mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inr.hq
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
A : Disjoint (Ioo a b) {b}
D : ↑f b - ↑f a = ↑f b - leftLim (↑f) b + (leftLim (↑f) b - ↑f a)
this : f.measure (Ioo a b) + ofReal (↑f b - leftLim (↑f) b) = ofReal (↑f b - leftLim (↑f) b + (leftLim (↑f) b - ↑f a))
⊢ 0 ≤ leftLim (↑f) b - ↑f a

Tactic:
· simp only [f.mono.le_leftLim hab, sub_nonneg]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
μ : Measure X
inst✝ : NormedSpace ℝ E
s : Set X
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun f ↦ ∫ (x : X) in s, ↑↑f x ∂μ

Tactic:
have : Fact ((1 : ℝ≥0∞) ≤ 1) := ⟨le_rfl⟩

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
μ : Measure X
inst✝ : NormedSpace ℝ E
s : Set X
this : Fact (1 ≤ 1)
⊢ Continuous[_, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun f ↦ ∫ (x : X) in s, ↑↑f x ∂μ
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
M : Type u_5
inst✝¹ : NormedAddCommGroup M
inst✝ : NormedSpace ℝ M
mX : MeasurableSpace X
ν : Measure X
F : X → M
⊢ ∫ (x : X) in support F, F x ∂ν = ∫ (x : X) in univ, F x ∂ν

Tactic:
setIntegral_eq_of_subset_of_forall_sdiff_eq_zero MeasurableSet.univ (subset_univ (support F))

Goal state after:
X : Type u_1
M : Type u_5
inst✝¹ : NormedAddCommGroup M
inst✝ : NormedSpace ℝ M
mX : MeasurableSpace X
ν : Measure X
F : X → M
⊢ ∫ (x : X) in support F, F x ∂ν = ∫ (x : X) in support F, F x ∂ν

X : Type u_1
M : Type u_5
inst✝¹ : NormedAddCommGroup M
inst✝ : NormedSpace ℝ M
mX : MeasurableSpace X
ν : Measure X
F : X → M
⊢ ∀ x ∈ univ \ support F, F x = 0
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
cd : Core 𝕜 F
tF : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜 F
h : ContinuousAt (fun v ↦ ⟪v, v⟫) 0
h' : IsVonNBounded 𝕜 {v | re ⟪v, v⟫ < 1}
p : Seminorm 𝕜 F := normSeminorm 𝕜 F
v : F
⊢ √(re ⟪v, v⟫) < 1 ↔ re ⟪v, v⟫ < 1

Tactic:
conv_lhs => rw [show (1 : ℝ) = √1 by simp]

Goal state after:
𝕜 : Type u_1
F : Type u_3
inst✝⁴ : RCLike 𝕜
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
cd : Core 𝕜 F
tF : TopologicalSpace F
inst✝¹ : IsTopologicalAddGroup F
inst✝ : ContinuousConstSMul 𝕜 F
h : ContinuousAt (fun v ↦ ⟪v, v⟫) 0
h' : IsVonNBounded 𝕜 {v | re ⟪v, v⟫ < 1}
p : Seminorm 𝕜 F := normSeminorm 𝕜 F
v : F
⊢ √(re ⟪v, v⟫) < √1 ↔ re ⟪v, v⟫ < 1
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: Y : Type u_2
E : Type u_3
X : Type u_5
inst✝⁹ : TopologicalSpace X
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : MeasurableSpace Y
inst✝⁶ : OpensMeasurableSpace Y
μ : Measure Y
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FirstCountableTopology X
inst✝² : LocallyCompactSpace X
inst✝¹ : SecondCountableTopologyEither Y E
inst✝ : IsLocallyFiniteMeasure μ
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
s : Set Y
hs : IsCompact s
x₀ : X
U : Set X
U_cpct : IsCompact U
U_nhds : U ∈ 𝓝 x₀
M : ℝ
hM : M ∈ upperBounds ((fun x ↦ ‖uncurry f x‖) '' U ×ˢ s)
x : X
x_in : x ∈ U
⊢ ∀ᵐ (a : Y) ∂μ.restrict s, ‖f x a‖ ≤ ?bound a

Tactic:
rw [ae_restrict_iff]

Goal state after:
Y : Type u_2
E : Type u_3
X : Type u_5
inst✝⁹ : TopologicalSpace X
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : MeasurableSpace Y
inst✝⁶ : OpensMeasurableSpace Y
μ : Measure Y
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FirstCountableTopology X
inst✝² : LocallyCompactSpace X
inst✝¹ : SecondCountableTopologyEither Y E
inst✝ : IsLocallyFiniteMeasure μ
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
s : Set Y
hs : IsCompact s
x₀ : X
U : Set X
U_cpct : IsCompact U
U_nhds : U ∈ 𝓝 x₀
M : ℝ
hM : M ∈ upperBounds ((fun x ↦ ‖uncurry f x‖) '' U ×ˢ s)
x : X
x_in : x ∈ U
⊢ ∀ᵐ (x_1 : Y) ∂μ, x_1 ∈ s → ‖f x x_1‖ ≤ ?bound x_1

Y : Type u_2
E : Type u_3
X : Type u_5
inst✝⁹ : TopologicalSpace X
inst✝⁸ : TopologicalSpace Y
inst✝⁷ : MeasurableSpace Y
inst✝⁶ : OpensMeasurableSpace Y
μ : Measure Y
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FirstCountableTopology X
inst✝² : LocallyCompactSpace X
inst✝¹ : SecondCountableTopologyEither Y E
inst✝ : IsLocallyFiniteMeasure μ
f : X → Y → E
hf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (uncurry f)
s : Set Y
hs : IsCompact s
x₀ : X
U : Set X
U_cpct : IsCompact U
U_nhds : U ∈ 𝓝 x₀
M : ℝ
hM : M ∈ upperBounds ((fun x ↦ ‖uncurry f x‖) '' U ×ˢ s)
x : X
x_in : x ∈ U
⊢ MeasurableSet {x_1 | ‖f x x_1‖ ≤ ?bound x_1}
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: Y : Type u_2
E : Type u_3
F : Type u_4
X : Type u_5
G : Type u_6
𝕜 : Type u_7
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : TopologicalSpace Y
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : OpensMeasurableSpace Y
μ : Measure Y
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : NormedSpace 𝕜 E
L : F →L[𝕜] G →L[𝕜] E
f : X → Y → G
s : Set X
k : Set Y
g : Y → F
hk : IsCompact k
hf : ContinuousOn (uncurry f) (s ×ˢ univ)
hfs : ∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0
hg : IntegrableOn g k μ
A : ∀ p ∈ s, Continuous[inst✝¹⁰, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f p)
q : X
hq : q ∈ s
ε : ℝ
εpos : ε > 0
⊢ ∃ δ, 0 < δ ∧ ∫ (x : Y) in k, ‖L‖ * ‖g x‖ * δ ∂μ < ε

Tactic:
by
    simpa [integral_mul_const] using exists_pos_mul_lt εpos _

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: Y : Type u_2
E : Type u_3
F : Type u_4
X : Type u_5
G : Type u_6
𝕜 : Type u_7
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : TopologicalSpace Y
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : OpensMeasurableSpace Y
μ : Measure Y
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : NormedSpace 𝕜 E
L : F →L[𝕜] G →L[𝕜] E
f : X → Y → G
s : Set X
k : Set Y
g : Y → F
hk : IsCompact k
hf : ContinuousOn (uncurry f) (s ×ˢ univ)
hfs : ∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0
hg : IntegrableOn g k μ
A : ∀ p ∈ s, Continuous[inst✝¹⁰, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f p)
q : X
hq : q ∈ s
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : 0 < δ
hδ : ∫ (x : Y) in k, ‖L‖ * ‖g x‖ * δ ∂μ < ε
v : Set X
v_mem : v ∈ 𝓝[s] q
hv : ∀ p ∈ v, ∀ x ∈ k, ‖f p x - f q x‖ < δ
p : X
hp : p ∈ s
C : ℝ
hC : ∀ (y : Y), ‖f p y‖ ≤ C
this : IntegrableOn (fun y ↦ ‖L‖ * ‖g y‖ * C) k μ
this✝¹ : MeasurableSpace G := ⋯
this✝ : BorelSpace G
⊢ AEStronglyMeasurable (f p) (μ.restrict k)

Tactic:
apply StronglyMeasurable.aestronglyMeasurable

Goal state after:
Y : Type u_2
E : Type u_3
F : Type u_4
X : Type u_5
G : Type u_6
𝕜 : Type u_7
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : TopologicalSpace Y
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : OpensMeasurableSpace Y
μ : Measure Y
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : NormedSpace 𝕜 E
L : F →L[𝕜] G →L[𝕜] E
f : X → Y → G
s : Set X
k : Set Y
g : Y → F
hk : IsCompact k
hf : ContinuousOn (uncurry f) (s ×ˢ univ)
hfs : ∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0
hg : IntegrableOn g k μ
A : ∀ p ∈ s, Continuous[inst✝¹⁰, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f p)
q : X
hq : q ∈ s
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : 0 < δ
hδ : ∫ (x : Y) in k, ‖L‖ * ‖g x‖ * δ ∂μ < ε
v : Set X
v_mem : v ∈ 𝓝[s] q
hv : ∀ p ∈ v, ∀ x ∈ k, ‖f p x - f q x‖ < δ
p : X
hp : p ∈ s
C : ℝ
hC : ∀ (y : Y), ‖f p y‖ ≤ C
this : IntegrableOn (fun y ↦ ‖L‖ * ‖g y‖ * C) k μ
this✝¹ : MeasurableSpace G := ⋯
this✝ : BorelSpace G
⊢ StronglyMeasurable (f p)
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
hx₀ : ¬x = 0
this✝ : ‖x‖ ^ 2 ≠ 0
this : InnerProductSpace.Core 𝕜 E := InnerProductSpace.toCore
h : ⟪x, y⟫ • x = ⟪x, x⟫ • y
⊢ y = (⟪x, y⟫ / ⟪x, x⟫) • x

Tactic:
rw [div_eq_inv_mul, mul_smul, h, inv_smul_smul₀]

Goal state after:
case ha
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
hx₀ : ¬x = 0
this✝ : ‖x‖ ^ 2 ≠ 0
this : InnerProductSpace.Core 𝕜 E := InnerProductSpace.toCore
h : ⟪x, y⟫ • x = ⟪x, x⟫ • y
⊢ ⟪x, x⟫ ≠ 0
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: Y : Type u_2
E : Type u_3
F : Type u_4
X : Type u_5
G : Type u_6
𝕜 : Type u_7
inst✝¹¹ : TopologicalSpace X
inst✝¹⁰ : TopologicalSpace Y
inst✝⁹ : MeasurableSpace Y
inst✝⁸ : OpensMeasurableSpace Y
μ : Measure Y
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : NormedSpace 𝕜 E
L : F →L[𝕜] G →L[𝕜] E
f : X → Y → G
s : Set X
k : Set Y
g : Y → F
hk : IsCompact k
hf : ContinuousOn (uncurry f) (s ×ˢ univ)
hfs : ∀ (p : X) (x : Y), p ∈ s → x ∉ k → f p x = 0
hg : IntegrableOn g k μ
A : ∀ p ∈ s, Continuous[inst✝¹⁰, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (f p)
q : X
hq : q ∈ s
ε : ℝ
εpos : ε > 0
δ : ℝ
δpos : 0 < δ
hδ : ∫ (x : Y) in k, ‖L‖ * ‖g x‖ * δ ∂μ < ε
v : Set X
v_mem : v ∈ 𝓝[s] q
I : ∀ p ∈ s, IntegrableOn (fun y ↦ (L (g y)) (f p y)) k μ
p : X
hp : p ∈ v
h'p : p ∈ s
y : Y
hv : ‖f p y - f q y‖ < δ
hy : y ∈ k
⊢ ‖(L (g y)) (f p y) - (L (g y)) (f q y)‖ = ‖(L (g y)) (f p y - f q y)‖

Tactic:
by simp only [map_sub]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: case inr
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
h₀ : x ≠ 0
⊢ ⟪x, y⟫ = ↑‖x‖ * ↑‖y‖ ↔ ↑‖y‖ • x = ↑‖x‖ • y

Tactic:
· rw [inner_eq_norm_mul_iff_div h₀, div_eq_inv_mul, mul_smul, inv_smul_eq_iff₀]
    rwa [Ne, ofReal_eq_zero, norm_eq_zero]

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Subspace

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : DecidableEq ι
f : (i : ι) → G i
s₁ s₂ : Finset ι
F : (i : ι) → G i := fun i ↦ if i ∈ s₁ then f i else -f i
hF₁ : ∀ i ∈ s₁ \ s₂, F i = f i
hF₂ : ∀ i ∈ s₂ \ s₁, F i = -f i
⊢ ‖∑ x ∈ s₁ \ s₂, (V x) (f x) + ∑ x ∈ s₂ \ s₁, -(V x) (f x)‖ ^ 2 = ∑ i ∈ s₁ \ s₂, ‖f i‖ ^ 2 + ∑ i ∈ s₂ \ s₁, ‖f i‖ ^ 2

Tactic:
have hF : ∀ i, ‖F i‖ = ‖f i‖ := by
    intro i
    dsimp only [F]
    split_ifs <;> simp only [norm_neg]

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝² : (i : ι) → NormedAddCommGroup (G i)
inst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
inst✝ : DecidableEq ι
f : (i : ι) → G i
s₁ s₂ : Finset ι
F : (i : ι) → G i := fun i ↦ if i ∈ s₁ then f i else -f i
hF₁ : ∀ i ∈ s₁ \ s₂, F i = f i
hF₂ : ∀ i ∈ s₂ \ s₁, F i = -f i
hF : ∀ (i : ι), ‖F i‖ = ‖f i‖
⊢ ‖∑ x ∈ s₁ \ s₂, (V x) (f x) + ∑ x ∈ s₂ \ s₁, -(V x) (f x)‖ ^ 2 = ∑ i ∈ s₁ \ s₂, ‖f i‖ ^ 2 + ∑ i ∈ s₂ \ s₁, ‖f i‖ ^ 2
Module: Mathlib.Analysis.InnerProductSpace.LinearMap

Goal state before:
pp: 𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
F : Type u_8
H : Type u_9
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
a c : F
b d : H
ha : a ≠ 0
hb : b ≠ 0
h : ∀ (x : H), ⟪b, x⟫_𝕜 • a = ⟪d, x⟫_𝕜 • c
h₂ : ∀ (x : F), ⟪a, x⟫_𝕜 • b = ⟪c, x⟫_𝕜 • d
h₃ : a = (⟪d, b⟫_𝕜 / ⟪b, b⟫_𝕜) • c
h₄ : b = (⟪d, b⟫_𝕜 / ⟪b, b⟫_𝕜 * (⟪c, c⟫_𝕜 / ⟪a, a⟫_𝕜)) • d
h₅ : ⟪d, b⟫_𝕜 ≠ 0
h₆ : c ≠ 0
⊢ 0 < ‖c‖ ^ 2 / ‖a‖ ^ 2 ∧ True

Tactic:
and_true

Goal state after:
𝕜 : Type u_4
inst✝⁴ : RCLike 𝕜
F : Type u_8
H : Type u_9
inst✝³ : NormedAddCommGroup F
inst✝² : InnerProductSpace 𝕜 F
inst✝¹ : NormedAddCommGroup H
inst✝ : InnerProductSpace 𝕜 H
a c : F
b d : H
ha : a ≠ 0
hb : b ≠ 0
h : ∀ (x : H), ⟪b, x⟫_𝕜 • a = ⟪d, x⟫_𝕜 • c
h₂ : ∀ (x : F), ⟪a, x⟫_𝕜 • b = ⟪c, x⟫_𝕜 • d
h₃ : a = (⟪d, b⟫_𝕜 / ⟪b, b⟫_𝕜) • c
h₄ : b = (⟪d, b⟫_𝕜 / ⟪b, b⟫_𝕜 * (⟪c, c⟫_𝕜 / ⟪a, a⟫_𝕜)) • d
h₅ : ⟪d, b⟫_𝕜 ≠ 0
h₆ : c ≠ 0
⊢ 0 < ‖c‖ ^ 2 / ‖a‖ ^ 2
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝³ : RingHomInvPair σ σ'
inst✝² : RingHomIsometric σ
inst✝¹ : RingHomIsometric σ'
inst✝ : CompleteSpace F
surj : Surjective ⇑f
y : F
x✝ : y ∈ Set.univ
x : E
hx : f x = y
n : ℕ
hn : ‖x‖ < ↑n
⊢ y ∈ ⋃ n, closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n)

Tactic:
refine mem_iUnion.2 ⟨n, subset_closure ?_⟩

Goal state after:
𝕜 : Type u_1
𝕜' : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝³ : RingHomInvPair σ σ'
inst✝² : RingHomIsometric σ
inst✝¹ : RingHomIsometric σ'
inst✝ : CompleteSpace F
surj : Surjective ⇑f
y : F
x✝ : y ∈ Set.univ
x : E
hx : f x = y
n : ℕ
hn : ‖x‖ < ↑n
⊢ y ∈ ⇑f '' ball 0 ↑n
Module: Mathlib.Analysis.InnerProductSpace.Projection.Reflection

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
v w : F
h : ‖v‖ = ‖w‖
R : F ≃ₗᵢ[ℝ] F := ⋯
h₁ : R (v - w) = -(v - w)
⊢ v + w ∈ (ℝ ∙ (v - w))ᗮ

Tactic:
Submodule.mem_orthogonal_singleton_iff_inner_left

Goal state after:
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
v w : F
h : ‖v‖ = ‖w‖
R : F ≃ₗᵢ[ℝ] F := (ℝ ∙ (v - w))ᗮ.reflection
h₁ : R (v - w) = -(v - w)
⊢ inner ℝ (v + w) (v - w) = 0
Module: Mathlib.Analysis.InnerProductSpace.Symmetric

Goal state before:
pp: case refine_1
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
U V : Submodule 𝕜 E
hUV : IsCompl U V
h : (U.subtype ∘ₗ U.projectionOnto V hUV).IsSymmetric
u : E
hu : u ∈ U
v : E
hv : v ∈ V
⊢ ⟪v, ↑⟨u, hu⟩⟫ = 0

Tactic:
← Subtype.coe_mk v hv,

Goal state after:
case refine_1
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
U V : Submodule 𝕜 E
hUV : IsCompl U V
h : (U.subtype ∘ₗ U.projectionOnto V hUV).IsSymmetric
u : E
hu : u ∈ U
v : E
hv : v ∈ V
⊢ ⟪↑⟨v, hv⟩, ↑⟨u, hu⟩⟫ = 0
Module: Mathlib.Analysis.InnerProductSpace.Projection.Submodule

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
K₀ K₁ : Submodule 𝕜 E
inst✝¹ : K₀.HasOrthogonalProjection
inst✝ : K₁.HasOrthogonalProjection
h : K₀ᗮ ≤ K₁ᗮ
⊢ K₁ ≤ K₀

Tactic:
simpa using orthogonal_le h

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝³ : RingHomInvPair σ σ'
inst✝² : RingHomIsometric σ
inst✝¹ : RingHomIsometric σ'
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n) = Set.univ
n : ℕ
a : F
ε : ℝ
εpos : ε > 0
H : ball a ε ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n)
c : 𝕜
hc : 1 < ‖c‖
y : F
hy : y ≠ 0
hc' : 1 < ‖σ c‖
d : 𝕜'
hd : d ≠ 0
ydlt : ‖d • y‖ < ε / 2
dinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖σ c‖ * ‖y‖
δ : ℝ := ‖d‖ * ‖y‖ / 4
δpos : 0 < δ
this✝ : a + d • y ∈ ball a ε
z₁ : F
z₁im : z₁ ∈ ⇑f '' ball 0 ↑n
x₁ : E
h₁ : dist (a + d • y) (f x₁) < δ
hx₁ : ‖x₁‖ < ↑n
xz₁ : f x₁ = z₁
this : a ∈ ball a ε
z₂ : F
z₂im : z₂ ∈ ⇑f '' ball 0 ↑n
x₂ : E
h₂ : dist a (f x₂) < δ
hx₂ : ‖x₂‖ < ↑n
xz₂ : f x₂ = z₂
x : E := x₁ - x₂
I : ‖f x - d • y‖ ≤ 2 * δ
⊢ ‖d⁻¹ • f x - d⁻¹ • d • y‖ = ‖d⁻¹ • (f x - d • y)‖

Tactic:
smul_sub

Goal state after:
𝕜 : Type u_1
𝕜' : Type u_2
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜' F
f : E →SL[σ] F
σ' : 𝕜' →+* 𝕜
inst✝³ : RingHomInvPair σ σ'
inst✝² : RingHomIsometric σ
inst✝¹ : RingHomIsometric σ'
inst✝ : CompleteSpace F
surj : Surjective ⇑f
A : ⋃ n, closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n) = Set.univ
n : ℕ
a : F
ε : ℝ
εpos : ε > 0
H : ball a ε ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (⇑f '' ball 0 ↑n)
c : 𝕜
hc : 1 < ‖c‖
y : F
hy : y ≠ 0
hc' : 1 < ‖σ c‖
d : 𝕜'
hd : d ≠ 0
ydlt : ‖d • y‖ < ε / 2
dinv : ‖d‖⁻¹ ≤ (ε / 2)⁻¹ * ‖σ c‖ * ‖y‖
δ : ℝ := ‖d‖ * ‖y‖ / 4
δpos : 0 < δ
this✝ : a + d • y ∈ ball a ε
z₁ : F
z₁im : z₁ ∈ ⇑f '' ball 0 ↑n
x₁ : E
h₁ : dist (a + d • y) (f x₁) < δ
hx₁ : ‖x₁‖ < ↑n
xz₁ : f x₁ = z₁
this : a ∈ ball a ε
z₂ : F
z₂im : z₂ ∈ ⇑f '' ball 0 ↑n
x₂ : E
h₂ : dist a (f x₂) < δ
hx₂ : ‖x₂‖ < ↑n
xz₂ : f x₂ = z₂
x : E := x₁ - x₂
I : ‖f x - d • y‖ ≤ 2 * δ
⊢ ‖d⁻¹ • f x - d⁻¹ • d • y‖ = ‖d⁻¹ • f x - d⁻¹ • d • y‖
Module: Mathlib.Analysis.InnerProductSpace.Projection.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
U : Submodule 𝕜 E
inst✝ : U.HasOrthogonalProjection
⊢ (LinearMap.id - ↑U.starProjection).range = (↑(ContinuousLinearMap.id 𝕜 E - U.starProjection)).range

Tactic:
toLinearMap_sub,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
U : Submodule 𝕜 E
inst✝ : U.HasOrthogonalProjection
⊢ (LinearMap.id - ↑U.starProjection).range = (↑(ContinuousLinearMap.id 𝕜 E) - ↑U.starProjection).range
Module: Mathlib.Analysis.InnerProductSpace.Projection.Basic

Goal state before:
pp: case hvo
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v w : E
⊢ ↑‖v‖ ^ 2 • w - ⟪v, w⟫ • v ∈ (𝕜 ∙ v)ᗮ

Tactic:
Submodule.mem_orthogonal_singleton_iff_inner_left

Goal state after:
case hvo
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v w : E
⊢ ⟪↑‖v‖ ^ 2 • w - ⟪v, w⟫ • v, v⟫ = 0
Module: Mathlib.Analysis.Normed.Operator.Banach

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NontriviallyNormedField 𝕜'
σ : 𝕜 →+* 𝕜'
E : Type u_3
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜 E
F : Type u_4
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace 𝕜' F
σ' : 𝕜' →+* 𝕜
inst✝⁵ : RingHomInvPair σ σ'
inst✝⁴ : RingHomIsometric σ
inst✝³ : RingHomIsometric σ'
inst✝² : CompleteSpace F
inst✝¹ : CompleteSpace E
inst✝ : RingHomInvPair σ' σ
f : E →SL[σ] F
hinj : Injective ⇑f
hclo : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (Set.range ⇑f)
x : E
⊢ (equivRange hinj hclo).symm ⟨f x, ⋯⟩ = x

Tactic:
by
  simp [ContinuousLinearEquiv.symm_apply_eq, Subtype.ext_iff]

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
ι : Type u_2
β : ι → Type u_4
inst✝² : Fact (1 ≤ p)
inst✝¹ : (i : ι) → PseudoEMetricSpace (β i)
inst✝ : Fintype ι
⊢ AntilipschitzWith (↑(Fintype.card ι) ^ (1 / p).toReal) ofLp

Tactic:
by
  intro x y
  rcases p.dichotomy with (rfl | h)
  · simp only [edist_eq_iSup, ENNReal.div_top, ENNReal.toReal_zero, NNReal.rpow_zero,
      ENNReal.coe_one, one_mul, iSup_le_iff]
    -- Porting note: `Finset.le_sup` needed some help
    exact fun i => Finset.le_sup (f := fun i => edist (x i) (y i)) (Finset.mem_univ i)
  · have pos : 0 < p.toReal := zero_lt_one.trans_le h
    have nonneg : 0 ≤ 1 / p.toReal := one_div_nonneg.2 (le_of_lt pos)
    have cancel : p.toReal * (1 / p.toReal) = 1 := mul_div_cancel₀ 1 (ne_of_gt pos)
    rw [edist_eq_sum pos, ENNReal.toReal_div 1 p]
    simp only [edist, ENNReal.toReal_one]
    calc
      (∑ i, edist (x i) (y i) ^ p.toReal) ^ (1 / p.toReal) ≤
          (∑ _i, edist (ofLp x) (ofLp y) ^ p.toReal) ^ (1 / p.toReal) := by
        gcongr with i
        exact Finset.le_sup (f := fun i => edist (x i) (y i)) (Finset.mem_univ i)
      _ =
          ((Fintype.card ι : ℝ≥0) ^ (1 / p.toReal) : ℝ≥0) *
            edist (ofLp x) (ofLp y) := by
        simp only [nsmul_eq_mul, Finset.card_univ, ENNReal.rpow_one, Finset.sum_const,
          ENNReal.mul_rpow_of_nonneg _ _ nonneg, ← ENNReal.rpow_mul, cancel]
        have : (Fintype.card ι : ℝ≥0∞) = (Fintype.card ι : ℝ≥0) :=
          (ENNReal.coe_natCast (Fintype.card ι)).symm
        rw [this, ENNReal.coe_rpow_of_nonneg _ nonneg]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.OfBasis

Goal state before:
pp: ι : Type u_1
E : Type u_3
inst✝² : Fintype ι
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
v : ι → E
x✝ : E
⊢ x✝ ∈ parallelepiped v ↔ x✝ ∈ ∑ i, segment ℝ 0 (v i)

Tactic:
simp only [mem_parallelepiped_iff, Set.mem_finsetSum, Finset.mem_univ, forall_true_left,
    segment_eq_image, smul_zero, zero_add, ← Set.pi_univ_Icc, Set.mem_univ_pi]

Goal state after:
ι : Type u_1
E : Type u_3
inst✝² : Fintype ι
inst✝¹ : AddCommGroup E
inst✝ : Module ℝ E
v : ι → E
x✝ : E
⊢ (∃ t, (∀ (i : ι), t i ∈ Icc (0 i) (1 i)) ∧ x✝ = ∑ i, t i • v i) ↔
    ∃ g, ∃ (_ : ∀ {i : ι}, g i ∈ (fun a ↦ a • v i) '' Icc 0 1), ∑ i, g i = x✝
Module: Mathlib.MeasureTheory.Measure.Haar.OfBasis

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
E : Type u_3
inst✝⁵ : Fintype ι
inst✝⁴ : Fintype ι'
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
b : Basis ι ℝ E
e : ι ≃ ι'
⊢ addHaarMeasure b.parallelepiped = b.addHaar

Tactic:
← Basis.addHaar

Goal state after:
ι : Type u_1
ι' : Type u_2
E : Type u_3
inst✝⁵ : Fintype ι
inst✝⁴ : Fintype ι'
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
b : Basis ι ℝ E
e : ι ≃ ι'
⊢ b.addHaar = b.addHaar
Module: Mathlib.MeasureTheory.Measure.Haar.OfBasis

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
E : Type u_3
F : Type u_4
inst✝¹⁰ : Fintype ι
inst✝⁹ : Fintype ι'
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : MeasurableSpace E
inst✝³ : BorelSpace E
inst✝² : MeasurableSpace F
inst✝¹ : BorelSpace F
inst✝ : SecondCountableTopologyEither E F
v : Basis ι ℝ E
w : Basis ι' ℝ F
⊢ (v.prod w).addHaar = v.addHaar.prod w.addHaar

Tactic:
have : FiniteDimensional ℝ E := v.finiteDimensional_of_finite
  have : FiniteDimensional ℝ F := w.finiteDimensional_of_finite
  simp [(v.prod w).addHaar_eq_iff, Basis.prod_parallelepiped, Basis.addHaar_self]

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.PiLp

Goal state before:
pp: p : ℝ≥0∞
𝕜 : Type u_1
ι : Type u_2
inst✝² : Finite ι
inst✝¹ : Ring 𝕜
inst✝ : DecidableEq ι
i : ι
⊢ (basisFun p 𝕜 ι) i = single p i 1

Tactic:
simp_rw [basisFun, Basis.coe_ofEquivFun, WithLp.coe_symm_linearEquiv, toLp_single]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Lebesgue.Basic

Goal state before:
pp: case pos
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
f g : α → ℝ
s : Set α
hf : Measurable f
hg : Measurable g
hs : MeasurableSet s
x : α
h : x ∈ s
hx : {a | x ∈ s ∧ a ∈ Ioo (f x) (g x)} = Ioo (f x) (g x)
⊢ volume {a | x ∈ s ∧ a ∈ Ioo (f x) (g x)} = ofReal (g x - f x)

Tactic:
simp only [hx, Real.volume_Ioo]

Goal state after: