mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Module.Ball.Pointwise

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
⊢ closedBall 0 ‖c‖ = closedBall 0 ‖c‖

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ball.Pointwise

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : 0 ≤ r
⊢ closedBall 0 r = closedBall 0 r

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ball.Pointwise

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : 0 < r
x : E
⊢ ball x r = ball x r

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ball.Pointwise

Goal state before:
pp: E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
r : ℝ
hr : 0 ≤ r
x : E
⊢ closedBall x r = closedBall x r

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: α : Type u_1
inst✝ : SeminormedAddCommGroup α
r C : ℝ
f : ℕ → α
hf : ∀ (n : ℕ), ‖f n‖ ≤ C * r ^ n
n : ℕ
⊢ ‖f n‖ ≤ C * r ^ n

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : SeminormedAddCommGroup α
r C : ℝ
f : ℕ → α
hf : ∀ (n : ℕ), ‖f n‖ ≤ C * r ^ n
n : ℕ
⊢ ‖f n‖ ≤ C * r ^ n
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: α : Type u_1
inst✝ : SeminormedAddCommGroup α
r C : ℝ
f : ℕ → α
hr : r < 1
hf : ∀ (n : ℕ), ‖f n‖ ≤ C * r ^ n
a : α
ha : HasSum f a
n : ℕ
⊢ dist (∑ x ∈ Finset.range n, f x) a ≤ C * r ^ n / (1 - r)

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : SeminormedAddCommGroup α
r C : ℝ
f : ℕ → α
hr : r < 1
hf : ∀ (n : ℕ), ‖f n‖ ≤ C * r ^ n
a : α
ha : HasSum f a
n : ℕ
⊢ dist (∑ x ∈ Finset.range n, f x) a ≤ C * r ^ n / (1 - r)
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: case pos
α : Type u_1
inst✝ : SeminormedAddCommGroup α
C : ℝ
u : ℕ → α
N : ℕ
r : ℝ
hr₀ : 0 < r
hr₁ : r < 1
h : ∀ n ≥ N, ‖u n‖ ≤ C * r ^ n
v : ℕ → α := fun n ↦ if n < N then 0 else u n
hC : 0 ≤ C
this : ∀ n ≥ N, u n = v n
n : ℕ
H : n < N
⊢ 0 ≤ C * r ^ n

Tactic:
]

Goal state after:
case pos
α : Type u_1
inst✝ : SeminormedAddCommGroup α
C : ℝ
u : ℕ → α
N : ℕ
r : ℝ
hr₀ : 0 < r
hr₁ : r < 1
h : ∀ n ≥ N, ‖u n‖ ≤ C * r ^ n
v : ℕ → α := fun n ↦ if n < N then 0 else u n
hC : 0 ≤ C
this : ∀ n ≥ N, u n = v n
n : ℕ
H : n < N
⊢ 0 ≤ C * r ^ n
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: α : Type u_1
inst✝ : SeminormedAddCommGroup α
f : ℕ → α
h : CauchySeq fun n ↦ ∑ k ∈ Finset.range n, f k
⊢ ∃ C, ∀ (n : ℕ), ‖f n‖ ≤ C

Tactic:
obtain ⟨b, ⟨_, key, _⟩⟩ := cauchySeq_iff_le_tendsto_0.mp h

Goal state after:
α : Type u_1
inst✝ : SeminormedAddCommGroup α
f : ℕ → α
h : CauchySeq fun n ↦ ∑ k ∈ Finset.range n, f k
b : ℕ → ℝ
left✝ : ∀ (n : ℕ), 0 ≤ b n
key : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (∑ k ∈ Finset.range n, f k) (∑ k ∈ Finset.range m, f k) ≤ b N
right✝ : Tendsto b atTop (𝓝 0)
⊢ ∃ C, ∀ (n : ℕ), ‖f n‖ ≤ C
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: case pos
α : Type u_4
inst✝¹ : SeminormedAddCommGroup α
inst✝ : CompleteSpace α
f : ℕ → α
r : ℝ
hr₁ : r < 1
h : ∃ a, ∀ (b : ℕ), a ≤ b → ‖f (b + 1)‖ ≤ r * ‖f b‖
hr₀ : 0 ≤ r
⊢ Summable f

Tactic:
]

Goal state after:
case pos
α : Type u_4
inst✝¹ : SeminormedAddCommGroup α
inst✝ : CompleteSpace α
f : ℕ → α
r : ℝ
hr₁ : r < 1
h : ∃ a, ∀ (b : ℕ), a ≤ b → ‖f (b + 1)‖ ≤ r * ‖f b‖
hr₀ : 0 ≤ r
⊢ Summable f
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: case pos
α : Type u_4
inst✝¹ : SeminormedAddCommGroup α
inst✝ : CompleteSpace α
f : ℕ → α
r : ℝ
hr₁ : r < 1
hr₀ : 0 ≤ r
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → ‖f (b + 1)‖ ≤ r * ‖f b‖
⊢ Summable fun n ↦ f (n + N)

Tactic:
]

Goal state after:
case pos
α : Type u_4
inst✝¹ : SeminormedAddCommGroup α
inst✝ : CompleteSpace α
f : ℕ → α
r : ℝ
hr₁ : r < 1
hr₀ : 0 ≤ r
N : ℕ
hN : ∀ (b : ℕ), N ≤ b → ‖f (b + 1)‖ ≤ r * ‖f b‖
⊢ Summable fun n ↦ f (n + N)
Module: Mathlib.Topology.Connected.PathConnected

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y : X
F : Set X
f : X → Y
hf : IsInducing f
hx : x ∈ F
hy : y ∈ F
γ : Path (f x) (f y)
γ' : ↑I → X
hγ'F : ∀ (t : ↑I), γ' t ∈ F
hγ' : ∀ (t : ↑I), f (γ' t) = γ t
⊢ f x ⤳ f x

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Connected.PathConnected

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
x y : X
F : Set X
f : X → Y
hf : IsInducing f
hx : x ∈ F
hy : y ∈ F
γ : Path (f x) (f y)
γ' : ↑I → X
hγ'F : ∀ (t : ↑I), γ' t ∈ F
hγ' : ∀ (t : ↑I), f (γ' t) = γ t
h₀ : x ⤳ γ' 0
⊢ f y ⤳ f y

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Connected.PathConnected

Goal state before:
pp: case mp
X : Type u_1
inst✝ : TopologicalSpace X
x y : X
h : x ∈ pathComponent y
z : X
h' : z ∈ pathComponent x
⊢ y ∈ pathComponent z

Tactic:
]

Goal state after:
case mp
X : Type u_1
inst✝ : TopologicalSpace X
x y : X
h : x ∈ pathComponent y
z : X
h' : z ∈ pathComponent x
⊢ y ∈ pathComponent z
Module: Mathlib.Topology.Connected.PathConnected

Goal state before:
pp: case mpr
X : Type u_1
inst✝ : TopologicalSpace X
x y : X
h : x ∈ pathComponent y
z : X
h' : y ∈ pathComponent z
⊢ x ∈ pathComponent z

Tactic:
]

Goal state after:
case mpr
X : Type u_1
inst✝ : TopologicalSpace X
x y : X
h : x ∈ pathComponent y
z : X
h' : y ∈ pathComponent z
⊢ x ∈ pathComponent z
Module: Mathlib.Topology.CompactOpen

Goal state before:
pp: X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : LocallyCompactPair Y Z
f : C(X, Y)
g : C(Y, Z)
K : Set X
hK : IsCompact K
U : Set Z
hU : IsOpen[inst✝¹] U
hKU : MapsTo (⇑g ∘ ⇑f) K U
L : Set Y
hKL : MapsTo (⇑f) K (interior L)
hLc : IsCompact L
hLU : MapsTo (⇑g) L U
⊢ ∀ᶠ (a : C(X, Y) × C(Y, Z)) in 𝓝 (f, g), MapsTo (⇑(a.2.comp a.1)) K U

Tactic:
]

Goal state after:
X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : LocallyCompactPair Y Z
f : C(X, Y)
g : C(Y, Z)
K : Set X
hK : IsCompact K
U : Set Z
hU : IsOpen[inst✝¹] U
hKU : MapsTo (⇑g ∘ ⇑f) K U
L : Set Y
hKL : MapsTo (⇑f) K (interior L)
hLc : IsCompact L
hLU : MapsTo (⇑g) L U
⊢ ∀ᶠ (a : C(X, Y) × C(Y, Z)) in 𝓝 (f, g), MapsTo (⇑(a.2.comp a.1)) K U
Module: Mathlib.Topology.Connected.PathConnected

Goal state before:
pp: case right
X : Type u_1
inst✝ : TopologicalSpace X
F : Set X
x : X
x_in : x ∈ F
h : pathComponentIn F x = F
y : X
y_in : y ∈ pathComponentIn F x
⊢ JoinedIn F x y

Tactic:
]

Goal state after:
case right
X : Type u_1
inst✝ : TopologicalSpace X
F : Set X
x : X
x_in : x ∈ F
h : pathComponentIn F x = F
y : X
y_in : y ∈ pathComponentIn F x
⊢ JoinedIn F x y
Module: Mathlib.Topology.Connected.PathConnected

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
F : Set X
f : X → Y
hf : IsInducing f
x : X
hx : x ∈ F
y : X
hy : y ∈ F
⊢ JoinedIn F x y ↔ JoinedIn F x y

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Connected.PathConnected

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set Y
h : X ≃ₜ Y
⊢ IsPathConnected (⇑h.symm '' s) ↔ IsPathConnected s

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
s : Set Y
h : X ≃ₜ Y
⊢ IsPathConnected (⇑h.symm '' s) ↔ IsPathConnected s
Module: Mathlib.Topology.CompactOpen

Goal state before:
pp: α : Type u_1
X : Type u_2
Y : Type u_3
Z : Type u_4
T : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Z
inst✝¹ : TopologicalSpace T
K : Set X
U : Set Y
inst✝ : R0Space Y
f g : C(X, Y)
h : ⇑f ⤳ ⇑g
⊢ ⇑g ⤳ ⇑f

Tactic:
]

Goal state after:
α : Type u_1
X : Type u_2
Y : Type u_3
Z : Type u_4
T : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Z
inst✝¹ : TopologicalSpace T
K : Set X
U : Set Y
inst✝ : R0Space Y
f g : C(X, Y)
h : ⇑f ⤳ ⇑g
⊢ ⇑g ⤳ ⇑f
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: α : Type u_4
inst✝¹ : NormedAddCommGroup α
inst✝ : CompleteSpace α
f : ℕ → α
l : ℝ
hl₁ : l < 1
hf : ∀ᶠ (n : ℕ) in atTop, f n ≠ 0
h : Tendsto (fun n ↦ ‖f (n + 1)‖ / ‖f n‖) atTop (𝓝 l)
r : ℝ
hr₀ : l < r
hr₁ : r < 1
a✝¹ : ℕ
a✝ : ‖f (a✝¹ + 1)‖ / ‖f a✝¹‖ ≤ r
h₁ : f a✝¹ ≠ 0
⊢ ‖f (a✝¹ + 1)‖ / ‖f a✝¹‖ ≤ r

Tactic:
]

Goal state after:
α : Type u_4
inst✝¹ : NormedAddCommGroup α
inst✝ : CompleteSpace α
f : ℕ → α
l : ℝ
hl₁ : l < 1
hf : ∀ᶠ (n : ℕ) in atTop, f n ≠ 0
h : Tendsto (fun n ↦ ‖f (n + 1)‖ / ‖f n‖) atTop (𝓝 l)
r : ℝ
hr₀ : l < r
hr₁ : r < 1
a✝¹ : ℕ
a✝ : ‖f (a✝¹ + 1)‖ / ‖f a✝¹‖ ≤ r
h₁ : f a✝¹ ≠ 0
⊢ ‖f (a✝¹ + 1)‖ / ‖f a✝¹‖ ≤ r
Module: Mathlib.Topology.CompactOpen

Goal state before:
pp: α : Type u_1
X : Type u_2
Y : Type u_3
Z : Type u_4
T : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Z
inst✝¹ : TopologicalSpace T
K : Set X
U : Set Y
inst✝ : RegularSpace Y
f : C(X, Y)
⊢ ∀ (K : Set X),
    IsCompact K →
      ∀ (U : Set Y), IsOpen[inst✝³] U → MapsTo (⇑f) K U → ∀ᶠ (a : C(X, Y)) in (𝓝 f).lift' closure, MapsTo (⇑(id a)) K U

Tactic:
]

Goal state after:
α : Type u_1
X : Type u_2
Y : Type u_3
Z : Type u_4
T : Type u_5
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Z
inst✝¹ : TopologicalSpace T
K : Set X
U : Set Y
inst✝ : RegularSpace Y
f : C(X, Y)
⊢ ∀ (K : Set X),
    IsCompact K →
      ∀ (U : Set Y), IsOpen[inst✝³] U → MapsTo (⇑f) K U → ∀ᶠ (a : C(X, Y)) in (𝓝 f).lift' closure, MapsTo (⇑(id a)) K U
Module: Mathlib.Topology.Connected.PathConnected

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
x : X
⊢ IsPathConnected (pathComponentIn univ x)

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
x : X
⊢ IsPathConnected (pathComponentIn univ x)
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: α : Type u_4
inst✝ : SeminormedAddCommGroup α
f : ℕ → α
r : ℝ
hr : 1 < r
hf : ∃ᶠ (n : ℕ) in atTop, ‖f n‖ ≠ 0
h : ∃ a, ∀ (b : ℕ), a ≤ b → r * ‖f b‖ ≤ ‖f (b + 1)‖
⊢ ¬Summable f

Tactic:
]

Goal state after:
α : Type u_4
inst✝ : SeminormedAddCommGroup α
f : ℕ → α
r : ℝ
hr : 1 < r
hf : ∃ᶠ (n : ℕ) in atTop, ‖f n‖ ≠ 0
h : ∃ a, ∀ (b : ℕ), a ≤ b → r * ‖f b‖ ≤ ‖f (b + 1)‖
⊢ ¬Summable f
Module: Mathlib.Topology.Connected.PathConnected

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
U W : Set X
hW : IsPathConnected W
hWU : W ⊆ U
⊢ IsPathConnected W

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
U W : Set X
hW : IsPathConnected W
hWU : W ⊆ U
⊢ IsPathConnected W
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: α : Type u_4
inst✝ : SeminormedAddCommGroup α
f : ℕ → α
r : ℝ
hr : 1 < r
hf : ∀ (a : ℕ), ∃ b, a ≤ b ∧ ‖f b‖ ≠ 0
N₀ : ℕ
hN₀ : ∀ (b : ℕ), N₀ ≤ b → r * ‖f b‖ ≤ ‖f (b + 1)‖
⊢ ¬Summable f

Tactic:
]

Goal state after:
α : Type u_4
inst✝ : SeminormedAddCommGroup α
f : ℕ → α
r : ℝ
hr : 1 < r
hf : ∀ (a : ℕ), ∃ b, a ≤ b ∧ ‖f b‖ ≠ 0
N₀ : ℕ
hN₀ : ∀ (b : ℕ), N₀ ≤ b → r * ‖f b‖ ≤ ‖f (b + 1)‖
⊢ ¬Summable f
Module: Mathlib.Topology.CompactOpen

Goal state before:
pp: X : Type u_2
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : C(X, Y)
⊢ 𝓝 f = ⨅ s, ⨅ (_ : IsCompact s), comap (restrict s) (𝓝 (restrict s f))

Tactic:
]

Goal state after:
X : Type u_2
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : C(X, Y)
⊢ 𝓝 f = ⨅ s, ⨅ (_ : IsCompact s), comap (restrict s) (𝓝 (restrict s f))
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: α : Type u_4
inst✝ : SeminormedAddCommGroup α
f : ℕ → α
r : ℝ
hr : 1 < r
hf : ∀ (a : ℕ), ∃ b, a ≤ b ∧ ‖f b‖ ≠ 0
N₀ : ℕ
hN₀ : ∀ (b : ℕ), N₀ ≤ b → r * ‖f b‖ ≤ ‖f (b + 1)‖
N : ℕ
hNN₀ : N₀ ≤ N
hN : ‖f N‖ ≠ 0
⊢ ¬Summable fun n ↦ f (n + N)

Tactic:
]

Goal state after:
α : Type u_4
inst✝ : SeminormedAddCommGroup α
f : ℕ → α
r : ℝ
hr : 1 < r
hf : ∀ (a : ℕ), ∃ b, a ≤ b ∧ ‖f b‖ ≠ 0
N₀ : ℕ
hN₀ : ∀ (b : ℕ), N₀ ≤ b → r * ‖f b‖ ≤ ‖f (b + 1)‖
N : ℕ
hNN₀ : N₀ ≤ N
hN : ‖f N‖ ≠ 0
⊢ ¬Summable fun n ↦ f (n + N)
Module: Mathlib.Topology.CompactOpen

Goal state before:
pp: X : Type u_2
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
ι : Type u_6
l : Filter ι
F : ι → C(X, Y)
f : C(X, Y)
⊢ Tendsto F l (𝓝 f) ↔ ∀ (K : Set X), IsCompact K → Tendsto (fun i ↦ restrict K (F i)) l (𝓝 (restrict K f))

Tactic:
]

Goal state after:
X : Type u_2
Y : Type u_3
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
ι : Type u_6
l : Filter ι
F : ι → C(X, Y)
f : C(X, Y)
⊢ Tendsto F l (𝓝 f) ↔ ∀ (K : Set X), IsCompact K → Tendsto (fun i ↦ restrict K (F i)) l (𝓝 (restrict K f))
Module: Mathlib.Topology.Path

Goal state before:
pp: case pos
X : Type u_1
inst✝ : TopologicalSpace X
x y z : X
γ : Path x y
γ' : Path y z
t : ↑I
h✝ : ↑t ≤ 1 / 2
⊢ γ ⟨2 * ↑t, ?pos.ht✝⟩ = γ ⟨2 * ↑t, ⋯⟩

pp: case pos.ht
X : Type u_1
inst✝ : TopologicalSpace X
x y z : X
γ : Path x y
γ' : Path y z
t : ↑I
h✝ : ↑t ≤ 1 / 2
⊢ 2 * ↑t ∈ Icc 0 1

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Path

Goal state before:
pp: case neg
X : Type u_1
inst✝ : TopologicalSpace X
x y z : X
γ : Path x y
γ' : Path y z
t : ↑I
h✝ : ¬↑t ≤ 1 / 2
⊢ γ' ⟨2 * ↑t - 1, ?neg.ht✝⟩ = γ' ⟨2 * ↑t - 1, ⋯⟩

pp: case neg.ht
X : Type u_1
inst✝ : TopologicalSpace X
x y z : X
γ : Path x y
γ' : Path y z
t : ↑I
h✝ : ¬↑t ≤ 1 / 2
⊢ 2 * ↑t - 1 ∈ Icc 0 1

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Connected.PathConnected

Goal state before:
pp: case snoc.snoc
X : Type u_1
inst✝ : TopologicalSpace X
n✝ : ℕ
s : Set X
h : IsPathConnected s
n : ℕ
p : Fin n → X
y x : X
hx : x ∈ s
hp : ∀ (i : Fin n), p i ∈ s
hy : y ∈ s
γ₀ : Path (snoc p y 0) y
hγ₀s : range ⇑γ₀ ⊆ s
hγ₀p : ∀ (i : Fin n), p i ∈ range ⇑γ₀
γ₁ : Path y x
hγ₁ : range ⇑γ₁ ⊆ s
⊢ ∃ γ, range ⇑γ ⊆ s ∧ (∀ (i : Fin n), p i ∈ range ⇑γ) ∧ y ∈ range ⇑γ

Tactic:
]

Goal state after:
case snoc.snoc
X : Type u_1
inst✝ : TopologicalSpace X
n✝ : ℕ
s : Set X
h : IsPathConnected s
n : ℕ
p : Fin n → X
y x : X
hx : x ∈ s
hp : ∀ (i : Fin n), p i ∈ s
hy : y ∈ s
γ₀ : Path (snoc p y 0) y
hγ₀s : range ⇑γ₀ ⊆ s
hγ₀p : ∀ (i : Fin n), p i ∈ range ⇑γ₀
γ₁ : Path y x
hγ₁ : range ⇑γ₁ ⊆ s
⊢ ∃ γ, range ⇑γ ⊆ s ∧ (∀ (i : Fin n), p i ∈ range ⇑γ) ∧ y ∈ range ⇑γ
Module: Mathlib.Topology.CompactOpen

Goal state before:
pp: case mpr
X : Type u_2
Y : Type u_3
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : WeaklyLocallyCompactSpace X
inst✝¹ : T2Space Y
ι : Type u_6
l : Filter ι
inst✝ : l.NeBot
F : ι → C(X, Y)
f : (s : Set X) → IsCompact s → C(↑s, Y)
hf : ∀ (s : Set X) (a : IsCompact s), Tendsto (fun i ↦ restrict s (F i)) l (𝓝 (f s a))
h :
  ∀ (s₁ : Set X) (hs₁ : IsCompact s₁) (s₂ : Set X) (hs₂ : IsCompact s₂) (x : X) (hxs₁ : x ∈ s₁) (hxs₂ : x ∈ s₂),
    (f s₁ hs₁) ⟨x, hxs₁⟩ = (f s₂ hs₂) ⟨x, hxs₂⟩
⊢ ∀ (K : Set X),
    IsCompact K →
      Tendsto (fun i ↦ restrict K (F i)) l
        (𝓝 (restrict K (liftCover' (fun s ↦ ∀ ⦃f : Filter X⦄ [f.NeBot], f ≤ 𝓟 s → ∃ x ∈ s, ClusterPt x f) f h ⋯)))

Tactic:
]

Goal state after:
case mpr
X : Type u_2
Y : Type u_3
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : WeaklyLocallyCompactSpace X
inst✝¹ : T2Space Y
ι : Type u_6
l : Filter ι
inst✝ : l.NeBot
F : ι → C(X, Y)
f : (s : Set X) → IsCompact s → C(↑s, Y)
hf : ∀ (s : Set X) (a : IsCompact s), Tendsto (fun i ↦ restrict s (F i)) l (𝓝 (f s a))
h :
  ∀ (s₁ : Set X) (hs₁ : IsCompact s₁) (s₂ : Set X) (hs₂ : IsCompact s₂) (x : X) (hxs₁ : x ∈ s₁) (hxs₂ : x ∈ s₂),
    (f s₁ hs₁) ⟨x, hxs₁⟩ = (f s₂ hs₂) ⟨x, hxs₂⟩
⊢ ∀ (K : Set X),
    IsCompact K →
      Tendsto (fun i ↦ restrict K (F i)) l
        (𝓝 (restrict K (liftCover' (fun s ↦ ∀ ⦃f : Filter X⦄ [f.NeBot], f ≤ 𝓟 s → ∃ x ∈ s, ClusterPt x f) f h ⋯)))
Module: Mathlib.Topology.Path

Goal state before:
pp: case pos
X : Type u_1
inst✝ : TopologicalSpace X
x y z : X
γ : Path x y
γ' : Path y z
t : ↑I
h✝ : ↑(σ t) ≤ 1 / 2
h : 1 - ↑t ≤ 1 / 2
h₁ : ↑t ≤ 1 / 2
⊢ γ ⟨2 * ↑(σ t), ⋯⟩ = γ' (σ ⟨2 * ↑t, ⋯⟩)

Tactic:
]

Goal state after:
case pos
X : Type u_1
inst✝ : TopologicalSpace X
x y z : X
γ : Path x y
γ' : Path y z
t : ↑I
h✝ : ↑(σ t) ≤ 1 / 2
h : 1 - ↑t ≤ 1 / 2
h₁ : ↑t ≤ 1 / 2
⊢ γ ⟨2 * ↑(σ t), ⋯⟩ = γ' (σ ⟨2 * ↑t, ⋯⟩)
Module: Mathlib.Topology.Path

Goal state before:
pp: case neg
X : Type u_1
inst✝ : TopologicalSpace X
x y z : X
γ : Path x y
γ' : Path y z
t : ↑I
h✝ : ↑(σ t) ≤ 1 / 2
h : 1 - ↑t ≤ 1 / 2
h₁ : ¬↑t ≤ 1 / 2
⊢ γ ⟨2 * ↑(σ t), ⋯⟩ = γ (σ ⟨2 * ↑t - 1, ⋯⟩)

Tactic:
]

Goal state after:
case neg
X : Type u_1
inst✝ : TopologicalSpace X
x y z : X
γ : Path x y
γ' : Path y z
t : ↑I
h✝ : ↑(σ t) ≤ 1 / 2
h : 1 - ↑t ≤ 1 / 2
h₁ : ¬↑t ≤ 1 / 2
⊢ γ ⟨2 * ↑(σ t), ⋯⟩ = γ (σ ⟨2 * ↑t - 1, ⋯⟩)
Module: Mathlib.Topology.Path

Goal state before:
pp: case pos
X : Type u_1
inst✝ : TopologicalSpace X
x y z : X
γ : Path x y
γ' : Path y z
t : ↑I
h✝ : ¬↑(σ t) ≤ 1 / 2
h : ¬1 - ↑t ≤ 1 / 2
h₂ : ↑t ≤ 1 / 2
⊢ γ' ⟨2 * ↑(σ t) - 1, ⋯⟩ = γ' (σ ⟨2 * ↑t, ⋯⟩)

Tactic:
]

Goal state after:
case pos
X : Type u_1
inst✝ : TopologicalSpace X
x y z : X
γ : Path x y
γ' : Path y z
t : ↑I
h✝ : ¬↑(σ t) ≤ 1 / 2
h : ¬1 - ↑t ≤ 1 / 2
h₂ : ↑t ≤ 1 / 2
⊢ γ' ⟨2 * ↑(σ t) - 1, ⋯⟩ = γ' (σ ⟨2 * ↑t, ⋯⟩)
Module: Mathlib.Topology.Path

Goal state before:
pp: case neg
X : Type u_1
inst✝ : TopologicalSpace X
x y z : X
γ : Path x y
γ' : Path y z
t : ↑I
h✝ : ¬↑(σ t) ≤ 1 / 2
h : ¬1 - ↑t ≤ 1 / 2
h₂ : ¬↑t ≤ 1 / 2
⊢ γ' ⟨2 * ↑(σ t) - 1, ⋯⟩ = γ (σ ⟨2 * ↑t - 1, ⋯⟩)

Tactic:
]

Goal state after:
case neg
X : Type u_1
inst✝ : TopologicalSpace X
x y z : X
γ : Path x y
γ' : Path y z
t : ↑I
h✝ : ¬↑(σ t) ≤ 1 / 2
h : ¬1 - ↑t ≤ 1 / 2
h₂ : ¬↑t ≤ 1 / 2
⊢ γ' ⟨2 * ↑(σ t) - 1, ⋯⟩ = γ (σ ⟨2 * ↑t - 1, ⋯⟩)
Module: Mathlib.Topology.CompactOpen

Goal state before:
pp: case mpr
X : Type u_2
Y : Type u_3
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : WeaklyLocallyCompactSpace X
inst✝¹ : T2Space Y
ι : Type u_6
l : Filter ι
inst✝ : l.NeBot
F : ι → C(X, Y)
f : (s : Set X) → IsCompact s → C(↑s, Y)
hf : ∀ (s : Set X) (a : IsCompact s), Tendsto (fun i ↦ restrict s (F i)) l (𝓝 (f s a))
h :
  ∀ (s₁ : Set X) (hs₁ : IsCompact s₁) (s₂ : Set X) (hs₂ : IsCompact s₂) (x : X) (hxs₁ : x ∈ s₁) (hxs₂ : x ∈ s₂),
    (f s₁ hs₁) ⟨x, hxs₁⟩ = (f s₂ hs₂) ⟨x, hxs₂⟩
s : Set X
hs : IsCompact s
⊢ Tendsto (fun i ↦ restrict s (F i)) l (𝓝 (f s ?m.230))

pp: X : Type u_2
Y : Type u_3
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : WeaklyLocallyCompactSpace X
inst✝¹ : T2Space Y
ι : Type u_6
l : Filter ι
inst✝ : l.NeBot
F : ι → C(X, Y)
f : (s : Set X) → IsCompact s → C(↑s, Y)
hf : ∀ (s : Set X) (a : IsCompact s), Tendsto (fun i ↦ restrict s (F i)) l (𝓝 (f s a))
h :
  ∀ (s₁ : Set X) (hs₁ : IsCompact s₁) (s₂ : Set X) (hs₂ : IsCompact s₂) (x : X) (hxs₁ : x ∈ s₁) (hxs₂ : x ∈ s₂),
    (f s₁ hs₁) ⟨x, hxs₁⟩ = (f s₂ hs₂) ⟨x, hxs₂⟩
s : Set X
hs : IsCompact s
⊢ s ∈ fun s ↦ ∀ ⦃f : Filter X⦄ [f.NeBot], f ≤ 𝓟 s → ∃ x ∈ s, ClusterPt x f

Tactic:
]

Goal state after:
case mpr
X : Type u_2
Y : Type u_3
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : WeaklyLocallyCompactSpace X
inst✝¹ : T2Space Y
ι : Type u_6
l : Filter ι
inst✝ : l.NeBot
F : ι → C(X, Y)
f : (s : Set X) → IsCompact s → C(↑s, Y)
hf : ∀ (s : Set X) (a : IsCompact s), Tendsto (fun i ↦ restrict s (F i)) l (𝓝 (f s a))
h :
  ∀ (s₁ : Set X) (hs₁ : IsCompact s₁) (s₂ : Set X) (hs₂ : IsCompact s₂) (x : X) (hxs₁ : x ∈ s₁) (hxs₂ : x ∈ s₂),
    (f s₁ hs₁) ⟨x, hxs₁⟩ = (f s₂ hs₂) ⟨x, hxs₂⟩
s : Set X
hs : IsCompact s
⊢ Tendsto (fun i ↦ restrict s (F i)) l (𝓝 (f s ?m.230))

X : Type u_2
Y : Type u_3
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : WeaklyLocallyCompactSpace X
inst✝¹ : T2Space Y
ι : Type u_6
l : Filter ι
inst✝ : l.NeBot
F : ι → C(X, Y)
f : (s : Set X) → IsCompact s → C(↑s, Y)
hf : ∀ (s : Set X) (a : IsCompact s), Tendsto (fun i ↦ restrict s (F i)) l (𝓝 (f s a))
h :
  ∀ (s₁ : Set X) (hs₁ : IsCompact s₁) (s₂ : Set X) (hs₂ : IsCompact s₂) (x : X) (hxs₁ : x ∈ s₁) (hxs₂ : x ∈ s₂),
    (f s₁ hs₁) ⟨x, hxs₁⟩ = (f s₂ hs₂) ⟨x, hxs₂⟩
s : Set X
hs : IsCompact s
⊢ s ∈ fun s ↦ ∀ ⦃f : Filter X⦄ [f.NeBot], f ≤ 𝓟 s → ∃ x ∈ s, ClusterPt x f
Module: Mathlib.Topology.Connected.PathConnected

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
n : ℕ
s : Set X
h : IsPathConnected s
p : Fin (n + 1) → X
hp : ∀ (i : Fin (n + 1)), p i ∈ s
γ : Path (p 0) (p (last n))
h₁ : ∀ (y : ↑I), γ y ∈ s
h₂ : ∀ (i : Fin (n + 1)), ∃ y, γ y = p i
⊢ ∃ γ t, (∀ (t : ↑I), γ t ∈ s) ∧ ∀ (i : Fin (n + 1)), γ (t i) = p i

Tactic:
]

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
n : ℕ
s : Set X
h : IsPathConnected s
p : Fin (n + 1) → X
hp : ∀ (i : Fin (n + 1)), p i ∈ s
γ : Path (p 0) (p (last n))
h₁ : ∀ (y : ↑I), γ y ∈ s
h₂ : ∀ (i : Fin (n + 1)), ∃ y, γ y = p i
⊢ ∃ γ t, (∀ (t : ↑I), γ t ∈ s) ∧ ∀ (i : Fin (n + 1)), γ (t i) = p i
Module: Mathlib.Topology.Connected.PathConnected

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
n : ℕ
s : Set X
h : IsPathConnected s
p : Fin (n + 1) → X
hp : ∀ (i : Fin (n + 1)), p i ∈ s
γ : Path (p 0) (p (last n))
h₁ : range ⇑γ ⊆ s
h₂ : ∀ (i : Fin (n + 1)), ∃ y, γ y = p i
⊢ ∃ γ t, (∀ (t : ↑I), γ t ∈ s) ∧ ∀ (i : Fin (n + 1)), γ (t i) = p i

Tactic:
rw [range_subset_iff] at h₁

Goal state after:
X : Type u_1
inst✝ : TopologicalSpace X
n : ℕ
s : Set X
h : IsPathConnected s
p : Fin (n + 1) → X
hp : ∀ (i : Fin (n + 1)), p i ∈ s
γ : Path (p 0) (p (last n))
h₁ : ∀ (y : ↑I), γ y ∈ s
h₂ : ∀ (i : Fin (n + 1)), ∃ y, γ y = p i
⊢ ∃ γ t, (∀ (t : ↑I), γ t ∈ s) ∧ ∀ (i : Fin (n + 1)), γ (t i) = p i
Module: Mathlib.Analysis.SpecificLimits.Normed

Goal state before:
pp: α : Type u_4
inst✝ : SeminormedAddCommGroup α
f : ℕ → α
l : ℝ
hl : 1 < l
h : Tendsto (fun n ↦ ‖f (n + 1)‖ / ‖f n‖) atTop (𝓝 l)
a✝ : ℕ
hn : 1 ≤ 0
hc : ‖f a✝‖ = 0
⊢ False

Tactic:
]

Goal state after:
α : Type u_4
inst✝ : SeminormedAddCommGroup α
f : ℕ → α
l : ℝ
hl : 1 < l
h : Tendsto (fun n ↦ ‖f (n + 1)‖ / ‖f n‖) atTop (𝓝 l)
a✝ : ℕ
hn : 1 ≤ 0
hc : ‖f a✝‖ = 0
⊢ False
Module: Mathlib.Topology.CompactOpen

Goal state before:
pp: X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : LocallyCompactSpace (X × Y)
⊢ Continuous[instTopologicalSpaceProd, inst✝¹]
    ((Function.uncurry fun x y ↦ (Function.uncurry (fun x y ↦ x.curry y) x) y) ∘
      ⇑(Homeomorph.prodAssoc C(X × Y, Z) X Y).symm)

Tactic:
]

Goal state after:
X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : TopologicalSpace Z
inst✝ : LocallyCompactSpace (X × Y)
⊢ Continuous[instTopologicalSpaceProd, inst✝¹]
    ((Function.uncurry fun x y ↦ (Function.uncurry (fun x y ↦ x.curry y) x) y) ∘
      ⇑(Homeomorph.prodAssoc C(X × Y, Z) X Y).symm)
Module: Mathlib.Topology.Connected.PathConnected

Goal state before:
pp: X : Type u_1
inst✝ : TopologicalSpace X
F : Set X
⊢ IsPathConnected F ↔ IsPathConnected F

Tactic:
]

Goal state after:
Module: Mathlib.Topology.Connected.PathConnected

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
hf : Continuous[inst✝², inst✝¹] f
⊢ IsPathConnected (f '' univ)

Tactic:
]

Goal state after:
X : Type u_1
Y : Type u_2
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : PathConnectedSpace X
f : X → Y
hf : Continuous[inst✝², inst✝¹] f
⊢ IsPathConnected (f '' univ)
Module: Mathlib.Topology.CompactOpen

Goal state before:
pp: X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Z
inst✝¹ : LocallyCompactSpace X
inst✝ : LocallyCompactSpace Y
⊢ Continuous[instTopologicalSpaceProd, inst✝²]
    ((Function.uncurry fun x y ↦ x.uncurry y) ∘ ⇑(Homeomorph.prodAssoc C(X, C(Y, Z)) X Y))

Tactic:
]

Goal state after:
X : Type u_2
Y : Type u_3
Z : Type u_4
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : TopologicalSpace Z
inst✝¹ : LocallyCompactSpace X
inst✝ : LocallyCompactSpace Y
⊢ Continuous[instTopologicalSpaceProd, inst✝²]
    ((Function.uncurry fun x y ↦ x.uncurry y) ∘ ⇑(Homeomorph.prodAssoc C(X, C(Y, Z)) X Y))