url
stringclasses 147
values | commit
stringclasses 147
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
94
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
| input
stringlengths 73
2.09M
|
|---|---|---|---|---|---|---|---|---|---|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.homToPerm_injective
|
[264, 1]
|
[275, 12]
|
calc
a = a * 1 := by simp
_ = (homToPerm G a) 1 := by
sorry
_ = 1 := by
sorry
|
G : Type
inst✝ : Group G
a : G
h : (homToPerm G) a = 1
⊢ a = 1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a : G
h : (homToPerm G) a = 1
⊢ a = 1
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.homToPerm_injective
|
[264, 1]
|
[275, 12]
|
simp
|
G : Type
inst✝ : Group G
a : G
h : (homToPerm G) a = 1
⊢ a = a * 1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a : G
h : (homToPerm G) a = 1
⊢ a = a * 1
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.homToPerm_injective
|
[264, 1]
|
[275, 12]
|
sorry
|
G : Type
inst✝ : Group G
a : G
h : (homToPerm G) a = 1
⊢ a * 1 = ((homToPerm G) a) 1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a : G
h : (homToPerm G) a = 1
⊢ a * 1 = ((homToPerm G) a) 1
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture2.lean
|
Tutorial.homToPerm_injective
|
[264, 1]
|
[275, 12]
|
sorry
|
G : Type
inst✝ : Group G
a : G
h : (homToPerm G) a = 1
⊢ ((homToPerm G) a) 1 = 1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
G : Type
inst✝ : Group G
a : G
h : (homToPerm G) a = 1
⊢ ((homToPerm G) a) 1 = 1
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_isLittleO
|
[49, 1]
|
[51, 6]
|
rfl
|
f : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
⊢ HasDerivAt f f' a ↔ (fun x => f x - f a - (x - a) * f') =o[𝓝 a] fun x => x - a
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/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 → 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 → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
⊢ HasDerivAt f f' a ↔ (fun h => f (a + h) - f a - h * f') =o[𝓝 0] fun h => h
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/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[≠] a) (𝓝 0) := ?iff2
|
f : ℝ → 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 iff1
f : ℝ → 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 → 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[≠] a) (𝓝 0)
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
⊢ HasDerivAt f f' a ↔ Tendsto (fun x => (f x - f a - (x - a) * f') / (x - a)) (𝓝[≠] a) (𝓝 0)
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/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 → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
⊢ HasDerivAt f f' a ↔ Tendsto (fun x => (f x - f a - (x - a) * f') / (x - a)) (𝓝 a) (𝓝 0)
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/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 → 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[≠] a) (𝓝 0)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → 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[≠] a) (𝓝 0)
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/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 → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
⊢ HasDerivAt f f' a ↔ Tendsto (fun x => (f x - f a - (x - a) * f') / (x - a)) (𝓝 a) (𝓝 0)
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_tendsto
|
[60, 1]
|
[66, 85]
|
intro _ h
|
f : ℝ → ℝ
f' a : ℝ
⊢ ∀ (x : ℝ), x - a = 0 → f x - f a - (x - a) * f' = 0
|
f : ℝ → ℝ
f' a x✝ : ℝ
h : x✝ - a = 0
⊢ f x✝ - f a - (x✝ - a) * f' = 0
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
⊢ ∀ (x : ℝ), x - a = 0 → f x - f a - (x - a) * f' = 0
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_tendsto
|
[60, 1]
|
[66, 85]
|
simp [sub_eq_zero.1 h]
|
f : ℝ → ℝ
f' a x✝ : ℝ
h : x✝ - a = 0
⊢ f x✝ - f a - (x✝ - a) * f' = 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a x✝ : ℝ
h : x✝ - a = 0
⊢ f x✝ - f a - (x✝ - a) * f' = 0
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/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 → 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[≠] a) (𝓝 0)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → 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[≠] a) (𝓝 0)
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_tendsto
|
[60, 1]
|
[66, 85]
|
simp
|
f : ℝ → ℝ
f' a : ℝ
⊢ ∀ a_1 ∉ {a}ᶜ, (f a_1 - f a - (a_1 - a) * f') / (a_1 - a) = 0
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
⊢ ∀ a_1 ∉ {a}ᶜ, (f a_1 - f a - (a_1 - a) * f') / (a_1 - a) = 0
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/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[≠] a) (𝓝 0) := ?iff1
_ ↔ Tendsto (fun x ↦ (f x - f a) / (x - a) - f') (𝓝[≠] a) (𝓝 0) := ?iff2
_ ↔ Tendsto (fun x ↦ (f x - f a) / (x - a)) (𝓝[≠] a) (𝓝 f') := ?iff3
|
f : ℝ → ℝ
f' a : ℝ
⊢ HasDerivAt f f' a ↔ Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[≠] a) (𝓝 f')
|
case iff1
f : ℝ → ℝ
f' a : ℝ
⊢ HasDerivAt f f' a ↔ Tendsto (fun x => (f x - f a) / (x - a) - (x - a) / (x - a) * f') (𝓝[≠] a) (𝓝 0)
case iff2
f : ℝ → ℝ
f' a : ℝ
⊢ Tendsto (fun x => (f x - f a) / (x - a) - (x - a) / (x - a) * f') (𝓝[≠] a) (𝓝 0) ↔
Tendsto (fun x => (f x - f a) / (x - a) - f') (𝓝[≠] a) (𝓝 0)
case iff3
f : ℝ → ℝ
f' a : ℝ
⊢ Tendsto (fun x => (f x - f a) / (x - a) - f') (𝓝[≠] a) (𝓝 0) ↔
Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[≠] a) (𝓝 f')
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
⊢ HasDerivAt f f' a ↔ Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[≠] a) (𝓝 f')
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/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 → ℝ
f' a : ℝ
⊢ HasDerivAt f f' a ↔ Tendsto (fun x => (f x - f a) / (x - a) - (x - a) / (x - a) * f') (𝓝[≠] a) (𝓝 0)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
⊢ HasDerivAt f f' a ↔ Tendsto (fun x => (f x - f a) / (x - a) - (x - a) / (x - a) * f') (𝓝[≠] a) (𝓝 0)
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/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 → ℝ
f' a : ℝ
⊢ Tendsto (fun x => (f x - f a) / (x - a) - (x - a) / (x - a) * f') (𝓝[≠] a) (𝓝 0) ↔
Tendsto (fun x => (f x - f a) / (x - a) - f') (𝓝[≠] a) (𝓝 0)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
⊢ Tendsto (fun x => (f x - f a) / (x - a) - (x - a) / (x - a) * f') (𝓝[≠] a) (𝓝 0) ↔
Tendsto (fun x => (f x - f a) / (x - a) - f') (𝓝[≠] a) (𝓝 0)
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/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 → ℝ
f' a : ℝ
⊢ Tendsto (fun x => (f x - f a) / (x - a) - f') (𝓝[≠] a) (𝓝 0) ↔
Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[≠] a) (𝓝 f')
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
⊢ Tendsto (fun x => (f x - f a) / (x - a) - f') (𝓝[≠] a) (𝓝 0) ↔
Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[≠] a) (𝓝 f')
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/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 → ℝ
f' a : ℝ
⊢ HasDerivAt f f' a ↔ Tendsto (fun x => (f x - f a) / (x - a) - (x - a) / (x - a) * f') (𝓝[≠] a) (𝓝 0)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
⊢ HasDerivAt f f' a ↔ Tendsto (fun x => (f x - f a) / (x - a) - (x - a) / (x - a) * f') (𝓝[≠] a) (𝓝 0)
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/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 → ℝ
f' a : ℝ
⊢ Tendsto (fun x => (f x - f a) / (x - a) - (x - a) / (x - a) * f') (𝓝[≠] a) (𝓝 0) ↔
Tendsto (fun x => (f x - f a) / (x - a) - f') (𝓝[≠] a) (𝓝 0)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
⊢ Tendsto (fun x => (f x - f a) / (x - a) - (x - a) / (x - a) * f') (𝓝[≠] a) (𝓝 0) ↔
Tendsto (fun x => (f x - f a) / (x - a) - f') (𝓝[≠] a) (𝓝 0)
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_tendsto_slope
|
[69, 1]
|
[77, 70]
|
field_simp [sub_ne_zero.2 h]
|
f : ℝ → ℝ
f' a x✝ : ℝ
h : x✝ ∈ {a}ᶜ
⊢ (f x✝ - f a) / (x✝ - a) - (x✝ - a) / (x✝ - a) * f' = (f x✝ - f a) / (x✝ - a) - f'
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a x✝ : ℝ
h : x✝ ∈ {a}ᶜ
⊢ (f x✝ - f a) / (x✝ - a) - (x✝ - a) / (x✝ - a) * f' = (f x✝ - f a) / (x✝ - a) - f'
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_tendsto_slope
|
[69, 1]
|
[77, 70]
|
rw [← nhds_translation_sub f', tendsto_comap_iff]
|
f : ℝ → ℝ
f' a : ℝ
⊢ Tendsto (fun x => (f x - f a) / (x - a) - f') (𝓝[≠] a) (𝓝 0) ↔
Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[≠] a) (𝓝 f')
|
f : ℝ → ℝ
f' a : ℝ
⊢ Tendsto (fun x => (f x - f a) / (x - a) - f') (𝓝[≠] a) (𝓝 0) ↔
Tendsto ((fun x => x - f') ∘ fun x => (f x - f a) / (x - a)) (𝓝[≠] a) (𝓝 0)
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
⊢ Tendsto (fun x => (f x - f a) / (x - a) - f') (𝓝[≠] a) (𝓝 0) ↔
Tendsto (fun x => (f x - f a) / (x - a)) (𝓝[≠] a) (𝓝 f')
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_iff_tendsto_slope
|
[69, 1]
|
[77, 70]
|
rfl
|
f : ℝ → ℝ
f' a : ℝ
⊢ Tendsto (fun x => (f x - f a) / (x - a) - f') (𝓝[≠] a) (𝓝 0) ↔
Tendsto ((fun x => x - f') ∘ fun x => (f x - f a) / (x - a)) (𝓝[≠] a) (𝓝 0)
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
⊢ Tendsto (fun x => (f x - f a) / (x - a) - f') (𝓝[≠] a) (𝓝 0) ↔
Tendsto ((fun x => x - f') ∘ fun x => (f x - f a) / (x - a)) (𝓝[≠] a) (𝓝 0)
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_const
|
[135, 1]
|
[138, 8]
|
rw [hasDerivAt_iff_isLittleO]
|
f : ℝ → ℝ
f' a c : ℝ
⊢ HasDerivAt (fun x => c) 0 a
|
f : ℝ → ℝ
f' a c : ℝ
⊢ (fun x => c - c - (x - a) * 0) =o[𝓝 a] fun x => x - a
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a c : ℝ
⊢ HasDerivAt (fun x => c) 0 a
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_const
|
[135, 1]
|
[138, 8]
|
sorry
|
f : ℝ → ℝ
f' a c : ℝ
⊢ (fun x => c - c - (x - a) * 0) =o[𝓝 a] fun x => x - a
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a c : ℝ
⊢ (fun x => c - c - (x - a) * 0) =o[𝓝 a] fun x => x - a
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_id
|
[140, 1]
|
[142, 8]
|
rw [hasDerivAt_iff_isLittleO]
|
f : ℝ → ℝ
f' a✝ a : ℝ
⊢ HasDerivAt id 1 a
|
f : ℝ → ℝ
f' a✝ a : ℝ
⊢ (fun x => id x - id a - (x - a) * 1) =o[𝓝 a] fun x => x - a
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a✝ a : ℝ
⊢ HasDerivAt id 1 a
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_id
|
[140, 1]
|
[142, 8]
|
sorry
|
f : ℝ → ℝ
f' a✝ a : ℝ
⊢ (fun x => id x - id a - (x - a) * 1) =o[𝓝 a] fun x => x - a
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a✝ a : ℝ
⊢ (fun x => id x - id a - (x - a) * 1) =o[𝓝 a] fun x => x - a
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.add
|
[144, 1]
|
[155, 10]
|
rw [hasDerivAt_iff_isLittleO] at *
|
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
hf : HasDerivAt f f' a
hg : HasDerivAt g g' a
⊢ HasDerivAt (fun x => f x + g x) (f' + g') a
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.add
|
[144, 1]
|
[155, 10]
|
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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.add
|
[144, 1]
|
[155, 10]
|
case eq1 =>
sorry
|
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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')
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.add
|
[144, 1]
|
[155, 10]
|
case eq2 =>
sorry
|
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.add
|
[144, 1]
|
[155, 10]
|
sorry
|
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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')
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.add
|
[144, 1]
|
[155, 10]
|
sorry
|
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.const_mul
|
[157, 1]
|
[161, 8]
|
rw [hasDerivAt_iff_isLittleO] at *
|
f : ℝ → ℝ
f' a c : ℝ
hf : HasDerivAt f f' a
⊢ HasDerivAt (fun x => c * f x) (c * f') a
|
f : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a c : ℝ
hf : HasDerivAt f f' a
⊢ HasDerivAt (fun x => c * f x) (c * f') a
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.const_mul
|
[157, 1]
|
[161, 8]
|
sorry
|
f : ℝ → 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
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.neg
|
[164, 1]
|
[167, 20]
|
simpa using this
|
f : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.sub
|
[170, 1]
|
[173, 18]
|
simpa [sub_eq_add_neg] using this
|
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.isBigO_sub
|
[180, 1]
|
[192, 10]
|
rw [hasDerivAt_iff_isLittleO] at h
|
f : ℝ → ℝ
f' a : ℝ
h : HasDerivAt f f' a
⊢ (fun x => f x - f a) =O[𝓝 a] fun x => x - a
|
f : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
h : HasDerivAt f f' a
⊢ (fun x => f x - f a) =O[𝓝 a] fun x => x - a
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.isBigO_sub
|
[180, 1]
|
[192, 10]
|
rw [h.isBigO.congr_of_sub]
|
f : ℝ → 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 → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.isBigO_sub
|
[180, 1]
|
[192, 10]
|
calc (fun x ↦ (x - a) * f')
_ = fun x ↦ f' * (x - a) := ?eq1
_ =O[𝓝 a] fun x ↦ x - a := ?eq2
|
f : ℝ → 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 → 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 → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.isBigO_sub
|
[180, 1]
|
[192, 10]
|
case eq1 =>
sorry
|
f : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → 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)
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.isBigO_sub
|
[180, 1]
|
[192, 10]
|
case eq2 =>
sorry
|
f : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.isBigO_sub
|
[180, 1]
|
[192, 10]
|
sorry
|
f : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → 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)
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.isBigO_sub
|
[180, 1]
|
[192, 10]
|
sorry
|
f : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.continuousAt
|
[195, 1]
|
[203, 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 → ℝ
f' a : ℝ
h : HasDerivAt f f' a
⊢ Tendsto f (𝓝 a) (𝓝 (f a))
|
f : ℝ → 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))
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
h : HasDerivAt f f' a
⊢ Tendsto f (𝓝 a) (𝓝 (f a))
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.continuousAt
|
[195, 1]
|
[203, 29]
|
rw [zero_add] at this
|
f : ℝ → 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 → 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))
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → 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))
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.continuousAt
|
[195, 1]
|
[203, 29]
|
exact this.congr (by simp)
|
f : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → 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))
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.continuousAt
|
[195, 1]
|
[203, 29]
|
apply Tendsto.add _ tendsto_const_nhds
|
f : ℝ → 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 → ℝ
f' a : ℝ
h : HasDerivAt f f' a
⊢ Tendsto (fun x => f x - f a) (𝓝 a) (𝓝 0)
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
h : HasDerivAt f f' a
⊢ Tendsto (fun x => f x - f a + f a) (𝓝 a) (𝓝 (0 + f a))
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.continuousAt
|
[195, 1]
|
[203, 29]
|
apply h.isBigO_sub.trans_tendsto
|
f : ℝ → ℝ
f' a : ℝ
h : HasDerivAt f f' a
⊢ Tendsto (fun x => f x - f a) (𝓝 a) (𝓝 0)
|
f : ℝ → ℝ
f' a : ℝ
h : HasDerivAt f f' a
⊢ Tendsto (fun x => x - a) (𝓝 a) (𝓝 0)
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
h : HasDerivAt f f' a
⊢ Tendsto (fun x => f x - f a) (𝓝 a) (𝓝 0)
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.continuousAt
|
[195, 1]
|
[203, 29]
|
rw [← sub_self a]
|
f : ℝ → ℝ
f' a : ℝ
h : HasDerivAt f f' a
⊢ Tendsto (fun x => x - a) (𝓝 a) (𝓝 0)
|
f : ℝ → ℝ
f' a : ℝ
h : HasDerivAt f f' a
⊢ Tendsto (fun x => x - a) (𝓝 a) (𝓝 (a - a))
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
h : HasDerivAt f f' a
⊢ Tendsto (fun x => x - a) (𝓝 a) (𝓝 0)
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.continuousAt
|
[195, 1]
|
[203, 29]
|
apply tendsto_id.sub tendsto_const_nhds
|
f : ℝ → ℝ
f' a : ℝ
h : HasDerivAt f f' a
⊢ Tendsto (fun x => x - a) (𝓝 a) (𝓝 (a - a))
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
h : HasDerivAt f f' a
⊢ Tendsto (fun x => x - a) (𝓝 a) (𝓝 (a - a))
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.continuousAt
|
[195, 1]
|
[203, 29]
|
simp
|
f : ℝ → ℝ
f' a : ℝ
h : HasDerivAt f f' a
this : Tendsto (fun x => f x - f a + f a) (𝓝 a) (𝓝 (f a))
⊢ ∀ (x : ℝ), f x - f a + f a = f x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
h : HasDerivAt f f' a
this : Tendsto (fun x => f x - f a + f a) (𝓝 a) (𝓝 (f a))
⊢ ∀ (x : ℝ), f x - f a + f a = f x
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[212, 1]
|
[234, 10]
|
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 → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
⊢ HasDerivAt (g ∘ f) (g' * f') a
|
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
hf : HasDerivAt f f' a
hg : HasDerivAt g g' (f a)
⊢ HasDerivAt (g ∘ f) (g' * f') a
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[212, 1]
|
[234, 10]
|
have h₂ :=
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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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
h₂ : (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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[212, 1]
|
[234, 10]
|
apply h₁.triangle h₂
|
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
h₂ : (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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
h₂ : (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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[212, 1]
|
[234, 10]
|
case eq1 =>
sorry
|
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[212, 1]
|
[234, 10]
|
case eq2 =>
sorry
|
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[212, 1]
|
[234, 10]
|
case eq3 =>
sorry
|
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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')
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[212, 1]
|
[234, 10]
|
case eq4 =>
sorry
|
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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'
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[212, 1]
|
[234, 10]
|
case eq5 =>
sorry
|
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[212, 1]
|
[234, 10]
|
sorry
|
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[212, 1]
|
[234, 10]
|
sorry
|
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[212, 1]
|
[234, 10]
|
sorry
|
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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')
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[212, 1]
|
[234, 10]
|
sorry
|
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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'
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.comp
|
[212, 1]
|
[234, 10]
|
sorry
|
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a : ℝ
g : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[240, 1]
|
[266, 12]
|
rw [hasDerivAt_iff_isLittleO]
|
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[240, 1]
|
[266, 12]
|
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 → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[240, 1]
|
[266, 12]
|
case eq1 =>
sorry
|
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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))
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[240, 1]
|
[266, 12]
|
case eq2 =>
have hf' : (fun x ↦ g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x ↦ x - a := by
sorry
have hg' : (fun x ↦ f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x ↦ x - a := by
sorry
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
sorry
case eq3 =>
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]
sorry
sorry
case eq4 =>
sorry
|
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[240, 1]
|
[266, 12]
|
sorry
|
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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))
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[240, 1]
|
[266, 12]
|
have hf' : (fun x ↦ g a * (f x - f a - (x - a) * f')) =o[𝓝 a] fun x ↦ x - a := by
sorry
|
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[240, 1]
|
[266, 12]
|
have hg' : (fun x ↦ f a * (g x - g a - (x - a) * g')) =o[𝓝 a] fun x ↦ x - a := by
sorry
|
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[240, 1]
|
[266, 12]
|
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 → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[240, 1]
|
[266, 12]
|
sorry
|
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[240, 1]
|
[266, 12]
|
case eq3 =>
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]
sorry
sorry
|
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[240, 1]
|
[266, 12]
|
case eq4 =>
sorry
|
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[240, 1]
|
[266, 12]
|
sorry
|
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[240, 1]
|
[266, 12]
|
sorry
|
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[240, 1]
|
[266, 12]
|
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]
sorry
|
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[240, 1]
|
[266, 12]
|
sorry
|
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[240, 1]
|
[266, 12]
|
rw [Asymptotics.isLittleO_one_iff, tendsto_sub_nhds_zero_iff]
|
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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 → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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))
|
Please generate a tactic in lean4 to solve the state.
STATE:
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[240, 1]
|
[266, 12]
|
sorry
|
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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))
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.HasDerivAt.mul
|
[240, 1]
|
[266, 12]
|
sorry
|
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
|
Please generate a tactic in lean4 to solve the state.
STATE:
f✝ : ℝ → ℝ
f' a : ℝ
g : ℝ → ℝ
g' : ℝ
f : ℝ → 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
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Analysis/Lecture1.lean
|
Tutorial.hasDerivAt_pow
|
[272, 1]
|
[275, 8]
|
sorry
|
f : ℝ → ℝ
f' a✝ : ℝ
g : ℝ → ℝ
g' : ℝ
n : ℕ
a : ℝ
⊢ HasDerivAt (fun x => x ^ (n + 1)) ((↑n + 1) * a ^ n) a
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
f : ℝ → ℝ
f' a✝ : ℝ
g : ℝ → ℝ
g' : ℝ
n : ℕ
a : ℝ
⊢ HasDerivAt (fun x => x ^ (n + 1)) ((↑n + 1) * a ^ n) a
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_mul_cancel_left
|
[119, 1]
|
[126, 12]
|
calc
a⁻¹ * (a * b) = (a⁻¹ * a) * b := by
sorry
_ = 1 * b := by
sorry
_ = b := by
sorry
|
A : Type ?u.1517
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a⁻¹ * (a * b) = b
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
A : Type ?u.1517
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a⁻¹ * (a * b) = b
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_mul_cancel_left
|
[119, 1]
|
[126, 12]
|
sorry
|
A : Type ?u.1517
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a⁻¹ * (a * b) = a⁻¹ * a * b
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
A : Type ?u.1517
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a⁻¹ * (a * b) = a⁻¹ * a * b
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_mul_cancel_left
|
[119, 1]
|
[126, 12]
|
sorry
|
A : Type ?u.1517
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a⁻¹ * a * b = 1 * b
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
A : Type ?u.1517
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a⁻¹ * a * b = 1 * b
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_mul_cancel_left
|
[119, 1]
|
[126, 12]
|
sorry
|
A : Type ?u.1517
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ 1 * b = b
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
A : Type ?u.1517
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ 1 * b = b
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_left_cancel
|
[133, 1]
|
[136, 8]
|
sorry
|
A : Type ?u.2016
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
⊢ a * x = a * y → x = y
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
A : Type ?u.2016
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
⊢ a * x = a * y → x = y
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_one
|
[140, 1]
|
[148, 8]
|
sorry
|
A : Type ?u.2142
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a : G
⊢ a * 1 = a
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
A : Type ?u.2142
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a : G
⊢ a * 1 = a
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_self
|
[152, 1]
|
[153, 8]
|
sorry
|
A : Type ?u.2284
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a : G
⊢ a * a⁻¹ = 1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
A : Type ?u.2284
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a : G
⊢ a * a⁻¹ = 1
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_cancel_left
|
[157, 1]
|
[159, 8]
|
rw [← mul_assoc]
|
A : Type ?u.2445
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a * (a⁻¹ * b) = b
|
A : Type ?u.2445
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a * a⁻¹ * b = b
|
Please generate a tactic in lean4 to solve the state.
STATE:
A : Type ?u.2445
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a * (a⁻¹ * b) = b
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_cancel_left
|
[157, 1]
|
[159, 8]
|
sorry
|
A : Type ?u.2445
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a * a⁻¹ * b = b
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
A : Type ?u.2445
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a * a⁻¹ * b = b
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_inv_cancel_right
|
[162, 1]
|
[163, 8]
|
sorry
|
A : Type ?u.2639
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a * b * b⁻¹ = a
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
A : Type ?u.2639
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a * b * b⁻¹ = a
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_mul_cancel_right
|
[166, 1]
|
[167, 8]
|
sorry
|
A : Type ?u.2795
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a * b⁻¹ * b = a
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
A : Type ?u.2795
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a b : G
⊢ a * b⁻¹ * b = a
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture1.lean
|
Tutorial.mul_right_cancel
|
[170, 1]
|
[171, 8]
|
sorry
|
A : Type ?u.2951
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
⊢ x * a = y * a → x = y
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
A : Type ?u.2951
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x y : G
⊢ x * a = y * a → x = y
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_eq_of_mul_eq_one_left
|
[174, 1]
|
[175, 8]
|
sorry
|
A : Type ?u.3077
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x : G
⊢ x * a = 1 → a⁻¹ = x
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
A : Type ?u.3077
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a x : G
⊢ x * a = 1 → a⁻¹ = x
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_one
|
[182, 1]
|
[184, 8]
|
apply inv_eq_of_mul_eq_one_left
|
A : Type ?u.3411
G : Type
inst✝¹ : Ring A
inst✝ : Group G
⊢ 1⁻¹ = 1
|
case a
A : Type ?u.3411
G : Type
inst✝¹ : Ring A
inst✝ : Group G
⊢ 1 * 1 = 1
|
Please generate a tactic in lean4 to solve the state.
STATE:
A : Type ?u.3411
G : Type
inst✝¹ : Ring A
inst✝ : Group G
⊢ 1⁻¹ = 1
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_one
|
[182, 1]
|
[184, 8]
|
sorry
|
case a
A : Type ?u.3411
G : Type
inst✝¹ : Ring A
inst✝ : Group G
⊢ 1 * 1 = 1
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
case a
A : Type ?u.3411
G : Type
inst✝¹ : Ring A
inst✝ : Group G
⊢ 1 * 1 = 1
TACTIC:
|
https://github.com/yuma-mizuno/lean-math-workshop.git
|
4a69b0130b276b45212e2b12b90032b146b56d67
|
Tutorial/Advanced/Algebra/Lecture1.lean
|
Tutorial.inv_inv
|
[187, 1]
|
[188, 8]
|
sorry
|
A : Type ?u.3529
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a : G
⊢ a⁻¹⁻¹ = a
|
no goals
|
Please generate a tactic in lean4 to solve the state.
STATE:
A : Type ?u.3529
G : Type
inst✝¹ : Ring A
inst✝ : Group G
a : G
⊢ a⁻¹⁻¹ = a
TACTIC:
|
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