Datasets:
internlm
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Dataset card Data Studio Files Community
1
url
stringclasses
147 values
commit
stringclasses
147 values
file_path
stringlengths
7
101
full_name
stringlengths
1
94
start
stringlengths
6
10
end
stringlengths
6
11
tactic
stringlengths
1
11.2k
state_before
stringlengths
3
2.09M
state_after
stringlengths
6
2.09M
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/VehicleSpeedScheduling.lean
VehicleSpeedSched.aβ‚šdβ‚š2_nonneg
[191, 1]
[193, 55]
fin_cases i <;> (dsimp [aβ‚š, dβ‚š]; norm_num)
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ i : Fin nβ‚š ⊒ 0 i ≀ (aβ‚š β€’ dβ‚š ^ 2) i
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/VehicleSpeedScheduling.lean
VehicleSpeedSched.aβ‚šdβ‚š2_nonneg
[191, 1]
[193, 55]
dsimp [aβ‚š, dβ‚š]
case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ ⊒ 0 ⟨9, β‹―βŸ© ≀ (aβ‚š β€’ dβ‚š ^ 2) ⟨9, β‹―βŸ©
case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ ⊒ 0 ≀ 1 * ![1.9501, 1.2311, 1.6068, 1.4860, 1.8913, 1.7621, 1.4565, 1.0185, 1.8214, 1.4447] ⟨9, β‹―βŸ© ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/VehicleSpeedScheduling.lean
VehicleSpeedSched.aβ‚šdβ‚š2_nonneg
[191, 1]
[193, 55]
norm_num
case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ ⊒ 0 ≀ 1 * ![1.9501, 1.2311, 1.6068, 1.4860, 1.8913, 1.7621, 1.4565, 1.0185, 1.8214, 1.4447] ⟨9, β‹―βŸ© ^ 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 : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ h_d_pos : StrongLT 0 d s : Fin n β†’ ℝ i : Fin n ⊒ d i / (d i / s i) = s i
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ h_d_pos : StrongLT 0 d s : Fin n β†’ 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 : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ h_d_pos : StrongLT 0 d s : Fin n β†’ ℝ i : Fin n h_di_pos : 0 i < d i ⊒ d i / (d i / s i) = s i
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ h_d_pos : StrongLT 0 d s : Fin n β†’ 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 β‰  0 := by linarith
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ h_d_pos : StrongLT 0 d s : Fin n β†’ ℝ i : Fin n h_di_pos : 0 < d i ⊒ d i / (d i / s i) = s i
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ h_d_pos : StrongLT 0 d s : Fin n β†’ ℝ i : Fin n h_di_pos : 0 < d i h_di_nonzero : d i β‰  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 : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ h_d_pos : StrongLT 0 d s : Fin n β†’ ℝ i : Fin n h_di_pos : 0 < d i h_di_nonzero : d i β‰  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 : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ h_d_pos : StrongLT 0 d s : Fin n β†’ ℝ i : Fin n h_di_pos : 0 < d i ⊒ d i β‰  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 : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ hn : Fact (0 < n) t : Fin n β†’ ℝ i : Fin n ⊒ βˆ‘ j ∈ [[0, i]], t j = Vec.cumsum t i
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ hn : Fact (0 < n) t : Fin n β†’ ℝ i : Fin n ⊒ βˆ‘ j ∈ [[0, i]], t j = if h : 0 < n then βˆ‘ 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 : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ hn : Fact (0 < n) t : Fin n β†’ ℝ i : Fin n ⊒ βˆ‘ j ∈ [[0, i]], t j = if h : 0 < n then βˆ‘ j ∈ [[⟨0, h⟩, i]], t j else 0
case pos n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ hn : Fact (0 < n) t : Fin n β†’ ℝ i : Fin n h✝ : 0 < n ⊒ βˆ‘ j ∈ [[0, i]], t j = βˆ‘ j ∈ [[⟨0, h✝⟩, i]], t j case neg n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ hn : Fact (0 < n) t : Fin n β†’ ℝ i : Fin n h✝ : Β¬0 < n ⊒ βˆ‘ 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 : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ hn : Fact (0 < n) t : Fin n β†’ ℝ i : Fin n h✝ : 0 < n ⊒ βˆ‘ j ∈ [[0, i]], t j = βˆ‘ 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 : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ hn : Fact (0 < n) t : Fin n β†’ ℝ i : Fin n h✝ : Β¬0 < n ⊒ βˆ‘ j ∈ [[0, i]], t j = 0
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/VehicleSpeedScheduling.lean
VehicleSpeedSched.nβ‚š_pos
[148, 1]
[148, 48]
unfold nβ‚š
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ ⊒ 0 < nβ‚š
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ ⊒ 0 < 10
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/VehicleSpeedScheduling.lean
VehicleSpeedSched.nβ‚š_pos
[148, 1]
[148, 48]
norm_num
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ ⊒ 0 < 10
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/VehicleSpeedScheduling.lean
VehicleSpeedSched.dβ‚š_pos
[154, 1]
[155, 50]
intro i
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ ⊒ StrongLT 0 dβ‚š
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ i : Fin nβ‚š ⊒ 0 i < dβ‚š i
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/VehicleSpeedScheduling.lean
VehicleSpeedSched.dβ‚š_pos
[154, 1]
[155, 50]
fin_cases i <;> (dsimp [dβ‚š]; norm_num)
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ i : Fin nβ‚š ⊒ 0 i < dβ‚š i
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/VehicleSpeedScheduling.lean
VehicleSpeedSched.dβ‚š_pos
[154, 1]
[155, 50]
dsimp [dβ‚š]
case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ ⊒ 0 ⟨9, β‹―βŸ© < dβ‚š ⟨9, β‹―βŸ©
case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ ⊒ 0 < ![1.9501, 1.2311, 1.6068, 1.4860, 1.8913, 1.7621, 1.4565, 1.0185, 1.8214, 1.4447] ⟨9, β‹―βŸ©
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/VehicleSpeedScheduling.lean
VehicleSpeedSched.dβ‚š_pos
[154, 1]
[155, 50]
norm_num
case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ ⊒ 0 < ![1.9501, 1.2311, 1.6068, 1.4860, 1.8913, 1.7621, 1.4565, 1.0185, 1.8214, 1.4447] ⟨9, β‹―βŸ©
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/VehicleSpeedScheduling.lean
VehicleSpeedSched.sminβ‚š_pos
[173, 1]
[174, 36]
intro i
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ ⊒ StrongLT 0 sminβ‚š
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ i : Fin nβ‚š ⊒ 0 i < sminβ‚š i
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/VehicleSpeedScheduling.lean
VehicleSpeedSched.sminβ‚š_pos
[173, 1]
[174, 36]
fin_cases i <;> norm_num
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ i : Fin nβ‚š ⊒ 0 i < sminβ‚š i
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/VehicleSpeedScheduling.lean
VehicleSpeedSched.sminβ‚š_le_smaxβ‚š
[179, 1]
[180, 60]
intro i
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ ⊒ sminβ‚š ≀ smaxβ‚š
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ i : Fin nβ‚š ⊒ sminβ‚š i ≀ smaxβ‚š i
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/VehicleSpeedScheduling.lean
VehicleSpeedSched.sminβ‚š_le_smaxβ‚š
[179, 1]
[180, 60]
fin_cases i <;> (dsimp [sminβ‚š, smaxβ‚š]; norm_num)
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ i : Fin nβ‚š ⊒ sminβ‚š i ≀ smaxβ‚š i
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/VehicleSpeedScheduling.lean
VehicleSpeedSched.sminβ‚š_le_smaxβ‚š
[179, 1]
[180, 60]
dsimp [sminβ‚š, smaxβ‚š]
case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ ⊒ sminβ‚š ⟨9, β‹―βŸ© ≀ smaxβ‚š ⟨9, β‹―βŸ©
case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ ⊒ ![0.7828, 0.6235, 0.7155, 0.5340, 0.6329, 0.4259, 0.7798, 0.9604, 0.7298, 0.8405] ⟨9, β‹―βŸ© ≀ ![1.9624, 1.6036, 1.6439, 1.5641, 1.7194, 1.9090, 1.3193, 1.3366, 1.9470, 2.8803] ⟨9, β‹―βŸ©
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/VehicleSpeedScheduling.lean
VehicleSpeedSched.sminβ‚š_le_smaxβ‚š
[179, 1]
[180, 60]
norm_num
case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ ⊒ ![0.7828, 0.6235, 0.7155, 0.5340, 0.6329, 0.4259, 0.7798, 0.9604, 0.7298, 0.8405] ⟨9, β‹―βŸ© ≀ ![1.9624, 1.6036, 1.6439, 1.5641, 1.7194, 1.9090, 1.3193, 1.3366, 1.9470, 2.8803] ⟨9, β‹―βŸ©
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/VehicleSpeedScheduling.lean
VehicleSpeedSched.aβ‚š_nonneg
[188, 1]
[189, 51]
unfold aβ‚š
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ ⊒ 0 ≀ aβ‚š
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ ⊒ 0 ≀ 1
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/VehicleSpeedScheduling.lean
VehicleSpeedSched.aβ‚š_nonneg
[188, 1]
[189, 51]
norm_num
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ ⊒ 0 ≀ 1
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/VehicleSpeedScheduling.lean
VehicleSpeedSched.aβ‚šdβ‚š2_nonneg
[191, 1]
[193, 55]
intros i
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ ⊒ 0 ≀ aβ‚š β€’ dβ‚š ^ 2
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ i : Fin nβ‚š ⊒ 0 i ≀ (aβ‚š β€’ dβ‚š ^ 2) i
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/VehicleSpeedScheduling.lean
VehicleSpeedSched.aβ‚šdβ‚š2_nonneg
[191, 1]
[193, 55]
fin_cases i <;> (dsimp [aβ‚š, dβ‚š]; norm_num)
n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ i : Fin nβ‚š ⊒ 0 i ≀ (aβ‚š β€’ dβ‚š ^ 2) i
no goals
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/VehicleSpeedScheduling.lean
VehicleSpeedSched.aβ‚šdβ‚š2_nonneg
[191, 1]
[193, 55]
dsimp [aβ‚š, dβ‚š]
case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ ⊒ 0 ⟨9, β‹―βŸ© ≀ (aβ‚š β€’ dβ‚š ^ 2) ⟨9, β‹―βŸ©
case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ ⊒ 0 ≀ 1 * ![1.9501, 1.2311, 1.6068, 1.4860, 1.8913, 1.7621, 1.4565, 1.0185, 1.8214, 1.4447] ⟨9, β‹―βŸ© ^ 2
https://github.com/verified-optimization/CvxLean.git
c62c2f292c6420f31a12e738ebebdfed50f6f840
CvxLean/Examples/VehicleSpeedScheduling.lean
VehicleSpeedSched.aβ‚šdβ‚š2_nonneg
[191, 1]
[193, 55]
norm_num
case tail.tail.tail.tail.tail.tail.tail.tail.tail.head n : β„• d Ο„min Ο„max smin smax : Fin n β†’ ℝ F : ℝ β†’ ℝ ⊒ 0 ≀ 1 * ![1.9501, 1.2311, 1.6068, 1.4860, 1.8913, 1.7621, 1.4565, 1.0185, 1.8214, 1.4447] ⟨9, β‹―βŸ© ^ 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 : βˆ€ (w : EuclideanSpace π•œ (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))
π•œ : Type u_2 inst✝⁴ : RCLike π•œ inst✝³ : DecidableEq π•œ E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace π•œ E inst✝ : FiniteDimensional π•œ E n : β„• hn : FiniteDimensional.finrank π•œ E = n T : E β†’β‚—[π•œ] E v : E i : Fin n xs : OrthonormalBasis (Fin n) π•œ E as : Fin n β†’ ℝ hxs : βˆ€ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j) ⊒ xs.repr (T v) i = ↑(as i) * xs.repr v i
π•œ : Type u_2 inst✝⁴ : RCLike π•œ inst✝³ : DecidableEq π•œ E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace π•œ E inst✝ : FiniteDimensional π•œ E n : β„• hn : FiniteDimensional.finrank π•œ E = n T : E β†’β‚—[π•œ] E v : E i : Fin n xs : OrthonormalBasis (Fin n) π•œ E as : Fin n β†’ ℝ hxs : βˆ€ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j) ⊒ βˆ€ (w : EuclideanSpace π•œ (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
π•œ : Type u_2 inst✝⁴ : RCLike π•œ inst✝³ : DecidableEq π•œ E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace π•œ E inst✝ : FiniteDimensional π•œ E n : β„• hn : FiniteDimensional.finrank π•œ E = n T : E β†’β‚—[π•œ] E v : E i : Fin n xs : OrthonormalBasis (Fin n) π•œ E as : Fin n β†’ ℝ hxs : βˆ€ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j) ⊒ βˆ€ (w : EuclideanSpace π•œ (Fin n)), T (xs.repr.symm w) = xs.repr.symm fun i => ↑(as i) * w i
π•œ : Type u_2 inst✝⁴ : RCLike π•œ inst✝³ : DecidableEq π•œ E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace π•œ E inst✝ : FiniteDimensional π•œ E n : β„• hn : FiniteDimensional.finrank π•œ E = n T : E β†’β‚—[π•œ] E v : E i : Fin n xs : OrthonormalBasis (Fin n) π•œ E as : Fin n β†’ ℝ hxs : βˆ€ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j) w : EuclideanSpace π•œ (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]
π•œ : Type u_2 inst✝⁴ : RCLike π•œ inst✝³ : DecidableEq π•œ E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace π•œ E inst✝ : FiniteDimensional π•œ E n : β„• hn : FiniteDimensional.finrank π•œ E = n T : E β†’β‚—[π•œ] E v : E i : Fin n xs : OrthonormalBasis (Fin n) π•œ E as : Fin n β†’ ℝ hxs : βˆ€ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j) w : EuclideanSpace π•œ (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))
π•œ : Type u_2 inst✝⁴ : RCLike π•œ inst✝³ : DecidableEq π•œ E : Type u_1 inst✝² : NormedAddCommGroup E inst✝¹ : InnerProductSpace π•œ E inst✝ : FiniteDimensional π•œ E n : β„• hn : FiniteDimensional.finrank π•œ E = n T : E β†’β‚—[π•œ] E v : E i : Fin n xs : OrthonormalBasis (Fin n) π•œ E as : Fin n β†’ ℝ hxs : βˆ€ (j : Fin n), Module.End.HasEigenvector T (↑(as j)) (xs j) hsuff : βˆ€ (w : EuclideanSpace π•œ (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 : β„• β†’ R n : β„• ⊒ βˆ‘ p ∈ n.primesBelow, f p = βˆ‘ m ∈ Finset.range n, {p | p.Prime}.indicator f m
case h.e'_3.a R : Type u_1 inst✝ : AddCommMonoid R f : β„• β†’ R n m : β„• 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 : β„• β†’ R n m : β„• 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 : β„• β†’ R n m : β„• 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 hβ‚‚ h₃
case h.e'_3.a R : Type u_1 inst✝ : AddCommMonoid R f : β„• β†’ R n m : β„• 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 : β„• β†’ R n m : β„• hm : m ∈ Finset.range n h₁ : m.Prime hβ‚‚ : m < n ∧ m.Prime ⊒ f m = f m case neg R : Type u_1 inst✝ : AddCommMonoid R f : β„• β†’ R n m : β„• hm : m ∈ Finset.range n h₁ : m.Prime hβ‚‚ : Β¬(m < n ∧ m.Prime) ⊒ f m = 0 case pos R : Type u_1 inst✝ : AddCommMonoid R f : β„• β†’ R n m : β„• hm : m ∈ Finset.range n h₁ : Β¬m.Prime h₃ : m < n ∧ m.Prime ⊒ 0 = f m case neg R : Type u_1 inst✝ : AddCommMonoid R f : β„• β†’ R n m : β„• hm : m ∈ Finset.range n h₁ : Β¬m.Prime h₃ : Β¬(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 : β„• β†’ R n m : β„• hm : m ∈ Finset.range n h₁ : m.Prime hβ‚‚ : 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 (hβ‚‚ ⟨Finset.mem_range.mp hm, hβ‚βŸ©).elim
case neg R : Type u_1 inst✝ : AddCommMonoid R f : β„• β†’ R n m : β„• hm : m ∈ Finset.range n h₁ : m.Prime hβ‚‚ : Β¬(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 : β„• β†’ R n m : β„• hm : m ∈ Finset.range n h₁ : Β¬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 : β„• β†’ R n m : β„• hm : m ∈ Finset.range n h₁ : Β¬m.Prime h₃ : Β¬(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 βˆ‘' p : Nat.Primes, f p = βˆ‘' p : {p : β„• | 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 : β„• β†’ R hsum : Summable f ⊒ Tendsto (fun n => βˆ‘ p ∈ n.primesBelow, f p) atTop (𝓝 (βˆ‘' (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 : β„• β†’ R hsum : Summable f ⊒ Tendsto (fun n => βˆ‘ p ∈ n.primesBelow, f p) atTop (𝓝 (βˆ‘' (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 : β„• β†’ R hsum : Summable f ⊒ Tendsto (fun n => βˆ‘ p ∈ n.primesBelow, f p) atTop (𝓝 (βˆ‘' (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 : β„• β†’ R hsum : Summable f ⊒ Tendsto (fun n => βˆ‘ m ∈ Finset.range n, {p | p.Prime}.indicator (fun p => f p) m) atTop (𝓝 (βˆ‘' (x : β„•), {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 : β„• β†’ R hsum : Summable f ⊒ Tendsto (fun n => βˆ‘ m ∈ Finset.range n, {p | p.Prime}.indicator (fun p => f p) m) atTop (𝓝 (βˆ‘' (x : β„•), {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 : β„• β†’ β„‚ hsum : Summable f ⊒ Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (f p)) atTop (𝓝 (cexp (βˆ‘' (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 : β„‚) : β„‚ := -log (1 - z)
Ξ± : Type u_1 f : Ξ± β†’ β„‚ hsum : Summable f ⊒ Summable fun n => -(1 - f n).log
Ξ± : Type u_1 f : Ξ± β†’ β„‚ hsum : Summable f g : β„‚ β†’ β„‚ := 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 β„‚ g 0 := DifferentiableAt.neg <| ((differentiableAt_const 1).sub differentiableAt_id').clog <| by simp only [sub_zero, one_mem_slitPlane]
Ξ± : Type u_1 f : Ξ± β†’ β„‚ hsum : Summable f g : β„‚ β†’ β„‚ := fun z => -(1 - z).log ⊒ Summable fun n => -(1 - f n).log
Ξ± : Type u_1 f : Ξ± β†’ β„‚ hsum : Summable f g : β„‚ β†’ β„‚ := fun z => -(1 - z).log hg : DifferentiableAt β„‚ 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
Ξ± : Type u_1 f : Ξ± β†’ β„‚ hsum : Summable f g : β„‚ β†’ β„‚ := fun z => -(1 - z).log hg : DifferentiableAt β„‚ g 0 ⊒ Summable fun n => -(1 - f n).log
Ξ± : Type u_1 f : Ξ± β†’ β„‚ hsum : Summable f g : β„‚ β†’ β„‚ := fun z => -(1 - z).log hg : DifferentiableAt β„‚ 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
Ξ± : Type u_1 f : Ξ± β†’ β„‚ hsum : Summable f g : β„‚ β†’ β„‚ := fun z => -(1 - z).log hg : DifferentiableAt β„‚ 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]
Ξ± : Type u_1 f : Ξ± β†’ β„‚ hsum : Summable f g : β„‚ β†’ β„‚ := 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
Ξ± : Type u_1 f : Ξ± β†’ β„‚ hsum : Summable f g : β„‚ β†’ β„‚ := fun z => -(1 - z).log hg : DifferentiableAt β„‚ 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 : β„•} (hp : 1 < p) : β€–f pβ€– < 1 := hsum.of_norm.norm_lt_one (f := f.toMonoidHom) hp
f : β„• β†’*β‚€ β„‚ hsum : Summable fun x => β€–f xβ€– ⊒ cexp (βˆ‘' (p : Nat.Primes), -(1 - f ↑p).log) = βˆ‘' (n : β„•), f n
f : β„• β†’*β‚€ β„‚ hsum : Summable fun x => β€–f xβ€– hs : βˆ€ {p : β„•}, 1 < p β†’ β€–f pβ€– < 1 ⊒ cexp (βˆ‘' (p : Nat.Primes), -(1 - f ↑p).log) = βˆ‘' (n : β„•), 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 : β„• β†’*β‚€ β„‚ hsum : Summable fun x => β€–f xβ€– hs : βˆ€ {p : β„•}, 1 < p β†’ β€–f pβ€– < 1 ⊒ cexp (βˆ‘' (p : Nat.Primes), -(1 - f ↑p).log) = βˆ‘' (n : β„•), f n
f : β„• β†’*β‚€ β„‚ hsum : Summable fun x => β€–f xβ€– hs : βˆ€ {p : β„•}, 1 < p β†’ β€–f pβ€– < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (βˆ‘' (p : Nat.Primes), -(1 - f ↑p).log))) ⊒ cexp (βˆ‘' (p : Nat.Primes), -(1 - f ↑p).log) = βˆ‘' (n : β„•), 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 : β„•) : 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 (Ξ± := β„‚) β–Έ h.symm β–Έ hs (Nat.prime_of_mem_primesBelow hp).one_lt).false
f : β„• β†’*β‚€ β„‚ hsum : Summable fun x => β€–f xβ€– hs : βˆ€ {p : β„•}, 1 < p β†’ β€–f pβ€– < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (βˆ‘' (p : Nat.Primes), -(1 - f ↑p).log))) ⊒ cexp (βˆ‘' (p : Nat.Primes), -(1 - f ↑p).log) = βˆ‘' (n : β„•), f n
f : β„• β†’*β‚€ β„‚ hsum : Summable fun x => β€–f xβ€– hs : βˆ€ {p : β„•}, 1 < p β†’ β€–f pβ€– < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (βˆ‘' (p : Nat.Primes), -(1 - f ↑p).log))) help : βˆ€ (n : β„•), ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log) = ∏ p ∈ n.primesBelow, (1 - f p)⁻¹ ⊒ cexp (βˆ‘' (p : Nat.Primes), -(1 - f ↑p).log) = βˆ‘' (n : β„•), 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 : β„• β†’*β‚€ β„‚ hsum : Summable fun x => β€–f xβ€– hs : βˆ€ {p : β„•}, 1 < p β†’ β€–f pβ€– < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (βˆ‘' (p : Nat.Primes), -(1 - f ↑p).log))) help : βˆ€ (n : β„•), ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log) = ∏ p ∈ n.primesBelow, (1 - f p)⁻¹ ⊒ cexp (βˆ‘' (p : Nat.Primes), -(1 - f ↑p).log) = βˆ‘' (n : β„•), f n
f : β„• β†’*β‚€ β„‚ hsum : Summable fun x => β€–f xβ€– hs : βˆ€ {p : β„•}, 1 < p β†’ β€–f pβ€– < 1 help : βˆ€ (n : β„•), ∏ 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 (βˆ‘' (p : Nat.Primes), -(1 - f ↑p).log))) ⊒ cexp (βˆ‘' (p : Nat.Primes), -(1 - f ↑p).log) = βˆ‘' (n : β„•), 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 : β„• β†’*β‚€ β„‚ hsum : Summable fun x => β€–f xβ€– hs : βˆ€ {p : β„•}, 1 < p β†’ β€–f pβ€– < 1 help : βˆ€ (n : β„•), ∏ 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 (βˆ‘' (p : Nat.Primes), -(1 - f ↑p).log))) ⊒ cexp (βˆ‘' (p : Nat.Primes), -(1 - f ↑p).log) = βˆ‘' (n : β„•), 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 : β„• β†’*β‚€ β„‚ hsum : Summable fun x => β€–f xβ€– hs : βˆ€ {p : β„•}, 1 < p β†’ β€–f pβ€– < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (βˆ‘' (p : Nat.Primes), -(1 - f ↑p).log))) n : β„• ⊒ ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log) = ∏ p ∈ n.primesBelow, (1 - f p)⁻¹
f : β„• β†’*β‚€ β„‚ hsum : Summable fun x => β€–f xβ€– hs : βˆ€ {p : β„•}, 1 < p β†’ β€–f pβ€– < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (βˆ‘' (p : Nat.Primes), -(1 - f ↑p).log))) n p : β„• 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 : β„• β†’*β‚€ β„‚ hsum : Summable fun x => β€–f xβ€– hs : βˆ€ {p : β„•}, 1 < p β†’ β€–f pβ€– < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (βˆ‘' (p : Nat.Primes), -(1 - f ↑p).log))) n p : β„• hp : p ∈ n.primesBelow ⊒ cexp (-(1 - f p).log) = (1 - f p)⁻¹
f : β„• β†’*β‚€ β„‚ hsum : Summable fun x => β€–f xβ€– hs : βˆ€ {p : β„•}, 1 < p β†’ β€–f pβ€– < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (βˆ‘' (p : Nat.Primes), -(1 - f ↑p).log))) n p : β„• hp : p ∈ n.primesBelow ⊒ 1 - f p β‰  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 : β„• β†’*β‚€ β„‚ hsum : Summable fun x => β€–f xβ€– hs : βˆ€ {p : β„•}, 1 < p β†’ β€–f pβ€– < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (βˆ‘' (p : Nat.Primes), -(1 - f ↑p).log))) n p : β„• hp : p ∈ n.primesBelow ⊒ 1 - f p β‰  0
f : β„• β†’*β‚€ β„‚ hsum : Summable fun x => β€–f xβ€– hs : βˆ€ {p : β„•}, 1 < p β†’ β€–f pβ€– < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (βˆ‘' (p : Nat.Primes), -(1 - f ↑p).log))) n p : β„• hp : p ∈ n.primesBelow ⊒ 1 β‰  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 (Ξ± := β„‚) β–Έ h.symm β–Έ hs (Nat.prime_of_mem_primesBelow hp).one_lt).false
f : β„• β†’*β‚€ β„‚ hsum : Summable fun x => β€–f xβ€– hs : βˆ€ {p : β„•}, 1 < p β†’ β€–f pβ€– < 1 H : Tendsto (fun n => ∏ p ∈ n.primesBelow, cexp (-(1 - f p).log)) atTop (𝓝 (cexp (βˆ‘' (p : Nat.Primes), -(1 - f ↑p).log))) n p : β„• hp : p ∈ n.primesBelow ⊒ 1 β‰  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']
Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 ⊒ (fun x => f x * g x) =O[l] h ↔ g =O[l] fun x => h x / f x
Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 ⊒ (βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–f x * g xβ€– ≀ c * β€–h xβ€–) ↔ βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€–
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Asymptotics.isBigO_mul_iff_isBigO_div
[31, 1]
[39, 36]
refine ⟨fun ⟨c, hc, H⟩ ↦ ⟨c, hc, ?_⟩, fun ⟨c, hc, H⟩ ↦ ⟨c, hc, ?_⟩⟩ <;> { refine H.congr <| Eventually.mp hf <| eventually_of_forall fun x hx ↦ ?_ rw [norm_mul, norm_div, ← mul_div_assoc, mul_comm] have hx' : β€–f xβ€– > 0 := norm_pos_iff.mpr hx rw [le_div_iff hx', mul_comm] }
Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 ⊒ (βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–f x * g xβ€– ≀ c * β€–h xβ€–) ↔ βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€–
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Asymptotics.isBigO_mul_iff_isBigO_div
[31, 1]
[39, 36]
refine H.congr <| Eventually.mp hf <| eventually_of_forall fun x hx ↦ ?_
case refine_2 Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 x✝ : βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– c : ℝ hc : c > 0 H : βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– ⊒ βˆ€αΆ  (x : Ξ±) in l, β€–f x * g xβ€– ≀ c * β€–h xβ€–
case refine_2 Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 x✝ : βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– c : ℝ hc : c > 0 H : βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– x : Ξ± hx : f x β‰  0 ⊒ β€–g xβ€– ≀ c * β€–h x / f xβ€– ↔ β€–f x * g xβ€– ≀ c * β€–h xβ€–
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Asymptotics.isBigO_mul_iff_isBigO_div
[31, 1]
[39, 36]
rw [norm_mul, norm_div, ← mul_div_assoc, mul_comm]
case refine_2 Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 x✝ : βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– c : ℝ hc : c > 0 H : βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– x : Ξ± hx : f x β‰  0 ⊒ β€–g xβ€– ≀ c * β€–h x / f xβ€– ↔ β€–f x * g xβ€– ≀ c * β€–h xβ€–
case refine_2 Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 x✝ : βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– c : ℝ hc : c > 0 H : βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– x : Ξ± hx : f x β‰  0 ⊒ β€–g xβ€– ≀ β€–h xβ€– * c / β€–f xβ€– ↔ β€–f xβ€– * β€–g xβ€– ≀ β€–h xβ€– * c
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Asymptotics.isBigO_mul_iff_isBigO_div
[31, 1]
[39, 36]
have hx' : β€–f xβ€– > 0 := norm_pos_iff.mpr hx
case refine_2 Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 x✝ : βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– c : ℝ hc : c > 0 H : βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– x : Ξ± hx : f x β‰  0 ⊒ β€–g xβ€– ≀ β€–h xβ€– * c / β€–f xβ€– ↔ β€–f xβ€– * β€–g xβ€– ≀ β€–h xβ€– * c
case refine_2 Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 x✝ : βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– c : ℝ hc : c > 0 H : βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– x : Ξ± hx : f x β‰  0 hx' : β€–f xβ€– > 0 ⊒ β€–g xβ€– ≀ β€–h xβ€– * c / β€–f xβ€– ↔ β€–f xβ€– * β€–g xβ€– ≀ β€–h xβ€– * c
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Asymptotics.isBigO_mul_iff_isBigO_div
[31, 1]
[39, 36]
rw [le_div_iff hx', mul_comm]
case refine_2 Ξ± : Type u_1 F : Type u_2 inst✝ : NormedField F l : Filter Ξ± f g h : Ξ± β†’ F hf : βˆ€αΆ  (x : Ξ±) in l, f x β‰  0 x✝ : βˆƒ c > 0, βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– c : ℝ hc : c > 0 H : βˆ€αΆ  (x : Ξ±) in l, β€–g xβ€– ≀ c * β€–h x / f xβ€– x : Ξ± hx : f x β‰  0 hx' : β€–f xβ€– > 0 ⊒ β€–g xβ€– ≀ β€–h xβ€– * c / β€–f xβ€– ↔ β€–f xβ€– * β€–g xβ€– ≀ β€–h xβ€– * c
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
DifferentiableAt.isBigO_of_eq_zero
[50, 1]
[54, 73]
rw [← zero_add z] at hf
f : β„‚ β†’ β„‚ z : β„‚ hf : DifferentiableAt β„‚ f z hz : f z = 0 ⊒ (fun w => f (w + z)) =O[𝓝 0] id
f : β„‚ β†’ β„‚ z : β„‚ hf : DifferentiableAt β„‚ f (0 + z) hz : f z = 0 ⊒ (fun w => f (w + z)) =O[𝓝 0] id
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
DifferentiableAt.isBigO_of_eq_zero
[50, 1]
[54, 73]
simpa only [zero_add, hz, sub_zero] using (hf.hasDerivAt.comp_add_const 0 z).differentiableAt.isBigO_sub
f : β„‚ β†’ β„‚ z : β„‚ hf : DifferentiableAt β„‚ f (0 + z) hz : f z = 0 ⊒ (fun w => f (w + z)) =O[𝓝 0] id
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
ContinuousAt.isBigO
[56, 1]
[70, 46]
rw [isBigO_iff']
f : β„‚ β†’ β„‚ z : β„‚ hf : ContinuousAt f z ⊒ (fun w => f (w + z)) =O[𝓝 0] fun x => 1
f : β„‚ β†’ β„‚ z : β„‚ hf : ContinuousAt f z ⊒ βˆƒ c > 0, βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z)β€– ≀ c * β€–1β€–
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
ContinuousAt.isBigO
[56, 1]
[70, 46]
simp_rw [Metric.continuousAt_iff', dist_eq_norm_sub, zero_add] at hf
f : β„‚ β†’ β„‚ z : β„‚ hf : ContinuousAt (fun w => f (w + z)) 0 ⊒ βˆƒ c > 0, βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z)β€– ≀ c * β€–1β€–
f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€ Ξ΅ > 0, βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < Ξ΅ ⊒ βˆƒ c > 0, βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z)β€– ≀ c * β€–1β€–
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
ContinuousAt.isBigO
[56, 1]
[70, 46]
specialize hf 1 zero_lt_one
f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€ Ξ΅ > 0, βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < Ξ΅ ⊒ βˆƒ c > 0, βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z)β€– ≀ c * β€–1β€–
f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < 1 ⊒ βˆƒ c > 0, βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z)β€– ≀ c * β€–1β€–
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
ContinuousAt.isBigO
[56, 1]
[70, 46]
refine βŸ¨β€–f zβ€– + 1, by positivity, ?_⟩
f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < 1 ⊒ βˆƒ c > 0, βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z)β€– ≀ c * β€–1β€–
f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < 1 ⊒ βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z)β€– ≀ (β€–f zβ€– + 1) * β€–1β€–
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
ContinuousAt.isBigO
[56, 1]
[70, 46]
refine Eventually.mp hf <| eventually_of_forall fun w hw ↦ le_of_lt ?_
f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < 1 ⊒ βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z)β€– ≀ (β€–f zβ€– + 1) * β€–1β€–
f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < 1 w : β„‚ hw : β€–f (w + z) - f zβ€– < 1 ⊒ β€–f (w + z)β€– < (β€–f zβ€– + 1) * β€–1β€–
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
ContinuousAt.isBigO
[56, 1]
[70, 46]
calc β€–f (w + z)β€– _ ≀ β€–f zβ€– + β€–f (w + z) - f zβ€– := norm_le_insert' .. _ < β€–f zβ€– + 1 := add_lt_add_left hw _ _ = _ := by simp only [norm_one, mul_one]
f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < 1 w : β„‚ hw : β€–f (w + z) - f zβ€– < 1 ⊒ β€–f (w + z)β€– < (β€–f zβ€– + 1) * β€–1β€–
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
ContinuousAt.isBigO
[56, 1]
[70, 46]
convert (Homeomorph.comp_continuousAt_iff' (Homeomorph.addLeft (-z)) _ z).mp ?_
f : β„‚ β†’ β„‚ z : β„‚ hf : ContinuousAt f z ⊒ ContinuousAt (fun w => f (w + z)) 0
case h.e'_1 f : β„‚ β†’ β„‚ z : β„‚ hf : ContinuousAt f z ⊒ 0 = (Homeomorph.addLeft (-z)) z case convert_4 f : β„‚ β†’ β„‚ z : β„‚ hf : ContinuousAt f z ⊒ ContinuousAt ((fun w => f (w + z)) ∘ ⇑(Homeomorph.addLeft (-z))) z
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
ContinuousAt.isBigO
[56, 1]
[70, 46]
simp
case h.e'_1 f : β„‚ β†’ β„‚ z : β„‚ hf : ContinuousAt f z ⊒ 0 = (Homeomorph.addLeft (-z)) z
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
ContinuousAt.isBigO
[56, 1]
[70, 46]
simp [Function.comp_def, hf]
case convert_4 f : β„‚ β†’ β„‚ z : β„‚ hf : ContinuousAt f z ⊒ ContinuousAt ((fun w => f (w + z)) ∘ ⇑(Homeomorph.addLeft (-z))) z
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
ContinuousAt.isBigO
[56, 1]
[70, 46]
positivity
f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < 1 ⊒ β€–f zβ€– + 1 > 0
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
ContinuousAt.isBigO
[56, 1]
[70, 46]
simp only [norm_one, mul_one]
f : β„‚ β†’ β„‚ z : β„‚ hf : βˆ€αΆ  (x : β„‚) in 𝓝 0, β€–f (x + z) - f zβ€– < 1 w : β„‚ hw : β€–f (w + z) - f zβ€– < 1 ⊒ β€–f zβ€– + 1 = (β€–f zβ€– + 1) * β€–1β€–
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
HasDerivAt.of_hasDerivAt_ofReal_comp
[125, 1]
[134, 80]
lift u to ℝ
z : ℝ f : ℝ β†’ ℝ u : β„‚ hf : HasDerivAt (fun y => ↑(f y)) u z ⊒ βˆƒ u', u = ↑u' ∧ HasDerivAt f u' z
z : ℝ f : ℝ β†’ ℝ u : β„‚ hf : HasDerivAt (fun y => ↑(f y)) u z ⊒ u.im = 0 case intro z : ℝ f : ℝ β†’ ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊒ βˆƒ u', ↑u = ↑u' ∧ HasDerivAt f u' z
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
HasDerivAt.of_hasDerivAt_ofReal_comp
[125, 1]
[134, 80]
refine ⟨u, rfl, ?_⟩
case intro z : ℝ f : ℝ β†’ ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊒ βˆƒ u', ↑u = ↑u' ∧ HasDerivAt f u' z
case intro z : ℝ f : ℝ β†’ ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊒ HasDerivAt f u z
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
HasDerivAt.of_hasDerivAt_ofReal_comp
[125, 1]
[134, 80]
convert (reCLM.hasFDerivAt.comp z hf.hasFDerivAt).hasDerivAt
case intro z : ℝ f : ℝ β†’ ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊒ HasDerivAt f u z
case h.e'_7 z : ℝ f : ℝ β†’ ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊒ u = (reCLM.comp (smulRight 1 ↑u)) 1
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
HasDerivAt.of_hasDerivAt_ofReal_comp
[125, 1]
[134, 80]
rw [comp_apply, smulRight_apply, one_apply, one_smul, reCLM_apply, ofReal_re]
case h.e'_7 z : ℝ f : ℝ β†’ ℝ u : ℝ hf : HasDerivAt (fun y => ↑(f y)) (↑u) z ⊒ u = (reCLM.comp (smulRight 1 ↑u)) 1
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
HasDerivAt.of_hasDerivAt_ofReal_comp
[125, 1]
[134, 80]
have H := (imCLM.hasFDerivAt.comp z hf.hasFDerivAt).hasDerivAt.deriv
z : ℝ f : ℝ β†’ ℝ u : β„‚ hf : HasDerivAt (fun y => ↑(f y)) u z ⊒ u.im = 0
z : ℝ f : ℝ β†’ ℝ u : β„‚ hf : HasDerivAt (fun y => ↑(f y)) u z H : _root_.deriv (⇑imCLM ∘ fun y => ↑(f y)) z = (imCLM.comp (smulRight 1 u)) 1 ⊒ u.im = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
HasDerivAt.of_hasDerivAt_ofReal_comp
[125, 1]
[134, 80]
simp only [Function.comp_def, imCLM_apply, ofReal_im, deriv_const] at H
z : ℝ f : ℝ β†’ ℝ u : β„‚ hf : HasDerivAt (fun y => ↑(f y)) u z H : _root_.deriv (⇑imCLM ∘ fun y => ↑(f y)) z = (imCLM.comp (smulRight 1 u)) 1 ⊒ u.im = 0
z : ℝ f : ℝ β†’ ℝ u : β„‚ hf : HasDerivAt (fun y => ↑(f y)) u z H : 0 = (imCLM.comp (smulRight 1 u)) 1 ⊒ u.im = 0
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
HasDerivAt.of_hasDerivAt_ofReal_comp
[125, 1]
[134, 80]
rwa [eq_comm, comp_apply, imCLM_apply, smulRight_apply, one_apply, one_smul] at H
z : ℝ f : ℝ β†’ ℝ u : β„‚ hf : HasDerivAt (fun y => ↑(f y)) u z H : 0 = (imCLM.comp (smulRight 1 u)) 1 ⊒ u.im = 0
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
DifferentiableAt.ofReal_comp_iff
[136, 1]
[140, 40]
refine ⟨fun H ↦ ?_, ofReal_comp⟩
z : ℝ f : ℝ β†’ ℝ ⊒ DifferentiableAt ℝ (fun y => ↑(f y)) z ↔ DifferentiableAt ℝ f z
z : ℝ f : ℝ β†’ ℝ H : DifferentiableAt ℝ (fun y => ↑(f y)) z ⊒ DifferentiableAt ℝ f z
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
DifferentiableAt.ofReal_comp_iff
[136, 1]
[140, 40]
obtain ⟨u, _, huβ‚‚βŸ© := H.hasDerivAt.of_hasDerivAt_ofReal_comp
z : ℝ f : ℝ β†’ ℝ H : DifferentiableAt ℝ (fun y => ↑(f y)) z ⊒ DifferentiableAt ℝ f z
case intro.intro z : ℝ f : ℝ β†’ ℝ H : DifferentiableAt ℝ (fun y => ↑(f y)) z u : ℝ left✝ : deriv (fun y => ↑(f y)) z = ↑u huβ‚‚ : HasDerivAt (fun y => f y) u z ⊒ DifferentiableAt ℝ f z
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
DifferentiableAt.ofReal_comp_iff
[136, 1]
[140, 40]
exact HasDerivAt.differentiableAt huβ‚‚
case intro.intro z : ℝ f : ℝ β†’ ℝ H : DifferentiableAt ℝ (fun y => ↑(f y)) z u : ℝ left✝ : deriv (fun y => ↑(f y)) z = ↑u huβ‚‚ : HasDerivAt (fun y => f y) u z ⊒ DifferentiableAt ℝ f z
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
deriv.ofReal_comp
[146, 1]
[152, 27]
by_cases hf : DifferentiableAt ℝ f z
z : ℝ f : ℝ β†’ ℝ ⊒ deriv (fun y => ↑(f y)) z = ↑(deriv f z)
case pos z : ℝ f : ℝ β†’ ℝ hf : DifferentiableAt ℝ f z ⊒ deriv (fun y => ↑(f y)) z = ↑(deriv f z) case neg z : ℝ f : ℝ β†’ ℝ hf : Β¬DifferentiableAt ℝ f z ⊒ deriv (fun y => ↑(f y)) z = ↑(deriv f z)
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
deriv.ofReal_comp
[146, 1]
[152, 27]
exact hf.hasDerivAt.ofReal_comp.deriv
case pos z : ℝ f : ℝ β†’ ℝ hf : DifferentiableAt ℝ f z ⊒ deriv (fun y => ↑(f y)) z = ↑(deriv f z)
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
deriv.ofReal_comp
[146, 1]
[152, 27]
have hf' := mt DifferentiableAt.ofReal_comp_iff.mp hf
case neg z : ℝ f : ℝ β†’ ℝ hf : Β¬DifferentiableAt ℝ f z ⊒ deriv (fun y => ↑(f y)) z = ↑(deriv f z)
case neg z : ℝ f : ℝ β†’ ℝ hf : Β¬DifferentiableAt ℝ f z hf' : Β¬DifferentiableAt ℝ (fun y => ↑(f y)) z ⊒ deriv (fun y => ↑(f y)) z = ↑(deriv f z)
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
deriv.ofReal_comp
[146, 1]
[152, 27]
rw [deriv_zero_of_not_differentiableAt hf, deriv_zero_of_not_differentiableAt hf', Complex.ofReal_zero]
case neg z : ℝ f : ℝ β†’ ℝ hf : Β¬DifferentiableAt ℝ f z hf' : Β¬DifferentiableAt ℝ (fun y => ↑(f y)) z ⊒ deriv (fun y => ↑(f y)) z = ↑(deriv f z)
no goals
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real_on_ball
[159, 1]
[183, 8]
have Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), (x : β„‚) ∈ Metric.ball (c : β„‚) r := by intro x hx refine Metric.mem_ball.mpr ?_ rw [dist_eq, ← ofReal_sub, abs_ofReal, abs_sub_lt_iff, sub_lt_iff_lt_add', sub_lt_comm] exact and_comm.mpr hx
f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) ⊒ βˆƒ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))
f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r ⊒ βˆƒ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real_on_ball
[159, 1]
[183, 8]
have H ⦃z : ℂ⦄ (hz : z ∈ Metric.ball (c : β„‚) r) := taylorSeries_eq_on_ball' hz hf
f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r ⊒ βˆƒ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))
f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z ⊒ βˆƒ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real_on_ball
[159, 1]
[183, 8]
refine ⟨fun x ↦ βˆ‘' (n : β„•), (↑n !)⁻¹ * (D n) * (x - c) ^ n, fun x hx ↦ ?_, fun x hx ↦ ?_⟩
f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z ⊒ βˆƒ F, DifferentiableOn ℝ F (Set.Ioo (c - r) (c + r)) ∧ Set.EqOn (f ∘ ofReal') (ofReal' ∘ F) (Set.Ioo (c - r) (c + r))
case refine_1 f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊒ DifferentiableWithinAt ℝ (fun x => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x case refine_2 f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) Hz : βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r H : βˆ€ ⦃z : ℂ⦄, z ∈ Metric.ball (↑c) r β†’ βˆ‘' (n : β„•), (↑n !)⁻¹ * iteratedDeriv n f ↑c * (z - ↑c) ^ n = f z x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊒ (f ∘ ofReal') x = (ofReal' ∘ fun x => βˆ‘' (n : β„•), (↑n !)⁻¹ * D n * (x - c) ^ n) x
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real_on_ball
[159, 1]
[183, 8]
intro x hx
f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) ⊒ βˆ€ x ∈ Set.Ioo (c - r) (c + r), ↑x ∈ Metric.ball (↑c) r
f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊒ ↑x ∈ Metric.ball (↑c) r
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real_on_ball
[159, 1]
[183, 8]
refine Metric.mem_ball.mpr ?_
f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊒ ↑x ∈ Metric.ball (↑c) r
f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊒ dist ↑x ↑c < r
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real_on_ball
[159, 1]
[183, 8]
rw [dist_eq, ← ofReal_sub, abs_ofReal, abs_sub_lt_iff, sub_lt_iff_lt_add', sub_lt_comm]
f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊒ dist ↑x ↑c < r
f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊒ x < c + r ∧ c - r < x
https://github.com/MichaelStollBayreuth/EulerProducts.git
21e07835d1a467b99b5c3c9390d61ae69404445d
EulerProducts/Auxiliary.lean
Complex.realValued_of_iteratedDeriv_real_on_ball
[159, 1]
[183, 8]
exact and_comm.mpr hx
f : β„‚ β†’ β„‚ r c : ℝ hf : DifferentiableOn β„‚ f (Metric.ball (↑c) r) D : β„• β†’ ℝ hd : βˆ€ (n : β„•), iteratedDeriv n f ↑c = ↑(D n) x : ℝ hx : x ∈ Set.Ioo (c - r) (c + r) ⊒ x < c + r ∧ c - r < x
no goals