Datasets:
internlm
/
Lean-Github

Dataset card Data Studio Files Community
1
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 : ℝ β†’ ℝ f' a : ℝ ⊒ HasDerivAt f f' a ↔ (fun x => f x - f a - (x - a) * f') =o[𝓝 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, ← map_add_left_nhds_zero a, Asymptotics.isLittleO_map]
f : ℝ β†’ ℝ f' a : ℝ ⊒ HasDerivAt f f' a ↔ (fun h => f (a + h) - f a - h * f') =o[𝓝 0] fun h => h
f : ℝ β†’ ℝ f' a : ℝ ⊒ ((fun x => f x - f a - (x - a) * f') ∘ fun x => a + x) =o[𝓝 0] ((fun x => x - a) ∘ fun x => a + x) ↔ (fun h => f (a + h) - f a - h * f') =o[𝓝 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 _ ↔ Tendsto (fun x ↦ (f x - f a - (x - a) * f') / (x - a)) (𝓝 a) (𝓝 0) := ?iff1 _ ↔ Tendsto (fun x ↦ (f x - f a - (x - a) * f') / (x - a)) (𝓝[β‰ ] a) (𝓝 0) := ?iff2
f : ℝ β†’ ℝ f' a : ℝ ⊒ HasDerivAt f f' a ↔ Tendsto (fun x => (f x - f a - (x - a) * f') / (x - a)) (𝓝[β‰ ] a) (𝓝 0)
case iff1 f : ℝ β†’ ℝ f' a : ℝ ⊒ HasDerivAt f f' a ↔ Tendsto (fun x => (f x - f a - (x - a) * f') / (x - a)) (𝓝 a) (𝓝 0) case iff2 f : ℝ β†’ ℝ f' a : ℝ ⊒ Tendsto (fun x => (f x - f a - (x - a) * f') / (x - a)) (𝓝 a) (𝓝 0) ↔ Tendsto (fun x => (f x - f a - (x - a) * f') / (x - a)) (𝓝[β‰ ] a) (𝓝 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 : ℝ β†’ ℝ f' a : ℝ ⊒ HasDerivAt f f' a ↔ Tendsto (fun x => (f x - f a - (x - a) * f') / (x - a)) (𝓝 a) (𝓝 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 : ℝ β†’ ℝ f' a : ℝ ⊒ Tendsto (fun x => (f x - f a - (x - a) * f') / (x - a)) (𝓝 a) (𝓝 0) ↔ Tendsto (fun x => (f x - f a - (x - a) * f') / (x - a)) (𝓝[β‰ ] a) (𝓝 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 : ℝ β†’ ℝ f' a : ℝ ⊒ HasDerivAt f f' a ↔ Tendsto (fun x => (f x - f a - (x - a) * f') / (x - a)) (𝓝 a) (𝓝 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 : ℝ β†’ ℝ f' a : ℝ ⊒ βˆ€ (x : ℝ), x - a = 0 β†’ f x - f a - (x - a) * f' = 0
f : ℝ β†’ ℝ f' a x✝ : ℝ h : x✝ - a = 0 ⊒ f x✝ - f a - (x✝ - 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 : ℝ β†’ ℝ f' a x✝ : ℝ h : x✝ - a = 0 ⊒ f x✝ - f a - (x✝ - 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 : ℝ β†’ ℝ f' a : ℝ ⊒ Tendsto (fun x => (f x - f a - (x - a) * f') / (x - a)) (𝓝 a) (𝓝 0) ↔ Tendsto (fun x => (f x - f a - (x - a) * f') / (x - a)) (𝓝[β‰ ] a) (𝓝 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 : ℝ β†’ ℝ f' a : ℝ ⊒ βˆ€ a_1 βˆ‰ {a}ᢜ, (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 _ ↔ Tendsto (fun x ↦ (f x - f a) / (x - a) - (x - a) / (x - a) * f') (𝓝[β‰ ] a) (𝓝 0) := ?iff1 _ ↔ Tendsto (fun x ↦ (f x - f a) / (x - a) - f') (𝓝[β‰ ] a) (𝓝 0) := ?iff2 _ ↔ Tendsto (fun x ↦ (f x - f a) / (x - a)) (𝓝[β‰ ] a) (𝓝 f') := ?iff3
f : ℝ β†’ ℝ f' a : ℝ ⊒ HasDerivAt f f' a ↔ Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[β‰ ] a) (𝓝 f')
case iff1 f : ℝ β†’ ℝ f' a : ℝ ⊒ HasDerivAt f f' a ↔ Tendsto (fun x => (f x - f a) / (x - a) - (x - a) / (x - a) * f') (𝓝[β‰ ] a) (𝓝 0) case iff2 f : ℝ β†’ ℝ f' a : ℝ ⊒ Tendsto (fun x => (f x - f a) / (x - a) - (x - a) / (x - a) * f') (𝓝[β‰ ] a) (𝓝 0) ↔ Tendsto (fun x => (f x - f a) / (x - a) - f') (𝓝[β‰ ] a) (𝓝 0) case iff3 f : ℝ β†’ ℝ f' a : ℝ ⊒ Tendsto (fun x => (f x - f a) / (x - a) - f') (𝓝[β‰ ] a) (𝓝 0) ↔ Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[β‰ ] a) (𝓝 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 : ℝ β†’ ℝ f' a : ℝ ⊒ HasDerivAt f f' a ↔ Tendsto (fun x => (f x - f a) / (x - a) - (x - a) / (x - a) * f') (𝓝[β‰ ] a) (𝓝 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 ↦ (by field_simp [sub_ne_zero.2 h])).filter_mono inf_le_right
f : ℝ β†’ ℝ f' a : ℝ ⊒ Tendsto (fun x => (f x - f a) / (x - a) - (x - a) / (x - a) * f') (𝓝[β‰ ] a) (𝓝 0) ↔ Tendsto (fun x => (f x - f a) / (x - a) - f') (𝓝[β‰ ] a) (𝓝 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 [← nhds_translation_sub f', tendsto_comap_iff]; rfl
f : ℝ β†’ ℝ f' a : ℝ ⊒ Tendsto (fun x => (f x - f a) / (x - a) - f') (𝓝[β‰ ] a) (𝓝 0) ↔ Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[β‰ ] a) (𝓝 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 : ℝ β†’ ℝ f' a : ℝ ⊒ HasDerivAt f f' a ↔ Tendsto (fun x => (f x - f a) / (x - a) - (x - a) / (x - a) * f') (𝓝[β‰ ] a) (𝓝 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 ↦ (by field_simp [sub_ne_zero.2 h])).filter_mono inf_le_right
f : ℝ β†’ ℝ f' a : ℝ ⊒ Tendsto (fun x => (f x - f a) / (x - a) - (x - a) / (x - a) * f') (𝓝[β‰ ] a) (𝓝 0) ↔ Tendsto (fun x => (f x - f a) / (x - a) - f') (𝓝[β‰ ] a) (𝓝 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 : ℝ β†’ ℝ f' a x✝ : ℝ h : x✝ ∈ {a}ᢜ ⊒ (f x✝ - f a) / (x✝ - a) - (x✝ - a) / (x✝ - a) * f' = (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_iff_tendsto_slope
[69, 1]
[77, 70]
rw [← nhds_translation_sub f', tendsto_comap_iff]
f : ℝ β†’ ℝ f' a : ℝ ⊒ Tendsto (fun x => (f x - f a) / (x - a) - f') (𝓝[β‰ ] a) (𝓝 0) ↔ Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[β‰ ] a) (𝓝 f')
f : ℝ β†’ ℝ f' a : ℝ ⊒ Tendsto (fun x => (f x - f a) / (x - a) - f') (𝓝[β‰ ] a) (𝓝 0) ↔ Tendsto ((fun x => x - f') ∘ fun x => (f x - f a) / (x - a)) (𝓝[β‰ ] a) (𝓝 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 : ℝ β†’ ℝ f' a : ℝ ⊒ Tendsto (fun x => (f x - f a) / (x - a) - f') (𝓝[β‰ ] a) (𝓝 0) ↔ Tendsto ((fun x => x - f') ∘ fun x => (f x - f a) / (x - a)) (𝓝[β‰ ] a) (𝓝 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 : ℝ β†’ ℝ f' a c : ℝ ⊒ HasDerivAt (fun x => c) 0 a
f : ℝ β†’ ℝ f' a c : ℝ ⊒ (fun x => c - c - (x - a) * 0) =o[𝓝 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 : ℝ β†’ ℝ f' a c : ℝ ⊒ (fun x => c - c - (x - a) * 0) =o[𝓝 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 : ℝ β†’ ℝ f' a✝ a : ℝ ⊒ HasDerivAt id 1 a
f : ℝ β†’ ℝ f' a✝ a : ℝ ⊒ (fun x => id x - id a - (x - a) * 1) =o[𝓝 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 : ℝ β†’ ℝ f' a✝ a : ℝ ⊒ (fun x => id x - id a - (x - a) * 1) =o[𝓝 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 : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a ⊒ HasDerivAt (fun x => f x + g x) (f' + g') a
f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a hg : (fun x => g x - g a - (x - a) * g') =o[𝓝 a] fun x => x - a ⊒ (fun x => f x + g x - (f a + g a) - (x - a) * (f' + g')) =o[𝓝 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 ↦ 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') := ?eq1 _ =o[𝓝 a] fun x ↦ x - a := ?eq2
f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a hg : (fun x => g x - g a - (x - a) * g') =o[𝓝 a] fun x => x - a ⊒ (fun x => f x + g x - (f a + g a) - (x - a) * (f' + g')) =o[𝓝 a] fun x => x - a
case eq1 f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a hg : (fun x => g x - g a - (x - a) * g') =o[𝓝 a] fun x => x - a ⊒ (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 : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a hg : (fun x => g x - g a - (x - a) * g') =o[𝓝 a] fun x => x - a ⊒ (fun x => f x - f a - (x - a) * f' + (g x - g a - (x - a) * g')) =o[𝓝 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 : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a hg : (fun x => g x - g a - (x - a) * g') =o[𝓝 a] fun x => x - a ⊒ (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 : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a hg : (fun x => g x - g a - (x - a) * g') =o[𝓝 a] fun x => x - a ⊒ (fun x => f x - f a - (x - a) * f' + (g x - g a - (x - a) * g')) =o[𝓝 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 : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a hg : (fun x => g x - g a - (x - a) * g') =o[𝓝 a] fun x => x - a ⊒ (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 : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a hg : (fun x => g x - g a - (x - a) * g') =o[𝓝 a] fun x => x - a x : ℝ ⊒ 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 : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a hg : (fun x => g x - g a - (x - a) * g') =o[𝓝 a] fun x => x - a x : ℝ ⊒ 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 : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a hg : (fun x => g x - g a - (x - a) * g') =o[𝓝 a] fun x => x - a ⊒ (fun x => f x - f a - (x - a) * f' + (g x - g a - (x - a) * g')) =o[𝓝 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 : ℝ β†’ ℝ f' a c : ℝ hf : HasDerivAt f f' a ⊒ HasDerivAt (fun x => c * f x) (c * f') a
f : ℝ β†’ ℝ f' a c : ℝ hf : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a ⊒ (fun x => c * f x - c * f a - (x - a) * (c * f')) =o[𝓝 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 ↦ c * f x - c * f a - (x - a) * (c * f')) _ = fun x ↦ c * (f x - f a - (x - a) * f') := ?eq1 _ =o[𝓝 a] fun x ↦ x - a := ?eq2
f : ℝ β†’ ℝ f' a c : ℝ hf : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a ⊒ (fun x => c * f x - c * f a - (x - a) * (c * f')) =o[𝓝 a] fun x => x - a
case eq1 f : ℝ β†’ ℝ f' a c : ℝ hf : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a ⊒ (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 : ℝ β†’ ℝ f' a c : ℝ hf : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a ⊒ (fun x => c * (f x - f a - (x - a) * f')) =o[𝓝 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 : ℝ β†’ ℝ f' a c : ℝ hf : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a ⊒ (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 : ℝ β†’ ℝ f' a c : ℝ hf : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a ⊒ (fun x => c * (f x - f a - (x - a) * f')) =o[𝓝 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 : ℝ β†’ ℝ f' a c : ℝ hf : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a ⊒ (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 : ℝ β†’ ℝ f' a c : ℝ hf : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a x : ℝ ⊒ 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 : ℝ β†’ ℝ f' a c : ℝ hf : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a x : ℝ ⊒ 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 : ℝ β†’ ℝ f' a c : ℝ hf : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a ⊒ (fun x => c * (f x - f a - (x - a) * f')) =o[𝓝 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 : ℝ β†’ ℝ f' a : ℝ hf : HasDerivAt f f' a this : HasDerivAt (fun x => -1 * f x) (-1 * f') a ⊒ 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 : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a this : HasDerivAt (fun x => f x + -g x) (f' + -g') a ⊒ 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 : ℝ β†’ ℝ f' a : ℝ h : HasDerivAt f f' a ⊒ (fun x => f x - f a) =O[𝓝 a] fun x => x - a
f : ℝ β†’ ℝ f' a : ℝ h : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a ⊒ (fun x => f x - f a) =O[𝓝 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 : ℝ β†’ ℝ f' a : ℝ h : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a ⊒ (fun x => f x - f a) =O[𝓝 a] fun x => x - a
f : ℝ β†’ ℝ f' a : ℝ h : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a ⊒ (fun x => (x - a) * f') =O[𝓝 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 ↦ (x - a) * f') _ = fun x ↦ f' * (x - a) := ?eq1 _ =O[𝓝 a] fun x ↦ x - a := ?eq2
f : ℝ β†’ ℝ f' a : ℝ h : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a ⊒ (fun x => (x - a) * f') =O[𝓝 a] fun x => x - a
case eq1 f : ℝ β†’ ℝ f' a : ℝ h : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a ⊒ (fun x => (x - a) * f') = fun x => f' * (x - a) case eq2 f : ℝ β†’ ℝ f' a : ℝ h : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a ⊒ (fun x => f' * (x - a)) =O[𝓝 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 : ℝ β†’ ℝ f' a : ℝ h : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a ⊒ (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 : ℝ β†’ ℝ f' a : ℝ h : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a ⊒ (fun x => f' * (x - a)) =O[𝓝 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 : ℝ β†’ ℝ f' a : ℝ h : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a ⊒ (fun x => (x - a) * f') = fun x => f' * (x - a)
case h f : ℝ β†’ ℝ f' a : ℝ h : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a x : ℝ ⊒ (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 : ℝ β†’ ℝ f' a : ℝ h : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a x : ℝ ⊒ (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 : ℝ β†’ ℝ f' a : ℝ h : (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a ⊒ (fun x => f' * (x - a)) =O[𝓝 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 ↦ f x - f a + f a) (𝓝 a) (𝓝 (0 + f a)) := by apply Tendsto.add _ tendsto_const_nhds apply h.isBigO_sub.trans_tendsto rw [← sub_self a] apply tendsto_id.sub tendsto_const_nhds
f : ℝ β†’ ℝ f' a : ℝ h : HasDerivAt f f' a ⊒ Tendsto f (𝓝 a) (𝓝 (f a))
f : ℝ β†’ ℝ f' a : ℝ h : HasDerivAt f f' a this : Tendsto (fun x => f x - f a + f a) (𝓝 a) (𝓝 (0 + f a)) ⊒ Tendsto f (𝓝 a) (𝓝 (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 : ℝ β†’ ℝ f' a : ℝ h : HasDerivAt f f' a this : Tendsto (fun x => f x - f a + f a) (𝓝 a) (𝓝 (0 + f a)) ⊒ Tendsto f (𝓝 a) (𝓝 (f a))
f : ℝ β†’ ℝ f' a : ℝ h : HasDerivAt f f' a this : Tendsto (fun x => f x - f a + f a) (𝓝 a) (𝓝 (f a)) ⊒ Tendsto f (𝓝 a) (𝓝 (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 : ℝ β†’ ℝ f' a : ℝ h : HasDerivAt f f' a this : Tendsto (fun x => f x - f a + f a) (𝓝 a) (𝓝 (f a)) ⊒ Tendsto f (𝓝 a) (𝓝 (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 : ℝ β†’ ℝ f' a : ℝ h : HasDerivAt f f' a ⊒ Tendsto (fun x => f x - f a + f a) (𝓝 a) (𝓝 (0 + f a))
f : ℝ β†’ ℝ f' a : ℝ h : HasDerivAt f f' a ⊒ Tendsto (fun x => f x - f a) (𝓝 a) (𝓝 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 : ℝ β†’ ℝ f' a : ℝ h : HasDerivAt f f' a ⊒ Tendsto (fun x => f x - f a) (𝓝 a) (𝓝 0)
f : ℝ β†’ ℝ f' a : ℝ h : HasDerivAt f f' a ⊒ Tendsto (fun x => x - a) (𝓝 a) (𝓝 0)
https://github.com/yuma-mizuno/lean-math-workshop.git
4a69b0130b276b45212e2b12b90032b146b56d67
Solution/Advanced/Analysis/Lecture1.lean
Tutorial.HasDerivAt.continuousAt
[223, 1]
[231, 29]
rw [← sub_self a]
f : ℝ β†’ ℝ f' a : ℝ h : HasDerivAt f f' a ⊒ Tendsto (fun x => x - a) (𝓝 a) (𝓝 0)
f : ℝ β†’ ℝ f' a : ℝ h : HasDerivAt f f' a ⊒ Tendsto (fun x => x - a) (𝓝 a) (𝓝 (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 : ℝ β†’ ℝ f' a : ℝ h : HasDerivAt f f' a ⊒ Tendsto (fun x => x - a) (𝓝 a) (𝓝 (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 : ℝ β†’ ℝ f' a : ℝ h : HasDerivAt f f' a this : Tendsto (fun x => f x - f a + f a) (𝓝 a) (𝓝 (f a)) ⊒ βˆ€ (x : ℝ), 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 h₁ := calc (fun x ↦ g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x ↦ f x - f a := ?eq1 _ =O[𝓝 a] fun x ↦ x - a := ?eq2
f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) ⊒ HasDerivAt (g ∘ f) (g' * f') a
f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) h₁ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => x - a ⊒ HasDerivAt (g ∘ f) (g' * f') a case eq1 f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) ⊒ (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => f x - f a case eq2 f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) ⊒ (fun x => f x - f a) =O[𝓝 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 ↦ (f x - f a) * g' - (x - a) * (g' * f')) _ = fun x ↦ g' * (f x - f a - (x - a) * f') := ?eq3 _ =O[𝓝 a] fun x ↦ f x - f a - (x - a) * f' := ?eq4 _ =o[𝓝 a] fun x ↦ x - a := ?eq5
f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) h₁ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => x - a ⊒ HasDerivAt (g ∘ f) (g' * f') a case eq1 f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) ⊒ (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => f x - f a case eq2 f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) ⊒ (fun x => f x - f a) =O[𝓝 a] fun x => x - a
f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) h₁ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => x - a hβ‚‚ : (fun x => (f x - f a) * g' - (x - a) * (g' * f')) =o[𝓝 a] fun x => x - a ⊒ HasDerivAt (g ∘ f) (g' * f') a case eq3 f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) h₁ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => x - a ⊒ (fun x => (f x - f a) * g' - (x - a) * (g' * f')) = fun x => g' * (f x - f a - (x - a) * f') case eq4 f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) h₁ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => x - a ⊒ (fun x => g' * (f x - f a - (x - a) * f')) =O[𝓝 a] fun x => f x - f a - (x - a) * f' case eq5 f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) h₁ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => x - a ⊒ (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a case eq1 f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) ⊒ (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => f x - f a case eq2 f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) ⊒ (fun x => f x - f a) =O[𝓝 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 h₁.triangle hβ‚‚
f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) h₁ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => x - a hβ‚‚ : (fun x => (f x - f a) * g' - (x - a) * (g' * f')) =o[𝓝 a] fun x => x - a ⊒ HasDerivAt (g ∘ f) (g' * f') a case eq3 f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) h₁ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => x - a ⊒ (fun x => (f x - f a) * g' - (x - a) * (g' * f')) = fun x => g' * (f x - f a - (x - a) * f') case eq4 f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) h₁ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => x - a ⊒ (fun x => g' * (f x - f a - (x - a) * f')) =O[𝓝 a] fun x => f x - f a - (x - a) * f' case eq5 f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) h₁ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => x - a ⊒ (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a case eq1 f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) ⊒ (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => f x - f a case eq2 f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) ⊒ (fun x => f x - f a) =O[𝓝 a] fun x => x - a
case eq3 f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) h₁ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => x - a ⊒ (fun x => (f x - f a) * g' - (x - a) * (g' * f')) = fun x => g' * (f x - f a - (x - a) * f') case eq4 f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) h₁ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => x - a ⊒ (fun x => g' * (f x - f a - (x - a) * f')) =O[𝓝 a] fun x => f x - f a - (x - a) * f' case eq5 f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) h₁ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => x - a ⊒ (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a case eq1 f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) ⊒ (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => f x - f a case eq2 f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) ⊒ (fun x => f x - f a) =O[𝓝 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 : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) ⊒ (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 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 : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) ⊒ (fun x => f x - f a) =O[𝓝 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 : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) h₁ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => x - a ⊒ (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 : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) h₁ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => x - a ⊒ (fun x => g' * (f x - f a - (x - a) * f')) =O[𝓝 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 : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) h₁ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => x - a ⊒ (fun x => f x - f a - (x - a) * f') =o[𝓝 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 : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) ⊒ (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => f x - f a
f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) ⊒ Tendsto (fun x => f x) (𝓝 a) (𝓝 (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 : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) ⊒ Tendsto (fun x => f x) (𝓝 a) (𝓝 (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 : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) ⊒ (fun x => f x - f a) =O[𝓝 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 : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) h₁ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => x - a ⊒ (fun x => (f x - f a) * g' - (x - a) * (g' * f')) = fun x => g' * (f x - f a - (x - a) * f')
case h f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) h₁ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => x - a x✝ : ℝ ⊒ (f x✝ - f a) * g' - (x✝ - a) * (g' * f') = g' * (f x✝ - f a - (x✝ - 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 : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) h₁ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => x - a x✝ : ℝ ⊒ (f x✝ - f a) * g' - (x✝ - a) * (g' * f') = 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]
apply isBigO_const_mul_self
f : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) h₁ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => x - a ⊒ (fun x => g' * (f x - f a - (x - a) * f')) =O[𝓝 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 : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' (f a) h₁ : (fun x => g (f x) - g (f a) - (f x - f a) * g') =o[𝓝 a] fun x => x - a ⊒ (fun x => f x - f a - (x - a) * f') =o[𝓝 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]
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a ⊒ HasDerivAt (fun x => f x * g x) (f' * g a + f a * g') a
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a ⊒ (fun x => f x * g x - f a * g a - (x - a) * (f' * g a + f a * g')) =o[𝓝 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 ↦ 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)) := ?eq1 _ =o[𝓝 a] fun x ↦ x - a := ?eq2
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a ⊒ (fun x => f x * g x - f a * g a - (x - a) * (f' * g a + f a * g')) =o[𝓝 a] fun x => x - a
case eq1 f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a ⊒ (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 f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a ⊒ (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[𝓝 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
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a ⊒ (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
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a ⊒ (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 f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a x✝ : ℝ ⊒ f x✝ * g x✝ - f a * g a - (x✝ - a) * (f' * g a + f a * g') = 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))
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 f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a x✝ : ℝ ⊒ f x✝ * g x✝ - f a * g a - (x✝ - a) * (f' * g a + f a * g') = 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]
have hf' : (fun x ↦ g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x ↦ x - a := by apply IsLittleO.const_mul_left hf
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a ⊒ (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[𝓝 a] fun x => x - a
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a ⊒ (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[𝓝 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 ↦ f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x ↦ x - a := by apply IsLittleO.const_mul_left hg
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a ⊒ (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[𝓝 a] fun x => x - a
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a ⊒ (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[𝓝 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 ↦ (f x - f a) * (g x - g a)) _ =o[𝓝 a] fun x ↦ (x - a) * 1 := ?eq3 _ = fun x ↦ x - a := ?eq4
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a ⊒ (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[𝓝 a] fun x => x - a
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a hfg : (fun x => (f x - f a) * (g x - g a)) =o[𝓝 a] fun x => x - a ⊒ (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[𝓝 a] fun x => x - a case eq3 f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a ⊒ (fun x => (f x - f a) * (g x - g a)) =o[𝓝 a] fun x => (x - a) * 1 case eq4 f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a ⊒ (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'
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a hfg : (fun x => (f x - f a) * (g x - g a)) =o[𝓝 a] fun x => x - a ⊒ (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[𝓝 a] fun x => x - a case eq3 f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a ⊒ (fun x => (f x - f a) * (g x - g a)) =o[𝓝 a] fun x => (x - a) * 1 case eq4 f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a ⊒ (fun x => (x - a) * 1) = fun x => x - a
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a hfg : (fun x => (f x - f a) * (g x - g a)) =o[𝓝 a] fun x => x - a ⊒ (fun x => f a * (g x - g a - (x - a) * g') + (f x - f a) * (g x - g a)) =o[𝓝 a] fun x => x - a case eq3 f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a ⊒ (fun x => (f x - f a) * (g x - g a)) =o[𝓝 a] fun x => (x - a) * 1 case eq4 f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a ⊒ (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'
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a hfg : (fun x => (f x - f a) * (g x - g a)) =o[𝓝 a] fun x => x - a ⊒ (fun x => f a * (g x - g a - (x - a) * g') + (f x - f a) * (g x - g a)) =o[𝓝 a] fun x => x - a case eq3 f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a ⊒ (fun x => (f x - f a) * (g x - g a)) =o[𝓝 a] fun x => (x - a) * 1 case eq4 f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a ⊒ (fun x => (x - a) * 1) = fun x => x - a
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a hfg : (fun x => (f x - f a) * (g x - g a)) =o[𝓝 a] fun x => x - a ⊒ (fun x => (f x - f a) * (g x - g a)) =o[𝓝 a] fun x => x - a case eq3 f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a ⊒ (fun x => (f x - f a) * (g x - g a)) =o[𝓝 a] fun x => (x - a) * 1 case eq4 f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a ⊒ (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
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a hfg : (fun x => (f x - f a) * (g x - g a)) =o[𝓝 a] fun x => x - a ⊒ (fun x => (f x - f a) * (g x - g a)) =o[𝓝 a] fun x => x - a case eq3 f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a ⊒ (fun x => (f x - f a) * (g x - g a)) =o[𝓝 a] fun x => (x - a) * 1 case eq4 f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a ⊒ (fun x => (x - a) * 1) = fun x => x - a
case eq3 f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a ⊒ (fun x => (f x - f a) * (g x - g a)) =o[𝓝 a] fun x => (x - a) * 1 case eq4 f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a ⊒ (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
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a ⊒ (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
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a ⊒ (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 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
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a ⊒ (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 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 ↦ g x - g a) =o[𝓝 a] fun _ ↦ (1 : ℝ) := by rw [Asymptotics.isLittleO_one_iff, tendsto_sub_nhds_zero_iff] apply hg.continuousAt
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a ⊒ (fun x => (f x - f a) * (g x - g a)) =o[𝓝 a] fun x => (x - a) * 1
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a hg'' : (fun x => g x - g a) =o[𝓝 a] fun x => 1 ⊒ (fun x => (f x - f a) * (g x - g a)) =o[𝓝 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
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a hg'' : (fun x => g x - g a) =o[𝓝 a] fun x => 1 ⊒ (fun x => (f x - f a) * (g x - g a)) =o[𝓝 a] fun x => (x - a) * 1
case h₁ f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a hg'' : (fun x => g x - g a) =o[𝓝 a] fun x => 1 ⊒ (fun x => f x - f a) =O[𝓝 a] fun x => x - a case hβ‚‚ f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a hg'' : (fun x => g x - g a) =o[𝓝 a] fun x => 1 ⊒ (fun x => g x - g a) =o[𝓝 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]
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a ⊒ (fun x => g x - g a) =o[𝓝 a] fun x => 1
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a ⊒ Tendsto (fun x => g x) (𝓝 a) (𝓝 (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
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a ⊒ Tendsto (fun x => g x) (𝓝 a) (𝓝 (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 h₁ f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a hg'' : (fun x => g x - g a) =o[𝓝 a] fun x => 1 ⊒ (fun x => f x - f a) =O[𝓝 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‚‚ f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a hg'' : (fun x => g x - g a) =o[𝓝 a] fun x => 1 ⊒ (fun x => g x - g a) =o[𝓝 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
f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a ⊒ (fun x => (x - a) * 1) = fun x => x - a
case h f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a x✝ : ℝ ⊒ (x✝ - a) * 1 = x✝ - 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 f✝ : ℝ β†’ ℝ f' a : ℝ g : ℝ β†’ ℝ g' : ℝ f : ℝ β†’ ℝ hf : HasDerivAt f f' a hg : HasDerivAt g g' a hf' : (fun x => g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x => x - a hg' : (fun x => f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x => x - a x✝ : ℝ ⊒ (x✝ - a) * 1 = x✝ - 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 : ℝ β†’ ℝ f' a✝ : ℝ g : ℝ β†’ ℝ g' : ℝ n : β„• a : ℝ ⊒ HasDerivAt (fun x => x ^ (n + 1)) ((↑n + 1) * a ^ n) a
case zero f : ℝ β†’ ℝ f' a✝ : ℝ g : ℝ β†’ ℝ g' a : ℝ ⊒ HasDerivAt (fun x => x ^ (Nat.zero + 1)) ((↑Nat.zero + 1) * a ^ Nat.zero) a case succ f : ℝ β†’ ℝ f' a✝ : ℝ g : ℝ β†’ ℝ g' a : ℝ n✝ : β„• n_ih✝ : HasDerivAt (fun x => x ^ (n✝ + 1)) ((↑n✝ + 1) * a ^ n✝) a ⊒ HasDerivAt (fun x => x ^ (Nat.succ n✝ + 1)) ((↑(Nat.succ n✝) + 1) * a ^ Nat.succ n✝) 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 : ℝ β†’ ℝ f' a✝ : ℝ g : ℝ β†’ ℝ g' a : ℝ ⊒ HasDerivAt (fun x => x ^ (Nat.zero + 1)) ((↑Nat.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 : ℝ β†’ ℝ f' a✝ : ℝ g : ℝ β†’ ℝ g' a : ℝ n : β„• ih : HasDerivAt (fun x => x ^ (n + 1)) ((↑n + 1) * a ^ n) a ⊒ HasDerivAt (fun x => x ^ (Nat.succ n + 1)) ((↑(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 : ℝ β†’ ℝ f' a✝ : ℝ g : ℝ β†’ ℝ g' a : ℝ ⊒ HasDerivAt (fun x => x ^ (Nat.zero + 1)) ((↑Nat.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 : ℝ β†’ ℝ f' a✝ : ℝ g : ℝ β†’ ℝ g' a : ℝ n : β„• ih : HasDerivAt (fun x => x ^ (n + 1)) ((↑n + 1) * a ^ n) a ⊒ HasDerivAt (fun x => x ^ (Nat.succ n + 1)) ((↑(Nat.succ n) + 1) * a ^ Nat.succ n) a
f : ℝ β†’ ℝ f' a✝ : ℝ g : ℝ β†’ ℝ g' a : ℝ n : β„• ih : HasDerivAt (fun x => x ^ (n + 1)) ((↑n + 1) * a ^ n) a ⊒ HasDerivAt (fun x => x ^ (n + 1 + 1)) ((↑(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 : ℝ β†’ ℝ f' a✝ : ℝ g : ℝ β†’ ℝ g' a : ℝ n : β„• ih : HasDerivAt (fun x => x ^ (n + 1)) ((↑n + 1) * a ^ n) a ⊒ HasDerivAt (fun x => x ^ (n + 1 + 1)) ((↑(n + 1) + 1) * a ^ (n + 1)) a
f : ℝ β†’ ℝ f' a✝ : ℝ g : ℝ β†’ ℝ g' a : ℝ n : β„• ih : HasDerivAt (fun x => x ^ (n + 1)) ((↑n + 1) * a ^ n) a ⊒ HasDerivAt (fun x => x ^ (n + 1) * x) ((↑n + 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 : ℝ β†’ ℝ f' a✝ : ℝ g : ℝ β†’ ℝ g' a : ℝ n : β„• ih : HasDerivAt (fun x => x ^ (n + 1)) ((↑n + 1) * a ^ n) a ⊒ HasDerivAt (fun x => x ^ (n + 1) * x) ((↑n + 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 : ℝ β†’ ℝ f' a✝ : ℝ g : ℝ β†’ ℝ g' a : ℝ n : β„• ih : HasDerivAt (fun x => x ^ (n + 1)) ((↑n + 1) * a ^ n) a this : HasDerivAt (fun x => x ^ (n + 1) * x) ((↑n + 1) * a ^ n * a + a ^ (n + 1) * 1) a ⊒ HasDerivAt (fun x => x ^ (n + 1 + 1)) ((↑(n + 1) + 1) * a ^ (n + 1)) a
f : ℝ β†’ ℝ f' a✝ : ℝ g : ℝ β†’ ℝ g' a : ℝ n : β„• ih : HasDerivAt (fun x => x ^ (n + 1)) ((↑n + 1) * a ^ n) a this : HasDerivAt (fun x => x ^ (n + 1) * x) ((↑n + 1) * a ^ n * a + a ^ (n + 1) * 1) a ⊒ βˆ€ (x : ℝ), (fun x => x ^ (n + 1) * x) x - (fun x => x ^ (n + 1) * x) a - (x - a) * ((↑n + 1) * a ^ n * a + a ^ (n + 1) * 1) = (fun x => x ^ (n + 1 + 1)) x - (fun x => x ^ (n + 1 + 1)) a - (x - a) * ((↑(n + 1) + 1) * a ^ (n + 1))