mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Group.GeometryOfNumbers

Goal state before:
pp: E : Type u_1
inst✝⁸ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : Nontrivial E
inst✝² : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝¹ : Countable ↥L
inst✝ : DiscreteTopology ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h_cpt : IsCompact s
h : μ F * 2 ^ finrank ℝ E ≤ μ s
h_mes : μ s ≠ 0
h_nemp : s.Nonempty
u : ℕ → ℝ≥0 := ⋯.choose
K : ConvexBody E := { carrier := s, convex' := h_conv, isCompact' := h_cpt, nonempty' := h_nemp }
S : ℕ → ConvexBody E := fun n ↦ (1 + u n) • K
Z : ℕ → Set E := fun n ↦ ↑(S n) ∩ (↑L \ {0})
h_zero : 0 ∈ K
h_clos : IsClosed (↑L \ {0})
n : ℕ
this : μ F * 2 ^ finrank ℝ E < μ ↑(S n)
h_symm' : ∀ x ∈ S n, -x ∈ S n
x : ↥L
hx_nz : x ≠ 0
hx_mem : ↑x ∈ ↑(S n)
⊢ ↑x ∈ ↑L \ {0}

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Group.GeometryOfNumbers

Goal state before:
pp: E : Type u_1
inst✝⁸ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : Nontrivial E
inst✝² : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝¹ : Countable ↥L
inst✝ : DiscreteTopology ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h_cpt : IsCompact s
h : μ F * 2 ^ finrank ℝ E ≤ μ s
h_mes : μ s ≠ 0
h_nemp : s.Nonempty
u : ℕ → ℝ≥0 := ⋯.choose
K : ConvexBody E := { carrier := s, convex' := h_conv, isCompact' := h_cpt, nonempty' := h_nemp }
S : ℕ → ConvexBody E := fun n ↦ (1 + u n) • K
Z : ℕ → Set E := fun n ↦ ↑(S n) ∩ (↑L \ {0})
h_zero : 0 ∈ K
h_clos : IsClosed (↑L \ {0})
n : ℕ
this : μ F * 2 ^ finrank ℝ E < μ ↑(S n)
⊢ (Z n).Nonempty

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.UnifTight

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
hfinε : ε ≠ ∞
hμ : μ ≠ 0
hε' : 0 < ε / 3
Eg : Set α
hmEg : MeasurableSet Eg
hμEg : μ Eg < ∞
hgε : eLpNorm (Egᶜ.indicator g) p μ < ε / 3
Ef : Set α
hmEf : MeasurableSet Ef
hμEf : μ Ef < ∞
hfε : ∀ (i : ℕ), eLpNorm (Efᶜ.indicator (f i)) p μ ≤ ε / 3
E : Set α := Ef ∪ Eg
hmE : MeasurableSet E
hfmE : μ E < ∞
hgE' : MemLp g p (μ.restrict E)
huiE : UnifIntegrable f p (μ.restrict E)
hfgE : ∀ᵐ (x : α) ∂μ.restrict E, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ffmE : Fact (μ E < ∞)
hInner : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p (μ.restrict E) ≤ ε
N : ℕ
hfngε : ∀ n ≥ N, eLpNorm (f n - g) p (μ.restrict E) ≤ ε / 3
n : ℕ
hn : n ≥ N
hmfngE : AEStronglyMeasurable (E.indicator (f n - g)) μ
hfngEε : eLpNorm (E.indicator (f n - g)) p μ ≤ ε / 3
hmgEc : AEStronglyMeasurable (Eᶜ.indicator g) μ
hgEcε : eLpNorm (Eᶜ.indicator g) p μ ≤ ε / 3
hmfnEc : AEStronglyMeasurable (Eᶜ.indicator (f n)) μ
hfnEcε : eLpNorm (Eᶜ.indicator (f n)) p μ ≤ ε / 3
hmfngEc : AEStronglyMeasurable (Eᶜ.indicator (f n - g)) μ
hfngEcε : eLpNorm (Eᶜ.indicator (f n - g)) p μ ≤ ε / 3 + ε / 3
⊢ eLpNorm (f n - g) p μ = eLpNorm (Eᶜ.indicator (f n - g) + E.indicator (f n - g)) p μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.UnifTight

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
hfinε : ε ≠ ∞
hμ : μ ≠ 0
hε' : 0 < ε / 3
Eg : Set α
hmEg : MeasurableSet Eg
hμEg : μ Eg < ∞
hgε : eLpNorm (Egᶜ.indicator g) p μ < ε / 3
Ef : Set α
hmEf : MeasurableSet Ef
hμEf : μ Ef < ∞
hfε : ∀ (i : ℕ), eLpNorm (Efᶜ.indicator (f i)) p μ ≤ ε / 3
E : Set α := Ef ∪ Eg
hmE : MeasurableSet E
hfmE : μ E < ∞
hgE' : MemLp g p (μ.restrict E)
huiE : UnifIntegrable f p (μ.restrict E)
hfgE : ∀ᵐ (x : α) ∂μ.restrict E, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ffmE : Fact (μ E < ∞)
hInner : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p (μ.restrict E) ≤ ε
N : ℕ
hfngε : ∀ n ≥ N, eLpNorm (f n - g) p (μ.restrict E) ≤ ε / 3
n : ℕ
hn : n ≥ N
hmfngE : AEStronglyMeasurable (E.indicator (f n - g)) μ
hfngEε : eLpNorm (E.indicator (f n - g)) p μ ≤ ε / 3
hmgEc : AEStronglyMeasurable (Eᶜ.indicator g) μ
hgEcε : eLpNorm (Eᶜ.indicator g) p μ ≤ ε / 3
hmfnEc : AEStronglyMeasurable (Eᶜ.indicator (f n)) μ
hfnEcε : eLpNorm (Eᶜ.indicator (f n)) p μ ≤ ε / 3
hmfngEc : AEStronglyMeasurable (Eᶜ.indicator (f n - g)) μ
hfngEcε : eLpNorm (Eᶜ.indicator (f n - g)) p μ ≤ ε / 3 + ε / 3
⊢ AEStronglyMeasurable (Eᶜ.indicator (f n - g)) μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.UnifTight

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
hfinε : ε ≠ ∞
hμ : μ ≠ 0
hε' : 0 < ε / 3
Eg : Set α
hmEg : MeasurableSet Eg
hμEg : μ Eg < ∞
hgε : eLpNorm (Egᶜ.indicator g) p μ < ε / 3
Ef : Set α
hmEf : MeasurableSet Ef
hμEf : μ Ef < ∞
hfε : ∀ (i : ℕ), eLpNorm (Efᶜ.indicator (f i)) p μ ≤ ε / 3
E : Set α := Ef ∪ Eg
hmE : MeasurableSet E
hfmE : μ E < ∞
hgE' : MemLp g p (μ.restrict E)
huiE : UnifIntegrable f p (μ.restrict E)
hfgE : ∀ᵐ (x : α) ∂μ.restrict E, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ffmE : Fact (μ E < ∞)
hInner : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p (μ.restrict E) ≤ ε
N : ℕ
hfngε : ∀ n ≥ N, eLpNorm (f n - g) p (μ.restrict E) ≤ ε / 3
n : ℕ
hn : n ≥ N
hmfngE : AEStronglyMeasurable (E.indicator (f n - g)) μ
hfngEε : eLpNorm (E.indicator (f n - g)) p μ ≤ ε / 3
hmgEc : AEStronglyMeasurable (Eᶜ.indicator g) μ
hgEcε : eLpNorm (Eᶜ.indicator g) p μ ≤ ε / 3
hmfnEc : AEStronglyMeasurable (Eᶜ.indicator (f n)) μ
hfnEcε : eLpNorm (Eᶜ.indicator (f n)) p μ ≤ ε / 3
hmfngEc : AEStronglyMeasurable (Eᶜ.indicator (f n - g)) μ
hfngEcε : eLpNorm (Eᶜ.indicator (f n - g)) p μ ≤ ε / 3 + ε / 3
⊢ AEStronglyMeasurable (E.indicator (f n - g)) μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.UnifTight

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
hfinε : ε ≠ ∞
hμ : μ ≠ 0
hε' : 0 < ε / 3
Eg : Set α
hmEg : MeasurableSet Eg
hμEg : μ Eg < ∞
hgε : eLpNorm (Egᶜ.indicator g) p μ < ε / 3
Ef : Set α
hmEf : MeasurableSet Ef
hμEf : μ Ef < ∞
hfε : ∀ (i : ℕ), eLpNorm (Efᶜ.indicator (f i)) p μ ≤ ε / 3
E : Set α := Ef ∪ Eg
hmE : MeasurableSet E
hfmE : μ E < ∞
hgE' : MemLp g p (μ.restrict E)
huiE : UnifIntegrable f p (μ.restrict E)
hfgE : ∀ᵐ (x : α) ∂μ.restrict E, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ffmE : Fact (μ E < ∞)
hInner : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p (μ.restrict E) ≤ ε
N : ℕ
hfngε : ∀ n ≥ N, eLpNorm (f n - g) p (μ.restrict E) ≤ ε / 3
n : ℕ
hn : n ≥ N
hmfngE : AEStronglyMeasurable (E.indicator (f n - g)) μ
hfngEε : eLpNorm (E.indicator (f n - g)) p μ ≤ ε / 3
hmgEc : AEStronglyMeasurable (Eᶜ.indicator g) μ
hgEcε : eLpNorm (Eᶜ.indicator g) p μ ≤ ε / 3
hmfnEc : AEStronglyMeasurable (Eᶜ.indicator (f n)) μ
hfnEcε : eLpNorm (Eᶜ.indicator (f n)) p μ ≤ ε / 3
hmfngEc : AEStronglyMeasurable (Eᶜ.indicator (f n - g)) μ
hfngEcε : eLpNorm (Eᶜ.indicator (f n - g)) p μ ≤ ε / 3 + ε / 3
⊢ eLpNorm (Eᶜ.indicator (f n - g) + E.indicator (f n - g)) p μ ≤
    eLpNorm (Eᶜ.indicator (f n - g)) p μ + eLpNorm (E.indicator (f n - g)) p μ

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.UnifTight

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : ε > 0
hfinε : ε ≠ ∞
hμ : μ ≠ 0
hε' : 0 < ε / 3
Eg : Set α
hmEg : MeasurableSet Eg
hμEg : μ Eg < ∞
hgε : eLpNorm (Egᶜ.indicator g) p μ < ε / 3
Ef : Set α
hmEf : MeasurableSet Ef
hμEf : μ Ef < ∞
hfε : ∀ (i : ℕ), eLpNorm (Efᶜ.indicator (f i)) p μ ≤ ε / 3
E : Set α := Ef ∪ Eg
hmE : MeasurableSet E
hfmE : μ E < ∞
hgE' : MemLp g p (μ.restrict E)
huiE : UnifIntegrable f p (μ.restrict E)
hfgE : ∀ᵐ (x : α) ∂μ.restrict E, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ffmE : Fact (μ E < ∞)
hInner : ∀ ε > 0, ∃ N, ∀ n ≥ N, eLpNorm (f n - g) p (μ.restrict E) ≤ ε
N : ℕ
hfngε : ∀ n ≥ N, eLpNorm (f n - g) p (μ.restrict E) ≤ ε / 3
n : ℕ
hn : n ≥ N
hmfngE : AEStronglyMeasurable (E.indicator (f n - g)) μ
hfngEε : eLpNorm (E.indicator (f n - g)) p μ ≤ ε / 3
hmgEc : AEStronglyMeasurable (Eᶜ.indicator g) μ
hgEcε : eLpNorm (Eᶜ.indicator g) p μ ≤ ε / 3
hmfnEc : AEStronglyMeasurable (Eᶜ.indicator (f n)) μ
hfnEcε : eLpNorm (Eᶜ.indicator (f n)) p μ ≤ ε / 3
hmfngEc : AEStronglyMeasurable (Eᶜ.indicator (f n - g)) μ
hfngEcε : eLpNorm (Eᶜ.indicator (f n - g)) p μ ≤ ε / 3 + ε / 3
⊢ ε / 3 + ε / 3 + ε / 3 = ε

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.UnifTight

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
hf : ∀ (n : ℕ), StronglyMeasurable (f n)
hg : StronglyMeasurable g
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
⊢ Tendsto (fun n ↦ eLpNorm (f n - g) p μ) atTop (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Group.GeometryOfNumbers

Goal state before:
pp: E : Type u_1
inst✝⁸ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : Nontrivial E
inst✝² : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝¹ : Countable ↥L
inst✝ : DiscreteTopology ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h_cpt : IsCompact s
h : μ F * 2 ^ finrank ℝ E ≤ μ s
h_mes : μ s ≠ 0
h_nemp : s.Nonempty
u : ℕ → ℝ≥0 := ⋯.choose
K : ConvexBody E := { carrier := s, convex' := h_conv, isCompact' := h_cpt, nonempty' := h_nemp }
S : ℕ → ConvexBody E := fun n ↦ (1 + u n) • K
Z : ℕ → Set E := fun n ↦ ↑(S n) ∩ (↑L \ {0})
h_zero : 0 ∈ K
h_clos : IsClosed (↑L \ {0})
n : ℕ
⊢ 0 ≤ 1 + ↑(u n)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.UnifTight

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
f f' : ℕ → α → β
g g' : α → β
hff' : ∀ (n : ℕ), f n =ᵐ[μ] f' n
hgg' : g =ᵐ[μ] g'
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
⊢ ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f' n x) atTop (𝓝 (g' x))

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Group.GeometryOfNumbers

Goal state before:
pp: E : Type u_1
inst✝⁸ : MeasurableSpace E
μ : Measure E
F s : Set E
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : BorelSpace E
inst✝⁴ : FiniteDimensional ℝ E
inst✝³ : Nontrivial E
inst✝² : μ.IsAddHaarMeasure
L : AddSubgroup E
inst✝¹ : Countable ↥L
inst✝ : DiscreteTopology ↥L
fund : IsAddFundamentalDomain (↥L) F μ
h_symm : ∀ x ∈ s, -x ∈ s
h_conv : Convex ℝ s
h_cpt : IsCompact s
h : μ F * 2 ^ finrank ℝ E ≤ μ s
⊢ ∃ x, x ≠ 0 ∧ ↑x ∈ s

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Homotopy.Affine

Goal state before:
pp: X : Type u_1
E : Type u_2
inst✝⁵ : TopologicalSpace X
inst✝⁴ : AddCommGroup E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : Module ℝ E
inst✝ : ContinuousSMul ℝ E
f g : C(X, E)
⊢ Continuous fun x ↦ (Path.segment (f x.2) (g x.2)) x.1

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Homotopy.Affine

Goal state before:
pp: X : Type u_1
E : Type u_2
inst✝⁵ : TopologicalSpace X
inst✝⁴ : AddCommGroup E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : Module ℝ E
inst✝ : ContinuousSMul ℝ E
f g : C(X, E)
⊢ ∀ (x : X), (Path.segment (f (0, x).2) (g (0, x).2)) (0, x).1 = f x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Homotopy.Affine

Goal state before:
pp: X : Type u_1
E : Type u_2
inst✝⁵ : TopologicalSpace X
inst✝⁴ : AddCommGroup E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : Module ℝ E
inst✝ : ContinuousSMul ℝ E
f g : C(X, E)
⊢ ∀ (x : X), (Path.segment (f (1, x).2) (g (1, x).2)) (1, x).1 = g x

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
g : α →C_c β
toFun✝ : α → β
continuous_toFun✝ : Continuous toFun✝
hasCompactSupport'✝ : HasCompactSupport { toFun := toFun✝, continuous_toFun := continuous_toFun✝ }.toFun
h :
  (fun f ↦ f.toFun)
      { toFun := toFun✝, continuous_toFun := continuous_toFun✝, hasCompactSupport' := hasCompactSupport'✝ } =
    (fun f ↦ f.toFun) g
⊢ { toFun := toFun✝, continuous_toFun := continuous_toFun✝, hasCompactSupport' := hasCompactSupport'✝ } = g

Tactic:
obtain ⟨⟨_, _⟩, _⟩ := g

Goal state after:
F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
toFun✝¹ : α → β
continuous_toFun✝¹ : Continuous toFun✝¹
hasCompactSupport'✝¹ : HasCompactSupport { toFun := toFun✝¹, continuous_toFun := continuous_toFun✝¹ }.toFun
toFun✝ : α → β
continuous_toFun✝ : Continuous toFun✝
hasCompactSupport'✝ : HasCompactSupport { toFun := toFun✝, continuous_toFun := continuous_toFun✝ }.toFun
h :
  (fun f ↦ f.toFun)
      { toFun := toFun✝¹, continuous_toFun := continuous_toFun✝¹, hasCompactSupport' := hasCompactSupport'✝¹ } =
    (fun f ↦ f.toFun)
      { toFun := toFun✝, continuous_toFun := continuous_toFun✝, hasCompactSupport' := hasCompactSupport'✝ }
⊢ { toFun := toFun✝¹, continuous_toFun := continuous_toFun✝¹, hasCompactSupport' := hasCompactSupport'✝¹ } =
    { toFun := toFun✝, continuous_toFun := continuous_toFun✝, hasCompactSupport' := hasCompactSupport'✝ }
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
f g : α →C_c β
h : (fun f ↦ f.toFun) f = (fun f ↦ f.toFun) g
⊢ f = g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
f : α →C_c β
f' : α → β
h : f' = ⇑f
⊢ Continuous f'

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
inst✝ : Zero β
f : α →C_c β
f' : α → β
h : f' = ⇑f
⊢ HasCompactSupport f'

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ✝ : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace β
inst✝² : Zero β
γ : Type u_5
inst✝¹ : TopologicalSpace γ
inst✝ : Zero γ
g : C(β, γ)
f : α →C_c β
⊢ C(α, γ)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.UnifTight

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
hp : 1 ≤ p
hp' : p ≠ ∞
f : ℕ → α → β
g : α → β
haef : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg' : MemLp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
⊢ Tendsto (fun n ↦ eLpNorm (f n - g) p μ) atTop (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ✝ : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace β
inst✝² : Zero β
γ : Type u_5
inst✝¹ : TopologicalSpace γ
inst✝ : Zero γ
g : C(β, γ)
f : α →C_c β
⊢ HasCompactSupport (if g 0 = 0 then g.comp ↑f else 0).toFun

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace β
inst✝² : Zero β
γ : Type u_5
inst✝¹ : TopologicalSpace γ
inst✝ : Zero γ
g : C(β, γ)
hg : g 0 = 0
f : α →C_c β
⊢ ⇑(compLeft g f) = ⇑g ∘ ⇑f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace β
inst✝² : Zero β
γ : Type u_5
inst✝¹ : TopologicalSpace γ
inst✝ : Zero γ
g : C(β, γ)
hg : g 0 = 0
f : α →C_c β
a : α
⊢ (compLeft g f) a = g (f a)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β✝ : Type u_3
γ✝ : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : TopologicalSpace β✝
inst✝⁴ : Zero β✝
γ : Type u_5
inst✝³ : TopologicalSpace γ
inst✝² : Zero γ
β : Type u_6
inst✝¹ : PseudoMetricSpace β
inst✝ : Zero β
f : α →C_c β
this : IsCompact (Set.range ⇑f)
C : ℝ
hC : ∀ ⦃x : β⦄, x ∈ Set.range ⇑f → ∀ ⦃y : β⦄, y ∈ Set.range ⇑f → dist x y ≤ C
⊢ ∀ (x y : α), dist (f x) (f y) ≤ C

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β✝ : Type u_3
γ✝ : Type u_4
inst✝⁶ : TopologicalSpace α
inst✝⁵ : TopologicalSpace β✝
inst✝⁴ : Zero β✝
γ : Type u_5
inst✝³ : TopologicalSpace γ
inst✝² : Zero γ
β : Type u_6
inst✝¹ : PseudoMetricSpace β
inst✝ : Zero β
f : α →C_c β
⊢ ∃ C, ∀ (x y : α), dist (f x) (f y) ≤ C

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : TopologicalSpace α
inst✝¹ : TopologicalSpace β
x : α
inst✝ : Zero β
⊢ HasCompactSupport ⇑0

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Function.UnifTight

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝ : NormedAddCommGroup β
μ : Measure α
p : ℝ≥0∞
f : ℕ → α → β
g : α → β
hp : 1 ≤ p
hp' : p ≠ ∞
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
hg : MemLp g p μ
hui : UnifIntegrable f p μ
hut : UnifTight f p μ
hfg : TendstoInMeasure μ f atTop g
⊢ Tendsto (fun n ↦ eLpNorm (f n - g) p μ) atTop (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : AddCommMonoid β
inst✝ : ContinuousAdd β
ι : Type u_5
s : Finset ι
f : ι → α →C_c β
a : α
⊢ (∑ i ∈ s, f i) a = ∑ i ∈ s, (f i) a

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
x : α
inst✝¹ : AddGroup β
inst✝ : IsTopologicalAddGroup β
f✝ g f : α →C_c β
⊢ HasCompactSupport (-⇑f.toContinuousMap)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
x : α
inst✝¹ : AddGroup β
inst✝ : IsTopologicalAddGroup β
f✝ g✝ f g : α →C_c β
⊢ HasCompactSupport (⇑f.toContinuousMap - ⇑g.toContinuousMap)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
x : α
R : Type u_5
inst✝⁵ : Semiring R
inst✝⁴ : NonUnitalNonAssocSemiring β
inst✝³ : IsTopologicalSemiring β
inst✝² : Module R β
inst✝¹ : ContinuousConstSMul R β
inst✝ : IsScalarTower R β β
r : R
f g : α →C_c β
⊢ (r • f) • g = r • f • g

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Gamma

Goal state before:
pp: p q : ℝ
hp : 0 < p
hq : -1 < q
⊢ ∫ (x : ℝ) in Ioi 0, x ^ q * rexp (-x ^ p) =
    ∫ (x : ℝ) in Ioi 0, (1 / p * x ^ (1 / p - 1)) • ((x ^ (1 / p)) ^ q * rexp (-x))

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁷ : TopologicalSpace α
inst✝⁶ : TopologicalSpace β
x : α
R : Type u_5
inst✝⁵ : Semiring R
inst✝⁴ : NonUnitalNonAssocSemiring β
inst✝³ : IsTopologicalSemiring β
inst✝² : Module R β
inst✝¹ : ContinuousConstSMul R β
inst✝ : SMulCommClass R β β
r : R
f g : α →C_c β
⊢ r • f • g = f • r • g

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace β
inst✝² : AddMonoid β
inst✝¹ : StarAddMonoid β
inst✝ : ContinuousStar β
f : α →C_c β
⊢ (Function.support fun x ↦ star (f x)) = Function.support ⇑f

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace β
inst✝² : AddMonoid β
inst✝¹ : StarAddMonoid β
inst✝ : ContinuousStar β
f : α →C_c β
⊢ HasCompactSupport fun x ↦ star (f x)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Gamma

Goal state before:
pp: p q : ℝ
hp : 0 < p
hq : -1 < q
⊢ ∫ (x : ℝ) in Ioi 0, (1 / p * x ^ (1 / p - 1)) • ((x ^ (1 / p)) ^ q * rexp (-x)) =
    ∫ (x : ℝ) in Ioi 0, 1 / p * rexp (-x) * x ^ (1 / p - 1 + q / p)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁴ : TopologicalSpace α
inst✝³ : SemilatticeSup β
inst✝² : Zero β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousSup β
ι : Type u_5
s : Finset ι
H : s.Nonempty
f : ι → α →C_c β
⊢ ⇑(s.sup' H f) = s.sup' H fun i ↦ ⇑(f i)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁴ : TopologicalSpace α
inst✝³ : SemilatticeInf β
inst✝² : Zero β
inst✝¹ : TopologicalSpace β
inst✝ : ContinuousInf β
ι : Type u_5
s : Finset ι
H : s.Nonempty
f : ι → α →C_c β
⊢ ⇑(s.inf' H f) = s.inf' H fun i ↦ ⇑(f i)

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Gamma

Goal state before:
pp: p q : ℝ
hp : 0 < p
hq : -1 < q
⊢ ∫ (x : ℝ) in Ioi 0, 1 / p * rexp (-x) * x ^ (1 / p - 1 + q / p) = 1 / p * Gamma ((q + 1) / p)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: F : Type u_1
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝⁴ : TopologicalSpace α
δ : Type u_5
inst✝³ : TopologicalSpace β
inst✝² : TopologicalSpace γ
inst✝¹ : TopologicalSpace δ
inst✝ : Zero δ
f : γ →C_c δ
g : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
⊢ HasCompactSupport ((↑f).comp ↑g).toFun

Tactic:
by

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Gamma

Goal state before:
pp: p q : ℝ
hp : 0 < p
hq : -1 < q
⊢ ∫ (x : ℝ) in Ioi 0, x ^ q * rexp (-x ^ p) = 1 / p * Gamma ((q + 1) / p)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.ContinuousMap.CompactlySupported

Goal state before:
pp: α : Type u_2
β : Type u_3
inst✝⁵ : TopologicalSpace α
inst✝⁴ : Zero β
inst✝³ : TopologicalSpace β
G : Type u_5
inst✝² : FunLike G α β
inst✝¹ : ContinuousMapClass G α β
inst✝ : CompactSpace α
⊢ ∀ (f : G), HasCompactSupport ⇑f

Tactic:
by

Goal state after: