Datasets:
pkuAI4M
/
lean_github

Dataset card Data Studio Files Community
1
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
input
stringlengths
73
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 : β„‚), natCast_mul_natCast_cpow]
a b : β„‚ m n : β„• s : β„‚ ⊒ a / ↑m ^ s * (b / ↑n ^ s) = a * b / (↑m * ↑n) ^ s
no goals
Please generate a tactic in lean4 to solve the state. STATE: a b : β„‚ m n : β„• s : β„‚ ⊒ a / ↑m ^ s * (b / ↑n ^ s) = a * b / (↑m * ↑n) ^ s TACTIC:
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 fβ‚‚ f : β„• β†’ β„‚ m n : β„• h : f (m * n) = f₁ m * fβ‚‚ n s : β„‚ ⊒ term f s (m * n) = term f₁ s m * term fβ‚‚ s n
case inl f₁ fβ‚‚ f : β„• β†’ β„‚ m n : β„• h : f (m * n) = f₁ m * fβ‚‚ n s : β„‚ H : m * n = 0 ⊒ term f s (m * n) = term f₁ s m * term fβ‚‚ s n case inr f₁ fβ‚‚ f : β„• β†’ β„‚ m n : β„• h : f (m * n) = f₁ m * fβ‚‚ n s : β„‚ H : m * n β‰  0 ⊒ term f s (m * n) = term f₁ s m * term fβ‚‚ s n
Please generate a tactic in lean4 to solve the state. STATE: f₁ fβ‚‚ f : β„• β†’ β„‚ m n : β„• h : f (m * n) = f₁ m * fβ‚‚ n s : β„‚ ⊒ term f s (m * n) = term f₁ s m * term fβ‚‚ s n TACTIC:
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 fβ‚‚ f : β„• β†’ β„‚ m n : β„• h : f (m * n) = f₁ m * fβ‚‚ n s : β„‚ H : m * n = 0 ⊒ term f s (m * n) = term f₁ s m * term fβ‚‚ s n
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inl f₁ fβ‚‚ f : β„• β†’ β„‚ m n : β„• h : f (m * n) = f₁ m * fβ‚‚ n s : β„‚ H : m * n = 0 ⊒ term f s (m * n) = term f₁ s m * term fβ‚‚ s n TACTIC:
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 fβ‚‚ f : β„• β†’ β„‚ m n : β„• h : f (m * n) = f₁ m * fβ‚‚ n s : β„‚ H : m * n β‰  0 ⊒ term f s (m * n) = term f₁ s m * term fβ‚‚ s n
case inr.intro f₁ fβ‚‚ f : β„• β†’ β„‚ m n : β„• h : f (m * n) = f₁ m * fβ‚‚ n s : β„‚ H : m * n β‰  0 hm : m β‰  0 hn : n β‰  0 ⊒ term f s (m * n) = term f₁ s m * term fβ‚‚ s n
Please generate a tactic in lean4 to solve the state. STATE: case inr f₁ fβ‚‚ f : β„• β†’ β„‚ m n : β„• h : f (m * n) = f₁ m * fβ‚‚ n s : β„‚ H : m * n β‰  0 ⊒ term f s (m * n) = term f₁ s m * term fβ‚‚ s n TACTIC:
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 fβ‚‚ f : β„• β†’ β„‚ m n : β„• h : f (m * n) = f₁ m * fβ‚‚ n s : β„‚ H : m * n β‰  0 hm : m β‰  0 hn : n β‰  0 ⊒ term f s (m * n) = term f₁ s m * term fβ‚‚ s n
no goals
Please generate a tactic in lean4 to solve the state. STATE: case inr.intro f₁ fβ‚‚ f : β„• β†’ β„‚ m n : β„• h : f (m * n) = f₁ m * fβ‚‚ n s : β„‚ H : m * n β‰  0 hm : m β‰  0 hn : n β‰  0 ⊒ term f s (m * n) = term f₁ s m * term fβ‚‚ s n TACTIC:
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 : β„• β†’ β„‚ h₁ : f 1 = 1 s : β„‚ ⊒ term f s 1 = 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„• β†’ β„‚ h₁ : f 1 = 1 s : β„‚ ⊒ term f s 1 = 1 TACTIC:
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 : β„• Ο‡ : DirichletCharacter β„‚ N ⊒ (fun n => Ο‡ ↑n) 1 = 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: N : β„• Ο‡ : DirichletCharacter β„‚ N ⊒ (fun n => Ο‡ ↑n) 1 = 1 TACTIC:
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 : β„• Ο‡ : DirichletCharacter β„‚ N m n : β„• ⊒ (fun n => Ο‡ ↑n) (m * n) = (fun n => Ο‡ ↑n) m * (fun n => Ο‡ ↑n) n
no goals
Please generate a tactic in lean4 to solve the state. STATE: N : β„• Ο‡ : DirichletCharacter β„‚ N m n : β„• ⊒ (fun n => Ο‡ ↑n) (m * n) = (fun n => Ο‡ ↑n) m * (fun n => Ο‡ ↑n) n TACTIC:
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 Ο‡) (toFun_on_nat_map_mul Ο‡) <| LSeriesSummable_of_one_lt_re Ο‡ hs
N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re ⊒ Tendsto (fun n => ∏ p ∈ n.primesBelow, (1 - Ο‡ ↑p * ↑p ^ (-s))⁻¹) atTop (𝓝 (L (fun n => Ο‡ ↑n) s))
N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re n p : β„• hp : p ∈ n.primesBelow ⊒ (1 - term (fun n => Ο‡ ↑n) s p)⁻¹ = (1 - Ο‡ ↑p * ↑p ^ (-s))⁻¹
Please generate a tactic in lean4 to solve the state. STATE: N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re ⊒ Tendsto (fun n => ∏ p ∈ n.primesBelow, (1 - Ο‡ ↑p * ↑p ^ (-s))⁻¹) atTop (𝓝 (L (fun n => Ο‡ ↑n) s)) TACTIC:
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 : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re n p : β„• hp : p ∈ n.primesBelow ⊒ (1 - term (fun n => Ο‡ ↑n) s p)⁻¹ = (1 - Ο‡ ↑p * ↑p ^ (-s))⁻¹
no goals
Please generate a tactic in lean4 to solve the state. STATE: N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re n p : β„• hp : p ∈ n.primesBelow ⊒ (1 - term (fun n => Ο‡ ↑n) s p)⁻¹ = (1 - Ο‡ ↑p * ↑p ^ (-s))⁻¹ TACTIC:
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/EulerProduct.lean
DirichletCharacter.LSeries_eulerProduct'
[102, 1]
[110, 61]
rw [LSeries]
N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re ⊒ cexp (βˆ‘' (p : Primes), -(1 - Ο‡ ↑↑p * ↑↑p ^ (-s)).log) = L (fun n => Ο‡ ↑n) s
N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re ⊒ cexp (βˆ‘' (p : Primes), -(1 - Ο‡ ↑↑p * ↑↑p ^ (-s)).log) = βˆ‘' (n : β„•), term (fun n => Ο‡ ↑n) s n
Please generate a tactic in lean4 to solve the state. STATE: N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re ⊒ cexp (βˆ‘' (p : Primes), -(1 - Ο‡ ↑↑p * ↑↑p ^ (-s)).log) = L (fun n => Ο‡ ↑n) s TACTIC:
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 Ο‡ <| ne_zero_of_one_lt_re hs) <| summable_dirichletSummand Ο‡ hs
N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re ⊒ cexp (βˆ‘' (p : Primes), -(1 - Ο‡ ↑↑p * ↑↑p ^ (-s)).log) = βˆ‘' (n : β„•), term (fun n => Ο‡ ↑n) s n
case h.e'_3.h.e'_5.h.h.e N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re x✝ : β„• ⊒ term (fun n => Ο‡ ↑n) s = ⇑(dirichletSummandHom Ο‡ β‹―)
Please generate a tactic in lean4 to solve the state. STATE: N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re ⊒ cexp (βˆ‘' (p : Primes), -(1 - Ο‡ ↑↑p * ↑↑p ^ (-s)).log) = βˆ‘' (n : β„•), term (fun n => Ο‡ ↑n) s n TACTIC:
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 : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re x✝ : β„• ⊒ term (fun n => Ο‡ ↑n) s = ⇑(dirichletSummandHom Ο‡ β‹―)
case h.e'_3.h.e'_5.h.h.e.h N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re x✝ n : β„• ⊒ term (fun n => Ο‡ ↑n) s n = (dirichletSummandHom Ο‡ β‹―) n
Please generate a tactic in lean4 to solve the state. STATE: case h.e'_3.h.e'_5.h.h.e N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re x✝ : β„• ⊒ term (fun n => Ο‡ ↑n) s = ⇑(dirichletSummandHom Ο‡ β‹―) TACTIC:
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 : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re x✝ n : β„• ⊒ term (fun n => Ο‡ ↑n) s n = (dirichletSummandHom Ο‡ β‹―) n
case h.e'_3.h.e'_5.h.h.e.h.inl N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re x✝ : β„• ⊒ term (fun n => Ο‡ ↑n) s 0 = (dirichletSummandHom Ο‡ β‹―) 0 case h.e'_3.h.e'_5.h.h.e.h.inr N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re x✝ n : β„• hn : n β‰  0 ⊒ term (fun n => Ο‡ ↑n) s n = (dirichletSummandHom Ο‡ β‹―) n
Please generate a tactic in lean4 to solve the state. STATE: case h.e'_3.h.e'_5.h.h.e.h N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re x✝ n : β„• ⊒ term (fun n => Ο‡ ↑n) s n = (dirichletSummandHom Ο‡ β‹―) n TACTIC:
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 : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re x✝ : β„• ⊒ term (fun n => Ο‡ ↑n) s 0 = (dirichletSummandHom Ο‡ β‹―) 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.e'_3.h.e'_5.h.h.e.h.inl N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re x✝ : β„• ⊒ term (fun n => Ο‡ ↑n) s 0 = (dirichletSummandHom Ο‡ β‹―) 0 TACTIC:
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 : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re x✝ n : β„• hn : n β‰  0 ⊒ term (fun n => Ο‡ ↑n) s n = (dirichletSummandHom Ο‡ β‹―) n
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.e'_3.h.e'_5.h.h.e.h.inr N : β„• Ο‡ : DirichletCharacter β„‚ N s : β„‚ hs : 1 < s.re x✝ n : β„• hn : n β‰  0 ⊒ term (fun n => Ο‡ ↑n) s n = (dirichletSummandHom Ο‡ β‹―) n TACTIC:
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/EulerProduct.lean
ArithmeticFunction.LSeries_zeta_eulerProduct'
[117, 1]
[120, 62]
convert modOne_eq_one (R := β„‚) β–Έ LSeries_eulerProduct' (1 : DirichletCharacter β„‚ 1) hs using 7
s : β„‚ hs : 1 < s.re ⊒ cexp (βˆ‘' (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 : β„‚ hs : 1 < s.re x✝ : Primes ⊒ ↑↑x✝ ^ (-s) = 1 ↑↑x✝ * ↑↑x✝ ^ (-s)
Please generate a tactic in lean4 to solve the state. STATE: s : β„‚ hs : 1 < s.re ⊒ cexp (βˆ‘' (p : Primes), -(1 - ↑↑p ^ (-s)).log) = L 1 s TACTIC:
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 : β„‚ hs : 1 < s.re x✝ : Primes ⊒ ↑↑x✝ ^ (-s) = 1 ↑↑x✝ * ↑↑x✝ ^ (-s)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.e'_2.h.e'_1.h.e'_5.h.h.e'_3.h.e'_1.h.e'_6 s : β„‚ hs : 1 < s.re x✝ : Primes ⊒ ↑↑x✝ ^ (-s) = 1 ↑↑x✝ * ↑↑x✝ ^ (-s) TACTIC:
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']
Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 ⊒ (fun x => f x * g x) =O[l] h ↔ g =O[l] fun x => h x / f x
Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 ⊒ (βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–f x * g xβ€– ≀ c * β€–h xβ€–) ↔ βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€–
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 ⊒ (fun x => f x * g x) =O[l] h ↔ g =O[l] fun x => h x / f x TACTIC:
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' : β€–f xβ€– > 0 := norm_pos_iff.mpr hx rw [le_div_iff hx', mul_comm] }
Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 ⊒ (βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–f x * g xβ€– ≀ c * β€–h xβ€–) ↔ βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€–
no goals
Please generate a tactic in lean4 to solve the state. STATE: Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 ⊒ (βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–f x * g xβ€– ≀ c * β€–h xβ€–) ↔ βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– TACTIC:
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 Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 x✝ : βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– c : ℝ hc : c > 0 H : βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– ⊒ βˆ€αΆ  (x : Ξ±) in l, β€–f x * g xβ€– ≀ c * β€–h xβ€–
case refine_2 Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 x✝ : βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– c : ℝ hc : c > 0 H : βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– x : Ξ± hx : f x β‰  0 ⊒ β€–g xβ€– ≀ c * β€–h x / f xβ€– ↔ β€–f x * g xβ€– ≀ c * β€–h xβ€–
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 x✝ : βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– c : ℝ hc : c > 0 H : βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– ⊒ βˆ€αΆ  (x : Ξ±) in l, β€–f x * g xβ€– ≀ c * β€–h xβ€– TACTIC:
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 Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 x✝ : βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– c : ℝ hc : c > 0 H : βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– x : Ξ± hx : f x β‰  0 ⊒ β€–g xβ€– ≀ c * β€–h x / f xβ€– ↔ β€–f x * g xβ€– ≀ c * β€–h xβ€–
case refine_2 Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 x✝ : βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– c : ℝ hc : c > 0 H : βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– x : Ξ± hx : f x β‰  0 ⊒ β€–g xβ€– ≀ β€–h xβ€– * c / β€–f xβ€– ↔ β€–f xβ€– * β€–g xβ€– ≀ β€–h xβ€– * c
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 x✝ : βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– c : ℝ hc : c > 0 H : βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– x : Ξ± hx : f x β‰  0 ⊒ β€–g xβ€– ≀ c * β€–h x / f xβ€– ↔ β€–f x * g xβ€– ≀ c * β€–h xβ€– TACTIC:
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Asymptotics.isBigO_mul_iff_isBigO_div
[31, 1]
[39, 36]
have hx' : β€–f xβ€– > 0 := norm_pos_iff.mpr hx
case refine_2 Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 x✝ : βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– c : ℝ hc : c > 0 H : βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– x : Ξ± hx : f x β‰  0 ⊒ β€–g xβ€– ≀ β€–h xβ€– * c / β€–f xβ€– ↔ β€–f xβ€– * β€–g xβ€– ≀ β€–h xβ€– * c
case refine_2 Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 x✝ : βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– c : ℝ hc : c > 0 H : βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– x : Ξ± hx : f x β‰  0 hx' : β€–f xβ€– > 0 ⊒ β€–g xβ€– ≀ β€–h xβ€– * c / β€–f xβ€– ↔ β€–f xβ€– * β€–g xβ€– ≀ β€–h xβ€– * c
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 x✝ : βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– c : ℝ hc : c > 0 H : βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– x : Ξ± hx : f x β‰  0 ⊒ β€–g xβ€– ≀ β€–h xβ€– * c / β€–f xβ€– ↔ β€–f xβ€– * β€–g xβ€– ≀ β€–h xβ€– * c TACTIC:
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 Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 x✝ : βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– c : ℝ hc : c > 0 H : βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– x : Ξ± hx : f x β‰  0 hx' : β€–f xβ€– > 0 ⊒ β€–g xβ€– ≀ β€–h xβ€– * c / β€–f xβ€– ↔ β€–f xβ€– * β€–g xβ€– ≀ β€–h xβ€– * c
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 x✝ : βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– c : ℝ hc : c > 0 H : βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– x : Ξ± hx : f x β‰  0 hx' : β€–f xβ€– > 0 ⊒ β€–g xβ€– ≀ β€–h xβ€– * c / β€–f xβ€– ↔ β€–f xβ€– * β€–g xβ€– ≀ β€–h xβ€– * c TACTIC:
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 : β„‚ β†’ β„‚ z : β„‚ hf : DifferentiableAt β„‚ f z hz : f z = 0 ⊒ (fun w => f (w + z)) =O[𝓝 0] id
f : β„‚ β†’ β„‚ z : β„‚ hf : DifferentiableAt β„‚ f (0 + z) hz : f z = 0 ⊒ (fun w => f (w + z)) =O[𝓝 0] id
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ hf : DifferentiableAt β„‚ f z hz : f z = 0 ⊒ (fun w => f (w + z)) =O[𝓝 0] id TACTIC:
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 : β„‚ β†’ β„‚ z : β„‚ hf : DifferentiableAt β„‚ f (0 + z) hz : f z = 0 ⊒ (fun w => f (w + z)) =O[𝓝 0] id
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ hf : DifferentiableAt β„‚ f (0 + z) hz : f z = 0 ⊒ (fun w => f (w + z)) =O[𝓝 0] id TACTIC:
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
ContinuousAt.isBigO
[56, 1]
[70, 46]
rw [isBigO_iff']
f : β„‚ β†’ β„‚ z : β„‚ hf : ContinuousAt f z ⊒ (fun w => f (w + z)) =O[𝓝 0] fun x => 1
f : β„‚ β†’ β„‚ z : β„‚ hf : ContinuousAt f z ⊒ βˆƒ c > 0, βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z)β€– ≀ c * β€–1β€–
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ hf : ContinuousAt f z ⊒ (fun w => f (w + z)) =O[𝓝 0] fun x => 1 TACTIC:
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 : β„‚ β†’ β„‚ z : β„‚ hf : ContinuousAt (fun w => f (w + z)) 0 ⊒ βˆƒ c > 0, βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z)β€– ≀ c * β€–1β€–
f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€ Ξ΅ > 0, βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < Ξ΅ ⊒ βˆƒ c > 0, βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z)β€– ≀ c * β€–1β€–
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ hf : ContinuousAt (fun w => f (w + z)) 0 ⊒ βˆƒ c > 0, βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z)β€– ≀ c * β€–1β€– TACTIC:
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
ContinuousAt.isBigO
[56, 1]
[70, 46]
specialize hf 1 zero_lt_one
f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€ Ξ΅ > 0, βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < Ξ΅ ⊒ βˆƒ c > 0, βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z)β€– ≀ c * β€–1β€–
f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < 1 ⊒ βˆƒ c > 0, βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z)β€– ≀ c * β€–1β€–
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€ Ξ΅ > 0, βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < Ξ΅ ⊒ βˆƒ c > 0, βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z)β€– ≀ c * β€–1β€– TACTIC:
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
ContinuousAt.isBigO
[56, 1]
[70, 46]
refine βŸ¨β€–f zβ€– + 1, by positivity, ?_⟩
f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < 1 ⊒ βˆƒ c > 0, βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z)β€– ≀ c * β€–1β€–
f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < 1 ⊒ βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z)β€– ≀ (β€–f zβ€– + 1) * β€–1β€–
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < 1 ⊒ βˆƒ c > 0, βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z)β€– ≀ c * β€–1β€– TACTIC:
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 : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < 1 ⊒ βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z)β€– ≀ (β€–f zβ€– + 1) * β€–1β€–
f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < 1 w : β„‚ hw : β€–f (w + z) - f zβ€– < 1 ⊒ β€–f (w + z)β€– < (β€–f zβ€– + 1) * β€–1β€–
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < 1 ⊒ βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z)β€– ≀ (β€–f zβ€– + 1) * β€–1β€– TACTIC:
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
ContinuousAt.isBigO
[56, 1]
[70, 46]
calc β€–f (w + z)β€– _ ≀ β€–f zβ€– + β€–f (w + z) - f zβ€– := norm_le_insert' .. _ < β€–f zβ€– + 1 := add_lt_add_left hw _ _ = _ := by simp only [norm_one, mul_one]
f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < 1 w : β„‚ hw : β€–f (w + z) - f zβ€– < 1 ⊒ β€–f (w + z)β€– < (β€–f zβ€– + 1) * β€–1β€–
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < 1 w : β„‚ hw : β€–f (w + z) - f zβ€– < 1 ⊒ β€–f (w + z)β€– < (β€–f zβ€– + 1) * β€–1β€– TACTIC:
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 : β„‚ β†’ β„‚ z : β„‚ hf : ContinuousAt f z ⊒ ContinuousAt (fun w => f (w + z)) 0
case h.e'_1 f : β„‚ β†’ β„‚ z : β„‚ hf : ContinuousAt f z ⊒ 0 = (Homeomorph.addLeft (-z)) z case convert_4 f : β„‚ β†’ β„‚ z : β„‚ hf : ContinuousAt f z ⊒ ContinuousAt ((fun w => f (w + z)) ∘ ⇑(Homeomorph.addLeft (-z))) z
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ hf : ContinuousAt f z ⊒ ContinuousAt (fun w => f (w + z)) 0 TACTIC:
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
ContinuousAt.isBigO
[56, 1]
[70, 46]
simp
case h.e'_1 f : β„‚ β†’ β„‚ z : β„‚ hf : ContinuousAt f z ⊒ 0 = (Homeomorph.addLeft (-z)) z
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.e'_1 f : β„‚ β†’ β„‚ z : β„‚ hf : ContinuousAt f z ⊒ 0 = (Homeomorph.addLeft (-z)) z TACTIC:
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 : β„‚ β†’ β„‚ z : β„‚ hf : ContinuousAt f z ⊒ ContinuousAt ((fun w => f (w + z)) ∘ ⇑(Homeomorph.addLeft (-z))) z
no goals
Please generate a tactic in lean4 to solve the state. STATE: case convert_4 f : β„‚ β†’ β„‚ z : β„‚ hf : ContinuousAt f z ⊒ ContinuousAt ((fun w => f (w + z)) ∘ ⇑(Homeomorph.addLeft (-z))) z TACTIC:
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
ContinuousAt.isBigO
[56, 1]
[70, 46]
positivity
f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < 1 ⊒ β€–f zβ€– + 1 > 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < 1 ⊒ β€–f zβ€– + 1 > 0 TACTIC:
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
ContinuousAt.isBigO
[56, 1]
[70, 46]
simp only [norm_one, mul_one]
f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < 1 w : β„‚ hw : β€–f (w + z) - f zβ€– < 1 ⊒ β€–f zβ€– + 1 = (β€–f zβ€– + 1) * β€–1β€–
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < 1 w : β„‚ hw : β€–f (w + z) - f zβ€– < 1 ⊒ β€–f zβ€– + 1 = (β€–f zβ€– + 1) * β€–1β€– TACTIC:
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 β†’ ℝ u : β„‚ hf : HasDerivAt (fun y => ↑(f y)) u z ⊒ βˆƒ u', u = ↑u' ∧ HasDerivAt f u' z
z : ℝ f : ℝ β†’ ℝ u : β„‚ hf : HasDerivAt (fun y => ↑(f y)) u z ⊒ u.im = 0 case intro z : ℝ f : ℝ β†’ ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊒ βˆƒ u', ↑u = ↑u' ∧ HasDerivAt f u' z
Please generate a tactic in lean4 to solve the state. STATE: z : ℝ f : ℝ β†’ ℝ u : β„‚ hf : HasDerivAt (fun y => ↑(f y)) u z ⊒ βˆƒ u', u = ↑u' ∧ HasDerivAt f u' z TACTIC:
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 β†’ ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊒ βˆƒ u', ↑u = ↑u' ∧ HasDerivAt f u' z
case intro z : ℝ f : ℝ β†’ ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊒ HasDerivAt f u z
Please generate a tactic in lean4 to solve the state. STATE: case intro z : ℝ f : ℝ β†’ ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊒ βˆƒ u', ↑u = ↑u' ∧ HasDerivAt f u' z TACTIC:
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 β†’ ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊒ HasDerivAt f u z
case h.e'_7 z : ℝ f : ℝ β†’ ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊒ u = (reCLM.comp (smulRight 1 ↑u)) 1
Please generate a tactic in lean4 to solve the state. STATE: case intro z : ℝ f : ℝ β†’ ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊒ HasDerivAt f u z TACTIC:
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 β†’ ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊒ u = (reCLM.comp (smulRight 1 ↑u)) 1
no goals
Please generate a tactic in lean4 to solve the state. STATE: case h.e'_7 z : ℝ f : ℝ β†’ ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊒ u = (reCLM.comp (smulRight 1 ↑u)) 1 TACTIC:
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 β†’ ℝ u : β„‚ hf : HasDerivAt (fun y => ↑(f y)) u z ⊒ u.im = 0
z : ℝ f : ℝ β†’ ℝ u : β„‚ 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
Please generate a tactic in lean4 to solve the state. STATE: z : ℝ f : ℝ β†’ ℝ u : β„‚ hf : HasDerivAt (fun y => ↑(f y)) u z ⊒ u.im = 0 TACTIC:
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 β†’ ℝ u : β„‚ 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 β†’ ℝ u : β„‚ hf : HasDerivAt (fun y => ↑(f y)) u z H : 0 = (imCLM.comp (smulRight 1 u)) 1 ⊒ u.im = 0
Please generate a tactic in lean4 to solve the state. STATE: z : ℝ f : ℝ β†’ ℝ u : β„‚ 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 TACTIC:
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 β†’ ℝ u : β„‚ hf : HasDerivAt (fun y => ↑(f y)) u z H : 0 = (imCLM.comp (smulRight 1 u)) 1 ⊒ u.im = 0
no goals
Please generate a tactic in lean4 to solve the state. STATE: z : ℝ f : ℝ β†’ ℝ u : β„‚ hf : HasDerivAt (fun y => ↑(f y)) u z H : 0 = (imCLM.comp (smulRight 1 u)) 1 ⊒ u.im = 0 TACTIC:
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 β†’ ℝ ⊒ DifferentiableAt ℝ (fun y => ↑(f y)) z ↔ DifferentiableAt ℝ f z
z : ℝ f : ℝ β†’ ℝ H : DifferentiableAt ℝ (fun y => ↑(f y)) z ⊒ DifferentiableAt ℝ f z
Please generate a tactic in lean4 to solve the state. STATE: z : ℝ f : ℝ β†’ ℝ ⊒ DifferentiableAt ℝ (fun y => ↑(f y)) z ↔ DifferentiableAt ℝ f z TACTIC:
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
DifferentiableAt.ofReal_comp_iff
[136, 1]
[140, 40]
obtain ⟨u, _, huβ‚‚βŸ© := H.hasDerivAt.of_hasDerivAt_ofReal_comp
z : ℝ f : ℝ β†’ ℝ H : DifferentiableAt ℝ (fun y => ↑(f y)) z ⊒ DifferentiableAt ℝ f z
case intro.intro z : ℝ f : ℝ β†’ ℝ H : DifferentiableAt ℝ (fun y => ↑(f y)) z u : ℝ left✝ : deriv (fun y => ↑(f y)) z = ↑u huβ‚‚ : HasDerivAt (fun y => f y) u z ⊒ DifferentiableAt ℝ f z
Please generate a tactic in lean4 to solve the state. STATE: z : ℝ f : ℝ β†’ ℝ H : DifferentiableAt ℝ (fun y => ↑(f y)) z ⊒ DifferentiableAt ℝ f z TACTIC:
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
DifferentiableAt.ofReal_comp_iff
[136, 1]
[140, 40]
exact HasDerivAt.differentiableAt huβ‚‚
case intro.intro z : ℝ f : ℝ β†’ ℝ H : DifferentiableAt ℝ (fun y => ↑(f y)) z u : ℝ left✝ : deriv (fun y => ↑(f y)) z = ↑u huβ‚‚ : HasDerivAt (fun y => f y) u z ⊒ DifferentiableAt ℝ f z
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro z : ℝ f : ℝ β†’ ℝ H : DifferentiableAt ℝ (fun y => ↑(f y)) z u : ℝ left✝ : deriv (fun y => ↑(f y)) z = ↑u huβ‚‚ : HasDerivAt (fun y => f y) u z ⊒ DifferentiableAt ℝ f z TACTIC:
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 β†’ ℝ ⊒ deriv (fun y => ↑(f y)) z = ↑(deriv f z)
case pos z : ℝ f : ℝ β†’ ℝ hf : DifferentiableAt ℝ f z ⊒ deriv (fun y => ↑(f y)) z = ↑(deriv f z) case neg z : ℝ f : ℝ β†’ ℝ hf : Β¬DifferentiableAt ℝ f z ⊒ deriv (fun y => ↑(f y)) z = ↑(deriv f z)
Please generate a tactic in lean4 to solve the state. STATE: z : ℝ f : ℝ β†’ ℝ ⊒ deriv (fun y => ↑(f y)) z = ↑(deriv f z) TACTIC:
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 β†’ ℝ hf : DifferentiableAt ℝ f z ⊒ deriv (fun y => ↑(f y)) z = ↑(deriv f z)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case pos z : ℝ f : ℝ β†’ ℝ hf : DifferentiableAt ℝ f z ⊒ deriv (fun y => ↑(f y)) z = ↑(deriv f z) TACTIC:
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 β†’ ℝ hf : Β¬DifferentiableAt ℝ f z ⊒ deriv (fun y => ↑(f y)) z = ↑(deriv f z)
case neg z : ℝ f : ℝ β†’ ℝ hf : Β¬DifferentiableAt ℝ f z hf' : Β¬DifferentiableAt ℝ (fun y => ↑(f y)) z ⊒ deriv (fun y => ↑(f y)) z = ↑(deriv f z)
Please generate a tactic in lean4 to solve the state. STATE: case neg z : ℝ f : ℝ β†’ ℝ hf : Β¬DifferentiableAt ℝ f z ⊒ deriv (fun y => ↑(f y)) z = ↑(deriv f z) TACTIC:
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 β†’ ℝ hf : Β¬DifferentiableAt ℝ f z hf' : Β¬DifferentiableAt ℝ (fun y => ↑(f y)) z ⊒ deriv (fun y => ↑(f y)) z = ↑(deriv f z)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case neg z : ℝ f : ℝ β†’ ℝ hf : Β¬DifferentiableAt ℝ f z hf' : Β¬DifferentiableAt ℝ (fun y => ↑(f y)) z ⊒ deriv (fun y => ↑(f y)) z = ↑(deriv f z) TACTIC:
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real_on_ball
[159, 1]
[183, 8]
have Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), (x : β„‚) ∈ Metric.ball (c : β„‚) 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 : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) ⊒ βˆƒ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))
f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r ⊒ βˆƒ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) ⊒ βˆƒ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r)) TACTIC:
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 : β„‚) r) := taylorSeries_eq_on_ball' hz hf
f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r ⊒ βˆƒ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))
f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z ⊒ βˆƒ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r ⊒ βˆƒ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r)) TACTIC:
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real_on_ball
[159, 1]
[183, 8]
refine ⟨fun x ↦ βˆ‘' (n : β„•), (↑n !)⁻¹ * (D n) * (x - c) ^ n, fun x hx ↦ ?_, fun x hx ↦ ?_⟩
f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z ⊒ βˆƒ 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 : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑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 => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x case refine_2 f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑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 => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n) x
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z ⊒ βˆƒ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r)) TACTIC:
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 : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) ⊒ βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r
f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊒ ↑x ∈ Metric.ball (↑c) r
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) ⊒ βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r TACTIC:
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 : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊒ ↑x ∈ Metric.ball (↑c) r
f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊒ dist ↑x ↑c < r
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊒ ↑x ∈ Metric.ball (↑c) r TACTIC:
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 : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊒ dist ↑x ↑c < r
f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊒ x < c + r ∧ c - r < x
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊒ dist ↑x ↑c < r TACTIC:
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 : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊒ x < c + r ∧ c - r < x
no goals
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊒ x < c + r ∧ c - r < x TACTIC:
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 : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑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 => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x
case refine_1 f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑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 => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑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 => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x TACTIC:
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 : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑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 => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x
case refine_1 f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑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 => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n) x
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑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 => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x TACTIC:
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 : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑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 => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n) x
case refine_1 f : β„‚ β†’ β„‚ r c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑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 => βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (↑x - ↑c) ^ n) x ⊒ DifferentiableAt ℝ (fun x => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n) x
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑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 => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n) x TACTIC:
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 : β„‚ β†’ β„‚ r c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑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 => βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (↑x - ↑c) ^ n) x ⊒ DifferentiableAt ℝ (fun x => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n) x
case refine_1 f : β„‚ β†’ β„‚ r c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑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 => ↑(βˆ‘' (a : β„•), (↑a !)⁻¹ * D a * (x - c) ^ a)) x ⊒ DifferentiableAt ℝ (fun x => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n) x
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 f : β„‚ β†’ β„‚ r c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑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 => βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (↑x - ↑c) ^ n) x ⊒ DifferentiableAt ℝ (fun x => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n) x TACTIC:
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 : β„‚ β†’ β„‚ r c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑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 => ↑(βˆ‘' (a : β„•), (↑a !)⁻¹ * D a * (x - c) ^ a)) x ⊒ DifferentiableAt ℝ (fun x => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n) x
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 f : β„‚ β†’ β„‚ r c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑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 => ↑(βˆ‘' (a : β„•), (↑a !)⁻¹ * D a * (x - c) ^ a)) x ⊒ DifferentiableAt ℝ (fun x => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n) x TACTIC:
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 : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑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 => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n) x
case refine_2 f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊒ βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (↑x - ↑c) ^ n = βˆ‘' (a : β„•), ↑((↑a !)⁻¹ * D a * (x - c) ^ a)
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑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 => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n) x TACTIC:
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 : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊒ βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (↑x - ↑c) ^ n = βˆ‘' (a : β„•), ↑((↑a !)⁻¹ * D a * (x - c) ^ a)
case refine_2 f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊒ βˆ‘' (a : β„•), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a = βˆ‘' (a : β„•), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊒ βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (↑x - ↑c) ^ n = βˆ‘' (a : β„•), ↑((↑a !)⁻¹ * D a * (x - c) ^ a) TACTIC:
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 : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊒ βˆ‘' (a : β„•), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a = βˆ‘' (a : β„•), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊒ βˆ‘' (a : β„•), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a = βˆ‘' (a : β„•), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a TACTIC:
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real
[185, 1]
[201, 8]
have H (z : β„‚) := taylorSeries_eq_of_entire' c z hf
f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) ⊒ βˆƒ F, Differentiable ℝ F ∧ f ∘ ofReal' = ofReal' ∘ F
f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z ⊒ βˆƒ F, Differentiable ℝ F ∧ f ∘ ofReal' = ofReal' ∘ F
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) ⊒ βˆƒ F, Differentiable ℝ F ∧ f ∘ ofReal' = ofReal' ∘ F TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z ⊒ βˆƒ F, Differentiable ℝ F ∧ f ∘ ofReal' = ofReal' ∘ F
f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊒ βˆƒ F, Differentiable ℝ F ∧ f ∘ ofReal' = ofReal' ∘ F
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z ⊒ βˆƒ F, Differentiable ℝ F ∧ f ∘ ofReal' = ofReal' ∘ F TACTIC:
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real
[185, 1]
[201, 8]
refine ⟨fun x ↦ βˆ‘' (n : β„•), (↑n !)⁻¹ * (D n) * (x - c) ^ n, ?_, ?_⟩
f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊒ βˆƒ F, Differentiable ℝ F ∧ f ∘ ofReal' = ofReal' ∘ F
case refine_1 f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊒ Differentiable ℝ fun x => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n case refine_2 f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊒ f ∘ ofReal' = ofReal' ∘ fun x => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊒ βˆƒ F, Differentiable ℝ F ∧ f ∘ ofReal' = ofReal' ∘ F TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊒ Differentiable ℝ fun x => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n
case refine_1 f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z this : Differentiable ℝ fun x => f ↑x ⊒ Differentiable ℝ fun x => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊒ Differentiable ℝ fun x => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z this : Differentiable ℝ fun x => f ↑x ⊒ Differentiable ℝ fun x => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n
case refine_1 f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z this : Differentiable ℝ fun x => ↑(βˆ‘' (a : β„•), (↑a !)⁻¹ * D a * (x - c) ^ a) ⊒ Differentiable ℝ fun x => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z this : Differentiable ℝ fun x => f ↑x ⊒ Differentiable ℝ fun x => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z this : Differentiable ℝ fun x => ↑(βˆ‘' (a : β„•), (↑a !)⁻¹ * D a * (x - c) ^ a) ⊒ Differentiable ℝ fun x => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z this : Differentiable ℝ fun x => ↑(βˆ‘' (a : β„•), (↑a !)⁻¹ * D a * (x - c) ^ a) ⊒ Differentiable ℝ fun x => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊒ f ∘ ofReal' = ofReal' ∘ fun x => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n
case refine_2.h f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊒ (f ∘ ofReal') x = (ofReal' ∘ fun x => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n) x
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z ⊒ f ∘ ofReal' = ofReal' ∘ fun x => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊒ (f ∘ ofReal') x = (ofReal' ∘ fun x => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n) x
case refine_2.h f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊒ βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (↑x - ↑c) ^ n = βˆ‘' (a : β„•), ↑((↑a !)⁻¹ * D a * (x - c) ^ a)
Please generate a tactic in lean4 to solve the state. STATE: case refine_2.h f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊒ (f ∘ ofReal') x = (ofReal' ∘ fun x => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n) x TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊒ βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (↑x - ↑c) ^ n = βˆ‘' (a : β„•), ↑((↑a !)⁻¹ * D a * (x - c) ^ a)
case refine_2.h f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊒ βˆ‘' (a : β„•), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a = βˆ‘' (a : β„•), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a
Please generate a tactic in lean4 to solve the state. STATE: case refine_2.h f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊒ βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (↑x - ↑c) ^ n = βˆ‘' (a : β„•), ↑((↑a !)⁻¹ * D a * (x - c) ^ a) TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊒ βˆ‘' (a : β„•), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a = βˆ‘' (a : β„•), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_2.h f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f c : ℝ D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) H : βˆ€ (z : β„‚), βˆ‘' (n : β„•), (↑n !)⁻¹ * ↑(D n) * (z - ↑c) ^ n = f z x : ℝ ⊒ βˆ‘' (a : β„•), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a = βˆ‘' (a : β„•), (↑a !)⁻¹ * ↑(D a) * (↑x - ↑c) ^ a TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ z ⊒ 0 ≀ f z
f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ z H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (z - 0) ^ n = f z ⊒ 0 ≀ f z
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ z ⊒ 0 ≀ f z TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ z H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (z - 0) ^ n = f z ⊒ 0 ≀ f z
f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ z H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (z - 0) ^ n = f z hz' : z = ↑z.re ⊒ 0 ≀ f z
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ z H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (z - 0) ^ n = f z ⊒ 0 ≀ f z TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ z H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (z - 0) ^ n = f z hz' : z = ↑z.re ⊒ 0 ≀ f z
f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re ⊒ 0 ≀ f ↑z.re
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ z H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (z - 0) ^ n = f z hz' : z = ↑z.re ⊒ 0 ≀ f z TACTIC:
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.nonneg_of_iteratedDeriv_nonneg
[207, 1]
[223, 13]
obtain ⟨D, hD⟩ : βˆƒ D : β„• β†’ ℝ, βˆ€ n, 0 ≀ D n ∧ iteratedDeriv n f 0 = D n
f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re ⊒ 0 ≀ f ↑z.re
f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re ⊒ βˆƒ D, βˆ€ (n : β„•), 0 ≀ D n ∧ iteratedDeriv n f 0 = ↑(D n) case intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : β„• β†’ ℝ hD : βˆ€ (n : β„•), 0 ≀ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊒ 0 ≀ f ↑z.re
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re ⊒ 0 ≀ f ↑z.re TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : β„• β†’ ℝ hD : βˆ€ (n : β„•), 0 ≀ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊒ 0 ≀ f ↑z.re
case intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : β„• β†’ ℝ hD : βˆ€ (n : β„•), 0 ≀ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊒ 0 ≀ ↑(βˆ‘' (a : β„•), (↑a !)⁻¹ * D a * z.re ^ a)
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : β„• β†’ ℝ hD : βˆ€ (n : β„•), 0 ≀ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊒ 0 ≀ f ↑z.re TACTIC:
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.nonneg_of_iteratedDeriv_nonneg
[207, 1]
[223, 13]
norm_cast
case intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : β„• β†’ ℝ hD : βˆ€ (n : β„•), 0 ≀ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊒ 0 ≀ ↑(βˆ‘' (a : β„•), (↑a !)⁻¹ * D a * z.re ^ a)
case intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : β„• β†’ ℝ hD : βˆ€ (n : β„•), 0 ≀ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊒ 0 ≀ βˆ‘' (a : β„•), (↑a !)⁻¹ * D a * z.re ^ a
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : β„• β†’ ℝ hD : βˆ€ (n : β„•), 0 ≀ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊒ 0 ≀ ↑(βˆ‘' (a : β„•), (↑a !)⁻¹ * D a * z.re ^ a) TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : β„• β†’ ℝ hD : βˆ€ (n : β„•), 0 ≀ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊒ 0 ≀ βˆ‘' (a : β„•), (↑a !)⁻¹ * D a * z.re ^ a
case intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : β„• β†’ ℝ hD : βˆ€ (n : β„•), 0 ≀ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : β„• ⊒ 0 ≀ (↑n !)⁻¹ * D n * z.re ^ n
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : β„• β†’ ℝ hD : βˆ€ (n : β„•), 0 ≀ D n ∧ iteratedDeriv n f 0 = ↑(D n) ⊒ 0 ≀ βˆ‘' (a : β„•), (↑a !)⁻¹ * D a * z.re ^ a TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : β„• β†’ ℝ hD : βˆ€ (n : β„•), 0 ≀ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : β„• ⊒ 0 ≀ (↑n !)⁻¹ * D n * z.re ^ n
case intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : β„• β†’ ℝ hD : βˆ€ (n : β„•), 0 ≀ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : β„• hz : 0 ≀ z.re ⊒ 0 ≀ (↑n !)⁻¹ * D n * z.re ^ n
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : β„• β†’ ℝ hD : βˆ€ (n : β„•), 0 ≀ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : β„• ⊒ 0 ≀ (↑n !)⁻¹ * D n * z.re ^ n TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : β„• β†’ ℝ hD : βˆ€ (n : β„•), 0 ≀ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : β„• hz : 0 ≀ z.re ⊒ 0 ≀ (↑n !)⁻¹ * D n * z.re ^ n
case intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : β„• β†’ ℝ hD : βˆ€ (n : β„•), 0 ≀ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : β„• hz : 0 ≀ z.re this : 0 ≀ D n ⊒ 0 ≀ (↑n !)⁻¹ * D n * z.re ^ n
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : β„• β†’ ℝ hD : βˆ€ (n : β„•), 0 ≀ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : β„• hz : 0 ≀ z.re ⊒ 0 ≀ (↑n !)⁻¹ * D n * z.re ^ n TACTIC:
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.nonneg_of_iteratedDeriv_nonneg
[207, 1]
[223, 13]
positivity
case intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : β„• β†’ ℝ hD : βˆ€ (n : β„•), 0 ≀ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : β„• hz : 0 ≀ z.re this : 0 ≀ D n ⊒ 0 ≀ (↑n !)⁻¹ * D n * z.re ^ n
no goals
Please generate a tactic in lean4 to solve the state. STATE: case intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re D : β„• β†’ ℝ hD : βˆ€ (n : β„•), 0 ≀ D n ∧ iteratedDeriv n f 0 = ↑(D n) n : β„• hz : 0 ≀ z.re this : 0 ≀ D n ⊒ 0 ≀ (↑n !)⁻¹ * D n * z.re ^ n TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re ⊒ βˆƒ D, βˆ€ (n : β„•), 0 ≀ D n ∧ iteratedDeriv n f 0 = ↑(D n)
case refine_1 f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : β„• ⊒ 0 ≀ (fun n => (iteratedDeriv n f 0).re) n case refine_2 f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : β„• ⊒ iteratedDeriv n f 0 = ↑((fun n => (iteratedDeriv n f 0).re) n)
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re ⊒ βˆƒ D, βˆ€ (n : β„•), 0 ≀ D n ∧ iteratedDeriv n f 0 = ↑(D n) TACTIC:
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) β–Έ h n
case refine_1 f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : β„• ⊒ 0 ≀ (fun n => (iteratedDeriv n f 0).re) n
case refine_1 f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : β„• this : 0 ≀ ↑(iteratedDeriv n f 0).re ⊒ 0 ≀ (fun n => (iteratedDeriv n f 0).re) n
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : β„• ⊒ 0 ≀ (fun n => (iteratedDeriv n f 0).re) n TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : β„• this : 0 ≀ ↑(iteratedDeriv n f 0).re ⊒ 0 ≀ (fun n => (iteratedDeriv n f 0).re) n
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_1 f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : β„• this : 0 ≀ ↑(iteratedDeriv n f 0).re ⊒ 0 ≀ (fun n => (iteratedDeriv n f 0).re) n TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : β„• ⊒ iteratedDeriv n f 0 = ↑((fun n => (iteratedDeriv n f 0).re) n)
no goals
Please generate a tactic in lean4 to solve the state. STATE: case refine_2 f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 z : β„‚ hz : 0 ≀ ↑z.re H : βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f 0 * (↑z.re - 0) ^ n = f ↑z.re hz' : z = ↑z.re n : β„• ⊒ iteratedDeriv n f 0 = ↑((fun n => (iteratedDeriv n f 0).re) n) TACTIC:
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
let D : β„• β†’ ℝ := fun n ↦ (iteratedDeriv n f 0).re
f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 ⊒ MonotoneOn (f ∘ ofReal') (Set.Ici 0)
f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re ⊒ MonotoneOn (f ∘ ofReal') (Set.Ici 0)
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 ⊒ MonotoneOn (f ∘ ofReal') (Set.Ici 0) TACTIC:
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
have hD (n : β„•) : iteratedDeriv n f 0 = D n := by refine Complex.ext rfl ?_ simp only [ofReal_im] exact (le_def.mp (h n)).2.symm
f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re ⊒ MonotoneOn (f ∘ ofReal') (Set.Ici 0)
f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) ⊒ MonotoneOn (f ∘ ofReal') (Set.Ici 0)
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re ⊒ MonotoneOn (f ∘ ofReal') (Set.Ici 0) TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) ⊒ MonotoneOn (f ∘ ofReal') (Set.Ici 0)
case intro.intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) F : ℝ β†’ ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F ⊒ MonotoneOn (f ∘ ofReal') (Set.Ici 0)
Please generate a tactic in lean4 to solve the state. STATE: f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) ⊒ MonotoneOn (f ∘ ofReal') (Set.Ici 0) TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) F : ℝ β†’ ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F ⊒ MonotoneOn (f ∘ ofReal') (Set.Ici 0)
case intro.intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) F : ℝ β†’ ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F ⊒ MonotoneOn (ofReal' ∘ F) (Set.Ici 0)
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) F : ℝ β†’ ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F ⊒ MonotoneOn (f ∘ ofReal') (Set.Ici 0) TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) F : ℝ β†’ ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F ⊒ MonotoneOn (ofReal' ∘ F) (Set.Ici 0)
case intro.intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) F : ℝ β†’ ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) ⊒ 0 ≀ deriv F x
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) F : ℝ β†’ ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F ⊒ MonotoneOn (ofReal' ∘ F) (Set.Ici 0) TACTIC:
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
have hD' (n : β„•) : 0 ≀ iteratedDeriv n (deriv f) 0 := by rw [← iteratedDeriv_succ'] exact h (n + 1)
case intro.intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) F : ℝ β†’ 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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) F : ℝ β†’ ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) hD' : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n (deriv f) 0 ⊒ 0 ≀ deriv F x
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) F : ℝ β†’ ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) ⊒ 0 ≀ deriv F x TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) F : ℝ β†’ ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) hD' : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n (deriv f) 0 ⊒ 0 ≀ deriv F x
case intro.intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) F : ℝ β†’ ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) hD' : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n (deriv f) 0 hf' : Differentiable β„‚ (deriv f) ⊒ 0 ≀ deriv F x
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) F : ℝ β†’ ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) hD' : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n (deriv f) 0 ⊒ 0 ≀ deriv F x TACTIC:
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.monotoneOn_of_iteratedDeriv_nonneg
[227, 1]
[250, 17]
have hx : (0 : β„‚) ≀ x := by norm_cast simp only [Set.nonempty_Iio, interior_Ici', Set.mem_Ioi] at hx exact hx.le
case intro.intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) F : ℝ β†’ ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) hD' : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n (deriv f) 0 hf' : Differentiable β„‚ (deriv f) ⊒ 0 ≀ deriv F x
case intro.intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) F : ℝ β†’ ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n (deriv f) 0 hf' : Differentiable β„‚ (deriv f) hx : 0 ≀ ↑x ⊒ 0 ≀ deriv F x
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) F : ℝ β†’ ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx : x ∈ interior (Set.Ici 0) hD' : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n (deriv f) 0 hf' : Differentiable β„‚ (deriv f) ⊒ 0 ≀ deriv F x TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) F : ℝ β†’ ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n (deriv f) 0 hf' : Differentiable β„‚ (deriv f) hx : 0 ≀ ↑x ⊒ 0 ≀ deriv F x
case intro.intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) F : ℝ β†’ ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n (deriv f) 0 hf' : Differentiable β„‚ (deriv f) hx : 0 ≀ ↑x H : 0 ≀ deriv f ↑x ⊒ 0 ≀ deriv F x
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) F : ℝ β†’ ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n (deriv f) 0 hf' : Differentiable β„‚ (deriv f) hx : 0 ≀ ↑x ⊒ 0 ≀ deriv F x TACTIC:
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 : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) F : ℝ β†’ ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n (deriv f) 0 hf' : Differentiable β„‚ (deriv f) hx : 0 ≀ ↑x H : 0 ≀ deriv f ↑x ⊒ 0 ≀ deriv F x
case intro.intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) F : ℝ β†’ ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n (deriv f) 0 hf' : Differentiable β„‚ (deriv f) hx : 0 ≀ ↑x H : 0 ≀ deriv (fun x => f ↑x) x ⊒ 0 ≀ deriv F x
Please generate a tactic in lean4 to solve the state. STATE: case intro.intro f : β„‚ β†’ β„‚ hf : Differentiable β„‚ f h : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n f 0 D : β„• β†’ ℝ := fun n => (iteratedDeriv n f 0).re hD : βˆ€ (n : β„•), iteratedDeriv n f 0 = ↑(D n) F : ℝ β†’ ℝ hFd : Differentiable ℝ F hF : f ∘ ofReal' = ofReal' ∘ F x : ℝ hx✝ : x ∈ interior (Set.Ici 0) hD' : βˆ€ (n : β„•), 0 ≀ iteratedDeriv n (deriv f) 0 hf' : Differentiable β„‚ (deriv f) hx : 0 ≀ ↑x H : 0 ≀ deriv f ↑x ⊒ 0 ≀ deriv F x TACTIC: