url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
| input
stringlengths 73
2.09M
|
|---|---|---|---|---|---|---|---|---|---|
https://github.com/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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