Theorem: Sieve.prod_factors_sum_pow_compMult (PrimeNumberTheoremAnd/Mathlib/NumberTheory/Sieve/SelbergBounds.lean)

State before:
case h.ha
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
hi :
  ∀ (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
h✝ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : b ∈ Finset.filter (fun x => d ∣ x) (d ^ M).divisors
h : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
⊢ i (fun p x => b.factorization p) h ≠ 0
Tactic:
apply hi_ne_zero _ h
State after:
no goals

State before:
case h.hb
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
hi :
  ∀ (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
h✝ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : b ∈ Finset.filter (fun x => d ∣ x) (d ^ M).divisors
h : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
⊢ b ≠ 0
Tactic:
exact ne_of_gt $ Nat.pos_of_mem_divisors (Finset.mem_filter.mp hb).1
State after:
no goals

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
hi :
  ∀ (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
h✝ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : p ∈ d.primeFactors
⊢ (if hp : p ∈ d.primeFactors then b.factorization p else 0) = b.factorization p
Tactic:
rw [dif_pos hp]
State after:
no goals

State before:
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
hi :
  ∀ (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
h✝ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : p ∉ d.primeFactors
⊢ (if hp : p ∈ d.primeFactors then b.factorization p else 0) = b.factorization p
Tactic:
rw [dif_neg hp, eq_comm, Nat.factorization_eq_zero_iff, ←or_assoc]
State after:
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
hi :
  ∀ (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
h✝ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : p ∉ d.primeFactors
⊢ (¬p.Prime ∨ ¬p ∣ b) ∨ b = 0

State before:
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
hi :
  ∀ (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
h✝ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : p ∉ d.primeFactors
⊢ (¬p.Prime ∨ ¬p ∣ b) ∨ b = 0
Tactic:
rw [Nat.mem_primeFactors] at hp
State after:
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
hi :
  ∀ (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
h✝ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : ¬(p.Prime ∧ p ∣ d ∧ d ≠ 0)
⊢ (¬p.Prime ∨ ¬p ∣ b) ∨ b = 0

State before:
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
hi :
  ∀ (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
h✝ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : ¬(p.Prime ∧ p ∣ d ∧ d ≠ 0)
⊢ (¬p.Prime ∨ ¬p ∣ b) ∨ b = 0
Tactic:
left
State after:
case neg.h
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
hi :
  ∀ (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
h✝ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : ¬(p.Prime ∧ p ∣ d ∧ d ≠ 0)
⊢ ¬p.Prime ∨ ¬p ∣ b

State before:
case neg.h
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
hi :
  ∀ (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
h✝ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : ¬(p.Prime ∧ p ∣ d ∧ d ≠ 0)
⊢ ¬p.Prime ∨ ¬p ∣ b
Tactic:
push_neg at hp
State after:
case neg.h
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
hi :
  ∀ (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
h✝ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : p.Prime → p ∣ d → d = 0
⊢ ¬p.Prime ∨ ¬p ∣ b

State before:
case neg.h
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
hi :
  ∀ (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
h✝ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : p.Prime → p ∣ d → d = 0
⊢ ¬p.Prime ∨ ¬p ∣ b
Tactic:
by_cases hpp : p.Prime
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
hi :
  ∀ (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
h✝ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : p.Prime → p ∣ d → d = 0
hpp : p.Prime
⊢ ¬p.Prime ∨ ¬p ∣ b

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
hi :
  ∀ (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
h✝ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : p.Prime → p ∣ d → d = 0
hpp : ¬p.Prime
⊢ ¬p.Prime ∨ ¬p ∣ b

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
hi :
  ∀ (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
h✝ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : p.Prime → p ∣ d → d = 0
hpp : p.Prime
⊢ ¬p.Prime ∨ ¬p ∣ b
Tactic:
right
State after:
case pos.h
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
hi :
  ∀ (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
h✝ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : p.Prime → p ∣ d → d = 0
hpp : p.Prime
⊢ ¬p ∣ b

State before:
case pos.h
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
hi :
  ∀ (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
h✝ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : p.Prime → p ∣ d → d = 0
hpp : p.Prime
⊢ ¬p ∣ b
Tactic:
intro h
State after:
case pos.h
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
hi :
  ∀ (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
h✝¹ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h✝ : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : p.Prime → p ∣ d → d = 0
hpp : p.Prime
h : p ∣ b
⊢ False

State before:
case pos.h
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
hi :
  ∀ (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
h✝¹ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h✝ : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : p.Prime → p ∣ d → d = 0
hpp : p.Prime
h : p ∣ b
⊢ False
Tactic:
apply absurd (hp hpp)
State after:
case pos.h
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
hi :
  ∀ (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
h✝¹ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h✝ : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : p.Prime → p ∣ d → d = 0
hpp : p.Prime
h : p ∣ b
⊢ ¬(p ∣ d → d = 0)

State before:
case pos.h
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
hi :
  ∀ (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
h✝¹ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h✝ : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : p.Prime → p ∣ d → d = 0
hpp : p.Prime
h : p ∣ b
⊢ ¬(p ∣ d → d = 0)
Tactic:
push_neg
State after:
case pos.h
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
hi :
  ∀ (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
h✝¹ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h✝ : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : p.Prime → p ∣ d → d = 0
hpp : p.Prime
h : p ∣ b
⊢ p ∣ d ∧ d ≠ 0

State before:
case pos.h
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
hi :
  ∀ (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
h✝¹ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h✝ : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : p.Prime → p ∣ d → d = 0
hpp : p.Prime
h : p ∣ b
⊢ p ∣ d ∧ d ≠ 0
Tactic:
exact ⟨hpp.dvd_of_dvd_pow (h.trans hb.1.1), hd.ne_zero⟩
State after:
no goals

State before:
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
hi :
  ∀ (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
h✝ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : p.Prime → p ∣ d → d = 0
hpp : ¬p.Prime
⊢ ¬p.Prime ∨ ¬p ∣ b
Tactic:
left
State after:
case neg.h
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
hi :
  ∀ (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
h✝ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : p.Prime → p ∣ d → d = 0
hpp : ¬p.Prime
⊢ ¬p.Prime

State before:
case neg.h
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
hi :
  ∀ (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
h✝ :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun _p => Finset.Icc 1 M),
    ∏ p ∈ d.primeFactors.attach, f (↑p ^ a ↑p ⋯) = f (i a ha)
i_inj :
  ∀ (a : (a : ℕ) → a ∈ d.primeFactors → ℕ) (ha : a ∈ d.primeFactors.pi fun p => Finset.Icc 1 M)
    (b : (a : ℕ) → a ∈ d.primeFactors → ℕ) (hb : b ∈ d.primeFactors.pi fun p => Finset.Icc 1 M), i a ha = i b hb → a = b
b : ℕ
hb : (b ∣ d ^ M ∧ d ^ M ≠ 0) ∧ d ∣ b
h : (fun p x => b.factorization p) ∈ d.primeFactors.pi fun p => Finset.Icc 1 M
p : ℕ
hp : p.Prime → p ∣ d → d = 0
hpp : ¬p.Prime
⊢ ¬p.Prime
Tactic:
exact hpp
State after:
no goals