ≤ x then ∏ i : Fin k, (↑(a i))⁻¹ else 0
Theorem: Aux.sum_pow_cardDistinctFactors_div_self_le_log_pow
Proof state before:
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
a : Fin k → ℕ
a✝ : a ∈ Fintype.piFinset fun _i => P.divisors
⊢ ∏ i : Fin k, a i ∣ P → ∏ i : Fin k, a i = ∏ i : Fin k, a i

P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
a : Fin k → ℕ
a✝ : a ∈ Fintype.piFinset fun _i => P.divisors
⊢ ∏ i : Fin k, a i ∣ P →
    (if ↑(∏ i : Fin k, a i) ≤ x then (↑(∏ i : Fin k, a i))⁻¹ else 0) =
      if ↑(∏ i : Fin k, a i) ≤ x then ∏ i : Fin k, (↑(a i))⁻¹ else 0
Tactic:
exact fun _ => rfl
Proof state after:
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
a : Fin k → ℕ
a✝ : a ∈ Fintype.piFinset fun _i => P.divisors
⊢ ∏ i : Fin k, a i ∣ P →
    (if ↑(∏ i : Fin k, a i) ≤ x then (↑(∏ i : Fin k, a i))⁻¹ else 0) =
      if ↑(∏ i : Fin k, a i) ≤ x then ∏ i : Fin k, (↑(a i))⁻¹ else 0
Theorem: Aux.sum_pow_cardDistinctFactors_div_self_le_log_pow
Proof state before:
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
a : Fin k → ℕ
a✝ : a ∈ Fintype.piFinset fun _i => P.divisors
⊢ ∏ i : Fin k, a i ∣ P →
    (if ↑(∏ i : Fin k, a i) ≤ x then (↑(∏ i : Fin k, a i))⁻¹ else 0) =
      if ↑(∏ i : Fin k, a i) ≤ x then ∏ i : Fin k, (↑(a i))⁻¹ else 0
Tactic:
intro
Proof state after:
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
a : Fin k → ℕ
a✝¹ : a ∈ Fintype.piFinset fun _i => P.divisors
a✝ : ∏ i : Fin k, a i ∣ P
⊢ (if ↑(∏ i : Fin k, a i) ≤ x then (↑(∏ i : Fin k, a i))⁻¹ else 0) =
    if ↑(∏ i : Fin k, a i) ≤ x then ∏ i : Fin k, (↑(a i))⁻¹ else 0
Theorem: Aux.sum_pow_cardDistinctFactors_div_self_le_log_pow
Proof state before:
P k : ℕ
x : ℝ
hx : 1 ≤ x
hP : Squarefree P
hx_pos : 0 < x
a : Fin k → ℕ
a✝¹ : a ∈ Fintype.piFinset fun _i => P.divisors
a✝ : ∏ i : Fin k, a i ∣ P
⊢ (if ↑(∏ i : Fin k, a i) ≤ x then (↑(∏ i : Fin k, a i))⁻¹ else 0) =
    if ↑(∏ i : Fin k, a i) ≤ x then ∏ i : Fin k, (↑(a i))⁻¹ else 0
Tactic:
rw [cast_prod, ← prod_inv_distrib]
Proof state after:
no goals