l3lab/ntp-mathlib huggingface dataset:
/-- The tautological cocone with point `P` is a colimit cocone, exhibiting `P` as a colimit of
    representables. -/
def isColimitTautologicalCocone : IsColimit (tautologicalCocone P) where
  desc := fun s => by
    refine' ⟨fun X t => yonedaEquiv (s.ι.app (CostructuredArrow.mk (yonedaEquiv.symm t))), _⟩
    intros X Y f
    ext t
    dsimp
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    erw [yonedaEquiv_naturality', yonedaEquiv_symm_map]
    simpa using (s.ι.naturality
      (CostructuredArrow.homMk' (CostructuredArrow.mk (yonedaEquiv.symm t)) f.unop)).symm
  fac := by
    intro s t
    dsimp
    apply yonedaEquiv.injective
    rw [yonedaEquiv_comp]
    

Proof state:
case a
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P : Cᵒᵖ ⥤ Type v₁
s : Cocone (CostructuredArrow.proj yoneda P ⋙ yoneda)
t : CostructuredArrow yoneda P
⊢ (NatTrans.mk fun X t => yonedaEquiv (s.ι.app (CostructuredArrow.mk (yonedaEquiv.symm t)))).app (Opposite.op t.left)
      (yonedaEquiv t.hom) =
    yonedaEquiv (s.ι.app t)

Next tactic:
dsimp only
/-- The tautological cocone with point `P` is a colimit cocone, exhibiting `P` as a colimit of
    representables. -/
def isColimitTautologicalCocone : IsColimit (tautologicalCocone P) where
  desc := fun s => by
    refine' ⟨fun X t => yonedaEquiv (s.ι.app (CostructuredArrow.mk (yonedaEquiv.symm t))), _⟩
    intros X Y f
    ext t
    dsimp
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    erw [yonedaEquiv_naturality', yonedaEquiv_symm_map]
    simpa using (s.ι.naturality
      (CostructuredArrow.homMk' (CostructuredArrow.mk (yonedaEquiv.symm t)) f.unop)).symm
  fac := by
    intro s t
    dsimp
    apply yonedaEquiv.injective
    rw [yonedaEquiv_comp]
    dsimp only
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    

Proof state:
case a
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P : Cᵒᵖ ⥤ Type v₁
s : Cocone (CostructuredArrow.proj yoneda P ⋙ yoneda)
t : CostructuredArrow yoneda P
⊢ yonedaEquiv (s.ι.app (CostructuredArrow.mk (yonedaEquiv.symm (yonedaEquiv t.hom)))) = yonedaEquiv (s.ι.app t)

Next tactic:
erw [Equiv.symm_apply_apply]
/-- The tautological cocone with point `P` is a colimit cocone, exhibiting `P` as a colimit of
    representables. -/
def isColimitTautologicalCocone : IsColimit (tautologicalCocone P) where
  desc := fun s => by
    refine' ⟨fun X t => yonedaEquiv (s.ι.app (CostructuredArrow.mk (yonedaEquiv.symm t))), _⟩
    intros X Y f
    ext t
    dsimp
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    erw [yonedaEquiv_naturality', yonedaEquiv_symm_map]
    simpa using (s.ι.naturality
      (CostructuredArrow.homMk' (CostructuredArrow.mk (yonedaEquiv.symm t)) f.unop)).symm
  fac := by
    intro s t
    dsimp
    apply yonedaEquiv.injective
    rw [yonedaEquiv_comp]
    dsimp only
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    erw [Equiv.symm_apply_apply]
    

Proof state:
case a
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P : Cᵒᵖ ⥤ Type v₁
s : Cocone (CostructuredArrow.proj yoneda P ⋙ yoneda)
t : CostructuredArrow yoneda P
⊢ yonedaEquiv (s.ι.app (CostructuredArrow.mk t.hom)) = yonedaEquiv (s.ι.app t)

Next tactic:
rfl
/-- The tautological cocone with point `P` is a colimit cocone, exhibiting `P` as a colimit of
    representables. -/
def isColimitTautologicalCocone : IsColimit (tautologicalCocone P) where
  desc := fun s => by
    refine' ⟨fun X t => yonedaEquiv (s.ι.app (CostructuredArrow.mk (yonedaEquiv.symm t))), _⟩
    intros X Y f
    ext t
    dsimp
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    erw [yonedaEquiv_naturality', yonedaEquiv_symm_map]
    simpa using (s.ι.naturality
      (CostructuredArrow.homMk' (CostructuredArrow.mk (yonedaEquiv.symm t)) f.unop)).symm
  fac := by
    intro s t
    dsimp
    apply yonedaEquiv.injective
    rw [yonedaEquiv_comp]
    dsimp only
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    erw [Equiv.symm_apply_apply]
    rfl
  uniq := by
    

Proof state:
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P : Cᵒᵖ ⥤ Type v₁
⊢ ∀ (s : Cocone (CostructuredArrow.proj yoneda P ⋙ yoneda)) (m : (tautologicalCocone P).pt ⟶ s.pt),
    (∀ (j : CostructuredArrow yoneda P), (tautologicalCocone P).ι.app j ≫ m = s.ι.app j) →
      m = (fun s => NatTrans.mk fun X t => yonedaEquiv (s.ι.app (CostructuredArrow.mk (yonedaEquiv.symm t)))) s

Next tactic:
intro s j h
/-- The tautological cocone with point `P` is a colimit cocone, exhibiting `P` as a colimit of
    representables. -/
def isColimitTautologicalCocone : IsColimit (tautologicalCocone P) where
  desc := fun s => by
    refine' ⟨fun X t => yonedaEquiv (s.ι.app (CostructuredArrow.mk (yonedaEquiv.symm t))), _⟩
    intros X Y f
    ext t
    dsimp
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    erw [yonedaEquiv_naturality', yonedaEquiv_symm_map]
    simpa using (s.ι.naturality
      (CostructuredArrow.homMk' (CostructuredArrow.mk (yonedaEquiv.symm t)) f.unop)).symm
  fac := by
    intro s t
    dsimp
    apply yonedaEquiv.injective
    rw [yonedaEquiv_comp]
    dsimp only
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    erw [Equiv.symm_apply_apply]
    rfl
  uniq := by
    intro s j h
    

Proof state:
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P : Cᵒᵖ ⥤ Type v₁
s : Cocone (CostructuredArrow.proj yoneda P ⋙ yoneda)
j : (tautologicalCocone P).pt ⟶ s.pt
h : ∀ (j_1 : CostructuredArrow yoneda P), (tautologicalCocone P).ι.app j_1 ≫ j = s.ι.app j_1
⊢ j = (fun s => NatTrans.mk fun X t => yonedaEquiv (s.ι.app (CostructuredArrow.mk (yonedaEquiv.symm t)))) s

Next tactic:
ext V x
/-- The tautological cocone with point `P` is a colimit cocone, exhibiting `P` as a colimit of
    representables. -/
def isColimitTautologicalCocone : IsColimit (tautologicalCocone P) where
  desc := fun s => by
    refine' ⟨fun X t => yonedaEquiv (s.ι.app (CostructuredArrow.mk (yonedaEquiv.symm t))), _⟩
    intros X Y f
    ext t
    dsimp
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    erw [yonedaEquiv_naturality', yonedaEquiv_symm_map]
    simpa using (s.ι.naturality
      (CostructuredArrow.homMk' (CostructuredArrow.mk (yonedaEquiv.symm t)) f.unop)).symm
  fac := by
    intro s t
    dsimp
    apply yonedaEquiv.injective
    rw [yonedaEquiv_comp]
    dsimp only
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    erw [Equiv.symm_apply_apply]
    rfl
  uniq := by
    intro s j h
    ext V x
    

Proof state:
case w.h.h
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P : Cᵒᵖ ⥤ Type v₁
s : Cocone (CostructuredArrow.proj yoneda P ⋙ yoneda)
j : (tautologicalCocone P).pt ⟶ s.pt
h : ∀ (j_1 : CostructuredArrow yoneda P), (tautologicalCocone P).ι.app j_1 ≫ j = s.ι.app j_1
V : Cᵒᵖ
x : (tautologicalCocone P).pt.obj V
⊢ j.app V x =
    ((fun s => NatTrans.mk fun X t => yonedaEquiv (s.ι.app (CostructuredArrow.mk (yonedaEquiv.symm t)))) s).app V x

Next tactic:
obtain ⟨t, rfl⟩ := yonedaEquiv.surjective x
/-- The tautological cocone with point `P` is a colimit cocone, exhibiting `P` as a colimit of
    representables. -/
def isColimitTautologicalCocone : IsColimit (tautologicalCocone P) where
  desc := fun s => by
    refine' ⟨fun X t => yonedaEquiv (s.ι.app (CostructuredArrow.mk (yonedaEquiv.symm t))), _⟩
    intros X Y f
    ext t
    dsimp
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    erw [yonedaEquiv_naturality', yonedaEquiv_symm_map]
    simpa using (s.ι.naturality
      (CostructuredArrow.homMk' (CostructuredArrow.mk (yonedaEquiv.symm t)) f.unop)).symm
  fac := by
    intro s t
    dsimp
    apply yonedaEquiv.injective
    rw [yonedaEquiv_comp]
    dsimp only
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    erw [Equiv.symm_apply_apply]
    rfl
  uniq := by
    intro s j h
    ext V x
    obtain ⟨t, rfl⟩ := yonedaEquiv.surjective x
    

Proof state:
case w.h.h.intro
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P : Cᵒᵖ ⥤ Type v₁
s : Cocone (CostructuredArrow.proj yoneda P ⋙ yoneda)
j : (tautologicalCocone P).pt ⟶ s.pt
h : ∀ (j_1 : CostructuredArrow yoneda P), (tautologicalCocone P).ι.app j_1 ≫ j = s.ι.app j_1
V : Cᵒᵖ
t : yoneda.obj V.unop ⟶ (tautologicalCocone P).pt
⊢ j.app V (yonedaEquiv t) =
    ((fun s => NatTrans.mk fun X t => yonedaEquiv (s.ι.app (CostructuredArrow.mk (yonedaEquiv.symm t)))) s).app V
      (yonedaEquiv t)

Next tactic:
dsimp
/-- The tautological cocone with point `P` is a colimit cocone, exhibiting `P` as a colimit of
    representables. -/
def isColimitTautologicalCocone : IsColimit (tautologicalCocone P) where
  desc := fun s => by
    refine' ⟨fun X t => yonedaEquiv (s.ι.app (CostructuredArrow.mk (yonedaEquiv.symm t))), _⟩
    intros X Y f
    ext t
    dsimp
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    erw [yonedaEquiv_naturality', yonedaEquiv_symm_map]
    simpa using (s.ι.naturality
      (CostructuredArrow.homMk' (CostructuredArrow.mk (yonedaEquiv.symm t)) f.unop)).symm
  fac := by
    intro s t
    dsimp
    apply yonedaEquiv.injective
    rw [yonedaEquiv_comp]
    dsimp only
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    erw [Equiv.symm_apply_apply]
    rfl
  uniq := by
    intro s j h
    ext V x
    obtain ⟨t, rfl⟩ := yonedaEquiv.surjective x
    dsimp
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    

Proof state:
case w.h.h.intro
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P : Cᵒᵖ ⥤ Type v₁
s : Cocone (CostructuredArrow.proj yoneda P ⋙ yoneda)
j : (tautologicalCocone P).pt ⟶ s.pt
h : ∀ (j_1 : CostructuredArrow yoneda P), (tautologicalCocone P).ι.app j_1 ≫ j = s.ι.app j_1
V : Cᵒᵖ
t : yoneda.obj V.unop ⟶ (tautologicalCocone P).pt
⊢ j.app V (yonedaEquiv t) = yonedaEquiv (s.ι.app (CostructuredArrow.mk (yonedaEquiv.symm (yonedaEquiv t))))

Next tactic:
erw [Equiv.symm_apply_apply, ← yonedaEquiv_comp']
/-- The tautological cocone with point `P` is a colimit cocone, exhibiting `P` as a colimit of
    representables. -/
def isColimitTautologicalCocone : IsColimit (tautologicalCocone P) where
  desc := fun s => by
    refine' ⟨fun X t => yonedaEquiv (s.ι.app (CostructuredArrow.mk (yonedaEquiv.symm t))), _⟩
    intros X Y f
    ext t
    dsimp
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    erw [yonedaEquiv_naturality', yonedaEquiv_symm_map]
    simpa using (s.ι.naturality
      (CostructuredArrow.homMk' (CostructuredArrow.mk (yonedaEquiv.symm t)) f.unop)).symm
  fac := by
    intro s t
    dsimp
    apply yonedaEquiv.injective
    rw [yonedaEquiv_comp]
    dsimp only
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    erw [Equiv.symm_apply_apply]
    rfl
  uniq := by
    intro s j h
    ext V x
    obtain ⟨t, rfl⟩ := yonedaEquiv.surjective x
    dsimp
    -- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
    erw [Equiv.symm_apply_apply, ← yonedaEquiv_comp']
    

Proof state:
case w.h.h.intro
C : Type u₁
inst✝ : Category.{v₁, u₁} C
P : Cᵒᵖ ⥤ Type v₁
s : Cocone (CostructuredArrow.proj yoneda P ⋙ yoneda)
j : (tautologicalCocone P).pt ⟶ s.pt
h : ∀ (j_1 : CostructuredArrow yoneda P), (tautologicalCocone P).ι.app j_1 ≫ j = s.ι.app j_1
V : Cᵒᵖ
t : yoneda.obj V.unop ⟶ (tautologicalCocone P).pt
⊢ yonedaEquiv (t ≫ j) = yonedaEquiv (s.ι.app (CostructuredArrow.mk t))

Next tactic:
exact congr_arg _ (h (CostructuredArrow.mk t))
/-- `log` as an arithmetic function `ℕ → ℝ`. Note this is in the `Nat.ArithmeticFunction`
namespace to indicate that it is bundled as an `ArithmeticFunction` rather than being the usual
real logarithm. -/
noncomputable def log : ArithmeticFunction ℝ :=
  ⟨fun n => Real.log n, by 

Proof state:
⊢ (fun n => Real.log ↑n) 0 = 0

Next tactic:
simp
@[simp]
theorem vonMangoldt_apply_one : Λ 1 = 0 := by 

Proof state:
⊢ Λ 1 = 0

Next tactic:
simp [vonMangoldt_apply]
@[simp]
theorem vonMangoldt_nonneg {n : ℕ} : 0 ≤ Λ n := by
  

Proof state:
n : ℕ
⊢ 0 ≤ Λ n

Next tactic:
rw [vonMangoldt_apply]
@[simp]
theorem vonMangoldt_nonneg {n : ℕ} : 0 ≤ Λ n := by
  rw [vonMangoldt_apply]
  

Proof state:
n : ℕ
⊢ 0 ≤ if IsPrimePow n then Real.log ↑(minFac n) else 0

Next tactic:
split_ifs
@[simp]
theorem vonMangoldt_nonneg {n : ℕ} : 0 ≤ Λ n := by
  rw [vonMangoldt_apply]
  split_ifs
  · 

Proof state:
case pos
n : ℕ
h✝ : IsPrimePow n
⊢ 0 ≤ Real.log ↑(minFac n)

Next tactic:
exact Real.log_nonneg (one_le_cast.2 (Nat.minFac_pos n))
@[simp]
theorem vonMangoldt_nonneg {n : ℕ} : 0 ≤ Λ n := by
  rw [vonMangoldt_apply]
  split_ifs
  · exact Real.log_nonneg (one_le_cast.2 (Nat.minFac_pos n))
  

Proof state:
case neg
n : ℕ
h✝ : ¬IsPrimePow n
⊢ 0 ≤ 0

Next tactic:
rfl
theorem vonMangoldt_apply_pow {n k : ℕ} (hk : k ≠ 0) : Λ (n ^ k) = Λ n := by
  

Proof state:
n k : ℕ
hk : k ≠ 0
⊢ Λ (n ^ k) = Λ n

Next tactic:
simp only [vonMangoldt_apply, isPrimePow_pow_iff hk, pow_minFac hk]
theorem vonMangoldt_apply_prime {p : ℕ} (hp : p.Prime) : Λ p = Real.log p := by
  

Proof state:
p : ℕ
hp : Prime p
⊢ Λ p = Real.log ↑p

Next tactic:
rw [vonMangoldt_apply, Prime.minFac_eq hp, if_pos hp.prime.isPrimePow]
theorem vonMangoldt_ne_zero_iff {n : ℕ} : Λ n ≠ 0 ↔ IsPrimePow n := by
  

Proof state:
n : ℕ
⊢ Λ n ≠ 0 ↔ IsPrimePow n

Next tactic:
rcases eq_or_ne n 1 with (rfl | hn)
theorem vonMangoldt_ne_zero_iff {n : ℕ} : Λ n ≠ 0 ↔ IsPrimePow n := by
  rcases eq_or_ne n 1 with (rfl | hn); · 

Proof state:
case inl
⊢ Λ 1 ≠ 0 ↔ IsPrimePow 1

Next tactic:
simp [not_isPrimePow_one]
theorem vonMangoldt_ne_zero_iff {n : ℕ} : Λ n ≠ 0 ↔ IsPrimePow n := by
  rcases eq_or_ne n 1 with (rfl | hn); · simp [not_isPrimePow_one]
  

Proof state:
case inr
n : ℕ
hn : n ≠ 1
⊢ Λ n ≠ 0 ↔ IsPrimePow n

Next tactic:
exact (Real.log_pos (one_lt_cast.2 (minFac_prime hn).one_lt)).ne'.ite_ne_right_iff
theorem vonMangoldt_sum {n : ℕ} : ∑ i in n.divisors, Λ i = Real.log n := by
  

Proof state:
n : ℕ
⊢ ∑ i in divisors n, Λ i = Real.log ↑n

Next tactic:
refine' recOnPrimeCoprime _ _ _ n
theorem vonMangoldt_sum {n : ℕ} : ∑ i in n.divisors, Λ i = Real.log n := by
  refine' recOnPrimeCoprime _ _ _ n
  · 

Proof state:
case refine'_1
n : ℕ
⊢ ∑ i in divisors 0, Λ i = Real.log ↑0

Next tactic:
simp
theorem vonMangoldt_sum {n : ℕ} : ∑ i in n.divisors, Λ i = Real.log n := by
  refine' recOnPrimeCoprime _ _ _ n
  · simp
  · 

Proof state:
case refine'_2
n : ℕ
⊢ ∀ (p n : ℕ), Prime p → ∑ i in divisors (p ^ n), Λ i = Real.log ↑(p ^ n)

Next tactic:
intro p k hp
theorem vonMangoldt_sum {n : ℕ} : ∑ i in n.divisors, Λ i = Real.log n := by
  refine' recOnPrimeCoprime _ _ _ n
  · simp
  · intro p k hp
    

Proof state:
case refine'_2
n p k : ℕ
hp : Prime p
⊢ ∑ i in divisors (p ^ k), Λ i = Real.log ↑(p ^ k)

Next tactic:
rw [sum_divisors_prime_pow hp, cast_pow, Real.log_pow, Finset.sum_range_succ', pow_zero,
      vonMangoldt_apply_one]
theorem vonMangoldt_sum {n : ℕ} : ∑ i in n.divisors, Λ i = Real.log n := by
  refine' recOnPrimeCoprime _ _ _ n
  · simp
  · intro p k hp
    rw [sum_divisors_prime_pow hp, cast_pow, Real.log_pow, Finset.sum_range_succ', pow_zero,
      vonMangoldt_apply_one]
    

Proof state:
case refine'_2
n p k : ℕ
hp : Prime p
⊢ ∑ k in range k, Λ (p ^ (k + 1)) + 0 = ↑k * Real.log ↑p

Next tactic:
simp [vonMangoldt_apply_pow (Nat.succ_ne_zero _), vonMangoldt_apply_prime hp]
theorem vonMangoldt_sum {n : ℕ} : ∑ i in n.divisors, Λ i = Real.log n := by
  refine' recOnPrimeCoprime _ _ _ n
  · simp
  · intro p k hp
    rw [sum_divisors_prime_pow hp, cast_pow, Real.log_pow, Finset.sum_range_succ', pow_zero,
      vonMangoldt_apply_one]
    simp [vonMangoldt_apply_pow (Nat.succ_ne_zero _), vonMangoldt_apply_prime hp]
  

Proof state:
case refine'_3
n : ℕ
⊢ ∀ (a b : ℕ),
    1 < a →
      1 < b →
        Coprime a b →
          ∑ i in divisors a, Λ i = Real.log ↑a →
            ∑ i in divisors b, Λ i = Real.log ↑b → ∑ i in divisors (a * b), Λ i = Real.log ↑(a * b)

Next tactic:
intro a b ha' hb' hab ha hb
theorem vonMangoldt_sum {n : ℕ} : ∑ i in n.divisors, Λ i = Real.log n := by
  refine' recOnPrimeCoprime _ _ _ n
  · simp
  · intro p k hp
    rw [sum_divisors_prime_pow hp, cast_pow, Real.log_pow, Finset.sum_range_succ', pow_zero,
      vonMangoldt_apply_one]
    simp [vonMangoldt_apply_pow (Nat.succ_ne_zero _), vonMangoldt_apply_prime hp]
  intro a b ha' hb' hab ha hb
  

Proof state:
case refine'_3
n a b : ℕ
ha' : 1 < a
hb' : 1 < b
hab : Coprime a b
ha : ∑ i in divisors a, Λ i = Real.log ↑a
hb : ∑ i in divisors b, Λ i = Real.log ↑b
⊢ ∑ i in divisors (a * b), Λ i = Real.log ↑(a * b)

Next tactic:
simp only [vonMangoldt_apply, ← sum_filter] at ha hb ⊢
theorem vonMangoldt_sum {n : ℕ} : ∑ i in n.divisors, Λ i = Real.log n := by
  refine' recOnPrimeCoprime _ _ _ n
  · simp
  · intro p k hp
    rw [sum_divisors_prime_pow hp, cast_pow, Real.log_pow, Finset.sum_range_succ', pow_zero,
      vonMangoldt_apply_one]
    simp [vonMangoldt_apply_pow (Nat.succ_ne_zero _), vonMangoldt_apply_prime hp]
  intro a b ha' hb' hab ha hb
  simp only [vonMangoldt_apply, ← sum_filter] at ha hb ⊢
  

Proof state:
case refine'_3
n a b : ℕ
ha' : 1 < a
hb' : 1 < b
hab : Coprime a b
ha : ∑ a in filter (fun a => IsPrimePow a) (divisors a), Real.log ↑(minFac a) = Real.log ↑a
hb : ∑ a in filter (fun a => IsPrimePow a) (divisors b), Real.log ↑(minFac a) = Real.log ↑b
⊢ ∑ a in filter (fun a => IsPrimePow a) (divisors (a * b)), Real.log ↑(minFac a) = Real.log ↑(a * b)

Next tactic:
rw [mul_divisors_filter_prime_pow hab, filter_union,
    sum_union (disjoint_divisors_filter_isPrimePow hab), ha, hb, Nat.cast_mul,
    Real.log_mul (cast_ne_zero.2 (pos_of_gt ha').ne') (cast_ne_zero.2 (pos_of_gt hb').ne')]
@[simp]
theorem vonMangoldt_mul_zeta : Λ * ζ = log := by
  

Proof state:
⊢ Λ * ↑ζ = log

Next tactic:
ext n
@[simp]
theorem vonMangoldt_mul_zeta : Λ * ζ = log := by
  ext n; 

Proof state:
case h
n : ℕ
⊢ (Λ * ↑ζ) n = log n

Next tactic:
rw [coe_mul_zeta_apply, vonMangoldt_sum]
@[simp]
theorem vonMangoldt_mul_zeta : Λ * ζ = log := by
  ext n; rw [coe_mul_zeta_apply, vonMangoldt_sum]; 

Proof state:
case h
n : ℕ
⊢ Real.log ↑n = log n

Next tactic:
rfl
@[simp]
theorem zeta_mul_vonMangoldt : (ζ : ArithmeticFunction ℝ) * Λ = log := by 

Proof state:
⊢ ↑ζ * Λ = log

Next tactic:
rw [mul_comm]
@[simp]
theorem zeta_mul_vonMangoldt : (ζ : ArithmeticFunction ℝ) * Λ = log := by rw [mul_comm]; 

Proof state:
⊢ Λ * ↑ζ = log

Next tactic:
simp
@[simp]
theorem log_mul_moebius_eq_vonMangoldt : log * μ = Λ := by
  

Proof state:
⊢ log * ↑μ = Λ

Next tactic:
rw [← vonMangoldt_mul_zeta, mul_assoc, coe_zeta_mul_coe_moebius, mul_one]
@[simp]
theorem moebius_mul_log_eq_vonMangoldt : (μ : ArithmeticFunction ℝ) * log = Λ := by
  

Proof state:
⊢ ↑μ * log = Λ

Next tactic:
rw [mul_comm]
@[simp]
theorem moebius_mul_log_eq_vonMangoldt : (μ : ArithmeticFunction ℝ) * log = Λ := by
  rw [mul_comm]; 

Proof state:
⊢ log * ↑μ = Λ

Next tactic:
simp
theorem sum_moebius_mul_log_eq {n : ℕ} : (∑ d in n.divisors, (μ d : ℝ) * log d) = -Λ n := by
  

Proof state:
n : ℕ
⊢ ∑ d in divisors n, ↑(μ d) * log d = -Λ n

Next tactic:
simp only [← log_mul_moebius_eq_vonMangoldt, mul_comm log, mul_apply, log_apply, intCoe_apply, ←
    Finset.sum_neg_distrib, neg_mul_eq_mul_neg]
theorem sum_moebius_mul_log_eq {n : ℕ} : (∑ d in n.divisors, (μ d : ℝ) * log d) = -Λ n := by
  simp only [← log_mul_moebius_eq_vonMangoldt, mul_comm log, mul_apply, log_apply, intCoe_apply, ←
    Finset.sum_neg_distrib, neg_mul_eq_mul_neg]
  

Proof state:
n : ℕ
⊢ ∑ x in divisors n, ↑(μ x) * Real.log ↑x = ∑ x in divisorsAntidiagonal n, ↑(μ x.1) * -Real.log ↑x.2

Next tactic:
rw [sum_divisorsAntidiagonal fun i j => (μ i : ℝ) * -Real.log j]
theorem sum_moebius_mul_log_eq {n : ℕ} : (∑ d in n.divisors, (μ d : ℝ) * log d) = -Λ n := by
  simp only [← log_mul_moebius_eq_vonMangoldt, mul_comm log, mul_apply, log_apply, intCoe_apply, ←
    Finset.sum_neg_distrib, neg_mul_eq_mul_neg]
  rw [sum_divisorsAntidiagonal fun i j => (μ i : ℝ) * -Real.log j]
  

Proof state:
n : ℕ
⊢ ∑ x in divisors n, ↑(μ x) * Real.log ↑x = ∑ i in divisors n, ↑(μ i) * -Real.log ↑(n / i)

Next tactic:
have : (∑ i : ℕ in n.divisors, (μ i : ℝ) * -Real.log (n / i : ℕ)) =
      ∑ i : ℕ in n.divisors, ((μ i : ℝ) * Real.log i - μ i * Real.log n) := by
    apply sum_congr rfl
    simp only [and_imp, Int.cast_eq_zero, mul_eq_mul_left_iff, Ne.def, neg_inj, mem_divisors]
    intro m mn hn
    have : (m : ℝ) ≠ 0 := by
      rw [cast_ne_zero]
      rintro rfl
      exact hn (by simpa using mn)
    rw [Nat.cast_div mn this, Real.log_div (cast_ne_zero.2 hn) this, neg_sub, mul_sub]
theorem sum_moebius_mul_log_eq {n : ℕ} : (∑ d in n.divisors, (μ d : ℝ) * log d) = -Λ n := by
  simp only [← log_mul_moebius_eq_vonMangoldt, mul_comm log, mul_apply, log_apply, intCoe_apply, ←
    Finset.sum_neg_distrib, neg_mul_eq_mul_neg]
  rw [sum_divisorsAntidiagonal fun i j => (μ i : ℝ) * -Real.log j]
  have : (∑ i : ℕ in n.divisors, (μ i : ℝ) * -Real.log (n / i : ℕ)) =
      ∑ i : ℕ in n.divisors, ((μ i : ℝ) * Real.log i - μ i * Real.log n) := by
    

Proof state:
n : ℕ
⊢ ∑ i in divisors n, ↑(μ i) * -Real.log ↑(n / i) = ∑ i in divisors n, (↑(μ i) * Real.log ↑i - ↑(μ i) * Real.log ↑n)

Next tactic:
apply sum_congr rfl
theorem sum_moebius_mul_log_eq {n : ℕ} : (∑ d in n.divisors, (μ d : ℝ) * log d) = -Λ n := by
  simp only [← log_mul_moebius_eq_vonMangoldt, mul_comm log, mul_apply, log_apply, intCoe_apply, ←
    Finset.sum_neg_distrib, neg_mul_eq_mul_neg]
  rw [sum_divisorsAntidiagonal fun i j => (μ i : ℝ) * -Real.log j]
  have : (∑ i : ℕ in n.divisors, (μ i : ℝ) * -Real.log (n / i : ℕ)) =
      ∑ i : ℕ in n.divisors, ((μ i : ℝ) * Real.log i - μ i * Real.log n) := by
    apply sum_congr rfl
    

Proof state:
n : ℕ
⊢ ∀ x ∈ divisors n, ↑(μ x) * -Real.log ↑(n / x) = ↑(μ x) * Real.log ↑x - ↑(μ x) * Real.log ↑n

Next tactic:
simp only [and_imp, Int.cast_eq_zero, mul_eq_mul_left_iff, Ne.def, neg_inj, mem_divisors]
theorem sum_moebius_mul_log_eq {n : ℕ} : (∑ d in n.divisors, (μ d : ℝ) * log d) = -Λ n := by
  simp only [← log_mul_moebius_eq_vonMangoldt, mul_comm log, mul_apply, log_apply, intCoe_apply, ←
    Finset.sum_neg_distrib, neg_mul_eq_mul_neg]
  rw [sum_divisorsAntidiagonal fun i j => (μ i : ℝ) * -Real.log j]
  have : (∑ i : ℕ in n.divisors, (μ i : ℝ) * -Real.log (n / i : ℕ)) =
      ∑ i : ℕ in n.divisors, ((μ i : ℝ) * Real.log i - μ i * Real.log n) := by
    apply sum_congr rfl
    simp only [and_imp, Int.cast_eq_zero, mul_eq_mul_left_iff, Ne.def, neg_inj, mem_divisors]
    

Proof state:
n : ℕ
⊢ ∀ (x : ℕ), x ∣ n → ¬n = 0 → ↑(μ x) * -Real.log ↑(n / x) = ↑(μ x) * Real.log ↑x - ↑(μ x) * Real.log ↑n

Next tactic:
intro m mn hn
theorem sum_moebius_mul_log_eq {n : ℕ} : (∑ d in n.divisors, (μ d : ℝ) * log d) = -Λ n := by
  simp only [← log_mul_moebius_eq_vonMangoldt, mul_comm log, mul_apply, log_apply, intCoe_apply, ←
    Finset.sum_neg_distrib, neg_mul_eq_mul_neg]
  rw [sum_divisorsAntidiagonal fun i j => (μ i : ℝ) * -Real.log j]
  have : (∑ i : ℕ in n.divisors, (μ i : ℝ) * -Real.log (n / i : ℕ)) =
      ∑ i : ℕ in n.divisors, ((μ i : ℝ) * Real.log i - μ i * Real.log n) := by
    apply sum_congr rfl
    simp only [and_imp, Int.cast_eq_zero, mul_eq_mul_left_iff, Ne.def, neg_inj, mem_divisors]
    intro m mn hn
    

Proof state:
n m : ℕ
mn : m ∣ n
hn : ¬n = 0
⊢ ↑(μ m) * -Real.log ↑(n / m) = ↑(μ m) * Real.log ↑m - ↑(μ m) * Real.log ↑n

Next tactic:
have : (m : ℝ) ≠ 0 := by
      rw [cast_ne_zero]
      rintro rfl
      exact hn (by simpa using mn)
theorem sum_moebius_mul_log_eq {n : ℕ} : (∑ d in n.divisors, (μ d : ℝ) * log d) = -Λ n := by
  simp only [← log_mul_moebius_eq_vonMangoldt, mul_comm log, mul_apply, log_apply, intCoe_apply, ←
    Finset.sum_neg_distrib, neg_mul_eq_mul_neg]
  rw [sum_divisorsAntidiagonal fun i j => (μ i : ℝ) * -Real.log j]
  have : (∑ i : ℕ in n.divisors, (μ i : ℝ) * -Real.log (n / i : ℕ)) =
      ∑ i : ℕ in n.divisors, ((μ i : ℝ) * Real.log i - μ i * Real.log n) := by
    apply sum_congr rfl
    simp only [and_imp, Int.cast_eq_zero, mul_eq_mul_left_iff, Ne.def, neg_inj, mem_divisors]
    intro m mn hn
    have : (m : ℝ) ≠ 0 := by
      

Proof state:
n m : ℕ
mn : m ∣ n
hn : ¬n = 0
⊢ ↑m ≠ 0

Next tactic:
rw [cast_ne_zero]