internlm/Lean-Github · Datasets at Fast360

url stringclasses 147 values	commit stringclasses 147 values	file_path stringlengths 7 101	full_name stringlengths 1 94	start stringlengths 6 10	end stringlengths 6 11	tactic stringlengths 1 11.2k	state_before stringlengths 3 2.09M	state_after stringlengths 6 2.09M
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.aₚdₚ2_nonneg	[191, 1]	[193, 55]	fin_cases i <;> (dsimp [aₚ, dₚ]; norm_num)	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ i : Fin nₚ ⊢ 0 i ≤ (aₚ • dₚ ^ 2) i	no goals
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.aₚdₚ2_nonneg	[191, 1]	[193, 55]	dsimp [aₚ, dₚ]	case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 ⟨9, ⋯⟩ ≤ (aₚ • dₚ ^ 2) ⟨9, ⋯⟩	case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 ≤ 1 * ![1.9501, 1.2311, 1.6068, 1.4860, 1.8913, 1.7621, 1.4565, 1.0185, 1.8214, 1.4447] ⟨9, ⋯⟩ ^ 2
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.aₚdₚ2_nonneg	[191, 1]	[193, 55]	norm_num	case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 ≤ 1 * ![1.9501, 1.2311, 1.6068, 1.4860, 1.8913, 1.7621, 1.4565, 1.0185, 1.8214, 1.4447] ⟨9, ⋯⟩ ^ 2	no goals
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.simp_vec_fraction	[43, 1]	[47, 49]	have h_di_pos := h_d_pos i	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ h_d_pos : StrongLT 0 d s : Fin n → ℝ i : Fin n ⊢ d i / (d i / s i) = s i	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ h_d_pos : StrongLT 0 d s : Fin n → ℝ i : Fin n h_di_pos : 0 i < d i ⊢ d i / (d i / s i) = s i
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.simp_vec_fraction	[43, 1]	[47, 49]	simp at h_di_pos	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ h_d_pos : StrongLT 0 d s : Fin n → ℝ i : Fin n h_di_pos : 0 i < d i ⊢ d i / (d i / s i) = s i	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ h_d_pos : StrongLT 0 d s : Fin n → ℝ i : Fin n h_di_pos : 0 < d i ⊢ d i / (d i / s i) = s i
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.simp_vec_fraction	[43, 1]	[47, 49]	have h_di_nonzero : d i ≠ 0 := by linarith	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ h_d_pos : StrongLT 0 d s : Fin n → ℝ i : Fin n h_di_pos : 0 < d i ⊢ d i / (d i / s i) = s i	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ h_d_pos : StrongLT 0 d s : Fin n → ℝ i : Fin n h_di_pos : 0 < d i h_di_nonzero : d i ≠ 0 ⊢ d i / (d i / s i) = s i
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.simp_vec_fraction	[43, 1]	[47, 49]	rw [← div_mul, div_self h_di_nonzero, one_mul]	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ h_d_pos : StrongLT 0 d s : Fin n → ℝ i : Fin n h_di_pos : 0 < d i h_di_nonzero : d i ≠ 0 ⊢ d i / (d i / s i) = s i	no goals
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.simp_vec_fraction	[43, 1]	[47, 49]	linarith	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ h_d_pos : StrongLT 0 d s : Fin n → ℝ i : Fin n h_di_pos : 0 < d i ⊢ d i ≠ 0	no goals
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.fold_partial_sum	[49, 1]	[53, 22]	simp [Vec.cumsum]	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ hn : Fact (0 < n) t : Fin n → ℝ i : Fin n ⊢ ∑ j ∈ [[0, i]], t j = Vec.cumsum t i	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ hn : Fact (0 < n) t : Fin n → ℝ i : Fin n ⊢ ∑ j ∈ [[0, i]], t j = if h : 0 < n then ∑ j ∈ [[⟨0, h⟩, i]], t j else 0
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.fold_partial_sum	[49, 1]	[53, 22]	split_ifs	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ hn : Fact (0 < n) t : Fin n → ℝ i : Fin n ⊢ ∑ j ∈ [[0, i]], t j = if h : 0 < n then ∑ j ∈ [[⟨0, h⟩, i]], t j else 0	case pos n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ hn : Fact (0 < n) t : Fin n → ℝ i : Fin n h✝ : 0 < n ⊢ ∑ j ∈ [[0, i]], t j = ∑ j ∈ [[⟨0, h✝⟩, i]], t j case neg n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ hn : Fact (0 < n) t : Fin n → ℝ i : Fin n h✝ : ¬0 < n ⊢ ∑ j ∈ [[0, i]], t j = 0
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.fold_partial_sum	[49, 1]	[53, 22]	rfl	case pos n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ hn : Fact (0 < n) t : Fin n → ℝ i : Fin n h✝ : 0 < n ⊢ ∑ j ∈ [[0, i]], t j = ∑ j ∈ [[⟨0, h✝⟩, i]], t j	no goals
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.fold_partial_sum	[49, 1]	[53, 22]	linarith [hn.out]	case neg n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ hn : Fact (0 < n) t : Fin n → ℝ i : Fin n h✝ : ¬0 < n ⊢ ∑ j ∈ [[0, i]], t j = 0	no goals
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.nₚ_pos	[148, 1]	[148, 48]	unfold nₚ	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 < nₚ	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 < 10
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.nₚ_pos	[148, 1]	[148, 48]	norm_num	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 < 10	no goals
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.dₚ_pos	[154, 1]	[155, 50]	intro i	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ StrongLT 0 dₚ	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ i : Fin nₚ ⊢ 0 i < dₚ i
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.dₚ_pos	[154, 1]	[155, 50]	fin_cases i <;> (dsimp [dₚ]; norm_num)	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ i : Fin nₚ ⊢ 0 i < dₚ i	no goals
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.dₚ_pos	[154, 1]	[155, 50]	dsimp [dₚ]	case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 ⟨9, ⋯⟩ < dₚ ⟨9, ⋯⟩	case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 < ![1.9501, 1.2311, 1.6068, 1.4860, 1.8913, 1.7621, 1.4565, 1.0185, 1.8214, 1.4447] ⟨9, ⋯⟩
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.dₚ_pos	[154, 1]	[155, 50]	norm_num	case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 < ![1.9501, 1.2311, 1.6068, 1.4860, 1.8913, 1.7621, 1.4565, 1.0185, 1.8214, 1.4447] ⟨9, ⋯⟩	no goals
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.sminₚ_pos	[173, 1]	[174, 36]	intro i	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ StrongLT 0 sminₚ	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ i : Fin nₚ ⊢ 0 i < sminₚ i
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.sminₚ_pos	[173, 1]	[174, 36]	fin_cases i <;> norm_num	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ i : Fin nₚ ⊢ 0 i < sminₚ i	no goals
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.sminₚ_le_smaxₚ	[179, 1]	[180, 60]	intro i	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ sminₚ ≤ smaxₚ	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ i : Fin nₚ ⊢ sminₚ i ≤ smaxₚ i
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.sminₚ_le_smaxₚ	[179, 1]	[180, 60]	fin_cases i <;> (dsimp [sminₚ, smaxₚ]; norm_num)	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ i : Fin nₚ ⊢ sminₚ i ≤ smaxₚ i	no goals
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.sminₚ_le_smaxₚ	[179, 1]	[180, 60]	dsimp [sminₚ, smaxₚ]	case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ sminₚ ⟨9, ⋯⟩ ≤ smaxₚ ⟨9, ⋯⟩	case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ ![0.7828, 0.6235, 0.7155, 0.5340, 0.6329, 0.4259, 0.7798, 0.9604, 0.7298, 0.8405] ⟨9, ⋯⟩ ≤ ![1.9624, 1.6036, 1.6439, 1.5641, 1.7194, 1.9090, 1.3193, 1.3366, 1.9470, 2.8803] ⟨9, ⋯⟩
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.sminₚ_le_smaxₚ	[179, 1]	[180, 60]	norm_num	case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ ![0.7828, 0.6235, 0.7155, 0.5340, 0.6329, 0.4259, 0.7798, 0.9604, 0.7298, 0.8405] ⟨9, ⋯⟩ ≤ ![1.9624, 1.6036, 1.6439, 1.5641, 1.7194, 1.9090, 1.3193, 1.3366, 1.9470, 2.8803] ⟨9, ⋯⟩	no goals
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.aₚ_nonneg	[188, 1]	[189, 51]	unfold aₚ	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 ≤ aₚ	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 ≤ 1
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.aₚ_nonneg	[188, 1]	[189, 51]	norm_num	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 ≤ 1	no goals
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.aₚdₚ2_nonneg	[191, 1]	[193, 55]	intros i	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 ≤ aₚ • dₚ ^ 2	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ i : Fin nₚ ⊢ 0 i ≤ (aₚ • dₚ ^ 2) i
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.aₚdₚ2_nonneg	[191, 1]	[193, 55]	fin_cases i <;> (dsimp [aₚ, dₚ]; norm_num)	n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ i : Fin nₚ ⊢ 0 i ≤ (aₚ • dₚ ^ 2) i	no goals
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.aₚdₚ2_nonneg	[191, 1]	[193, 55]	dsimp [aₚ, dₚ]	case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 ⟨9, ⋯⟩ ≤ (aₚ • dₚ ^ 2) ⟨9, ⋯⟩	case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 ≤ 1 * ![1.9501, 1.2311, 1.6068, 1.4860, 1.8913, 1.7621, 1.4565, 1.0185, 1.8214, 1.4447] ⟨9, ⋯⟩ ^ 2
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Examples/VehicleSpeedScheduling.lean	VehicleSpeedSched.aₚdₚ2_nonneg	[191, 1]	[193, 55]	norm_num	case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 ≤ 1 * ![1.9501, 1.2311, 1.6068, 1.4860, 1.8913, 1.7621, 1.4565, 1.0185, 1.8214, 1.4447] ⟨9, ⋯⟩ ^ 2	no goals
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Lib/Math/Analysis/InnerProductSpace/Spectrum.lean	LinearMap.spectral_theorem'	[15, 1]	[29, 78]	suffices hsuff : ∀ (w : EuclideanSpace 𝕜 (Fin n)), T (xs.repr.symm w) = xs.repr.symm (fun i => as i * w i) by simpa only [LinearIsometryEquiv.symm_apply_apply, LinearIsometryEquiv.apply_symm_apply] using congr_arg (fun (v : E) => (xs.repr) v i) (hsuff ((xs.repr) v))	𝕜 : Type u_2 inst✝⁴ : RCLike 𝕜 inst✝³ : DecidableEq 𝕜 E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : FiniteDimensional 𝕜 E n : ℕ hn : FiniteDimensional.finrank 𝕜 E = n T : E →ₗ[𝕜] E v : E i : Fin n xs : OrthonormalBasis (Fin n) 𝕜 E as : Fin n → ℝ hxs : ∀ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j) ⊢ xs.repr (T v) i = ↑(as i) * xs.repr v i	𝕜 : Type u_2 inst✝⁴ : RCLike 𝕜 inst✝³ : DecidableEq 𝕜 E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : FiniteDimensional 𝕜 E n : ℕ hn : FiniteDimensional.finrank 𝕜 E = n T : E →ₗ[𝕜] E v : E i : Fin n xs : OrthonormalBasis (Fin n) 𝕜 E as : Fin n → ℝ hxs : ∀ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j) ⊢ ∀ (w : EuclideanSpace 𝕜 (Fin n)), T (xs.repr.symm w) = xs.repr.symm fun i => ↑(as i) * w i
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Lib/Math/Analysis/InnerProductSpace/Spectrum.lean	LinearMap.spectral_theorem'	[15, 1]	[29, 78]	intros w	𝕜 : Type u_2 inst✝⁴ : RCLike 𝕜 inst✝³ : DecidableEq 𝕜 E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : FiniteDimensional 𝕜 E n : ℕ hn : FiniteDimensional.finrank 𝕜 E = n T : E →ₗ[𝕜] E v : E i : Fin n xs : OrthonormalBasis (Fin n) 𝕜 E as : Fin n → ℝ hxs : ∀ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j) ⊢ ∀ (w : EuclideanSpace 𝕜 (Fin n)), T (xs.repr.symm w) = xs.repr.symm fun i => ↑(as i) * w i	𝕜 : Type u_2 inst✝⁴ : RCLike 𝕜 inst✝³ : DecidableEq 𝕜 E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : FiniteDimensional 𝕜 E n : ℕ hn : FiniteDimensional.finrank 𝕜 E = n T : E →ₗ[𝕜] E v : E i : Fin n xs : OrthonormalBasis (Fin n) 𝕜 E as : Fin n → ℝ hxs : ∀ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j) w : EuclideanSpace 𝕜 (Fin n) ⊢ T (xs.repr.symm w) = xs.repr.symm fun i => ↑(as i) * w i
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Lib/Math/Analysis/InnerProductSpace/Spectrum.lean	LinearMap.spectral_theorem'	[15, 1]	[29, 78]	simp_rw [← OrthonormalBasis.sum_repr_symm, map_sum, LinearMap.map_smul, fun j => Module.End.mem_eigenspace_iff.mp (hxs j).1, smul_smul, mul_comm]	𝕜 : Type u_2 inst✝⁴ : RCLike 𝕜 inst✝³ : DecidableEq 𝕜 E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : FiniteDimensional 𝕜 E n : ℕ hn : FiniteDimensional.finrank 𝕜 E = n T : E →ₗ[𝕜] E v : E i : Fin n xs : OrthonormalBasis (Fin n) 𝕜 E as : Fin n → ℝ hxs : ∀ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j) w : EuclideanSpace 𝕜 (Fin n) ⊢ T (xs.repr.symm w) = xs.repr.symm fun i => ↑(as i) * w i	no goals
https://github.com/verified-optimization/CvxLean.git	c62c2f292c6420f31a12e738ebebdfed50f6f840	CvxLean/Lib/Math/Analysis/InnerProductSpace/Spectrum.lean	LinearMap.spectral_theorem'	[15, 1]	[29, 78]	simpa only [LinearIsometryEquiv.symm_apply_apply, LinearIsometryEquiv.apply_symm_apply] using congr_arg (fun (v : E) => (xs.repr) v i) (hsuff ((xs.repr) v))	𝕜 : Type u_2 inst✝⁴ : RCLike 𝕜 inst✝³ : DecidableEq 𝕜 E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : FiniteDimensional 𝕜 E n : ℕ hn : FiniteDimensional.finrank 𝕜 E = n T : E →ₗ[𝕜] E v : E i : Fin n xs : OrthonormalBasis (Fin n) 𝕜 E as : Fin n → ℝ hxs : ∀ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j) hsuff : ∀ (w : EuclideanSpace 𝕜 (Fin n)), T (xs.repr.symm w) = xs.repr.symm fun i => ↑(as i) * w i ⊢ xs.repr (T v) i = ↑(as i) * xs.repr v i	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	sum_primesBelow_eq_sum_range_indicator	[49, 1]	[58, 8]	convert (Finset.sum_indicator_subset f Finset.mem_of_mem_filter).symm using 2 with _ _ m hm	R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n : ℕ ⊢ ∑ p ∈ n.primesBelow, f p = ∑ m ∈ Finset.range n, {p \| p.Prime}.indicator f m	case h.e'_3.a R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n ⊢ {p \| p.Prime}.indicator f m = (↑(Finset.filter (fun p => p.Prime) (Finset.range n))).indicator f m
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	sum_primesBelow_eq_sum_range_indicator	[49, 1]	[58, 8]	simp only [Set.mem_setOf_eq, Finset.mem_range, Finset.coe_filter, not_and, Set.indicator_apply]	case h.e'_3.a R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n ⊢ {p \| p.Prime}.indicator f m = (↑(Finset.filter (fun p => p.Prime) (Finset.range n))).indicator f m	case h.e'_3.a R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n ⊢ (if m.Prime then f m else 0) = if m < n ∧ m.Prime then f m else 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	sum_primesBelow_eq_sum_range_indicator	[49, 1]	[58, 8]	split_ifs with h₁ h₂ h₃	case h.e'_3.a R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n ⊢ (if m.Prime then f m else 0) = if m < n ∧ m.Prime then f m else 0	case pos R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : m.Prime h₂ : m < n ∧ m.Prime ⊢ f m = f m case neg R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : m.Prime h₂ : ¬(m < n ∧ m.Prime) ⊢ f m = 0 case pos R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : ¬m.Prime h₃ : m < n ∧ m.Prime ⊢ 0 = f m case neg R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : ¬m.Prime h₃ : ¬(m < n ∧ m.Prime) ⊢ 0 = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	sum_primesBelow_eq_sum_range_indicator	[49, 1]	[58, 8]	rfl	case pos R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : m.Prime h₂ : m < n ∧ m.Prime ⊢ f m = f m	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	sum_primesBelow_eq_sum_range_indicator	[49, 1]	[58, 8]	exact (h₂ ⟨Finset.mem_range.mp hm, h₁⟩).elim	case neg R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : m.Prime h₂ : ¬(m < n ∧ m.Prime) ⊢ f m = 0	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	sum_primesBelow_eq_sum_range_indicator	[49, 1]	[58, 8]	exact (h₁ h₃.2).elim	case pos R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : ¬m.Prime h₃ : m < n ∧ m.Prime ⊢ 0 = f m	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	sum_primesBelow_eq_sum_range_indicator	[49, 1]	[58, 8]	rfl	case neg R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : ¬m.Prime h₃ : ¬(m < n ∧ m.Prime) ⊢ 0 = 0	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	tendsto_sum_primesBelow_tsum	[62, 1]	[69, 94]	rw [(show ∑' p : Nat.Primes, f p = ∑' p : {p : ℕ \| p.Prime}, f p from rfl)]	R : Type u_1 inst✝⁴ : AddCommGroup R inst✝³ : UniformSpace R inst✝² : UniformAddGroup R inst✝¹ : CompleteSpace R inst✝ : T2Space R f : ℕ → R hsum : Summable f ⊢ Tendsto (fun n => ∑ p ∈ n.primesBelow, f p) atTop (𝓝 (∑' (p : Nat.Primes), f ↑p))	R : Type u_1 inst✝⁴ : AddCommGroup R inst✝³ : UniformSpace R inst✝² : UniformAddGroup R inst✝¹ : CompleteSpace R inst✝ : T2Space R f : ℕ → R hsum : Summable f ⊢ Tendsto (fun n => ∑ p ∈ n.primesBelow, f p) atTop (𝓝 (∑' (p : ↑{p \| p.Prime}), f ↑p))
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	tendsto_sum_primesBelow_tsum	[62, 1]	[69, 94]	simp_rw [tsum_subtype, sum_primesBelow_eq_sum_range_indicator]	R : Type u_1 inst✝⁴ : AddCommGroup R inst✝³ : UniformSpace R inst✝² : UniformAddGroup R inst✝¹ : CompleteSpace R inst✝ : T2Space R f : ℕ → R hsum : Summable f ⊢ Tendsto (fun n => ∑ p ∈ n.primesBelow, f p) atTop (𝓝 (∑' (p : ↑{p \| p.Prime}), f ↑p))	R : Type u_1 inst✝⁴ : AddCommGroup R inst✝³ : UniformSpace R inst✝² : UniformAddGroup R inst✝¹ : CompleteSpace R inst✝ : T2Space R f : ℕ → R hsum : Summable f ⊢ Tendsto (fun n => ∑ m ∈ Finset.range n, {p \| p.Prime}.indicator (fun p => f p) m) atTop (𝓝 (∑' (x : ℕ), {p \| p.Prime}.indicator f x))
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	tendsto_sum_primesBelow_tsum	[62, 1]	[69, 94]	exact (Summable.hasSum_iff_tendsto_nat <\| hsum.indicator _).mp <\| (hsum.indicator _).hasSum	R : Type u_1 inst✝⁴ : AddCommGroup R inst✝³ : UniformSpace R inst✝² : UniformAddGroup R inst✝¹ : CompleteSpace R inst✝ : T2Space R f : ℕ → R hsum : Summable f ⊢ Tendsto (fun n => ∑ m ∈ Finset.range n, {p \| p.Prime}.indicator (fun p => f p) m) atTop (𝓝 (∑' (x : ℕ), {p \| p.Prime}.indicator f x))	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	Complex.exp_tsum_primes	[71, 1]	[77, 81]	simpa only [← exp_sum] using Tendsto.cexp <\| tendsto_sum_primesBelow_tsum hsum	f : ℕ → ℂ hsum : Summable f ⊢ Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (f p)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), f ↑p)))	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	Summable.neg_clog_one_sub	[82, 1]	[91, 51]	let g (z : ℂ) : ℂ := -log (1 - z)	α : Type u_1 f : α → ℂ hsum : Summable f ⊢ Summable fun n => -(1 - f n).log	α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log ⊢ Summable fun n => -(1 - f n).log
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	Summable.neg_clog_one_sub	[82, 1]	[91, 51]	have hg : DifferentiableAt ℂ g 0 := DifferentiableAt.neg <\| ((differentiableAt_const 1).sub differentiableAt_id').clog <\| by simp only [sub_zero, one_mem_slitPlane]	α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log ⊢ Summable fun n => -(1 - f n).log	α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log hg : DifferentiableAt ℂ g 0 ⊢ Summable fun n => -(1 - f n).log
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	Summable.neg_clog_one_sub	[82, 1]	[91, 51]	have : g =O[𝓝 0] id := by simpa only [g, sub_zero, log_one, neg_zero] using DifferentiableAt.isBigO_sub hg	α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log hg : DifferentiableAt ℂ g 0 ⊢ Summable fun n => -(1 - f n).log	α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log hg : DifferentiableAt ℂ g 0 this : g =O[𝓝 0] id ⊢ Summable fun n => -(1 - f n).log
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	Summable.neg_clog_one_sub	[82, 1]	[91, 51]	exact Asymptotics.IsBigO.comp_summable this hsum	α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log hg : DifferentiableAt ℂ g 0 this : g =O[𝓝 0] id ⊢ Summable fun n => -(1 - f n).log	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	Summable.neg_clog_one_sub	[82, 1]	[91, 51]	simp only [sub_zero, one_mem_slitPlane]	α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log ⊢ 1 - 0 ∈ slitPlane	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	Summable.neg_clog_one_sub	[82, 1]	[91, 51]	simpa only [g, sub_zero, log_one, neg_zero] using DifferentiableAt.isBigO_sub hg	α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log hg : DifferentiableAt ℂ g 0 ⊢ g =O[𝓝 0] id	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	EulerProduct.exp_sum_primes_log_eq_tsum	[96, 1]	[107, 77]	have hs {p : ℕ} (hp : 1 < p) : ‖f p‖ < 1 := hsum.of_norm.norm_lt_one (f := f.toMonoidHom) hp	f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n	f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	EulerProduct.exp_sum_primes_log_eq_tsum	[96, 1]	[107, 77]	have H := Complex.exp_tsum_primes hsum.of_norm.neg_clog_one_sub	f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n	f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	EulerProduct.exp_sum_primes_log_eq_tsum	[96, 1]	[107, 77]	have help (n : ℕ) : n.primesBelow.prod (fun p ↦ cexp (-log (1 - f p))) = n.primesBelow.prod fun p ↦ (1 - f p)⁻¹ := by refine Finset.prod_congr rfl (fun p hp ↦ ?_) rw [exp_neg, exp_log ?_] rw [ne_eq, sub_eq_zero, ← ne_eq] exact fun h ↦ (norm_one (α := ℂ) ▸ h.symm ▸ hs (Nat.prime_of_mem_primesBelow hp).one_lt).false	f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n	f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) help : ∀ (n : ℕ), ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log) = ∏ p ∈ n.primesBelow, (1 - f p)⁻¹ ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	EulerProduct.exp_sum_primes_log_eq_tsum	[96, 1]	[107, 77]	simp_rw [help] at H	f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) help : ∀ (n : ℕ), ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log) = ∏ p ∈ n.primesBelow, (1 - f p)⁻¹ ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n	f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 help : ∀ (n : ℕ), ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log) = ∏ p ∈ n.primesBelow, (1 - f p)⁻¹ H : Tendsto (fun n => ∏ p ∈ n.primesBelow, (1 - f p)⁻¹) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	EulerProduct.exp_sum_primes_log_eq_tsum	[96, 1]	[107, 77]	exact tendsto_nhds_unique H <\| eulerProduct_completely_multiplicative hsum	f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 help : ∀ (n : ℕ), ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log) = ∏ p ∈ n.primesBelow, (1 - f p)⁻¹ H : Tendsto (fun n => ∏ p ∈ n.primesBelow, (1 - f p)⁻¹) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	EulerProduct.exp_sum_primes_log_eq_tsum	[96, 1]	[107, 77]	refine Finset.prod_congr rfl (fun p hp ↦ ?_)	f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) n : ℕ ⊢ ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log) = ∏ p ∈ n.primesBelow, (1 - f p)⁻¹	f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) n p : ℕ hp : p ∈ n.primesBelow ⊢ cexp (-(1 - f p).log) = (1 - f p)⁻¹
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	EulerProduct.exp_sum_primes_log_eq_tsum	[96, 1]	[107, 77]	rw [exp_neg, exp_log ?_]	f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) n p : ℕ hp : p ∈ n.primesBelow ⊢ cexp (-(1 - f p).log) = (1 - f p)⁻¹	f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) n p : ℕ hp : p ∈ n.primesBelow ⊢ 1 - f p ≠ 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	EulerProduct.exp_sum_primes_log_eq_tsum	[96, 1]	[107, 77]	rw [ne_eq, sub_eq_zero, ← ne_eq]	f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) n p : ℕ hp : p ∈ n.primesBelow ⊢ 1 - f p ≠ 0	f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) n p : ℕ hp : p ∈ n.primesBelow ⊢ 1 ≠ f p
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Logarithm.lean	EulerProduct.exp_sum_primes_log_eq_tsum	[96, 1]	[107, 77]	exact fun h ↦ (norm_one (α := ℂ) ▸ h.symm ▸ hs (Nat.prime_of_mem_primesBelow hp).one_lt).false	f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) n p : ℕ hp : p ∈ n.primesBelow ⊢ 1 ≠ f p	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Asymptotics.isBigO_mul_iff_isBigO_div	[31, 1]	[39, 36]	rw [isBigO_iff', isBigO_iff']	α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 ⊢ (fun x => f x * g x) =O[l] h ↔ g =O[l] fun x => h x / f x	α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 ⊢ (∃ c > 0, ∀ᶠ (x : α) in l, ‖f x * g x‖ ≤ c * ‖h x‖) ↔ ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Asymptotics.isBigO_mul_iff_isBigO_div	[31, 1]	[39, 36]	refine ⟨fun ⟨c, hc, H⟩ ↦ ⟨c, hc, ?_⟩, fun ⟨c, hc, H⟩ ↦ ⟨c, hc, ?_⟩⟩ <;> { refine H.congr <\| Eventually.mp hf <\| eventually_of_forall fun x hx ↦ ?_ rw [norm_mul, norm_div, ← mul_div_assoc, mul_comm] have hx' : ‖f x‖ > 0 := norm_pos_iff.mpr hx rw [le_div_iff hx', mul_comm] }	α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 ⊢ (∃ c > 0, ∀ᶠ (x : α) in l, ‖f x * g x‖ ≤ c * ‖h x‖) ↔ ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Asymptotics.isBigO_mul_iff_isBigO_div	[31, 1]	[39, 36]	refine H.congr <\| Eventually.mp hf <\| eventually_of_forall fun x hx ↦ ?_	case refine_2 α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ c : ℝ hc : c > 0 H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ ⊢ ∀ᶠ (x : α) in l, ‖f x * g x‖ ≤ c * ‖h x‖	case refine_2 α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ c : ℝ hc : c > 0 H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ x : α hx : f x ≠ 0 ⊢ ‖g x‖ ≤ c * ‖h x / f x‖ ↔ ‖f x * g x‖ ≤ c * ‖h x‖
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Asymptotics.isBigO_mul_iff_isBigO_div	[31, 1]	[39, 36]	rw [norm_mul, norm_div, ← mul_div_assoc, mul_comm]	case refine_2 α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ c : ℝ hc : c > 0 H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ x : α hx : f x ≠ 0 ⊢ ‖g x‖ ≤ c * ‖h x / f x‖ ↔ ‖f x * g x‖ ≤ c * ‖h x‖	case refine_2 α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ c : ℝ hc : c > 0 H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ x : α hx : f x ≠ 0 ⊢ ‖g x‖ ≤ ‖h x‖ * c / ‖f x‖ ↔ ‖f x‖ * ‖g x‖ ≤ ‖h x‖ * c
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Asymptotics.isBigO_mul_iff_isBigO_div	[31, 1]	[39, 36]	have hx' : ‖f x‖ > 0 := norm_pos_iff.mpr hx	case refine_2 α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ c : ℝ hc : c > 0 H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ x : α hx : f x ≠ 0 ⊢ ‖g x‖ ≤ ‖h x‖ * c / ‖f x‖ ↔ ‖f x‖ * ‖g x‖ ≤ ‖h x‖ * c	case refine_2 α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ c : ℝ hc : c > 0 H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ x : α hx : f x ≠ 0 hx' : ‖f x‖ > 0 ⊢ ‖g x‖ ≤ ‖h x‖ * c / ‖f x‖ ↔ ‖f x‖ * ‖g x‖ ≤ ‖h x‖ * c
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Asymptotics.isBigO_mul_iff_isBigO_div	[31, 1]	[39, 36]	rw [le_div_iff hx', mul_comm]	case refine_2 α : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter α f g h : α → F hf : ∀ᶠ (x : α) in l, f x ≠ 0 x✝ : ∃ c > 0, ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ c : ℝ hc : c > 0 H : ∀ᶠ (x : α) in l, ‖g x‖ ≤ c * ‖h x / f x‖ x : α hx : f x ≠ 0 hx' : ‖f x‖ > 0 ⊢ ‖g x‖ ≤ ‖h x‖ * c / ‖f x‖ ↔ ‖f x‖ * ‖g x‖ ≤ ‖h x‖ * c	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	DifferentiableAt.isBigO_of_eq_zero	[50, 1]	[54, 73]	rw [← zero_add z] at hf	f : ℂ → ℂ z : ℂ hf : DifferentiableAt ℂ f z hz : f z = 0 ⊢ (fun w => f (w + z)) =O[𝓝 0] id	f : ℂ → ℂ z : ℂ hf : DifferentiableAt ℂ f (0 + z) hz : f z = 0 ⊢ (fun w => f (w + z)) =O[𝓝 0] id
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	DifferentiableAt.isBigO_of_eq_zero	[50, 1]	[54, 73]	simpa only [zero_add, hz, sub_zero] using (hf.hasDerivAt.comp_add_const 0 z).differentiableAt.isBigO_sub	f : ℂ → ℂ z : ℂ hf : DifferentiableAt ℂ f (0 + z) hz : f z = 0 ⊢ (fun w => f (w + z)) =O[𝓝 0] id	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	ContinuousAt.isBigO	[56, 1]	[70, 46]	rw [isBigO_iff']	f : ℂ → ℂ z : ℂ hf : ContinuousAt f z ⊢ (fun w => f (w + z)) =O[𝓝 0] fun x => 1	f : ℂ → ℂ z : ℂ hf : ContinuousAt f z ⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	ContinuousAt.isBigO	[56, 1]	[70, 46]	simp_rw [Metric.continuousAt_iff', dist_eq_norm_sub, zero_add] at hf	f : ℂ → ℂ z : ℂ hf : ContinuousAt (fun w => f (w + z)) 0 ⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖	f : ℂ → ℂ z : ℂ hf : ∀ ε > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < ε ⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	ContinuousAt.isBigO	[56, 1]	[70, 46]	specialize hf 1 zero_lt_one	f : ℂ → ℂ z : ℂ hf : ∀ ε > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < ε ⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖	f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 ⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	ContinuousAt.isBigO	[56, 1]	[70, 46]	refine ⟨‖f z‖ + 1, by positivity, ?_⟩	f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 ⊢ ∃ c > 0, ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ c * ‖1‖	f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 ⊢ ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ (‖f z‖ + 1) * ‖1‖
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	ContinuousAt.isBigO	[56, 1]	[70, 46]	refine Eventually.mp hf <\| eventually_of_forall fun w hw ↦ le_of_lt ?_	f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 ⊢ ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z)‖ ≤ (‖f z‖ + 1) * ‖1‖	f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 w : ℂ hw : ‖f (w + z) - f z‖ < 1 ⊢ ‖f (w + z)‖ < (‖f z‖ + 1) * ‖1‖
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	ContinuousAt.isBigO	[56, 1]	[70, 46]	calc ‖f (w + z)‖ _ ≤ ‖f z‖ + ‖f (w + z) - f z‖ := norm_le_insert' .. _ < ‖f z‖ + 1 := add_lt_add_left hw _ _ = _ := by simp only [norm_one, mul_one]	f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 w : ℂ hw : ‖f (w + z) - f z‖ < 1 ⊢ ‖f (w + z)‖ < (‖f z‖ + 1) * ‖1‖	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	ContinuousAt.isBigO	[56, 1]	[70, 46]	convert (Homeomorph.comp_continuousAt_iff' (Homeomorph.addLeft (-z)) _ z).mp ?_	f : ℂ → ℂ z : ℂ hf : ContinuousAt f z ⊢ ContinuousAt (fun w => f (w + z)) 0	case h.e'_1 f : ℂ → ℂ z : ℂ hf : ContinuousAt f z ⊢ 0 = (Homeomorph.addLeft (-z)) z case convert_4 f : ℂ → ℂ z : ℂ hf : ContinuousAt f z ⊢ ContinuousAt ((fun w => f (w + z)) ∘ ⇑(Homeomorph.addLeft (-z))) z
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	ContinuousAt.isBigO	[56, 1]	[70, 46]	simp	case h.e'_1 f : ℂ → ℂ z : ℂ hf : ContinuousAt f z ⊢ 0 = (Homeomorph.addLeft (-z)) z	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	ContinuousAt.isBigO	[56, 1]	[70, 46]	simp [Function.comp_def, hf]	case convert_4 f : ℂ → ℂ z : ℂ hf : ContinuousAt f z ⊢ ContinuousAt ((fun w => f (w + z)) ∘ ⇑(Homeomorph.addLeft (-z))) z	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	ContinuousAt.isBigO	[56, 1]	[70, 46]	positivity	f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 ⊢ ‖f z‖ + 1 > 0	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	ContinuousAt.isBigO	[56, 1]	[70, 46]	simp only [norm_one, mul_one]	f : ℂ → ℂ z : ℂ hf : ∀ᶠ (x : ℂ) in 𝓝 0, ‖f (x + z) - f z‖ < 1 w : ℂ hw : ‖f (w + z) - f z‖ < 1 ⊢ ‖f z‖ + 1 = (‖f z‖ + 1) * ‖1‖	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	HasDerivAt.of_hasDerivAt_ofReal_comp	[125, 1]	[134, 80]	lift u to ℝ	z : ℝ f : ℝ → ℝ u : ℂ hf : HasDerivAt (fun y => ↑(f y)) u z ⊢ ∃ u', u = ↑u' ∧ HasDerivAt f u' z	z : ℝ f : ℝ → ℝ u : ℂ hf : HasDerivAt (fun y => ↑(f y)) u z ⊢ u.im = 0 case intro z : ℝ f : ℝ → ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊢ ∃ u', ↑u = ↑u' ∧ HasDerivAt f u' z
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	HasDerivAt.of_hasDerivAt_ofReal_comp	[125, 1]	[134, 80]	refine ⟨u, rfl, ?_⟩	case intro z : ℝ f : ℝ → ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊢ ∃ u', ↑u = ↑u' ∧ HasDerivAt f u' z	case intro z : ℝ f : ℝ → ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊢ HasDerivAt f u z
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	HasDerivAt.of_hasDerivAt_ofReal_comp	[125, 1]	[134, 80]	convert (reCLM.hasFDerivAt.comp z hf.hasFDerivAt).hasDerivAt	case intro z : ℝ f : ℝ → ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊢ HasDerivAt f u z	case h.e'_7 z : ℝ f : ℝ → ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊢ u = (reCLM.comp (smulRight 1 ↑u)) 1
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	HasDerivAt.of_hasDerivAt_ofReal_comp	[125, 1]	[134, 80]	rw [comp_apply, smulRight_apply, one_apply, one_smul, reCLM_apply, ofReal_re]	case h.e'_7 z : ℝ f : ℝ → ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊢ u = (reCLM.comp (smulRight 1 ↑u)) 1	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	HasDerivAt.of_hasDerivAt_ofReal_comp	[125, 1]	[134, 80]	have H := (imCLM.hasFDerivAt.comp z hf.hasFDerivAt).hasDerivAt.deriv	z : ℝ f : ℝ → ℝ u : ℂ hf : HasDerivAt (fun y => ↑(f y)) u z ⊢ u.im = 0	z : ℝ f : ℝ → ℝ u : ℂ hf : HasDerivAt (fun y => ↑(f y)) u z H : _root_.deriv (⇑imCLM ∘ fun y => ↑(f y)) z = (imCLM.comp (smulRight 1 u)) 1 ⊢ u.im = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	HasDerivAt.of_hasDerivAt_ofReal_comp	[125, 1]	[134, 80]	simp only [Function.comp_def, imCLM_apply, ofReal_im, deriv_const] at H	z : ℝ f : ℝ → ℝ u : ℂ hf : HasDerivAt (fun y => ↑(f y)) u z H : _root_.deriv (⇑imCLM ∘ fun y => ↑(f y)) z = (imCLM.comp (smulRight 1 u)) 1 ⊢ u.im = 0	z : ℝ f : ℝ → ℝ u : ℂ hf : HasDerivAt (fun y => ↑(f y)) u z H : 0 = (imCLM.comp (smulRight 1 u)) 1 ⊢ u.im = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	HasDerivAt.of_hasDerivAt_ofReal_comp	[125, 1]	[134, 80]	rwa [eq_comm, comp_apply, imCLM_apply, smulRight_apply, one_apply, one_smul] at H	z : ℝ f : ℝ → ℝ u : ℂ hf : HasDerivAt (fun y => ↑(f y)) u z H : 0 = (imCLM.comp (smulRight 1 u)) 1 ⊢ u.im = 0	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	DifferentiableAt.ofReal_comp_iff	[136, 1]	[140, 40]	refine ⟨fun H ↦ ?_, ofReal_comp⟩	z : ℝ f : ℝ → ℝ ⊢ DifferentiableAt ℝ (fun y => ↑(f y)) z ↔ DifferentiableAt ℝ f z	z : ℝ f : ℝ → ℝ H : DifferentiableAt ℝ (fun y => ↑(f y)) z ⊢ DifferentiableAt ℝ f z
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	DifferentiableAt.ofReal_comp_iff	[136, 1]	[140, 40]	obtain ⟨u, _, hu₂⟩ := H.hasDerivAt.of_hasDerivAt_ofReal_comp	z : ℝ f : ℝ → ℝ H : DifferentiableAt ℝ (fun y => ↑(f y)) z ⊢ DifferentiableAt ℝ f z	case intro.intro z : ℝ f : ℝ → ℝ H : DifferentiableAt ℝ (fun y => ↑(f y)) z u : ℝ left✝ : deriv (fun y => ↑(f y)) z = ↑u hu₂ : HasDerivAt (fun y => f y) u z ⊢ DifferentiableAt ℝ f z
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	DifferentiableAt.ofReal_comp_iff	[136, 1]	[140, 40]	exact HasDerivAt.differentiableAt hu₂	case intro.intro z : ℝ f : ℝ → ℝ H : DifferentiableAt ℝ (fun y => ↑(f y)) z u : ℝ left✝ : deriv (fun y => ↑(f y)) z = ↑u hu₂ : HasDerivAt (fun y => f y) u z ⊢ DifferentiableAt ℝ f z	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	deriv.ofReal_comp	[146, 1]	[152, 27]	by_cases hf : DifferentiableAt ℝ f z	z : ℝ f : ℝ → ℝ ⊢ deriv (fun y => ↑(f y)) z = ↑(deriv f z)	case pos z : ℝ f : ℝ → ℝ hf : DifferentiableAt ℝ f z ⊢ deriv (fun y => ↑(f y)) z = ↑(deriv f z) case neg z : ℝ f : ℝ → ℝ hf : ¬DifferentiableAt ℝ f z ⊢ deriv (fun y => ↑(f y)) z = ↑(deriv f z)
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	deriv.ofReal_comp	[146, 1]	[152, 27]	exact hf.hasDerivAt.ofReal_comp.deriv	case pos z : ℝ f : ℝ → ℝ hf : DifferentiableAt ℝ f z ⊢ deriv (fun y => ↑(f y)) z = ↑(deriv f z)	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	deriv.ofReal_comp	[146, 1]	[152, 27]	have hf' := mt DifferentiableAt.ofReal_comp_iff.mp hf	case neg z : ℝ f : ℝ → ℝ hf : ¬DifferentiableAt ℝ f z ⊢ deriv (fun y => ↑(f y)) z = ↑(deriv f z)	case neg z : ℝ f : ℝ → ℝ hf : ¬DifferentiableAt ℝ f z hf' : ¬DifferentiableAt ℝ (fun y => ↑(f y)) z ⊢ deriv (fun y => ↑(f y)) z = ↑(deriv f z)
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	deriv.ofReal_comp	[146, 1]	[152, 27]	rw [deriv_zero_of_not_differentiableAt hf, deriv_zero_of_not_differentiableAt hf', Complex.ofReal_zero]	case neg z : ℝ f : ℝ → ℝ hf : ¬DifferentiableAt ℝ f z hf' : ¬DifferentiableAt ℝ (fun y => ↑(f y)) z ⊢ deriv (fun y => ↑(f y)) z = ↑(deriv f z)	no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real_on_ball	[159, 1]	[183, 8]	have Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), (x : ℂ) ∈ Metric.ball (c : ℂ) r := by intro x hx refine Metric.mem_ball.mpr ?_ rw [dist_eq, ← ofReal_sub, abs_ofReal, abs_sub_lt_iff, sub_lt_iff_lt_add', sub_lt_comm] exact and_comm.mpr hx	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) ⊢ ∃ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r ⊢ ∃ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real_on_ball	[159, 1]	[183, 8]	have H ⦃z : ℂ⦄ (hz : z ∈ Metric.ball (c : ℂ) r) := taylorSeries_eq_on_ball' hz hf	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r ⊢ ∃ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z ⊢ ∃ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real_on_ball	[159, 1]	[183, 8]	refine ⟨fun x ↦ ∑' (n : ℕ), (↑n !)⁻¹ * (D n) * (x - c) ^ n, fun x hx ↦ ?_, fun x hx ↦ ?_⟩	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z ⊢ ∃ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))	case refine_1 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ DifferentiableWithinAt ℝ (fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x case refine_2 f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) Hz : ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : ∀ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r → ∑' (n : ℕ), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ (f ∘ ofReal') x = (ofReal' ∘ fun x => ∑' (n : ℕ), (↑n !)⁻¹ * D n * (x - c) ^ n) x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real_on_ball	[159, 1]	[183, 8]	intro x hx	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) ⊢ ∀ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ ↑x ∈ Metric.ball (↑c) r
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real_on_ball	[159, 1]	[183, 8]	refine Metric.mem_ball.mpr ?_	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ ↑x ∈ Metric.ball (↑c) r	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ dist ↑x ↑c < r
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real_on_ball	[159, 1]	[183, 8]	rw [dist_eq, ← ofReal_sub, abs_ofReal, abs_sub_lt_iff, sub_lt_iff_lt_add', sub_lt_comm]	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ dist ↑x ↑c < r	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ x < c + r ∧ c - r < x
https://github.com/MichaelStollBayreuth/EulerProducts.git	21e07835d1a467b99b5c3c9390d61ae69404445d	EulerProducts/Auxiliary.lean	Complex.realValued_of_iteratedDeriv_real_on_ball	[159, 1]	[183, 8]	exact and_comm.mpr hx	f : ℂ → ℂ r c : ℝ hf : DifferentiableOn ℂ f (Metric.ball (↑c) r) D : ℕ → ℝ hd : ∀ (n : ℕ), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊢ x < c + r ∧ c - r < x	no goals

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.aₚdₚ2_nonneg

[191, 1]

[193, 55]

fin_cases i <;> (dsimp [aₚ, dₚ]; norm_num)

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ i : Fin nₚ ⊢ 0 i ≤ (aₚ • dₚ ^ 2) i

no goals

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.aₚdₚ2_nonneg

[191, 1]

[193, 55]

dsimp [aₚ, dₚ]

case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 ⟨9, ⋯⟩ ≤ (aₚ • dₚ ^ 2) ⟨9, ⋯⟩

case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 ≤ 1 * ![1.9501, 1.2311, 1.6068, 1.4860, 1.8913, 1.7621, 1.4565, 1.0185, 1.8214, 1.4447] ⟨9, ⋯⟩ ^ 2

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.aₚdₚ2_nonneg

[191, 1]

[193, 55]

norm_num

case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 ≤ 1 * ![1.9501, 1.2311, 1.6068, 1.4860, 1.8913, 1.7621, 1.4565, 1.0185, 1.8214, 1.4447] ⟨9, ⋯⟩ ^ 2

no goals

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.simp_vec_fraction

[43, 1]

[47, 49]

have h_di_pos := h_d_pos i

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ h_d_pos : StrongLT 0 d s : Fin n → ℝ i : Fin n ⊢ d i / (d i / s i) = s i

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ h_d_pos : StrongLT 0 d s : Fin n → ℝ i : Fin n h_di_pos : 0 i < d i ⊢ d i / (d i / s i) = s i

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.simp_vec_fraction

[43, 1]

[47, 49]

simp at h_di_pos

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ h_d_pos : StrongLT 0 d s : Fin n → ℝ i : Fin n h_di_pos : 0 i < d i ⊢ d i / (d i / s i) = s i

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ h_d_pos : StrongLT 0 d s : Fin n → ℝ i : Fin n h_di_pos : 0 < d i ⊢ d i / (d i / s i) = s i

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.simp_vec_fraction

[43, 1]

[47, 49]

have h_di_nonzero : d i ≠ 0 := by linarith

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ h_d_pos : StrongLT 0 d s : Fin n → ℝ i : Fin n h_di_pos : 0 < d i ⊢ d i / (d i / s i) = s i

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ h_d_pos : StrongLT 0 d s : Fin n → ℝ i : Fin n h_di_pos : 0 < d i h_di_nonzero : d i ≠ 0 ⊢ d i / (d i / s i) = s i

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.simp_vec_fraction

[43, 1]

[47, 49]

rw [← div_mul, div_self h_di_nonzero, one_mul]

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ h_d_pos : StrongLT 0 d s : Fin n → ℝ i : Fin n h_di_pos : 0 < d i h_di_nonzero : d i ≠ 0 ⊢ d i / (d i / s i) = s i

no goals

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.simp_vec_fraction

[43, 1]

[47, 49]

linarith

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ h_d_pos : StrongLT 0 d s : Fin n → ℝ i : Fin n h_di_pos : 0 < d i ⊢ d i ≠ 0

no goals

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.fold_partial_sum

[49, 1]

[53, 22]

simp [Vec.cumsum]

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ hn : Fact (0 < n) t : Fin n → ℝ i : Fin n ⊢ ∑ j ∈ [[0, i]], t j = Vec.cumsum t i

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ hn : Fact (0 < n) t : Fin n → ℝ i : Fin n ⊢ ∑ j ∈ [[0, i]], t j = if h : 0 < n then ∑ j ∈ [[⟨0, h⟩, i]], t j else 0

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.fold_partial_sum

[49, 1]

[53, 22]

split_ifs

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ hn : Fact (0 < n) t : Fin n → ℝ i : Fin n ⊢ ∑ j ∈ [[0, i]], t j = if h : 0 < n then ∑ j ∈ [[⟨0, h⟩, i]], t j else 0

case pos n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ hn : Fact (0 < n) t : Fin n → ℝ i : Fin n h✝ : 0 < n ⊢ ∑ j ∈ [[0, i]], t j = ∑ j ∈ [[⟨0, h✝⟩, i]], t j case neg n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ hn : Fact (0 < n) t : Fin n → ℝ i : Fin n h✝ : ¬0 < n ⊢ ∑ j ∈ [[0, i]], t j = 0

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.fold_partial_sum

[49, 1]

[53, 22]

rfl

case pos n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ hn : Fact (0 < n) t : Fin n → ℝ i : Fin n h✝ : 0 < n ⊢ ∑ j ∈ [[0, i]], t j = ∑ j ∈ [[⟨0, h✝⟩, i]], t j

no goals

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.fold_partial_sum

[49, 1]

[53, 22]

linarith [hn.out]

case neg n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ hn : Fact (0 < n) t : Fin n → ℝ i : Fin n h✝ : ¬0 < n ⊢ ∑ j ∈ [[0, i]], t j = 0

no goals

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.nₚ_pos

[148, 1]

[148, 48]

unfold nₚ

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 < nₚ

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 < 10

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.nₚ_pos

[148, 1]

[148, 48]

norm_num

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 < 10

no goals

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.dₚ_pos

[154, 1]

[155, 50]

intro i

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ StrongLT 0 dₚ

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ i : Fin nₚ ⊢ 0 i < dₚ i

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.dₚ_pos

[154, 1]

[155, 50]

fin_cases i <;> (dsimp [dₚ]; norm_num)

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ i : Fin nₚ ⊢ 0 i < dₚ i

no goals

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.dₚ_pos

[154, 1]

[155, 50]

dsimp [dₚ]

case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 ⟨9, ⋯⟩ < dₚ ⟨9, ⋯⟩

case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 < ![1.9501, 1.2311, 1.6068, 1.4860, 1.8913, 1.7621, 1.4565, 1.0185, 1.8214, 1.4447] ⟨9, ⋯⟩

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.dₚ_pos

[154, 1]

[155, 50]

norm_num

case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 < ![1.9501, 1.2311, 1.6068, 1.4860, 1.8913, 1.7621, 1.4565, 1.0185, 1.8214, 1.4447] ⟨9, ⋯⟩

no goals

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.sminₚ_pos

[173, 1]

[174, 36]

intro i

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ StrongLT 0 sminₚ

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ i : Fin nₚ ⊢ 0 i < sminₚ i

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.sminₚ_pos

[173, 1]

[174, 36]

fin_cases i <;> norm_num

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ i : Fin nₚ ⊢ 0 i < sminₚ i

no goals

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.sminₚ_le_smaxₚ

[179, 1]

[180, 60]

intro i

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ sminₚ ≤ smaxₚ

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ i : Fin nₚ ⊢ sminₚ i ≤ smaxₚ i

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.sminₚ_le_smaxₚ

[179, 1]

[180, 60]

fin_cases i <;> (dsimp [sminₚ, smaxₚ]; norm_num)

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ i : Fin nₚ ⊢ sminₚ i ≤ smaxₚ i

no goals

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.sminₚ_le_smaxₚ

[179, 1]

[180, 60]

dsimp [sminₚ, smaxₚ]

case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ sminₚ ⟨9, ⋯⟩ ≤ smaxₚ ⟨9, ⋯⟩

case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ ![0.7828, 0.6235, 0.7155, 0.5340, 0.6329, 0.4259, 0.7798, 0.9604, 0.7298, 0.8405] ⟨9, ⋯⟩ ≤ ![1.9624, 1.6036, 1.6439, 1.5641, 1.7194, 1.9090, 1.3193, 1.3366, 1.9470, 2.8803] ⟨9, ⋯⟩

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.sminₚ_le_smaxₚ

[179, 1]

[180, 60]

norm_num

case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ ![0.7828, 0.6235, 0.7155, 0.5340, 0.6329, 0.4259, 0.7798, 0.9604, 0.7298, 0.8405] ⟨9, ⋯⟩ ≤ ![1.9624, 1.6036, 1.6439, 1.5641, 1.7194, 1.9090, 1.3193, 1.3366, 1.9470, 2.8803] ⟨9, ⋯⟩

no goals

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.aₚ_nonneg

[188, 1]

[189, 51]

unfold aₚ

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 ≤ aₚ

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 ≤ 1

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.aₚ_nonneg

[188, 1]

[189, 51]

norm_num

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 ≤ 1

no goals

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.aₚdₚ2_nonneg

[191, 1]

[193, 55]

intros i

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 ≤ aₚ • dₚ ^ 2

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ i : Fin nₚ ⊢ 0 i ≤ (aₚ • dₚ ^ 2) i

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.aₚdₚ2_nonneg

[191, 1]

[193, 55]

fin_cases i <;> (dsimp [aₚ, dₚ]; norm_num)

n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ i : Fin nₚ ⊢ 0 i ≤ (aₚ • dₚ ^ 2) i

no goals

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.aₚdₚ2_nonneg

[191, 1]

[193, 55]

dsimp [aₚ, dₚ]

case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 ⟨9, ⋯⟩ ≤ (aₚ • dₚ ^ 2) ⟨9, ⋯⟩

case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 ≤ 1 * ![1.9501, 1.2311, 1.6068, 1.4860, 1.8913, 1.7621, 1.4565, 1.0185, 1.8214, 1.4447] ⟨9, ⋯⟩ ^ 2

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Examples/VehicleSpeedScheduling.lean

VehicleSpeedSched.aₚdₚ2_nonneg

[191, 1]

[193, 55]

norm_num

case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : ℕ d τmin τmax smin smax : Fin n → ℝ F : ℝ → ℝ ⊢ 0 ≤ 1 * ![1.9501, 1.2311, 1.6068, 1.4860, 1.8913, 1.7621, 1.4565, 1.0185, 1.8214, 1.4447] ⟨9, ⋯⟩ ^ 2

no goals

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Lib/Math/Analysis/InnerProductSpace/Spectrum.lean

LinearMap.spectral_theorem'

[15, 1]

[29, 78]

suffices hsuff : ∀ (w : EuclideanSpace 𝕜 (Fin n)), T (xs.repr.symm w) = xs.repr.symm (fun i => as i * w i) by simpa only [LinearIsometryEquiv.symm_apply_apply, LinearIsometryEquiv.apply_symm_apply] using congr_arg (fun (v : E) => (xs.repr) v i) (hsuff ((xs.repr) v))

𝕜 : Type u_2 inst✝⁴ : RCLike 𝕜 inst✝³ : DecidableEq 𝕜 E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : FiniteDimensional 𝕜 E n : ℕ hn : FiniteDimensional.finrank 𝕜 E = n T : E →ₗ[𝕜] E v : E i : Fin n xs : OrthonormalBasis (Fin n) 𝕜 E as : Fin n → ℝ hxs : ∀ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j) ⊢ xs.repr (T v) i = ↑(as i) * xs.repr v i

𝕜 : Type u_2 inst✝⁴ : RCLike 𝕜 inst✝³ : DecidableEq 𝕜 E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : FiniteDimensional 𝕜 E n : ℕ hn : FiniteDimensional.finrank 𝕜 E = n T : E →ₗ[𝕜] E v : E i : Fin n xs : OrthonormalBasis (Fin n) 𝕜 E as : Fin n → ℝ hxs : ∀ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j) ⊢ ∀ (w : EuclideanSpace 𝕜 (Fin n)), T (xs.repr.symm w) = xs.repr.symm fun i => ↑(as i) * w i

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Lib/Math/Analysis/InnerProductSpace/Spectrum.lean

LinearMap.spectral_theorem'

[15, 1]

[29, 78]

intros w

𝕜 : Type u_2 inst✝⁴ : RCLike 𝕜 inst✝³ : DecidableEq 𝕜 E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : FiniteDimensional 𝕜 E n : ℕ hn : FiniteDimensional.finrank 𝕜 E = n T : E →ₗ[𝕜] E v : E i : Fin n xs : OrthonormalBasis (Fin n) 𝕜 E as : Fin n → ℝ hxs : ∀ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j) ⊢ ∀ (w : EuclideanSpace 𝕜 (Fin n)), T (xs.repr.symm w) = xs.repr.symm fun i => ↑(as i) * w i

𝕜 : Type u_2 inst✝⁴ : RCLike 𝕜 inst✝³ : DecidableEq 𝕜 E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : FiniteDimensional 𝕜 E n : ℕ hn : FiniteDimensional.finrank 𝕜 E = n T : E →ₗ[𝕜] E v : E i : Fin n xs : OrthonormalBasis (Fin n) 𝕜 E as : Fin n → ℝ hxs : ∀ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j) w : EuclideanSpace 𝕜 (Fin n) ⊢ T (xs.repr.symm w) = xs.repr.symm fun i => ↑(as i) * w i

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Lib/Math/Analysis/InnerProductSpace/Spectrum.lean

LinearMap.spectral_theorem'

[15, 1]

[29, 78]

simp_rw [← OrthonormalBasis.sum_repr_symm, map_sum, LinearMap.map_smul, fun j => Module.End.mem_eigenspace_iff.mp (hxs j).1, smul_smul, mul_comm]

𝕜 : Type u_2 inst✝⁴ : RCLike 𝕜 inst✝³ : DecidableEq 𝕜 E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : FiniteDimensional 𝕜 E n : ℕ hn : FiniteDimensional.finrank 𝕜 E = n T : E →ₗ[𝕜] E v : E i : Fin n xs : OrthonormalBasis (Fin n) 𝕜 E as : Fin n → ℝ hxs : ∀ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j) w : EuclideanSpace 𝕜 (Fin n) ⊢ T (xs.repr.symm w) = xs.repr.symm fun i => ↑(as i) * w i

no goals

https://github.com/verified-optimization/CvxLean.git

c62c2f292c6420f31a12e738ebebdfed50f6f840

CvxLean/Lib/Math/Analysis/InnerProductSpace/Spectrum.lean

LinearMap.spectral_theorem'

[15, 1]

[29, 78]

simpa only [LinearIsometryEquiv.symm_apply_apply, LinearIsometryEquiv.apply_symm_apply] using congr_arg (fun (v : E) => (xs.repr) v i) (hsuff ((xs.repr) v))

𝕜 : Type u_2 inst✝⁴ : RCLike 𝕜 inst✝³ : DecidableEq 𝕜 E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace 𝕜 E inst✝ : FiniteDimensional 𝕜 E n : ℕ hn : FiniteDimensional.finrank 𝕜 E = n T : E →ₗ[𝕜] E v : E i : Fin n xs : OrthonormalBasis (Fin n) 𝕜 E as : Fin n → ℝ hxs : ∀ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j) hsuff : ∀ (w : EuclideanSpace 𝕜 (Fin n)), T (xs.repr.symm w) = xs.repr.symm fun i => ↑(as i) * w i ⊢ xs.repr (T v) i = ↑(as i) * xs.repr v i

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

sum_primesBelow_eq_sum_range_indicator

[49, 1]

[58, 8]

convert (Finset.sum_indicator_subset f Finset.mem_of_mem_filter).symm using 2 with _ _ m hm

R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n : ℕ ⊢ ∑ p ∈ n.primesBelow, f p = ∑ m ∈ Finset.range n, {p | p.Prime}.indicator f m

case h.e'_3.a R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n ⊢ {p | p.Prime}.indicator f m = (↑(Finset.filter (fun p => p.Prime) (Finset.range n))).indicator f m

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

sum_primesBelow_eq_sum_range_indicator

[49, 1]

[58, 8]

simp only [Set.mem_setOf_eq, Finset.mem_range, Finset.coe_filter, not_and, Set.indicator_apply]

case h.e'_3.a R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n ⊢ {p | p.Prime}.indicator f m = (↑(Finset.filter (fun p => p.Prime) (Finset.range n))).indicator f m

case h.e'_3.a R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n ⊢ (if m.Prime then f m else 0) = if m < n ∧ m.Prime then f m else 0

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

sum_primesBelow_eq_sum_range_indicator

[49, 1]

[58, 8]

split_ifs with h₁ h₂ h₃

case h.e'_3.a R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n ⊢ (if m.Prime then f m else 0) = if m < n ∧ m.Prime then f m else 0

case pos R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : m.Prime h₂ : m < n ∧ m.Prime ⊢ f m = f m case neg R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : m.Prime h₂ : ¬(m < n ∧ m.Prime) ⊢ f m = 0 case pos R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : ¬m.Prime h₃ : m < n ∧ m.Prime ⊢ 0 = f m case neg R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : ¬m.Prime h₃ : ¬(m < n ∧ m.Prime) ⊢ 0 = 0

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

sum_primesBelow_eq_sum_range_indicator

[49, 1]

[58, 8]

rfl

case pos R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : m.Prime h₂ : m < n ∧ m.Prime ⊢ f m = f m

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

sum_primesBelow_eq_sum_range_indicator

[49, 1]

[58, 8]

exact (h₂ ⟨Finset.mem_range.mp hm, h₁⟩).elim

case neg R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : m.Prime h₂ : ¬(m < n ∧ m.Prime) ⊢ f m = 0

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

sum_primesBelow_eq_sum_range_indicator

[49, 1]

[58, 8]

exact (h₁ h₃.2).elim

case pos R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : ¬m.Prime h₃ : m < n ∧ m.Prime ⊢ 0 = f m

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

sum_primesBelow_eq_sum_range_indicator

[49, 1]

[58, 8]

rfl

case neg R : Type u_1 inst✝ : AddCommMonoid R f : ℕ → R n m : ℕ hm : m ∈ Finset.range n h₁ : ¬m.Prime h₃ : ¬(m < n ∧ m.Prime) ⊢ 0 = 0

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

tendsto_sum_primesBelow_tsum

[62, 1]

[69, 94]

rw [(show ∑' p : Nat.Primes, f p = ∑' p : {p : ℕ | p.Prime}, f p from rfl)]

R : Type u_1 inst✝⁴ : AddCommGroup R inst✝³ : UniformSpace R inst✝² : UniformAddGroup R inst✝¹ : CompleteSpace R inst✝ : T2Space R f : ℕ → R hsum : Summable f ⊢ Tendsto (fun n => ∑ p ∈ n.primesBelow, f p) atTop (𝓝 (∑' (p : Nat.Primes), f ↑p))

R : Type u_1 inst✝⁴ : AddCommGroup R inst✝³ : UniformSpace R inst✝² : UniformAddGroup R inst✝¹ : CompleteSpace R inst✝ : T2Space R f : ℕ → R hsum : Summable f ⊢ Tendsto (fun n => ∑ p ∈ n.primesBelow, f p) atTop (𝓝 (∑' (p : ↑{p | p.Prime}), f ↑p))

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

tendsto_sum_primesBelow_tsum

[62, 1]

[69, 94]

simp_rw [tsum_subtype, sum_primesBelow_eq_sum_range_indicator]

R : Type u_1 inst✝⁴ : AddCommGroup R inst✝³ : UniformSpace R inst✝² : UniformAddGroup R inst✝¹ : CompleteSpace R inst✝ : T2Space R f : ℕ → R hsum : Summable f ⊢ Tendsto (fun n => ∑ p ∈ n.primesBelow, f p) atTop (𝓝 (∑' (p : ↑{p | p.Prime}), f ↑p))

R : Type u_1 inst✝⁴ : AddCommGroup R inst✝³ : UniformSpace R inst✝² : UniformAddGroup R inst✝¹ : CompleteSpace R inst✝ : T2Space R f : ℕ → R hsum : Summable f ⊢ Tendsto (fun n => ∑ m ∈ Finset.range n, {p | p.Prime}.indicator (fun p => f p) m) atTop (𝓝 (∑' (x : ℕ), {p | p.Prime}.indicator f x))

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

tendsto_sum_primesBelow_tsum

[62, 1]

[69, 94]

exact (Summable.hasSum_iff_tendsto_nat <| hsum.indicator _).mp <| (hsum.indicator _).hasSum

R : Type u_1 inst✝⁴ : AddCommGroup R inst✝³ : UniformSpace R inst✝² : UniformAddGroup R inst✝¹ : CompleteSpace R inst✝ : T2Space R f : ℕ → R hsum : Summable f ⊢ Tendsto (fun n => ∑ m ∈ Finset.range n, {p | p.Prime}.indicator (fun p => f p) m) atTop (𝓝 (∑' (x : ℕ), {p | p.Prime}.indicator f x))

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

Complex.exp_tsum_primes

[71, 1]

[77, 81]

simpa only [← exp_sum] using Tendsto.cexp <| tendsto_sum_primesBelow_tsum hsum

f : ℕ → ℂ hsum : Summable f ⊢ Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (f p)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), f ↑p)))

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

Summable.neg_clog_one_sub

[82, 1]

[91, 51]

let g (z : ℂ) : ℂ := -log (1 - z)

α : Type u_1 f : α → ℂ hsum : Summable f ⊢ Summable fun n => -(1 - f n).log

α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log ⊢ Summable fun n => -(1 - f n).log

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

Summable.neg_clog_one_sub

[82, 1]

[91, 51]

have hg : DifferentiableAt ℂ g 0 := DifferentiableAt.neg <| ((differentiableAt_const 1).sub differentiableAt_id').clog <| by simp only [sub_zero, one_mem_slitPlane]

α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log ⊢ Summable fun n => -(1 - f n).log

α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log hg : DifferentiableAt ℂ g 0 ⊢ Summable fun n => -(1 - f n).log

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

Summable.neg_clog_one_sub

[82, 1]

[91, 51]

have : g =O[𝓝 0] id := by simpa only [g, sub_zero, log_one, neg_zero] using DifferentiableAt.isBigO_sub hg

α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log hg : DifferentiableAt ℂ g 0 ⊢ Summable fun n => -(1 - f n).log

α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log hg : DifferentiableAt ℂ g 0 this : g =O[𝓝 0] id ⊢ Summable fun n => -(1 - f n).log

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

Summable.neg_clog_one_sub

[82, 1]

[91, 51]

exact Asymptotics.IsBigO.comp_summable this hsum

α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log hg : DifferentiableAt ℂ g 0 this : g =O[𝓝 0] id ⊢ Summable fun n => -(1 - f n).log

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

Summable.neg_clog_one_sub

[82, 1]

[91, 51]

simp only [sub_zero, one_mem_slitPlane]

α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log ⊢ 1 - 0 ∈ slitPlane

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

Summable.neg_clog_one_sub

[82, 1]

[91, 51]

simpa only [g, sub_zero, log_one, neg_zero] using DifferentiableAt.isBigO_sub hg

α : Type u_1 f : α → ℂ hsum : Summable f g : ℂ → ℂ := fun z => -(1 - z).log hg : DifferentiableAt ℂ g 0 ⊢ g =O[𝓝 0] id

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

EulerProduct.exp_sum_primes_log_eq_tsum

[96, 1]

[107, 77]

have hs {p : ℕ} (hp : 1 < p) : ‖f p‖ < 1 := hsum.of_norm.norm_lt_one (f := f.toMonoidHom) hp

f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n

f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

EulerProduct.exp_sum_primes_log_eq_tsum

[96, 1]

[107, 77]

have H := Complex.exp_tsum_primes hsum.of_norm.neg_clog_one_sub

f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 < p → ‖f p‖ < 1 ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n

f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

EulerProduct.exp_sum_primes_log_eq_tsum

[96, 1]

[107, 77]

have help (n : ℕ) : n.primesBelow.prod (fun p ↦ cexp (-log (1 - f p))) = n.primesBelow.prod fun p ↦ (1 - f p)⁻¹ := by refine Finset.prod_congr rfl (fun p hp ↦ ?_) rw [exp_neg, exp_log ?_] rw [ne_eq, sub_eq_zero, ← ne_eq] exact fun h ↦ (norm_one (α := ℂ) ▸ h.symm ▸ hs (Nat.prime_of_mem_primesBelow hp).one_lt).false

f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n

f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) help : ∀ (n : ℕ), ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log) = ∏ p ∈ n.primesBelow, (1 - f p)⁻¹ ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

EulerProduct.exp_sum_primes_log_eq_tsum

[96, 1]

[107, 77]

simp_rw [help] at H

f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) help : ∀ (n : ℕ), ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log) = ∏ p ∈ n.primesBelow, (1 - f p)⁻¹ ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n

f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 ∏ p ∈ n.primesBelow, (1 - f p)⁻¹) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

EulerProduct.exp_sum_primes_log_eq_tsum

[96, 1]

[107, 77]

exact tendsto_nhds_unique H <| eulerProduct_completely_multiplicative hsum

f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 ∏ p ∈ n.primesBelow, (1 - f p)⁻¹) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) ⊢ cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log) = ∑' (n : ℕ), f n

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

EulerProduct.exp_sum_primes_log_eq_tsum

[96, 1]

[107, 77]

refine Finset.prod_congr rfl (fun p hp ↦ ?_)

f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) n : ℕ ⊢ ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log) = ∏ p ∈ n.primesBelow, (1 - f p)⁻¹

f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) n p : ℕ hp : p ∈ n.primesBelow ⊢ cexp (-(1 - f p).log) = (1 - f p)⁻¹

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

EulerProduct.exp_sum_primes_log_eq_tsum

[96, 1]

[107, 77]

rw [exp_neg, exp_log ?_]

f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) n p : ℕ hp : p ∈ n.primesBelow ⊢ cexp (-(1 - f p).log) = (1 - f p)⁻¹

f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) n p : ℕ hp : p ∈ n.primesBelow ⊢ 1 - f p ≠ 0

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

EulerProduct.exp_sum_primes_log_eq_tsum

[96, 1]

[107, 77]

rw [ne_eq, sub_eq_zero, ← ne_eq]

f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) n p : ℕ hp : p ∈ n.primesBelow ⊢ 1 - f p ≠ 0

f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) n p : ℕ hp : p ∈ n.primesBelow ⊢ 1 ≠ f p

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Logarithm.lean

EulerProduct.exp_sum_primes_log_eq_tsum

[96, 1]

[107, 77]

exact fun h ↦ (norm_one (α := ℂ) ▸ h.symm ▸ hs (Nat.prime_of_mem_primesBelow hp).one_lt).false

f : ℕ →*₀ ℂ hsum : Summable fun x => ‖f x‖ hs : ∀ {p : ℕ}, 1 ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (∑' (p : Nat.Primes), -(1 - f ↑p).log))) n p : ℕ hp : p ∈ n.primesBelow ⊢ 1 ≠ f p

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Asymptotics.isBigO_mul_iff_isBigO_div

[31, 1]

[39, 36]

rw [isBigO_iff', isBigO_iff']

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Asymptotics.isBigO_mul_iff_isBigO_div

[31, 1]

[39, 36]

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

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

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Asymptotics.isBigO_mul_iff_isBigO_div

[31, 1]

[39, 36]

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

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Asymptotics.isBigO_mul_iff_isBigO_div

[31, 1]

[39, 36]

rw [norm_mul, norm_div, ← mul_div_assoc, mul_comm]

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Asymptotics.isBigO_mul_iff_isBigO_div

[31, 1]

[39, 36]

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

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Asymptotics.isBigO_mul_iff_isBigO_div

[31, 1]

[39, 36]

rw [le_div_iff hx', mul_comm]

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

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

DifferentiableAt.isBigO_of_eq_zero

[50, 1]

[54, 73]

rw [← zero_add z] at hf

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

DifferentiableAt.isBigO_of_eq_zero

[50, 1]

[54, 73]

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

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

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

ContinuousAt.isBigO

[56, 1]

[70, 46]

rw [isBigO_iff']

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

ContinuousAt.isBigO

[56, 1]

[70, 46]

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

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

ContinuousAt.isBigO

[56, 1]

[70, 46]

specialize hf 1 zero_lt_one

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

ContinuousAt.isBigO

[56, 1]

[70, 46]

refine ⟨‖f z‖ + 1, by positivity, ?_⟩

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

ContinuousAt.isBigO

[56, 1]

[70, 46]

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

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

ContinuousAt.isBigO

[56, 1]

[70, 46]

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

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

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

ContinuousAt.isBigO

[56, 1]

[70, 46]

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

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

ContinuousAt.isBigO

[56, 1]

[70, 46]

simp

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

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

ContinuousAt.isBigO

[56, 1]

[70, 46]

simp [Function.comp_def, hf]

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

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

ContinuousAt.isBigO

[56, 1]

[70, 46]

positivity

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

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

ContinuousAt.isBigO

[56, 1]

[70, 46]

simp only [norm_one, mul_one]

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

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

HasDerivAt.of_hasDerivAt_ofReal_comp

[125, 1]

[134, 80]

lift u to ℝ

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

HasDerivAt.of_hasDerivAt_ofReal_comp

[125, 1]

[134, 80]

refine ⟨u, rfl, ?_⟩

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

HasDerivAt.of_hasDerivAt_ofReal_comp

[125, 1]

[134, 80]

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

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

HasDerivAt.of_hasDerivAt_ofReal_comp

[125, 1]

[134, 80]

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

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

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

HasDerivAt.of_hasDerivAt_ofReal_comp

[125, 1]

[134, 80]

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

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

HasDerivAt.of_hasDerivAt_ofReal_comp

[125, 1]

[134, 80]

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

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

HasDerivAt.of_hasDerivAt_ofReal_comp

[125, 1]

[134, 80]

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

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

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

DifferentiableAt.ofReal_comp_iff

[136, 1]

[140, 40]

refine ⟨fun H ↦ ?_, ofReal_comp⟩

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

DifferentiableAt.ofReal_comp_iff

[136, 1]

[140, 40]

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

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

DifferentiableAt.ofReal_comp_iff

[136, 1]

[140, 40]

exact HasDerivAt.differentiableAt hu₂

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

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

deriv.ofReal_comp

[146, 1]

[152, 27]

by_cases hf : DifferentiableAt ℝ f z

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

deriv.ofReal_comp

[146, 1]

[152, 27]

exact hf.hasDerivAt.ofReal_comp.deriv

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

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

deriv.ofReal_comp

[146, 1]

[152, 27]

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

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

deriv.ofReal_comp

[146, 1]

[152, 27]

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

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

no goals

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real_on_ball

[159, 1]

[183, 8]

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

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real_on_ball

[159, 1]

[183, 8]

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

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real_on_ball

[159, 1]

[183, 8]

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

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real_on_ball

[159, 1]

[183, 8]

intro x hx

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real_on_ball

[159, 1]

[183, 8]

refine Metric.mem_ball.mpr ?_

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real_on_ball

[159, 1]

[183, 8]

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

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

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

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

21e07835d1a467b99b5c3c9390d61ae69404445d

EulerProducts/Auxiliary.lean

Complex.realValued_of_iteratedDeriv_real_on_ball

[159, 1]

[183, 8]

exact and_comm.mpr hx

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

no goals