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

Goal state before:
pp: case _a
α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
s : Set α
hs : IsCompact s
hf' : ContinuousOn f s
ε : (x : α) → x ∈ s → ℝ
hε : ∀ (x : α) (a : x ∈ s), ε x a > 0
K : (x : α) → x ∈ s → ℝ≥0
hf : ∀ (x : α) (a : x ∈ s), LipschitzOnWith (K x a) f (ball x (ε x a) ∩ s)
K' : (x : α) → x ∈ s → ℝ≥0
hK' : ∀ (x : α) (hx : x ∈ s), ∀ y ∈ s.diff (ball x (ε x hx)), edist (f x) (f y) ≤ ↑(K' x hx) * edist x y
t : Finset ↑s
ht : s ⊆ ⋃ x ∈ t, s ∩ ball (↑x) (ε ↑x ⋯ / 2)
x : α
hx : x ∈ s
y : α
hy : y ∈ s
z : ↑s
hz : z ∈ t
hx' : x ∈ s ∩ ball (↑z) (ε ↑z ⋯ / 2)
hy' : y ∉ ball (↑z) (ε ↑z ⋯)
⊢ K ↑z ⋯ + 2 * K' ↑z ⋯ ≤ t.sup fun i ↦ K ↑i ⋯ + 2 * K' ↑i ⋯

Tactic:
;

Goal state after:
case _a
α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
s : Set α
hs : IsCompact s
hf' : ContinuousOn f s
ε : (x : α) → x ∈ s → ℝ
hε : ∀ (x : α) (a : x ∈ s), ε x a > 0
K : (x : α) → x ∈ s → ℝ≥0
hf : ∀ (x : α) (a : x ∈ s), LipschitzOnWith (K x a) f (ball x (ε x a) ∩ s)
K' : (x : α) → x ∈ s → ℝ≥0
hK' : ∀ (x : α) (hx : x ∈ s), ∀ y ∈ s.diff (ball x (ε x hx)), edist (f x) (f y) ≤ ↑(K' x hx) * edist x y
t : Finset ↑s
ht : s ⊆ ⋃ x ∈ t, s ∩ ball (↑x) (ε ↑x ⋯ / 2)
x : α
hx : x ∈ s
y : α
hy : y ∈ s
z : ↑s
hz : z ∈ t
hx' : x ∈ s ∩ ball (↑z) (ε ↑z ⋯ / 2)
hy' : y ∉ ball (↑z) (ε ↑z ⋯)
⊢ K ↑z ⋯ + 2 * K' ↑z ⋯ ≤ t.sup fun i ↦ K ↑i ⋯ + 2 * K' ↑i ⋯
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: i : ↑I
hi : 0 < i
j k : ↑I
hjk : j < k
⊢ ↑i * ↑j < ↑i * ↑k

Tactic:
;

Goal state after:
i : ↑I
hi : 0 < i
j k : ↑I
hjk : j < k
⊢ ↑i * ↑j < ↑i * ↑k
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
Φ : Z → X
Ψ : Z → Y
ε : ℝ
x : Y
y : X
⊢ ε ≤ glueDist Φ Ψ ε (Sum.inl y) (Sum.inr x)

Tactic:
;

Goal state after:
X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
Φ : Z → X
Ψ : Z → Y
ε : ℝ
x : Y
y : X
⊢ ε ≤ glueDist Φ Ψ ε (Sum.inl y) (Sum.inr x)
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: case inr
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : LinearOrder α
inst✝ : IsOrderedAddMonoid α
a b c d : α
h : a ≤ b
this : ∀ {c d : α}, d ≤ c → |↑(projIcc a b h c) - ↑(projIcc a b h d)| ≤ |c - d|
hdc : ¬d ≤ c
⊢ |↑(projIcc a b h d) - ↑(projIcc a b h c)| ≤ |d - c|

Tactic:
;

Goal state after:
case inr
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : LinearOrder α
inst✝ : IsOrderedAddMonoid α
a b c d : α
h : a ≤ b
this : ∀ {c d : α}, d ≤ c → |↑(projIcc a b h c) - ↑(projIcc a b h d)| ≤ |c - d|
hdc : ¬d ≤ c
⊢ |↑(projIcc a b h d) - ↑(projIcc a b h c)| ≤ |d - c|
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : LinearOrder α
inst✝ : IsOrderedAddMonoid α
a b : α
h : a ≤ b
δ : α
hδ : 0 ≤ δ
t : ↑(Icc a b)
n : ℕ
ht : t ∈ Icc (addNSMul h δ n) (addNSMul h δ (n + 1))
⊢ ↑t ≤ ↑(projIcc a b h (a + (n + 1) • δ))

Tactic:
;

Goal state after:
α : Type u_1
inst✝² : AddCommGroup α
inst✝¹ : LinearOrder α
inst✝ : IsOrderedAddMonoid α
a b : α
h : a ≤ b
δ : α
hδ : 0 ≤ δ
t : ↑(Icc a b)
n : ℕ
ht : t ∈ Icc (addNSMul h δ n) (addNSMul h δ (n + 1))
⊢ ↑t ≤ ↑(projIcc a b h (a + (n + 1) • δ))
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
Φ : Z → X
Ψ : Z → Y
ε : ℝ
ε0 : 0 < ε
x : X
y : Y
h : glueDist Φ Ψ ε (Sum.inl x) (Sum.inr y) = 0
⊢ False

Tactic:
;

Goal state after:
X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
Φ : Z → X
Ψ : Z → Y
ε : ℝ
ε0 : 0 < ε
x : X
y : Y
h : glueDist Φ Ψ ε (Sum.inl x) (Sum.inr y) = 0
⊢ False
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
Φ : Z → X
Ψ : Z → Y
ε : ℝ
ε0 : 0 < ε
x : Y
y : X
h : glueDist Φ Ψ ε (Sum.inr x) (Sum.inl y) = 0
⊢ False

Tactic:
;

Goal state after:
X : Type u
Y : Type v
Z : Type w
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
Φ : Z → X
Ψ : Z → Y
ε : ℝ
ε0 : 0 < ε
x : Y
y : X
h : glueDist Φ Ψ ε (Sum.inr x) (Sum.inl y) = 0
⊢ False
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.inl x) (Sum.inr y)

Tactic:
;

Goal state after:
X : Type u
Y : Type v
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
x : X
y : Y
⊢ 1 ≤ Sum.dist (Sum.inl x) (Sum.inr y)
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
this : TopologicalSpace α := t i₀
⊢ T1Space α

Tactic:
;

Goal state after:
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
this : TopologicalSpace α := t i₀
⊢ T1Space α
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✝¹ : SecondCountableTopology α
this✝ : TopologicalSpace α := t i₀
this : PolishSpace α
⊢ T1Space α

Tactic:
;

Goal state after:
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✝¹ : SecondCountableTopology α
this✝ : TopologicalSpace α := t i₀
this : PolishSpace α
⊢ T1Space α
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: case mpr.succ
α : Type u_2
x y : ℕ → α
n : ℕ
ih : (∀ ⦃m : ℕ⦄, m < n → x m = y m) → res x n = res y n
h : ∀ ⦃m : ℕ⦄, m < n + 1 → x m = y m
m : ℕ
hm : m < n
⊢ x m = y m

Tactic:
exact h (hm.trans (Nat.lt_succ_self _))

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

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

pp: case inr
E : ℕ → Type u_1
x y : (n : ℕ) → E n
hxy : dist x y = 0
h : x ≠ y
⊢ x = y

Tactic:
;

Goal state after:
case inl
E : ℕ → Type u_1
x : (n : ℕ) → E n
hxy : dist x x = 0
⊢ x = x

case inr
E : ℕ → Type u_1
x y : (n : ℕ) → E n
hxy : dist x y = 0
h : x ≠ y
⊢ x = y
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: case mk
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
y : (i : ι) × E i
fst✝ : ι
snd✝ : E fst✝
h : dist ⟨fst✝, snd✝⟩ y < 1
⊢ ⟨fst✝, snd✝⟩.fst = y.fst

Tactic:
;

Goal state after:
case mk
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
y : (i : ι) × E i
fst✝ : ι
snd✝ : E fst✝
h : dist ⟨fst✝, snd✝⟩ y < 1
⊢ ⟨fst✝, snd✝⟩.fst = y.fst
Module: Mathlib.Topology.MetricSpace.Gluing

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

Tactic:
;

Goal state after:
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
y z : (i : ι) × E i
i : ι
x : E i
⊢ dist ⟨i, x⟩ z ≤ dist ⟨i, x⟩ y + dist y z
Module: Mathlib.Topology.MetricSpace.Gluing

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

Tactic:
;

Goal state after:
ι : Type u_1
E : ι → Type u_2
inst✝ : (i : ι) → MetricSpace (E i)
z : (i : ι) × E i
i : ι
x : E i
j : ι
y : E j
⊢ dist ⟨i, x⟩ z ≤ dist ⟨i, x⟩ ⟨j, y⟩ + dist ⟨j, y⟩ z
Module: Mathlib.Topology.MetricSpace.PiNat

Goal state before:
pp: case mpr.inr
E : ℕ → Type u_1
α : Type u_2
inst✝ : PseudoMetricSpace α
f : ((n : ℕ) → E n) → α
H : ∀ (x y : (n : ℕ) → E n) (n : ℕ), y ∈ cylinder x n → dist (f x) (f y) ≤ (1 / 2) ^ n
x y : (n : ℕ) → E n
hne : x ≠ y
⊢ y ∈ cylinder x (firstDiff y x)

Tactic:
exact mem_cylinder_firstDiff _ _

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

Goal state before:
pp: case inr.inr.inl
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:
calc
          dist x (Nonempty.some ⟨x⟩) + 1 + dist (Nonempty.some ⟨z⟩) z ≤
              dist x (Nonempty.some ⟨x⟩) + 1 + (dist (Nonempty.some ⟨z⟩) y + dist y z) := by
            apply_rules [add_le_add, le_rfl, dist_triangle]
          _ = _ := by abel

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: ⊢ Continuous fun i ↦ ⟨↑i, ⋯⟩

Tactic:
;

Goal state after:
⊢ Continuous fun i ↦ ⟨↑i, ⋯⟩
Module: Mathlib.Topology.UnitInterval

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

Tactic:
;

Goal state after:
x : ↑I
⊢ ↑(toNNReal x + toNNReal (σ x)) = ↑1
Module: Mathlib.Topology.UnitInterval

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

Tactic:
;

Goal state after:
x : ↑I
⊢ ↑(toNNReal (σ x) + toNNReal x) = ↑1
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
⊢ 0 < ε / 2

Tactic:
;

Goal state after:
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
⊢ 0 < ε / 2
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 ↔ y ∉ (↑s)ᶜ

Tactic:
;

Goal state after:
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 ↔ y ∉ (↑s)ᶜ
Module: Mathlib.Topology.MetricSpace.CantorScheme

Goal state before:
pp: case refine_1
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
y : ↑(inducedMap A).fst
hyx : res (↑y) n = res (↑x) n
⊢ dist ((inducedMap A).snd y) ((inducedMap A).snd x) < ε

Tactic:
apply hn

Goal state after:
case refine_1.x
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
y : ↑(inducedMap A).fst
hyx : res (↑y) n = res (↑x) n
⊢ (inducedMap A).snd y ∈ A (res (↑x) n)

case refine_1.x
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
y : ↑(inducedMap A).fst
hyx : res (↑y) n = res (↑x) n
⊢ (inducedMap A).snd x ∈ A (res (↑x) n)
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
⊢ 0 < 1

Tactic:
;

Goal state after:
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
⊢ 0 < 1
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: X : Type u
Y : Type v
Z : Type w
inst✝³ : Nonempty Z
inst✝² : MetricSpace Z
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
Φ : Z → X
Ψ : Z → Y
ε : ℝ
hΦ : Isometry Φ
hΨ : Isometry Ψ
p q : Z
⊢ |dist p q - dist p q| ≤ 2 * 0

Tactic:
;

Goal state after:
X : Type u
Y : Type v
Z : Type w
inst✝³ : Nonempty Z
inst✝² : MetricSpace Z
inst✝¹ : MetricSpace X
inst✝ : MetricSpace Y
Φ : Z → X
Ψ : Z → Y
ε : ℝ
hΦ : Isometry Φ
hΨ : Isometry Ψ
p q : Z
⊢ |dist p q - dist p q| ≤ 2 * 0
Module: Mathlib.Topology.MetricSpace.Gluing

Goal state before:
pp: X : ℕ → Type u
inst✝ : (n : ℕ) → MetricSpace (X n)
f : (n : ℕ) → X n → X (n + 1)
I : ∀ (n : ℕ), Isometry (f n)
y : (n : ℕ) × X n
hy : y.fst ≤ 0
i : ℕ
x : X i
hx : ⟨i, x⟩.fst ≤ 0
⊢ inductiveLimitDist f ⟨i, x⟩ y = dist (leRecOn hx (fun {k} ↦ f k) ⟨i, x⟩.snd) (leRecOn hy (fun {k} ↦ f k) y.snd)

Tactic:
;

Goal state after:
X : ℕ → Type u
inst✝ : (n : ℕ) → MetricSpace (X n)
f : (n : ℕ) → X n → X (n + 1)
I : ∀ (n : ℕ), Isometry (f n)
y : (n : ℕ) × X n
hy : y.fst ≤ 0
i : ℕ
x : X i
hx : ⟨i, x⟩.fst ≤ 0
⊢ inductiveLimitDist f ⟨i, x⟩ y = dist (leRecOn hx (fun {k} ↦ f k) ⟨i, x⟩.snd) (leRecOn hy (fun {k} ↦ f k) y.snd)
Module: Mathlib.Topology.MetricSpace.CantorScheme

Goal state before:
pp: case h
β : Type u_1
α : Type u_2
A : List β → Set α
inst✝¹ : PseudoMetricSpace α
inst✝ : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : ℕ → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
⊢ ∀ m ≥ n, ∀ n_1 ≥ n, dist (u m) (u n_1) < ε

Tactic:
intro m₀ hm₀ m₁ hm₁

Goal state after:
case h
β : Type u_1
α : Type u_2
A : List β → Set α
inst✝¹ : PseudoMetricSpace α
inst✝ : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : ℕ → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
m₀ : ℕ
hm₀ : m₀ ≥ n
m₁ : ℕ
hm₁ : m₁ ≥ n
⊢ dist (u m₀) (u m₁) < ε
Module: Mathlib.Topology.MetricSpace.CantorScheme

Goal state before:
pp: case h
β : Type u_1
α : Type u_2
A : List β → Set α
inst✝¹ : PseudoMetricSpace α
inst✝ : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : ℕ → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
m₀ : ℕ
hm₀ : m₀ ≥ n
m₁ : ℕ
hm₁ : m₁ ≥ n
⊢ dist (u m₀) (u m₁) < ε

Tactic:
apply hn

Goal state after:
case h.x
β : Type u_1
α : Type u_2
A : List β → Set α
inst✝¹ : PseudoMetricSpace α
inst✝ : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : ℕ → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
m₀ : ℕ
hm₀ : m₀ ≥ n
m₁ : ℕ
hm₁ : m₁ ≥ n
⊢ u m₀ ∈ A (res x n)

case h.x
β : Type u_1
α : Type u_2
A : List β → Set α
inst✝¹ : PseudoMetricSpace α
inst✝ : CompleteSpace α
hdiam : VanishingDiam A
hanti : ClosureAntitone A
hnonempty : ∀ (l : List β), (A l).Nonempty
x : ℕ → β
u : ℕ → α
hu : ∀ (n : ℕ), u n ∈ A (res x n)
umem : ∀ (n m : ℕ), n ≤ m → u m ∈ A (res x n)
ε : ℝ
ε_pos : ε > 0
n : ℕ
hn : ∀ y ∈ A (res x n), ∀ z ∈ A (res x n), dist y z < ε
m₀ : ℕ
hm₀ : m₀ ≥ n
m₁ : ℕ
hm₁ : m₁ ≥ n
⊢ u m₁ ∈ A (res x n)
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 : IsClosed[Pi.topologicalSpace] s
x y : (n : ℕ) → E n
hx : x ∉ s
hy : y ∈ s
A : ∃ n, Disjoint s (cylinder x n)
⊢ firstDiff x y < shortestPrefixDiff x s

Tactic:
rw [shortestPrefixDiff, dif_pos A]

Goal state after:
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 : IsClosed[Pi.topologicalSpace] s
x y : (n : ℕ) → E n
hx : x ∉ s
hy : y ∈ s
A : ∃ n, Disjoint s (cylinder x n)
⊢ firstDiff x y < Nat.find A
Module: Mathlib.Topology.MetricSpace.Perfect

Goal state before:
pp: α : Type u_1
inst✝¹ : MetricSpace α
C : Set α
hC : Perfect C
hnonempty : C.Nonempty
inst✝ : CompleteSpace α
u : ℕ → ℝ≥0∞
upos' : ∀ (n : ℕ), u n ∈ Ioo 0 1
hu : Tendsto u atTop (nhds 0)
upos : ∀ (n : ℕ), 0 < u n
P : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }
C0 C1 : {C : Set α} → Perfect C → C.Nonempty → {ε : ℝ≥0∞} → 0 < ε → Set α
h0 :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Perfect (C0 hC hnonempty hε) ∧
      (C0 hC hnonempty hε).Nonempty ∧ C0 hC hnonempty hε ⊆ C ∧ ediam (C0 hC hnonempty hε) ≤ ε
h1 :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Perfect (C1 hC hnonempty hε) ∧
      (C1 hC hnonempty hε).Nonempty ∧ C1 hC hnonempty hε ⊆ C ∧ ediam (C1 hC hnonempty hε) ≤ ε
hdisj :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)
DP : List Bool → P :=
  fun l ↦
    List.rec ⟨C, ⋯⟩
      (fun a l ih ↦ Bool.casesOn (motive := fun t ↦ a = t → P) a (fun h ↦ ⟨C0 ⋯ ⋯ ⋯, ⋯⟩) (fun h ↦ ⟨C1 ⋯ ⋯ ⋯, ⋯⟩) ⋯) l
⊢ ∃ f, range f ⊆ C ∧ Continuous f ∧ Injective f

Tactic:
let D : List Bool → Set α := fun l => (DP l).val

Goal state after:
α : Type u_1
inst✝¹ : MetricSpace α
C : Set α
hC : Perfect C
hnonempty : C.Nonempty
inst✝ : CompleteSpace α
u : ℕ → ℝ≥0∞
upos' : ∀ (n : ℕ), u n ∈ Ioo 0 1
hu : Tendsto u atTop (nhds 0)
upos : ∀ (n : ℕ), 0 < u n
P : Type (max 0 u_1) := { E // Perfect E ∧ E.Nonempty }
C0 C1 : {C : Set α} → Perfect C → C.Nonempty → {ε : ℝ≥0∞} → 0 < ε → Set α
h0 :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Perfect (C0 hC hnonempty hε) ∧
      (C0 hC hnonempty hε).Nonempty ∧ C0 hC hnonempty hε ⊆ C ∧ ediam (C0 hC hnonempty hε) ≤ ε
h1 :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Perfect (C1 hC hnonempty hε) ∧
      (C1 hC hnonempty hε).Nonempty ∧ C1 hC hnonempty hε ⊆ C ∧ ediam (C1 hC hnonempty hε) ≤ ε
hdisj :
  ∀ {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε),
    Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)
DP : List Bool → P :=
  fun l ↦
    List.rec ⟨C, ⋯⟩
      (fun a l ih ↦ Bool.casesOn (motive := fun t ↦ a = t → P) a (fun h ↦ ⟨C0 ⋯ ⋯ ⋯, ⋯⟩) (fun h ↦ ⟨C1 ⋯ ⋯ ⋯, ⋯⟩) ⋯) l
D : List Bool → Set α := fun l ↦ ↑(DP l)
⊢ ∃ f, range f ⊆ C ∧ Continuous f ∧ Injective f
Module: Mathlib.MeasureTheory.Integral.Lebesgue.Map

Goal state before:
pp: case e'_3
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
f : β → ℝ≥0∞
g : α → β
hf : Measurable f
hg : Measurable g
n : ℕ
x : α
⊢ (eapprox (f ∘ g) n) x = ((eapprox f n).comp g hg) x

Tactic:
;

Goal state after:
case e'_3
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
f : β → ℝ≥0∞
g : α → β
hf : Measurable f
hg : Measurable g
n : ℕ
x : α
⊢ (eapprox (f ∘ g) n) x = ((eapprox f n).comp g hg) x
Module: Mathlib.MeasureTheory.Measure.MutuallySingular

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
h : μ = 0
⊢ 0 ⟂ₘ 0

Tactic:
;

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
h : μ = 0
⊢ 0 ⟂ₘ 0
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s : Set α
inst✝ : IsProbabilityMeasure μ
p : α → Prop
f✝ : β → α
f : β ≃ᵐ α
⊢ IsProbabilityMeasure (Measure.map (⇑f.symm) μ)

Tactic:
;

Goal state after:
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
s : Set α
inst✝ : IsProbabilityMeasure μ
p : α → Prop
f✝ : β → α
f : β ≃ᵐ α
⊢ IsProbabilityMeasure (Measure.map (⇑f.symm) μ)
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
inst✝ : IsProbabilityMeasure μ
f : β → α
hf : Injective f
hf' : ∀ᵐ (a : α) ∂μ, a ∈ range f
hf'' : ∀ (s : Set β), MeasurableSet s → MeasurableSet (f '' s)
⊢ (Measure.comap f μ) univ = 1

Tactic:
comap_apply _ hf hf'' _ MeasurableSet.univ,

Goal state after:
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
μ : Measure α
inst✝ : IsProbabilityMeasure μ
f : β → α
hf : Injective f
hf' : ∀ᵐ (a : α) ∂μ, a ∈ range f
hf'' : ∀ (s : Set β), MeasurableSet s → MeasurableSet (f '' s)
⊢ μ (f '' univ) = 1
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
s : Set α
⊢ IsZeroOrProbabilityMeasure 0

Tactic:
;

Goal state after:
case inl
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
s : Set α
⊢ IsZeroOrProbabilityMeasure 0
Module: Mathlib.MeasureTheory.Measure.Typeclasses.Probability

Goal state before:
pp: case inr.inl
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s : Set α
h✝ : NeZero μ
h : μ univ = ∞
⊢ IsZeroOrProbabilityMeasure 0

Tactic:
;

Goal state after:
case inr.inl
α : Type u_1
β : Type u_2
m0 : MeasurableSpace α
inst✝ : MeasurableSpace β
μ : Measure α
s : Set α
h✝ : NeZero μ
h : μ univ = ∞
⊢ IsZeroOrProbabilityMeasure 0
Module: Mathlib.MeasureTheory.Measure.Dirac

Goal state before:
pp: case pos
α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
a : α
s : Set α
h : a ∈ s
⊢ (dirac a) s = s.indicator 1 a

pp: case neg
α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
a : α
s : Set α
h : a ∉ s
⊢ (dirac a) s = s.indicator 1 a

Tactic:
;

Goal state after:
case pos
α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
a : α
s : Set α
h : a ∈ s
⊢ (dirac a) s = s.indicator 1 a

case neg
α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : MeasurableSingletonClass α
a : α
s : Set α
h : a ∉ s
⊢ (dirac a) s = s.indicator 1 a
Module: Mathlib.MeasureTheory.Measure.Count

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
⊢ count ∅ = 0

Tactic:
;

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
⊢ count ∅ = 0
Module: Mathlib.MeasureTheory.Measure.Count

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
s : Set α
inst✝ : Nonempty α
h : count = 0
⊢ False

Tactic:
;

Goal state after:
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSpace β
s : Set α
inst✝ : Nonempty α
h : count = 0
⊢ False
Module: Mathlib.MeasureTheory.Measure.Count

Goal state before:
pp: case e_s
α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : Subsingleton α
i : α
⊢ univ = {i}

Tactic:
;

Goal state after:
case e_s
α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : Subsingleton α
i : α
⊢ univ = {i}
Module: Mathlib.MeasureTheory.Measure.Count

Goal state before:
pp: case e_s
α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : Subsingleton α
i j : α
⊢ j ∈ univ ↔ j ∈ {i}

Tactic:
;

Goal state after:
case e_s
α : Type u_1
inst✝¹ : MeasurableSpace α
inst✝ : Subsingleton α
i j : α
⊢ j ∈ univ ↔ j ∈ {i}
Module: Mathlib.MeasureTheory.Measure.Dirac

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
x y : α
h : ∀ (A : Set α), MeasurableSet A → (x ∈ A ↔ y ∈ A)
A : Set α
A_mble : MeasurableSet A
⊢ x ∈ A → y ∈ A

Tactic:
simp only [h A A_mble, imp_self]

Goal state after: