mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
x : X
y : Y
⊢ 1 ≤ Sum.dist (Sum.inr y) (Sum.inl x)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
x y : (n : ℕ) → E n
n : ℕ
⊢ y ∈ cylinder x n ↔ cylinder y n = cylinder x n

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
x y : (n : ℕ) → E n
n : ℕ
⊢ y ∈ cylinder x n ↔ x ∈ cylinder y n

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: a b : ℝ
x y z : ↑(Icc a b)
s t : ↑unitInterval
⊢ convexComb (convexComb x y s) z t =
    convexComb x (convexComb y z (convexComb_assoc_coeff₂' s t)) (convexComb_assoc_coeff₁' s t)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Metrizable.CompletelyMetrizable

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun x y ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
x : X
⊢ 0 < 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
x y : (n : ℕ) → E n
hne : x ≠ y
i : ℕ
⊢ x ∈ cylinder y i ↔ i ≤ firstDiff x y

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
x y : (n : ℕ) → E n
n : ℕ
hn : n ≤ firstDiff x y
⊢ cylinder x n = cylinder y n

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: a b : ℝ
x y z : ↑(Icc a b)
hxy : x ≤ y
hyz : y ≤ z
⊢ 0 ≤ (↑y - ↑x) / (↑z - ↑x)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Metrizable.CompletelyMetrizable

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
m : MetricSpace X :=
  { dist := fun x y ↦ if x = y then 0 else 1, dist_self := ⋯, dist_comm := ⋯, dist_triangle := ⋯, edist_dist := ⋯,
    uniformity_dist := ⋯, cobounded_sets := ⋯, eq_of_dist_eq_zero := ⋯ }
u : ℕ → X
hu : ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) (u N) < ε
⊢ 1 > 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: a b : ℝ
x y z : ↑(Icc a b)
hxy : x ≤ y
hyz : y ≤ z
⊢ (↑y - ↑x) / (↑z - ↑x) ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
x : (n : ℕ) → E n
n : ℕ
⊢ ⋃ k, cylinder (update x n k) (n + 1) = cylinder x n

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Metrizable.CompletelyMetrizable

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : DiscreteTopology X
⊢ IsCompletelyMetrizableSpace X

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
x : (n : ℕ) → E n
n : ℕ
y : E n
i : ℕ
hi : i < n
⊢ update x n y i = x i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: α : Type u_2
x : ℕ → α
n : ℕ
⊢ (res x n).length = n

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Polish

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : SeparableSpace α
inst✝ : IsCompletelyMetrizableSpace α
⊢ PolishSpace α

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Polish

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PolishSpace β
f : α → β
hf : IsClosedEmbedding f
this✝¹ : UpgradedIsCompletelyMetrizableSpace β := upgradeIsCompletelyMetrizable β
this✝ : MetricSpace α := IsEmbedding.comapMetricSpace f ⋯
this : SecondCountableTopology α
⊢ CompleteSpace α

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Polish

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : PolishSpace β
f : α → β
hf : IsClosedEmbedding f
⊢ PolishSpace α

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Polish

Goal state before:
pp: α : Type u_1
ι : Type u_3
inst✝ : Countable ι
t : ι → TopologicalSpace α
ht₀ : ∃ t₀, T2Space α ∧ ∀ (i : ι), t i ≤ t₀
ht : ∀ (i : ι), ∃ u, CompleteSpace α ∧ (𝓤 α).IsCountablyGenerated ∧ u.toTopologicalSpace = t i
⊢ ∃ u, CompleteSpace α ∧ (𝓤 α).IsCountablyGenerated ∧ u.toTopologicalSpace = ⨅ i, t i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: a b : ℝ
x y z : ↑(Icc a b)
hxy : x ≤ y
hyz : y ≤ z
⊢ y = convexComb x z ⟨(↑y - ↑x) / (↑z - ↑x), ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Polish

Goal state before:
pp: α : Type u_1
ι : Type u_3
inst✝ : Countable ι
t : ι → TopologicalSpace α
ht : ∀ (i : ι), PolishSpace α
i₀ : ι
hi₀ : ∀ (i : ι), t i ≤ t i₀
u : UniformSpace α
hcomp : CompleteSpace α
hcount : (𝓤 α).IsCountablyGenerated
htop : u.toTopologicalSpace = ⨅ i, t i
this : SecondCountableTopology α
⊢ T1Space α

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Polish

Goal state before:
pp: α : Type u_1
ι : Type u_3
inst✝ : Countable ι
t : ι → TopologicalSpace α
ht₀ : ∃ i₀, ∀ (i : ι), t i ≤ t i₀
ht : ∀ (i : ι), PolishSpace α
⊢ PolishSpace α

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: α : Type u_2
x y : ℕ → α
n : ℕ
⊢ res x n = res y n ↔ ∀ ⦃m : ℕ⦄, m < n → x m = y m

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: α : Type u_2
⊢ Injective res

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: α : Type u_2
x : ℕ → α
n : ℕ
⊢ cylinder x n = {y | res y n = res x n}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
x y : (n : ℕ) → E n
h : x ≠ y
⊢ dist x y = (1 / 2) ^ firstDiff x y

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
x : (n : ℕ) → E n
⊢ dist x x = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
x y : (n : ℕ) → E n
⊢ dist x y = dist y x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: ι : Sort u_1
a b : ℝ
h : a ≤ b
c : ι → Set ↑(Icc a b)
hc₁ : ∀ (i : ι), IsOpen (c i)
hc₂ : univ ⊆ ⋃ i, c i
⊢ ∃ t, ↑(t 0) = a ∧ Monotone t ∧ (∃ m, ∀ n ≥ m, ↑(t n) = b) ∧ ∀ (n : ℕ), ∃ i, Icc (t n) (t (n + 1)) ⊆ c i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
x y : (n : ℕ) → E n
⊢ 0 ≤ dist x y

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
x y : (n : ℕ) → E n
⊢ dist x y ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
s : Set ((X ⊕ Y) × (X ⊕ Y))
⊢ s ∈ 𝓤 (X ⊕ Y) ↔ ∃ ε > 0, ∀ (a b : X ⊕ Y), Sum.dist a b < ε → (a, b) ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
x y z : (n : ℕ) → E n
⊢ dist x z ≤ max (dist x y) (dist y z)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
x : X ⊕ Y
⊢ Sum.dist x x = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
x y : (n : ℕ) → E n
hxy : dist x y = 0
⊢ x = y

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
p : X
q : Y
x✝ : X ⊕ Y
⊢ ∀ (p_1 q_1 : Unit), |dist ⋯.some ⋯.some - dist ⋯.some ⋯.some| ≤ 2 * 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
p : X
q : Y
x✝ : X ⊕ Y
⊢ Sum.dist (Sum.inl p) x✝ ≤ Sum.dist (Sum.inl p) (Sum.inr q) + Sum.dist (Sum.inr q) x✝

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
x y : (n : ℕ) → E n
n : ℕ
hne : y ≠ x
this : (∀ i < n, y i = x i) ↔ n ≤ firstDiff y x
⊢ y ∈ cylinder x n ↔ dist y x ≤ (1 / 2) ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: ι : Sort u_1
c : ι → Set ↑I
hc₁ : ∀ (i : ι), IsOpen (c i)
hc₂ : univ ⊆ ⋃ i, c i
⊢ ∃ t, t 0 = 0 ∧ Monotone t ∧ (∃ n, ∀ m ≥ n, t m = 1) ∧ ∀ (n : ℕ), ∃ i, Icc (t n) (t (n + 1)) ⊆ c i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
x✝ : X ⊕ Y
q : X
r : Y
⊢ ∀ (p q_1 : Unit), |dist ⋯.some ⋯.some - dist ⋯.some ⋯.some| ≤ 2 * 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
x✝ : X ⊕ Y
q : X
r : Y
⊢ Sum.dist x✝ (Sum.inr r) ≤ Sum.dist x✝ (Sum.inl q) + Sum.dist (Sum.inl q) (Sum.inr r)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
x y : (n : ℕ) → E n
n : ℕ
⊢ y ∈ cylinder x n ↔ dist y x ≤ (1 / 2) ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
p : Y
x✝ : X ⊕ Y
r : X
⊢ ∀ (p_1 q : Unit), |dist ⋯.some ⋯.some - dist ⋯.some ⋯.some| ≤ 2 * 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
x y : (n : ℕ) → E n
n : ℕ
h : dist x y < (1 / 2) ^ n
i : ℕ
hi : i ≤ n
hne : x ≠ y
⊢ n < firstDiff x y

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
p : Y
x✝ : X ⊕ Y
r : X
⊢ Sum.dist (Sum.inr p) (Sum.inl r) ≤ Sum.dist (Sum.inr p) x✝ + Sum.dist x✝ (Sum.inl r)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
x y : (n : ℕ) → E n
n : ℕ
h : dist x y < (1 / 2) ^ n
i : ℕ
hi : i ≤ n
⊢ x i = y i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
x y : (n : ℕ) → E n
n : ℕ
h : dist x y < (1 / 2) ^ n
i : ℕ
hi : i ≤ n
⊢ x i = y i

Tactic:
by
  rcases eq_or_ne x y with (rfl | hne)
  · rfl
  have : n < firstDiff x y := by
    simpa [dist_eq_of_ne hne, inv_lt_inv₀, pow_lt_pow_iff_right₀, one_lt_two] using h
  exact apply_eq_of_lt_firstDiff (hi.trans_lt this)

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
p q : X ⊕ Y
h : Sum.dist p q = 0
⊢ p = q

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
h : i = j
⊢ E j = E i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x y : E i
⊢ dist ⟨i, x⟩ ⟨i, y⟩ = dist x y

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i j : ι
h : i ≠ j
x : E i
y : E j
⊢ 1 ≤ dist ⟨i, x⟩ ⟨j, y⟩

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
x y : (i : ι) × E i
h : dist x y < 1
⊢ x.fst = y.fst

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Polish

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : MetricSpace α
s : Opens α
t : Set s.CompleteCopy
h : ∀ x ∈ t, ∃ ε > 0, ∀ (y : s.CompleteCopy), dist x y < ε → y ∈ t
x : s.CompleteCopy
hx : x ∈ t
ε : ℝ
ε0 : ε > 0
hε : ∀ (y : s.CompleteCopy), dist ↑x ↑y + |(infDist (↑x) (↑s)ᶜ)⁻¹ - (infDist (↑y) (↑s)ᶜ)⁻¹| < ε → y ∈ t
⊢ Tendsto (fun y ↦ dist ↑x ↑y + |(infDist (↑x) (↑s)ᶜ)⁻¹ - (infDist (↑y) (↑s)ᶜ)⁻¹|) (𝓝 x)
    (𝓝 (dist ↑x ↑x + |(infDist (↑x) (↑s)ᶜ)⁻¹ - (infDist (↑x) (↑s)ᶜ)⁻¹|))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
z : E i
hij : i ≠ j
⊢ dist x z ≤ dist x ⋯.some + 0 + 0 + (0 + 0 + dist ⋯.some z)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
z : E i
hij : i ≠ j
⊢ dist x ⋯.some + 0 + 0 + (0 + 0 + dist ⋯.some z) ≤
    dist x ⋯.some + 1 + dist ⋯.some y + (dist y ⋯.some + 1 + dist ⋯.some z)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: ι : Sort u_1
c : ι → Set (↑I × ↑I)
hc₁ : ∀ (i : ι), IsOpen (c i)
hc₂ : univ ⊆ ⋃ i, c i
⊢ ∃ t,
    t 0 = 0 ∧
      Monotone t ∧ (∃ n, ∀ m ≥ n, t m = 1) ∧ ∀ (n m : ℕ), ∃ i, Icc (t n) (t (n + 1)) ×ˢ Icc (t m) (t (m + 1)) ⊆ c i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: x : ↑unitInterval
⊢ 0 ≤ ↑x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Polish

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : MetricSpace α
s : Opens α
⊢ MetricSpace s.CompleteCopy

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : IsTopologicalRing 𝕜
a b : 𝕜
h : 0 < a
⊢ ⇑(affineHomeomorph a b ⋯) '' Icc 0 1 = Icc b (a + b)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
k : ι
z : E k
hik : i ≠ k
y : E i
⊢ dist x ⋯.some + 1 + dist ⋯.some z ≤ dist x y + dist y ⋯.some + 1 + dist ⋯.some z

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
k : ι
z : E k
hik : i ≠ k
y : E i
⊢ dist x y + dist y ⋯.some + 1 + dist ⋯.some z = dist x y + (dist y ⋯.some + 1 + dist ⋯.some z)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Polish

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
⊢ ∀ (n : ℕ), 0 < (1 / 2) ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
hij : i ≠ j
z : E j
hik : i ≠ j
⊢ dist x ⋯.some + 1 + dist ⋯.some z ≤ dist x ⋯.some + 1 + (dist ⋯.some y + dist y z)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
hij : i ≠ j
z : E j
hik : i ≠ j
⊢ dist x ⋯.some + 1 + (dist ⋯.some y + dist y z) = dist x ⋯.some + 1 + dist ⋯.some y + dist y z

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
α : Type u_2
inst✝ : PseudoMetricSpace α
f : ((n : ℕ) → E n) → α
⊢ (∀ (x y : (n : ℕ) → E n), dist (f x) (f y) ≤ dist x y) ↔
    ∀ (x y : (n : ℕ) → E n) (n : ℕ), y ∈ cylinder x n → dist (f x) (f y) ≤ (1 / 2) ^ n

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : IsTopologicalRing 𝕜
a b : 𝕜
h : a < b
⊢ ↑(Icc a b) ≃ₜ ↑(Icc 0 1)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: ⊢ Continuous toNNReal

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
k : ι
z : E k
hik : i ≠ k
hij : i ≠ j
hjk : j ≠ k
⊢ dist x ⋯.some + 1 + dist ⋯.some z = dist x ⋯.some + 1 + 0 + (0 + 0 + dist ⋯.some z)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: x : ↑I
⊢ toNNReal x + toNNReal (σ x) = 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
x : (n : ℕ) → E n
n : ℕ
⊢ IsOpen[Pi.topologicalSpace] (cylinder x n)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Polish

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → s.CompleteCopy
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (1 / 2) ^ N
⊢ 0 ≤ 1 / 2

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
k : ι
z : E k
hik : i ≠ k
hij : i ≠ j
hjk : j ≠ k
⊢ dist x ⋯.some + 1 + 0 + (0 + 0 + dist ⋯.some z) ≤
    dist x ⋯.some + 1 + dist ⋯.some y + (dist y ⋯.some + 1 + dist ⋯.some z)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: x : ↑I
⊢ toNNReal (σ x) + toNNReal x = 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Polish

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → s.CompleteCopy
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (1 / 2) ^ N
⊢ 1 / 2 < 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
x y z : (i : ι) × E i
⊢ dist x z ≤ dist x y + dist y z

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Polish

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → s.CompleteCopy
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (1 / 2) ^ N
⊢ CauchySeq fun n ↦ ↑(u n)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.CantorScheme

Goal state before:
pp: β : Type u_1
α : Type u_2
A : List β → Set α
x : ↑(inducedMap A).fst
n : ℕ
⊢ (inducedMap A).snd x ∈ A (res (↑x) n)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.CantorScheme

Goal state before:
pp: β : Type u_1
α : Type u_2
A : List β → Set α
hA : CantorScheme.Disjoint A
⊢ Injective (inducedMap A).snd

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.CantorScheme

Goal state before:
pp: β : Type u_1
α : Type u_2
A : List β → Set α
inst✝ : PseudoMetricSpace α
ε : ℝ
ε_pos : 0 < ε
x : ℕ → β
hA : ∀ ε > 0, ∃ N, ∀ n ≥ N, Metric.ediam (A (res x n)) ≤ ε
⊢ ENNReal.ofReal (ε / 2) > 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Polish

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → s.CompleteCopy
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (1 / 2) ^ N
A : CauchySeq fun n ↦ ↑(u n)
x : α
xlim : Tendsto (fun n ↦ ↑(u n)) atTop (𝓝 x)
xs : x ∉ s
⊢ ∃ C, ∀ (n : ℕ), 1 / infDist (↑(u n)) (↑s)ᶜ < C

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
x : (n : ℕ) → E n
u : Set ((n : ℕ) → E n)
hx : x ∈ u
u_open : IsOpen[Pi.topologicalSpace] u
U : (i : ℕ) → Set (E i)
F : Finset ℕ
xU : x ∈ (↑F).pi U
Uu : (↑F).pi U ⊆ u
n : ℕ
hn : n ∈ upperBounds ↑F
y : (n : ℕ) → E n
hy : y ∈ cylinder x (n + 1)
this : ∀ i ∈ F, y i ∈ U i
⊢ y ∈ (↑F).pi U

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.CantorScheme

Goal state before:
pp: β : Type u_1
α : Type u_2
A : List β → Set α
inst✝ : PseudoMetricSpace α
hA : VanishingDiam A
ε : ℝ
ε_pos : 0 < ε
x : ℕ → β
⊢ ∃ n, ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Polish

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → s.CompleteCopy
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (1 / 2) ^ N
A : CauchySeq fun n ↦ ↑(u n)
x : α
xlim : Tendsto (fun n ↦ ↑(u n)) atTop (𝓝 x)
xs : x ∉ s
C : ℝ
hC : ∀ (n : ℕ), 1 / infDist (↑(u n)) (↑s)ᶜ < C
Cpos : 0 < C
y : α
⊢ y ∈ s ↔ 0 < infDist y (↑s)ᶜ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
s : Set ((i : ι) × E i)
⊢ IsOpen[instTopologicalSpaceSigma] s ↔ ∀ x ∈ s, ∃ ε > 0, ∀ (y : (i : ι) × E i), dist x y < ε → y ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.CantorScheme

Goal state before:
pp: β : Type u_1
α : Type u_2
A : List β → Set α
inst✝² : PseudoMetricSpace α
inst✝¹ : TopologicalSpace β
inst✝ : DiscreteTopology β
hA : VanishingDiam A
x : ↑(inducedMap A).fst
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res (↑x) n), ∀ z ∈ A (res (↑x) n), dist y z < ε
⊢ x ∈ Subtype.val ⁻¹' cylinder (↑x) n

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: case h_nhds
E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
x : (n : ℕ) → E n
u : Set ((n : ℕ) → E n)
hx : x ∈ u
u_open : IsOpen[Pi.topologicalSpace] u
U : (i : ℕ) → Set (E i)
F : Finset ℕ
Uu : (↑F).pi U ⊆ u
n : ℕ
hn : n ∈ upperBounds ↑F
y : (n : ℕ) → E n
hy : y ∈ cylinder x (n + 1)
i : ℕ
hi : i ∈ F
this : y i = x i
xU : ∀ i ∈ F, x i ∈ U i
⊢ x i ∈ U i

Tactic:
exact xU i hi

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
⊢ IsTopologicalBasis {s | ∃ x n, s = cylinder x n}

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Polish

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
u : ℕ → s.CompleteCopy
hu : ∀ (N n m : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) < (1 / 2) ^ N
A : CauchySeq fun n ↦ ↑(u n)
x : α
xlim : Tendsto (fun n ↦ ↑(u n)) atTop (𝓝 x)
xs : x ∉ s
C : ℝ
hC : ∀ (n : ℕ), 1 / infDist (↑(u n)) (↑s)ᶜ < C
Cpos : 0 < C
Hmem : ∀ {y : α}, y ∈ s ↔ 0 < infDist y (↑s)ᶜ
n : ℕ
⊢ 1 / C ≤ infDist (↑(u n)) (↑s)ᶜ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
i : ι
x : E i
j : ι
y : E j
hij : i ≠ j
h : Sigma.dist ⟨i, x⟩ ⟨j, y⟩ = 0
⊢ Sigma.dist ⟨i, x⟩ ⟨j, y⟩ < 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Polish

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MetricSpace α
s : Opens α
inst✝ : CompleteSpace α
⊢ CompleteSpace s.CompleteCopy

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
⊢ MetricSpace ((i : ι) × E i)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.Polish

Goal state before:
pp: α : Type u_3
inst✝¹ : TopologicalSpace α
inst✝ : PolishSpace α
s : Set α
hs : IsOpen[inst✝¹] s
⊢ PolishSpace ↑s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: E : ℕ → Type u_1
inst✝¹ : (n : ℕ) → TopologicalSpace (E n)
inst✝ : ∀ (n : ℕ), DiscreteTopology (E n)
s : Set ((n : ℕ) → E n)
hs : IsOpen[Pi.topologicalSpace] s
x : (n : ℕ) → E n
hx : x ∈ s
y : (n : ℕ) → E n
n : ℕ
h'x : x ∈ cylinder y n
h's : cylinder x n ⊆ s
⊢ (1 / 2) ^ n > 0

Tactic:
by

Goal state after: