url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | have e1 := Real.add_one_le_exp (x / y - 1) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
e : y⁻¹ * (x - y) = x / y - 1
⊢ x / y ≤ (x / y - 1).exp | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
e : y⁻¹ * (x - y) = x / y - 1
e1 : x / y - 1 + 1 ≤ (x / y - 1).exp
⊢ x / y ≤ (x / y - 1).exp | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
e : y⁻¹ * (x - y) = x / y - 1
⊢ x / y ≤ (x / y - 1).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | simp at e1 | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
e : y⁻¹ * (x - y) = x / y - 1
e1 : x / y - 1 + 1 ≤ (x / y - 1).exp
⊢ x / y ≤ (x / y - 1).exp | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
e : y⁻¹ * (x - y) = x / y - 1
e1 : x / y ≤ (x / y - 1).exp
⊢ x / y ≤ (x / y - 1).exp | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
e : y⁻¹ * (x - y) = x / y - 1
e1 : x / y - 1 + 1 ≤ (x / y - 1).exp
⊢ x / y ≤ (x / y - 1).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | exact e1 | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
e : y⁻¹ * (x - y) = x / y - 1
e1 : x / y ≤ (x / y - 1).exp
⊢ x / y ≤ (x / y - 1).exp | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
e : y⁻¹ * (x - y) = x / y - 1
e1 : x / y ≤ (x / y - 1).exp
⊢ x / y ≤ (x / y - 1).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | rw [Real.exp_neg] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
⊢ y⁻¹ ≤ (-b).exp | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
⊢ y⁻¹ ≤ b.exp⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
⊢ y⁻¹ ≤ (-b).exp
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | bound | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
⊢ y⁻¹ ≤ b.exp⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
⊢ y⁻¹ ≤ b.exp⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | field_simp [yp.ne'] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ y⁻¹ * (x - y) = x / y - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b x y : ℝ
yb : b.exp ≤ y
xy : y ≤ x
yp : y > 0
xp : x > 0
yi : y⁻¹ ≤ (-b).exp
⊢ y⁻¹ * (x - y) = x / y - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzOnWith.log | [67, 1] | [91, 26] | exact half x y ys xy | case pos
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
xy : x ≥ y
⊢ |x.log - y.log| ≤ (-b).exp * |x - y| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
half : ∀ (x y : ℝ), b.exp ≤ y → y ≤ x → |x.log - y.log| ≤ (-b).exp * |x - y|
x y : ℝ
xs : b.exp ≤ x
ys : b.exp ≤ y
xy : x ≥ y
⊢ |x.log - y.log| ≤ (-b).exp * |x - y|
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzWith.maxLog | [94, 1] | [100, 70] | rw [← lipschitzOn_univ] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ LipschitzWith (-b).exp.toNNReal (_root_.maxLog b) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ LipschitzOnWith (-b).exp.toNNReal (_root_.maxLog b) univ | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ LipschitzWith (-b).exp.toNNReal (_root_.maxLog b)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzWith.maxLog | [94, 1] | [100, 70] | have h := (LipschitzOnWith.log b).comp ((LipschitzWith.id.const_max b.exp).lipschitzOnWith univ)
(by simp only [id_eq, Set.mapsTo_univ_iff, Set.mem_Ici, le_max_iff, le_refl, true_or,
forall_const]) | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ LipschitzOnWith (-b).exp.toNNReal (_root_.maxLog b) univ | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
h : LipschitzOnWith ((-b).exp.toNNReal * 1) (Real.log ∘ fun x => max b.exp (id x)) univ
⊢ LipschitzOnWith (-b).exp.toNNReal (_root_.maxLog b) univ | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ LipschitzOnWith (-b).exp.toNNReal (_root_.maxLog b) univ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzWith.maxLog | [94, 1] | [100, 70] | have e : Real.log ∘ max (Real.exp b) = _root_.maxLog b := by funext x; simp [_root_.maxLog] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
h : LipschitzOnWith ((-b).exp.toNNReal * 1) (Real.log ∘ fun x => max b.exp (id x)) univ
⊢ LipschitzOnWith (-b).exp.toNNReal (_root_.maxLog b) univ | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
h : LipschitzOnWith ((-b).exp.toNNReal * 1) (Real.log ∘ fun x => max b.exp (id x)) univ
e : Real.log ∘ max b.exp = _root_.maxLog b
⊢ LipschitzOnWith (-b).exp.toNNReal (_root_.maxLog b) univ | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
h : LipschitzOnWith ((-b).exp.toNNReal * 1) (Real.log ∘ fun x => max b.exp (id x)) univ
⊢ LipschitzOnWith (-b).exp.toNNReal (_root_.maxLog b) univ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzWith.maxLog | [94, 1] | [100, 70] | simpa only [e, mul_one, id_eq, ge_iff_le, lipschitzOn_univ] using h | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
h : LipschitzOnWith ((-b).exp.toNNReal * 1) (Real.log ∘ fun x => max b.exp (id x)) univ
e : Real.log ∘ max b.exp = _root_.maxLog b
⊢ LipschitzOnWith (-b).exp.toNNReal (_root_.maxLog b) univ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
h : LipschitzOnWith ((-b).exp.toNNReal * 1) (Real.log ∘ fun x => max b.exp (id x)) univ
e : Real.log ∘ max b.exp = _root_.maxLog b
⊢ LipschitzOnWith (-b).exp.toNNReal (_root_.maxLog b) univ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzWith.maxLog | [94, 1] | [100, 70] | simp only [id_eq, Set.mapsTo_univ_iff, Set.mem_Ici, le_max_iff, le_refl, true_or,
forall_const] | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ Set.MapsTo (fun x => max b.exp (id x)) univ (Ici b.exp) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
⊢ Set.MapsTo (fun x => max b.exp (id x)) univ (Ici b.exp)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzWith.maxLog | [94, 1] | [100, 70] | funext x | E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
h : LipschitzOnWith ((-b).exp.toNNReal * 1) (Real.log ∘ fun x => max b.exp (id x)) univ
⊢ Real.log ∘ max b.exp = _root_.maxLog b | case h
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
h : LipschitzOnWith ((-b).exp.toNNReal * 1) (Real.log ∘ fun x => max b.exp (id x)) univ
x : ℝ
⊢ (Real.log ∘ max b.exp) x = _root_.maxLog b x | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
h : LipschitzOnWith ((-b).exp.toNNReal * 1) (Real.log ∘ fun x => max b.exp (id x)) univ
⊢ Real.log ∘ max b.exp = _root_.maxLog b
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/MaxLog.lean | LipschitzWith.maxLog | [94, 1] | [100, 70] | simp [_root_.maxLog] | case h
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
h : LipschitzOnWith ((-b).exp.toNNReal * 1) (Real.log ∘ fun x => max b.exp (id x)) univ
x : ℝ
⊢ (Real.log ∘ max b.exp) x = _root_.maxLog b x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
E : Type
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
b : ℝ
h : LipschitzOnWith ((-b).exp.toNNReal * 1) (Real.log ∘ fun x => max b.exp (id x)) univ
x : ℝ
⊢ (Real.log ∘ max b.exp) x = _root_.maxLog b x
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_basis | [25, 1] | [28, 93] | apply Filter.hasBasis_iInf_principal | X : Type
inst✝ : Norm X
⊢ atInf.HasBasis (fun x => True) fun r => {x | r < ‖x‖} | case h
X : Type
inst✝ : Norm X
⊢ Directed (fun x x_1 => x ≥ x_1) fun i => {x | i < ‖x‖} | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : Norm X
⊢ atInf.HasBasis (fun x => True) fun r => {x | r < ‖x‖}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_basis | [25, 1] | [28, 93] | apply directed_of_isDirected_le | case h
X : Type
inst✝ : Norm X
⊢ Directed (fun x x_1 => x ≥ x_1) fun i => {x | i < ‖x‖} | case h.H
X : Type
inst✝ : Norm X
⊢ ∀ ⦃i j : ℝ⦄, i ≤ j → {x | i < ‖x‖} ≥ {x | j < ‖x‖} | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝ : Norm X
⊢ Directed (fun x x_1 => x ≥ x_1) fun i => {x | i < ‖x‖}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_basis | [25, 1] | [28, 93] | intro a b ab | case h.H
X : Type
inst✝ : Norm X
⊢ ∀ ⦃i j : ℝ⦄, i ≤ j → {x | i < ‖x‖} ≥ {x | j < ‖x‖} | case h.H
X : Type
inst✝ : Norm X
a b : ℝ
ab : a ≤ b
⊢ {x | a < ‖x‖} ≥ {x | b < ‖x‖} | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H
X : Type
inst✝ : Norm X
⊢ ∀ ⦃i j : ℝ⦄, i ≤ j → {x | i < ‖x‖} ≥ {x | j < ‖x‖}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_basis | [25, 1] | [28, 93] | simp only [ge_iff_le, le_eq_subset, setOf_subset_setOf] | case h.H
X : Type
inst✝ : Norm X
a b : ℝ
ab : a ≤ b
⊢ {x | a < ‖x‖} ≥ {x | b < ‖x‖} | case h.H
X : Type
inst✝ : Norm X
a b : ℝ
ab : a ≤ b
⊢ ∀ (a_1 : X), b < ‖a_1‖ → a < ‖a_1‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H
X : Type
inst✝ : Norm X
a b : ℝ
ab : a ≤ b
⊢ {x | a < ‖x‖} ≥ {x | b < ‖x‖}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_basis | [25, 1] | [28, 93] | intro x h | case h.H
X : Type
inst✝ : Norm X
a b : ℝ
ab : a ≤ b
⊢ ∀ (a_1 : X), b < ‖a_1‖ → a < ‖a_1‖ | case h.H
X : Type
inst✝ : Norm X
a b : ℝ
ab : a ≤ b
x : X
h : b < ‖x‖
⊢ a < ‖x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H
X : Type
inst✝ : Norm X
a b : ℝ
ab : a ≤ b
⊢ ∀ (a_1 : X), b < ‖a_1‖ → a < ‖a_1‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_basis | [25, 1] | [28, 93] | linarith | case h.H
X : Type
inst✝ : Norm X
a b : ℝ
ab : a ≤ b
x : X
h : b < ‖x‖
⊢ a < ‖x‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.H
X : Type
inst✝ : Norm X
a b : ℝ
ab : a ≤ b
x : X
h : b < ‖x‖
⊢ a < ‖x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf | [36, 1] | [38, 74] | rw [atInf_basis.tendsto_right_iff] | X Y : Type
inst✝ : Norm Y
f : X → Y
l : Filter X
⊢ Tendsto f l atInf ↔ ∀ (r : ℝ), ∀ᶠ (x : X) in l, r < ‖f x‖ | X Y : Type
inst✝ : Norm Y
f : X → Y
l : Filter X
⊢ (∀ (i : ℝ), True → ∀ᶠ (x : X) in l, f x ∈ {x | i < ‖x‖}) ↔ ∀ (r : ℝ), ∀ᶠ (x : X) in l, r < ‖f x‖ | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
inst✝ : Norm Y
f : X → Y
l : Filter X
⊢ Tendsto f l atInf ↔ ∀ (r : ℝ), ∀ᶠ (x : X) in l, r < ‖f x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf | [36, 1] | [38, 74] | simp only [true_imp_iff, mem_setOf] | X Y : Type
inst✝ : Norm Y
f : X → Y
l : Filter X
⊢ (∀ (i : ℝ), True → ∀ᶠ (x : X) in l, f x ∈ {x | i < ‖x‖}) ↔ ∀ (r : ℝ), ∀ᶠ (x : X) in l, r < ‖f x‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
inst✝ : Norm Y
f : X → Y
l : Filter X
⊢ (∀ (i : ℝ), True → ∀ᶠ (x : X) in l, f x ∈ {x | i < ‖x‖}) ↔ ∀ (r : ℝ), ∀ᶠ (x : X) in l, r < ‖f x‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atTop_atInf | [41, 1] | [45, 12] | have h := Filter.HasBasis.tendsto_iff (f := f) Filter.atTop_basis atInf_basis | X : Type
inst✝ : Norm X
f : ℕ → X
⊢ Tendsto f atTop atInf ↔ ∀ (r : ℝ), ∃ N, ∀ (n : ℕ), N ≤ n → r < ‖f n‖ | X : Type
inst✝ : Norm X
f : ℕ → X
h : Tendsto f atTop atInf ↔ ∀ (ib : ℝ), True → ∃ ia, True ∧ ∀ x ∈ Ici ia, f x ∈ {x | ib < ‖x‖}
⊢ Tendsto f atTop atInf ↔ ∀ (r : ℝ), ∃ N, ∀ (n : ℕ), N ≤ n → r < ‖f n‖ | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : Norm X
f : ℕ → X
⊢ Tendsto f atTop atInf ↔ ∀ (r : ℝ), ∃ N, ∀ (n : ℕ), N ≤ n → r < ‖f n‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atTop_atInf | [41, 1] | [45, 12] | simpa only [mem_Ici, ge_iff_le, mem_setOf_eq, exists_true_left, forall_true_left, true_and]
using h | X : Type
inst✝ : Norm X
f : ℕ → X
h : Tendsto f atTop atInf ↔ ∀ (ib : ℝ), True → ∃ ia, True ∧ ∀ x ∈ Ici ia, f x ∈ {x | ib < ‖x‖}
⊢ Tendsto f atTop atInf ↔ ∀ (r : ℝ), ∃ N, ∀ (n : ℕ), N ≤ n → r < ‖f n‖ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : Norm X
f : ℕ → X
h : Tendsto f atTop atInf ↔ ∀ (ib : ℝ), True → ∃ ia, True ∧ ∀ x ∈ Ici ia, f x ∈ {x | ib < ‖x‖}
⊢ Tendsto f atTop atInf ↔ ∀ (r : ℝ), ∃ N, ∀ (n : ℕ), N ≤ n → r < ‖f n‖
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_norm_tendsto_atTop | [48, 1] | [51, 78] | rw [Filter.atTop_basis_Ioi.tendsto_right_iff] | X Y : Type
inst✝ : Norm Y
f : Filter X
g : X → Y
⊢ Tendsto (fun x => g x) f atInf ↔ Tendsto (fun x => ‖g x‖) f atTop | X Y : Type
inst✝ : Norm Y
f : Filter X
g : X → Y
⊢ Tendsto (fun x => g x) f atInf ↔ ∀ (i : ℝ), True → ∀ᶠ (x : X) in f, ‖g x‖ ∈ Ioi i | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
inst✝ : Norm Y
f : Filter X
g : X → Y
⊢ Tendsto (fun x => g x) f atInf ↔ Tendsto (fun x => ‖g x‖) f atTop
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_norm_tendsto_atTop | [48, 1] | [51, 78] | simp only [atInf_basis.tendsto_right_iff, true_imp_iff, mem_setOf, mem_Ioi] | X Y : Type
inst✝ : Norm Y
f : Filter X
g : X → Y
⊢ Tendsto (fun x => g x) f atInf ↔ ∀ (i : ℝ), True → ∀ᶠ (x : X) in f, ‖g x‖ ∈ Ioi i | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X Y : Type
inst✝ : Norm Y
f : Filter X
g : X → Y
⊢ Tendsto (fun x => g x) f atInf ↔ ∀ (i : ℝ), True → ∀ᶠ (x : X) in f, ‖g x‖ ∈ Ioi i
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | mem_atInf_iff | [54, 1] | [56, 79] | simp only [Filter.hasBasis_iff.mp atInf_basis s, exists_true_left, true_and] | X : Type
inst✝ : Norm X
s : Set X
⊢ s ∈ atInf ↔ ∃ r, {x | ‖x‖ > r} ⊆ s | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : Norm X
s : Set X
⊢ s ∈ atInf ↔ ∃ r, {x | ‖x‖ > r} ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | eventually_atInf | [59, 1] | [60, 51] | rw [Filter.eventually_iff, mem_atInf_iff] | X : Type
inst✝ : Norm X
r : ℝ
⊢ ∀ᶠ (x : X) in atInf, ‖x‖ > r | X : Type
inst✝ : Norm X
r : ℝ
⊢ ∃ r_1, {x | ‖x‖ > r_1} ⊆ {x | ‖x‖ > r} | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : Norm X
r : ℝ
⊢ ∀ᶠ (x : X) in atInf, ‖x‖ > r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | eventually_atInf | [59, 1] | [60, 51] | use r | X : Type
inst✝ : Norm X
r : ℝ
⊢ ∃ r_1, {x | ‖x‖ > r_1} ⊆ {x | ‖x‖ > r} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : Norm X
r : ℝ
⊢ ∃ r_1, {x | ‖x‖ > r_1} ⊆ {x | ‖x‖ > r}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | rw [Filter.HasBasis.tendsto_left_iff atInf_basis, Metric.nhdsWithin_basis_ball.tendsto_left_iff] | 𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
⊢ Tendsto f atInf l ↔ Tendsto (fun x => f x⁻¹) (𝓝[≠] 0) l | 𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
⊢ (∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t) ↔ ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
⊢ Tendsto f atInf l ↔ Tendsto (fun x => f x⁻¹) (𝓝[≠] 0) l
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | constructor | 𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
⊢ (∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t) ↔ ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t | case mp
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
⊢ (∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t) → ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
case mpr
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
⊢ (∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t) → ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
⊢ (∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t) ↔ ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | intro h t tl | case mp
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
⊢ (∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t) → ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t | case mp
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
⊢ ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
⊢ (∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t) → ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | rcases h t tl with ⟨r, _, m⟩ | case mp
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
⊢ ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t | case mp.intro.intro
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
⊢ ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t | Please generate a tactic in lean4 to solve the state.
STATE:
case mp
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
⊢ ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | by_cases rp : 0 < r | case mp.intro.intro
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
⊢ ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t | case pos
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : 0 < r
⊢ ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
case neg
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : ¬0 < r
⊢ ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t | Please generate a tactic in lean4 to solve the state.
STATE:
case mp.intro.intro
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
⊢ ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | use r⁻¹ | case pos
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : 0 < r
⊢ ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : 0 < r
⊢ 0 < r⁻¹ ∧ MapsTo (fun x => f x⁻¹) (ball 0 r⁻¹ ∩ {0}ᶜ) t | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : 0 < r
⊢ ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | simp only [rp, inv_pos, true_and_iff] | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : 0 < r
⊢ 0 < r⁻¹ ∧ MapsTo (fun x => f x⁻¹) (ball 0 r⁻¹ ∩ {0}ᶜ) t | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : 0 < r
⊢ MapsTo (fun x => f x⁻¹) (ball 0 r⁻¹ ∩ {0}ᶜ) t | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : 0 < r
⊢ 0 < r⁻¹ ∧ MapsTo (fun x => f x⁻¹) (ball 0 r⁻¹ ∩ {0}ᶜ) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | intro x xs | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : 0 < r
⊢ MapsTo (fun x => f x⁻¹) (ball 0 r⁻¹ ∩ {0}ᶜ) t | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : 0 < r
x : 𝕜
xs : x ∈ ball 0 r⁻¹ ∩ {0}ᶜ
⊢ (fun x => f x⁻¹) x ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : 0 < r
⊢ MapsTo (fun x => f x⁻¹) (ball 0 r⁻¹ ∩ {0}ᶜ) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | refine m ?_ | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : 0 < r
x : 𝕜
xs : x ∈ ball 0 r⁻¹ ∩ {0}ᶜ
⊢ (fun x => f x⁻¹) x ∈ t | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : 0 < r
x : 𝕜
xs : x ∈ ball 0 r⁻¹ ∩ {0}ᶜ
⊢ x⁻¹ ∈ {x | r < ‖x‖} | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : 0 < r
x : 𝕜
xs : x ∈ ball 0 r⁻¹ ∩ {0}ᶜ
⊢ (fun x => f x⁻¹) x ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | simp only [mem_inter_iff, mem_ball_zero_iff, mem_compl_iff, mem_singleton_iff] at xs | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : 0 < r
x : 𝕜
xs : x ∈ ball 0 r⁻¹ ∩ {0}ᶜ
⊢ x⁻¹ ∈ {x | r < ‖x‖} | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : 0 < r
x : 𝕜
xs : ‖x‖ < r⁻¹ ∧ ¬x = 0
⊢ x⁻¹ ∈ {x | r < ‖x‖} | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : 0 < r
x : 𝕜
xs : x ∈ ball 0 r⁻¹ ∩ {0}ᶜ
⊢ x⁻¹ ∈ {x | r < ‖x‖}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | simp only [← lt_inv (norm_pos_iff.mpr xs.2) rp, xs.1, mem_setOf_eq, norm_inv] | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : 0 < r
x : 𝕜
xs : ‖x‖ < r⁻¹ ∧ ¬x = 0
⊢ x⁻¹ ∈ {x | r < ‖x‖} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : 0 < r
x : 𝕜
xs : ‖x‖ < r⁻¹ ∧ ¬x = 0
⊢ x⁻¹ ∈ {x | r < ‖x‖}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | use 1 | case neg
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : ¬0 < r
⊢ ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : ¬0 < r
⊢ 0 < 1 ∧ MapsTo (fun x => f x⁻¹) (ball 0 1 ∩ {0}ᶜ) t | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : ¬0 < r
⊢ ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | simp only [zero_lt_one, true_and_iff] | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : ¬0 < r
⊢ 0 < 1 ∧ MapsTo (fun x => f x⁻¹) (ball 0 1 ∩ {0}ᶜ) t | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : ¬0 < r
⊢ MapsTo (fun x => f x⁻¹) (ball 0 1 ∩ {0}ᶜ) t | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : ¬0 < r
⊢ 0 < 1 ∧ MapsTo (fun x => f x⁻¹) (ball 0 1 ∩ {0}ᶜ) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | intro x xs | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : ¬0 < r
⊢ MapsTo (fun x => f x⁻¹) (ball 0 1 ∩ {0}ᶜ) t | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : ¬0 < r
x : 𝕜
xs : x ∈ ball 0 1 ∩ {0}ᶜ
⊢ (fun x => f x⁻¹) x ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : ¬0 < r
⊢ MapsTo (fun x => f x⁻¹) (ball 0 1 ∩ {0}ᶜ) t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | refine m ?_ | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : ¬0 < r
x : 𝕜
xs : x ∈ ball 0 1 ∩ {0}ᶜ
⊢ (fun x => f x⁻¹) x ∈ t | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : ¬0 < r
x : 𝕜
xs : x ∈ ball 0 1 ∩ {0}ᶜ
⊢ x⁻¹ ∈ {x | r < ‖x‖} | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : ¬0 < r
x : 𝕜
xs : x ∈ ball 0 1 ∩ {0}ᶜ
⊢ (fun x => f x⁻¹) x ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | simp only [mem_inter_iff, mem_ball_zero_iff, mem_compl_iff, mem_singleton_iff] at xs | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : ¬0 < r
x : 𝕜
xs : x ∈ ball 0 1 ∩ {0}ᶜ
⊢ x⁻¹ ∈ {x | r < ‖x‖} | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : ¬0 < r
x : 𝕜
xs : ‖x‖ < 1 ∧ ¬x = 0
⊢ x⁻¹ ∈ {x | r < ‖x‖} | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : ¬0 < r
x : 𝕜
xs : x ∈ ball 0 1 ∩ {0}ᶜ
⊢ x⁻¹ ∈ {x | r < ‖x‖}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | simp only [mem_setOf_eq, norm_inv] | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : ¬0 < r
x : 𝕜
xs : ‖x‖ < 1 ∧ ¬x = 0
⊢ x⁻¹ ∈ {x | r < ‖x‖} | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : ¬0 < r
x : 𝕜
xs : ‖x‖ < 1 ∧ ¬x = 0
⊢ r < ‖x‖⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : ¬0 < r
x : 𝕜
xs : ‖x‖ < 1 ∧ ¬x = 0
⊢ x⁻¹ ∈ {x | r < ‖x‖}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | simp only [not_lt] at rp | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : ¬0 < r
x : 𝕜
xs : ‖x‖ < 1 ∧ ¬x = 0
⊢ r < ‖x‖⁻¹ | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
x : 𝕜
xs : ‖x‖ < 1 ∧ ¬x = 0
rp : r ≤ 0
⊢ r < ‖x‖⁻¹ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
rp : ¬0 < r
x : 𝕜
xs : ‖x‖ < 1 ∧ ¬x = 0
⊢ r < ‖x‖⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | exact lt_of_le_of_lt rp (inv_pos.mpr (norm_pos_iff.mpr xs.2)) | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
x : 𝕜
xs : ‖x‖ < 1 ∧ ¬x = 0
rp : r ≤ 0
⊢ r < ‖x‖⁻¹ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
t : Set X
tl : t ∈ l
r : ℝ
left✝ : True
m : MapsTo f {x | r < ‖x‖} t
x : 𝕜
xs : ‖x‖ < 1 ∧ ¬x = 0
rp : r ≤ 0
⊢ r < ‖x‖⁻¹
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | intro h t tl | case mpr
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
⊢ (∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t) → ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t | case mpr
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
⊢ ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
⊢ (∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t) → ∀ t ∈ l, ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | rcases h t tl with ⟨r, rp, m⟩ | case mpr
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
⊢ ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t | case mpr.intro.intro
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
⊢ ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
⊢ ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | use r⁻¹ | case mpr.intro.intro
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
⊢ ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
⊢ True ∧ MapsTo f {x | r⁻¹ < ‖x‖} t | Please generate a tactic in lean4 to solve the state.
STATE:
case mpr.intro.intro
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
⊢ ∃ i, True ∧ MapsTo f {x | i < ‖x‖} t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | simp only [true_and_iff] | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
⊢ True ∧ MapsTo f {x | r⁻¹ < ‖x‖} t | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
⊢ MapsTo f {x | r⁻¹ < ‖x‖} t | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
⊢ True ∧ MapsTo f {x | r⁻¹ < ‖x‖} t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | intro x xs | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
⊢ MapsTo f {x | r⁻¹ < ‖x‖} t | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
x : 𝕜
xs : x ∈ {x | r⁻¹ < ‖x‖}
⊢ f x ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
⊢ MapsTo f {x | r⁻¹ < ‖x‖} t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | simp only [mem_setOf_eq] at xs | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
x : 𝕜
xs : x ∈ {x | r⁻¹ < ‖x‖}
⊢ f x ∈ t | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
x : 𝕜
xs : r⁻¹ < ‖x‖
⊢ f x ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
x : 𝕜
xs : x ∈ {x | r⁻¹ < ‖x‖}
⊢ f x ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | have m := @m x⁻¹ ?_ | case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
x : 𝕜
xs : r⁻¹ < ‖x‖
⊢ f x ∈ t | case h.refine_2
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m✝ : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
x : 𝕜
xs : r⁻¹ < ‖x‖
m : (fun x => f x⁻¹) x⁻¹ ∈ t
⊢ f x ∈ t
case h.refine_1
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
x : 𝕜
xs : r⁻¹ < ‖x‖
⊢ x⁻¹ ∈ ball 0 r ∩ {0}ᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
x : 𝕜
xs : r⁻¹ < ‖x‖
⊢ f x ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | simp only [mem_inter_iff, mem_ball_zero_iff, norm_inv, mem_compl_iff, mem_singleton_iff,
inv_eq_zero] | case h.refine_1
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
x : 𝕜
xs : r⁻¹ < ‖x‖
⊢ x⁻¹ ∈ ball 0 r ∩ {0}ᶜ | case h.refine_1
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
x : 𝕜
xs : r⁻¹ < ‖x‖
⊢ ‖x‖⁻¹ < r ∧ ¬x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
x : 𝕜
xs : r⁻¹ < ‖x‖
⊢ x⁻¹ ∈ ball 0 r ∩ {0}ᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | have np : 0 < ‖x‖ := _root_.trans (inv_pos.mpr rp) xs | case h.refine_1
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
x : 𝕜
xs : r⁻¹ < ‖x‖
⊢ ‖x‖⁻¹ < r ∧ ¬x = 0 | case h.refine_1
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
x : 𝕜
xs : r⁻¹ < ‖x‖
np : 0 < ‖x‖
⊢ ‖x‖⁻¹ < r ∧ ¬x = 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
x : 𝕜
xs : r⁻¹ < ‖x‖
⊢ ‖x‖⁻¹ < r ∧ ¬x = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | simp [inv_lt np rp, xs, norm_pos_iff.mp np] | case h.refine_1
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
x : 𝕜
xs : r⁻¹ < ‖x‖
np : 0 < ‖x‖
⊢ ‖x‖⁻¹ < r ∧ ¬x = 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_1
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
x : 𝕜
xs : r⁻¹ < ‖x‖
np : 0 < ‖x‖
⊢ ‖x‖⁻¹ < r ∧ ¬x = 0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | simp only [inv_inv] at m | case h.refine_2
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m✝ : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
x : 𝕜
xs : r⁻¹ < ‖x‖
m : (fun x => f x⁻¹) x⁻¹ ∈ t
⊢ f x ∈ t | case h.refine_2
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m✝ : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
x : 𝕜
xs : r⁻¹ < ‖x‖
m : f x ∈ t
⊢ f x ∈ t | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_2
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m✝ : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
x : 𝕜
xs : r⁻¹ < ‖x‖
m : (fun x => f x⁻¹) x⁻¹ ∈ t
⊢ f x ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | tendsto_atInf_iff_tendsto_nhds_zero | [63, 1] | [82, 48] | exact m | case h.refine_2
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m✝ : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
x : 𝕜
xs : r⁻¹ < ‖x‖
m : f x ∈ t
⊢ f x ∈ t | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.refine_2
𝕜 X : Type
inst✝ : NontriviallyNormedField 𝕜
l : Filter X
f : 𝕜 → X
h : ∀ t ∈ l, ∃ i, 0 < i ∧ MapsTo (fun x => f x⁻¹) (ball 0 i ∩ {0}ᶜ) t
t : Set X
tl : t ∈ l
r : ℝ
rp : 0 < r
m✝ : MapsTo (fun x => f x⁻¹) (ball 0 r ∩ {0}ᶜ) t
x : 𝕜
xs : r⁻¹ < ‖x‖
m : f x ∈ t
⊢ f x ∈ t
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_le_cocompact | [85, 1] | [95, 15] | rw [Filter.le_def] | X : Type
inst✝ : NormedAddCommGroup X
⊢ atInf ≤ Filter.cocompact X | X : Type
inst✝ : NormedAddCommGroup X
⊢ ∀ x ∈ Filter.cocompact X, x ∈ atInf | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : NormedAddCommGroup X
⊢ atInf ≤ Filter.cocompact X
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_le_cocompact | [85, 1] | [95, 15] | intro s m | X : Type
inst✝ : NormedAddCommGroup X
⊢ ∀ x ∈ Filter.cocompact X, x ∈ atInf | X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m : s ∈ Filter.cocompact X
⊢ s ∈ atInf | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : NormedAddCommGroup X
⊢ ∀ x ∈ Filter.cocompact X, x ∈ atInf
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_le_cocompact | [85, 1] | [95, 15] | rcases Filter.mem_cocompact.mp m with ⟨t, tc, ts⟩ | X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m : s ∈ Filter.cocompact X
⊢ s ∈ atInf | case intro.intro
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
⊢ s ∈ atInf | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m : s ∈ Filter.cocompact X
⊢ s ∈ atInf
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_le_cocompact | [85, 1] | [95, 15] | rcases tc.bddAbove_image continuousOn_id.norm with ⟨r, rh⟩ | case intro.intro
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
⊢ s ∈ atInf | case intro.intro.intro
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : r ∈ upperBounds ((fun x => ‖id x‖) '' t)
⊢ s ∈ atInf | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
⊢ s ∈ atInf
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_le_cocompact | [85, 1] | [95, 15] | simp only [id_eq, mem_upperBounds, mem_image, forall_exists_index, and_imp,
forall_apply_eq_imp_iff₂] at rh | case intro.intro.intro
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : r ∈ upperBounds ((fun x => ‖id x‖) '' t)
⊢ s ∈ atInf | case intro.intro.intro
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
⊢ s ∈ atInf | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : r ∈ upperBounds ((fun x => ‖id x‖) '' t)
⊢ s ∈ atInf
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_le_cocompact | [85, 1] | [95, 15] | rw [mem_atInf_iff] | case intro.intro.intro
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
⊢ s ∈ atInf | case intro.intro.intro
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
⊢ ∃ r, {x | ‖x‖ > r} ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
⊢ s ∈ atInf
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_le_cocompact | [85, 1] | [95, 15] | use r | case intro.intro.intro
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
⊢ ∃ r, {x | ‖x‖ > r} ⊆ s | case h
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
⊢ {x | ‖x‖ > r} ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro.intro
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
⊢ ∃ r, {x | ‖x‖ > r} ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_le_cocompact | [85, 1] | [95, 15] | intro x m | case h
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
⊢ {x | ‖x‖ > r} ⊆ s | case h
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m✝ : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
x : X
m : x ∈ {x | ‖x‖ > r}
⊢ x ∈ s | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
⊢ {x | ‖x‖ > r} ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_le_cocompact | [85, 1] | [95, 15] | apply ts | case h
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m✝ : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
x : X
m : x ∈ {x | ‖x‖ > r}
⊢ x ∈ s | case h.a
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m✝ : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
x : X
m : x ∈ {x | ‖x‖ > r}
⊢ x ∈ tᶜ | Please generate a tactic in lean4 to solve the state.
STATE:
case h
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m✝ : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
x : X
m : x ∈ {x | ‖x‖ > r}
⊢ x ∈ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_le_cocompact | [85, 1] | [95, 15] | contrapose m | case h.a
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m✝ : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
x : X
m : x ∈ {x | ‖x‖ > r}
⊢ x ∈ tᶜ | case h.a
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m✝ : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
x : X
m : x ∉ tᶜ
⊢ x ∉ {x | ‖x‖ > r} | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m✝ : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
x : X
m : x ∈ {x | ‖x‖ > r}
⊢ x ∈ tᶜ
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_le_cocompact | [85, 1] | [95, 15] | simp only [mem_compl_iff, not_not_mem] at m | case h.a
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m✝ : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
x : X
m : x ∉ tᶜ
⊢ x ∉ {x | ‖x‖ > r} | case h.a
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m✝ : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
x : X
m : x ∈ t
⊢ x ∉ {x | ‖x‖ > r} | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m✝ : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
x : X
m : x ∉ tᶜ
⊢ x ∉ {x | ‖x‖ > r}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_le_cocompact | [85, 1] | [95, 15] | simp only [mem_setOf_eq, not_lt] | case h.a
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m✝ : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
x : X
m : x ∈ t
⊢ x ∉ {x | ‖x‖ > r} | case h.a
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m✝ : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
x : X
m : x ∈ t
⊢ ‖x‖ ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m✝ : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
x : X
m : x ∈ t
⊢ x ∉ {x | ‖x‖ > r}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_le_cocompact | [85, 1] | [95, 15] | exact rh _ m | case h.a
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m✝ : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
x : X
m : x ∈ t
⊢ ‖x‖ ≤ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case h.a
X : Type
inst✝ : NormedAddCommGroup X
s : Set X
m✝ : s ∈ Filter.cocompact X
t : Set X
tc : IsCompact t
ts : tᶜ ⊆ s
r : ℝ
rh : ∀ a ∈ t, ‖a‖ ≤ r
x : X
m : x ∈ t
⊢ ‖x‖ ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_eq_cocompact | [98, 1] | [104, 77] | apply le_antisymm atInf_le_cocompact | X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
⊢ atInf = Filter.cocompact X | X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
⊢ Filter.cocompact X ≤ atInf | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
⊢ atInf = Filter.cocompact X
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_eq_cocompact | [98, 1] | [104, 77] | rw [Filter.le_def] | X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
⊢ Filter.cocompact X ≤ atInf | X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
⊢ ∀ x ∈ atInf, x ∈ Filter.cocompact X | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
⊢ Filter.cocompact X ≤ atInf
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_eq_cocompact | [98, 1] | [104, 77] | intro s m | X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
⊢ ∀ x ∈ atInf, x ∈ Filter.cocompact X | X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
s : Set X
m : s ∈ atInf
⊢ s ∈ Filter.cocompact X | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
⊢ ∀ x ∈ atInf, x ∈ Filter.cocompact X
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_eq_cocompact | [98, 1] | [104, 77] | rcases mem_atInf_iff.mp m with ⟨r, h⟩ | X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
s : Set X
m : s ∈ atInf
⊢ s ∈ Filter.cocompact X | case intro
X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
s : Set X
m : s ∈ atInf
r : ℝ
h : {x | ‖x‖ > r} ⊆ s
⊢ s ∈ Filter.cocompact X | Please generate a tactic in lean4 to solve the state.
STATE:
X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
s : Set X
m : s ∈ atInf
⊢ s ∈ Filter.cocompact X
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_eq_cocompact | [98, 1] | [104, 77] | rw [Filter.mem_cocompact] | case intro
X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
s : Set X
m : s ∈ atInf
r : ℝ
h : {x | ‖x‖ > r} ⊆ s
⊢ s ∈ Filter.cocompact X | case intro
X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
s : Set X
m : s ∈ atInf
r : ℝ
h : {x | ‖x‖ > r} ⊆ s
⊢ ∃ t, IsCompact t ∧ tᶜ ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
s : Set X
m : s ∈ atInf
r : ℝ
h : {x | ‖x‖ > r} ⊆ s
⊢ s ∈ Filter.cocompact X
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_eq_cocompact | [98, 1] | [104, 77] | use closedBall 0 r, isCompact_closedBall _ _ | case intro
X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
s : Set X
m : s ∈ atInf
r : ℝ
h : {x | ‖x‖ > r} ⊆ s
⊢ ∃ t, IsCompact t ∧ tᶜ ⊆ s | case right
X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
s : Set X
m : s ∈ atInf
r : ℝ
h : {x | ‖x‖ > r} ⊆ s
⊢ (closedBall 0 r)ᶜ ⊆ s | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
s : Set X
m : s ∈ atInf
r : ℝ
h : {x | ‖x‖ > r} ⊆ s
⊢ ∃ t, IsCompact t ∧ tᶜ ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_eq_cocompact | [98, 1] | [104, 77] | refine _root_.trans ?_ h | case right
X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
s : Set X
m : s ∈ atInf
r : ℝ
h : {x | ‖x‖ > r} ⊆ s
⊢ (closedBall 0 r)ᶜ ⊆ s | case right
X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
s : Set X
m : s ∈ atInf
r : ℝ
h : {x | ‖x‖ > r} ⊆ s
⊢ (closedBall 0 r)ᶜ ⊆ {x | ‖x‖ > r} | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
s : Set X
m : s ∈ atInf
r : ℝ
h : {x | ‖x‖ > r} ⊆ s
⊢ (closedBall 0 r)ᶜ ⊆ s
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_eq_cocompact | [98, 1] | [104, 77] | intro x xs | case right
X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
s : Set X
m : s ∈ atInf
r : ℝ
h : {x | ‖x‖ > r} ⊆ s
⊢ (closedBall 0 r)ᶜ ⊆ {x | ‖x‖ > r} | case right
X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
s : Set X
m : s ∈ atInf
r : ℝ
h : {x | ‖x‖ > r} ⊆ s
x : X
xs : x ∈ (closedBall 0 r)ᶜ
⊢ x ∈ {x | ‖x‖ > r} | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
s : Set X
m : s ∈ atInf
r : ℝ
h : {x | ‖x‖ > r} ⊆ s
⊢ (closedBall 0 r)ᶜ ⊆ {x | ‖x‖ > r}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_eq_cocompact | [98, 1] | [104, 77] | simp only [mem_compl_iff, mem_closedBall_zero_iff, not_le] at xs | case right
X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
s : Set X
m : s ∈ atInf
r : ℝ
h : {x | ‖x‖ > r} ⊆ s
x : X
xs : x ∈ (closedBall 0 r)ᶜ
⊢ x ∈ {x | ‖x‖ > r} | case right
X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
s : Set X
m : s ∈ atInf
r : ℝ
h : {x | ‖x‖ > r} ⊆ s
x : X
xs : r < ‖x‖
⊢ x ∈ {x | ‖x‖ > r} | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
s : Set X
m : s ∈ atInf
r : ℝ
h : {x | ‖x‖ > r} ⊆ s
x : X
xs : x ∈ (closedBall 0 r)ᶜ
⊢ x ∈ {x | ‖x‖ > r}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | atInf_eq_cocompact | [98, 1] | [104, 77] | exact xs | case right
X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
s : Set X
m : s ∈ atInf
r : ℝ
h : {x | ‖x‖ > r} ⊆ s
x : X
xs : r < ‖x‖
⊢ x ∈ {x | ‖x‖ > r} | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
X : Type
inst✝¹ : NormedAddCommGroup X
inst✝ : ProperSpace X
s : Set X
m : s ∈ atInf
r : ℝ
h : {x | ‖x‖ > r} ⊆ s
x : X
xs : r < ‖x‖
⊢ x ∈ {x | ‖x‖ > r}
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | inv_tendsto_atInf | [107, 1] | [109, 90] | rw [←tendsto_atInf_iff_tendsto_nhds_zero (f := fun x : 𝕜 ↦ x)] | 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
⊢ Tendsto (fun x => x⁻¹) (𝓝[≠] 0) atInf | 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
⊢ Tendsto (fun x => x) atInf atInf | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
⊢ Tendsto (fun x => x⁻¹) (𝓝[≠] 0) atInf
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | inv_tendsto_atInf | [107, 1] | [109, 90] | exact Filter.tendsto_id | 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
⊢ Tendsto (fun x => x) atInf atInf | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
⊢ Tendsto (fun x => x) atInf atInf
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | inv_tendsto_atInf' | [112, 1] | [115, 55] | simp only [tendsto_atInf_iff_tendsto_nhds_zero, inv_inv] | 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
⊢ Tendsto (fun x => x⁻¹) atInf (𝓝 0) | 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
⊢ Tendsto (fun x => x) (𝓝[≠] 0) (𝓝 0) | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
⊢ Tendsto (fun x => x⁻¹) atInf (𝓝 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Misc/AtInf.lean | inv_tendsto_atInf' | [112, 1] | [115, 55] | exact Filter.tendsto_id.mono_left nhdsWithin_le_nhds | 𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
⊢ Tendsto (fun x => x) (𝓝[≠] 0) (𝓝 0) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
𝕜 : Type
inst✝ : NontriviallyNormedField 𝕜
⊢ Tendsto (fun x => x) (𝓝[≠] 0) (𝓝 0)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_large_z3 | [42, 1] | [46, 45] | rw [(by norm_num : (2:ℝ) = 3-1)] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 2 ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ (3 - 1) ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 2 ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_large_z3 | [42, 1] | [46, 45] | exact iter_large d 3 (by norm_num) z3 cz n | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ (3 - 1) ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ (3 - 1) ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_large_z3 | [42, 1] | [46, 45] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 2 = 3 - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 2 = 3 - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_large_z3 | [42, 1] | [46, 45] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 2 ≤ 3 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 2 ≤ 3
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_large_z4 | [48, 1] | [52, 45] | rw [(by norm_num : (3:ℝ) = 4-1)] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 3 ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ (4 - 1) ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z) | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 3 ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_large_z4 | [48, 1] | [52, 45] | exact iter_large d 4 (by norm_num) z4 cz n | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ (4 - 1) ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ (4 - 1) ^ n * Complex.abs z ≤ Complex.abs ((f' d c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_large_z4 | [48, 1] | [52, 45] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 3 = 4 - 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 3 = 4 - 1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | iter_large_z4 | [48, 1] | [52, 45] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 2 ≤ 4 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z4 : 4 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 2 ≤ 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | le_self_iter | [54, 1] | [58, 92] | refine le_trans ?_ (iter_large_z3 d z3 cz n) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ Complex.abs z ≤ Complex.abs ((f' d c)^[n] z) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ Complex.abs z ≤ 2 ^ n * Complex.abs z | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ Complex.abs z ≤ Complex.abs ((f' d c)^[n] z)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | le_self_iter | [54, 1] | [58, 92] | exact le_mul_of_one_le_left (Complex.abs.nonneg _) (one_le_pow_of_one_le (by norm_num) _) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ Complex.abs z ≤ 2 ^ n * Complex.abs z | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ Complex.abs z ≤ 2 ^ n * Complex.abs z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | le_self_iter | [54, 1] | [58, 92] | norm_num | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 1 ≤ 2 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
n : ℕ
⊢ 1 ≤ 2
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | tendsto_iter_atInf | [61, 1] | [65, 100] | simp only [tendsto_atInf_iff_norm_tendsto_atTop, Complex.norm_eq_abs] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun n => (f' d c)^[n] z) atTop atInf | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun x => Complex.abs ((f' d c)^[x] z)) atTop atTop | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun n => (f' d c)^[n] z) atTop atInf
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Iterates.lean | tendsto_iter_atInf | [61, 1] | [65, 100] | refine Filter.tendsto_atTop_mono (iter_large_z3 d z3 cz) ?_ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun x => Complex.abs ((f' d c)^[x] z)) atTop atTop | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun n => 2 ^ n * Complex.abs z) atTop atTop | Please generate a tactic in lean4 to solve the state.
STATE:
c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c z : ℂ
z3 : 3 ≤ Complex.abs z
cz : Complex.abs c ≤ Complex.abs z
⊢ Tendsto (fun x => Complex.abs ((f' d c)^[x] z)) atTop atTop
TACTIC:
|
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