internlm/Lean-Github huggingface dataset:
Theorem: Sieve.prod_factors_sum_pow_compMult
Proof state before:
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
⊢ ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    i a ha ∈ Finset.filter (fun x => d ∣ x) (d ^ M).divisors
Tactic:
intro a ha
Proof state after:
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
⊢ i a ha ∈ Finset.filter (fun x => d ∣ x) (d ^ M).divisors
Theorem: Sieve.prod_factors_sum_pow_compMult
Proof state before:
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
⊢ i a ha ∈ Finset.filter (fun x => d ∣ x) (d ^ M).divisors
Tactic:
rw [Finset.mem_filter, Nat.mem_divisors, ←Nat.factorization_le_iff_dvd hd.ne_zero (hi_ne_zero a ha),
  ←Nat.factorization_le_iff_dvd (hi_ne_zero a ha) (pow_ne_zero _ hd.ne_zero)]
Proof state after:
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
⊢ ((i a ha).factorization ≤ (d ^ M).factorization ∧ d ^ M ≠ 0) ∧ d.factorization ≤ (i a ha).factorization
Theorem: Sieve.prod_factors_sum_pow_compMult
Proof state before:
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
⊢ ((i a ha).factorization ≤ (d ^ M).factorization ∧ d ^ M ≠ 0) ∧ d.factorization ≤ (i a ha).factorization
Tactic:
constructor
Proof state after:
case left
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
⊢ (i a ha).factorization ≤ (d ^ M).factorization ∧ d ^ M ≠ 0

case right
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
⊢ d.factorization ≤ (i a ha).factorization
Theorem: Sieve.prod_factors_sum_pow_compMult
Proof state before:
case left.left
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
⊢ (i a ha).factorization ≤ (d ^ M).factorization
Tactic:
rw [Finsupp.le_iff]
Proof state after:
case left.left
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
⊢ ∀ i_1 ∈ (i a ha).factorization.support, (i a ha).factorization i_1 ≤ (d ^ M).factorization i_1
Theorem: Sieve.prod_factors_sum_pow_compMult
Proof state before:
case left.left
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
⊢ ∀ i_1 ∈ (i a ha).factorization.support, (i a ha).factorization i_1 ≤ (d ^ M).factorization i_1
Tactic:
intro p _
Proof state after:
case left.left
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
p : ℕ
a✝ : p ∈ (i a ha).factorization.support
⊢ (i a ha).factorization p ≤ (d ^ M).factorization p
Theorem: Sieve.prod_factors_sum_pow_compMult
Proof state before:
case left.left
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
p : ℕ
a✝ : p ∈ (i a ha).factorization.support
⊢ (i a ha).factorization p ≤ (d ^ M).factorization p
Tactic:
rw [hfact_i a ha]
Proof state after:
case left.left
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
p : ℕ
a✝ : p ∈ (i a ha).factorization.support
⊢ (if hp : p ∈ d.primeFactors then a p hp else 0) ≤ (d ^ M).factorization p
Theorem: Sieve.prod_factors_sum_pow_compMult
Proof state before:
case left.left
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
p : ℕ
a✝ : p ∈ (i a ha).factorization.support
⊢ (if hp : p ∈ d.primeFactors then a p hp else 0) ≤ (d ^ M).factorization p
Tactic:
by_cases hp :  p ∈ d.primeFactors
Proof state after:
case pos
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
p : ℕ
a✝ : p ∈ (i a ha).factorization.support
hp : p ∈ d.primeFactors
⊢ (if hp : p ∈ d.primeFactors then a p hp else 0) ≤ (d ^ M).factorization p

case neg
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
p : ℕ
a✝ : p ∈ (i a ha).factorization.support
hp : p ∉ d.primeFactors
⊢ (if hp : p ∈ d.primeFactors then a p hp else 0) ≤ (d ^ M).factorization p
Theorem: Sieve.prod_factors_sum_pow_compMult
Proof state before:
case pos
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
p : ℕ
a✝ : p ∈ (i a ha).factorization.support
hp : p ∈ d.primeFactors
⊢ (if hp : p ∈ d.primeFactors then a p hp else 0) ≤ (d ^ M).factorization p
Tactic:
rw [dif_pos hp]
Proof state after:
case pos
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
p : ℕ
a✝ : p ∈ (i a ha).factorization.support
hp : p ∈ d.primeFactors
⊢ a p hp ≤ (d ^ M).factorization p
Theorem: Sieve.prod_factors_sum_pow_compMult
Proof state before:
case pos
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
p : ℕ
a✝ : p ∈ (i a ha).factorization.support
hp : p ∈ d.primeFactors
⊢ a p hp ≤ (d ^ M).factorization p
Tactic:
rw [Nat.factorization_pow, Finsupp.smul_apply]
Proof state after:
case pos
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
p : ℕ
a✝ : p ∈ (i a ha).factorization.support
hp : p ∈ d.primeFactors
⊢ a p hp ≤ M • d.factorization p
Theorem: Sieve.prod_factors_sum_pow_compMult
Proof state before:
case pos
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
p : ℕ
a✝ : p ∈ (i a ha).factorization.support
hp : p ∈ d.primeFactors
⊢ a p hp ≤ M • d.factorization p
Tactic:
simp_rw [Finset.mem_pi, Finset.mem_Icc] at ha
Proof state after:
case pos
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha✝ : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
p : ℕ
a✝ : p ∈ (i a ha✝).factorization.support
hp : p ∈ d.primeFactors
ha : ∀ (a_1 : ℕ) (h : a_1 ∈ d.primeFactors), 1 ≤ a a_1 h ∧ a a_1 h ≤ M
⊢ a p hp ≤ M • d.factorization p
Theorem: Sieve.prod_factors_sum_pow_compMult
Proof state before:
case pos
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha✝ : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
p : ℕ
a✝ : p ∈ (i a ha✝).factorization.support
hp : p ∈ d.primeFactors
ha : ∀ (a_1 : ℕ) (h : a_1 ∈ d.primeFactors), 1 ≤ a a_1 h ∧ a a_1 h ≤ M
⊢ a p hp ≤ M • d.factorization p
Tactic:
trans (M • 1)
Proof state after:
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha✝ : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
p : ℕ
a✝ : p ∈ (i a ha✝).factorization.support
hp : p ∈ d.primeFactors
ha : ∀ (a_1 : ℕ) (h : a_1 ∈ d.primeFactors), 1 ≤ a a_1 h ∧ a a_1 h ≤ M
⊢ a p hp ≤ M • 1

M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha✝ : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
p : ℕ
a✝ : p ∈ (i a ha✝).factorization.support
hp : p ∈ d.primeFactors
ha : ∀ (a_1 : ℕ) (h : a_1 ∈ d.primeFactors), 1 ≤ a a_1 h ∧ a a_1 h ≤ M
⊢ M • 1 ≤ M • d.factorization p
Theorem: Sieve.prod_factors_sum_pow_compMult
Proof state before:
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha✝ : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
p : ℕ
a✝ : p ∈ (i a ha✝).factorization.support
hp : p ∈ d.primeFactors
ha : ∀ (a_1 : ℕ) (h : a_1 ∈ d.primeFactors), 1 ≤ a a_1 h ∧ a a_1 h ≤ M
⊢ a p hp ≤ M • 1
Tactic:
norm_num
Proof state after:
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha✝ : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
p : ℕ
a✝ : p ∈ (i a ha✝).factorization.support
hp : p ∈ d.primeFactors
ha : ∀ (a_1 : ℕ) (h : a_1 ∈ d.primeFactors), 1 ≤ a a_1 h ∧ a a_1 h ≤ M
⊢ a p hp ≤ M
Theorem: Sieve.prod_factors_sum_pow_compMult
Proof state before:
M : ℕ
hM : M ≠ 0
f : ArithmeticFunction ℝ
hf : CompletelyMultiplicative f
d : ℕ
hd : Squarefree d
i : (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) → (a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) → ℕ :=
  fun a x => ∏ p ∈ d.primeFactors.attach, ↑p ^ a ↑p ⋯
hfact_i :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M) (p : ℕ),
    (i a ha).factorization p = if hp : p ∈ d.primeFactors then a p hp else 0
hi_ne_zero : ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M), i a ha ≠ 0
a : (a : ℕ) → a ∈ d.primeFactors → ℕ
ha✝ : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M
p : ℕ
a✝ : p ∈ (i a ha✝).factorization.support
hp : p ∈ d.primeFactors
ha : ∀ (a_1 : ℕ) (h : a_1 ∈ d.primeFactors), 1 ≤ a a_1 h ∧ a a_1 h ≤ M
⊢ a p hp ≤ M
Tactic:
exact (ha p hp).2
Proof state after:
no goals