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/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_ball | [296, 1] | [305, 8] | rw [←Metric.closedBall_zero, ←annulus_oc] | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ ∫ (x : ℂ) in closedBall c r \ {c}, f x = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ ∫ (x : ℂ) in annulus_oc c 0 r, f x = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_ball | [296, 1] | [305, 8] | apply fubini_annulus | E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ ∫ (x : ℂ) in annulus_oc c 0 r, f x = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), f (circleMap c s t) | case fc
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ ContinuousOn (fun z => f z) (annulus_cc c 0 r)
case r0p
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ 0 ≤ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_ball | [296, 1] | [305, 8] | simpa only [annulus_cc, Metric.ball_zero, diff_empty] | case fc
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ ContinuousOn (fun z => f z) (annulus_cc c 0 r) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_ball | [296, 1] | [305, 8] | rfl | case r0p
E : Type
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : CompleteSpace E
f : ℂ → E
c : ℂ
r : ℝ
fc : ContinuousOn f (closedBall c r)
⊢ 0 ≤ 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Complex.volume_closedBall' | [308, 1] | [317, 26] | have c : ContinuousOn (fun _ : ℂ ↦ (1 : ℝ)) (closedBall c r) := continuousOn_const | c : ℂ
r : ℝ
rp : r ≥ 0
⊢ (↑volume (closedBall c r)).toReal = π * r ^ 2 | c✝ : ℂ
r : ℝ
rp : r ≥ 0
c : ContinuousOn (fun x => 1) (closedBall c✝ r)
⊢ (↑volume (closedBall c✝ r)).toReal = π * r ^ 2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Complex.volume_closedBall' | [308, 1] | [317, 26] | have f := fubini_ball c | c✝ : ℂ
r : ℝ
rp : r ≥ 0
c : ContinuousOn (fun x => 1) (closedBall c✝ r)
⊢ (↑volume (closedBall c✝ r)).toReal = π * r ^ 2 | c✝ : ℂ
r : ℝ
rp : r ≥ 0
c : ContinuousOn (fun x => 1) (closedBall c✝ r)
f : ∫ (z : ℂ) in closedBall c✝ r, 1 = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), 1
⊢ (↑volume (closedBall c✝ r)).toReal = π * r ^ 2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Complex.volume_closedBall' | [308, 1] | [317, 26] | clear c | c✝ : ℂ
r : ℝ
rp : r ≥ 0
c : ContinuousOn (fun x => 1) (closedBall c✝ r)
f : ∫ (z : ℂ) in closedBall c✝ r, 1 = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), 1
⊢ (↑volume (closedBall c✝ r)).toReal = π * r ^ 2 | c : ℂ
r : ℝ
rp : r ≥ 0
f : ∫ (z : ℂ) in closedBall c r, 1 = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), 1
⊢ (↑volume (closedBall c r)).toReal = π * r ^ 2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Complex.volume_closedBall' | [308, 1] | [317, 26] | simp only [ENNReal.toReal_ofReal Real.two_pi_pos.le, ←
intervalIntegral.integral_of_le rp, integral_const, Measure.restrict_apply, MeasurableSet.univ,
univ_inter, Algebra.id.smul_eq_mul, mul_one, Real.volume_Ioc, tsub_zero,
intervalIntegral.integral_mul_const, integral_id, zero_pow, Ne, bit0_eq_zero,
Nat.one_ne_zero, not_false_iff] at f | c : ℂ
r : ℝ
rp : r ≥ 0
f : ∫ (z : ℂ) in closedBall c r, 1 = ∫ (s : ℝ) in Ioc 0 r, s • ∫ (t : ℝ) in Ioc 0 (2 * π), 1
⊢ (↑volume (closedBall c r)).toReal = π * r ^ 2 | c : ℂ
r : ℝ
rp : r ≥ 0
f : (↑volume (closedBall c r)).toReal = r ^ 2 / 2 * (2 * π)
⊢ (↑volume (closedBall c r)).toReal = π * r ^ 2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Complex.volume_closedBall' | [308, 1] | [317, 26] | ring_nf at f ⊢ | c : ℂ
r : ℝ
rp : r ≥ 0
f : (↑volume (closedBall c r)).toReal = r ^ 2 / 2 * (2 * π)
⊢ (↑volume (closedBall c r)).toReal = π * r ^ 2 | c : ℂ
r : ℝ
rp : r ≥ 0
f : (↑volume (closedBall c r)).toReal = r ^ 2 * π
⊢ (↑volume (closedBall c r)).toReal = r ^ 2 * π |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Complex.volume_closedBall' | [308, 1] | [317, 26] | exact f | c : ℂ
r : ℝ
rp : r ≥ 0
f : (↑volume (closedBall c r)).toReal = r ^ 2 * π
⊢ (↑volume (closedBall c r)).toReal = r ^ 2 * π | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | NiceVolume.closedBall | [320, 1] | [332, 14] | simp only [Complex.volume_closedBall] | c : ℂ
r : ℝ
rp : r > 0
⊢ ↑volume (Metric.closedBall c r) < ⊤ | c : ℂ
r : ℝ
rp : r > 0
⊢ ENNReal.ofReal r ^ 2 * ↑NNReal.pi < ⊤ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | NiceVolume.closedBall | [320, 1] | [332, 14] | apply ENNReal.mul_lt_top | c : ℂ
r : ℝ
rp : r > 0
⊢ ENNReal.ofReal r ^ 2 * ↑NNReal.pi < ⊤ | case a
c : ℂ
r : ℝ
rp : r > 0
⊢ ENNReal.ofReal r ^ 2 ≠ ⊤
case a
c : ℂ
r : ℝ
rp : r > 0
⊢ ↑NNReal.pi ≠ ⊤ |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | NiceVolume.closedBall | [320, 1] | [332, 14] | simp only [ne_eq, ENNReal.pow_eq_top_iff, ENNReal.ofReal_ne_top, OfNat.ofNat_ne_zero,
not_false_eq_true, and_true] | case a
c : ℂ
r : ℝ
rp : r > 0
⊢ ENNReal.ofReal r ^ 2 ≠ ⊤ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | NiceVolume.closedBall | [320, 1] | [332, 14] | simp only [ne_eq, ENNReal.coe_ne_top, not_false_eq_true] | case a
c : ℂ
r : ℝ
rp : r > 0
⊢ ↑NNReal.pi ≠ ⊤ | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | NiceVolume.closedBall | [320, 1] | [332, 14] | simp only [Complex.volume_closedBall, gt_iff_lt, CanonicallyOrderedCommSemiring.mul_pos,
ENNReal.coe_pos, NNReal.pi_pos, and_true] | c : ℂ
r : ℝ
rp : r > 0
⊢ ↑volume (Metric.closedBall c r) > 0 | c : ℂ
r : ℝ
rp : r > 0
⊢ 0 < ENNReal.ofReal r ^ 2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | NiceVolume.closedBall | [320, 1] | [332, 14] | apply ENNReal.pow_pos | c : ℂ
r : ℝ
rp : r > 0
⊢ 0 < ENNReal.ofReal r ^ 2 | case a
c : ℂ
r : ℝ
rp : r > 0
⊢ 0 < ENNReal.ofReal r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | NiceVolume.closedBall | [320, 1] | [332, 14] | bound | case a
c : ℂ
r : ℝ
rp : r > 0
⊢ 0 < ENNReal.ofReal r | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | LocalVolume.closedBall | [335, 1] | [343, 73] | apply LocalVolume.closure_interior | c : ℂ
r : ℝ
rp : r > 0
⊢ LocalVolumeSet (Metric.closedBall c r) | case bp
c : ℂ
r : ℝ
rp : r > 0
⊢ ∀ (x : ℂ), ∀ r > 0, ↑volume (ball x r) > 0
case ci
c : ℂ
r : ℝ
rp : r > 0
⊢ Metric.closedBall c r ⊆ closure (interior (Metric.closedBall c r)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | LocalVolume.closedBall | [335, 1] | [343, 73] | intro x r rp | case bp
c : ℂ
r : ℝ
rp : r > 0
⊢ ∀ (x : ℂ), ∀ r > 0, ↑volume (ball x r) > 0 | case bp
c : ℂ
r✝ : ℝ
rp✝ : r✝ > 0
x : ℂ
r : ℝ
rp : r > 0
⊢ ↑volume (ball x r) > 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | LocalVolume.closedBall | [335, 1] | [343, 73] | simp only [Complex.volume_ball, gt_iff_lt, CanonicallyOrderedCommSemiring.mul_pos,
ENNReal.coe_pos, NNReal.pi_pos, and_true] | case bp
c : ℂ
r✝ : ℝ
rp✝ : r✝ > 0
x : ℂ
r : ℝ
rp : r > 0
⊢ ↑volume (ball x r) > 0 | case bp
c : ℂ
r✝ : ℝ
rp✝ : r✝ > 0
x : ℂ
r : ℝ
rp : r > 0
⊢ 0 < ENNReal.ofReal r ^ 2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | LocalVolume.closedBall | [335, 1] | [343, 73] | apply ENNReal.pow_pos | case bp
c : ℂ
r✝ : ℝ
rp✝ : r✝ > 0
x : ℂ
r : ℝ
rp : r > 0
⊢ 0 < ENNReal.ofReal r ^ 2 | case bp.a
c : ℂ
r✝ : ℝ
rp✝ : r✝ > 0
x : ℂ
r : ℝ
rp : r > 0
⊢ 0 < ENNReal.ofReal r |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | LocalVolume.closedBall | [335, 1] | [343, 73] | bound | case bp.a
c : ℂ
r✝ : ℝ
rp✝ : r✝ > 0
x : ℂ
r : ℝ
rp : r > 0
⊢ 0 < ENNReal.ofReal r | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | LocalVolume.closedBall | [335, 1] | [343, 73] | have rz := rp.ne' | case ci
c : ℂ
r : ℝ
rp : r > 0
⊢ Metric.closedBall c r ⊆ closure (interior (Metric.closedBall c r)) | case ci
c : ℂ
r : ℝ
rp : r > 0
rz : r ≠ 0
⊢ Metric.closedBall c r ⊆ closure (interior (Metric.closedBall c r)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | LocalVolume.closedBall | [335, 1] | [343, 73] | simp only [interior_closedBall c rz, closure_ball c rz, subset_refl] | case ci
c : ℂ
r : ℝ
rp : r > 0
rz : r ≠ 0
⊢ Metric.closedBall c r ⊆ closure (interior (Metric.closedBall c r)) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | set fl := fun n z ↦ log (f n z) | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have near1 : ∀ n z, z ∈ s → abs (f n z - 1) ≤ 1 / 2 := by
intro n z zs
calc abs (f n z - 1)
_ ≤ c * a ^ n := hf n z zs
_ ≤ (1 / 2 : ℝ) * (1:ℝ) ^ n := by bound
_ = 1 / 2 := by norm_num | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have near1' : ∀ n z, z ∈ s → abs (f n z - 1) < 1 := fun n z zs ↦
lt_of_le_of_lt (near1 n z zs) (by linarith) | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have expfl : ∀ n z, z ∈ s → exp (fl n z) = f n z := by
intro n z zs; refine Complex.exp_log ?_
exact near_one_avoids_zero (near1' n z zs) | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have hl : ∀ n, AnalyticOn ℂ (fl n) s := fun n ↦
(h n).log (fun z m ↦ mem_slitPlane_of_near_one (near1' n z m)) | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | set c2 := 2 * c | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have hfl : ∀ n z, z ∈ s → abs (fl n z) ≤ c2 * a ^ n := by
intro n z zs
calc abs (fl n z)
_ = abs (log (f n z)) := rfl
_ ≤ 2 * abs (f n z - 1) := (log_small (near1 n z zs))
_ ≤ 2 * (c * a ^ n) := by linarith [hf n z zs]
_ = 2 * c * a ^ n := by ring
_ = c2 * a ^ n := rfl | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | rcases fast_series_converge o a0 a1 hl hfl with ⟨gl, gla, us⟩ | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 | case intro.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
us : HasSumOn (fun n => fl n) gl s
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | generalize hg : (fun z ↦ exp (gl z)) = g | case intro.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
us : HasSumOn (fun n => fl n) gl s
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 | case intro.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
us : HasSumOn (fun n => fl n) gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | use g | case intro.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
us : HasSumOn (fun n => fl n) gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
⊢ ∃ g, HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
us : HasSumOn (fun n => fl n) gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
⊢ HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | refine ⟨?_, ?_, ?_⟩ | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
us : HasSumOn (fun n => fl n) gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
⊢ HasProdOn f g s ∧ AnalyticOn ℂ g s ∧ ∀ z ∈ s, g z ≠ 0 | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
us : HasSumOn (fun n => fl n) gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
⊢ HasProdOn f g s
case h.refine_2
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
us : HasSumOn (fun n => fl n) gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
⊢ AnalyticOn ℂ g s
case h.refine_3
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
us : HasSumOn (fun n => fl n) gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
⊢ ∀ z ∈ s, g z ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | intro n z zs | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
⊢ ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2 | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
n : ℕ
z : ℂ
zs : z ∈ s
⊢ Complex.abs (f n z - 1) ≤ 1 / 2 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | calc abs (f n z - 1)
_ ≤ c * a ^ n := hf n z zs
_ ≤ (1 / 2 : ℝ) * (1:ℝ) ^ n := by bound
_ = 1 / 2 := by norm_num | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
n : ℕ
z : ℂ
zs : z ∈ s
⊢ Complex.abs (f n z - 1) ≤ 1 / 2 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | bound | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
n : ℕ
z : ℂ
zs : z ∈ s
⊢ c * a ^ n ≤ 1 / 2 * 1 ^ n | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | norm_num | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
n : ℕ
z : ℂ
zs : z ∈ s
⊢ 1 / 2 * 1 ^ n = 1 / 2 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | linarith | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
n : ℕ
z : ℂ
zs : z ∈ s
⊢ 1 / 2 < 1 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | intro n z zs | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
⊢ ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
n : ℕ
z : ℂ
zs : z ∈ s
⊢ (fl n z).exp = f n z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | refine Complex.exp_log ?_ | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
n : ℕ
z : ℂ
zs : z ∈ s
⊢ (fl n z).exp = f n z | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
n : ℕ
z : ℂ
zs : z ∈ s
⊢ f n z ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | exact near_one_avoids_zero (near1' n z zs) | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
n : ℕ
z : ℂ
zs : z ∈ s
⊢ f n z ≠ 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | intro n z zs | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
⊢ ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
n : ℕ
z : ℂ
zs : z ∈ s
⊢ Complex.abs (fl n z) ≤ c2 * a ^ n |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | calc abs (fl n z)
_ = abs (log (f n z)) := rfl
_ ≤ 2 * abs (f n z - 1) := (log_small (near1 n z zs))
_ ≤ 2 * (c * a ^ n) := by linarith [hf n z zs]
_ = 2 * c * a ^ n := by ring
_ = c2 * a ^ n := rfl | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
n : ℕ
z : ℂ
zs : z ∈ s
⊢ Complex.abs (fl n z) ≤ c2 * a ^ n | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | linarith [hf n z zs] | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
n : ℕ
z : ℂ
zs : z ∈ s
⊢ 2 * Complex.abs (f n z - 1) ≤ 2 * (c * a ^ n) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | ring | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
n : ℕ
z : ℂ
zs : z ∈ s
⊢ 2 * (c * a ^ n) = 2 * c * a ^ n | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | intro z zs | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
us : HasSumOn (fun n => fl n) gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
⊢ HasProdOn f g s | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
us : HasSumOn (fun n => fl n) gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
z : ℂ
zs : z ∈ s
⊢ HasProd (fun n => f n z) (g z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | specialize us z zs | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
us : HasSumOn (fun n => fl n) gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
z : ℂ
zs : z ∈ s
⊢ HasProd (fun n => f n z) (g z) | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
z : ℂ
zs : z ∈ s
us : HasSum (fun n => (fun n => fl n) n z) (gl z)
⊢ HasProd (fun n => f n z) (g z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | simp at us | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
z : ℂ
zs : z ∈ s
us : HasSum (fun n => (fun n => fl n) n z) (gl z)
⊢ HasProd (fun n => f n z) (g z) | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
z : ℂ
zs : z ∈ s
us : HasSum (fun n => fl n z) (gl z)
⊢ HasProd (fun n => f n z) (g z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have comp :
Filter.Tendsto (exp ∘ fun N : Finset ℕ ↦ N.sum fun n ↦ fl n z) atTop (𝓝 (exp (gl z))) :=
Filter.Tendsto.comp (Continuous.tendsto Complex.continuous_exp _) us | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
z : ℂ
zs : z ∈ s
us : HasSum (fun n => fl n z) (gl z)
⊢ HasProd (fun n => f n z) (g z) | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
z : ℂ
zs : z ∈ s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp ∘ fun N => N.sum fun n => fl n z) atTop (𝓝 (gl z).exp)
⊢ HasProd (fun n => f n z) (g z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have expsum0 : (exp ∘ fun N : Finset ℕ ↦ N.sum fun n ↦ fl n z) = fun N : Finset ℕ ↦
N.prod fun n ↦ f n z := by
apply funext; intro N; simp; rw [Complex.exp_sum]; simp_rw [expfl _ z zs] | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
z : ℂ
zs : z ∈ s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp ∘ fun N => N.sum fun n => fl n z) atTop (𝓝 (gl z).exp)
⊢ HasProd (fun n => f n z) (g z) | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
z : ℂ
zs : z ∈ s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp ∘ fun N => N.sum fun n => fl n z) atTop (𝓝 (gl z).exp)
expsum0 : (exp ∘ fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
⊢ HasProd (fun n => f n z) (g z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | rw [expsum0] at comp | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
z : ℂ
zs : z ∈ s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp ∘ fun N => N.sum fun n => fl n z) atTop (𝓝 (gl z).exp)
expsum0 : (exp ∘ fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
⊢ HasProd (fun n => f n z) (g z) | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
z : ℂ
zs : z ∈ s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (fun N => N.prod fun n => f n z) atTop (𝓝 (gl z).exp)
expsum0 : (exp ∘ fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
⊢ HasProd (fun n => f n z) (g z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | rw [← hg] | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
z : ℂ
zs : z ∈ s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (fun N => N.prod fun n => f n z) atTop (𝓝 (gl z).exp)
expsum0 : (exp ∘ fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
⊢ HasProd (fun n => f n z) (g z) | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
z : ℂ
zs : z ∈ s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (fun N => N.prod fun n => f n z) atTop (𝓝 (gl z).exp)
expsum0 : (exp ∘ fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
⊢ HasProd (fun n => f n z) ((fun z => (gl z).exp) z) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | assumption | case h.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
z : ℂ
zs : z ∈ s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (fun N => N.prod fun n => f n z) atTop (𝓝 (gl z).exp)
expsum0 : (exp ∘ fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
⊢ HasProd (fun n => f n z) ((fun z => (gl z).exp) z) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | apply funext | f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
z : ℂ
zs : z ∈ s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp ∘ fun N => N.sum fun n => fl n z) atTop (𝓝 (gl z).exp)
⊢ (exp ∘ fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
z : ℂ
zs : z ∈ s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp ∘ fun N => N.sum fun n => fl n z) atTop (𝓝 (gl z).exp)
⊢ ∀ (x : Finset ℕ), (exp ∘ fun N => N.sum fun n => fl n z) x = x.prod fun n => f n z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | intro N | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
z : ℂ
zs : z ∈ s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp ∘ fun N => N.sum fun n => fl n z) atTop (𝓝 (gl z).exp)
⊢ ∀ (x : Finset ℕ), (exp ∘ fun N => N.sum fun n => fl n z) x = x.prod fun n => f n z | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
z : ℂ
zs : z ∈ s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp ∘ fun N => N.sum fun n => fl n z) atTop (𝓝 (gl z).exp)
N : Finset ℕ
⊢ (exp ∘ fun N => N.sum fun n => fl n z) N = N.prod fun n => f n z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | simp | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
z : ℂ
zs : z ∈ s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp ∘ fun N => N.sum fun n => fl n z) atTop (𝓝 (gl z).exp)
N : Finset ℕ
⊢ (exp ∘ fun N => N.sum fun n => fl n z) N = N.prod fun n => f n z | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
z : ℂ
zs : z ∈ s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp ∘ fun N => N.sum fun n => fl n z) atTop (𝓝 (gl z).exp)
N : Finset ℕ
⊢ (N.sum fun n => fl n z).exp = N.prod fun n => f n z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | rw [Complex.exp_sum] | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
z : ℂ
zs : z ∈ s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp ∘ fun N => N.sum fun n => fl n z) atTop (𝓝 (gl z).exp)
N : Finset ℕ
⊢ (N.sum fun n => fl n z).exp = N.prod fun n => f n z | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
z : ℂ
zs : z ∈ s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp ∘ fun N => N.sum fun n => fl n z) atTop (𝓝 (gl z).exp)
N : Finset ℕ
⊢ (N.prod fun x => (fl x z).exp) = N.prod fun n => f n z |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | simp_rw [expfl _ z zs] | case h
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
z : ℂ
zs : z ∈ s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp ∘ fun N => N.sum fun n => fl n z) atTop (𝓝 (gl z).exp)
N : Finset ℕ
⊢ (N.prod fun x => (fl x z).exp) = N.prod fun n => f n z | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | rw [← hg] | case h.refine_2
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
us : HasSumOn (fun n => fl n) gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
⊢ AnalyticOn ℂ g s | case h.refine_2
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
us : HasSumOn (fun n => fl n) gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
⊢ AnalyticOn ℂ (fun z => (gl z).exp) s |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | exact fun z zs ↦ AnalyticAt.exp.comp (gla z zs) | case h.refine_2
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
us : HasSumOn (fun n => fl n) gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
⊢ AnalyticOn ℂ (fun z => (gl z).exp) s | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | simp only [Complex.exp_ne_zero, Ne, not_false_iff, imp_true_iff, ← hg] | case h.refine_3
f : ℕ → ℂ → ℂ
s : Set ℂ
a c : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : a ≥ 0
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
fl : ℕ → ℂ → ℂ := fun n z => (f n z).log
near1 : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ 1 / 2
near1' : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) < 1
expfl : ∀ (n : ℕ), ∀ z ∈ s, (fl n z).exp = f n z
hl : ∀ (n : ℕ), AnalyticOn ℂ (fl n) s
c2 : ℝ := 2 * c
hfl : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (fl n z) ≤ c2 * a ^ n
gl : ℂ → ℂ
gla : AnalyticOn ℂ gl s
us : HasSumOn (fun n => fl n) gl s
g : ℂ → ℂ
hg : (fun z => (gl z).exp) = g
⊢ ∀ z ∈ s, g z ≠ 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | rcases fast_products_converge o c12 a0 a1 h hf with ⟨g, gp, ga, g0⟩ | f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
⊢ ProdExistsOn f s ∧ AnalyticOn ℂ (tprodOn f) s ∧ ∀ z ∈ s, tprodOn f z ≠ 0 | case intro.intro.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
g : ℂ → ℂ
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn ℂ g s
g0 : ∀ z ∈ s, g z ≠ 0
⊢ ProdExistsOn f s ∧ AnalyticOn ℂ (tprodOn f) s ∧ ∀ z ∈ s, tprodOn f z ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | refine ⟨?_, ?_, ?_⟩ | case intro.intro.intro
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
g : ℂ → ℂ
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn ℂ g s
g0 : ∀ z ∈ s, g z ≠ 0
⊢ ProdExistsOn f s ∧ AnalyticOn ℂ (tprodOn f) s ∧ ∀ z ∈ s, tprodOn f z ≠ 0 | case intro.intro.intro.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
g : ℂ → ℂ
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn ℂ g s
g0 : ∀ z ∈ s, g z ≠ 0
⊢ ProdExistsOn f s
case intro.intro.intro.refine_2
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
g : ℂ → ℂ
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn ℂ g s
g0 : ∀ z ∈ s, g z ≠ 0
⊢ AnalyticOn ℂ (tprodOn f) s
case intro.intro.intro.refine_3
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
g : ℂ → ℂ
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn ℂ g s
g0 : ∀ z ∈ s, g z ≠ 0
⊢ ∀ z ∈ s, tprodOn f z ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | exact fun z zs ↦ ⟨g z, gp z zs⟩ | case intro.intro.intro.refine_1
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
g : ℂ → ℂ
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn ℂ g s
g0 : ∀ z ∈ s, g z ≠ 0
⊢ ProdExistsOn f s | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | rwa [← analyticOn_congr o fun z zs ↦ (gp.tprodOn_eq z zs).symm] | case intro.intro.intro.refine_2
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
g : ℂ → ℂ
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn ℂ g s
g0 : ∀ z ∈ s, g z ≠ 0
⊢ AnalyticOn ℂ (tprodOn f) s | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | intro z zs | case intro.intro.intro.refine_3
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
g : ℂ → ℂ
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn ℂ g s
g0 : ∀ z ∈ s, g z ≠ 0
⊢ ∀ z ∈ s, tprodOn f z ≠ 0 | case intro.intro.intro.refine_3
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
g : ℂ → ℂ
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn ℂ g s
g0 : ∀ z ∈ s, g z ≠ 0
z : ℂ
zs : z ∈ s
⊢ tprodOn f z ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | rw [gp.tprodOn_eq z zs] | case intro.intro.intro.refine_3
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
g : ℂ → ℂ
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn ℂ g s
g0 : ∀ z ∈ s, g z ≠ 0
z : ℂ
zs : z ∈ s
⊢ tprodOn f z ≠ 0 | case intro.intro.intro.refine_3
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
g : ℂ → ℂ
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn ℂ g s
g0 : ∀ z ∈ s, g z ≠ 0
z : ℂ
zs : z ∈ s
⊢ g z ≠ 0 |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | exact g0 z zs | case intro.intro.intro.refine_3
f : ℕ → ℂ → ℂ
s : Set ℂ
c a : ℝ
o : IsOpen s
c12 : c ≤ 1 / 2
a0 : 0 ≤ a
a1 : a < 1
h : ∀ (n : ℕ), AnalyticOn ℂ (f n) s
hf : ∀ (n : ℕ), ∀ z ∈ s, Complex.abs (f n z - 1) ≤ c * a ^ n
g : ℂ → ℂ
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn ℂ g s
g0 : ∀ z ∈ s, g z ≠ 0
z : ℂ
zs : z ∈ s
⊢ g z ≠ 0 | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_pow | [112, 1] | [115, 72] | rw [HasProd] | f : ℕ → ℂ
g : ℂ
p : ℕ
h : HasProd f g
⊢ HasProd (fun n => f n ^ p) (g ^ p) | f : ℕ → ℂ
g : ℂ
p : ℕ
h : HasProd f g
⊢ Filter.Tendsto (fun s => s.prod fun b => f b ^ p) atTop (𝓝 (g ^ p)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_pow | [112, 1] | [115, 72] | simp_rw [Finset.prod_pow] | f : ℕ → ℂ
g : ℂ
p : ℕ
h : HasProd f g
⊢ Filter.Tendsto (fun s => s.prod fun b => f b ^ p) atTop (𝓝 (g ^ p)) | f : ℕ → ℂ
g : ℂ
p : ℕ
h : HasProd f g
⊢ Filter.Tendsto (fun s => (s.prod fun x => f x) ^ p) atTop (𝓝 (g ^ p)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_pow | [112, 1] | [115, 72] | exact Filter.Tendsto.comp (Continuous.tendsto (continuous_pow p) g) h | f : ℕ → ℂ
g : ℂ
p : ℕ
h : HasProd f g
⊢ Filter.Tendsto (fun s => (s.prod fun x => f x) ^ p) atTop (𝓝 (g ^ p)) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_pow' | [118, 1] | [120, 96] | rcases h with ⟨g, h⟩ | f : ℕ → ℂ
p : ℕ
h : ProdExists f
⊢ tprod f ^ p = ∏' (n : ℕ), f n ^ p | case intro
f : ℕ → ℂ
p : ℕ
g : ℂ
h : HasProd f g
⊢ tprod f ^ p = ∏' (n : ℕ), f n ^ p |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_pow' | [118, 1] | [120, 96] | rw [HasProd.tprod_eq h] | case intro
f : ℕ → ℂ
p : ℕ
g : ℂ
h : HasProd f g
⊢ tprod f ^ p = ∏' (n : ℕ), f n ^ p | case intro
f : ℕ → ℂ
p : ℕ
g : ℂ
h : HasProd f g
⊢ g ^ p = ∏' (n : ℕ), f n ^ p |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_pow' | [118, 1] | [120, 96] | rw [HasProd.tprod_eq _] | case intro
f : ℕ → ℂ
p : ℕ
g : ℂ
h : HasProd f g
⊢ g ^ p = ∏' (n : ℕ), f n ^ p | f : ℕ → ℂ
p : ℕ
g : ℂ
h : HasProd f g
⊢ HasProd (fun n => f n ^ p) (g ^ p) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_pow' | [118, 1] | [120, 96] | exact product_pow p h | f : ℕ → ℂ
p : ℕ
g : ℂ
h : HasProd f g
⊢ HasProd (fun n => f n ^ p) (g ^ p) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | rw [HasProd] at h ⊢ | a g : ℂ
f : ℕ → ℂ
h : HasProd f g
⊢ HasProd (Stream'.cons a f) (a * g) | a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
⊢ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (𝓝 (a * g)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | have ha := Filter.Tendsto.comp (Continuous.tendsto (continuous_mul_left a) g) h | a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
⊢ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (𝓝 (a * g)) | a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a * b) ∘ fun s => s.prod fun b => f b) atTop (𝓝 (a * g))
⊢ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (𝓝 (a * g)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | have s : ((fun z ↦ a * z) ∘ fun N : Finset ℕ ↦ N.prod f) =
(fun N : Finset ℕ ↦ N.prod (Stream'.cons a f)) ∘ push := by
apply funext; intro N; simp; exact push_prod | a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a * b) ∘ fun s => s.prod fun b => f b) atTop (𝓝 (a * g))
⊢ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (𝓝 (a * g)) | a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a * b) ∘ fun s => s.prod fun b => f b) atTop (𝓝 (a * g))
s : ((fun z => a * z) ∘ fun N => N.prod f) = (fun N => N.prod (Stream'.cons a f)) ∘ push
⊢ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (𝓝 (a * g)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | rw [s] at ha | a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a * b) ∘ fun s => s.prod fun b => f b) atTop (𝓝 (a * g))
s : ((fun z => a * z) ∘ fun N => N.prod f) = (fun N => N.prod (Stream'.cons a f)) ∘ push
⊢ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (𝓝 (a * g)) | a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun N => N.prod (Stream'.cons a f)) ∘ push) atTop (𝓝 (a * g))
s : ((fun z => a * z) ∘ fun N => N.prod f) = (fun N => N.prod (Stream'.cons a f)) ∘ push
⊢ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (𝓝 (a * g)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | exact tendsto_comp_push.mp ha | a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun N => N.prod (Stream'.cons a f)) ∘ push) atTop (𝓝 (a * g))
s : ((fun z => a * z) ∘ fun N => N.prod f) = (fun N => N.prod (Stream'.cons a f)) ∘ push
⊢ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (𝓝 (a * g)) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | apply funext | a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a * b) ∘ fun s => s.prod fun b => f b) atTop (𝓝 (a * g))
⊢ ((fun z => a * z) ∘ fun N => N.prod f) = (fun N => N.prod (Stream'.cons a f)) ∘ push | case h
a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a * b) ∘ fun s => s.prod fun b => f b) atTop (𝓝 (a * g))
⊢ ∀ (x : Finset ℕ), ((fun z => a * z) ∘ fun N => N.prod f) x = ((fun N => N.prod (Stream'.cons a f)) ∘ push) x |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | intro N | case h
a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a * b) ∘ fun s => s.prod fun b => f b) atTop (𝓝 (a * g))
⊢ ∀ (x : Finset ℕ), ((fun z => a * z) ∘ fun N => N.prod f) x = ((fun N => N.prod (Stream'.cons a f)) ∘ push) x | case h
a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a * b) ∘ fun s => s.prod fun b => f b) atTop (𝓝 (a * g))
N : Finset ℕ
⊢ ((fun z => a * z) ∘ fun N => N.prod f) N = ((fun N => N.prod (Stream'.cons a f)) ∘ push) N |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | simp | case h
a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a * b) ∘ fun s => s.prod fun b => f b) atTop (𝓝 (a * g))
N : Finset ℕ
⊢ ((fun z => a * z) ∘ fun N => N.prod f) N = ((fun N => N.prod (Stream'.cons a f)) ∘ push) N | case h
a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a * b) ∘ fun s => s.prod fun b => f b) atTop (𝓝 (a * g))
N : Finset ℕ
⊢ a * N.prod f = (push N).prod (Stream'.cons a f) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | exact push_prod | case h
a g : ℂ
f : ℕ → ℂ
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (𝓝 g)
ha : Filter.Tendsto ((fun b => a * b) ∘ fun s => s.prod fun b => f b) atTop (𝓝 (a * g))
N : Finset ℕ
⊢ a * N.prod f = (push N).prod (Stream'.cons a f) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons' | [134, 1] | [136, 95] | rcases h with ⟨g, h⟩ | a : ℂ
f : ℕ → ℂ
h : ProdExists f
⊢ tprod (Stream'.cons a f) = a * tprod f | case intro
a : ℂ
f : ℕ → ℂ
g : ℂ
h : HasProd f g
⊢ tprod (Stream'.cons a f) = a * tprod f |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons' | [134, 1] | [136, 95] | rw [HasProd.tprod_eq h] | case intro
a : ℂ
f : ℕ → ℂ
g : ℂ
h : HasProd f g
⊢ tprod (Stream'.cons a f) = a * tprod f | case intro
a : ℂ
f : ℕ → ℂ
g : ℂ
h : HasProd f g
⊢ tprod (Stream'.cons a f) = a * g |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons' | [134, 1] | [136, 95] | rw [HasProd.tprod_eq _] | case intro
a : ℂ
f : ℕ → ℂ
g : ℂ
h : HasProd f g
⊢ tprod (Stream'.cons a f) = a * g | a : ℂ
f : ℕ → ℂ
g : ℂ
h : HasProd f g
⊢ HasProd (fun b => Stream'.cons a f b) (a * g) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons' | [134, 1] | [136, 95] | exact product_cons h | a : ℂ
f : ℕ → ℂ
g : ℂ
h : HasProd f g
⊢ HasProd (fun b => Stream'.cons a f b) (a * g) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | have c := @product_cons (f 0)⁻¹ _ _ h | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
⊢ HasProd (fun n => f (n + 1)) (g / f 0) | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
c : HasProd (Stream'.cons (f 0)⁻¹ f) ((f 0)⁻¹ * g)
⊢ HasProd (fun n => f (n + 1)) (g / f 0) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | rw [HasProd] | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
c : HasProd (Stream'.cons (f 0)⁻¹ f) ((f 0)⁻¹ * g)
⊢ HasProd (fun n => f (n + 1)) (g / f 0) | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
c : HasProd (Stream'.cons (f 0)⁻¹ f) ((f 0)⁻¹ * g)
⊢ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (𝓝 (g / f 0)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | rw [inv_mul_eq_div, HasProd, ← tendsto_comp_push, ← tendsto_comp_push] at c | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
c : HasProd (Stream'.cons (f 0)⁻¹ f) ((f 0)⁻¹ * g)
⊢ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (𝓝 (g / f 0)) | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
c : Filter.Tendsto (((fun s => s.prod fun b => Stream'.cons (f 0)⁻¹ f b) ∘ push) ∘ push) atTop (𝓝 (g / f 0))
⊢ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (𝓝 (g / f 0)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | have s : ((fun N : Finset ℕ ↦ N.prod fun n ↦ (Stream'.cons (f 0)⁻¹ f) n) ∘ push) ∘ push =
fun N : Finset ℕ ↦ N.prod fun n ↦ f (n + 1) := by
clear c h g; apply funext; intro N; simp
nth_rw 2 [← Stream'.eta f]
simp only [←push_prod, Stream'.head, Stream'.tail, Stream'.get, ←mul_assoc, inv_mul_cancel f0,
one_mul] | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
c : Filter.Tendsto (((fun s => s.prod fun b => Stream'.cons (f 0)⁻¹ f b) ∘ push) ∘ push) atTop (𝓝 (g / f 0))
⊢ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (𝓝 (g / f 0)) | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
c : Filter.Tendsto (((fun s => s.prod fun b => Stream'.cons (f 0)⁻¹ f b) ∘ push) ∘ push) atTop (𝓝 (g / f 0))
s : ((fun N => N.prod fun n => Stream'.cons (f 0)⁻¹ f n) ∘ push) ∘ push = fun N => N.prod fun n => f (n + 1)
⊢ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (𝓝 (g / f 0)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | rw [s] at c | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
c : Filter.Tendsto (((fun s => s.prod fun b => Stream'.cons (f 0)⁻¹ f b) ∘ push) ∘ push) atTop (𝓝 (g / f 0))
s : ((fun N => N.prod fun n => Stream'.cons (f 0)⁻¹ f n) ∘ push) ∘ push = fun N => N.prod fun n => f (n + 1)
⊢ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (𝓝 (g / f 0)) | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
c : Filter.Tendsto (fun N => N.prod fun n => f (n + 1)) atTop (𝓝 (g / f 0))
s : ((fun N => N.prod fun n => Stream'.cons (f 0)⁻¹ f n) ∘ push) ∘ push = fun N => N.prod fun n => f (n + 1)
⊢ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (𝓝 (g / f 0)) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | assumption | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
c : Filter.Tendsto (fun N => N.prod fun n => f (n + 1)) atTop (𝓝 (g / f 0))
s : ((fun N => N.prod fun n => Stream'.cons (f 0)⁻¹ f n) ∘ push) ∘ push = fun N => N.prod fun n => f (n + 1)
⊢ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (𝓝 (g / f 0)) | no goals |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | clear c h g | f : ℕ → ℂ
g : ℂ
f0 : f 0 ≠ 0
h : HasProd f g
c : Filter.Tendsto (((fun s => s.prod fun b => Stream'.cons (f 0)⁻¹ f b) ∘ push) ∘ push) atTop (𝓝 (g / f 0))
⊢ ((fun N => N.prod fun n => Stream'.cons (f 0)⁻¹ f n) ∘ push) ∘ push = fun N => N.prod fun n => f (n + 1) | f : ℕ → ℂ
f0 : f 0 ≠ 0
⊢ ((fun N => N.prod fun n => Stream'.cons (f 0)⁻¹ f n) ∘ push) ∘ push = fun N => N.prod fun n => f (n + 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | apply funext | f : ℕ → ℂ
f0 : f 0 ≠ 0
⊢ ((fun N => N.prod fun n => Stream'.cons (f 0)⁻¹ f n) ∘ push) ∘ push = fun N => N.prod fun n => f (n + 1) | case h
f : ℕ → ℂ
f0 : f 0 ≠ 0
⊢ ∀ (x : Finset ℕ), (((fun N => N.prod fun n => Stream'.cons (f 0)⁻¹ f n) ∘ push) ∘ push) x = x.prod fun n => f (n + 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | intro N | case h
f : ℕ → ℂ
f0 : f 0 ≠ 0
⊢ ∀ (x : Finset ℕ), (((fun N => N.prod fun n => Stream'.cons (f 0)⁻¹ f n) ∘ push) ∘ push) x = x.prod fun n => f (n + 1) | case h
f : ℕ → ℂ
f0 : f 0 ≠ 0
N : Finset ℕ
⊢ (((fun N => N.prod fun n => Stream'.cons (f 0)⁻¹ f n) ∘ push) ∘ push) N = N.prod fun n => f (n + 1) |
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | simp | case h
f : ℕ → ℂ
f0 : f 0 ≠ 0
N : Finset ℕ
⊢ (((fun N => N.prod fun n => Stream'.cons (f 0)⁻¹ f n) ∘ push) ∘ push) N = N.prod fun n => f (n + 1) | case h
f : ℕ → ℂ
f0 : f 0 ≠ 0
N : Finset ℕ
⊢ ((push (push N)).prod fun n => Stream'.cons (f 0)⁻¹ f n) = N.prod fun n => f (n + 1) |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.