mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Group.FunctionSeries

Goal state before:
pp: α : Type u_1
β : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : CompleteSpace F
u : α → ℝ
f : α → β → F
hu : Summable u
s : Set β
hfu : ∀ (n : α), ∀ x ∈ s, ‖f n x‖ ≤ u n
⊢ TendstoUniformlyOn (fun t x ↦ ∑ n ∈ t, f n x) (fun x ↦ ∑' (n : α), f n x) atTop s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.InfiniteSum

Goal state before:
pp: ι : Type u_1
E : Type u_3
inst✝¹ : SeminormedAddCommGroup E
inst✝ : CompleteSpace E
f : ι → E
g : ι → ℝ
hg : Summable g
h : ∀ (i : ι), ‖f i‖ ≤ g i
⊢ Summable f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.InfiniteSum

Goal state before:
pp: ι : Type u_1
ε : Type u_5
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddCommMonoid ε
f : ι → ε
g : ι → ℝ≥0∞
a : ε
b : ℝ≥0∞
hf : HasSum f a
hg : HasSum g b
h : ∀ (i : ι), ‖f i‖ₑ ≤ g i
⊢ ‖a‖ₑ ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.InfiniteSum

Goal state before:
pp: ι : Type u_1
E : Type u_3
inst✝ : SeminormedAddCommGroup E
f : ι → E
g : ι → ℝ
a : E
b : ℝ
hf : HasSum f a
hg : HasSum g b
h : ∀ (i : ι), ‖f i‖ ≤ g i
⊢ ‖a‖ ≤ b

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
mβ : MeasurableSpace β
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : BorelSpace β
m : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous[inst✝⁷, inst✝²] fun i ↦ u i x
h : ∀ (i : ι), Measurable (u i)
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n ↦ u ((t_sf n) j) x) atTop (𝓝 (u j x))
U : ℕ → ι × α → β := fun n p ↦ u ((t_sf n) p.1) p.2
h_tendsto : Tendsto U atTop (𝓝 fun p ↦ u p.1 p.2)
n : ℕ
⊢ Measurable fun p ↦ u (↑p.1) p.2

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.InfiniteSum

Goal state before:
pp: ι : Type u_1
ε : Type u_5
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddCommMonoid ε
f : ι → ε
g : ι → ℝ≥0∞
a : ℝ≥0∞
hg : HasSum g a
h : ∀ (i : ι), ‖f i‖ₑ ≤ g i
⊢ ‖∑' (i : ι), f i‖ₑ ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.FunctionSeries

Goal state before:
pp: β : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : CompleteSpace F
ι : Type u_4
f : ι → β → F
u : ι → ℝ
hu : Summable u
s : Set β
ε : ℝ
εpos : ε > 0
t : Finset ι
ht : ∀ (b : Finset ι), t ⊆ b → ∑' (a : { x // x ∉ b }), u ↑a < ε
N : Set ι
hN : N ∈ cofinite
HN : ∀ y ∈ N, ∀ x ∈ s, ‖f y x‖ ≤ u y
n : Finset ι
hn : Finite.toFinset hN ∪ t ⊆ n
x : β
hx : x ∈ s
⊢ Summable fun n ↦ ‖f n x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.InfiniteSum

Goal state before:
pp: ι : Type u_1
E : Type u_3
inst✝ : SeminormedAddCommGroup E
f : ι → E
g : ι → ℝ
a : ℝ
hg : HasSum g a
h : ∀ (i : ι), ‖f i‖ ≤ g i
⊢ ‖∑' (i : ι), f i‖ ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.InfiniteSum

Goal state before:
pp: ι : Type u_1
E : Type u_3
inst✝ : SeminormedAddCommGroup E
f : ι → E
g : ι → ℝ≥0
a : ℝ≥0
hg : HasSum g a
h : ∀ (i : ι), ‖f i‖₊ ≤ g i
⊢ ‖∑' (i : ι), f i‖₊ ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.InfiniteSum

Goal state before:
pp: E : Type u_3
inst✝ : SeminormedAddCommGroup E
ι : Type u_6
f : ι → E
⊢ ∑' (i : ι), ‖f i‖ₑ ≠ ∞ ↔ Summable fun i ↦ ‖f i‖₊

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.InfiniteSum

Goal state before:
pp: E : Type u_3
inst✝ : SeminormedAddCommGroup E
ι : Type u_6
f : ι → E
⊢ ∑' (i : ι), ‖f i‖ₑ ≠ ∞ ↔ Summable fun i ↦ ‖f i‖

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
mβ : MeasurableSpace β
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : BorelSpace β
m : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous[inst✝⁷, inst✝²] fun i ↦ u i x
h : ∀ (i : ι), Measurable (u i)
⊢ Measurable (uncurry u)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.FunctionSeries

Goal state before:
pp: β : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : CompleteSpace F
ι : Type u_4
f : ι → β → F
u : ι → ℝ
hu : Summable u
s : Set β
hfu : ∀ᶠ (n : ι) in cofinite, ∀ x ∈ s, ‖f n x‖ ≤ u n
⊢ TendstoUniformlyOn (fun t x ↦ ∑ n ∈ t, f n x) (fun x ↦ ∑' (n : ι), f n x) atTop s

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence

Goal state before:
pp: α : Type u_2
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ∞
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t ↦ Nat.rec (motive := fun x ↦ α → ℝ≥0∞) (g 0) (fun n I x ↦ max (g (n + 1) x) (I x)) t
M : ∀ (n : ℕ), Measurable (g' n)
I : ∀ (n : ℕ) (x : α), g n x ≤ g' n x
⊢ ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), g' n x ≤ f n x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.FunctionSeries

Goal state before:
pp: α : Type u_1
β : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : CompleteSpace F
u : α → ℝ
f : α → β → F
hu : Summable u
hfu : ∀ (n : α) (x : β), ‖f n x‖ ≤ u n
⊢ TendstoUniformly (fun t x ↦ ∑ n ∈ t, f n x) (fun x ↦ ∑' (n : α), f n x) atTop

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous[inst✝⁷, inst✝²] fun i ↦ u i x
h : ∀ (i : ι), StronglyMeasurable (u i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
⊢ ∃ t, ∀ (j : ι) (x : α), Tendsto (fun n ↦ u ((t n) j) x) atTop (𝓝 (u j x))

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.FunctionSeries

Goal state before:
pp: β : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : CompleteSpace F
ι : Type u_4
f : ι → β → F
u : ι → ℝ
hu : Summable u
hfu : ∀ᶠ (n : ι) in cofinite, ∀ (x : β), ‖f n x‖ ≤ u n
⊢ TendstoUniformly (fun t x ↦ ∑ n ∈ t, f n x) (fun x ↦ ∑' (n : ι), f n x) atTop

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous[inst✝⁷, inst✝²] fun i ↦ u i x
h : ∀ (i : ι), StronglyMeasurable (u i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n ↦ u ((t_sf n) j) x) atTop (𝓝 (u j x))
U : ℕ → ι × α → β := fun n p ↦ u ((t_sf n) p.1) p.2
⊢ Tendsto U atTop (𝓝 fun p ↦ u p.1 p.2)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.FunctionSeries

Goal state before:
pp: α : Type u_1
β : Type u_2
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : TopologicalSpace β
f : α → β → F
s : Set β
hf : ∀ (i : α), ContinuousOn (f i) s
hu : Summable u
hfu : ∀ (n : α), ∀ x ∈ s, ‖f n x‖ ≤ u n
⊢ ContinuousOn (fun x ↦ ∑' (n : α), f n x) s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Group.FunctionSeries

Goal state before:
pp: α : Type u_1
β : Type u_2
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : TopologicalSpace β
f : α → β → F
hf : ∀ (i : α), Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f i)
hu : Summable u
hfu : ∀ (n : α) (x : β), ‖f n x‖ ≤ u n
⊢ Continuous[inst✝, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ ∑' (n : α), f n x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Order.ProjIcc

Goal state before:
pp: α : Type u_1
inst✝² : LinearOrder α
a b : α
h : a ≤ b
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
⊢ Continuous[inst✝¹, inst✝¹] fun x ↦ max a (min b x)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Order.ProjIcc

Goal state before:
pp: α : Type u_1
inst✝² : LinearOrder α
a b : α
h : a ≤ b
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
s : Set ↑(Icc a b)
hs : IsOpen[inst✝¹] (projIcc a b h ⁻¹' s)
⊢ Subtype.val ⁻¹' projIcc a b h ⁻¹' s = s

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous[inst✝⁷, inst✝²] fun i ↦ u i x
h : ∀ (i : ι), StronglyMeasurable (u i)
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
t_sf : ℕ → ι →ₛ ι
ht_sf : ∀ (j : ι) (x : α), Tendsto (fun n ↦ u ((t_sf n) j) x) atTop (𝓝 (u j x))
U : ℕ → ι × α → β := fun n p ↦ u ((t_sf n) p.1) p.2
h_tendsto : Tendsto U atTop (𝓝 fun p ↦ u p.1 p.2)
n : ℕ
⊢ StronglyMeasurable fun p ↦ u (↑p.1) p.2

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
K : ℝ≥0
f : α → β
hf : LipschitzWith K f
u : ℕ → α
hu : CauchySeq u
b : ℕ → ℝ
b_nonneg : ∀ (n : ℕ), 0 ≤ b n
hb : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) ≤ b N
blim : Tendsto b atTop (nhds 0)
n : ℕ
⊢ 0 ≤ ↑K

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic

Goal state before:
pp: α : Type u_5
β : Type u_6
ι : Type u_7
inst✝⁷ : TopologicalSpace ι
inst✝⁶ : MetrizableSpace ι
inst✝⁵ : MeasurableSpace ι
inst✝⁴ : SecondCountableTopology ι
inst✝³ : OpensMeasurableSpace ι
inst✝² : TopologicalSpace β
inst✝¹ : PseudoMetrizableSpace β
inst✝ : MeasurableSpace α
u : ι → α → β
hu_cont : ∀ (x : α), Continuous[inst✝⁷, inst✝²] fun i ↦ u i x
h : ∀ (i : ι), StronglyMeasurable (u i)
⊢ StronglyMeasurable (uncurry u)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence

Goal state before:
pp: α : Type u_2
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ∞
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t ↦ Nat.rec (motive := fun x ↦ α → ℝ≥0∞) (g 0) (fun n I x ↦ max (g (n + 1) x) (I x)) t
M : ∀ (n : ℕ), Measurable (g' n)
I : ∀ (n : ℕ) (x : α), g n x ≤ g' n x
I' : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), g' n x ≤ f n x
n : ℕ
⊢ ∫⁻ (x : α), g' n x ∂μ = ∫⁻ (x : α), f n x ∂μ

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
K : ℝ≥0
f : α → β
hf : LipschitzWith K f
u : ℕ → α
hu : CauchySeq u
⊢ CauchySeq (f ∘ u)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
K : ℝ≥0
s : Set α
f : α → β
hf : LipschitzOnWith K f s
u : ℕ → α
hu : CauchySeq u
h'u : range u ⊆ s
b : ℕ → ℝ
b_nonneg : ∀ (n : ℕ), 0 ≤ b n
hb : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (u n) (u m) ≤ b N
blim : Tendsto b atTop (nhds 0)
n : ℕ
⊢ 0 ≤ ↑K

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
K : ℝ≥0
s : Set α
f : α → β
hf : LipschitzOnWith K f s
u : ℕ → α
hu : CauchySeq u
h'u : range u ⊆ s
⊢ CauchySeq (f ∘ u)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
x : α
r : ℝ
hr : 0 < r
K : ℝ
h : ∀ (y : α), dist y x < r → dist (f y) (f x) ≤ K * dist y x
⊢ Continuous fun a ↦ K * dist a x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
x : α
r : ℝ
hr : 0 < r
K : ℝ
h : ∀ (y : α), dist y x < r → dist (f y) (f x) ≤ K * dist y x
⊢ K * dist x x = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : PseudoMetricSpace α
inst✝ : PseudoMetricSpace β
f : α → β
x : α
r : ℝ
hr : 0 < r
K : ℝ
h : ∀ (y : α), dist y x < r → dist (f y) (f x) ≤ K * dist y x
⊢ ContinuousAt f x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: 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 : LocallyLipschitzOn s f
hf' : ContinuousOn f s
x : α
hx : x ∈ s
⊢ ∃ ε > 0, ∃ K, LipschitzOnWith K f (ball x ε ∩ s)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: 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)
x : α
hx : x ∈ s
⊢ ContinuousOn (fun y ↦ dist (f x) (f y)) s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: 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)
x : α
hx : x ∈ s
⊢ ContinuousOn (dist x) (s \ ball x (ε x hx))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: 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)
x : α
hx : x ∈ s
⊢ ∃ K', ∀ y ∈ s.diff (ball x (ε x hx)), edist (f x) (f y) ≤ ↑K' * edist x y

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence

Goal state before:
pp: α : Type u_2
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ∞
g : ℕ → α → ℝ≥0∞
gmeas : ∀ (n : ℕ), Measurable (g n)
gf : ∀ (n : ℕ), g n ≤ f n
hg : ∀ (n : ℕ), ∫⁻ (a : α), f n a ∂μ = ∫⁻ (a : α), g n a ∂μ
g' : ℕ → α → ℝ≥0∞ := fun t ↦ Nat.rec (motive := fun x ↦ α → ℝ≥0∞) (g 0) (fun n I x ↦ max (g (n + 1) x) (I x)) t
M : ∀ (n : ℕ), Measurable (g' n)
I : ∀ (n : ℕ) (x : α), g n x ≤ g' n x
I' : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), g' n x ≤ f n x
Int_eq : ∀ (n : ℕ), ∫⁻ (x : α), g' n x ∂μ = ∫⁻ (x : α), f n x ∂μ
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ g' i a) atTop (𝓝 (F a))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: 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 ⋯) * edist (↑z) x + ↑(K' ↑z ⋯) * edist (↑z) y ≤
    ↑(K ↑z ⋯) * edist (↑z) x + ↑(K' ↑z ⋯) * (edist x ↑z + edist x y)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: 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 ⋯) * edist (↑z) x + ↑(K' ↑z ⋯) * (edist x ↑z + edist x y) ≤
    ↑(K ↑z ⋯) * edist x y + ↑(K' ↑z ⋯) * (edist x y + edist x y)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: ⊢ I = Metric.closedBall 2⁻¹ 2⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: ⊢ ↑(σ 0) = ↑1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: ⊢ ↑(σ 1) = ↑0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

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

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: s : Set ↑I
⊢ Subtype.val '' σ ⁻¹' s = (fun x ↦ 1 - x) ⁻¹' Subtype.val '' s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: 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 ⋯) * edist x y + ↑(K' ↑z ⋯) * (edist x y + edist x y) = ↑(K ↑z ⋯ + 2 * K' ↑z ⋯) * edist x y

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: x : ℝ
⊢ σ (projIcc 0 1 ⋯ x) = projIcc 0 1 ⋯ (1 - x)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: ⊢ Continuous fun x ↦ 1 - ↑x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: t : ↑I
⊢ 1 / 2 ≤ ↑(σ t) ↔ ↑t ≤ 1 / 2

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: i : ↑I
⊢ σ i = 1 ↔ i = 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: i : ↑I
⊢ σ i = 0 ↔ i = 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: i j : ↑I
⊢ σ i ≤ σ j ↔ j ≤ i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: i j : ↑I
⊢ i ≤ σ j ↔ j ≤ σ i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: i j : ↑I
⊢ σ i ≤ j ↔ σ j ≤ i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: i j : ↑I
⊢ σ i < σ j ↔ j < i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: i j : ↑I
⊢ i < σ j ↔ j < σ i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: i j : ↑I
⊢ σ i < j ↔ σ j < i

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: ⊢ CompactSpace ↑I

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: 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 ⋯ ≤ K ↑z ⋯ + 2 * K' ↑z ⋯

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: x : ↑I
⊢ 0 ≤ 1 - ↑x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: x : ↑I
⊢ 1 - ↑x ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: 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 ⋯) * edist x y ≤ ↑(t.sup fun i ↦ K ↑i ⋯ + 2 * K' ↑i ⋯) * edist x y

Tactic:
by

Goal state after:
Module: Mathlib.Topology.UnitInterval

Goal state before:
pp: i j : ↑I
h : i ≤ j * i
⊢ i = 0 ∨ j = 1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.MetricSpace.Lipschitz

Goal state before:
pp: 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 : LocallyLipschitzOn s f
⊢ ∃ K, LipschitzOnWith K f s

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence

Goal state before:
pp: α : Type u_2
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hF_meas : AEMeasurable F μ
hf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
h_int_finite : ∫⁻ (a : α), F a ∂μ ≠ ∞
⊢ ∀ᵐ (a : α) ∂μ, Tendsto (fun i ↦ f i a) atTop (𝓝 (F a))

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Lebesgue.DominatedConvergence

Goal state before:
pp: α : Type u_2
mα : MeasurableSpace α
f : ℕ → α → ℝ≥0∞
F : α → ℝ≥0∞
μ : Measure α
hf_meas : ∀ (n : ℕ), AEMeasurable (f n) μ
hf_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Antitone fun i ↦ f i a
h_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), F a ≤ f i a
h0 : ∫⁻ (a : α), f 0 a ∂μ ≠ ∞
⊢ ∫⁻ (a : α), F a ∂μ ≠ ∞

Tactic:
by

Goal state after: