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stringclasses 147
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stringlengths 1
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stringlengths 6
10
| end
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|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_ball | [296, 1] | [305, 8] | rw [βMetric.closedBall_zero, βannulus_oc] | E : Type
instβΒ² : NormedAddCommGroup E
instβΒΉ : NormedSpace β E
instβ : CompleteSpace E
f : β β E
c : β
r : β
fc : ContinuousOn f (closedBall c r)
β’ β« (x : β) in closedBall c r \ {c}, f x = β« (s : β) in Ioc 0 r, s β’ β« (t : β) in Ioc 0 (2 * Ο), f (circleMap c s t) | E : Type
instβΒ² : NormedAddCommGroup E
instβΒΉ : NormedSpace β E
instβ : CompleteSpace E
f : β β E
c : β
r : β
fc : ContinuousOn f (closedBall c r)
β’ β« (x : β) in annulus_oc c 0 r, f x = β« (s : β) in Ioc 0 r, s β’ β« (t : β) in Ioc 0 (2 * Ο), f (circleMap c s t) | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ² : NormedAddCommGroup E
instβΒΉ : NormedSpace β E
instβ : CompleteSpace E
f : β β E
c : β
r : β
fc : ContinuousOn f (closedBall c r)
β’ β« (x : β) in closedBall c r \ {c}, f x = β« (s : β) in Ioc 0 r, s β’ β« (t : β) in Ioc 0 (2 * Ο), f (circleMap c s t)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_ball | [296, 1] | [305, 8] | apply fubini_annulus | E : Type
instβΒ² : NormedAddCommGroup E
instβΒΉ : NormedSpace β E
instβ : CompleteSpace E
f : β β E
c : β
r : β
fc : ContinuousOn f (closedBall c r)
β’ β« (x : β) in annulus_oc c 0 r, f x = β« (s : β) in Ioc 0 r, s β’ β« (t : β) in Ioc 0 (2 * Ο), f (circleMap c s t) | case fc
E : Type
instβΒ² : NormedAddCommGroup E
instβΒΉ : NormedSpace β E
instβ : CompleteSpace E
f : β β E
c : β
r : β
fc : ContinuousOn f (closedBall c r)
β’ ContinuousOn (fun z => f z) (annulus_cc c 0 r)
case r0p
E : Type
instβΒ² : NormedAddCommGroup E
instβΒΉ : NormedSpace β E
instβ : CompleteSpace E
f : β β E
c : β
r : β
fc : ContinuousOn f (closedBall c r)
β’ 0 β€ 0 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
instβΒ² : NormedAddCommGroup E
instβΒΉ : NormedSpace β E
instβ : CompleteSpace E
f : β β E
c : β
r : β
fc : ContinuousOn f (closedBall c r)
β’ β« (x : β) in annulus_oc c 0 r, f x = β« (s : β) in Ioc 0 r, s β’ β« (t : β) in Ioc 0 (2 * Ο), f (circleMap c s t)
TACTIC:
|
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
instβΒ² : NormedAddCommGroup E
instβΒΉ : NormedSpace β E
instβ : CompleteSpace E
f : β β E
c : β
r : β
fc : ContinuousOn f (closedBall c r)
β’ ContinuousOn (fun z => f z) (annulus_cc c 0 r) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case fc
E : Type
instβΒ² : NormedAddCommGroup E
instβΒΉ : NormedSpace β E
instβ : CompleteSpace E
f : β β E
c : β
r : β
fc : ContinuousOn f (closedBall c r)
β’ ContinuousOn (fun z => f z) (annulus_cc c 0 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | fubini_ball | [296, 1] | [305, 8] | rfl | case r0p
E : Type
instβΒ² : NormedAddCommGroup E
instβΒΉ : NormedSpace β E
instβ : CompleteSpace E
f : β β E
c : β
r : β
fc : ContinuousOn f (closedBall c r)
β’ 0 β€ 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case r0p
E : Type
instβΒ² : NormedAddCommGroup E
instβΒΉ : NormedSpace β E
instβ : CompleteSpace E
f : β β E
c : β
r : β
fc : ContinuousOn f (closedBall c r)
β’ 0 β€ 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Complex.volume_closedBall' | [308, 1] | [317, 26] | have c : ContinuousOn (fun _ : β β¦ (1 : β)) (closedBall c r) := continuousOn_const | c : β
r : β
rp : r β₯ 0
β’ (βvolume (closedBall c r)).toReal = Ο * r ^ 2 | cβ : β
r : β
rp : r β₯ 0
c : ContinuousOn (fun x => 1) (closedBall cβ r)
β’ (βvolume (closedBall cβ r)).toReal = Ο * r ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r : β
rp : r β₯ 0
β’ (βvolume (closedBall c r)).toReal = Ο * r ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Complex.volume_closedBall' | [308, 1] | [317, 26] | have f := fubini_ball c | cβ : β
r : β
rp : r β₯ 0
c : ContinuousOn (fun x => 1) (closedBall cβ r)
β’ (βvolume (closedBall cβ r)).toReal = Ο * r ^ 2 | cβ : β
r : β
rp : r β₯ 0
c : ContinuousOn (fun x => 1) (closedBall cβ r)
f : β« (z : β) in closedBall cβ r, 1 = β« (s : β) in Ioc 0 r, s β’ β« (t : β) in Ioc 0 (2 * Ο), 1
β’ (βvolume (closedBall cβ r)).toReal = Ο * r ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
r : β
rp : r β₯ 0
c : ContinuousOn (fun x => 1) (closedBall cβ r)
β’ (βvolume (closedBall cβ r)).toReal = Ο * r ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Complex.volume_closedBall' | [308, 1] | [317, 26] | clear c | cβ : β
r : β
rp : r β₯ 0
c : ContinuousOn (fun x => 1) (closedBall cβ r)
f : β« (z : β) in closedBall cβ r, 1 = β« (s : β) in Ioc 0 r, s β’ β« (t : β) in Ioc 0 (2 * Ο), 1
β’ (βvolume (closedBall cβ r)).toReal = Ο * r ^ 2 | c : β
r : β
rp : r β₯ 0
f : β« (z : β) in closedBall c r, 1 = β« (s : β) in Ioc 0 r, s β’ β« (t : β) in Ioc 0 (2 * Ο), 1
β’ (βvolume (closedBall c r)).toReal = Ο * r ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
cβ : β
r : β
rp : r β₯ 0
c : ContinuousOn (fun x => 1) (closedBall cβ r)
f : β« (z : β) in closedBall cβ r, 1 = β« (s : β) in Ioc 0 r, s β’ β« (t : β) in Ioc 0 (2 * Ο), 1
β’ (βvolume (closedBall cβ r)).toReal = Ο * r ^ 2
TACTIC:
|
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, β
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 : β
r : β
rp : r β₯ 0
f : β« (z : β) in closedBall c r, 1 = β« (s : β) in Ioc 0 r, s β’ β« (t : β) in Ioc 0 (2 * Ο), 1
β’ (βvolume (closedBall c r)).toReal = Ο * r ^ 2 | c : β
r : β
rp : r β₯ 0
f : (βvolume (closedBall c r)).toReal = r ^ 2 / 2 * (2 * Ο)
β’ (βvolume (closedBall c r)).toReal = Ο * r ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r : β
rp : r β₯ 0
f : β« (z : β) in closedBall c r, 1 = β« (s : β) in Ioc 0 r, s β’ β« (t : β) in Ioc 0 (2 * Ο), 1
β’ (βvolume (closedBall c r)).toReal = Ο * r ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Complex.volume_closedBall' | [308, 1] | [317, 26] | ring_nf at f β’ | c : β
r : β
rp : r β₯ 0
f : (βvolume (closedBall c r)).toReal = r ^ 2 / 2 * (2 * Ο)
β’ (βvolume (closedBall c r)).toReal = Ο * r ^ 2 | c : β
r : β
rp : r β₯ 0
f : (βvolume (closedBall c r)).toReal = r ^ 2 * Ο
β’ (βvolume (closedBall c r)).toReal = r ^ 2 * Ο | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r : β
rp : r β₯ 0
f : (βvolume (closedBall c r)).toReal = r ^ 2 / 2 * (2 * Ο)
β’ (βvolume (closedBall c r)).toReal = Ο * r ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | Complex.volume_closedBall' | [308, 1] | [317, 26] | exact f | c : β
r : β
rp : r β₯ 0
f : (βvolume (closedBall c r)).toReal = r ^ 2 * Ο
β’ (βvolume (closedBall c r)).toReal = r ^ 2 * Ο | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r : β
rp : r β₯ 0
f : (βvolume (closedBall c r)).toReal = r ^ 2 * Ο
β’ (βvolume (closedBall c r)).toReal = r ^ 2 * Ο
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | NiceVolume.closedBall | [320, 1] | [332, 14] | simp only [Complex.volume_closedBall] | c : β
r : β
rp : r > 0
β’ βvolume (Metric.closedBall c r) < β€ | c : β
r : β
rp : r > 0
β’ ENNReal.ofReal r ^ 2 * βNNReal.pi < β€ | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r : β
rp : r > 0
β’ βvolume (Metric.closedBall c r) < β€
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | NiceVolume.closedBall | [320, 1] | [332, 14] | apply ENNReal.mul_lt_top | c : β
r : β
rp : r > 0
β’ ENNReal.ofReal r ^ 2 * βNNReal.pi < β€ | case a
c : β
r : β
rp : r > 0
β’ ENNReal.ofReal r ^ 2 β β€
case a
c : β
r : β
rp : r > 0
β’ βNNReal.pi β β€ | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r : β
rp : r > 0
β’ ENNReal.ofReal r ^ 2 * βNNReal.pi < β€
TACTIC:
|
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 : β
r : β
rp : r > 0
β’ ENNReal.ofReal r ^ 2 β β€ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
c : β
r : β
rp : r > 0
β’ ENNReal.ofReal r ^ 2 β β€
TACTIC:
|
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 : β
r : β
rp : r > 0
β’ βNNReal.pi β β€ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
c : β
r : β
rp : r > 0
β’ βNNReal.pi β β€
TACTIC:
|
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 : β
r : β
rp : r > 0
β’ βvolume (Metric.closedBall c r) > 0 | c : β
r : β
rp : r > 0
β’ 0 < ENNReal.ofReal r ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r : β
rp : r > 0
β’ βvolume (Metric.closedBall c r) > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | NiceVolume.closedBall | [320, 1] | [332, 14] | apply ENNReal.pow_pos | c : β
r : β
rp : r > 0
β’ 0 < ENNReal.ofReal r ^ 2 | case a
c : β
r : β
rp : r > 0
β’ 0 < ENNReal.ofReal r | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r : β
rp : r > 0
β’ 0 < ENNReal.ofReal r ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | NiceVolume.closedBall | [320, 1] | [332, 14] | bound | case a
c : β
r : β
rp : r > 0
β’ 0 < ENNReal.ofReal r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case a
c : β
r : β
rp : r > 0
β’ 0 < ENNReal.ofReal r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | LocalVolume.closedBall | [335, 1] | [343, 73] | apply LocalVolume.closure_interior | c : β
r : β
rp : r > 0
β’ LocalVolumeSet (Metric.closedBall c r) | case bp
c : β
r : β
rp : r > 0
β’ β (x : β), β r > 0, βvolume (ball x r) > 0
case ci
c : β
r : β
rp : r > 0
β’ Metric.closedBall c r β closure (interior (Metric.closedBall c r)) | Please generate a tactic in lean4 to solve the state.
STATE:
c : β
r : β
rp : r > 0
β’ LocalVolumeSet (Metric.closedBall c r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | LocalVolume.closedBall | [335, 1] | [343, 73] | intro x r rp | case bp
c : β
r : β
rp : r > 0
β’ β (x : β), β r > 0, βvolume (ball x r) > 0 | case bp
c : β
rβ : β
rpβ : rβ > 0
x : β
r : β
rp : r > 0
β’ βvolume (ball x r) > 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case bp
c : β
r : β
rp : r > 0
β’ β (x : β), β r > 0, βvolume (ball x r) > 0
TACTIC:
|
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 : β
rβ : β
rpβ : rβ > 0
x : β
r : β
rp : r > 0
β’ βvolume (ball x r) > 0 | case bp
c : β
rβ : β
rpβ : rβ > 0
x : β
r : β
rp : r > 0
β’ 0 < ENNReal.ofReal r ^ 2 | Please generate a tactic in lean4 to solve the state.
STATE:
case bp
c : β
rβ : β
rpβ : rβ > 0
x : β
r : β
rp : r > 0
β’ βvolume (ball x r) > 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | LocalVolume.closedBall | [335, 1] | [343, 73] | apply ENNReal.pow_pos | case bp
c : β
rβ : β
rpβ : rβ > 0
x : β
r : β
rp : r > 0
β’ 0 < ENNReal.ofReal r ^ 2 | case bp.a
c : β
rβ : β
rpβ : rβ > 0
x : β
r : β
rp : r > 0
β’ 0 < ENNReal.ofReal r | Please generate a tactic in lean4 to solve the state.
STATE:
case bp
c : β
rβ : β
rpβ : rβ > 0
x : β
r : β
rp : r > 0
β’ 0 < ENNReal.ofReal r ^ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | LocalVolume.closedBall | [335, 1] | [343, 73] | bound | case bp.a
c : β
rβ : β
rpβ : rβ > 0
x : β
r : β
rp : r > 0
β’ 0 < ENNReal.ofReal r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case bp.a
c : β
rβ : β
rpβ : rβ > 0
x : β
r : β
rp : r > 0
β’ 0 < ENNReal.ofReal r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/FubiniBall.lean | LocalVolume.closedBall | [335, 1] | [343, 73] | have rz := rp.ne' | case ci
c : β
r : β
rp : r > 0
β’ Metric.closedBall c r β closure (interior (Metric.closedBall c r)) | case ci
c : β
r : β
rp : r > 0
rz : r β 0
β’ Metric.closedBall c r β closure (interior (Metric.closedBall c r)) | Please generate a tactic in lean4 to solve the state.
STATE:
case ci
c : β
r : β
rp : r > 0
β’ Metric.closedBall c r β closure (interior (Metric.closedBall c r))
TACTIC:
|
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 : β
r : β
rp : r > 0
rz : r β 0
β’ Metric.closedBall c r β closure (interior (Metric.closedBall c r)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case ci
c : β
r : β
rp : r > 0
rz : r β 0
β’ Metric.closedBall c r β closure (interior (Metric.closedBall c r))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | set fl := fun n z β¦ log (f n z) | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have near1 : β n z, z β s β abs (f n z - 1) β€ 1 / 2 := by
intro n z zs
calc abs (f n z - 1)
_ β€ c * a ^ n := hf n z zs
_ β€ (1 / 2 : β) * (1:β) ^ n := by bound
_ = 1 / 2 := by norm_num | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have near1' : β n z, z β s β abs (f n z - 1) < 1 := fun n z zs β¦
lt_of_le_of_lt (near1 n z zs) (by linarith) | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have expfl : β n z, z β s β 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 : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have hl : β n, AnalyticOn β (fl n) s := fun n β¦
(h n).log (fun z m β¦ mem_slitPlane_of_near_one (near1' n z m)) | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | set c2 := 2 * c | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have hfl : β n z, z β s β abs (fl n z) β€ c2 * a ^ n := by
intro n z zs
calc abs (fl n z)
_ = abs (log (f n z)) := rfl
_ β€ 2 * abs (f n z - 1) := (log_small (near1 n z zs))
_ β€ 2 * (c * a ^ n) := by linarith [hf n z zs]
_ = 2 * c * a ^ n := by ring
_ = c2 * a ^ n := rfl | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0
TACTIC:
|
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 β¨gl, gla, usβ© | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | case intro.intro
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | generalize hg : (fun z β¦ exp (gl z)) = g | case intro.intro
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | case intro.intro
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | use g | case intro.intro
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ β g, HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | refine β¨?_, ?_, ?_β© | case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0 | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ HasProdOn f g s
case h.refine_2
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ AnalyticOn β g s
case h.refine_3
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ β z β s, g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ HasProdOn f g s β§ AnalyticOn β g s β§ β z β s, g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | intro n z zs | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
β’ β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2 | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
n : β
z : β
zs : z β s
β’ Complex.abs (f n z - 1) β€ 1 / 2 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
β’ β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | calc abs (f n z - 1)
_ β€ c * a ^ n := hf n z zs
_ β€ (1 / 2 : β) * (1:β) ^ n := by bound
_ = 1 / 2 := by norm_num | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
n : β
z : β
zs : z β s
β’ Complex.abs (f n z - 1) β€ 1 / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
n : β
z : β
zs : z β s
β’ Complex.abs (f n z - 1) β€ 1 / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | bound | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
n : β
z : β
zs : z β s
β’ c * a ^ n β€ 1 / 2 * 1 ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
n : β
z : β
zs : z β s
β’ c * a ^ n β€ 1 / 2 * 1 ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | norm_num | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
n : β
z : β
zs : z β s
β’ 1 / 2 * 1 ^ n = 1 / 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
n : β
z : β
zs : z β s
β’ 1 / 2 * 1 ^ n = 1 / 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | linarith | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
n : β
z : β
zs : z β s
β’ 1 / 2 < 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
n : β
z : β
zs : z β s
β’ 1 / 2 < 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | intro n z zs | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
β’ β (n : β), β z β s, (fl n z).exp = f n z | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
n : β
z : β
zs : z β s
β’ (fl n z).exp = f n z | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
β’ β (n : β), β z β s, (fl n z).exp = f n z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | refine Complex.exp_log ?_ | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
n : β
z : β
zs : z β s
β’ (fl n z).exp = f n z | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
n : β
z : β
zs : z β s
β’ f n z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
n : β
z : β
zs : z β s
β’ (fl n z).exp = f n z
TACTIC:
|
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 : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
n : β
z : β
zs : z β s
β’ f n z β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
n : β
z : β
zs : z β s
β’ f n z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | intro n z zs | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
β’ β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
n : β
z : β
zs : z β s
β’ Complex.abs (fl n z) β€ c2 * a ^ n | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
β’ β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
TACTIC:
|
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
_ β€ 2 * abs (f n z - 1) := (log_small (near1 n z zs))
_ β€ 2 * (c * a ^ n) := by linarith [hf n z zs]
_ = 2 * c * a ^ n := by ring
_ = c2 * a ^ n := rfl | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
n : β
z : β
zs : z β s
β’ Complex.abs (fl n z) β€ c2 * a ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
n : β
z : β
zs : z β s
β’ Complex.abs (fl n z) β€ c2 * a ^ n
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | linarith [hf n z zs] | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
n : β
z : β
zs : z β s
β’ 2 * Complex.abs (f n z - 1) β€ 2 * (c * a ^ n) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
n : β
z : β
zs : z β s
β’ 2 * Complex.abs (f n z - 1) β€ 2 * (c * a ^ n)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | ring | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
n : β
z : β
zs : z β s
β’ 2 * (c * a ^ n) = 2 * c * a ^ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
n : β
z : β
zs : z β s
β’ 2 * (c * a ^ n) = 2 * c * a ^ n
TACTIC:
|
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 : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ HasProdOn f g s | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
β’ HasProd (fun n => f n z) (g z) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ HasProdOn f g s
TACTIC:
|
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 : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
β’ HasProd (fun n => f n z) (g z) | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => (fun n => fl n) n z) (gl z)
β’ HasProd (fun n => f n z) (g z) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
β’ HasProd (fun n => f n z) (g z)
TACTIC:
|
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 : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => (fun n => fl n) n z) (gl z)
β’ HasProd (fun n => f n z) (g z) | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
β’ HasProd (fun n => f n z) (g z) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => (fun n => fl n) n z) (gl z)
β’ HasProd (fun n => f n z) (g z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have comp :
Filter.Tendsto (exp β fun N : Finset β β¦ N.sum fun n β¦ fl n z) atTop (π (exp (gl z))) :=
Filter.Tendsto.comp (Continuous.tendsto Complex.continuous_exp _) us | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
β’ HasProd (fun n => f n z) (g z) | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
β’ HasProd (fun n => f n z) (g z) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
β’ HasProd (fun n => f n z) (g z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | have expsum0 : (exp β fun N : Finset β β¦ N.sum fun n β¦ fl n z) = fun N : Finset β β¦
N.prod fun n β¦ f n z := by
apply funext; intro N; simp; rw [Complex.exp_sum]; simp_rw [expfl _ z zs] | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
β’ HasProd (fun n => f n z) (g z) | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
expsum0 : (exp β fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
β’ HasProd (fun n => f n z) (g z) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
β’ HasProd (fun n => f n z) (g z)
TACTIC:
|
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 : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
expsum0 : (exp β fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
β’ HasProd (fun n => f n z) (g z) | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (fun N => N.prod fun n => f n z) atTop (π (gl z).exp)
expsum0 : (exp β fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
β’ HasProd (fun n => f n z) (g z) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
expsum0 : (exp β fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
β’ HasProd (fun n => f n z) (g z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | rw [β hg] | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (fun N => N.prod fun n => f n z) atTop (π (gl z).exp)
expsum0 : (exp β fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
β’ HasProd (fun n => f n z) (g z) | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (fun N => N.prod fun n => f n z) atTop (π (gl z).exp)
expsum0 : (exp β fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
β’ HasProd (fun n => f n z) ((fun z => (gl z).exp) z) | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (fun N => N.prod fun n => f n z) atTop (π (gl z).exp)
expsum0 : (exp β fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
β’ HasProd (fun n => f n z) (g z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | assumption | case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (fun N => N.prod fun n => f n z) atTop (π (gl z).exp)
expsum0 : (exp β fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
β’ HasProd (fun n => f n z) ((fun z => (gl z).exp) z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (fun N => N.prod fun n => f n z) atTop (π (gl z).exp)
expsum0 : (exp β fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
β’ HasProd (fun n => f n z) ((fun z => (gl z).exp) z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | apply funext | f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
β’ (exp β fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z | case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
β’ β (x : Finset β), (exp β fun N => N.sum fun n => fl n z) x = x.prod fun n => f n z | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
β’ (exp β fun N => N.sum fun n => fl n z) = fun N => N.prod fun n => f n z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | intro N | case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
β’ β (x : Finset β), (exp β fun N => N.sum fun n => fl n z) x = x.prod fun n => f n z | case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
N : Finset β
β’ (exp β fun N => N.sum fun n => fl n z) N = N.prod fun n => f n z | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
β’ β (x : Finset β), (exp β fun N => N.sum fun n => fl n z) x = x.prod fun n => f n z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | simp | case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
N : Finset β
β’ (exp β fun N => N.sum fun n => fl n z) N = N.prod fun n => f n z | case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
N : Finset β
β’ (N.sum fun n => fl n z).exp = N.prod fun n => f n z | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
N : Finset β
β’ (exp β fun N => N.sum fun n => fl n z) N = N.prod fun n => f n z
TACTIC:
|
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 : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
N : Finset β
β’ (N.sum fun n => fl n z).exp = N.prod fun n => f n z | case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
N : Finset β
β’ (N.prod fun x => (fl x z).exp) = N.prod fun n => f n z | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
N : Finset β
β’ (N.sum fun n => fl n z).exp = N.prod fun n => f n z
TACTIC:
|
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 : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
N : Finset β
β’ (N.prod fun x => (fl x z).exp) = N.prod fun n => f n z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
g : β β β
hg : (fun z => (gl z).exp) = g
z : β
zs : z β s
us : HasSum (fun n => fl n z) (gl z)
comp : Filter.Tendsto (exp β fun N => N.sum fun n => fl n z) atTop (π (gl z).exp)
N : Finset β
β’ (N.prod fun x => (fl x z).exp) = N.prod fun n => f n z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | rw [β hg] | case h.refine_2
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ AnalyticOn β g s | case h.refine_2
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ AnalyticOn β (fun z => (gl z).exp) s | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_2
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ AnalyticOn β g s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge | [57, 1] | [98, 75] | exact fun z zs β¦ AnalyticAt.exp.comp (gla z zs) | case h.refine_2
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ AnalyticOn β (fun z => (gl z).exp) s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_2
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ AnalyticOn β (fun z => (gl z).exp) s
TACTIC:
|
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, β hg] | case h.refine_3
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ β z β s, g z β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_3
f : β β β β β
s : Set β
a c : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : a β₯ 0
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
fl : β β β β β := fun n z => (f n z).log
near1 : β (n : β), β z β s, Complex.abs (f n z - 1) β€ 1 / 2
near1' : β (n : β), β z β s, Complex.abs (f n z - 1) < 1
expfl : β (n : β), β z β s, (fl n z).exp = f n z
hl : β (n : β), AnalyticOn β (fl n) s
c2 : β := 2 * c
hfl : β (n : β), β z β s, Complex.abs (fl n z) β€ c2 * a ^ n
gl : β β β
gla : AnalyticOn β gl s
us : HasSumOn (fun n => fl n) gl s
g : β β β
hg : (fun z => (gl z).exp) = g
β’ β z β s, g z β 0
TACTIC:
|
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 β¨g, gp, ga, g0β© | f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
β’ ProdExistsOn f s β§ AnalyticOn β (tprodOn f) s β§ β z β s, tprodOn f z β 0 | case intro.intro.intro
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ ProdExistsOn f s β§ AnalyticOn β (tprodOn f) s β§ β z β s, tprodOn f z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
β’ ProdExistsOn f s β§ AnalyticOn β (tprodOn f) s β§ β z β s, tprodOn f z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | refine β¨?_, ?_, ?_β© | case intro.intro.intro
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ ProdExistsOn f s β§ AnalyticOn β (tprodOn f) s β§ β z β s, tprodOn f z β 0 | case intro.intro.intro.refine_1
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ ProdExistsOn f s
case intro.intro.intro.refine_2
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ AnalyticOn β (tprodOn f) s
case intro.intro.intro.refine_3
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ β z β s, tprodOn f z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ ProdExistsOn f s β§ AnalyticOn β (tprodOn f) s β§ β z β s, tprodOn f z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | exact fun z zs β¦ β¨g z, gp z zsβ© | case intro.intro.intro.refine_1
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ ProdExistsOn f s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_1
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ ProdExistsOn f s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | fast_products_converge' | [101, 1] | [109, 55] | rwa [β analyticOn_congr o fun z zs β¦ (gp.tprodOn_eq z zs).symm] | case intro.intro.intro.refine_2
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ AnalyticOn β (tprodOn f) s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_2
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ AnalyticOn β (tprodOn f) s
TACTIC:
|
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 : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ β z β s, tprodOn f z β 0 | case intro.intro.intro.refine_3
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
z : β
zs : z β s
β’ tprodOn f z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
β’ β z β s, tprodOn f z β 0
TACTIC:
|
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 : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
z : β
zs : z β s
β’ tprodOn f z β 0 | case intro.intro.intro.refine_3
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
z : β
zs : z β s
β’ g z β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
z : β
zs : z β s
β’ tprodOn f z β 0
TACTIC:
|
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 : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
z : β
zs : z β s
β’ g z β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro.refine_3
f : β β β β β
s : Set β
c a : β
o : IsOpen s
c12 : c β€ 1 / 2
a0 : 0 β€ a
a1 : a < 1
h : β (n : β), AnalyticOn β (f n) s
hf : β (n : β), β z β s, Complex.abs (f n z - 1) β€ c * a ^ n
g : β β β
gp : HasProdOn (fun n => f n) g s
ga : AnalyticOn β g s
g0 : β z β s, g z β 0
z : β
zs : z β s
β’ g z β 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_pow | [112, 1] | [115, 72] | rw [HasProd] | f : β β β
g : β
p : β
h : HasProd f g
β’ HasProd (fun n => f n ^ p) (g ^ p) | f : β β β
g : β
p : β
h : HasProd f g
β’ Filter.Tendsto (fun s => s.prod fun b => f b ^ p) atTop (π (g ^ p)) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
g : β
p : β
h : HasProd f g
β’ HasProd (fun n => f n ^ p) (g ^ p)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_pow | [112, 1] | [115, 72] | simp_rw [Finset.prod_pow] | f : β β β
g : β
p : β
h : HasProd f g
β’ Filter.Tendsto (fun s => s.prod fun b => f b ^ p) atTop (π (g ^ p)) | f : β β β
g : β
p : β
h : HasProd f g
β’ Filter.Tendsto (fun s => (s.prod fun x => f x) ^ p) atTop (π (g ^ p)) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
g : β
p : β
h : HasProd f g
β’ Filter.Tendsto (fun s => s.prod fun b => f b ^ p) atTop (π (g ^ p))
TACTIC:
|
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 : β β β
g : β
p : β
h : HasProd f g
β’ Filter.Tendsto (fun s => (s.prod fun x => f x) ^ p) atTop (π (g ^ p)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
g : β
p : β
h : HasProd f g
β’ Filter.Tendsto (fun s => (s.prod fun x => f x) ^ p) atTop (π (g ^ p))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_pow' | [118, 1] | [120, 96] | rcases h with β¨g, hβ© | f : β β β
p : β
h : ProdExists f
β’ tprod f ^ p = β' (n : β), f n ^ p | case intro
f : β β β
p : β
g : β
h : HasProd f g
β’ tprod f ^ p = β' (n : β), f n ^ p | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
p : β
h : ProdExists f
β’ tprod f ^ p = β' (n : β), f n ^ p
TACTIC:
|
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 : β β β
p : β
g : β
h : HasProd f g
β’ tprod f ^ p = β' (n : β), f n ^ p | case intro
f : β β β
p : β
g : β
h : HasProd f g
β’ g ^ p = β' (n : β), f n ^ p | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β
p : β
g : β
h : HasProd f g
β’ tprod f ^ p = β' (n : β), f n ^ p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_pow' | [118, 1] | [120, 96] | rw [HasProd.tprod_eq _] | case intro
f : β β β
p : β
g : β
h : HasProd f g
β’ g ^ p = β' (n : β), f n ^ p | f : β β β
p : β
g : β
h : HasProd f g
β’ HasProd (fun n => f n ^ p) (g ^ p) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β
p : β
g : β
h : HasProd f g
β’ g ^ p = β' (n : β), f n ^ p
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_pow' | [118, 1] | [120, 96] | exact product_pow p h | f : β β β
p : β
g : β
h : HasProd f g
β’ HasProd (fun n => f n ^ p) (g ^ p) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
p : β
g : β
h : HasProd f g
β’ HasProd (fun n => f n ^ p) (g ^ p)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | rw [HasProd] at h β’ | a g : β
f : β β β
h : HasProd f g
β’ HasProd (Stream'.cons a f) (a * g) | a g : β
f : β β β
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (π g)
β’ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (π (a * g)) | Please generate a tactic in lean4 to solve the state.
STATE:
a g : β
f : β β β
h : HasProd f g
β’ HasProd (Stream'.cons a f) (a * g)
TACTIC:
|
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 : β
f : β β β
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (π g)
β’ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (π (a * g)) | a g : β
f : β β β
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a * b) β fun s => s.prod fun b => f b) atTop (π (a * g))
β’ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (π (a * g)) | Please generate a tactic in lean4 to solve the state.
STATE:
a g : β
f : β β β
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (π g)
β’ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (π (a * g))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | have s : ((fun z β¦ a * z) β fun N : Finset β β¦ N.prod f) =
(fun N : Finset β β¦ N.prod (Stream'.cons a f)) β push := by
apply funext; intro N; simp; exact push_prod | a g : β
f : β β β
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a * b) β fun s => s.prod fun b => f b) atTop (π (a * g))
β’ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (π (a * g)) | a g : β
f : β β β
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a * b) β fun s => s.prod fun b => f b) atTop (π (a * g))
s : ((fun z => a * z) β fun N => N.prod f) = (fun N => N.prod (Stream'.cons a f)) β push
β’ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (π (a * g)) | Please generate a tactic in lean4 to solve the state.
STATE:
a g : β
f : β β β
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a * b) β fun s => s.prod fun b => f b) atTop (π (a * g))
β’ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (π (a * g))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | rw [s] at ha | a g : β
f : β β β
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a * b) β fun s => s.prod fun b => f b) atTop (π (a * g))
s : ((fun z => a * z) β fun N => N.prod f) = (fun N => N.prod (Stream'.cons a f)) β push
β’ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (π (a * g)) | a g : β
f : β β β
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun N => N.prod (Stream'.cons a f)) β push) atTop (π (a * g))
s : ((fun z => a * z) β fun N => N.prod f) = (fun N => N.prod (Stream'.cons a f)) β push
β’ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (π (a * g)) | Please generate a tactic in lean4 to solve the state.
STATE:
a g : β
f : β β β
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a * b) β fun s => s.prod fun b => f b) atTop (π (a * g))
s : ((fun z => a * z) β fun N => N.prod f) = (fun N => N.prod (Stream'.cons a f)) β push
β’ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (π (a * g))
TACTIC:
|
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 : β
f : β β β
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun N => N.prod (Stream'.cons a f)) β push) atTop (π (a * g))
s : ((fun z => a * z) β fun N => N.prod f) = (fun N => N.prod (Stream'.cons a f)) β push
β’ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (π (a * g)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a g : β
f : β β β
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun N => N.prod (Stream'.cons a f)) β push) atTop (π (a * g))
s : ((fun z => a * z) β fun N => N.prod f) = (fun N => N.prod (Stream'.cons a f)) β push
β’ Filter.Tendsto (fun s => s.prod fun b => Stream'.cons a f b) atTop (π (a * g))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | apply funext | a g : β
f : β β β
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a * b) β fun s => s.prod fun b => f b) atTop (π (a * g))
β’ ((fun z => a * z) β fun N => N.prod f) = (fun N => N.prod (Stream'.cons a f)) β push | case h
a g : β
f : β β β
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a * b) β fun s => s.prod fun b => f b) atTop (π (a * g))
β’ β (x : Finset β), ((fun z => a * z) β fun N => N.prod f) x = ((fun N => N.prod (Stream'.cons a f)) β push) x | Please generate a tactic in lean4 to solve the state.
STATE:
a g : β
f : β β β
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a * b) β fun s => s.prod fun b => f b) atTop (π (a * g))
β’ ((fun z => a * z) β fun N => N.prod f) = (fun N => N.prod (Stream'.cons a f)) β push
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | intro N | case h
a g : β
f : β β β
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a * b) β fun s => s.prod fun b => f b) atTop (π (a * g))
β’ β (x : Finset β), ((fun z => a * z) β fun N => N.prod f) x = ((fun N => N.prod (Stream'.cons a f)) β push) x | case h
a g : β
f : β β β
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a * b) β fun s => s.prod fun b => f b) atTop (π (a * g))
N : Finset β
β’ ((fun z => a * z) β fun N => N.prod f) N = ((fun N => N.prod (Stream'.cons a f)) β push) N | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a g : β
f : β β β
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a * b) β fun s => s.prod fun b => f b) atTop (π (a * g))
β’ β (x : Finset β), ((fun z => a * z) β fun N => N.prod f) x = ((fun N => N.prod (Stream'.cons a f)) β push) x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | simp | case h
a g : β
f : β β β
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a * b) β fun s => s.prod fun b => f b) atTop (π (a * g))
N : Finset β
β’ ((fun z => a * z) β fun N => N.prod f) N = ((fun N => N.prod (Stream'.cons a f)) β push) N | case h
a g : β
f : β β β
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a * b) β fun s => s.prod fun b => f b) atTop (π (a * g))
N : Finset β
β’ a * N.prod f = (push N).prod (Stream'.cons a f) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a g : β
f : β β β
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a * b) β fun s => s.prod fun b => f b) atTop (π (a * g))
N : Finset β
β’ ((fun z => a * z) β fun N => N.prod f) N = ((fun N => N.prod (Stream'.cons a f)) β push) N
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons | [123, 1] | [131, 32] | exact push_prod | case h
a g : β
f : β β β
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a * b) β fun s => s.prod fun b => f b) atTop (π (a * g))
N : Finset β
β’ a * N.prod f = (push N).prod (Stream'.cons a f) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
a g : β
f : β β β
h : Filter.Tendsto (fun s => s.prod fun b => f b) atTop (π g)
ha : Filter.Tendsto ((fun b => a * b) β fun s => s.prod fun b => f b) atTop (π (a * g))
N : Finset β
β’ a * N.prod f = (push N).prod (Stream'.cons a f)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons' | [134, 1] | [136, 95] | rcases h with β¨g, hβ© | a : β
f : β β β
h : ProdExists f
β’ tprod (Stream'.cons a f) = a * tprod f | case intro
a : β
f : β β β
g : β
h : HasProd f g
β’ tprod (Stream'.cons a f) = a * tprod f | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
f : β β β
h : ProdExists f
β’ tprod (Stream'.cons a f) = a * tprod f
TACTIC:
|
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 : β
f : β β β
g : β
h : HasProd f g
β’ tprod (Stream'.cons a f) = a * tprod f | case intro
a : β
f : β β β
g : β
h : HasProd f g
β’ tprod (Stream'.cons a f) = a * g | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
a : β
f : β β β
g : β
h : HasProd f g
β’ tprod (Stream'.cons a f) = a * tprod f
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons' | [134, 1] | [136, 95] | rw [HasProd.tprod_eq _] | case intro
a : β
f : β β β
g : β
h : HasProd f g
β’ tprod (Stream'.cons a f) = a * g | a : β
f : β β β
g : β
h : HasProd f g
β’ HasProd (fun b => Stream'.cons a f b) (a * g) | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
a : β
f : β β β
g : β
h : HasProd f g
β’ tprod (Stream'.cons a f) = a * g
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_cons' | [134, 1] | [136, 95] | exact product_cons h | a : β
f : β β β
g : β
h : HasProd f g
β’ HasProd (fun b => Stream'.cons a f b) (a * g) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
f : β β β
g : β
h : HasProd f g
β’ HasProd (fun b => Stream'.cons a f b) (a * g)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | have c := @product_cons (f 0)β»ΒΉ _ _ h | f : β β β
g : β
f0 : f 0 β 0
h : HasProd f g
β’ HasProd (fun n => f (n + 1)) (g / f 0) | f : β β β
g : β
f0 : f 0 β 0
h : HasProd f g
c : HasProd (Stream'.cons (f 0)β»ΒΉ f) ((f 0)β»ΒΉ * g)
β’ HasProd (fun n => f (n + 1)) (g / f 0) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
g : β
f0 : f 0 β 0
h : HasProd f g
β’ HasProd (fun n => f (n + 1)) (g / f 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | rw [HasProd] | f : β β β
g : β
f0 : f 0 β 0
h : HasProd f g
c : HasProd (Stream'.cons (f 0)β»ΒΉ f) ((f 0)β»ΒΉ * g)
β’ HasProd (fun n => f (n + 1)) (g / f 0) | f : β β β
g : β
f0 : f 0 β 0
h : HasProd f g
c : HasProd (Stream'.cons (f 0)β»ΒΉ f) ((f 0)β»ΒΉ * g)
β’ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (π (g / f 0)) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
g : β
f0 : f 0 β 0
h : HasProd f g
c : HasProd (Stream'.cons (f 0)β»ΒΉ f) ((f 0)β»ΒΉ * g)
β’ HasProd (fun n => f (n + 1)) (g / f 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | rw [inv_mul_eq_div, HasProd, β tendsto_comp_push, β tendsto_comp_push] at c | f : β β β
g : β
f0 : f 0 β 0
h : HasProd f g
c : HasProd (Stream'.cons (f 0)β»ΒΉ f) ((f 0)β»ΒΉ * g)
β’ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (π (g / f 0)) | f : β β β
g : β
f0 : f 0 β 0
h : HasProd f g
c : Filter.Tendsto (((fun s => s.prod fun b => Stream'.cons (f 0)β»ΒΉ f b) β push) β push) atTop (π (g / f 0))
β’ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (π (g / f 0)) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
g : β
f0 : f 0 β 0
h : HasProd f g
c : HasProd (Stream'.cons (f 0)β»ΒΉ f) ((f 0)β»ΒΉ * g)
β’ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (π (g / f 0))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | have s : ((fun N : Finset β β¦ N.prod fun n β¦ (Stream'.cons (f 0)β»ΒΉ f) n) β push) β push =
fun N : Finset β β¦ N.prod fun n β¦ f (n + 1) := by
clear c h g; apply funext; intro N; simp
nth_rw 2 [β Stream'.eta f]
simp only [βpush_prod, Stream'.head, Stream'.tail, Stream'.get, βmul_assoc, inv_mul_cancel f0,
one_mul] | f : β β β
g : β
f0 : f 0 β 0
h : HasProd f g
c : Filter.Tendsto (((fun s => s.prod fun b => Stream'.cons (f 0)β»ΒΉ f b) β push) β push) atTop (π (g / f 0))
β’ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (π (g / f 0)) | f : β β β
g : β
f0 : f 0 β 0
h : HasProd f g
c : Filter.Tendsto (((fun s => s.prod fun b => Stream'.cons (f 0)β»ΒΉ f b) β push) β push) atTop (π (g / f 0))
s : ((fun N => N.prod fun n => Stream'.cons (f 0)β»ΒΉ f n) β push) β push = fun N => N.prod fun n => f (n + 1)
β’ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (π (g / f 0)) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
g : β
f0 : f 0 β 0
h : HasProd f g
c : Filter.Tendsto (((fun s => s.prod fun b => Stream'.cons (f 0)β»ΒΉ f b) β push) β push) atTop (π (g / f 0))
β’ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (π (g / f 0))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | rw [s] at c | f : β β β
g : β
f0 : f 0 β 0
h : HasProd f g
c : Filter.Tendsto (((fun s => s.prod fun b => Stream'.cons (f 0)β»ΒΉ f b) β push) β push) atTop (π (g / f 0))
s : ((fun N => N.prod fun n => Stream'.cons (f 0)β»ΒΉ f n) β push) β push = fun N => N.prod fun n => f (n + 1)
β’ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (π (g / f 0)) | f : β β β
g : β
f0 : f 0 β 0
h : HasProd f g
c : Filter.Tendsto (fun N => N.prod fun n => f (n + 1)) atTop (π (g / f 0))
s : ((fun N => N.prod fun n => Stream'.cons (f 0)β»ΒΉ f n) β push) β push = fun N => N.prod fun n => f (n + 1)
β’ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (π (g / f 0)) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
g : β
f0 : f 0 β 0
h : HasProd f g
c : Filter.Tendsto (((fun s => s.prod fun b => Stream'.cons (f 0)β»ΒΉ f b) β push) β push) atTop (π (g / f 0))
s : ((fun N => N.prod fun n => Stream'.cons (f 0)β»ΒΉ f n) β push) β push = fun N => N.prod fun n => f (n + 1)
β’ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (π (g / f 0))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | assumption | f : β β β
g : β
f0 : f 0 β 0
h : HasProd f g
c : Filter.Tendsto (fun N => N.prod fun n => f (n + 1)) atTop (π (g / f 0))
s : ((fun N => N.prod fun n => Stream'.cons (f 0)β»ΒΉ f n) β push) β push = fun N => N.prod fun n => f (n + 1)
β’ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (π (g / f 0)) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
g : β
f0 : f 0 β 0
h : HasProd f g
c : Filter.Tendsto (fun N => N.prod fun n => f (n + 1)) atTop (π (g / f 0))
s : ((fun N => N.prod fun n => Stream'.cons (f 0)β»ΒΉ f n) β push) β push = fun N => N.prod fun n => f (n + 1)
β’ Filter.Tendsto (fun s => s.prod fun b => f (b + 1)) atTop (π (g / f 0))
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | clear c h g | f : β β β
g : β
f0 : f 0 β 0
h : HasProd f g
c : Filter.Tendsto (((fun s => s.prod fun b => Stream'.cons (f 0)β»ΒΉ f b) β push) β push) atTop (π (g / f 0))
β’ ((fun N => N.prod fun n => Stream'.cons (f 0)β»ΒΉ f n) β push) β push = fun N => N.prod fun n => f (n + 1) | f : β β β
f0 : f 0 β 0
β’ ((fun N => N.prod fun n => Stream'.cons (f 0)β»ΒΉ f n) β push) β push = fun N => N.prod fun n => f (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
g : β
f0 : f 0 β 0
h : HasProd f g
c : Filter.Tendsto (((fun s => s.prod fun b => Stream'.cons (f 0)β»ΒΉ f b) β push) β push) atTop (π (g / f 0))
β’ ((fun N => N.prod fun n => Stream'.cons (f 0)β»ΒΉ f n) β push) β push = fun N => N.prod fun n => f (n + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | apply funext | f : β β β
f0 : f 0 β 0
β’ ((fun N => N.prod fun n => Stream'.cons (f 0)β»ΒΉ f n) β push) β push = fun N => N.prod fun n => f (n + 1) | case h
f : β β β
f0 : f 0 β 0
β’ β (x : Finset β), (((fun N => N.prod fun n => Stream'.cons (f 0)β»ΒΉ f n) β push) β push) x = x.prod fun n => f (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
f0 : f 0 β 0
β’ ((fun N => N.prod fun n => Stream'.cons (f 0)β»ΒΉ f n) β push) β push = fun N => N.prod fun n => f (n + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | intro N | case h
f : β β β
f0 : f 0 β 0
β’ β (x : Finset β), (((fun N => N.prod fun n => Stream'.cons (f 0)β»ΒΉ f n) β push) β push) x = x.prod fun n => f (n + 1) | case h
f : β β β
f0 : f 0 β 0
N : Finset β
β’ (((fun N => N.prod fun n => Stream'.cons (f 0)β»ΒΉ f n) β push) β push) N = N.prod fun n => f (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
f0 : f 0 β 0
β’ β (x : Finset β), (((fun N => N.prod fun n => Stream'.cons (f 0)β»ΒΉ f n) β push) β push) x = x.prod fun n => f (n + 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Analytic/Products.lean | product_drop | [139, 1] | [150, 26] | simp | case h
f : β β β
f0 : f 0 β 0
N : Finset β
β’ (((fun N => N.prod fun n => Stream'.cons (f 0)β»ΒΉ f n) β push) β push) N = N.prod fun n => f (n + 1) | case h
f : β β β
f0 : f 0 β 0
N : Finset β
β’ ((push (push N)).prod fun n => Stream'.cons (f 0)β»ΒΉ f n) = N.prod fun n => f (n + 1) | Please generate a tactic in lean4 to solve the state.
STATE:
case h
f : β β β
f0 : f 0 β 0
N : Finset β
β’ (((fun N => N.prod fun n => Stream'.cons (f 0)β»ΒΉ f n) β push) β push) N = N.prod fun n => f (n + 1)
TACTIC:
|
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