url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_isLittleO
|
[49, 1]
|
[51, 6]
|
rfl
|
f : β β β
f' a : β
β’ HasDerivAt f f' a β (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_isLittleO_nhds_zero
|
[54, 1]
|
[57, 17]
|
rw [hasDerivAt_iff_isLittleO, β map_add_left_nhds_zero a, Asymptotics.isLittleO_map]
|
f : β β β
f' a : β
β’ HasDerivAt f f' a β (fun h => f (a + h) - f a - h * f') =o[π 0] fun h => h
|
f : β β β
f' a : β
β’ ((fun x => f x - f a - (x - a) * f') β fun x => a + x) =o[π 0] ((fun x => x - a) β fun x => a + x) β
(fun h => f (a + h) - f a - h * f') =o[π 0] fun h => h
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_tendsto
|
[60, 1]
|
[66, 85]
|
calc HasDerivAt f f' a
_ β Tendsto (fun x β¦ (f x - f a - (x - a) * f') / (x - a)) (π a) (π 0) := ?iff1
_ β Tendsto (fun x β¦ (f x - f a - (x - a) * f') / (x - a)) (π[β ] a) (π 0) := ?iff2
|
f : β β β
f' a : β
β’ HasDerivAt f f' a β Tendsto (fun x => (f x - f a - (x - a) * f') / (x - a)) (π[β ] a) (π 0)
|
case iff1
f : β β β
f' a : β
β’ HasDerivAt f f' a β Tendsto (fun x => (f x - f a - (x - a) * f') / (x - a)) (π a) (π 0)
case iff2
f : β β β
f' a : β
β’ Tendsto (fun x => (f x - f a - (x - a) * f') / (x - a)) (π a) (π 0) β
Tendsto (fun x => (f x - f a - (x - a) * f') / (x - a)) (π[β ] a) (π 0)
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_tendsto
|
[60, 1]
|
[66, 85]
|
case iff1 => rw [hasDerivAt_iff_isLittleO, Asymptotics.isLittleO_iff_tendsto (by intro _ h; simp [sub_eq_zero.1 h])]
|
f : β β β
f' a : β
β’ HasDerivAt f f' a β Tendsto (fun x => (f x - f a - (x - a) * f') / (x - a)) (π a) (π 0)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_tendsto
|
[60, 1]
|
[66, 85]
|
case iff2 => exact .symm <| tendsto_inf_principal_nhds_iff_of_forall_eq <| by simp
|
f : β β β
f' a : β
β’ Tendsto (fun x => (f x - f a - (x - a) * f') / (x - a)) (π a) (π 0) β
Tendsto (fun x => (f x - f a - (x - a) * f') / (x - a)) (π[β ] a) (π 0)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_tendsto
|
[60, 1]
|
[66, 85]
|
rw [hasDerivAt_iff_isLittleO, Asymptotics.isLittleO_iff_tendsto (by intro _ h; simp [sub_eq_zero.1 h])]
|
f : β β β
f' a : β
β’ HasDerivAt f f' a β Tendsto (fun x => (f x - f a - (x - a) * f') / (x - a)) (π a) (π 0)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_tendsto
|
[60, 1]
|
[66, 85]
|
intro _ h
|
f : β β β
f' a : β
β’ β (x : β), x - a = 0 β f x - f a - (x - a) * f' = 0
|
f : β β β
f' a xβ : β
h : xβ - a = 0
β’ f xβ - f a - (xβ - a) * f' = 0
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_tendsto
|
[60, 1]
|
[66, 85]
|
simp [sub_eq_zero.1 h]
|
f : β β β
f' a xβ : β
h : xβ - a = 0
β’ f xβ - f a - (xβ - a) * f' = 0
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_tendsto
|
[60, 1]
|
[66, 85]
|
exact .symm <| tendsto_inf_principal_nhds_iff_of_forall_eq <| by simp
|
f : β β β
f' a : β
β’ Tendsto (fun x => (f x - f a - (x - a) * f') / (x - a)) (π a) (π 0) β
Tendsto (fun x => (f x - f a - (x - a) * f') / (x - a)) (π[β ] a) (π 0)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_tendsto
|
[60, 1]
|
[66, 85]
|
simp
|
f : β β β
f' a : β
β’ β a_1 β {a}αΆ, (f a_1 - f a - (a_1 - a) * f') / (a_1 - a) = 0
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_tendsto_slope
|
[69, 1]
|
[77, 70]
|
calc HasDerivAt f f' a
_ β Tendsto (fun x β¦ (f x - f a) / (x - a) - (x - a) / (x - a) * f') (π[β ] a) (π 0) := ?iff1
_ β Tendsto (fun x β¦ (f x - f a) / (x - a) - f') (π[β ] a) (π 0) := ?iff2
_ β Tendsto (fun x β¦ (f x - f a) / (x - a)) (π[β ] a) (π f') := ?iff3
|
f : β β β
f' a : β
β’ HasDerivAt f f' a β Tendsto (fun x => (f x - f a) / (x - a)) (π[β ] a) (π f')
|
case iff1
f : β β β
f' a : β
β’ HasDerivAt f f' a β Tendsto (fun x => (f x - f a) / (x - a) - (x - a) / (x - a) * f') (π[β ] a) (π 0)
case iff2
f : β β β
f' a : β
β’ Tendsto (fun x => (f x - f a) / (x - a) - (x - a) / (x - a) * f') (π[β ] a) (π 0) β
Tendsto (fun x => (f x - f a) / (x - a) - f') (π[β ] a) (π 0)
case iff3
f : β β β
f' a : β
β’ Tendsto (fun x => (f x - f a) / (x - a) - f') (π[β ] a) (π 0) β
Tendsto (fun x => (f x - f a) / (x - a)) (π[β ] a) (π f')
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_tendsto_slope
|
[69, 1]
|
[77, 70]
|
case iff1 => simp only [hasDerivAt_iff_tendsto, sub_div, mul_div_right_comm]
|
f : β β β
f' a : β
β’ HasDerivAt f f' a β Tendsto (fun x => (f x - f a) / (x - a) - (x - a) / (x - a) * f') (π[β ] a) (π 0)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_tendsto_slope
|
[69, 1]
|
[77, 70]
|
case iff2 => exact tendsto_congr' <| (Set.EqOn.eventuallyEq fun _ h β¦ (by field_simp [sub_ne_zero.2 h])).filter_mono inf_le_right
|
f : β β β
f' a : β
β’ Tendsto (fun x => (f x - f a) / (x - a) - (x - a) / (x - a) * f') (π[β ] a) (π 0) β
Tendsto (fun x => (f x - f a) / (x - a) - f') (π[β ] a) (π 0)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_tendsto_slope
|
[69, 1]
|
[77, 70]
|
case iff3 => rw [β nhds_translation_sub f', tendsto_comap_iff]; rfl
|
f : β β β
f' a : β
β’ Tendsto (fun x => (f x - f a) / (x - a) - f') (π[β ] a) (π 0) β
Tendsto (fun x => (f x - f a) / (x - a)) (π[β ] a) (π f')
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_tendsto_slope
|
[69, 1]
|
[77, 70]
|
simp only [hasDerivAt_iff_tendsto, sub_div, mul_div_right_comm]
|
f : β β β
f' a : β
β’ HasDerivAt f f' a β Tendsto (fun x => (f x - f a) / (x - a) - (x - a) / (x - a) * f') (π[β ] a) (π 0)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_tendsto_slope
|
[69, 1]
|
[77, 70]
|
exact tendsto_congr' <| (Set.EqOn.eventuallyEq fun _ h β¦ (by field_simp [sub_ne_zero.2 h])).filter_mono inf_le_right
|
f : β β β
f' a : β
β’ Tendsto (fun x => (f x - f a) / (x - a) - (x - a) / (x - a) * f') (π[β ] a) (π 0) β
Tendsto (fun x => (f x - f a) / (x - a) - f') (π[β ] a) (π 0)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_tendsto_slope
|
[69, 1]
|
[77, 70]
|
field_simp [sub_ne_zero.2 h]
|
f : β β β
f' a xβ : β
h : xβ β {a}αΆ
β’ (f xβ - f a) / (xβ - a) - (xβ - a) / (xβ - a) * f' = (f xβ - f a) / (xβ - a) - f'
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_tendsto_slope
|
[69, 1]
|
[77, 70]
|
rw [β nhds_translation_sub f', tendsto_comap_iff]
|
f : β β β
f' a : β
β’ Tendsto (fun x => (f x - f a) / (x - a) - f') (π[β ] a) (π 0) β
Tendsto (fun x => (f x - f a) / (x - a)) (π[β ] a) (π f')
|
f : β β β
f' a : β
β’ Tendsto (fun x => (f x - f a) / (x - a) - f') (π[β ] a) (π 0) β
Tendsto ((fun x => x - f') β fun x => (f x - f a) / (x - a)) (π[β ] a) (π 0)
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_tendsto_slope
|
[69, 1]
|
[77, 70]
|
rfl
|
f : β β β
f' a : β
β’ Tendsto (fun x => (f x - f a) / (x - a) - f') (π[β ] a) (π 0) β
Tendsto ((fun x => x - f') β fun x => (f x - f a) / (x - a)) (π[β ] a) (π 0)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_const
|
[140, 1]
|
[144, 7]
|
rw [hasDerivAt_iff_isLittleO]
|
f : β β β
f' a c : β
β’ HasDerivAt (fun x => c) 0 a
|
f : β β β
f' a c : β
β’ (fun x => c - c - (x - a) * 0) =o[π a] fun x => x - a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_const
|
[140, 1]
|
[144, 7]
|
simp
|
f : β β β
f' a c : β
β’ (fun x => c - c - (x - a) * 0) =o[π a] fun x => x - a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_id
|
[147, 1]
|
[150, 7]
|
rw [hasDerivAt_iff_isLittleO]
|
f : β β β
f' aβ a : β
β’ HasDerivAt id 1 a
|
f : β β β
f' aβ a : β
β’ (fun x => id x - id a - (x - a) * 1) =o[π a] fun x => x - a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_id
|
[147, 1]
|
[150, 7]
|
simp
|
f : β β β
f' aβ a : β
β’ (fun x => id x - id a - (x - a) * 1) =o[π a] fun x => x - a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.add
|
[153, 1]
|
[168, 30]
|
rw [hasDerivAt_iff_isLittleO] at *
|
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
β’ HasDerivAt (fun x => f x + g x) (f' + g') a
|
f : β β β
f' a : β
g : β β β
g' : β
hf : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
hg : (fun x => g x - g a - (x - a) * g') =o[π a] fun x => x - a
β’ (fun x => f x + g x - (f a + g a) - (x - a) * (f' + g')) =o[π a] fun x => x - a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.add
|
[153, 1]
|
[168, 30]
|
calc (fun x β¦ f x + g x - (f a + g a) - (x - a) * (f' + g'))
_ = fun x β¦ (f x - f a - (x - a) * f') + (g x - g a - (x - a) * g') := ?eq1
_ =o[π a] fun x β¦ x - a := ?eq2
|
f : β β β
f' a : β
g : β β β
g' : β
hf : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
hg : (fun x => g x - g a - (x - a) * g') =o[π a] fun x => x - a
β’ (fun x => f x + g x - (f a + g a) - (x - a) * (f' + g')) =o[π a] fun x => x - a
|
case eq1
f : β β β
f' a : β
g : β β β
g' : β
hf : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
hg : (fun x => g x - g a - (x - a) * g') =o[π a] fun x => x - a
β’ (fun x => f x + g x - (f a + g a) - (x - a) * (f' + g')) = fun x =>
f x - f a - (x - a) * f' + (g x - g a - (x - a) * g')
case eq2
f : β β β
f' a : β
g : β β β
g' : β
hf : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
hg : (fun x => g x - g a - (x - a) * g') =o[π a] fun x => x - a
β’ (fun x => f x - f a - (x - a) * f' + (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.add
|
[153, 1]
|
[168, 30]
|
case eq1 =>
funext x
ring
|
f : β β β
f' a : β
g : β β β
g' : β
hf : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
hg : (fun x => g x - g a - (x - a) * g') =o[π a] fun x => x - a
β’ (fun x => f x + g x - (f a + g a) - (x - a) * (f' + g')) = fun x =>
f x - f a - (x - a) * f' + (g x - g a - (x - a) * g')
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.add
|
[153, 1]
|
[168, 30]
|
case eq2 =>
apply IsLittleO.add hf hg
|
f : β β β
f' a : β
g : β β β
g' : β
hf : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
hg : (fun x => g x - g a - (x - a) * g') =o[π a] fun x => x - a
β’ (fun x => f x - f a - (x - a) * f' + (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.add
|
[153, 1]
|
[168, 30]
|
funext x
|
f : β β β
f' a : β
g : β β β
g' : β
hf : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
hg : (fun x => g x - g a - (x - a) * g') =o[π a] fun x => x - a
β’ (fun x => f x + g x - (f a + g a) - (x - a) * (f' + g')) = fun x =>
f x - f a - (x - a) * f' + (g x - g a - (x - a) * g')
|
case h
f : β β β
f' a : β
g : β β β
g' : β
hf : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
hg : (fun x => g x - g a - (x - a) * g') =o[π a] fun x => x - a
x : β
β’ f x + g x - (f a + g a) - (x - a) * (f' + g') = f x - f a - (x - a) * f' + (g x - g a - (x - a) * g')
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.add
|
[153, 1]
|
[168, 30]
|
ring
|
case h
f : β β β
f' a : β
g : β β β
g' : β
hf : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
hg : (fun x => g x - g a - (x - a) * g') =o[π a] fun x => x - a
x : β
β’ f x + g x - (f a + g a) - (x - a) * (f' + g') = f x - f a - (x - a) * f' + (g x - g a - (x - a) * g')
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.add
|
[153, 1]
|
[168, 30]
|
apply IsLittleO.add hf hg
|
f : β β β
f' a : β
g : β β β
g' : β
hf : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
hg : (fun x => g x - g a - (x - a) * g') =o[π a] fun x => x - a
β’ (fun x => f x - f a - (x - a) * f' + (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.const_mul
|
[171, 1]
|
[183, 40]
|
rw [hasDerivAt_iff_isLittleO] at *
|
f : β β β
f' a c : β
hf : HasDerivAt f f' a
β’ HasDerivAt (fun x => c * f x) (c * f') a
|
f : β β β
f' a c : β
hf : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
β’ (fun x => c * f x - c * f a - (x - a) * (c * f')) =o[π a] fun x => x - a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.const_mul
|
[171, 1]
|
[183, 40]
|
calc (fun x β¦ c * f x - c * f a - (x - a) * (c * f'))
_ = fun x β¦ c * (f x - f a - (x - a) * f') := ?eq1
_ =o[π a] fun x β¦ x - a := ?eq2
|
f : β β β
f' a c : β
hf : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
β’ (fun x => c * f x - c * f a - (x - a) * (c * f')) =o[π a] fun x => x - a
|
case eq1
f : β β β
f' a c : β
hf : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
β’ (fun x => c * f x - c * f a - (x - a) * (c * f')) = fun x => c * (f x - f a - (x - a) * f')
case eq2
f : β β β
f' a c : β
hf : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
β’ (fun x => c * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.const_mul
|
[171, 1]
|
[183, 40]
|
case eq1 =>
funext x
ring
|
f : β β β
f' a c : β
hf : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
β’ (fun x => c * f x - c * f a - (x - a) * (c * f')) = fun x => c * (f x - f a - (x - a) * f')
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.const_mul
|
[171, 1]
|
[183, 40]
|
case eq2 =>
apply IsLittleO.const_mul_left hf c
|
f : β β β
f' a c : β
hf : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
β’ (fun x => c * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.const_mul
|
[171, 1]
|
[183, 40]
|
funext x
|
f : β β β
f' a c : β
hf : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
β’ (fun x => c * f x - c * f a - (x - a) * (c * f')) = fun x => c * (f x - f a - (x - a) * f')
|
case h
f : β β β
f' a c : β
hf : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
x : β
β’ c * f x - c * f a - (x - a) * (c * f') = c * (f x - f a - (x - a) * f')
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.const_mul
|
[171, 1]
|
[183, 40]
|
ring
|
case h
f : β β β
f' a c : β
hf : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
x : β
β’ c * f x - c * f a - (x - a) * (c * f') = c * (f x - f a - (x - a) * f')
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.const_mul
|
[171, 1]
|
[183, 40]
|
apply IsLittleO.const_mul_left hf c
|
f : β β β
f' a c : β
hf : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
β’ (fun x => c * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.neg
|
[187, 1]
|
[190, 20]
|
simpa using this
|
f : β β β
f' a : β
hf : HasDerivAt f f' a
this : HasDerivAt (fun x => -1 * f x) (-1 * f') a
β’ HasDerivAt (fun x => -f x) (-f') a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.sub
|
[193, 1]
|
[196, 18]
|
simpa [sub_eq_add_neg] using this
|
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
this : HasDerivAt (fun x => f x + -g x) (f' + -g') a
β’ HasDerivAt (fun x => f x - g x) (f' - g') a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.isBigO_sub
|
[203, 1]
|
[219, 32]
|
rw [hasDerivAt_iff_isLittleO] at h
|
f : β β β
f' a : β
h : HasDerivAt f f' a
β’ (fun x => f x - f a) =O[π a] fun x => x - a
|
f : β β β
f' a : β
h : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
β’ (fun x => f x - f a) =O[π a] fun x => x - a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.isBigO_sub
|
[203, 1]
|
[219, 32]
|
rw [h.isBigO.congr_of_sub]
|
f : β β β
f' a : β
h : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
β’ (fun x => f x - f a) =O[π a] fun x => x - a
|
f : β β β
f' a : β
h : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
β’ (fun x => (x - a) * f') =O[π a] fun x => x - a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.isBigO_sub
|
[203, 1]
|
[219, 32]
|
calc (fun x β¦ (x - a) * f')
_ = fun x β¦ f' * (x - a) := ?eq1
_ =O[π a] fun x β¦ x - a := ?eq2
|
f : β β β
f' a : β
h : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
β’ (fun x => (x - a) * f') =O[π a] fun x => x - a
|
case eq1
f : β β β
f' a : β
h : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
β’ (fun x => (x - a) * f') = fun x => f' * (x - a)
case eq2
f : β β β
f' a : β
h : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
β’ (fun x => f' * (x - a)) =O[π a] fun x => x - a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.isBigO_sub
|
[203, 1]
|
[219, 32]
|
case eq1 =>
funext x
ring
|
f : β β β
f' a : β
h : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
β’ (fun x => (x - a) * f') = fun x => f' * (x - a)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.isBigO_sub
|
[203, 1]
|
[219, 32]
|
case eq2 =>
apply isBigO_const_mul_self
|
f : β β β
f' a : β
h : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
β’ (fun x => f' * (x - a)) =O[π a] fun x => x - a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.isBigO_sub
|
[203, 1]
|
[219, 32]
|
funext x
|
f : β β β
f' a : β
h : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
β’ (fun x => (x - a) * f') = fun x => f' * (x - a)
|
case h
f : β β β
f' a : β
h : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
x : β
β’ (x - a) * f' = f' * (x - a)
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.isBigO_sub
|
[203, 1]
|
[219, 32]
|
ring
|
case h
f : β β β
f' a : β
h : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
x : β
β’ (x - a) * f' = f' * (x - a)
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.isBigO_sub
|
[203, 1]
|
[219, 32]
|
apply isBigO_const_mul_self
|
f : β β β
f' a : β
h : (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
β’ (fun x => f' * (x - a)) =O[π a] fun x => x - a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.continuousAt
|
[223, 1]
|
[231, 29]
|
have : Tendsto (fun x β¦ f x - f a + f a) (π a) (π (0 + f a)) := by
apply Tendsto.add _ tendsto_const_nhds
apply h.isBigO_sub.trans_tendsto
rw [β sub_self a]
apply tendsto_id.sub tendsto_const_nhds
|
f : β β β
f' a : β
h : HasDerivAt f f' a
β’ Tendsto f (π a) (π (f a))
|
f : β β β
f' a : β
h : HasDerivAt f f' a
this : Tendsto (fun x => f x - f a + f a) (π a) (π (0 + f a))
β’ Tendsto f (π a) (π (f a))
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.continuousAt
|
[223, 1]
|
[231, 29]
|
rw [zero_add] at this
|
f : β β β
f' a : β
h : HasDerivAt f f' a
this : Tendsto (fun x => f x - f a + f a) (π a) (π (0 + f a))
β’ Tendsto f (π a) (π (f a))
|
f : β β β
f' a : β
h : HasDerivAt f f' a
this : Tendsto (fun x => f x - f a + f a) (π a) (π (f a))
β’ Tendsto f (π a) (π (f a))
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.continuousAt
|
[223, 1]
|
[231, 29]
|
exact this.congr (by simp)
|
f : β β β
f' a : β
h : HasDerivAt f f' a
this : Tendsto (fun x => f x - f a + f a) (π a) (π (f a))
β’ Tendsto f (π a) (π (f a))
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.continuousAt
|
[223, 1]
|
[231, 29]
|
apply Tendsto.add _ tendsto_const_nhds
|
f : β β β
f' a : β
h : HasDerivAt f f' a
β’ Tendsto (fun x => f x - f a + f a) (π a) (π (0 + f a))
|
f : β β β
f' a : β
h : HasDerivAt f f' a
β’ Tendsto (fun x => f x - f a) (π a) (π 0)
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.continuousAt
|
[223, 1]
|
[231, 29]
|
apply h.isBigO_sub.trans_tendsto
|
f : β β β
f' a : β
h : HasDerivAt f f' a
β’ Tendsto (fun x => f x - f a) (π a) (π 0)
|
f : β β β
f' a : β
h : HasDerivAt f f' a
β’ Tendsto (fun x => x - a) (π a) (π 0)
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.continuousAt
|
[223, 1]
|
[231, 29]
|
rw [β sub_self a]
|
f : β β β
f' a : β
h : HasDerivAt f f' a
β’ Tendsto (fun x => x - a) (π a) (π 0)
|
f : β β β
f' a : β
h : HasDerivAt f f' a
β’ Tendsto (fun x => x - a) (π a) (π (a - a))
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.continuousAt
|
[223, 1]
|
[231, 29]
|
apply tendsto_id.sub tendsto_const_nhds
|
f : β β β
f' a : β
h : HasDerivAt f f' a
β’ Tendsto (fun x => x - a) (π a) (π (a - a))
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.continuousAt
|
[223, 1]
|
[231, 29]
|
simp
|
f : β β β
f' a : β
h : HasDerivAt f f' a
this : Tendsto (fun x => f x - f a + f a) (π a) (π (f a))
β’ β (x : β), f x - f a + f a = f x
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[240, 1]
|
[273, 13]
|
have hβ :=
calc (fun x β¦ g (f x) - g (f a) - (f x - f a) * g')
=o[π a] fun x β¦ f x - f a := ?eq1
_ =O[π a] fun x β¦ x - a := ?eq2
|
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
β’ HasDerivAt (g β f) (g' * f') a
|
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
hβ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => x - a
β’ HasDerivAt (g β f) (g' * f') a
case eq1
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
β’ (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => f x - f a
case eq2
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
β’ (fun x => f x - f a) =O[π a] fun x => x - a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[240, 1]
|
[273, 13]
|
have hβ :=
calc (fun x β¦ (f x - f a) * g' - (x - a) * (g' * f'))
_ = fun x β¦ g' * (f x - f a - (x - a) * f') := ?eq3
_ =O[π a] fun x β¦ f x - f a - (x - a) * f' := ?eq4
_ =o[π a] fun x β¦ x - a := ?eq5
|
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
hβ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => x - a
β’ HasDerivAt (g β f) (g' * f') a
case eq1
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
β’ (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => f x - f a
case eq2
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
β’ (fun x => f x - f a) =O[π a] fun x => x - a
|
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
hβ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => x - a
hβ : (fun x => (f x - f a) * g' - (x - a) * (g' * f')) =o[π a] fun x => x - a
β’ HasDerivAt (g β f) (g' * f') a
case eq3
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
hβ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => x - a
β’ (fun x => (f x - f a) * g' - (x - a) * (g' * f')) = fun x => g' * (f x - f a - (x - a) * f')
case eq4
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
hβ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => x - a
β’ (fun x => g' * (f x - f a - (x - a) * f')) =O[π a] fun x => f x - f a - (x - a) * f'
case eq5
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
hβ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => x - a
β’ (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
case eq1
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
β’ (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => f x - f a
case eq2
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
β’ (fun x => f x - f a) =O[π a] fun x => x - a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[240, 1]
|
[273, 13]
|
apply hβ.triangle hβ
|
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
hβ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => x - a
hβ : (fun x => (f x - f a) * g' - (x - a) * (g' * f')) =o[π a] fun x => x - a
β’ HasDerivAt (g β f) (g' * f') a
case eq3
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
hβ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => x - a
β’ (fun x => (f x - f a) * g' - (x - a) * (g' * f')) = fun x => g' * (f x - f a - (x - a) * f')
case eq4
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
hβ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => x - a
β’ (fun x => g' * (f x - f a - (x - a) * f')) =O[π a] fun x => f x - f a - (x - a) * f'
case eq5
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
hβ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => x - a
β’ (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
case eq1
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
β’ (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => f x - f a
case eq2
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
β’ (fun x => f x - f a) =O[π a] fun x => x - a
|
case eq3
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
hβ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => x - a
β’ (fun x => (f x - f a) * g' - (x - a) * (g' * f')) = fun x => g' * (f x - f a - (x - a) * f')
case eq4
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
hβ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => x - a
β’ (fun x => g' * (f x - f a - (x - a) * f')) =O[π a] fun x => f x - f a - (x - a) * f'
case eq5
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
hβ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => x - a
β’ (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
case eq1
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
β’ (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => f x - f a
case eq2
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
β’ (fun x => f x - f a) =O[π a] fun x => x - a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[240, 1]
|
[273, 13]
|
case eq1 =>
apply hg.comp_tendsto
apply hf.continuousAt
|
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
β’ (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => f x - f a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[240, 1]
|
[273, 13]
|
case eq2 =>
apply hf.isBigO_sub
|
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
β’ (fun x => f x - f a) =O[π a] fun x => x - a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[240, 1]
|
[273, 13]
|
case eq3 =>
funext
ring
|
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
hβ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => x - a
β’ (fun x => (f x - f a) * g' - (x - a) * (g' * f')) = fun x => g' * (f x - f a - (x - a) * f')
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[240, 1]
|
[273, 13]
|
case eq4 =>
apply isBigO_const_mul_self
|
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
hβ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => x - a
β’ (fun x => g' * (f x - f a - (x - a) * f')) =O[π a] fun x => f x - f a - (x - a) * f'
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[240, 1]
|
[273, 13]
|
case eq5 =>
apply hf
|
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
hβ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => x - a
β’ (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[240, 1]
|
[273, 13]
|
apply hg.comp_tendsto
|
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
β’ (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => f x - f a
|
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
β’ Tendsto (fun x => f x) (π a) (π (f a))
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[240, 1]
|
[273, 13]
|
apply hf.continuousAt
|
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
β’ Tendsto (fun x => f x) (π a) (π (f a))
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[240, 1]
|
[273, 13]
|
apply hf.isBigO_sub
|
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
β’ (fun x => f x - f a) =O[π a] fun x => x - a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[240, 1]
|
[273, 13]
|
funext
|
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
hβ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => x - a
β’ (fun x => (f x - f a) * g' - (x - a) * (g' * f')) = fun x => g' * (f x - f a - (x - a) * f')
|
case h
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
hβ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => x - a
xβ : β
β’ (f xβ - f a) * g' - (xβ - a) * (g' * f') = g' * (f xβ - f a - (xβ - a) * f')
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[240, 1]
|
[273, 13]
|
ring
|
case h
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
hβ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => x - a
xβ : β
β’ (f xβ - f a) * g' - (xβ - a) * (g' * f') = g' * (f xβ - f a - (xβ - a) * f')
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[240, 1]
|
[273, 13]
|
apply isBigO_const_mul_self
|
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
hβ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => x - a
β’ (fun x => g' * (f x - f a - (x - a) * f')) =O[π a] fun x => f x - f a - (x - a) * f'
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[240, 1]
|
[273, 13]
|
apply hf
|
f : β β β
f' a : β
g : β β β
g' : β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
hβ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[π a] fun x => x - a
β’ (fun x => f x - f a - (x - a) * f') =o[π a] fun x => x - a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[280, 1]
|
[325, 11]
|
rw [hasDerivAt_iff_isLittleO]
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
β’ HasDerivAt (fun x => f x * g x) (f' * g a + f a * g') a
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
β’ (fun x => f x * g x - f a * g a - (x - a) * (f' * g a + f a * g')) =o[π a] fun x => x - a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[280, 1]
|
[325, 11]
|
calc (fun x β¦ f x * g x - f a * g a - (x - a) * (f' * g a + f a * g'))
_ = fun x β¦ g a * (f x - f a - (x - a) * f') +
(f a * (g x - g a - (x - a) * g') + (f x - f a) * (g x - g a)) := ?eq1
_ =o[π a] fun x β¦ x - a := ?eq2
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
β’ (fun x => f x * g x - f a * g a - (x - a) * (f' * g a + f a * g')) =o[π a] fun x => x - a
|
case eq1
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
β’ (fun x => f x * g x - f a * g a - (x - a) * (f' * g a + f a * g')) = fun x =>
g a * (f x - f a - (x - a) * f') + (f a * (g x - g a - (x - a) * g') + (f x - f a) * (g x - g a))
case eq2
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
β’ (fun x => g a * (f x - f a - (x - a) * f') + (f a * (g x - g a - (x - a) * g') + (f x - f a) * (g x - g a))) =o[π a]
fun x => x - a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[280, 1]
|
[325, 11]
|
case eq1 =>
funext
ring
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
β’ (fun x => f x * g x - f a * g a - (x - a) * (f' * g a + f a * g')) = fun x =>
g a * (f x - f a - (x - a) * f') + (f a * (g x - g a - (x - a) * g') + (f x - f a) * (g x - g a))
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[280, 1]
|
[325, 11]
|
funext
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
β’ (fun x => f x * g x - f a * g a - (x - a) * (f' * g a + f a * g')) = fun x =>
g a * (f x - f a - (x - a) * f') + (f a * (g x - g a - (x - a) * g') + (f x - f a) * (g x - g a))
|
case h
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
xβ : β
β’ f xβ * g xβ - f a * g a - (xβ - a) * (f' * g a + f a * g') =
g a * (f xβ - f a - (xβ - a) * f') + (f a * (g xβ - g a - (xβ - a) * g') + (f xβ - f a) * (g xβ - g a))
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[280, 1]
|
[325, 11]
|
ring
|
case h
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
xβ : β
β’ f xβ * g xβ - f a * g a - (xβ - a) * (f' * g a + f a * g') =
g a * (f xβ - f a - (xβ - a) * f') + (f a * (g xβ - g a - (xβ - a) * g') + (f xβ - f a) * (g xβ - g a))
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[280, 1]
|
[325, 11]
|
have hf' : (fun x β¦ g a * (f x - f a - (x - a) * f')) =o[π a] fun x β¦ x - a := by
apply IsLittleO.const_mul_left hf
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
β’ (fun x => g a * (f x - f a - (x - a) * f') + (f a * (g x - g a - (x - a) * g') + (f x - f a) * (g x - g a))) =o[π a]
fun x => x - a
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
β’ (fun x => g a * (f x - f a - (x - a) * f') + (f a * (g x - g a - (x - a) * g') + (f x - f a) * (g x - g a))) =o[π a]
fun x => x - a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[280, 1]
|
[325, 11]
|
have hg' : (fun x β¦ f a * (g x - g a - (x - a) * g')) =o[π a] fun x β¦ x - a := by
apply IsLittleO.const_mul_left hg
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
β’ (fun x => g a * (f x - f a - (x - a) * f') + (f a * (g x - g a - (x - a) * g') + (f x - f a) * (g x - g a))) =o[π a]
fun x => x - a
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
β’ (fun x => g a * (f x - f a - (x - a) * f') + (f a * (g x - g a - (x - a) * g') + (f x - f a) * (g x - g a))) =o[π a]
fun x => x - a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[280, 1]
|
[325, 11]
|
have hfg :=
calc (fun x β¦ (f x - f a) * (g x - g a))
_ =o[π a] fun x β¦ (x - a) * 1 := ?eq3
_ = fun x β¦ x - a := ?eq4
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
β’ (fun x => g a * (f x - f a - (x - a) * f') + (f a * (g x - g a - (x - a) * g') + (f x - f a) * (g x - g a))) =o[π a]
fun x => x - a
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
hfg : (fun x => (f x - f a) * (g x - g a)) =o[π a] fun x => x - a
β’ (fun x => g a * (f x - f a - (x - a) * f') + (f a * (g x - g a - (x - a) * g') + (f x - f a) * (g x - g a))) =o[π a]
fun x => x - a
case eq3
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
β’ (fun x => (f x - f a) * (g x - g a)) =o[π a] fun x => (x - a) * 1
case eq4
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
β’ (fun x => (x - a) * 1) = fun x => x - a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[280, 1]
|
[325, 11]
|
apply IsLittleO.add hf'
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
hfg : (fun x => (f x - f a) * (g x - g a)) =o[π a] fun x => x - a
β’ (fun x => g a * (f x - f a - (x - a) * f') + (f a * (g x - g a - (x - a) * g') + (f x - f a) * (g x - g a))) =o[π a]
fun x => x - a
case eq3
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
β’ (fun x => (f x - f a) * (g x - g a)) =o[π a] fun x => (x - a) * 1
case eq4
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
β’ (fun x => (x - a) * 1) = fun x => x - a
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
hfg : (fun x => (f x - f a) * (g x - g a)) =o[π a] fun x => x - a
β’ (fun x => f a * (g x - g a - (x - a) * g') + (f x - f a) * (g x - g a)) =o[π a] fun x => x - a
case eq3
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
β’ (fun x => (f x - f a) * (g x - g a)) =o[π a] fun x => (x - a) * 1
case eq4
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
β’ (fun x => (x - a) * 1) = fun x => x - a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[280, 1]
|
[325, 11]
|
apply IsLittleO.add hg'
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
hfg : (fun x => (f x - f a) * (g x - g a)) =o[π a] fun x => x - a
β’ (fun x => f a * (g x - g a - (x - a) * g') + (f x - f a) * (g x - g a)) =o[π a] fun x => x - a
case eq3
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
β’ (fun x => (f x - f a) * (g x - g a)) =o[π a] fun x => (x - a) * 1
case eq4
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
β’ (fun x => (x - a) * 1) = fun x => x - a
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
hfg : (fun x => (f x - f a) * (g x - g a)) =o[π a] fun x => x - a
β’ (fun x => (f x - f a) * (g x - g a)) =o[π a] fun x => x - a
case eq3
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
β’ (fun x => (f x - f a) * (g x - g a)) =o[π a] fun x => (x - a) * 1
case eq4
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
β’ (fun x => (x - a) * 1) = fun x => x - a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[280, 1]
|
[325, 11]
|
apply hfg
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
hfg : (fun x => (f x - f a) * (g x - g a)) =o[π a] fun x => x - a
β’ (fun x => (f x - f a) * (g x - g a)) =o[π a] fun x => x - a
case eq3
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
β’ (fun x => (f x - f a) * (g x - g a)) =o[π a] fun x => (x - a) * 1
case eq4
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
β’ (fun x => (x - a) * 1) = fun x => x - a
|
case eq3
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
β’ (fun x => (f x - f a) * (g x - g a)) =o[π a] fun x => (x - a) * 1
case eq4
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
β’ (fun x => (x - a) * 1) = fun x => x - a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[280, 1]
|
[325, 11]
|
case eq4 =>
funext
ring
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
β’ (fun x => (x - a) * 1) = fun x => x - a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[280, 1]
|
[325, 11]
|
apply IsLittleO.const_mul_left hf
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
β’ (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[280, 1]
|
[325, 11]
|
apply IsLittleO.const_mul_left hg
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
β’ (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[280, 1]
|
[325, 11]
|
have hg'' : (fun x β¦ g x - g a) =o[π a] fun _ β¦ (1 : β) := by
rw [Asymptotics.isLittleO_one_iff, tendsto_sub_nhds_zero_iff]
apply hg.continuousAt
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
β’ (fun x => (f x - f a) * (g x - g a)) =o[π a] fun x => (x - a) * 1
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
hg'' : (fun x => g x - g a) =o[π a] fun x => 1
β’ (fun x => (f x - f a) * (g x - g a)) =o[π a] fun x => (x - a) * 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[280, 1]
|
[325, 11]
|
apply IsBigO.mul_isLittleO
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
hg'' : (fun x => g x - g a) =o[π a] fun x => 1
β’ (fun x => (f x - f a) * (g x - g a)) =o[π a] fun x => (x - a) * 1
|
case hβ
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
hg'' : (fun x => g x - g a) =o[π a] fun x => 1
β’ (fun x => f x - f a) =O[π a] fun x => x - a
case hβ
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
hg'' : (fun x => g x - g a) =o[π a] fun x => 1
β’ (fun x => g x - g a) =o[π a] fun x => 1
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[280, 1]
|
[325, 11]
|
rw [Asymptotics.isLittleO_one_iff, tendsto_sub_nhds_zero_iff]
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
β’ (fun x => g x - g a) =o[π a] fun x => 1
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
β’ Tendsto (fun x => g x) (π a) (π (g a))
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[280, 1]
|
[325, 11]
|
apply hg.continuousAt
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
β’ Tendsto (fun x => g x) (π a) (π (g a))
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[280, 1]
|
[325, 11]
|
apply isBigO_sub hf
|
case hβ
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
hg'' : (fun x => g x - g a) =o[π a] fun x => 1
β’ (fun x => f x - f a) =O[π a] fun x => x - a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[280, 1]
|
[325, 11]
|
apply hg''
|
case hβ
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
hg'' : (fun x => g x - g a) =o[π a] fun x => 1
β’ (fun x => g x - g a) =o[π a] fun x => 1
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[280, 1]
|
[325, 11]
|
funext
|
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
β’ (fun x => (x - a) * 1) = fun x => x - a
|
case h
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
xβ : β
β’ (xβ - a) * 1 = xβ - a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[280, 1]
|
[325, 11]
|
ring
|
case h
fβ : β β β
f' a : β
g : β β β
g' : β
f : β β β
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[π a] fun x => x - a
hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[π a] fun x => x - a
xβ : β
β’ (xβ - a) * 1 = xβ - a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_pow
|
[332, 1]
|
[345, 35]
|
induction n
|
f : β β β
f' aβ : β
g : β β β
g' : β
n : β
a : β
β’ HasDerivAt (fun x => x ^ (n + 1)) ((βn + 1) * a ^ n) a
|
case zero
f : β β β
f' aβ : β
g : β β β
g' a : β
β’ HasDerivAt (fun x => x ^ (Nat.zero + 1)) ((βNat.zero + 1) * a ^ Nat.zero) a
case succ
f : β β β
f' aβ : β
g : β β β
g' a : β
nβ : β
n_ihβ : HasDerivAt (fun x => x ^ (nβ + 1)) ((βnβ + 1) * a ^ nβ) a
β’ HasDerivAt (fun x => x ^ (Nat.succ nβ + 1)) ((β(Nat.succ nβ) + 1) * a ^ Nat.succ nβ) a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_pow
|
[332, 1]
|
[345, 35]
|
case zero => simp [hasDerivAt_iff_isLittleO_nhds_zero]
|
f : β β β
f' aβ : β
g : β β β
g' a : β
β’ HasDerivAt (fun x => x ^ (Nat.zero + 1)) ((βNat.zero + 1) * a ^ Nat.zero) a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_pow
|
[332, 1]
|
[345, 35]
|
case succ n ih =>
rw [Nat.succ_eq_add_one]
suffices HasDerivAt (fun x => x ^ (n + 1) * x) (((n + 1) * a ^ n) * a + a ^ (n + 1) * 1) a by
apply IsLittleO.congr_left this
intro x
simp
ring
apply ih.mul (hasDerivAt_id a)
|
f : β β β
f' aβ : β
g : β β β
g' a : β
n : β
ih : HasDerivAt (fun x => x ^ (n + 1)) ((βn + 1) * a ^ n) a
β’ HasDerivAt (fun x => x ^ (Nat.succ n + 1)) ((β(Nat.succ n) + 1) * a ^ Nat.succ n) a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_pow
|
[332, 1]
|
[345, 35]
|
simp [hasDerivAt_iff_isLittleO_nhds_zero]
|
f : β β β
f' aβ : β
g : β β β
g' a : β
β’ HasDerivAt (fun x => x ^ (Nat.zero + 1)) ((βNat.zero + 1) * a ^ Nat.zero) a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_pow
|
[332, 1]
|
[345, 35]
|
rw [Nat.succ_eq_add_one]
|
f : β β β
f' aβ : β
g : β β β
g' a : β
n : β
ih : HasDerivAt (fun x => x ^ (n + 1)) ((βn + 1) * a ^ n) a
β’ HasDerivAt (fun x => x ^ (Nat.succ n + 1)) ((β(Nat.succ n) + 1) * a ^ Nat.succ n) a
|
f : β β β
f' aβ : β
g : β β β
g' a : β
n : β
ih : HasDerivAt (fun x => x ^ (n + 1)) ((βn + 1) * a ^ n) a
β’ HasDerivAt (fun x => x ^ (n + 1 + 1)) ((β(n + 1) + 1) * a ^ (n + 1)) a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_pow
|
[332, 1]
|
[345, 35]
|
suffices HasDerivAt (fun x => x ^ (n + 1) * x) (((n + 1) * a ^ n) * a + a ^ (n + 1) * 1) a by
apply IsLittleO.congr_left this
intro x
simp
ring
|
f : β β β
f' aβ : β
g : β β β
g' a : β
n : β
ih : HasDerivAt (fun x => x ^ (n + 1)) ((βn + 1) * a ^ n) a
β’ HasDerivAt (fun x => x ^ (n + 1 + 1)) ((β(n + 1) + 1) * a ^ (n + 1)) a
|
f : β β β
f' aβ : β
g : β β β
g' a : β
n : β
ih : HasDerivAt (fun x => x ^ (n + 1)) ((βn + 1) * a ^ n) a
β’ HasDerivAt (fun x => x ^ (n + 1) * x) ((βn + 1) * a ^ n * a + a ^ (n + 1) * 1) a
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_pow
|
[332, 1]
|
[345, 35]
|
apply ih.mul (hasDerivAt_id a)
|
f : β β β
f' aβ : β
g : β β β
g' a : β
n : β
ih : HasDerivAt (fun x => x ^ (n + 1)) ((βn + 1) * a ^ n) a
β’ HasDerivAt (fun x => x ^ (n + 1) * x) ((βn + 1) * a ^ n * a + a ^ (n + 1) * 1) a
|
no goals
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Solution/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_pow
|
[332, 1]
|
[345, 35]
|
apply IsLittleO.congr_left this
|
f : β β β
f' aβ : β
g : β β β
g' a : β
n : β
ih : HasDerivAt (fun x => x ^ (n + 1)) ((βn + 1) * a ^ n) a
this : HasDerivAt (fun x => x ^ (n + 1) * x) ((βn + 1) * a ^ n * a + a ^ (n + 1) * 1) a
β’ HasDerivAt (fun x => x ^ (n + 1 + 1)) ((β(n + 1) + 1) * a ^ (n + 1)) a
|
f : β β β
f' aβ : β
g : β β β
g' a : β
n : β
ih : HasDerivAt (fun x => x ^ (n + 1)) ((βn + 1) * a ^ n) a
this : HasDerivAt (fun x => x ^ (n + 1) * x) ((βn + 1) * a ^ n * a + a ^ (n + 1) * 1) a
β’ β (x : β),
(fun x => x ^ (n + 1) * x) x - (fun x => x ^ (n + 1) * x) a - (x - a) * ((βn + 1) * a ^ n * a + a ^ (n + 1) * 1) =
(fun x => x ^ (n + 1 + 1)) x - (fun x => x ^ (n + 1 + 1)) a - (x - a) * ((β(n + 1) + 1) * a ^ (n + 1))
|
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