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/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | rw [e] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
e✝ : ∀ᶠ (z : ℂ) in atInf, s.bottcher c ↑z = s.bottcherNear c ↑z
z : ℂ
e : s.bottcher c ↑z = s.bottcherNear c ↑z
⊢ s.bottcher c ↑z * z = s.bottcherNear c ↑z * 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 : ℂ
s : Super (f d) d ∞ := superF d
e✝ : ∀ᶠ (z : ℂ) in atInf, s.bottcher c ↑z = s.bottcherNear c ↑z
z : ℂ
e : s.bottcher c ↑z = s.bottcherNear c ↑z
⊢ s.bottcher c ↑z * z = s.bottcherNear c ↑z * z
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | simp only [one_div, map_inv₀] at zr ⊢ | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
m : ∀ ε > 0, ∃ δ > 0, ∀ {x : ℂ}, Complex.abs x < δ → (Complex.abs x)⁻¹ * Complex.abs (bottcherNear (s.fl c) d x - x) < ε
e : ℝ
ep : e > 0
r : ℝ
rp : r > 0
h : ∀ {x : ℂ}, Complex.abs x < r → (Complex.abs x)⁻¹ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
z : ℂ
zr : 1 / r < Complex.abs z
az0 : Complex.abs z ≠ 0
z0 : z ≠ 0
⊢ Complex.abs z⁻¹ < r | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
m : ∀ ε > 0, ∃ δ > 0, ∀ {x : ℂ}, Complex.abs x < δ → (Complex.abs x)⁻¹ * Complex.abs (bottcherNear (s.fl c) d x - x) < ε
e : ℝ
ep : e > 0
r : ℝ
rp : r > 0
h : ∀ {x : ℂ}, Complex.abs x < r → (Complex.abs x)⁻¹ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
z : ℂ
az0 : Complex.abs z ≠ 0
z0 : z ≠ 0
zr : r⁻¹ < Complex.abs z
⊢ (Complex.abs z)⁻¹ < r | 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 : ℂ
s : Super (f d) d ∞ := superF d
m : ∀ ε > 0, ∃ δ > 0, ∀ {x : ℂ}, Complex.abs x < δ → (Complex.abs x)⁻¹ * Complex.abs (bottcherNear (s.fl c) d x - x) < ε
e : ℝ
ep : e > 0
r : ℝ
rp : r > 0
h : ∀ {x : ℂ}, Complex.abs x < r → (Complex.abs x)⁻¹ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
z : ℂ
zr : 1 / r < Complex.abs z
az0 : Complex.abs z ≠ 0
z0 : z ≠ 0
⊢ Complex.abs z⁻¹ < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | bottcher_large_approx | [715, 1] | [741, 10] | exact inv_lt_of_inv_lt rp zr | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
m : ∀ ε > 0, ∃ δ > 0, ∀ {x : ℂ}, Complex.abs x < δ → (Complex.abs x)⁻¹ * Complex.abs (bottcherNear (s.fl c) d x - x) < ε
e : ℝ
ep : e > 0
r : ℝ
rp : r > 0
h : ∀ {x : ℂ}, Complex.abs x < r → (Complex.abs x)⁻¹ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
z : ℂ
az0 : Complex.abs z ≠ 0
z0 : z ≠ 0
zr : r⁻¹ < Complex.abs z
⊢ (Complex.abs z)⁻¹ < r | 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 : ℂ
s : Super (f d) d ∞ := superF d
m : ∀ ε > 0, ∃ δ > 0, ∀ {x : ℂ}, Complex.abs x < δ → (Complex.abs x)⁻¹ * Complex.abs (bottcherNear (s.fl c) d x - x) < ε
e : ℝ
ep : e > 0
r : ℝ
rp : r > 0
h : ∀ {x : ℂ}, Complex.abs x < r → (Complex.abs x)⁻¹ * Complex.abs (bottcherNear (s.fl c) d x - x) < e
z : ℂ
az0 : Complex.abs z ≠ 0
z0 : z ≠ 0
zr : r⁻¹ < Complex.abs z
⊢ (Complex.abs z)⁻¹ < r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_tendsto | [744, 1] | [748, 85] | set s := superF d | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
⊢ Tendsto (fun z => ⋯.potential c ↑z * Complex.abs z) atInf (𝓝 1) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
⊢ Tendsto (fun z => s.potential c ↑z * Complex.abs z) atInf (𝓝 1) | 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 : ℂ
⊢ Tendsto (fun z => ⋯.potential c ↑z * Complex.abs z) atInf (𝓝 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_tendsto | [744, 1] | [748, 85] | simp only [← s.abs_bottcher, ← Complex.abs.map_mul, ← Complex.abs.map_one] | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
⊢ Tendsto (fun z => s.potential c ↑z * Complex.abs z) atInf (𝓝 1) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
⊢ Tendsto (fun z => Complex.abs (s.bottcher c ↑z * z)) atInf (𝓝 (Complex.abs 1)) | 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 : ℂ
s : Super (f d) d ∞ := superF d
⊢ Tendsto (fun z => s.potential c ↑z * Complex.abs z) atInf (𝓝 1)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Dynamics/Multibrot/Basic.lean | potential_tendsto | [744, 1] | [748, 85] | exact Complex.continuous_abs.continuousAt.tendsto.comp (bottcher_large_approx d c) | c✝ : ℂ
d✝ : ℕ
inst✝¹ : Fact (2 ≤ d✝)
d : ℕ
inst✝ : Fact (2 ≤ d)
c : ℂ
s : Super (f d) d ∞ := superF d
⊢ Tendsto (fun z => Complex.abs (s.bottcher c ↑z * z)) atInf (𝓝 (Complex.abs 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 : ℂ
s : Super (f d) d ∞ := superF d
⊢ Tendsto (fun z => Complex.abs (s.bottcher c ↑z * z)) atInf (𝓝 (Complex.abs 1))
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/Hartogs/Hartogs.lean | Har.flip | [70, 1] | [77, 46] | intro c0 c1 cs | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
⊢ ∀ (c0 c1 : ℂ), (c0, c1) ∈ swap '' s → AnalyticAt ℂ (fun z0 => (f ∘ swap) (z0, c1)) c0 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z0 => (f ∘ swap) (z0, c1)) c0 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
⊢ ∀ (c0 c1 : ℂ), (c0, c1) ∈ swap '' s → AnalyticAt ℂ (fun z0 => (f ∘ swap) (z0, c1)) c0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.flip | [70, 1] | [77, 46] | simp only [Function.comp_apply, Prod.swap_prod_mk] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z0 => (f ∘ swap) (z0, c1)) c0 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z0 => f (c1, z0)) c0 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z0 => (f ∘ swap) (z0, c1)) c0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.flip | [70, 1] | [77, 46] | rw [swap_mem] at cs | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z0 => f (c1, z0)) c0 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c1, c0) ∈ s
⊢ AnalyticAt ℂ (fun z0 => f (c1, z0)) c0 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z0 => f (c1, z0)) c0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.flip | [70, 1] | [77, 46] | exact h.fa1 c1 c0 cs | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c1, c0) ∈ s
⊢ AnalyticAt ℂ (fun z0 => f (c1, z0)) c0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c1, c0) ∈ s
⊢ AnalyticAt ℂ (fun z0 => f (c1, z0)) c0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.flip | [70, 1] | [77, 46] | intro c0 c1 cs | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
⊢ ∀ (c0 c1 : ℂ), (c0, c1) ∈ swap '' s → AnalyticAt ℂ (fun z1 => (f ∘ swap) (c0, z1)) c1 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z1 => (f ∘ swap) (c0, z1)) c1 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
⊢ ∀ (c0 c1 : ℂ), (c0, c1) ∈ swap '' s → AnalyticAt ℂ (fun z1 => (f ∘ swap) (c0, z1)) c1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.flip | [70, 1] | [77, 46] | simp only [Function.comp_apply, Prod.swap_prod_mk] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z1 => (f ∘ swap) (c0, z1)) c1 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z1 => f (z1, c0)) c1 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z1 => (f ∘ swap) (c0, z1)) c1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.flip | [70, 1] | [77, 46] | rw [swap_mem] at cs | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z1 => f (z1, c0)) c1 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c1, c0) ∈ s
⊢ AnalyticAt ℂ (fun z1 => f (z1, c0)) c1 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c0, c1) ∈ swap '' s
⊢ AnalyticAt ℂ (fun z1 => f (z1, c0)) c1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.flip | [70, 1] | [77, 46] | exact h.fa0 c1 c0 cs | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c1, c0) ∈ s
⊢ AnalyticAt ℂ (fun z1 => f (z1, c0)) c1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0✝ c0' c1✝ z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f s
c0 c1 : ℂ
cs : (c1, c0) ∈ s
⊢ AnalyticAt ℂ (fun z1 => f (z1, c0)) c1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.on0 | [85, 1] | [90, 19] | intro z0 z0s | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
⊢ AnalyticOn ℂ (fun z0 => f (z0, z1)) (closedBall c0 r) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : z0 ∈ closedBall c0 r
⊢ AnalyticAt ℂ (fun z0 => f (z0, z1)) z0 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
⊢ AnalyticOn ℂ (fun z0 => f (z0, z1)) (closedBall c0 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.on0 | [85, 1] | [90, 19] | apply h.fa0 z0 z1 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : z0 ∈ closedBall c0 r
⊢ AnalyticAt ℂ (fun z0 => f (z0, z1)) z0 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : z0 ∈ closedBall c0 r
⊢ (z0, z1) ∈ closedBall (c0, c1) r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : z0 ∈ closedBall c0 r
⊢ AnalyticAt ℂ (fun z0 => f (z0, z1)) z0
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.on0 | [85, 1] | [90, 19] | rw [← closedBall_prod_same] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : z0 ∈ closedBall c0 r
⊢ (z0, z1) ∈ closedBall (c0, c1) r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : z0 ∈ closedBall c0 r
⊢ (z0, z1) ∈ closedBall c0 r ×ˢ closedBall c1 r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : z0 ∈ closedBall c0 r
⊢ (z0, z1) ∈ closedBall (c0, c1) r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.on0 | [85, 1] | [90, 19] | simp only [Set.prod_mk_mem_set_prod_eq, Metric.mem_closedBall] at z0s ⊢ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : z0 ∈ closedBall c0 r
⊢ (z0, z1) ∈ closedBall c0 r ×ˢ closedBall c1 r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : dist z0 c0 ≤ r
⊢ dist z0 c0 ≤ r ∧ dist z1 c1 ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : z0 ∈ closedBall c0 r
⊢ (z0, z1) ∈ closedBall c0 r ×ˢ closedBall c1 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.on0 | [85, 1] | [90, 19] | exact ⟨z0s, z1r⟩ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : dist z0 c0 ≤ r
⊢ dist z0 c0 ≤ r ∧ dist z1 c1 ≤ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0✝ z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z1r : z1 ∈ closedBall c1 r
z0 : ℂ
z0s : dist z0 c0 ≤ r
⊢ dist z0 c0 ≤ r ∧ dist z1 c1 ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.on1 | [93, 1] | [98, 19] | intro z1 z1s | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
⊢ AnalyticOn ℂ (fun z1 => f (z0, z1)) (closedBall c1 r) | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : z1 ∈ closedBall c1 r
⊢ AnalyticAt ℂ (fun z1 => f (z0, z1)) z1 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
⊢ AnalyticOn ℂ (fun z1 => f (z0, z1)) (closedBall c1 r)
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.on1 | [93, 1] | [98, 19] | apply h.fa1 z0 z1 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : z1 ∈ closedBall c1 r
⊢ AnalyticAt ℂ (fun z1 => f (z0, z1)) z1 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : z1 ∈ closedBall c1 r
⊢ (z0, z1) ∈ closedBall (c0, c1) r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : z1 ∈ closedBall c1 r
⊢ AnalyticAt ℂ (fun z1 => f (z0, z1)) z1
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.on1 | [93, 1] | [98, 19] | rw [← closedBall_prod_same] | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : z1 ∈ closedBall c1 r
⊢ (z0, z1) ∈ closedBall (c0, c1) r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : z1 ∈ closedBall c1 r
⊢ (z0, z1) ∈ closedBall c0 r ×ˢ closedBall c1 r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : z1 ∈ closedBall c1 r
⊢ (z0, z1) ∈ closedBall (c0, c1) r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.on1 | [93, 1] | [98, 19] | simp only [Set.prod_mk_mem_set_prod_eq, Metric.mem_closedBall] at z1s ⊢ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : z1 ∈ closedBall c1 r
⊢ (z0, z1) ∈ closedBall c0 r ×ˢ closedBall c1 r | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : dist z1 c1 ≤ r
⊢ dist z0 c0 ≤ r ∧ dist z1 c1 ≤ r | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : z1 ∈ closedBall c1 r
⊢ (z0, z1) ∈ closedBall c0 r ×ˢ closedBall c1 r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Har.on1 | [93, 1] | [98, 19] | exact ⟨z0r, z1s⟩ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : dist z1 c1 ≤ r
⊢ dist z0 c0 ≤ r ∧ dist z1 c1 ≤ r | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1✝ w0 w1 : ℂ
r r0 r1 b e : ℝ
h : Har f (closedBall (c0, c1) r)
z0r : z0 ∈ closedBall c0 r
z1 : ℂ
z1s : dist z1 c1 ≤ r
⊢ dist z0 c0 ≤ r ∧ dist z1 c1 ≤ r
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | generalize hu : min (r / 2) (e * r / b / 24) = u | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
hu : min (r / 2) (e * r / b / 24) = u
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | rw [hu] at wz | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
hu : min (r / 2) (e * r / b / 24) = u
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
wz : dist w z < min (r / 2) (e * r / b / 24)
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
hu : min (r / 2) (e * r / b / 24) = u
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4
TACTIC:
|
https://github.com/girving/ray.git | 0be790285dd0fce78913b0cb9bddaffa94bd25f9 | Ray/Hartogs/Hartogs.lean | Bounded.dist0 | [101, 1] | [136, 8] | have up : 0 < u := by
rw [← hu]; simp only [gt_iff_lt, lt_min_iff]
exact ⟨by bound, by bound⟩ | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
up : 0 < u
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4 | Please generate a tactic in lean4 to solve the state.
STATE:
E : Type
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℂ E
inst✝¹ : CompleteSpace E
inst✝ : SecondCountableTopology E
f : ℂ × ℂ → E
s : Set (ℂ × ℂ)
c0 c0' c1 z0 z1 w0 w1 : ℂ
r✝ r0 r1 b✝ e✝ : ℝ
h : Har f s
z w : ℂ × ℂ
b e r : ℝ
bp : 0 < b
ep : 0 < e
rp : 0 < r
rs : ball z (r / 2) ⊆ s
fb : ∀ z ∈ s, ‖f z‖ ≤ b
u : ℝ
wz : dist w z < u
hu : min (r / 2) (e * r / b / 24) = u
⊢ dist (f w) (f (z.1, w.2)) ≤ e / 4
TACTIC:
|
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