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/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.realValued_of_iteratedDeriv_real_on_ball
|
[159, 1]
|
[183, 8]
|
have Hx := Hz _ hx
|
case refine_1
f : β β β
r c : β
hf : DifferentiableOn β f (Metric.ball (βc) r)
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
Hz : β x β Set.Ioo (c - r) (c + r), βx β Metric.ball (βc) r
H : β β¦z : ββ¦, z β Metric.ball (βc) r β β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
x : β
hx : x β Set.Ioo (c - r) (c + r)
β’ DifferentiableWithinAt β (fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x
|
case refine_1
f : β β β
r c : β
hf : DifferentiableOn β f (Metric.ball (βc) r)
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
Hz : β x β Set.Ioo (c - r) (c + r), βx β Metric.ball (βc) r
H : β β¦z : ββ¦, z β Metric.ball (βc) r β β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
x : β
hx : x β Set.Ioo (c - r) (c + r)
Hx : βx β Metric.ball (βc) r
β’ DifferentiableWithinAt β (fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : β β β
r c : β
hf : DifferentiableOn β f (Metric.ball (βc) r)
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
Hz : β x β Set.Ioo (c - r) (c + r), βx β Metric.ball (βc) r
H : β β¦z : ββ¦, z β Metric.ball (βc) r β β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
x : β
hx : x β Set.Ioo (c - r) (c + r)
β’ DifferentiableWithinAt β (fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.realValued_of_iteratedDeriv_real_on_ball
|
[159, 1]
|
[183, 8]
|
refine DifferentiableAt.differentiableWithinAt ?_
|
case refine_1
f : β β β
r c : β
hf : DifferentiableOn β f (Metric.ball (βc) r)
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
Hz : β x β Set.Ioo (c - r) (c + r), βx β Metric.ball (βc) r
H : β β¦z : ββ¦, z β Metric.ball (βc) r β β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
x : β
hx : x β Set.Ioo (c - r) (c + r)
Hx : βx β Metric.ball (βc) r
β’ DifferentiableWithinAt β (fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x
|
case refine_1
f : β β β
r c : β
hf : DifferentiableOn β f (Metric.ball (βc) r)
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
Hz : β x β Set.Ioo (c - r) (c + r), βx β Metric.ball (βc) r
H : β β¦z : ββ¦, z β Metric.ball (βc) r β β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
x : β
hx : x β Set.Ioo (c - r) (c + r)
Hx : βx β Metric.ball (βc) r
β’ DifferentiableAt β (fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n) x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : β β β
r c : β
hf : DifferentiableOn β f (Metric.ball (βc) r)
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
Hz : β x β Set.Ioo (c - r) (c + r), βx β Metric.ball (βc) r
H : β β¦z : ββ¦, z β Metric.ball (βc) r β β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
x : β
hx : x β Set.Ioo (c - r) (c + r)
Hx : βx β Metric.ball (βc) r
β’ DifferentiableWithinAt β (fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n) (Set.Ioo (c - r) (c + r)) x
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.realValued_of_iteratedDeriv_real_on_ball
|
[159, 1]
|
[183, 8]
|
replace hf := ((hf x Hx).congr (fun _ hz β¦ H hz) (H Hx)).differentiableAt
(Metric.isOpen_ball.mem_nhds Hx) |>.comp_ofReal
|
case refine_1
f : β β β
r c : β
hf : DifferentiableOn β f (Metric.ball (βc) r)
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
Hz : β x β Set.Ioo (c - r) (c + r), βx β Metric.ball (βc) r
H : β β¦z : ββ¦, z β Metric.ball (βc) r β β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
x : β
hx : x β Set.Ioo (c - r) (c + r)
Hx : βx β Metric.ball (βc) r
β’ DifferentiableAt β (fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n) x
|
case refine_1
f : β β β
r c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
Hz : β x β Set.Ioo (c - r) (c + r), βx β Metric.ball (βc) r
H : β β¦z : ββ¦, z β Metric.ball (βc) r β β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
x : β
hx : x β Set.Ioo (c - r) (c + r)
Hx : βx β Metric.ball (βc) r
hf : DifferentiableAt β (fun x => β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (βx - βc) ^ n) x
β’ DifferentiableAt β (fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n) x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : β β β
r c : β
hf : DifferentiableOn β f (Metric.ball (βc) r)
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
Hz : β x β Set.Ioo (c - r) (c + r), βx β Metric.ball (βc) r
H : β β¦z : ββ¦, z β Metric.ball (βc) r β β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
x : β
hx : x β Set.Ioo (c - r) (c + r)
Hx : βx β Metric.ball (βc) r
β’ DifferentiableAt β (fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n) x
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.realValued_of_iteratedDeriv_real_on_ball
|
[159, 1]
|
[183, 8]
|
simp_rw [hd, β ofReal_sub, β ofReal_natCast, β ofReal_inv, β ofReal_pow, β ofReal_mul,
β ofReal_tsum] at hf
|
case refine_1
f : β β β
r c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
Hz : β x β Set.Ioo (c - r) (c + r), βx β Metric.ball (βc) r
H : β β¦z : ββ¦, z β Metric.ball (βc) r β β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
x : β
hx : x β Set.Ioo (c - r) (c + r)
Hx : βx β Metric.ball (βc) r
hf : DifferentiableAt β (fun x => β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (βx - βc) ^ n) x
β’ DifferentiableAt β (fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n) x
|
case refine_1
f : β β β
r c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
Hz : β x β Set.Ioo (c - r) (c + r), βx β Metric.ball (βc) r
H : β β¦z : ββ¦, z β Metric.ball (βc) r β β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
x : β
hx : x β Set.Ioo (c - r) (c + r)
Hx : βx β Metric.ball (βc) r
hf : DifferentiableAt β (fun x => β(β' (a : β), (βa !)β»ΒΉ * D a * (x - c) ^ a)) x
β’ DifferentiableAt β (fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n) x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : β β β
r c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
Hz : β x β Set.Ioo (c - r) (c + r), βx β Metric.ball (βc) r
H : β β¦z : ββ¦, z β Metric.ball (βc) r β β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
x : β
hx : x β Set.Ioo (c - r) (c + r)
Hx : βx β Metric.ball (βc) r
hf : DifferentiableAt β (fun x => β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (βx - βc) ^ n) x
β’ DifferentiableAt β (fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n) x
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.realValued_of_iteratedDeriv_real_on_ball
|
[159, 1]
|
[183, 8]
|
exact DifferentiableAt.ofReal_comp_iff.mp hf
|
case refine_1
f : β β β
r c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
Hz : β x β Set.Ioo (c - r) (c + r), βx β Metric.ball (βc) r
H : β β¦z : ββ¦, z β Metric.ball (βc) r β β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
x : β
hx : x β Set.Ioo (c - r) (c + r)
Hx : βx β Metric.ball (βc) r
hf : DifferentiableAt β (fun x => β(β' (a : β), (βa !)β»ΒΉ * D a * (x - c) ^ a)) x
β’ DifferentiableAt β (fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n) x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : β β β
r c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
Hz : β x β Set.Ioo (c - r) (c + r), βx β Metric.ball (βc) r
H : β β¦z : ββ¦, z β Metric.ball (βc) r β β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
x : β
hx : x β Set.Ioo (c - r) (c + r)
Hx : βx β Metric.ball (βc) r
hf : DifferentiableAt β (fun x => β(β' (a : β), (βa !)β»ΒΉ * D a * (x - c) ^ a)) x
β’ DifferentiableAt β (fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n) x
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.realValued_of_iteratedDeriv_real_on_ball
|
[159, 1]
|
[183, 8]
|
simp only [Function.comp_apply, β H (Hz _ hx), hd, ofReal_tsum]
|
case refine_2
f : β β β
r c : β
hf : DifferentiableOn β f (Metric.ball (βc) r)
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
Hz : β x β Set.Ioo (c - r) (c + r), βx β Metric.ball (βc) r
H : β β¦z : ββ¦, z β Metric.ball (βc) r β β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
x : β
hx : x β Set.Ioo (c - r) (c + r)
β’ (f β ofReal') x = (ofReal' β fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n) x
|
case refine_2
f : β β β
r c : β
hf : DifferentiableOn β f (Metric.ball (βc) r)
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
Hz : β x β Set.Ioo (c - r) (c + r), βx β Metric.ball (βc) r
H : β β¦z : ββ¦, z β Metric.ball (βc) r β β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
x : β
hx : x β Set.Ioo (c - r) (c + r)
β’ β' (n : β), (βn !)β»ΒΉ * β(D n) * (βx - βc) ^ n = β' (a : β), β((βa !)β»ΒΉ * D a * (x - c) ^ a)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f : β β β
r c : β
hf : DifferentiableOn β f (Metric.ball (βc) r)
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
Hz : β x β Set.Ioo (c - r) (c + r), βx β Metric.ball (βc) r
H : β β¦z : ββ¦, z β Metric.ball (βc) r β β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
x : β
hx : x β Set.Ioo (c - r) (c + r)
β’ (f β ofReal') x = (ofReal' β fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n) x
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.realValued_of_iteratedDeriv_real_on_ball
|
[159, 1]
|
[183, 8]
|
push_cast
|
case refine_2
f : β β β
r c : β
hf : DifferentiableOn β f (Metric.ball (βc) r)
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
Hz : β x β Set.Ioo (c - r) (c + r), βx β Metric.ball (βc) r
H : β β¦z : ββ¦, z β Metric.ball (βc) r β β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
x : β
hx : x β Set.Ioo (c - r) (c + r)
β’ β' (n : β), (βn !)β»ΒΉ * β(D n) * (βx - βc) ^ n = β' (a : β), β((βa !)β»ΒΉ * D a * (x - c) ^ a)
|
case refine_2
f : β β β
r c : β
hf : DifferentiableOn β f (Metric.ball (βc) r)
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
Hz : β x β Set.Ioo (c - r) (c + r), βx β Metric.ball (βc) r
H : β β¦z : ββ¦, z β Metric.ball (βc) r β β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
x : β
hx : x β Set.Ioo (c - r) (c + r)
β’ β' (a : β), (βa !)β»ΒΉ * β(D a) * (βx - βc) ^ a = β' (a : β), (βa !)β»ΒΉ * β(D a) * (βx - βc) ^ a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f : β β β
r c : β
hf : DifferentiableOn β f (Metric.ball (βc) r)
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
Hz : β x β Set.Ioo (c - r) (c + r), βx β Metric.ball (βc) r
H : β β¦z : ββ¦, z β Metric.ball (βc) r β β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
x : β
hx : x β Set.Ioo (c - r) (c + r)
β’ β' (n : β), (βn !)β»ΒΉ * β(D n) * (βx - βc) ^ n = β' (a : β), β((βa !)β»ΒΉ * D a * (x - c) ^ a)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.realValued_of_iteratedDeriv_real_on_ball
|
[159, 1]
|
[183, 8]
|
rfl
|
case refine_2
f : β β β
r c : β
hf : DifferentiableOn β f (Metric.ball (βc) r)
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
Hz : β x β Set.Ioo (c - r) (c + r), βx β Metric.ball (βc) r
H : β β¦z : ββ¦, z β Metric.ball (βc) r β β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
x : β
hx : x β Set.Ioo (c - r) (c + r)
β’ β' (a : β), (βa !)β»ΒΉ * β(D a) * (βx - βc) ^ a = β' (a : β), (βa !)β»ΒΉ * β(D a) * (βx - βc) ^ a
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f : β β β
r c : β
hf : DifferentiableOn β f (Metric.ball (βc) r)
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
Hz : β x β Set.Ioo (c - r) (c + r), βx β Metric.ball (βc) r
H : β β¦z : ββ¦, z β Metric.ball (βc) r β β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
x : β
hx : x β Set.Ioo (c - r) (c + r)
β’ β' (a : β), (βa !)β»ΒΉ * β(D a) * (βx - βc) ^ a = β' (a : β), (βa !)β»ΒΉ * β(D a) * (βx - βc) ^ a
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.realValued_of_iteratedDeriv_real
|
[185, 1]
|
[201, 8]
|
have H (z : β) := taylorSeries_eq_of_entire' c z hf
|
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
β’ β F, Differentiable β F β§ f β ofReal' = ofReal' β F
|
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
β’ β F, Differentiable β F β§ f β ofReal' = ofReal' β F
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
β’ β F, Differentiable β F β§ f β ofReal' = ofReal' β F
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.realValued_of_iteratedDeriv_real
|
[185, 1]
|
[201, 8]
|
simp_rw [hd] at H
|
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
β’ β F, Differentiable β F β§ f β ofReal' = ofReal' β F
|
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
β’ β F, Differentiable β F β§ f β ofReal' = ofReal' β F
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f βc * (z - βc) ^ n = f z
β’ β F, Differentiable β F β§ f β ofReal' = ofReal' β F
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.realValued_of_iteratedDeriv_real
|
[185, 1]
|
[201, 8]
|
refine β¨fun x β¦ β' (n : β), (βn !)β»ΒΉ * (D n) * (x - c) ^ n, ?_, ?_β©
|
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
β’ β F, Differentiable β F β§ f β ofReal' = ofReal' β F
|
case refine_1
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
β’ Differentiable β fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n
case refine_2
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
β’ f β ofReal' = ofReal' β fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
β’ β F, Differentiable β F β§ f β ofReal' = ofReal' β F
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.realValued_of_iteratedDeriv_real
|
[185, 1]
|
[201, 8]
|
have := hf.comp_ofReal
|
case refine_1
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
β’ Differentiable β fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n
|
case refine_1
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
this : Differentiable β fun x => f βx
β’ Differentiable β fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
β’ Differentiable β fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.realValued_of_iteratedDeriv_real
|
[185, 1]
|
[201, 8]
|
simp_rw [β H, β ofReal_sub, β ofReal_natCast, β ofReal_inv, β ofReal_pow, β ofReal_mul,
β ofReal_tsum] at this
|
case refine_1
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
this : Differentiable β fun x => f βx
β’ Differentiable β fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n
|
case refine_1
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
this : Differentiable β fun x => β(β' (a : β), (βa !)β»ΒΉ * D a * (x - c) ^ a)
β’ Differentiable β fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
this : Differentiable β fun x => f βx
β’ Differentiable β fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.realValued_of_iteratedDeriv_real
|
[185, 1]
|
[201, 8]
|
exact Differentiable.ofReal_comp_iff.mp this
|
case refine_1
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
this : Differentiable β fun x => β(β' (a : β), (βa !)β»ΒΉ * D a * (x - c) ^ a)
β’ Differentiable β fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
this : Differentiable β fun x => β(β' (a : β), (βa !)β»ΒΉ * D a * (x - c) ^ a)
β’ Differentiable β fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.realValued_of_iteratedDeriv_real
|
[185, 1]
|
[201, 8]
|
ext x
|
case refine_2
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
β’ f β ofReal' = ofReal' β fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n
|
case refine_2.h
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
x : β
β’ (f β ofReal') x = (ofReal' β fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n) x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
β’ f β ofReal' = ofReal' β fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.realValued_of_iteratedDeriv_real
|
[185, 1]
|
[201, 8]
|
simp only [Function.comp_apply, ofReal_eq_coe, β H, ofReal_tsum]
|
case refine_2.h
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
x : β
β’ (f β ofReal') x = (ofReal' β fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n) x
|
case refine_2.h
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
x : β
β’ β' (n : β), (βn !)β»ΒΉ * β(D n) * (βx - βc) ^ n = β' (a : β), β((βa !)β»ΒΉ * D a * (x - c) ^ a)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.h
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
x : β
β’ (f β ofReal') x = (ofReal' β fun x => β' (n : β), (βn !)β»ΒΉ * D n * (x - c) ^ n) x
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.realValued_of_iteratedDeriv_real
|
[185, 1]
|
[201, 8]
|
push_cast
|
case refine_2.h
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
x : β
β’ β' (n : β), (βn !)β»ΒΉ * β(D n) * (βx - βc) ^ n = β' (a : β), β((βa !)β»ΒΉ * D a * (x - c) ^ a)
|
case refine_2.h
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
x : β
β’ β' (a : β), (βa !)β»ΒΉ * β(D a) * (βx - βc) ^ a = β' (a : β), (βa !)β»ΒΉ * β(D a) * (βx - βc) ^ a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.h
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
x : β
β’ β' (n : β), (βn !)β»ΒΉ * β(D n) * (βx - βc) ^ n = β' (a : β), β((βa !)β»ΒΉ * D a * (x - c) ^ a)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.realValued_of_iteratedDeriv_real
|
[185, 1]
|
[201, 8]
|
rfl
|
case refine_2.h
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
x : β
β’ β' (a : β), (βa !)β»ΒΉ * β(D a) * (βx - βc) ^ a = β' (a : β), (βa !)β»ΒΉ * β(D a) * (βx - βc) ^ a
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2.h
f : β β β
hf : Differentiable β f
c : β
D : β β β
hd : β (n : β), iteratedDeriv n f βc = β(D n)
H : β (z : β), β' (n : β), (βn !)β»ΒΉ * β(D n) * (z - βc) ^ n = f z
x : β
β’ β' (a : β), (βa !)β»ΒΉ * β(D a) * (βx - βc) ^ a = β' (a : β), (βa !)β»ΒΉ * β(D a) * (βx - βc) ^ a
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.nonneg_of_iteratedDeriv_nonneg
|
[207, 1]
|
[223, 13]
|
have H := taylorSeries_eq_of_entire' 0 z hf
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ z
β’ 0 β€ f z
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ z
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (z - 0) ^ n = f z
β’ 0 β€ f z
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ z
β’ 0 β€ f z
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.nonneg_of_iteratedDeriv_nonneg
|
[207, 1]
|
[223, 13]
|
have hz' := eq_re_of_ofReal_le hz
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ z
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (z - 0) ^ n = f z
β’ 0 β€ f z
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ z
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (z - 0) ^ n = f z
hz' : z = βz.re
β’ 0 β€ f z
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ z
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (z - 0) ^ n = f z
β’ 0 β€ f z
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.nonneg_of_iteratedDeriv_nonneg
|
[207, 1]
|
[223, 13]
|
rw [hz'] at hz H β’
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ z
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (z - 0) ^ n = f z
hz' : z = βz.re
β’ 0 β€ f z
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
β’ 0 β€ f βz.re
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ z
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (z - 0) ^ n = f z
hz' : z = βz.re
β’ 0 β€ f z
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.nonneg_of_iteratedDeriv_nonneg
|
[207, 1]
|
[223, 13]
|
obtain β¨D, hDβ© : β D : β β β, β n, 0 β€ D n β§ iteratedDeriv n f 0 = D n
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
β’ 0 β€ f βz.re
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
β’ β D, β (n : β), 0 β€ D n β§ iteratedDeriv n f 0 = β(D n)
case intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
D : β β β
hD : β (n : β), 0 β€ D n β§ iteratedDeriv n f 0 = β(D n)
β’ 0 β€ f βz.re
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
β’ 0 β€ f βz.re
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.nonneg_of_iteratedDeriv_nonneg
|
[207, 1]
|
[223, 13]
|
simp_rw [β H, hD, β ofReal_natCast, sub_zero, β ofReal_pow, β ofReal_inv, β ofReal_mul,
β ofReal_tsum]
|
case intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
D : β β β
hD : β (n : β), 0 β€ D n β§ iteratedDeriv n f 0 = β(D n)
β’ 0 β€ f βz.re
|
case intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
D : β β β
hD : β (n : β), 0 β€ D n β§ iteratedDeriv n f 0 = β(D n)
β’ 0 β€ β(β' (a : β), (βa !)β»ΒΉ * D a * z.re ^ a)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
D : β β β
hD : β (n : β), 0 β€ D n β§ iteratedDeriv n f 0 = β(D n)
β’ 0 β€ f βz.re
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.nonneg_of_iteratedDeriv_nonneg
|
[207, 1]
|
[223, 13]
|
norm_cast
|
case intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
D : β β β
hD : β (n : β), 0 β€ D n β§ iteratedDeriv n f 0 = β(D n)
β’ 0 β€ β(β' (a : β), (βa !)β»ΒΉ * D a * z.re ^ a)
|
case intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
D : β β β
hD : β (n : β), 0 β€ D n β§ iteratedDeriv n f 0 = β(D n)
β’ 0 β€ β' (a : β), (βa !)β»ΒΉ * D a * z.re ^ a
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
D : β β β
hD : β (n : β), 0 β€ D n β§ iteratedDeriv n f 0 = β(D n)
β’ 0 β€ β(β' (a : β), (βa !)β»ΒΉ * D a * z.re ^ a)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.nonneg_of_iteratedDeriv_nonneg
|
[207, 1]
|
[223, 13]
|
refine tsum_nonneg fun n β¦ ?_
|
case intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
D : β β β
hD : β (n : β), 0 β€ D n β§ iteratedDeriv n f 0 = β(D n)
β’ 0 β€ β' (a : β), (βa !)β»ΒΉ * D a * z.re ^ a
|
case intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
D : β β β
hD : β (n : β), 0 β€ D n β§ iteratedDeriv n f 0 = β(D n)
n : β
β’ 0 β€ (βn !)β»ΒΉ * D n * z.re ^ n
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
D : β β β
hD : β (n : β), 0 β€ D n β§ iteratedDeriv n f 0 = β(D n)
β’ 0 β€ β' (a : β), (βa !)β»ΒΉ * D a * z.re ^ a
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.nonneg_of_iteratedDeriv_nonneg
|
[207, 1]
|
[223, 13]
|
norm_cast at hz
|
case intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
D : β β β
hD : β (n : β), 0 β€ D n β§ iteratedDeriv n f 0 = β(D n)
n : β
β’ 0 β€ (βn !)β»ΒΉ * D n * z.re ^ n
|
case intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
D : β β β
hD : β (n : β), 0 β€ D n β§ iteratedDeriv n f 0 = β(D n)
n : β
hz : 0 β€ z.re
β’ 0 β€ (βn !)β»ΒΉ * D n * z.re ^ n
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
D : β β β
hD : β (n : β), 0 β€ D n β§ iteratedDeriv n f 0 = β(D n)
n : β
β’ 0 β€ (βn !)β»ΒΉ * D n * z.re ^ n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.nonneg_of_iteratedDeriv_nonneg
|
[207, 1]
|
[223, 13]
|
have := (hD n).1
|
case intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
D : β β β
hD : β (n : β), 0 β€ D n β§ iteratedDeriv n f 0 = β(D n)
n : β
hz : 0 β€ z.re
β’ 0 β€ (βn !)β»ΒΉ * D n * z.re ^ n
|
case intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
D : β β β
hD : β (n : β), 0 β€ D n β§ iteratedDeriv n f 0 = β(D n)
n : β
hz : 0 β€ z.re
this : 0 β€ D n
β’ 0 β€ (βn !)β»ΒΉ * D n * z.re ^ n
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
D : β β β
hD : β (n : β), 0 β€ D n β§ iteratedDeriv n f 0 = β(D n)
n : β
hz : 0 β€ z.re
β’ 0 β€ (βn !)β»ΒΉ * D n * z.re ^ n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.nonneg_of_iteratedDeriv_nonneg
|
[207, 1]
|
[223, 13]
|
positivity
|
case intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
D : β β β
hD : β (n : β), 0 β€ D n β§ iteratedDeriv n f 0 = β(D n)
n : β
hz : 0 β€ z.re
this : 0 β€ D n
β’ 0 β€ (βn !)β»ΒΉ * D n * z.re ^ n
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
D : β β β
hD : β (n : β), 0 β€ D n β§ iteratedDeriv n f 0 = β(D n)
n : β
hz : 0 β€ z.re
this : 0 β€ D n
β’ 0 β€ (βn !)β»ΒΉ * D n * z.re ^ n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.nonneg_of_iteratedDeriv_nonneg
|
[207, 1]
|
[223, 13]
|
refine β¨fun n β¦ (iteratedDeriv n f 0).re, fun n β¦ β¨?_, ?_β©β©
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
β’ β D, β (n : β), 0 β€ D n β§ iteratedDeriv n f 0 = β(D n)
|
case refine_1
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
n : β
β’ 0 β€ (fun n => (iteratedDeriv n f 0).re) n
case refine_2
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
n : β
β’ iteratedDeriv n f 0 = β((fun n => (iteratedDeriv n f 0).re) n)
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
β’ β D, β (n : β), 0 β€ D n β§ iteratedDeriv n f 0 = β(D n)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.nonneg_of_iteratedDeriv_nonneg
|
[207, 1]
|
[223, 13]
|
have := eq_re_of_ofReal_le (h n) βΈ h n
|
case refine_1
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
n : β
β’ 0 β€ (fun n => (iteratedDeriv n f 0).re) n
|
case refine_1
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
n : β
this : 0 β€ β(iteratedDeriv n f 0).re
β’ 0 β€ (fun n => (iteratedDeriv n f 0).re) n
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
n : β
β’ 0 β€ (fun n => (iteratedDeriv n f 0).re) n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.nonneg_of_iteratedDeriv_nonneg
|
[207, 1]
|
[223, 13]
|
norm_cast at this
|
case refine_1
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
n : β
this : 0 β€ β(iteratedDeriv n f 0).re
β’ 0 β€ (fun n => (iteratedDeriv n f 0).re) n
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_1
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
n : β
this : 0 β€ β(iteratedDeriv n f 0).re
β’ 0 β€ (fun n => (iteratedDeriv n f 0).re) n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.nonneg_of_iteratedDeriv_nonneg
|
[207, 1]
|
[223, 13]
|
rw [eq_re_of_ofReal_le (h n)]
|
case refine_2
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
n : β
β’ iteratedDeriv n f 0 = β((fun n => (iteratedDeriv n f 0).re) n)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case refine_2
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ βz.re
H : β' (n : β), (βn !)β»ΒΉ * iteratedDeriv n f 0 * (βz.re - 0) ^ n = f βz.re
hz' : z = βz.re
n : β
β’ iteratedDeriv n f 0 = β((fun n => (iteratedDeriv n f 0).re) n)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
let D : β β β := fun n β¦ (iteratedDeriv n f 0).re
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
β’ MonotoneOn (f β ofReal') (Set.Ici 0)
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
β’ MonotoneOn (f β ofReal') (Set.Ici 0)
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
β’ MonotoneOn (f β ofReal') (Set.Ici 0)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
have hD (n : β) : iteratedDeriv n f 0 = D n := by
refine Complex.ext rfl ?_
simp only [ofReal_im]
exact (le_def.mp (h n)).2.symm
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
β’ MonotoneOn (f β ofReal') (Set.Ici 0)
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
β’ MonotoneOn (f β ofReal') (Set.Ici 0)
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
β’ MonotoneOn (f β ofReal') (Set.Ici 0)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
obtain β¨F, hFd, hFβ© := realValued_of_iteratedDeriv_real hf hD
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
β’ MonotoneOn (f β ofReal') (Set.Ici 0)
|
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
β’ MonotoneOn (f β ofReal') (Set.Ici 0)
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
β’ MonotoneOn (f β ofReal') (Set.Ici 0)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
rw [hF]
|
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
β’ MonotoneOn (f β ofReal') (Set.Ici 0)
|
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
β’ MonotoneOn (ofReal' β F) (Set.Ici 0)
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
β’ MonotoneOn (f β ofReal') (Set.Ici 0)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
refine monotone_ofReal.comp_monotoneOn <| monotoneOn_of_deriv_nonneg (convex_Ici 0)
hFd.continuous.continuousOn hFd.differentiableOn fun x hx β¦ ?_
|
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
β’ MonotoneOn (ofReal' β F) (Set.Ici 0)
|
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hx : x β interior (Set.Ici 0)
β’ 0 β€ deriv F x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
β’ MonotoneOn (ofReal' β F) (Set.Ici 0)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
have hD' (n : β) : 0 β€ iteratedDeriv n (deriv f) 0 := by
rw [β iteratedDeriv_succ']
exact h (n + 1)
|
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hx : x β interior (Set.Ici 0)
β’ 0 β€ deriv F x
|
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hx : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
β’ 0 β€ deriv F x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hx : x β interior (Set.Ici 0)
β’ 0 β€ deriv F x
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
have hf' := (contDiff_succ_iff_deriv.mp <| hf.contDiff (n := 2)).2.differentiable rfl.le
|
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hx : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
β’ 0 β€ deriv F x
|
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hx : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
β’ 0 β€ deriv F x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hx : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
β’ 0 β€ deriv F x
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
have hx : (0 : β) β€ x := by
norm_cast
simp only [Set.nonempty_Iio, interior_Ici', Set.mem_Ioi] at hx
exact hx.le
|
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hx : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
β’ 0 β€ deriv F x
|
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hxβ : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
hx : 0 β€ βx
β’ 0 β€ deriv F x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hx : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
β’ 0 β€ deriv F x
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
have H := nonneg_of_iteratedDeriv_nonneg hf' hD' hx
|
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hxβ : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
hx : 0 β€ βx
β’ 0 β€ deriv F x
|
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hxβ : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
hx : 0 β€ βx
H : 0 β€ deriv f βx
β’ 0 β€ deriv F x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hxβ : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
hx : 0 β€ βx
β’ 0 β€ deriv F x
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
rw [β deriv.comp_ofReal hf.differentiableAt] at H
|
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hxβ : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
hx : 0 β€ βx
H : 0 β€ deriv f βx
β’ 0 β€ deriv F x
|
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hxβ : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
hx : 0 β€ βx
H : 0 β€ deriv (fun x => f βx) x
β’ 0 β€ deriv F x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hxβ : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
hx : 0 β€ βx
H : 0 β€ deriv f βx
β’ 0 β€ deriv F x
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
change 0 β€ deriv (f β ofReal') x at H
|
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hxβ : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
hx : 0 β€ βx
H : 0 β€ deriv (fun x => f βx) x
β’ 0 β€ deriv F x
|
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hxβ : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
hx : 0 β€ βx
H : 0 β€ deriv (f β ofReal') x
β’ 0 β€ deriv F x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hxβ : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
hx : 0 β€ βx
H : 0 β€ deriv (fun x => f βx) x
β’ 0 β€ deriv F x
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
erw [hF, deriv.ofReal_comp] at H
|
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hxβ : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
hx : 0 β€ βx
H : 0 β€ deriv (f β ofReal') x
β’ 0 β€ deriv F x
|
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hxβ : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
hx : 0 β€ βx
H : 0 β€ β(deriv (fun y => F y) x)
β’ 0 β€ deriv F x
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hxβ : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
hx : 0 β€ βx
H : 0 β€ deriv (f β ofReal') x
β’ 0 β€ deriv F x
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
norm_cast at H
|
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hxβ : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
hx : 0 β€ βx
H : 0 β€ β(deriv (fun y => F y) x)
β’ 0 β€ deriv F x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hxβ : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
hx : 0 β€ βx
H : 0 β€ β(deriv (fun y => F y) x)
β’ 0 β€ deriv F x
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
refine Complex.ext rfl ?_
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
n : β
β’ iteratedDeriv n f 0 = β(D n)
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
n : β
β’ (iteratedDeriv n f 0).im = (β(D n)).im
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
n : β
β’ iteratedDeriv n f 0 = β(D n)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
simp only [ofReal_im]
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
n : β
β’ (iteratedDeriv n f 0).im = (β(D n)).im
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
n : β
β’ (iteratedDeriv n f 0).im = 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
n : β
β’ (iteratedDeriv n f 0).im = (β(D n)).im
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
exact (le_def.mp (h n)).2.symm
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
n : β
β’ (iteratedDeriv n f 0).im = 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
n : β
β’ (iteratedDeriv n f 0).im = 0
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
rw [β iteratedDeriv_succ']
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hx : x β interior (Set.Ici 0)
n : β
β’ 0 β€ iteratedDeriv n (deriv f) 0
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hx : x β interior (Set.Ici 0)
n : β
β’ 0 β€ iteratedDeriv (n + 1) f 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hx : x β interior (Set.Ici 0)
n : β
β’ 0 β€ iteratedDeriv n (deriv f) 0
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
exact h (n + 1)
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hx : x β interior (Set.Ici 0)
n : β
β’ 0 β€ iteratedDeriv (n + 1) f 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hx : x β interior (Set.Ici 0)
n : β
β’ 0 β€ iteratedDeriv (n + 1) f 0
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
norm_cast
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hx : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
β’ 0 β€ βx
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hx : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
β’ 0 β€ x
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hx : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
β’ 0 β€ βx
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
simp only [Set.nonempty_Iio, interior_Ici', Set.mem_Ioi] at hx
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hx : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
β’ 0 β€ x
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
hx : 0 < x
β’ 0 β€ x
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hx : x β interior (Set.Ici 0)
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
β’ 0 β€ x
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.monotoneOn_of_iteratedDeriv_nonneg
|
[227, 1]
|
[250, 17]
|
exact hx.le
|
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
hx : 0 < x
β’ 0 β€ x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
h : β (n : β), 0 β€ iteratedDeriv n f 0
D : β β β := fun n => (iteratedDeriv n f 0).re
hD : β (n : β), iteratedDeriv n f 0 = β(D n)
F : β β β
hFd : Differentiable β F
hF : f β ofReal' = ofReal' β F
x : β
hD' : β (n : β), 0 β€ iteratedDeriv n (deriv f) 0
hf' : Differentiable β (deriv f)
hx : 0 < x
β’ 0 β€ x
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_nonneg
|
[255, 1]
|
[266, 83]
|
exact sub_nonneg.mp <| nonneg_of_iteratedDeriv_nonneg (hf.sub_const (f 0)) h' hz
|
f : β β β
hf : Differentiable β f
h : β (n : β), n β 0 β 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ z
h' : β (n : β), 0 β€ iteratedDeriv n (fun x => f x - f 0) 0
β’ f 0 β€ f z
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
h : β (n : β), n β 0 β 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ z
h' : β (n : β), 0 β€ iteratedDeriv n (fun x => f x - f 0) 0
β’ f 0 β€ f z
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_nonneg
|
[255, 1]
|
[266, 83]
|
cases n with
| zero => simp only [iteratedDeriv_zero, sub_self, le_refl]
| succ n =>
specialize h n.succ <| succ_ne_zero n
rw [iteratedDeriv_succ'] at h β’
convert h using 2
ext w
exact deriv_sub_const (f 0)
|
f : β β β
hf : Differentiable β f
h : β (n : β), n β 0 β 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ z
n : β
β’ 0 β€ iteratedDeriv n (fun x => f x - f 0) 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
h : β (n : β), n β 0 β 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ z
n : β
β’ 0 β€ iteratedDeriv n (fun x => f x - f 0) 0
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_nonneg
|
[255, 1]
|
[266, 83]
|
simp only [iteratedDeriv_zero, sub_self, le_refl]
|
case zero
f : β β β
hf : Differentiable β f
h : β (n : β), n β 0 β 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ z
β’ 0 β€ iteratedDeriv 0 (fun x => f x - f 0) 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case zero
f : β β β
hf : Differentiable β f
h : β (n : β), n β 0 β 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ z
β’ 0 β€ iteratedDeriv 0 (fun x => f x - f 0) 0
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_nonneg
|
[255, 1]
|
[266, 83]
|
specialize h n.succ <| succ_ne_zero n
|
case succ
f : β β β
hf : Differentiable β f
h : β (n : β), n β 0 β 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ z
n : β
β’ 0 β€ iteratedDeriv (n + 1) (fun x => f x - f 0) 0
|
case succ
f : β β β
hf : Differentiable β f
z : β
hz : 0 β€ z
n : β
h : 0 β€ iteratedDeriv n.succ f 0
β’ 0 β€ iteratedDeriv (n + 1) (fun x => f x - f 0) 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
case succ
f : β β β
hf : Differentiable β f
h : β (n : β), n β 0 β 0 β€ iteratedDeriv n f 0
z : β
hz : 0 β€ z
n : β
β’ 0 β€ iteratedDeriv (n + 1) (fun x => f x - f 0) 0
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_nonneg
|
[255, 1]
|
[266, 83]
|
rw [iteratedDeriv_succ'] at h β’
|
case succ
f : β β β
hf : Differentiable β f
z : β
hz : 0 β€ z
n : β
h : 0 β€ iteratedDeriv n.succ f 0
β’ 0 β€ iteratedDeriv (n + 1) (fun x => f x - f 0) 0
|
case succ
f : β β β
hf : Differentiable β f
z : β
hz : 0 β€ z
n : β
h : 0 β€ iteratedDeriv n (deriv f) 0
β’ 0 β€ iteratedDeriv n (deriv fun x => f x - f 0) 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
case succ
f : β β β
hf : Differentiable β f
z : β
hz : 0 β€ z
n : β
h : 0 β€ iteratedDeriv n.succ f 0
β’ 0 β€ iteratedDeriv (n + 1) (fun x => f x - f 0) 0
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_nonneg
|
[255, 1]
|
[266, 83]
|
convert h using 2
|
case succ
f : β β β
hf : Differentiable β f
z : β
hz : 0 β€ z
n : β
h : 0 β€ iteratedDeriv n (deriv f) 0
β’ 0 β€ iteratedDeriv n (deriv fun x => f x - f 0) 0
|
case h.e'_4.h.e'_7
f : β β β
hf : Differentiable β f
z : β
hz : 0 β€ z
n : β
h : 0 β€ iteratedDeriv n (deriv f) 0
β’ (deriv fun x => f x - f 0) = deriv f
|
Please generate a tactic in lean4 to solve the state.
STATE:
case succ
f : β β β
hf : Differentiable β f
z : β
hz : 0 β€ z
n : β
h : 0 β€ iteratedDeriv n (deriv f) 0
β’ 0 β€ iteratedDeriv n (deriv fun x => f x - f 0) 0
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_nonneg
|
[255, 1]
|
[266, 83]
|
ext w
|
case h.e'_4.h.e'_7
f : β β β
hf : Differentiable β f
z : β
hz : 0 β€ z
n : β
h : 0 β€ iteratedDeriv n (deriv f) 0
β’ (deriv fun x => f x - f 0) = deriv f
|
case h.e'_4.h.e'_7.h
f : β β β
hf : Differentiable β f
z : β
hz : 0 β€ z
n : β
h : 0 β€ iteratedDeriv n (deriv f) 0
w : β
β’ deriv (fun x => f x - f 0) w = deriv f w
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.e'_7
f : β β β
hf : Differentiable β f
z : β
hz : 0 β€ z
n : β
h : 0 β€ iteratedDeriv n (deriv f) 0
β’ (deriv fun x => f x - f 0) = deriv f
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_nonneg
|
[255, 1]
|
[266, 83]
|
exact deriv_sub_const (f 0)
|
case h.e'_4.h.e'_7.h
f : β β β
hf : Differentiable β f
z : β
hz : 0 β€ z
n : β
h : 0 β€ iteratedDeriv n (deriv f) 0
w : β
β’ deriv (fun x => f x - f 0) w = deriv f w
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4.h.e'_7.h
f : β β β
hf : Differentiable β f
z : β
hz : 0 β€ z
n : β
h : 0 β€ iteratedDeriv n (deriv f) 0
w : β
β’ deriv (fun x => f x - f 0) w = deriv f w
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_alternating
|
[271, 1]
|
[278, 66]
|
let F : β β β := fun z β¦ f (-z)
|
f : β β β
hf : Differentiable β f
h : β (n : β), n β 0 β 0 β€ (-1) ^ n * iteratedDeriv n f 0
z : β
hz : z β€ 0
β’ f 0 β€ f z
|
f : β β β
hf : Differentiable β f
h : β (n : β), n β 0 β 0 β€ (-1) ^ n * iteratedDeriv n f 0
z : β
hz : z β€ 0
F : β β β := fun z => f (-z)
β’ f 0 β€ f z
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
h : β (n : β), n β 0 β 0 β€ (-1) ^ n * iteratedDeriv n f 0
z : β
hz : z β€ 0
β’ f 0 β€ f z
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_alternating
|
[271, 1]
|
[278, 66]
|
convert at_zero_le_of_iteratedDeriv_nonneg (f := F) (hf.comp <| differentiable_neg)
(fun n hn β¦ ?_) (neg_nonneg.mpr hz) using 1
|
f : β β β
hf : Differentiable β f
h : β (n : β), n β 0 β 0 β€ (-1) ^ n * iteratedDeriv n f 0
z : β
hz : z β€ 0
F : β β β := fun z => f (-z)
β’ f 0 β€ f z
|
case h.e'_3
f : β β β
hf : Differentiable β f
h : β (n : β), n β 0 β 0 β€ (-1) ^ n * iteratedDeriv n f 0
z : β
hz : z β€ 0
F : β β β := fun z => f (-z)
β’ f 0 = F 0
case h.e'_4
f : β β β
hf : Differentiable β f
h : β (n : β), n β 0 β 0 β€ (-1) ^ n * iteratedDeriv n f 0
z : β
hz : z β€ 0
F : β β β := fun z => f (-z)
β’ f z = F (-z)
f : β β β
hf : Differentiable β f
h : β (n : β), n β 0 β 0 β€ (-1) ^ n * iteratedDeriv n f 0
z : β
hz : z β€ 0
F : β β β := fun z => f (-z)
n : β
hn : n β 0
β’ 0 β€ iteratedDeriv n F 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
h : β (n : β), n β 0 β 0 β€ (-1) ^ n * iteratedDeriv n f 0
z : β
hz : z β€ 0
F : β β β := fun z => f (-z)
β’ f 0 β€ f z
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_alternating
|
[271, 1]
|
[278, 66]
|
simp only [F, neg_zero]
|
case h.e'_3
f : β β β
hf : Differentiable β f
h : β (n : β), n β 0 β 0 β€ (-1) ^ n * iteratedDeriv n f 0
z : β
hz : z β€ 0
F : β β β := fun z => f (-z)
β’ f 0 = F 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3
f : β β β
hf : Differentiable β f
h : β (n : β), n β 0 β 0 β€ (-1) ^ n * iteratedDeriv n f 0
z : β
hz : z β€ 0
F : β β β := fun z => f (-z)
β’ f 0 = F 0
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_alternating
|
[271, 1]
|
[278, 66]
|
simp only [F, neg_neg]
|
case h.e'_4
f : β β β
hf : Differentiable β f
h : β (n : β), n β 0 β 0 β€ (-1) ^ n * iteratedDeriv n f 0
z : β
hz : z β€ 0
F : β β β := fun z => f (-z)
β’ f z = F (-z)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_4
f : β β β
hf : Differentiable β f
h : β (n : β), n β 0 β 0 β€ (-1) ^ n * iteratedDeriv n f 0
z : β
hz : z β€ 0
F : β β β := fun z => f (-z)
β’ f z = F (-z)
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Auxiliary.lean
|
Complex.at_zero_le_of_iteratedDeriv_alternating
|
[271, 1]
|
[278, 66]
|
simpa only [F, iteratedDeriv_comp_neg, neg_zero] using h n hn
|
f : β β β
hf : Differentiable β f
h : β (n : β), n β 0 β 0 β€ (-1) ^ n * iteratedDeriv n f 0
z : β
hz : z β€ 0
F : β β β := fun z => f (-z)
n : β
hn : n β 0
β’ 0 β€ iteratedDeriv n F 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hf : Differentiable β f
h : β (n : β), n β 0 β 0 β€ (-1) ^ n * iteratedDeriv n f 0
z : β
hz : z β€ 0
F : β β β := fun z => f (-z)
n : β
hn : n β 0
β’ 0 β€ iteratedDeriv n F 0
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
sum_primesBelow_eq_sum_range_indicator
|
[49, 1]
|
[58, 8]
|
convert (Finset.sum_indicator_subset f Finset.mem_of_mem_filter).symm using 2 with _ _ m hm
|
R : Type u_1
instβ : AddCommMonoid R
f : β β R
n : β
β’ β p β n.primesBelow, f p = β m β Finset.range n, {p | p.Prime}.indicator f m
|
case h.e'_3.a
R : Type u_1
instβ : AddCommMonoid R
f : β β R
n m : β
hm : m β Finset.range n
β’ {p | p.Prime}.indicator f m = (β(Finset.filter (fun p => p.Prime) (Finset.range n))).indicator f m
|
Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instβ : AddCommMonoid R
f : β β R
n : β
β’ β p β n.primesBelow, f p = β m β Finset.range n, {p | p.Prime}.indicator f m
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
sum_primesBelow_eq_sum_range_indicator
|
[49, 1]
|
[58, 8]
|
simp only [Set.mem_setOf_eq, Finset.mem_range, Finset.coe_filter, not_and, Set.indicator_apply]
|
case h.e'_3.a
R : Type u_1
instβ : AddCommMonoid R
f : β β R
n m : β
hm : m β Finset.range n
β’ {p | p.Prime}.indicator f m = (β(Finset.filter (fun p => p.Prime) (Finset.range n))).indicator f m
|
case h.e'_3.a
R : Type u_1
instβ : AddCommMonoid R
f : β β R
n m : β
hm : m β Finset.range n
β’ (if m.Prime then f m else 0) = if m < n β§ m.Prime then f m else 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.a
R : Type u_1
instβ : AddCommMonoid R
f : β β R
n m : β
hm : m β Finset.range n
β’ {p | p.Prime}.indicator f m = (β(Finset.filter (fun p => p.Prime) (Finset.range n))).indicator f m
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
sum_primesBelow_eq_sum_range_indicator
|
[49, 1]
|
[58, 8]
|
split_ifs with hβ hβ hβ
|
case h.e'_3.a
R : Type u_1
instβ : AddCommMonoid R
f : β β R
n m : β
hm : m β Finset.range n
β’ (if m.Prime then f m else 0) = if m < n β§ m.Prime then f m else 0
|
case pos
R : Type u_1
instβ : AddCommMonoid R
f : β β R
n m : β
hm : m β Finset.range n
hβ : m.Prime
hβ : m < n β§ m.Prime
β’ f m = f m
case neg
R : Type u_1
instβ : AddCommMonoid R
f : β β R
n m : β
hm : m β Finset.range n
hβ : m.Prime
hβ : Β¬(m < n β§ m.Prime)
β’ f m = 0
case pos
R : Type u_1
instβ : AddCommMonoid R
f : β β R
n m : β
hm : m β Finset.range n
hβ : Β¬m.Prime
hβ : m < n β§ m.Prime
β’ 0 = f m
case neg
R : Type u_1
instβ : AddCommMonoid R
f : β β R
n m : β
hm : m β Finset.range n
hβ : Β¬m.Prime
hβ : Β¬(m < n β§ m.Prime)
β’ 0 = 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
case h.e'_3.a
R : Type u_1
instβ : AddCommMonoid R
f : β β R
n m : β
hm : m β Finset.range n
β’ (if m.Prime then f m else 0) = if m < n β§ m.Prime then f m else 0
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
sum_primesBelow_eq_sum_range_indicator
|
[49, 1]
|
[58, 8]
|
rfl
|
case pos
R : Type u_1
instβ : AddCommMonoid R
f : β β R
n m : β
hm : m β Finset.range n
hβ : m.Prime
hβ : m < n β§ m.Prime
β’ f m = f m
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_1
instβ : AddCommMonoid R
f : β β R
n m : β
hm : m β Finset.range n
hβ : m.Prime
hβ : m < n β§ m.Prime
β’ f m = f m
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
sum_primesBelow_eq_sum_range_indicator
|
[49, 1]
|
[58, 8]
|
exact (hβ β¨Finset.mem_range.mp hm, hββ©).elim
|
case neg
R : Type u_1
instβ : AddCommMonoid R
f : β β R
n m : β
hm : m β Finset.range n
hβ : m.Prime
hβ : Β¬(m < n β§ m.Prime)
β’ f m = 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_1
instβ : AddCommMonoid R
f : β β R
n m : β
hm : m β Finset.range n
hβ : m.Prime
hβ : Β¬(m < n β§ m.Prime)
β’ f m = 0
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
sum_primesBelow_eq_sum_range_indicator
|
[49, 1]
|
[58, 8]
|
exact (hβ hβ.2).elim
|
case pos
R : Type u_1
instβ : AddCommMonoid R
f : β β R
n m : β
hm : m β Finset.range n
hβ : Β¬m.Prime
hβ : m < n β§ m.Prime
β’ 0 = f m
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case pos
R : Type u_1
instβ : AddCommMonoid R
f : β β R
n m : β
hm : m β Finset.range n
hβ : Β¬m.Prime
hβ : m < n β§ m.Prime
β’ 0 = f m
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
sum_primesBelow_eq_sum_range_indicator
|
[49, 1]
|
[58, 8]
|
rfl
|
case neg
R : Type u_1
instβ : AddCommMonoid R
f : β β R
n m : β
hm : m β Finset.range n
hβ : Β¬m.Prime
hβ : Β¬(m < n β§ m.Prime)
β’ 0 = 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case neg
R : Type u_1
instβ : AddCommMonoid R
f : β β R
n m : β
hm : m β Finset.range n
hβ : Β¬m.Prime
hβ : Β¬(m < n β§ m.Prime)
β’ 0 = 0
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
tendsto_sum_primesBelow_tsum
|
[62, 1]
|
[69, 94]
|
rw [(show β' p : Nat.Primes, f p = β' p : {p : β | p.Prime}, f p from rfl)]
|
R : Type u_1
instββ΄ : AddCommGroup R
instβΒ³ : UniformSpace R
instβΒ² : UniformAddGroup R
instβΒΉ : CompleteSpace R
instβ : T2Space R
f : β β R
hsum : Summable f
β’ Tendsto (fun n => β p β n.primesBelow, f p) atTop (π (β' (p : Nat.Primes), f βp))
|
R : Type u_1
instββ΄ : AddCommGroup R
instβΒ³ : UniformSpace R
instβΒ² : UniformAddGroup R
instβΒΉ : CompleteSpace R
instβ : T2Space R
f : β β R
hsum : Summable f
β’ Tendsto (fun n => β p β n.primesBelow, f p) atTop (π (β' (p : β{p | p.Prime}), f βp))
|
Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instββ΄ : AddCommGroup R
instβΒ³ : UniformSpace R
instβΒ² : UniformAddGroup R
instβΒΉ : CompleteSpace R
instβ : T2Space R
f : β β R
hsum : Summable f
β’ Tendsto (fun n => β p β n.primesBelow, f p) atTop (π (β' (p : Nat.Primes), f βp))
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
tendsto_sum_primesBelow_tsum
|
[62, 1]
|
[69, 94]
|
simp_rw [tsum_subtype, sum_primesBelow_eq_sum_range_indicator]
|
R : Type u_1
instββ΄ : AddCommGroup R
instβΒ³ : UniformSpace R
instβΒ² : UniformAddGroup R
instβΒΉ : CompleteSpace R
instβ : T2Space R
f : β β R
hsum : Summable f
β’ Tendsto (fun n => β p β n.primesBelow, f p) atTop (π (β' (p : β{p | p.Prime}), f βp))
|
R : Type u_1
instββ΄ : AddCommGroup R
instβΒ³ : UniformSpace R
instβΒ² : UniformAddGroup R
instβΒΉ : CompleteSpace R
instβ : T2Space R
f : β β R
hsum : Summable f
β’ Tendsto (fun n => β m β Finset.range n, {p | p.Prime}.indicator (fun p => f p) m) atTop
(π (β' (x : β), {p | p.Prime}.indicator f x))
|
Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instββ΄ : AddCommGroup R
instβΒ³ : UniformSpace R
instβΒ² : UniformAddGroup R
instβΒΉ : CompleteSpace R
instβ : T2Space R
f : β β R
hsum : Summable f
β’ Tendsto (fun n => β p β n.primesBelow, f p) atTop (π (β' (p : β{p | p.Prime}), f βp))
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
tendsto_sum_primesBelow_tsum
|
[62, 1]
|
[69, 94]
|
exact (Summable.hasSum_iff_tendsto_nat <| hsum.indicator _).mp <| (hsum.indicator _).hasSum
|
R : Type u_1
instββ΄ : AddCommGroup R
instβΒ³ : UniformSpace R
instβΒ² : UniformAddGroup R
instβΒΉ : CompleteSpace R
instβ : T2Space R
f : β β R
hsum : Summable f
β’ Tendsto (fun n => β m β Finset.range n, {p | p.Prime}.indicator (fun p => f p) m) atTop
(π (β' (x : β), {p | p.Prime}.indicator f x))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
R : Type u_1
instββ΄ : AddCommGroup R
instβΒ³ : UniformSpace R
instβΒ² : UniformAddGroup R
instβΒΉ : CompleteSpace R
instβ : T2Space R
f : β β R
hsum : Summable f
β’ Tendsto (fun n => β m β Finset.range n, {p | p.Prime}.indicator (fun p => f p) m) atTop
(π (β' (x : β), {p | p.Prime}.indicator f x))
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
Complex.exp_tsum_primes
|
[71, 1]
|
[77, 81]
|
simpa only [β exp_sum] using Tendsto.cexp <| tendsto_sum_primesBelow_tsum hsum
|
f : β β β
hsum : Summable f
β’ Tendsto (fun n => β p β n.primesBelow, cexp (f p)) atTop (π (cexp (β' (p : Nat.Primes), f βp)))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hsum : Summable f
β’ Tendsto (fun n => β p β n.primesBelow, cexp (f p)) atTop (π (cexp (β' (p : Nat.Primes), f βp)))
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
Summable.neg_clog_one_sub
|
[82, 1]
|
[91, 51]
|
let g (z : β) : β := -log (1 - z)
|
Ξ± : Type u_1
f : Ξ± β β
hsum : Summable f
β’ Summable fun n => -(1 - f n).log
|
Ξ± : Type u_1
f : Ξ± β β
hsum : Summable f
g : β β β := fun z => -(1 - z).log
β’ Summable fun n => -(1 - f n).log
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type u_1
f : Ξ± β β
hsum : Summable f
β’ Summable fun n => -(1 - f n).log
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
Summable.neg_clog_one_sub
|
[82, 1]
|
[91, 51]
|
have hg : DifferentiableAt β g 0 :=
DifferentiableAt.neg <| ((differentiableAt_const 1).sub differentiableAt_id').clog <|
by simp only [sub_zero, one_mem_slitPlane]
|
Ξ± : Type u_1
f : Ξ± β β
hsum : Summable f
g : β β β := fun z => -(1 - z).log
β’ Summable fun n => -(1 - f n).log
|
Ξ± : Type u_1
f : Ξ± β β
hsum : Summable f
g : β β β := fun z => -(1 - z).log
hg : DifferentiableAt β g 0
β’ Summable fun n => -(1 - f n).log
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type u_1
f : Ξ± β β
hsum : Summable f
g : β β β := fun z => -(1 - z).log
β’ Summable fun n => -(1 - f n).log
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
Summable.neg_clog_one_sub
|
[82, 1]
|
[91, 51]
|
have : g =O[π 0] id := by
simpa only [g, sub_zero, log_one, neg_zero] using DifferentiableAt.isBigO_sub hg
|
Ξ± : Type u_1
f : Ξ± β β
hsum : Summable f
g : β β β := fun z => -(1 - z).log
hg : DifferentiableAt β g 0
β’ Summable fun n => -(1 - f n).log
|
Ξ± : Type u_1
f : Ξ± β β
hsum : Summable f
g : β β β := fun z => -(1 - z).log
hg : DifferentiableAt β g 0
this : g =O[π 0] id
β’ Summable fun n => -(1 - f n).log
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type u_1
f : Ξ± β β
hsum : Summable f
g : β β β := fun z => -(1 - z).log
hg : DifferentiableAt β g 0
β’ Summable fun n => -(1 - f n).log
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
Summable.neg_clog_one_sub
|
[82, 1]
|
[91, 51]
|
exact Asymptotics.IsBigO.comp_summable this hsum
|
Ξ± : Type u_1
f : Ξ± β β
hsum : Summable f
g : β β β := fun z => -(1 - z).log
hg : DifferentiableAt β g 0
this : g =O[π 0] id
β’ Summable fun n => -(1 - f n).log
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type u_1
f : Ξ± β β
hsum : Summable f
g : β β β := fun z => -(1 - z).log
hg : DifferentiableAt β g 0
this : g =O[π 0] id
β’ Summable fun n => -(1 - f n).log
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
Summable.neg_clog_one_sub
|
[82, 1]
|
[91, 51]
|
simp only [sub_zero, one_mem_slitPlane]
|
Ξ± : Type u_1
f : Ξ± β β
hsum : Summable f
g : β β β := fun z => -(1 - z).log
β’ 1 - 0 β slitPlane
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type u_1
f : Ξ± β β
hsum : Summable f
g : β β β := fun z => -(1 - z).log
β’ 1 - 0 β slitPlane
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
Summable.neg_clog_one_sub
|
[82, 1]
|
[91, 51]
|
simpa only [g, sub_zero, log_one, neg_zero] using DifferentiableAt.isBigO_sub hg
|
Ξ± : Type u_1
f : Ξ± β β
hsum : Summable f
g : β β β := fun z => -(1 - z).log
hg : DifferentiableAt β g 0
β’ g =O[π 0] id
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
Ξ± : Type u_1
f : Ξ± β β
hsum : Summable f
g : β β β := fun z => -(1 - z).log
hg : DifferentiableAt β g 0
β’ g =O[π 0] id
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
EulerProduct.exp_sum_primes_log_eq_tsum
|
[96, 1]
|
[107, 77]
|
have hs {p : β} (hp : 1 < p) : βf pβ < 1 := hsum.of_norm.norm_lt_one (f := f.toMonoidHom) hp
|
f : β β*β β
hsum : Summable fun x => βf xβ
β’ cexp (β' (p : Nat.Primes), -(1 - f βp).log) = β' (n : β), f n
|
f : β β*β β
hsum : Summable fun x => βf xβ
hs : β {p : β}, 1 < p β βf pβ < 1
β’ cexp (β' (p : Nat.Primes), -(1 - f βp).log) = β' (n : β), f n
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β*β β
hsum : Summable fun x => βf xβ
β’ cexp (β' (p : Nat.Primes), -(1 - f βp).log) = β' (n : β), f n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
EulerProduct.exp_sum_primes_log_eq_tsum
|
[96, 1]
|
[107, 77]
|
have H := Complex.exp_tsum_primes hsum.of_norm.neg_clog_one_sub
|
f : β β*β β
hsum : Summable fun x => βf xβ
hs : β {p : β}, 1 < p β βf pβ < 1
β’ cexp (β' (p : Nat.Primes), -(1 - f βp).log) = β' (n : β), f n
|
f : β β*β β
hsum : Summable fun x => βf xβ
hs : β {p : β}, 1 < p β βf pβ < 1
H :
Tendsto (fun n => β p β n.primesBelow, cexp (-(1 - f p).log)) atTop (π (cexp (β' (p : Nat.Primes), -(1 - f βp).log)))
β’ cexp (β' (p : Nat.Primes), -(1 - f βp).log) = β' (n : β), f n
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β*β β
hsum : Summable fun x => βf xβ
hs : β {p : β}, 1 < p β βf pβ < 1
β’ cexp (β' (p : Nat.Primes), -(1 - f βp).log) = β' (n : β), f n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
EulerProduct.exp_sum_primes_log_eq_tsum
|
[96, 1]
|
[107, 77]
|
have help (n : β) : n.primesBelow.prod (fun p β¦ cexp (-log (1 - f p))) =
n.primesBelow.prod fun p β¦ (1 - f p)β»ΒΉ := by
refine Finset.prod_congr rfl (fun p hp β¦ ?_)
rw [exp_neg, exp_log ?_]
rw [ne_eq, sub_eq_zero, β ne_eq]
exact fun h β¦ (norm_one (Ξ± := β) βΈ h.symm βΈ hs (Nat.prime_of_mem_primesBelow hp).one_lt).false
|
f : β β*β β
hsum : Summable fun x => βf xβ
hs : β {p : β}, 1 < p β βf pβ < 1
H :
Tendsto (fun n => β p β n.primesBelow, cexp (-(1 - f p).log)) atTop (π (cexp (β' (p : Nat.Primes), -(1 - f βp).log)))
β’ cexp (β' (p : Nat.Primes), -(1 - f βp).log) = β' (n : β), f n
|
f : β β*β β
hsum : Summable fun x => βf xβ
hs : β {p : β}, 1 < p β βf pβ < 1
H :
Tendsto (fun n => β p β n.primesBelow, cexp (-(1 - f p).log)) atTop (π (cexp (β' (p : Nat.Primes), -(1 - f βp).log)))
help : β (n : β), β p β n.primesBelow, cexp (-(1 - f p).log) = β p β n.primesBelow, (1 - f p)β»ΒΉ
β’ cexp (β' (p : Nat.Primes), -(1 - f βp).log) = β' (n : β), f n
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β*β β
hsum : Summable fun x => βf xβ
hs : β {p : β}, 1 < p β βf pβ < 1
H :
Tendsto (fun n => β p β n.primesBelow, cexp (-(1 - f p).log)) atTop (π (cexp (β' (p : Nat.Primes), -(1 - f βp).log)))
β’ cexp (β' (p : Nat.Primes), -(1 - f βp).log) = β' (n : β), f n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
EulerProduct.exp_sum_primes_log_eq_tsum
|
[96, 1]
|
[107, 77]
|
simp_rw [help] at H
|
f : β β*β β
hsum : Summable fun x => βf xβ
hs : β {p : β}, 1 < p β βf pβ < 1
H :
Tendsto (fun n => β p β n.primesBelow, cexp (-(1 - f p).log)) atTop (π (cexp (β' (p : Nat.Primes), -(1 - f βp).log)))
help : β (n : β), β p β n.primesBelow, cexp (-(1 - f p).log) = β p β n.primesBelow, (1 - f p)β»ΒΉ
β’ cexp (β' (p : Nat.Primes), -(1 - f βp).log) = β' (n : β), f n
|
f : β β*β β
hsum : Summable fun x => βf xβ
hs : β {p : β}, 1 < p β βf pβ < 1
help : β (n : β), β p β n.primesBelow, cexp (-(1 - f p).log) = β p β n.primesBelow, (1 - f p)β»ΒΉ
H : Tendsto (fun n => β p β n.primesBelow, (1 - f p)β»ΒΉ) atTop (π (cexp (β' (p : Nat.Primes), -(1 - f βp).log)))
β’ cexp (β' (p : Nat.Primes), -(1 - f βp).log) = β' (n : β), f n
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β*β β
hsum : Summable fun x => βf xβ
hs : β {p : β}, 1 < p β βf pβ < 1
H :
Tendsto (fun n => β p β n.primesBelow, cexp (-(1 - f p).log)) atTop (π (cexp (β' (p : Nat.Primes), -(1 - f βp).log)))
help : β (n : β), β p β n.primesBelow, cexp (-(1 - f p).log) = β p β n.primesBelow, (1 - f p)β»ΒΉ
β’ cexp (β' (p : Nat.Primes), -(1 - f βp).log) = β' (n : β), f n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
EulerProduct.exp_sum_primes_log_eq_tsum
|
[96, 1]
|
[107, 77]
|
exact tendsto_nhds_unique H <| eulerProduct_completely_multiplicative hsum
|
f : β β*β β
hsum : Summable fun x => βf xβ
hs : β {p : β}, 1 < p β βf pβ < 1
help : β (n : β), β p β n.primesBelow, cexp (-(1 - f p).log) = β p β n.primesBelow, (1 - f p)β»ΒΉ
H : Tendsto (fun n => β p β n.primesBelow, (1 - f p)β»ΒΉ) atTop (π (cexp (β' (p : Nat.Primes), -(1 - f βp).log)))
β’ cexp (β' (p : Nat.Primes), -(1 - f βp).log) = β' (n : β), f n
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β*β β
hsum : Summable fun x => βf xβ
hs : β {p : β}, 1 < p β βf pβ < 1
help : β (n : β), β p β n.primesBelow, cexp (-(1 - f p).log) = β p β n.primesBelow, (1 - f p)β»ΒΉ
H : Tendsto (fun n => β p β n.primesBelow, (1 - f p)β»ΒΉ) atTop (π (cexp (β' (p : Nat.Primes), -(1 - f βp).log)))
β’ cexp (β' (p : Nat.Primes), -(1 - f βp).log) = β' (n : β), f n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
EulerProduct.exp_sum_primes_log_eq_tsum
|
[96, 1]
|
[107, 77]
|
refine Finset.prod_congr rfl (fun p hp β¦ ?_)
|
f : β β*β β
hsum : Summable fun x => βf xβ
hs : β {p : β}, 1 < p β βf pβ < 1
H :
Tendsto (fun n => β p β n.primesBelow, cexp (-(1 - f p).log)) atTop (π (cexp (β' (p : Nat.Primes), -(1 - f βp).log)))
n : β
β’ β p β n.primesBelow, cexp (-(1 - f p).log) = β p β n.primesBelow, (1 - f p)β»ΒΉ
|
f : β β*β β
hsum : Summable fun x => βf xβ
hs : β {p : β}, 1 < p β βf pβ < 1
H :
Tendsto (fun n => β p β n.primesBelow, cexp (-(1 - f p).log)) atTop (π (cexp (β' (p : Nat.Primes), -(1 - f βp).log)))
n p : β
hp : p β n.primesBelow
β’ cexp (-(1 - f p).log) = (1 - f p)β»ΒΉ
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β*β β
hsum : Summable fun x => βf xβ
hs : β {p : β}, 1 < p β βf pβ < 1
H :
Tendsto (fun n => β p β n.primesBelow, cexp (-(1 - f p).log)) atTop (π (cexp (β' (p : Nat.Primes), -(1 - f βp).log)))
n : β
β’ β p β n.primesBelow, cexp (-(1 - f p).log) = β p β n.primesBelow, (1 - f p)β»ΒΉ
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
EulerProduct.exp_sum_primes_log_eq_tsum
|
[96, 1]
|
[107, 77]
|
rw [exp_neg, exp_log ?_]
|
f : β β*β β
hsum : Summable fun x => βf xβ
hs : β {p : β}, 1 < p β βf pβ < 1
H :
Tendsto (fun n => β p β n.primesBelow, cexp (-(1 - f p).log)) atTop (π (cexp (β' (p : Nat.Primes), -(1 - f βp).log)))
n p : β
hp : p β n.primesBelow
β’ cexp (-(1 - f p).log) = (1 - f p)β»ΒΉ
|
f : β β*β β
hsum : Summable fun x => βf xβ
hs : β {p : β}, 1 < p β βf pβ < 1
H :
Tendsto (fun n => β p β n.primesBelow, cexp (-(1 - f p).log)) atTop (π (cexp (β' (p : Nat.Primes), -(1 - f βp).log)))
n p : β
hp : p β n.primesBelow
β’ 1 - f p β 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β*β β
hsum : Summable fun x => βf xβ
hs : β {p : β}, 1 < p β βf pβ < 1
H :
Tendsto (fun n => β p β n.primesBelow, cexp (-(1 - f p).log)) atTop (π (cexp (β' (p : Nat.Primes), -(1 - f βp).log)))
n p : β
hp : p β n.primesBelow
β’ cexp (-(1 - f p).log) = (1 - f p)β»ΒΉ
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
EulerProduct.exp_sum_primes_log_eq_tsum
|
[96, 1]
|
[107, 77]
|
rw [ne_eq, sub_eq_zero, β ne_eq]
|
f : β β*β β
hsum : Summable fun x => βf xβ
hs : β {p : β}, 1 < p β βf pβ < 1
H :
Tendsto (fun n => β p β n.primesBelow, cexp (-(1 - f p).log)) atTop (π (cexp (β' (p : Nat.Primes), -(1 - f βp).log)))
n p : β
hp : p β n.primesBelow
β’ 1 - f p β 0
|
f : β β*β β
hsum : Summable fun x => βf xβ
hs : β {p : β}, 1 < p β βf pβ < 1
H :
Tendsto (fun n => β p β n.primesBelow, cexp (-(1 - f p).log)) atTop (π (cexp (β' (p : Nat.Primes), -(1 - f βp).log)))
n p : β
hp : p β n.primesBelow
β’ 1 β f p
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β*β β
hsum : Summable fun x => βf xβ
hs : β {p : β}, 1 < p β βf pβ < 1
H :
Tendsto (fun n => β p β n.primesBelow, cexp (-(1 - f p).log)) atTop (π (cexp (β' (p : Nat.Primes), -(1 - f βp).log)))
n p : β
hp : p β n.primesBelow
β’ 1 - f p β 0
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/Logarithm.lean
|
EulerProduct.exp_sum_primes_log_eq_tsum
|
[96, 1]
|
[107, 77]
|
exact fun h β¦ (norm_one (Ξ± := β) βΈ h.symm βΈ hs (Nat.prime_of_mem_primesBelow hp).one_lt).false
|
f : β β*β β
hsum : Summable fun x => βf xβ
hs : β {p : β}, 1 < p β βf pβ < 1
H :
Tendsto (fun n => β p β n.primesBelow, cexp (-(1 - f p).log)) atTop (π (cexp (β' (p : Nat.Primes), -(1 - f βp).log)))
n p : β
hp : p β n.primesBelow
β’ 1 β f p
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β*β β
hsum : Summable fun x => βf xβ
hs : β {p : β}, 1 < p β βf pβ < 1
H :
Tendsto (fun n => β p β n.primesBelow, cexp (-(1 - f p).log)) atTop (π (cexp (β' (p : Nat.Primes), -(1 - f βp).log)))
n p : β
hp : p β n.primesBelow
β’ 1 β f p
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/EulerProduct.lean
|
LSeries.term_mul_aux
|
[21, 1]
|
[23, 90]
|
rw [mul_comm_div, div_div, β mul_div_assoc, mul_comm (m : β), natCast_mul_natCast_cpow]
|
a b : β
m n : β
s : β
β’ a / βm ^ s * (b / βn ^ s) = a * b / (βm * βn) ^ s
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
a b : β
m n : β
s : β
β’ a / βm ^ s * (b / βn ^ s) = a * b / (βm * βn) ^ s
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/EulerProduct.lean
|
LSeries.term_mul
|
[25, 1]
|
[30, 100]
|
rcases eq_or_ne (m * n) 0 with H | H
|
fβ fβ f : β β β
m n : β
h : f (m * n) = fβ m * fβ n
s : β
β’ term f s (m * n) = term fβ s m * term fβ s n
|
case inl
fβ fβ f : β β β
m n : β
h : f (m * n) = fβ m * fβ n
s : β
H : m * n = 0
β’ term f s (m * n) = term fβ s m * term fβ s n
case inr
fβ fβ f : β β β
m n : β
h : f (m * n) = fβ m * fβ n
s : β
H : m * n β 0
β’ term f s (m * n) = term fβ s m * term fβ s n
|
Please generate a tactic in lean4 to solve the state.
STATE:
fβ fβ f : β β β
m n : β
h : f (m * n) = fβ m * fβ n
s : β
β’ term f s (m * n) = term fβ s m * term fβ s n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/EulerProduct.lean
|
LSeries.term_mul
|
[25, 1]
|
[30, 100]
|
rcases mul_eq_zero.mp H with rfl | rfl <;> simp only [term_zero, mul_zero, zero_mul]
|
case inl
fβ fβ f : β β β
m n : β
h : f (m * n) = fβ m * fβ n
s : β
H : m * n = 0
β’ term f s (m * n) = term fβ s m * term fβ s n
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inl
fβ fβ f : β β β
m n : β
h : f (m * n) = fβ m * fβ n
s : β
H : m * n = 0
β’ term f s (m * n) = term fβ s m * term fβ s n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/EulerProduct.lean
|
LSeries.term_mul
|
[25, 1]
|
[30, 100]
|
obtain β¨hm, hnβ© := mul_ne_zero_iff.mp H
|
case inr
fβ fβ f : β β β
m n : β
h : f (m * n) = fβ m * fβ n
s : β
H : m * n β 0
β’ term f s (m * n) = term fβ s m * term fβ s n
|
case inr.intro
fβ fβ f : β β β
m n : β
h : f (m * n) = fβ m * fβ n
s : β
H : m * n β 0
hm : m β 0
hn : n β 0
β’ term f s (m * n) = term fβ s m * term fβ s n
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inr
fβ fβ f : β β β
m n : β
h : f (m * n) = fβ m * fβ n
s : β
H : m * n β 0
β’ term f s (m * n) = term fβ s m * term fβ s n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/EulerProduct.lean
|
LSeries.term_mul
|
[25, 1]
|
[30, 100]
|
simp only [ne_eq, H, not_false_eq_true, term_of_ne_zero, Nat.cast_mul, hm, hn, h, term_mul_aux]
|
case inr.intro
fβ fβ f : β β β
m n : β
h : f (m * n) = fβ m * fβ n
s : β
H : m * n β 0
hm : m β 0
hn : n β 0
β’ term f s (m * n) = term fβ s m * term fβ s n
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case inr.intro
fβ fβ f : β β β
m n : β
h : f (m * n) = fβ m * fβ n
s : β
H : m * n β 0
hm : m β 0
hn : n β 0
β’ term f s (m * n) = term fβ s m * term fβ s n
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/EulerProduct.lean
|
LSeries.term_at_one
|
[44, 1]
|
[45, 72]
|
rw [term_of_ne_zero one_ne_zero, hβ, Nat.cast_one, one_cpow, div_one]
|
f : β β β
hβ : f 1 = 1
s : β
β’ term f s 1 = 1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : β β β
hβ : f 1 = 1
s : β
β’ term f s 1 = 1
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/EulerProduct.lean
|
DirichletCharacter.toFun_on_nat_map_one
|
[86, 1]
|
[87, 32]
|
simp only [cast_one, map_one]
|
N : β
Ο : DirichletCharacter β N
β’ (fun n => Ο βn) 1 = 1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
β’ (fun n => Ο βn) 1 = 1
TACTIC:
|
https://github.com/MichaelStollBayreuth/EulerProducts.git
|
21e07835d1a467b99b5c3c9390d61ae69404445d
|
EulerProducts/EulerProduct.lean
|
DirichletCharacter.toFun_on_nat_map_mul
|
[89, 1]
|
[91, 32]
|
simp only [cast_mul, map_mul]
|
N : β
Ο : DirichletCharacter β N
m n : β
β’ (fun n => Ο βn) (m * n) = (fun n => Ο βn) m * (fun n => Ο βn) n
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
N : β
Ο : DirichletCharacter β N
m n : β
β’ (fun n => Ο βn) (m * n) = (fun n => Ο βn) m * (fun n => Ο βn) n
TACTIC:
|
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