internlm/Lean-Github · Datasets at Fast360

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/EulerProduct.lean	LSeries.term_mul_aux	[21, 1]	[23, 90]	rw [mul_comm_div, div_div, ← mul_div_assoc, mul_comm (m : ℂ), natCast_mul_natCast_cpow]	a b : ℂ m n : ℕ s : ℂ ⊢ a / ↑m ^ s * (b / ↑n ^ s) = a * b / (↑m * ↑n) ^ s	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/EulerProduct.lean	LSeries.term_mul	[25, 1]	[30, 100]	rcases eq_or_ne (m * n) 0 with H \| H	f₁ f₂ f : ℕ → ℂ m n : ℕ h : f (m * n) = f₁ m * f₂ n s : ℂ ⊢ term f s (m * n) = term f₁ s m * term f₂ s n	case inl f₁ f₂ f : ℕ → ℂ m n : ℕ h : f (m * n) = f₁ m * f₂ n s : ℂ H : m * n = 0 ⊢ term f s (m * n) = term f₁ s m * term f₂ s n case inr f₁ f₂ f : ℕ → ℂ m n : ℕ h : f (m * n) = f₁ m * f₂ n s : ℂ H : m * n ≠ 0 ⊢ term f s (m * n) = term f₁ s m * term f₂ s n
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/EulerProduct.lean	LSeries.term_mul	[25, 1]	[30, 100]	rcases mul_eq_zero.mp H with rfl \| rfl <;> simp only [term_zero, mul_zero, zero_mul]	case inl f₁ f₂ f : ℕ → ℂ m n : ℕ h : f (m * n) = f₁ m * f₂ n s : ℂ H : m * n = 0 ⊢ term f s (m * n) = term f₁ s m * term f₂ s n	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/EulerProduct.lean	LSeries.term_mul	[25, 1]	[30, 100]	obtain ⟨hm, hn⟩ := mul_ne_zero_iff.mp H	case inr f₁ f₂ f : ℕ → ℂ m n : ℕ h : f (m * n) = f₁ m * f₂ n s : ℂ H : m * n ≠ 0 ⊢ term f s (m * n) = term f₁ s m * term f₂ s n	case inr.intro f₁ f₂ f : ℕ → ℂ m n : ℕ h : f (m * n) = f₁ m * f₂ n s : ℂ H : m * n ≠ 0 hm : m ≠ 0 hn : n ≠ 0 ⊢ term f s (m * n) = term f₁ s m * term f₂ s n
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/EulerProduct.lean	LSeries.term_mul	[25, 1]	[30, 100]	simp only [ne_eq, H, not_false_eq_true, term_of_ne_zero, Nat.cast_mul, hm, hn, h, term_mul_aux]	case inr.intro f₁ f₂ f : ℕ → ℂ m n : ℕ h : f (m * n) = f₁ m * f₂ n s : ℂ H : m * n ≠ 0 hm : m ≠ 0 hn : n ≠ 0 ⊢ term f s (m * n) = term f₁ s m * term f₂ s n	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/EulerProduct.lean	LSeries.term_at_one	[44, 1]	[45, 72]	rw [term_of_ne_zero one_ne_zero, h₁, Nat.cast_one, one_cpow, div_one]	f : ℕ → ℂ h₁ : f 1 = 1 s : ℂ ⊢ term f s 1 = 1	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/EulerProduct.lean	DirichletCharacter.toFun_on_nat_map_one	[86, 1]	[87, 32]	simp only [cast_one, map_one]	N : ℕ χ : DirichletCharacter ℂ N ⊢ (fun n => χ ↑n) 1 = 1	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/EulerProduct.lean	DirichletCharacter.toFun_on_nat_map_mul	[89, 1]	[91, 32]	simp only [cast_mul, map_mul]	N : ℕ χ : DirichletCharacter ℂ N m n : ℕ ⊢ (fun n => χ ↑n) (m * n) = (fun n => χ ↑n) m * (fun n => χ ↑n) n	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/EulerProduct.lean	DirichletCharacter.LSeries_eulerProduct	[94, 1]	[99, 87]	refine Tendsto.congr (fun n ↦ Finset.prod_congr rfl fun p hp ↦ ?_) <\| eulerProduct_of_completelyMultiplicative (toFun_on_nat_map_one χ) (toFun_on_nat_map_mul χ) <\| LSeriesSummable_of_one_lt_re χ hs	N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re ⊢ Tendsto (fun n => ∏ p ∈ n.primesBelow, (1 - χ ↑p * ↑p ^ (-s))⁻¹) atTop (𝓝 (L (fun n => χ ↑n) s))	N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re n p : ℕ hp : p ∈ n.primesBelow ⊢ (1 - term (fun n => χ ↑n) s p)⁻¹ = (1 - χ ↑p * ↑p ^ (-s))⁻¹
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/EulerProduct.lean	DirichletCharacter.LSeries_eulerProduct	[94, 1]	[99, 87]	rw [term_of_ne_zero (prime_of_mem_primesBelow hp).ne_zero, cpow_neg, div_eq_mul_inv]	N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re n p : ℕ hp : p ∈ n.primesBelow ⊢ (1 - term (fun n => χ ↑n) s p)⁻¹ = (1 - χ ↑p * ↑p ^ (-s))⁻¹	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/EulerProduct.lean	DirichletCharacter.LSeries_eulerProduct'	[102, 1]	[110, 61]	rw [LSeries]	N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re ⊢ cexp (∑' (p : Primes), -(1 - χ ↑↑p * ↑↑p ^ (-s)).log) = L (fun n => χ ↑n) s	N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re ⊢ cexp (∑' (p : Primes), -(1 - χ ↑↑p * ↑↑p ^ (-s)).log) = ∑' (n : ℕ), term (fun n => χ ↑n) s n
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/EulerProduct.lean	DirichletCharacter.LSeries_eulerProduct'	[102, 1]	[110, 61]	convert exp_sum_primes_log_eq_tsum (f := dirichletSummandHom χ <\| ne_zero_of_one_lt_re hs) <\| summable_dirichletSummand χ hs	N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re ⊢ cexp (∑' (p : Primes), -(1 - χ ↑↑p * ↑↑p ^ (-s)).log) = ∑' (n : ℕ), term (fun n => χ ↑n) s n	case h.e'_3.h.e'_5.h.h.e N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ : ℕ ⊢ term (fun n => χ ↑n) s = ⇑(dirichletSummandHom χ ⋯)
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/EulerProduct.lean	DirichletCharacter.LSeries_eulerProduct'	[102, 1]	[110, 61]	ext n	case h.e'_3.h.e'_5.h.h.e N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ : ℕ ⊢ term (fun n => χ ↑n) s = ⇑(dirichletSummandHom χ ⋯)	case h.e'_3.h.e'_5.h.h.e.h N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ n : ℕ ⊢ term (fun n => χ ↑n) s n = (dirichletSummandHom χ ⋯) n
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/EulerProduct.lean	DirichletCharacter.LSeries_eulerProduct'	[102, 1]	[110, 61]	rcases eq_or_ne n 0 with rfl \| hn	case h.e'_3.h.e'_5.h.h.e.h N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ n : ℕ ⊢ term (fun n => χ ↑n) s n = (dirichletSummandHom χ ⋯) n	case h.e'_3.h.e'_5.h.h.e.h.inl N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ : ℕ ⊢ term (fun n => χ ↑n) s 0 = (dirichletSummandHom χ ⋯) 0 case h.e'_3.h.e'_5.h.h.e.h.inr N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ n : ℕ hn : n ≠ 0 ⊢ term (fun n => χ ↑n) s n = (dirichletSummandHom χ ⋯) n
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/EulerProduct.lean	DirichletCharacter.LSeries_eulerProduct'	[102, 1]	[110, 61]	simp only [term_zero, map_zero]	case h.e'_3.h.e'_5.h.h.e.h.inl N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ : ℕ ⊢ term (fun n => χ ↑n) s 0 = (dirichletSummandHom χ ⋯) 0	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/EulerProduct.lean	DirichletCharacter.LSeries_eulerProduct'	[102, 1]	[110, 61]	simp [hn, dirichletSummandHom, div_eq_mul_inv, cpow_neg]	case h.e'_3.h.e'_5.h.h.e.h.inr N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ n : ℕ hn : n ≠ 0 ⊢ term (fun n => χ ↑n) s n = (dirichletSummandHom χ ⋯) n	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/EulerProduct.lean	ArithmeticFunction.LSeries_zeta_eulerProduct'	[117, 1]	[120, 62]	convert modOne_eq_one (R := ℂ) ▸ LSeries_eulerProduct' (1 : DirichletCharacter ℂ 1) hs using 7	s : ℂ hs : 1 < s.re ⊢ cexp (∑' (p : Primes), -(1 - ↑↑p ^ (-s)).log) = L 1 s	case h.e'_2.h.e'_1.h.e'_5.h.h.e'_3.h.e'_1.h.e'_6 s : ℂ hs : 1 < s.re x✝ : Primes ⊢ ↑↑x✝ ^ (-s) = 1 ↑↑x✝ * ↑↑x✝ ^ (-s)
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/EulerProduct.lean	ArithmeticFunction.LSeries_zeta_eulerProduct'	[117, 1]	[120, 62]	rw [MulChar.one_apply <\| isUnit_of_subsingleton _, one_mul]	case h.e'_2.h.e'_1.h.e'_5.h.h.e'_3.h.e'_1.h.e'_6 s : ℂ hs : 1 < s.re x✝ : Primes ⊢ ↑↑x✝ ^ (-s) = 1 ↑↑x✝ * ↑↑x✝ ^ (-s)	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Asymptotics.isBigO_mul_iff_isBigO_div	[31, 1]	[39, 36]	rw [isBigO_iff', isBigO_iff']	α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 ⊢ (fun x => f x * g x) =O[l] h ↔ g =O[l] fun x => h x / f x	α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 ⊢ (∃ c > 0, ∀ᶠ (x : α) in l, ‖f x * g x‖ ≤ c * ‖h x‖) ↔ ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Asymptotics.isBigO_mul_iff_isBigO_div	[31, 1]	[39, 36]	refine ⟨fun ⟨c, hc, H⟩ ↦ ⟨c, hc, ?_⟩, fun ⟨c, hc, H⟩ ↦ ⟨c, hc, ?_⟩⟩ <;> { refine H.congr <\| Eventually.mp hf <\| eventually_of_forall fun x hx ↦ ?_ rw [norm_mul, norm_div, ← mul_div_assoc, mul_comm] have hx' : ‖f x‖ > 0 := norm_pos_iff.mpr hx rw [le_div_iff hx', mul_comm] }	α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 ⊢ (∃ c > 0, ∀ᶠ (x : α) in l, ‖f x * g x‖ ≤ c * ‖h x‖) ↔ ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Asymptotics.isBigO_mul_iff_isBigO_div	[31, 1]	[39, 36]	refine H.congr <\| Eventually.mp hf <\| eventually_of_forall fun x hx ↦ ?_	case refine_2 α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ c : ℝ hc : c > 0 H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ ⊢ ∀ᶠ (x : α) in l, ‖f x * g x‖ ≤ c * ‖h x‖	case refine_2 α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ c : ℝ hc : c > 0 H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ x : α hx : f x ≠ 0 ⊢ ‖g x‖ ≤ c * ‖h x / f x‖ ↔ ‖f x * g x‖ ≤ c * ‖h x‖
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Asymptotics.isBigO_mul_iff_isBigO_div	[31, 1]	[39, 36]	rw [norm_mul, norm_div, ← mul_div_assoc, mul_comm]	case refine_2 α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ c : ℝ hc : c > 0 H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ x : α hx : f x ≠ 0 ⊢ ‖g x‖ ≤ c * ‖h x / f x‖ ↔ ‖f x * g x‖ ≤ c * ‖h x‖	case refine_2 α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ c : ℝ hc : c > 0 H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ x : α hx : f x ≠ 0 ⊢ ‖g x‖ ≤ ‖h x‖ * c / ‖f x‖ ↔ ‖f x‖ * ‖g x‖ ≤ ‖h x‖ * c
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Asymptotics.isBigO_mul_iff_isBigO_div	[31, 1]	[39, 36]	have hx' : ‖f x‖ > 0 := norm_pos_iff.mpr hx	case refine_2 α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ c : ℝ hc : c > 0 H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ x : α hx : f x ≠ 0 ⊢ ‖g x‖ ≤ ‖h x‖ * c / ‖f x‖ ↔ ‖f x‖ * ‖g x‖ ≤ ‖h x‖ * c	case refine_2 α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ c : ℝ hc : c > 0 H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ x : α hx : f x ≠ 0 hx' : ‖f x‖ > 0 ⊢ ‖g x‖ ≤ ‖h x‖ * c / ‖f x‖ ↔ ‖f x‖ * ‖g x‖ ≤ ‖h x‖ * c
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Asymptotics.isBigO_mul_iff_isBigO_div	[31, 1]	[39, 36]	rw [le_div_iff hx', mul_comm]	case refine_2 α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ c : ℝ hc : c > 0 H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ x : α hx : f x ≠ 0 hx' : ‖f x‖ > 0 ⊢ ‖g x‖ ≤ ‖h x‖ * c / ‖f x‖ ↔ ‖f x‖ * ‖g x‖ ≤ ‖h x‖ * c	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	DifferentiableAt.isBigO_of_eq_zero	[50, 1]	[54, 73]	rw [← zero_add z] at hf	f : ℂ → ℂ z : ℂ hf : DifferentiableAt ℂ f z hz : f z = 0 ⊢ (fun w => f (w + z)) =O[𝓝 0] id	f : ℂ → ℂ z : ℂ hf : DifferentiableAt ℂ f (0 + z) hz : f z = 0 ⊢ (fun w => f (w + z)) =O[𝓝 0] id
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	DifferentiableAt.isBigO_of_eq_zero	[50, 1]	[54, 73]	simpa only [zero_add, hz, sub_zero] using (hf.hasDerivAt.comp_add_const 0 z).differentiableAt.isBigO_sub	f : ℂ → ℂ z : ℂ hf : DifferentiableAt ℂ f (0 + z) hz : f z = 0 ⊢ (fun w => f (w + z)) =O[𝓝 0] id	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	ContinuousAt.isBigO	[56, 1]	[70, 46]	rw [isBigO_iff']	f : ℂ → ℂ z : ℂ hf : ContinuousAt f z ⊢ (fun w => f (w + z)) =O[𝓝 0] fun x => 1	f : ℂ → ℂ z : ℂ hf : ContinuousAt f z ⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	ContinuousAt.isBigO	[56, 1]	[70, 46]	simp_rw [Metric.continuousAt_iff', dist_eq_norm_sub, zero_add] at hf	f : ℂ → ℂ z : ℂ hf : ContinuousAt (fun w => f (w + z)) 0 ⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖	f : ℂ → ℂ z : ℂ hf : ∀ ε > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < ε ⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	ContinuousAt.isBigO	[56, 1]	[70, 46]	specialize hf 1 zero_lt_one	f : ℂ → ℂ z : ℂ hf : ∀ ε > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < ε ⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖	f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 ⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	ContinuousAt.isBigO	[56, 1]	[70, 46]	refine ⟨‖f z‖ + 1, by positivity, ?_⟩	f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 ⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖	f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 ⊢ ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ (‖f z‖ + 1) * ‖1‖
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	ContinuousAt.isBigO	[56, 1]	[70, 46]	refine Eventually.mp hf <\| eventually_of_forall fun w hw ↦ le_of_lt ?_	f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 ⊢ ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ (‖f z‖ + 1) * ‖1‖	f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 w : ℂ hw : ‖f (w + z) - f z‖ < 1 ⊢ ‖f (w + z)‖ < (‖f z‖ + 1) * ‖1‖
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	ContinuousAt.isBigO	[56, 1]	[70, 46]	calc ‖f (w + z)‖ _ ≤ ‖f z‖ + ‖f (w + z) - f z‖ := norm_le_insert' .. _ < ‖f z‖ + 1 := add_lt_add_left hw _ _ = _ := by simp only [norm_one, mul_one]	f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 w : ℂ hw : ‖f (w + z) - f z‖ < 1 ⊢ ‖f (w + z)‖ < (‖f z‖ + 1) * ‖1‖	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	ContinuousAt.isBigO	[56, 1]	[70, 46]	convert (Homeomorph.comp_continuousAt_iff' (Homeomorph.addLeft (-z)) _ z).mp ?_	f : ℂ → ℂ z : ℂ hf : ContinuousAt f z ⊢ ContinuousAt (fun w => f (w + z)) 0	case h.e'_1 f : ℂ → ℂ z : ℂ hf : ContinuousAt f z ⊢ 0 = (Homeomorph.addLeft (-z)) z case convert_4 f : ℂ → ℂ z : ℂ hf : ContinuousAt f z ⊢ ContinuousAt ((fun w => f (w + z)) ∘ ⇑(Homeomorph.addLeft (-z))) z
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	ContinuousAt.isBigO	[56, 1]	[70, 46]	simp	case h.e'_1 f : ℂ → ℂ z : ℂ hf : ContinuousAt f z ⊢ 0 = (Homeomorph.addLeft (-z)) z	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	ContinuousAt.isBigO	[56, 1]	[70, 46]	simp [Function.comp_def, hf]	case convert_4 f : ℂ → ℂ z : ℂ hf : ContinuousAt f z ⊢ ContinuousAt ((fun w => f (w + z)) ∘ ⇑(Homeomorph.addLeft (-z))) z	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	ContinuousAt.isBigO	[56, 1]	[70, 46]	positivity	f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 ⊢ ‖f z‖ + 1 > 0	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	ContinuousAt.isBigO	[56, 1]	[70, 46]	simp only [norm_one, mul_one]	f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 w : ℂ hw : ‖f (w + z) - f z‖ < 1 ⊢ ‖f z‖ + 1 = (‖f z‖ + 1) * ‖1‖	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	HasDerivAt.of_hasDerivAt_ofReal_comp	[125, 1]	[134, 80]	lift u to ℝ	z : ℝ f : ℝ → ℝ u : ℂ hf : HasDerivAt (fun y => ↑(f y)) u z ⊢ ∃ u', u = ↑u' ∧ HasDerivAt f u' z	z : ℝ f : ℝ → ℝ u : ℂ hf : HasDerivAt (fun y => ↑(f y)) u z ⊢ u.im = 0 case intro z : ℝ f : ℝ → ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊢ ∃ u', ↑u = ↑u' ∧ HasDerivAt f u' z
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	HasDerivAt.of_hasDerivAt_ofReal_comp	[125, 1]	[134, 80]	refine ⟨u, rfl, ?_⟩	case intro z : ℝ f : ℝ → ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊢ ∃ u', ↑u = ↑u' ∧ HasDerivAt f u' z	case intro z : ℝ f : ℝ → ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊢ HasDerivAt f u z
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	HasDerivAt.of_hasDerivAt_ofReal_comp	[125, 1]	[134, 80]	convert (reCLM.hasFDerivAt.comp z hf.hasFDerivAt).hasDerivAt	case intro z : ℝ f : ℝ → ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊢ HasDerivAt f u z	case h.e'_7 z : ℝ f : ℝ → ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊢ u = (reCLM.comp (smulRight 1 ↑u)) 1
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	HasDerivAt.of_hasDerivAt_ofReal_comp	[125, 1]	[134, 80]	rw [comp_apply, smulRight_apply, one_apply, one_smul, reCLM_apply, ofReal_re]	case h.e'_7 z : ℝ f : ℝ → ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊢ u = (reCLM.comp (smulRight 1 ↑u)) 1	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	HasDerivAt.of_hasDerivAt_ofReal_comp	[125, 1]	[134, 80]	have H := (imCLM.hasFDerivAt.comp z hf.hasFDerivAt).hasDerivAt.deriv	z : ℝ f : ℝ → ℝ u : ℂ hf : HasDerivAt (fun y => ↑(f y)) u z ⊢ u.im = 0	z : ℝ f : ℝ → ℝ u : ℂ hf : HasDerivAt (fun y => ↑(f y)) u z H : _root_.deriv (⇑imCLM ∘ fun y => ↑(f y)) z = (imCLM.comp (smulRight 1 u)) 1 ⊢ u.im = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	HasDerivAt.of_hasDerivAt_ofReal_comp	[125, 1]	[134, 80]	simp only [Function.comp_def, imCLM_apply, ofReal_im, deriv_const] at H	z : ℝ f : ℝ → ℝ u : ℂ hf : HasDerivAt (fun y => ↑(f y)) u z H : _root_.deriv (⇑imCLM ∘ fun y => ↑(f y)) z = (imCLM.comp (smulRight 1 u)) 1 ⊢ u.im = 0	z : ℝ f : ℝ → ℝ u : ℂ hf : HasDerivAt (fun y => ↑(f y)) u z H : 0 = (imCLM.comp (smulRight 1 u)) 1 ⊢ u.im = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	HasDerivAt.of_hasDerivAt_ofReal_comp	[125, 1]	[134, 80]	rwa [eq_comm, comp_apply, imCLM_apply, smulRight_apply, one_apply, one_smul] at H	z : ℝ f : ℝ → ℝ u : ℂ hf : HasDerivAt (fun y => ↑(f y)) u z H : 0 = (imCLM.comp (smulRight 1 u)) 1 ⊢ u.im = 0	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	DifferentiableAt.ofReal_comp_iff	[136, 1]	[140, 40]	refine ⟨fun H ↦ ?_, ofReal_comp⟩	z : ℝ f : ℝ → ℝ ⊢ DifferentiableAt ℝ (fun y => ↑(f y)) z ↔ DifferentiableAt ℝ f z	z : ℝ f : ℝ → ℝ H : DifferentiableAt ℝ (fun y => ↑(f y)) z ⊢ DifferentiableAt ℝ f z
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	DifferentiableAt.ofReal_comp_iff	[136, 1]	[140, 40]	obtain ⟨u, _, hu₂⟩ := H.hasDerivAt.of_hasDerivAt_ofReal_comp	z : ℝ f : ℝ → ℝ H : DifferentiableAt ℝ (fun y => ↑(f y)) z ⊢ DifferentiableAt ℝ f z	case intro.intro z : ℝ f : ℝ → ℝ H : DifferentiableAt ℝ (fun y => ↑(f y)) z u : ℝ left✝ : deriv (fun y => ↑(f y)) z = ↑u hu₂ : HasDerivAt (fun y => f y) u z ⊢ DifferentiableAt ℝ f z
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	DifferentiableAt.ofReal_comp_iff	[136, 1]	[140, 40]	exact HasDerivAt.differentiableAt hu₂	case intro.intro z : ℝ f : ℝ → ℝ H : DifferentiableAt ℝ (fun y => ↑(f y)) z u : ℝ left✝ : deriv (fun y => ↑(f y)) z = ↑u hu₂ : HasDerivAt (fun y => f y) u z ⊢ DifferentiableAt ℝ f z	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	deriv.ofReal_comp	[146, 1]	[152, 27]	by_cases hf : DifferentiableAt ℝ f z	z : ℝ f : ℝ → ℝ ⊢ deriv (fun y => ↑(f y)) z = ↑(deriv f z)	case pos z : ℝ f : ℝ → ℝ hf : DifferentiableAt ℝ f z ⊢ deriv (fun y => ↑(f y)) z = ↑(deriv f z) case neg z : ℝ f : ℝ → ℝ hf : ¬DifferentiableAt ℝ f z ⊢ deriv (fun y => ↑(f y)) z = ↑(deriv f z)
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	deriv.ofReal_comp	[146, 1]	[152, 27]	exact hf.hasDerivAt.ofReal_comp.deriv	case pos z : ℝ f : ℝ → ℝ hf : DifferentiableAt ℝ f z ⊢ deriv (fun y => ↑(f y)) z = ↑(deriv f z)	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	deriv.ofReal_comp	[146, 1]	[152, 27]	have hf' := mt DifferentiableAt.ofReal_comp_iff.mp hf	case neg z : ℝ f : ℝ → ℝ hf : ¬DifferentiableAt ℝ f z ⊢ deriv (fun y => ↑(f y)) z = ↑(deriv f z)	case neg z : ℝ f : ℝ → ℝ hf : ¬DifferentiableAt ℝ f z hf' : ¬DifferentiableAt ℝ (fun y => ↑(f y)) z ⊢ deriv (fun y => ↑(f y)) z = ↑(deriv f z)
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	deriv.ofReal_comp	[146, 1]	[152, 27]	rw [deriv_zero_of_not_differentiableAt hf, deriv_zero_of_not_differentiableAt hf', Complex.ofReal_zero]	case neg z : ℝ f : ℝ → ℝ hf : ¬DifferentiableAt ℝ f z hf' : ¬DifferentiableAt ℝ (fun y => ↑(f y)) z ⊢ deriv (fun y => ↑(f y)) z = ↑(deriv f z)	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real_on_ball	[159, 1]	[183, 8]	have Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), (x : ℂ) ∈ Metric.ball (c : ℂ) r := by intro x hx refine Metric.mem_ball.mpr ?_ rw [dist_eq, ← ofReal_sub, abs_ofReal, abs_sub_lt_iff, sub_lt_iff_lt_add', sub_lt_comm] exact and_comm.mpr hx	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) ⊢ ∃ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r ⊢ ∃ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real_on_ball	[159, 1]	[183, 8]	have H ⦃z : ℂ⦄ (hz : z ∈ Metric.ball (c : ℂ) r) := taylorSeries_eq_on_ball' hz hf	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r ⊢ ∃ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z ⊢ ∃ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real_on_ball	[159, 1]	[183, 8]	refine ⟨fun x ↦ ∑' (n : ℕ), (↑n !)⁻¹ * (D n) * (x - c) ^ n, fun x hx ↦ ?_, fun x hx ↦ ?_⟩	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z ⊢ ∃ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))	case refine_1 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ DifferentiableWithinAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x case refine_2 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ (f ∘ ofReal') x = (ofReal' ∘ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real_on_ball	[159, 1]	[183, 8]	intro x hx	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) ⊢ ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ ↑x ∈ Metric.ball (↑c) r
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real_on_ball	[159, 1]	[183, 8]	refine Metric.mem_ball.mpr ?_	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ ↑x ∈ Metric.ball (↑c) r	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ dist ↑x ↑c < r
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real_on_ball	[159, 1]	[183, 8]	rw [dist_eq, ← ofReal_sub, abs_ofReal, abs_sub_lt_iff, sub_lt_iff_lt_add', sub_lt_comm]	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ dist ↑x ↑c < r	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ x < c + r ∧ c - r < x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real_on_ball	[159, 1]	[183, 8]	exact and_comm.mpr hx	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ x < c + r ∧ c - r < x	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real_on_ball	[159, 1]	[183, 8]	have Hx := Hz _ hx	case refine_1 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ DifferentiableWithinAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x	case refine_1 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r ⊢ DifferentiableWithinAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real_on_ball	[159, 1]	[183, 8]	refine DifferentiableAt.differentiableWithinAt ?_	case refine_1 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r ⊢ DifferentiableWithinAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x	case refine_1 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r ⊢ DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real_on_ball	[159, 1]	[183, 8]	replace hf := ((hf x Hx).congr (fun _ hz ↦ H hz) (H Hx)).differentiableAt (Metric.isOpen_ball.mem_nhds Hx) \|>.comp_ofReal	case refine_1 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r ⊢ DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x	case refine_1 f : ℂ → ℂ r c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r hf : DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (↑x - ↑c) ^ n) x ⊢ DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real_on_ball	[159, 1]	[183, 8]	simp_rw [hd, ← ofReal_sub, ← ofReal_natCast, ← ofReal_inv, ← ofReal_pow, ← ofReal_mul, ← ofReal_tsum] at hf	case refine_1 f : ℂ → ℂ r c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r hf : DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (↑x - ↑c) ^ n) x ⊢ DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x	case refine_1 f : ℂ → ℂ r c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r hf : DifferentiableAt ℝ (fun x => ↑(∑' (a : ℕ), (↑a !)⁻¹ * D a * (x - c) ^ a)) x ⊢ DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real_on_ball	[159, 1]	[183, 8]	exact DifferentiableAt.ofReal_comp_iff.mp hf	case refine_1 f : ℂ → ℂ r c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r hf : DifferentiableAt ℝ (fun x => ↑(∑' (a : ℕ), (↑a !)⁻¹ * D a * (x - c) ^ a)) x ⊢ DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real_on_ball	[159, 1]	[183, 8]	simp only [Function.comp_apply, ← H (Hz _ hx), hd, ofReal_tsum]	case refine_2 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ (f ∘ ofReal') x = (ofReal' ∘ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x	case refine_2 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (↑x - ↑c) ^ n = ∑' (a : ℕ), ↑((↑a !)⁻¹ * D a * (x - c) ^ a)
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real_on_ball	[159, 1]	[183, 8]	push_cast	case refine_2 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (↑x - ↑c) ^ n = ∑' (a : ℕ), ↑((↑a !)⁻¹ * D a * (x - c) ^ a)	case refine_2 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a = ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real_on_ball	[159, 1]	[183, 8]	rfl	case refine_2 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a = ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real	[185, 1]	[201, 8]	have H (z : ℂ) := taylorSeries_eq_of_entire' c z hf	f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) ⊢ ∃ F, Differentiable ℝ F ∧ f ∘ ofReal' = ofReal' ∘ F	f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z ⊢ ∃ F, Differentiable ℝ F ∧ f ∘ ofReal' = ofReal' ∘ F
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real	[185, 1]	[201, 8]	simp_rw [hd] at H	f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z ⊢ ∃ F, Differentiable ℝ F ∧ f ∘ ofReal' = ofReal' ∘ F	f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊢ ∃ F, Differentiable ℝ F ∧ f ∘ ofReal' = ofReal' ∘ F
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real	[185, 1]	[201, 8]	refine ⟨fun x ↦ ∑' (n : ℕ), (↑n !)⁻¹ * (D n) * (x - c) ^ n, ?_, ?_⟩	f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊢ ∃ F, Differentiable ℝ F ∧ f ∘ ofReal' = ofReal' ∘ F	case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊢ Differentiable ℝ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n case refine_2 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊢ f ∘ ofReal' = ofReal' ∘ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real	[185, 1]	[201, 8]	have := hf.comp_ofReal	case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊢ Differentiable ℝ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n	case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z this : Differentiable ℝ fun x => f ↑x ⊢ Differentiable ℝ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real	[185, 1]	[201, 8]	simp_rw [← H, ← ofReal_sub, ← ofReal_natCast, ← ofReal_inv, ← ofReal_pow, ← ofReal_mul, ← ofReal_tsum] at this	case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z this : Differentiable ℝ fun x => f ↑x ⊢ Differentiable ℝ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n	case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z this : Differentiable ℝ fun x => ↑(∑' (a : ℕ), (↑a !)⁻¹ * D a * (x - c) ^ a) ⊢ Differentiable ℝ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real	[185, 1]	[201, 8]	exact Differentiable.ofReal_comp_iff.mp this	case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z this : Differentiable ℝ fun x => ↑(∑' (a : ℕ), (↑a !)⁻¹ * D a * (x - c) ^ a) ⊢ Differentiable ℝ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real	[185, 1]	[201, 8]	ext x	case refine_2 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊢ f ∘ ofReal' = ofReal' ∘ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n	case refine_2.h f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊢ (f ∘ ofReal') x = (ofReal' ∘ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real	[185, 1]	[201, 8]	simp only [Function.comp_apply, ofReal_eq_coe, ← H, ofReal_tsum]	case refine_2.h f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊢ (f ∘ ofReal') x = (ofReal' ∘ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x	case refine_2.h f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊢ ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (↑x - ↑c) ^ n = ∑' (a : ℕ), ↑((↑a !)⁻¹ * D a * (x - c) ^ a)
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real	[185, 1]	[201, 8]	push_cast	case refine_2.h f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊢ ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (↑x - ↑c) ^ n = ∑' (a : ℕ), ↑((↑a !)⁻¹ * D a * (x - c) ^ a)	case refine_2.h f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊢ ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a = ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real	[185, 1]	[201, 8]	rfl	case refine_2.h f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊢ ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a = ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.nonneg_of_iteratedDeriv_nonneg	[207, 1]	[223, 13]	have H := taylorSeries_eq_of_entire' 0 z hf	f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ z ⊢ 0 ≤ f z	f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ z H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (z - 0) ^ n = f z ⊢ 0 ≤ f z
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.nonneg_of_iteratedDeriv_nonneg	[207, 1]	[223, 13]	have hz' := eq_re_of_ofReal_le hz	f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ z H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (z - 0) ^ n = f z ⊢ 0 ≤ f z	f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ z H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (z - 0) ^ n = f z hz' : z = ↑z.re ⊢ 0 ≤ f z
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.nonneg_of_iteratedDeriv_nonneg	[207, 1]	[223, 13]	rw [hz'] at hz H ⊢	f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ z H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (z - 0) ^ n = f z hz' : z = ↑z.re ⊢ 0 ≤ f z	f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re ⊢ 0 ≤ f ↑z.re
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.nonneg_of_iteratedDeriv_nonneg	[207, 1]	[223, 13]	obtain ⟨D, hD⟩ : ∃ D : ℕ → ℝ, ∀ n, 0 ≤ D n ∧ iteratedDeriv n f 0 = D n	f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re ⊢ 0 ≤ f ↑z.re	f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re ⊢ ∃ D, ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊢ 0 ≤ f ↑z.re
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.nonneg_of_iteratedDeriv_nonneg	[207, 1]	[223, 13]	simp_rw [← H, hD, ← ofReal_natCast, sub_zero, ← ofReal_pow, ← ofReal_inv, ← ofReal_mul, ← ofReal_tsum]	case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊢ 0 ≤ f ↑z.re	case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊢ 0 ≤ ↑(∑' (a : ℕ), (↑a !)⁻¹ * D a * z.re ^ a)
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.nonneg_of_iteratedDeriv_nonneg	[207, 1]	[223, 13]	norm_cast	case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊢ 0 ≤ ↑(∑' (a : ℕ), (↑a !)⁻¹ * D a * z.re ^ a)	case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊢ 0 ≤ ∑' (a : ℕ), (↑a !)⁻¹ * D a * z.re ^ a
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.nonneg_of_iteratedDeriv_nonneg	[207, 1]	[223, 13]	refine tsum_nonneg fun n ↦ ?_	case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊢ 0 ≤ ∑' (a : ℕ), (↑a !)⁻¹ * D a * z.re ^ a	case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : ℕ ⊢ 0 ≤ (↑n !)⁻¹ * D n * z.re ^ n
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.nonneg_of_iteratedDeriv_nonneg	[207, 1]	[223, 13]	norm_cast at hz	case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : ℕ ⊢ 0 ≤ (↑n !)⁻¹ * D n * z.re ^ n	case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : ℕ hz : 0 ≤ z.re ⊢ 0 ≤ (↑n !)⁻¹ * D n * z.re ^ n
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.nonneg_of_iteratedDeriv_nonneg	[207, 1]	[223, 13]	have := (hD n).1	case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : ℕ hz : 0 ≤ z.re ⊢ 0 ≤ (↑n !)⁻¹ * D n * z.re ^ n	case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : ℕ hz : 0 ≤ z.re this : 0 ≤ D n ⊢ 0 ≤ (↑n !)⁻¹ * D n * z.re ^ n
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.nonneg_of_iteratedDeriv_nonneg	[207, 1]	[223, 13]	positivity	case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : ℕ hz : 0 ≤ z.re this : 0 ≤ D n ⊢ 0 ≤ (↑n !)⁻¹ * D n * z.re ^ n	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.nonneg_of_iteratedDeriv_nonneg	[207, 1]	[223, 13]	refine ⟨fun n ↦ (iteratedDeriv n f 0).re, fun n ↦ ⟨?_, ?_⟩⟩	f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re ⊢ ∃ D, ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n)	case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : ℕ ⊢ 0 ≤ (fun n => (iteratedDeriv n f 0).re) n case refine_2 f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : ℕ ⊢ iteratedDeriv n f 0 = ↑((fun n => (iteratedDeriv n f 0).re) n)
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.nonneg_of_iteratedDeriv_nonneg	[207, 1]	[223, 13]	have := eq_re_of_ofReal_le (h n) ▸ h n	case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : ℕ ⊢ 0 ≤ (fun n => (iteratedDeriv n f 0).re) n	case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : ℕ this : 0 ≤ ↑(iteratedDeriv n f 0).re ⊢ 0 ≤ (fun n => (iteratedDeriv n f 0).re) n
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.nonneg_of_iteratedDeriv_nonneg	[207, 1]	[223, 13]	norm_cast at this	case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : ℕ this : 0 ≤ ↑(iteratedDeriv n f 0).re ⊢ 0 ≤ (fun n => (iteratedDeriv n f 0).re) n	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.nonneg_of_iteratedDeriv_nonneg	[207, 1]	[223, 13]	rw [eq_re_of_ofReal_le (h n)]	case refine_2 f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : ℕ ⊢ iteratedDeriv n f 0 = ↑((fun n => (iteratedDeriv n f 0).re) n)	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.monotoneOn_of_iteratedDeriv_nonneg	[227, 1]	[250, 17]	let D : ℕ → ℝ := fun n ↦ (iteratedDeriv n f 0).re	f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 ⊢ MonotoneOn (f ∘ ofReal') (Set.Ici 0)	f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re ⊢ MonotoneOn (f ∘ ofReal') (Set.Ici 0)
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.monotoneOn_of_iteratedDeriv_nonneg	[227, 1]	[250, 17]	have hD (n : ℕ) : iteratedDeriv n f 0 = D n := by refine Complex.ext rfl ?_ simp only [ofReal_im] exact (le_def.mp (h n)).2.symm	f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re ⊢ MonotoneOn (f ∘ ofReal') (Set.Ici 0)	f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) ⊢ MonotoneOn (f ∘ ofReal') (Set.Ici 0)
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.monotoneOn_of_iteratedDeriv_nonneg	[227, 1]	[250, 17]	obtain ⟨F, hFd, hF⟩ := realValued_of_iteratedDeriv_real hf hD	f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) ⊢ MonotoneOn (f ∘ ofReal') (Set.Ici 0)	case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F ⊢ MonotoneOn (f ∘ ofReal') (Set.Ici 0)
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.monotoneOn_of_iteratedDeriv_nonneg	[227, 1]	[250, 17]	rw [hF]	case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F ⊢ MonotoneOn (f ∘ ofReal') (Set.Ici 0)	case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F ⊢ MonotoneOn (ofReal' ∘ F) (Set.Ici 0)
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.monotoneOn_of_iteratedDeriv_nonneg	[227, 1]	[250, 17]	refine monotone_ofReal.comp_monotoneOn <\| monotoneOn_of_deriv_nonneg (convex_Ici 0) hFd.continuous.continuousOn hFd.differentiableOn fun x hx ↦ ?_	case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F ⊢ MonotoneOn (ofReal' ∘ F) (Set.Ici 0)	case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) ⊢ 0 ≤ deriv F x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.monotoneOn_of_iteratedDeriv_nonneg	[227, 1]	[250, 17]	have hD' (n : ℕ) : 0 ≤ iteratedDeriv n (deriv f) 0 := by rw [← iteratedDeriv_succ'] exact h (n + 1)	case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) ⊢ 0 ≤ deriv F x	case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 ⊢ 0 ≤ deriv F x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.monotoneOn_of_iteratedDeriv_nonneg	[227, 1]	[250, 17]	have hf' := (contDiff_succ_iff_deriv.mp <\| hf.contDiff (n := 2)).2.differentiable rfl.le	case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 ⊢ 0 ≤ deriv F x	case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) ⊢ 0 ≤ deriv F x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.monotoneOn_of_iteratedDeriv_nonneg	[227, 1]	[250, 17]	have hx : (0 : ℂ) ≤ x := by norm_cast simp only [Set.nonempty_Iio, interior_Ici', Set.mem_Ioi] at hx exact hx.le	case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) ⊢ 0 ≤ deriv F x	case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) hx : 0 ≤ ↑x ⊢ 0 ≤ deriv F x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.monotoneOn_of_iteratedDeriv_nonneg	[227, 1]	[250, 17]	have H := nonneg_of_iteratedDeriv_nonneg hf' hD' hx	case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) hx : 0 ≤ ↑x ⊢ 0 ≤ deriv F x	case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) hx : 0 ≤ ↑x H : 0 ≤ deriv f ↑x ⊢ 0 ≤ deriv F x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.monotoneOn_of_iteratedDeriv_nonneg	[227, 1]	[250, 17]	rw [← deriv.comp_ofReal hf.differentiableAt] at H	case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) hx : 0 ≤ ↑x H : 0 ≤ deriv f ↑x ⊢ 0 ≤ deriv F x	case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) hx : 0 ≤ ↑x H : 0 ≤ deriv (fun x => f ↑x) x ⊢ 0 ≤ deriv F x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/EulerProduct.lean

LSeries.term_mul_aux

[21, 1]

[23, 90]

rw [mul_comm_div, div_div, ← mul_div_assoc, mul_comm (m : ℂ), natCast_mul_natCast_cpow]

a b : ℂ m n : ℕ s : ℂ ⊢ a / ↑m ^ s * (b / ↑n ^ s) = a * b / (↑m * ↑n) ^ s

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/EulerProduct.lean

LSeries.term_mul

[25, 1]

[30, 100]

rcases eq_or_ne (m * n) 0 with H | H

f₁ f₂ f : ℕ → ℂ m n : ℕ h : f (m * n) = f₁ m * f₂ n s : ℂ ⊢ term f s (m * n) = term f₁ s m * term f₂ s n

case inl f₁ f₂ f : ℕ → ℂ m n : ℕ h : f (m * n) = f₁ m * f₂ n s : ℂ H : m * n = 0 ⊢ term f s (m * n) = term f₁ s m * term f₂ s n case inr f₁ f₂ f : ℕ → ℂ m n : ℕ h : f (m * n) = f₁ m * f₂ n s : ℂ H : m * n ≠ 0 ⊢ term f s (m * n) = term f₁ s m * term f₂ s n

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/EulerProduct.lean

LSeries.term_mul

[25, 1]

[30, 100]

rcases mul_eq_zero.mp H with rfl | rfl <;> simp only [term_zero, mul_zero, zero_mul]

case inl f₁ f₂ f : ℕ → ℂ m n : ℕ h : f (m * n) = f₁ m * f₂ n s : ℂ H : m * n = 0 ⊢ term f s (m * n) = term f₁ s m * term f₂ s n

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/EulerProduct.lean

LSeries.term_mul

[25, 1]

[30, 100]

obtain ⟨hm, hn⟩ := mul_ne_zero_iff.mp H

case inr f₁ f₂ f : ℕ → ℂ m n : ℕ h : f (m * n) = f₁ m * f₂ n s : ℂ H : m * n ≠ 0 ⊢ term f s (m * n) = term f₁ s m * term f₂ s n

case inr.intro f₁ f₂ f : ℕ → ℂ m n : ℕ h : f (m * n) = f₁ m * f₂ n s : ℂ H : m * n ≠ 0 hm : m ≠ 0 hn : n ≠ 0 ⊢ term f s (m * n) = term f₁ s m * term f₂ s n

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/EulerProduct.lean

LSeries.term_mul

[25, 1]

[30, 100]

simp only [ne_eq, H, not_false_eq_true, term_of_ne_zero, Nat.cast_mul, hm, hn, h, term_mul_aux]

case inr.intro f₁ f₂ f : ℕ → ℂ m n : ℕ h : f (m * n) = f₁ m * f₂ n s : ℂ H : m * n ≠ 0 hm : m ≠ 0 hn : n ≠ 0 ⊢ term f s (m * n) = term f₁ s m * term f₂ s n

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/EulerProduct.lean

LSeries.term_at_one

[44, 1]

[45, 72]

rw [term_of_ne_zero one_ne_zero, h₁, Nat.cast_one, one_cpow, div_one]

f : ℕ → ℂ h₁ : f 1 = 1 s : ℂ ⊢ term f s 1 = 1

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/EulerProduct.lean

DirichletCharacter.toFun_on_nat_map_one

[86, 1]

[87, 32]

simp only [cast_one, map_one]

N : ℕ χ : DirichletCharacter ℂ N ⊢ (fun n => χ ↑n) 1 = 1

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/EulerProduct.lean

DirichletCharacter.toFun_on_nat_map_mul

[89, 1]

[91, 32]

simp only [cast_mul, map_mul]

N : ℕ χ : DirichletCharacter ℂ N m n : ℕ ⊢ (fun n => χ ↑n) (m * n) = (fun n => χ ↑n) m * (fun n => χ ↑n) n

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/EulerProduct.lean

DirichletCharacter.LSeries_eulerProduct

[94, 1]

[99, 87]

refine Tendsto.congr (fun n ↦ Finset.prod_congr rfl fun p hp ↦ ?_) <| eulerProduct_of_completelyMultiplicative (toFun_on_nat_map_one χ) (toFun_on_nat_map_mul χ) <| LSeriesSummable_of_one_lt_re χ hs

N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re ⊢ Tendsto (fun n => ∏ p ∈ n.primesBelow, (1 - χ ↑p * ↑p ^ (-s))⁻¹) atTop (𝓝 (L (fun n => χ ↑n) s))

N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re n p : ℕ hp : p ∈ n.primesBelow ⊢ (1 - term (fun n => χ ↑n) s p)⁻¹ = (1 - χ ↑p * ↑p ^ (-s))⁻¹

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/EulerProduct.lean

DirichletCharacter.LSeries_eulerProduct

[94, 1]

[99, 87]

rw [term_of_ne_zero (prime_of_mem_primesBelow hp).ne_zero, cpow_neg, div_eq_mul_inv]

N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re n p : ℕ hp : p ∈ n.primesBelow ⊢ (1 - term (fun n => χ ↑n) s p)⁻¹ = (1 - χ ↑p * ↑p ^ (-s))⁻¹

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/EulerProduct.lean

DirichletCharacter.LSeries_eulerProduct'

[102, 1]

[110, 61]

rw [LSeries]

N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re ⊢ cexp (∑' (p : Primes), -(1 - χ ↑↑p * ↑↑p ^ (-s)).log) = L (fun n => χ ↑n) s

N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re ⊢ cexp (∑' (p : Primes), -(1 - χ ↑↑p * ↑↑p ^ (-s)).log) = ∑' (n : ℕ), term (fun n => χ ↑n) s n

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/EulerProduct.lean

DirichletCharacter.LSeries_eulerProduct'

[102, 1]

[110, 61]

convert exp_sum_primes_log_eq_tsum (f := dirichletSummandHom χ <| ne_zero_of_one_lt_re hs) <| summable_dirichletSummand χ hs

N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re ⊢ cexp (∑' (p : Primes), -(1 - χ ↑↑p * ↑↑p ^ (-s)).log) = ∑' (n : ℕ), term (fun n => χ ↑n) s n

case h.e'_3.h.e'_5.h.h.e N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ : ℕ ⊢ term (fun n => χ ↑n) s = ⇑(dirichletSummandHom χ ⋯)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/EulerProduct.lean

DirichletCharacter.LSeries_eulerProduct'

[102, 1]

[110, 61]

ext n

case h.e'_3.h.e'_5.h.h.e N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ : ℕ ⊢ term (fun n => χ ↑n) s = ⇑(dirichletSummandHom χ ⋯)

case h.e'_3.h.e'_5.h.h.e.h N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ n : ℕ ⊢ term (fun n => χ ↑n) s n = (dirichletSummandHom χ ⋯) n

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/EulerProduct.lean

DirichletCharacter.LSeries_eulerProduct'

[102, 1]

[110, 61]

rcases eq_or_ne n 0 with rfl | hn

case h.e'_3.h.e'_5.h.h.e.h N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ n : ℕ ⊢ term (fun n => χ ↑n) s n = (dirichletSummandHom χ ⋯) n

case h.e'_3.h.e'_5.h.h.e.h.inl N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ : ℕ ⊢ term (fun n => χ ↑n) s 0 = (dirichletSummandHom χ ⋯) 0 case h.e'_3.h.e'_5.h.h.e.h.inr N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ n : ℕ hn : n ≠ 0 ⊢ term (fun n => χ ↑n) s n = (dirichletSummandHom χ ⋯) n

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/EulerProduct.lean

DirichletCharacter.LSeries_eulerProduct'

[102, 1]

[110, 61]

simp only [term_zero, map_zero]

case h.e'_3.h.e'_5.h.h.e.h.inl N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ : ℕ ⊢ term (fun n => χ ↑n) s 0 = (dirichletSummandHom χ ⋯) 0

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/EulerProduct.lean

DirichletCharacter.LSeries_eulerProduct'

[102, 1]

[110, 61]

simp [hn, dirichletSummandHom, div_eq_mul_inv, cpow_neg]

case h.e'_3.h.e'_5.h.h.e.h.inr N : ℕ χ : DirichletCharacter ℂ N s : ℂ hs : 1 < s.re x✝ n : ℕ hn : n ≠ 0 ⊢ term (fun n => χ ↑n) s n = (dirichletSummandHom χ ⋯) n

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/EulerProduct.lean

ArithmeticFunction.LSeries_zeta_eulerProduct'

[117, 1]

[120, 62]

convert modOne_eq_one (R := ℂ) ▸ LSeries_eulerProduct' (1 : DirichletCharacter ℂ 1) hs using 7

s : ℂ hs : 1 < s.re ⊢ cexp (∑' (p : Primes), -(1 - ↑↑p ^ (-s)).log) = L 1 s

case h.e'_2.h.e'_1.h.e'_5.h.h.e'_3.h.e'_1.h.e'_6 s : ℂ hs : 1 < s.re x✝ : Primes ⊢ ↑↑x✝ ^ (-s) = 1 ↑↑x✝ * ↑↑x✝ ^ (-s)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/EulerProduct.lean

ArithmeticFunction.LSeries_zeta_eulerProduct'

[117, 1]

[120, 62]

rw [MulChar.one_apply <| isUnit_of_subsingleton _, one_mul]

case h.e'_2.h.e'_1.h.e'_5.h.h.e'_3.h.e'_1.h.e'_6 s : ℂ hs : 1 < s.re x✝ : Primes ⊢ ↑↑x✝ ^ (-s) = 1 ↑↑x✝ * ↑↑x✝ ^ (-s)

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Asymptotics.isBigO_mul_iff_isBigO_div

[31, 1]

[39, 36]

rw [isBigO_iff', isBigO_iff']

α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 ⊢ (fun x => f x * g x) =O[l] h ↔ g =O[l] fun x => h x / f x

α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 ⊢ (∃ c > 0, ∀ᶠ (x : α) in l, ‖f x * g x‖ ≤ c * ‖h x‖) ↔ ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Asymptotics.isBigO_mul_iff_isBigO_div

[31, 1]

[39, 36]

refine ⟨fun ⟨c, hc, H⟩ ↦ ⟨c, hc, ?_⟩, fun ⟨c, hc, H⟩ ↦ ⟨c, hc, ?_⟩⟩ <;> { refine H.congr <| Eventually.mp hf <| eventually_of_forall fun x hx ↦ ?_ rw [norm_mul, norm_div, ← mul_div_assoc, mul_comm] have hx' : ‖f x‖ > 0 := norm_pos_iff.mpr hx rw [le_div_iff hx', mul_comm] }

α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 ⊢ (∃ c > 0, ∀ᶠ (x : α) in l, ‖f x * g x‖ ≤ c * ‖h x‖) ↔ ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Asymptotics.isBigO_mul_iff_isBigO_div

[31, 1]

[39, 36]

refine H.congr <| Eventually.mp hf <| eventually_of_forall fun x hx ↦ ?_

case refine_2 α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ c : ℝ hc : c > 0 H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ ⊢ ∀ᶠ (x : α) in l, ‖f x * g x‖ ≤ c * ‖h x‖

case refine_2 α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ c : ℝ hc : c > 0 H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ x : α hx : f x ≠ 0 ⊢ ‖g x‖ ≤ c * ‖h x / f x‖ ↔ ‖f x * g x‖ ≤ c * ‖h x‖

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Asymptotics.isBigO_mul_iff_isBigO_div

[31, 1]

[39, 36]

rw [norm_mul, norm_div, ← mul_div_assoc, mul_comm]

case refine_2 α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ c : ℝ hc : c > 0 H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ x : α hx : f x ≠ 0 ⊢ ‖g x‖ ≤ c * ‖h x / f x‖ ↔ ‖f x * g x‖ ≤ c * ‖h x‖

case refine_2 α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ c : ℝ hc : c > 0 H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ x : α hx : f x ≠ 0 ⊢ ‖g x‖ ≤ ‖h x‖ * c / ‖f x‖ ↔ ‖f x‖ * ‖g x‖ ≤ ‖h x‖ * c

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Asymptotics.isBigO_mul_iff_isBigO_div

[31, 1]

[39, 36]

have hx' : ‖f x‖ > 0 := norm_pos_iff.mpr hx

case refine_2 α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ c : ℝ hc : c > 0 H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ x : α hx : f x ≠ 0 ⊢ ‖g x‖ ≤ ‖h x‖ * c / ‖f x‖ ↔ ‖f x‖ * ‖g x‖ ≤ ‖h x‖ * c

case refine_2 α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ c : ℝ hc : c > 0 H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ x : α hx : f x ≠ 0 hx' : ‖f x‖ > 0 ⊢ ‖g x‖ ≤ ‖h x‖ * c / ‖f x‖ ↔ ‖f x‖ * ‖g x‖ ≤ ‖h x‖ * c

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Asymptotics.isBigO_mul_iff_isBigO_div

[31, 1]

[39, 36]

rw [le_div_iff hx', mul_comm]

case refine_2 α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ c : ℝ hc : c > 0 H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ x : α hx : f x ≠ 0 hx' : ‖f x‖ > 0 ⊢ ‖g x‖ ≤ ‖h x‖ * c / ‖f x‖ ↔ ‖f x‖ * ‖g x‖ ≤ ‖h x‖ * c

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

DifferentiableAt.isBigO_of_eq_zero

[50, 1]

[54, 73]

rw [← zero_add z] at hf

f : ℂ → ℂ z : ℂ hf : DifferentiableAt ℂ f z hz : f z = 0 ⊢ (fun w => f (w + z)) =O[𝓝 0] id

f : ℂ → ℂ z : ℂ hf : DifferentiableAt ℂ f (0 + z) hz : f z = 0 ⊢ (fun w => f (w + z)) =O[𝓝 0] id

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

DifferentiableAt.isBigO_of_eq_zero

[50, 1]

[54, 73]

simpa only [zero_add, hz, sub_zero] using (hf.hasDerivAt.comp_add_const 0 z).differentiableAt.isBigO_sub

f : ℂ → ℂ z : ℂ hf : DifferentiableAt ℂ f (0 + z) hz : f z = 0 ⊢ (fun w => f (w + z)) =O[𝓝 0] id

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

ContinuousAt.isBigO

[56, 1]

[70, 46]

rw [isBigO_iff']

f : ℂ → ℂ z : ℂ hf : ContinuousAt f z ⊢ (fun w => f (w + z)) =O[𝓝 0] fun x => 1

f : ℂ → ℂ z : ℂ hf : ContinuousAt f z ⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

ContinuousAt.isBigO

[56, 1]

[70, 46]

simp_rw [Metric.continuousAt_iff', dist_eq_norm_sub, zero_add] at hf

f : ℂ → ℂ z : ℂ hf : ContinuousAt (fun w => f (w + z)) 0 ⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖

f : ℂ → ℂ z : ℂ hf : ∀ ε > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < ε ⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

ContinuousAt.isBigO

[56, 1]

[70, 46]

specialize hf 1 zero_lt_one

f : ℂ → ℂ z : ℂ hf : ∀ ε > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < ε ⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖

f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 ⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

ContinuousAt.isBigO

[56, 1]

[70, 46]

refine ⟨‖f z‖ + 1, by positivity, ?_⟩

f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 ⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖

f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 ⊢ ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ (‖f z‖ + 1) * ‖1‖

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

ContinuousAt.isBigO

[56, 1]

[70, 46]

refine Eventually.mp hf <| eventually_of_forall fun w hw ↦ le_of_lt ?_

f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 ⊢ ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ (‖f z‖ + 1) * ‖1‖

f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 w : ℂ hw : ‖f (w + z) - f z‖ < 1 ⊢ ‖f (w + z)‖ < (‖f z‖ + 1) * ‖1‖

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

ContinuousAt.isBigO

[56, 1]

[70, 46]

calc ‖f (w + z)‖ _ ≤ ‖f z‖ + ‖f (w + z) - f z‖ := norm_le_insert' .. _ < ‖f z‖ + 1 := add_lt_add_left hw _ _ = _ := by simp only [norm_one, mul_one]

f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 w : ℂ hw : ‖f (w + z) - f z‖ < 1 ⊢ ‖f (w + z)‖ < (‖f z‖ + 1) * ‖1‖

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

ContinuousAt.isBigO

[56, 1]

[70, 46]

convert (Homeomorph.comp_continuousAt_iff' (Homeomorph.addLeft (-z)) _ z).mp ?_

f : ℂ → ℂ z : ℂ hf : ContinuousAt f z ⊢ ContinuousAt (fun w => f (w + z)) 0

case h.e'_1 f : ℂ → ℂ z : ℂ hf : ContinuousAt f z ⊢ 0 = (Homeomorph.addLeft (-z)) z case convert_4 f : ℂ → ℂ z : ℂ hf : ContinuousAt f z ⊢ ContinuousAt ((fun w => f (w + z)) ∘ ⇑(Homeomorph.addLeft (-z))) z

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

ContinuousAt.isBigO

[56, 1]

[70, 46]

simp

case h.e'_1 f : ℂ → ℂ z : ℂ hf : ContinuousAt f z ⊢ 0 = (Homeomorph.addLeft (-z)) z

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

ContinuousAt.isBigO

[56, 1]

[70, 46]

simp [Function.comp_def, hf]

case convert_4 f : ℂ → ℂ z : ℂ hf : ContinuousAt f z ⊢ ContinuousAt ((fun w => f (w + z)) ∘ ⇑(Homeomorph.addLeft (-z))) z

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

ContinuousAt.isBigO

[56, 1]

[70, 46]

positivity

f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 ⊢ ‖f z‖ + 1 > 0

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

ContinuousAt.isBigO

[56, 1]

[70, 46]

simp only [norm_one, mul_one]

f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 w : ℂ hw : ‖f (w + z) - f z‖ < 1 ⊢ ‖f z‖ + 1 = (‖f z‖ + 1) * ‖1‖

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

HasDerivAt.of_hasDerivAt_ofReal_comp

[125, 1]

[134, 80]

lift u to ℝ

z : ℝ f : ℝ → ℝ u : ℂ hf : HasDerivAt (fun y => ↑(f y)) u z ⊢ ∃ u', u = ↑u' ∧ HasDerivAt f u' z

z : ℝ f : ℝ → ℝ u : ℂ hf : HasDerivAt (fun y => ↑(f y)) u z ⊢ u.im = 0 case intro z : ℝ f : ℝ → ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊢ ∃ u', ↑u = ↑u' ∧ HasDerivAt f u' z

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

HasDerivAt.of_hasDerivAt_ofReal_comp

[125, 1]

[134, 80]

refine ⟨u, rfl, ?_⟩

case intro z : ℝ f : ℝ → ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊢ ∃ u', ↑u = ↑u' ∧ HasDerivAt f u' z

case intro z : ℝ f : ℝ → ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊢ HasDerivAt f u z

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

HasDerivAt.of_hasDerivAt_ofReal_comp

[125, 1]

[134, 80]

convert (reCLM.hasFDerivAt.comp z hf.hasFDerivAt).hasDerivAt

case intro z : ℝ f : ℝ → ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊢ HasDerivAt f u z

case h.e'_7 z : ℝ f : ℝ → ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊢ u = (reCLM.comp (smulRight 1 ↑u)) 1

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

HasDerivAt.of_hasDerivAt_ofReal_comp

[125, 1]

[134, 80]

rw [comp_apply, smulRight_apply, one_apply, one_smul, reCLM_apply, ofReal_re]

case h.e'_7 z : ℝ f : ℝ → ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊢ u = (reCLM.comp (smulRight 1 ↑u)) 1

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

HasDerivAt.of_hasDerivAt_ofReal_comp

[125, 1]

[134, 80]

have H := (imCLM.hasFDerivAt.comp z hf.hasFDerivAt).hasDerivAt.deriv

z : ℝ f : ℝ → ℝ u : ℂ hf : HasDerivAt (fun y => ↑(f y)) u z ⊢ u.im = 0

z : ℝ f : ℝ → ℝ u : ℂ hf : HasDerivAt (fun y => ↑(f y)) u z H : _root_.deriv (⇑imCLM ∘ fun y => ↑(f y)) z = (imCLM.comp (smulRight 1 u)) 1 ⊢ u.im = 0

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

HasDerivAt.of_hasDerivAt_ofReal_comp

[125, 1]

[134, 80]

simp only [Function.comp_def, imCLM_apply, ofReal_im, deriv_const] at H

z : ℝ f : ℝ → ℝ u : ℂ hf : HasDerivAt (fun y => ↑(f y)) u z H : _root_.deriv (⇑imCLM ∘ fun y => ↑(f y)) z = (imCLM.comp (smulRight 1 u)) 1 ⊢ u.im = 0

z : ℝ f : ℝ → ℝ u : ℂ hf : HasDerivAt (fun y => ↑(f y)) u z H : 0 = (imCLM.comp (smulRight 1 u)) 1 ⊢ u.im = 0

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

HasDerivAt.of_hasDerivAt_ofReal_comp

[125, 1]

[134, 80]

rwa [eq_comm, comp_apply, imCLM_apply, smulRight_apply, one_apply, one_smul] at H

z : ℝ f : ℝ → ℝ u : ℂ hf : HasDerivAt (fun y => ↑(f y)) u z H : 0 = (imCLM.comp (smulRight 1 u)) 1 ⊢ u.im = 0

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

DifferentiableAt.ofReal_comp_iff

[136, 1]

[140, 40]

refine ⟨fun H ↦ ?_, ofReal_comp⟩

z : ℝ f : ℝ → ℝ ⊢ DifferentiableAt ℝ (fun y => ↑(f y)) z ↔ DifferentiableAt ℝ f z

z : ℝ f : ℝ → ℝ H : DifferentiableAt ℝ (fun y => ↑(f y)) z ⊢ DifferentiableAt ℝ f z

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

DifferentiableAt.ofReal_comp_iff

[136, 1]

[140, 40]

obtain ⟨u, _, hu₂⟩ := H.hasDerivAt.of_hasDerivAt_ofReal_comp

z : ℝ f : ℝ → ℝ H : DifferentiableAt ℝ (fun y => ↑(f y)) z ⊢ DifferentiableAt ℝ f z

case intro.intro z : ℝ f : ℝ → ℝ H : DifferentiableAt ℝ (fun y => ↑(f y)) z u : ℝ left✝ : deriv (fun y => ↑(f y)) z = ↑u hu₂ : HasDerivAt (fun y => f y) u z ⊢ DifferentiableAt ℝ f z

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

DifferentiableAt.ofReal_comp_iff

[136, 1]

[140, 40]

exact HasDerivAt.differentiableAt hu₂

case intro.intro z : ℝ f : ℝ → ℝ H : DifferentiableAt ℝ (fun y => ↑(f y)) z u : ℝ left✝ : deriv (fun y => ↑(f y)) z = ↑u hu₂ : HasDerivAt (fun y => f y) u z ⊢ DifferentiableAt ℝ f z

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

deriv.ofReal_comp

[146, 1]

[152, 27]

by_cases hf : DifferentiableAt ℝ f z

z : ℝ f : ℝ → ℝ ⊢ deriv (fun y => ↑(f y)) z = ↑(deriv f z)

case pos z : ℝ f : ℝ → ℝ hf : DifferentiableAt ℝ f z ⊢ deriv (fun y => ↑(f y)) z = ↑(deriv f z) case neg z : ℝ f : ℝ → ℝ hf : ¬DifferentiableAt ℝ f z ⊢ deriv (fun y => ↑(f y)) z = ↑(deriv f z)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

deriv.ofReal_comp

[146, 1]

[152, 27]

exact hf.hasDerivAt.ofReal_comp.deriv

case pos z : ℝ f : ℝ → ℝ hf : DifferentiableAt ℝ f z ⊢ deriv (fun y => ↑(f y)) z = ↑(deriv f z)

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

deriv.ofReal_comp

[146, 1]

[152, 27]

have hf' := mt DifferentiableAt.ofReal_comp_iff.mp hf

case neg z : ℝ f : ℝ → ℝ hf : ¬DifferentiableAt ℝ f z ⊢ deriv (fun y => ↑(f y)) z = ↑(deriv f z)

case neg z : ℝ f : ℝ → ℝ hf : ¬DifferentiableAt ℝ f z hf' : ¬DifferentiableAt ℝ (fun y => ↑(f y)) z ⊢ deriv (fun y => ↑(f y)) z = ↑(deriv f z)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

deriv.ofReal_comp

[146, 1]

[152, 27]

rw [deriv_zero_of_not_differentiableAt hf, deriv_zero_of_not_differentiableAt hf', Complex.ofReal_zero]

case neg z : ℝ f : ℝ → ℝ hf : ¬DifferentiableAt ℝ f z hf' : ¬DifferentiableAt ℝ (fun y => ↑(f y)) z ⊢ deriv (fun y => ↑(f y)) z = ↑(deriv f z)

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real_on_ball

[159, 1]

[183, 8]

have Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), (x : ℂ) ∈ Metric.ball (c : ℂ) r := by intro x hx refine Metric.mem_ball.mpr ?_ rw [dist_eq, ← ofReal_sub, abs_ofReal, abs_sub_lt_iff, sub_lt_iff_lt_add', sub_lt_comm] exact and_comm.mpr hx

f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) ⊢ ∃ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))

f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r ⊢ ∃ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real_on_ball

[159, 1]

[183, 8]

have H ⦃z : ℂ⦄ (hz : z ∈ Metric.ball (c : ℂ) r) := taylorSeries_eq_on_ball' hz hf

f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r ⊢ ∃ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))

f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z ⊢ ∃ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real_on_ball

[159, 1]

[183, 8]

refine ⟨fun x ↦ ∑' (n : ℕ), (↑n !)⁻¹ * (D n) * (x - c) ^ n, fun x hx ↦ ?_, fun x hx ↦ ?_⟩

f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z ⊢ ∃ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))

case refine_1 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ DifferentiableWithinAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x case refine_2 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ (f ∘ ofReal') x = (ofReal' ∘ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real_on_ball

[159, 1]

[183, 8]

intro x hx

f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) ⊢ ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r

f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ ↑x ∈ Metric.ball (↑c) r

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real_on_ball

[159, 1]

[183, 8]

refine Metric.mem_ball.mpr ?_

f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ ↑x ∈ Metric.ball (↑c) r

f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ dist ↑x ↑c < r

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real_on_ball

[159, 1]

[183, 8]

rw [dist_eq, ← ofReal_sub, abs_ofReal, abs_sub_lt_iff, sub_lt_iff_lt_add', sub_lt_comm]

f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ dist ↑x ↑c < r

f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ x < c + r ∧ c - r < x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real_on_ball

[159, 1]

[183, 8]

exact and_comm.mpr hx

f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ x < c + r ∧ c - r < x

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real_on_ball

[159, 1]

[183, 8]

have Hx := Hz _ hx

case refine_1 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ DifferentiableWithinAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x

case refine_1 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r ⊢ DifferentiableWithinAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real_on_ball

[159, 1]

[183, 8]

refine DifferentiableAt.differentiableWithinAt ?_

case refine_1 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r ⊢ DifferentiableWithinAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x

case refine_1 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r ⊢ DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real_on_ball

[159, 1]

[183, 8]

replace hf := ((hf x Hx).congr (fun _ hz ↦ H hz) (H Hx)).differentiableAt (Metric.isOpen_ball.mem_nhds Hx) |>.comp_ofReal

case refine_1 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r ⊢ DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x

case refine_1 f : ℂ → ℂ r c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r hf : DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (↑x - ↑c) ^ n) x ⊢ DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real_on_ball

[159, 1]

[183, 8]

simp_rw [hd, ← ofReal_sub, ← ofReal_natCast, ← ofReal_inv, ← ofReal_pow, ← ofReal_mul, ← ofReal_tsum] at hf

case refine_1 f : ℂ → ℂ r c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r hf : DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (↑x - ↑c) ^ n) x ⊢ DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x

case refine_1 f : ℂ → ℂ r c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r hf : DifferentiableAt ℝ (fun x => ↑(∑' (a : ℕ), (↑a !)⁻¹ * D a * (x - c) ^ a)) x ⊢ DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real_on_ball

[159, 1]

[183, 8]

exact DifferentiableAt.ofReal_comp_iff.mp hf

case refine_1 f : ℂ → ℂ r c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) Hx : ↑x ∈ Metric.ball (↑c) r hf : DifferentiableAt ℝ (fun x => ↑(∑' (a : ℕ), (↑a !)⁻¹ * D a * (x - c) ^ a)) x ⊢ DifferentiableAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real_on_ball

[159, 1]

[183, 8]

simp only [Function.comp_apply, ← H (Hz _ hx), hd, ofReal_tsum]

case refine_2 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ (f ∘ ofReal') x = (ofReal' ∘ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x

case refine_2 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (↑x - ↑c) ^ n = ∑' (a : ℕ), ↑((↑a !)⁻¹ * D a * (x - c) ^ a)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real_on_ball

[159, 1]

[183, 8]

push_cast

case refine_2 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (↑x - ↑c) ^ n = ∑' (a : ℕ), ↑((↑a !)⁻¹ * D a * (x - c) ^ a)

case refine_2 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a = ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real_on_ball

[159, 1]

[183, 8]

rfl

case refine_2 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a = ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real

[185, 1]

[201, 8]

have H (z : ℂ) := taylorSeries_eq_of_entire' c z hf

f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) ⊢ ∃ F, Differentiable ℝ F ∧ f ∘ ofReal' = ofReal' ∘ F

f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z ⊢ ∃ F, Differentiable ℝ F ∧ f ∘ ofReal' = ofReal' ∘ F

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real

[185, 1]

[201, 8]

simp_rw [hd] at H

f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z ⊢ ∃ F, Differentiable ℝ F ∧ f ∘ ofReal' = ofReal' ∘ F

f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊢ ∃ F, Differentiable ℝ F ∧ f ∘ ofReal' = ofReal' ∘ F

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real

[185, 1]

[201, 8]

refine ⟨fun x ↦ ∑' (n : ℕ), (↑n !)⁻¹ * (D n) * (x - c) ^ n, ?_, ?_⟩

f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊢ ∃ F, Differentiable ℝ F ∧ f ∘ ofReal' = ofReal' ∘ F

case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊢ Differentiable ℝ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n case refine_2 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊢ f ∘ ofReal' = ofReal' ∘ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real

[185, 1]

[201, 8]

have := hf.comp_ofReal

case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊢ Differentiable ℝ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n

case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z this : Differentiable ℝ fun x => f ↑x ⊢ Differentiable ℝ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real

[185, 1]

[201, 8]

simp_rw [← H, ← ofReal_sub, ← ofReal_natCast, ← ofReal_inv, ← ofReal_pow, ← ofReal_mul, ← ofReal_tsum] at this

case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z this : Differentiable ℝ fun x => f ↑x ⊢ Differentiable ℝ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n

case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z this : Differentiable ℝ fun x => ↑(∑' (a : ℕ), (↑a !)⁻¹ * D a * (x - c) ^ a) ⊢ Differentiable ℝ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real

[185, 1]

[201, 8]

exact Differentiable.ofReal_comp_iff.mp this

case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z this : Differentiable ℝ fun x => ↑(∑' (a : ℕ), (↑a !)⁻¹ * D a * (x - c) ^ a) ⊢ Differentiable ℝ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real

[185, 1]

[201, 8]

ext x

case refine_2 f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊢ f ∘ ofReal' = ofReal' ∘ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n

case refine_2.h f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊢ (f ∘ ofReal') x = (ofReal' ∘ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real

[185, 1]

[201, 8]

simp only [Function.comp_apply, ofReal_eq_coe, ← H, ofReal_tsum]

case refine_2.h f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊢ (f ∘ ofReal') x = (ofReal' ∘ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x

case refine_2.h f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊢ ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (↑x - ↑c) ^ n = ∑' (a : ℕ), ↑((↑a !)⁻¹ * D a * (x - c) ^ a)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real

[185, 1]

[201, 8]

push_cast

case refine_2.h f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊢ ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (↑x - ↑c) ^ n = ∑' (a : ℕ), ↑((↑a !)⁻¹ * D a * (x - c) ^ a)

case refine_2.h f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊢ ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a = ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real

[185, 1]

[201, 8]

rfl

case refine_2.h f : ℂ → ℂ hf : Differentiable ℂ f c : ℝ D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) H : ∀ (z : ℂ), ∑' (n : ℕ), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊢ ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a = ∑' (a : ℕ), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.nonneg_of_iteratedDeriv_nonneg

[207, 1]

[223, 13]

have H := taylorSeries_eq_of_entire' 0 z hf

f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ z ⊢ 0 ≤ f z

f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ z H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (z - 0) ^ n = f z ⊢ 0 ≤ f z

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.nonneg_of_iteratedDeriv_nonneg

[207, 1]

[223, 13]

have hz' := eq_re_of_ofReal_le hz

f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ z H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (z - 0) ^ n = f z ⊢ 0 ≤ f z

f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ z H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (z - 0) ^ n = f z hz' : z = ↑z.re ⊢ 0 ≤ f z

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.nonneg_of_iteratedDeriv_nonneg

[207, 1]

[223, 13]

rw [hz'] at hz H ⊢

f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ z H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (z - 0) ^ n = f z hz' : z = ↑z.re ⊢ 0 ≤ f z

f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re ⊢ 0 ≤ f ↑z.re

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.nonneg_of_iteratedDeriv_nonneg

[207, 1]

[223, 13]

obtain ⟨D, hD⟩ : ∃ D : ℕ → ℝ, ∀ n, 0 ≤ D n ∧ iteratedDeriv n f 0 = D n

f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re ⊢ 0 ≤ f ↑z.re

f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re ⊢ ∃ D, ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊢ 0 ≤ f ↑z.re

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.nonneg_of_iteratedDeriv_nonneg

[207, 1]

[223, 13]

simp_rw [← H, hD, ← ofReal_natCast, sub_zero, ← ofReal_pow, ← ofReal_inv, ← ofReal_mul, ← ofReal_tsum]

case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊢ 0 ≤ f ↑z.re

case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊢ 0 ≤ ↑(∑' (a : ℕ), (↑a !)⁻¹ * D a * z.re ^ a)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.nonneg_of_iteratedDeriv_nonneg

[207, 1]

[223, 13]

norm_cast

case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊢ 0 ≤ ↑(∑' (a : ℕ), (↑a !)⁻¹ * D a * z.re ^ a)

case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊢ 0 ≤ ∑' (a : ℕ), (↑a !)⁻¹ * D a * z.re ^ a

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.nonneg_of_iteratedDeriv_nonneg

[207, 1]

[223, 13]

refine tsum_nonneg fun n ↦ ?_

case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊢ 0 ≤ ∑' (a : ℕ), (↑a !)⁻¹ * D a * z.re ^ a

case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : ℕ ⊢ 0 ≤ (↑n !)⁻¹ * D n * z.re ^ n

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.nonneg_of_iteratedDeriv_nonneg

[207, 1]

[223, 13]

norm_cast at hz

case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : ℕ ⊢ 0 ≤ (↑n !)⁻¹ * D n * z.re ^ n

case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : ℕ hz : 0 ≤ z.re ⊢ 0 ≤ (↑n !)⁻¹ * D n * z.re ^ n

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.nonneg_of_iteratedDeriv_nonneg

[207, 1]

[223, 13]

have := (hD n).1

case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : ℕ hz : 0 ≤ z.re ⊢ 0 ≤ (↑n !)⁻¹ * D n * z.re ^ n

case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : ℕ hz : 0 ≤ z.re this : 0 ≤ D n ⊢ 0 ≤ (↑n !)⁻¹ * D n * z.re ^ n

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.nonneg_of_iteratedDeriv_nonneg

[207, 1]

[223, 13]

positivity

case intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : ℕ → ℝ hD : ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : ℕ hz : 0 ≤ z.re this : 0 ≤ D n ⊢ 0 ≤ (↑n !)⁻¹ * D n * z.re ^ n

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.nonneg_of_iteratedDeriv_nonneg

[207, 1]

[223, 13]

refine ⟨fun n ↦ (iteratedDeriv n f 0).re, fun n ↦ ⟨?_, ?_⟩⟩

f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re ⊢ ∃ D, ∀ (n : ℕ), 0 ≤ D n ∧ iteratedDeriv n f 0 = ↑(D n)

case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : ℕ ⊢ 0 ≤ (fun n => (iteratedDeriv n f 0).re) n case refine_2 f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : ℕ ⊢ iteratedDeriv n f 0 = ↑((fun n => (iteratedDeriv n f 0).re) n)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.nonneg_of_iteratedDeriv_nonneg

[207, 1]

[223, 13]

have := eq_re_of_ofReal_le (h n) ▸ h n

case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : ℕ ⊢ 0 ≤ (fun n => (iteratedDeriv n f 0).re) n

case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : ℕ this : 0 ≤ ↑(iteratedDeriv n f 0).re ⊢ 0 ≤ (fun n => (iteratedDeriv n f 0).re) n

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.nonneg_of_iteratedDeriv_nonneg

[207, 1]

[223, 13]

norm_cast at this

case refine_1 f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : ℕ this : 0 ≤ ↑(iteratedDeriv n f 0).re ⊢ 0 ≤ (fun n => (iteratedDeriv n f 0).re) n

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.nonneg_of_iteratedDeriv_nonneg

[207, 1]

[223, 13]

rw [eq_re_of_ofReal_le (h n)]

case refine_2 f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 z : ℂ hz : 0 ≤ ↑z.re H : ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : ℕ ⊢ iteratedDeriv n f 0 = ↑((fun n => (iteratedDeriv n f 0).re) n)

no goals

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.monotoneOn_of_iteratedDeriv_nonneg

[227, 1]

[250, 17]

let D : ℕ → ℝ := fun n ↦ (iteratedDeriv n f 0).re

f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 ⊢ MonotoneOn (f ∘ ofReal') (Set.Ici 0)

f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re ⊢ MonotoneOn (f ∘ ofReal') (Set.Ici 0)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.monotoneOn_of_iteratedDeriv_nonneg

[227, 1]

[250, 17]

have hD (n : ℕ) : iteratedDeriv n f 0 = D n := by refine Complex.ext rfl ?_ simp only [ofReal_im] exact (le_def.mp (h n)).2.symm

f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re ⊢ MonotoneOn (f ∘ ofReal') (Set.Ici 0)

f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) ⊢ MonotoneOn (f ∘ ofReal') (Set.Ici 0)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.monotoneOn_of_iteratedDeriv_nonneg

[227, 1]

[250, 17]

obtain ⟨F, hFd, hF⟩ := realValued_of_iteratedDeriv_real hf hD

f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) ⊢ MonotoneOn (f ∘ ofReal') (Set.Ici 0)

case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F ⊢ MonotoneOn (f ∘ ofReal') (Set.Ici 0)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.monotoneOn_of_iteratedDeriv_nonneg

[227, 1]

[250, 17]

rw [hF]

case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F ⊢ MonotoneOn (f ∘ ofReal') (Set.Ici 0)

case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F ⊢ MonotoneOn (ofReal' ∘ F) (Set.Ici 0)

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.monotoneOn_of_iteratedDeriv_nonneg

[227, 1]

[250, 17]

refine monotone_ofReal.comp_monotoneOn <| monotoneOn_of_deriv_nonneg (convex_Ici 0) hFd.continuous.continuousOn hFd.differentiableOn fun x hx ↦ ?_

case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F ⊢ MonotoneOn (ofReal' ∘ F) (Set.Ici 0)

case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) ⊢ 0 ≤ deriv F x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.monotoneOn_of_iteratedDeriv_nonneg

[227, 1]

[250, 17]

have hD' (n : ℕ) : 0 ≤ iteratedDeriv n (deriv f) 0 := by rw [← iteratedDeriv_succ'] exact h (n + 1)

case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) ⊢ 0 ≤ deriv F x

case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 ⊢ 0 ≤ deriv F x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.monotoneOn_of_iteratedDeriv_nonneg

[227, 1]

[250, 17]

have hf' := (contDiff_succ_iff_deriv.mp <| hf.contDiff (n := 2)).2.differentiable rfl.le

case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 ⊢ 0 ≤ deriv F x

case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) ⊢ 0 ≤ deriv F x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.monotoneOn_of_iteratedDeriv_nonneg

[227, 1]

[250, 17]

have hx : (0 : ℂ) ≤ x := by norm_cast simp only [Set.nonempty_Iio, interior_Ici', Set.mem_Ioi] at hx exact hx.le

case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) ⊢ 0 ≤ deriv F x

case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) hx : 0 ≤ ↑x ⊢ 0 ≤ deriv F x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.monotoneOn_of_iteratedDeriv_nonneg

[227, 1]

[250, 17]

have H := nonneg_of_iteratedDeriv_nonneg hf' hD' hx

case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) hx : 0 ≤ ↑x ⊢ 0 ≤ deriv F x

case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) hx : 0 ≤ ↑x H : 0 ≤ deriv f ↑x ⊢ 0 ≤ deriv F x

https://github.com/MichaelStollBayreuth/EulerProducts.git

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.monotoneOn_of_iteratedDeriv_nonneg

[227, 1]

[250, 17]

rw [← deriv.comp_ofReal hf.differentiableAt] at H

case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) hx : 0 ≤ ↑x H : 0 ≤ deriv f ↑x ⊢ 0 ≤ deriv F x

case intro.intro f : ℂ → ℂ hf : Differentiable ℂ f h : ∀ (n : ℕ), 0 ≤ iteratedDeriv n f 0 D : ℕ → ℝ := fun n => (iteratedDeriv n f 0).re hD : ∀ (n : ℕ), iteratedDeriv n f 0 = ↑(D n) F : ℝ → ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : ∀ (n : ℕ), 0 ≤ iteratedDeriv n (deriv f) 0 hf' : Differentiable ℂ (deriv f) hx : 0 ≤ ↑x H : 0 ≤ deriv (fun x => f ↑x) x ⊢ 0 ≤ deriv F x