mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
F : Type u_3
F' : Type u_4
G : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
T : Set α → F →L[ℝ] F'
h_add : FinMeasAdditive μ T
f : α →ₛ G
hf : Integrable (⇑f) μ
g : G → F
hg : g 0 = 0
⊢ setToSimpleFunc T (map g f) = ∑ x ∈ f.range, (T (⇑f ⁻¹' {x})) (g x)

Tactic:
have T_empty : T ∅ = 0 := h_add.map_empty_eq_zero

Goal state after:
α : Type u_1
F : Type u_3
F' : Type u_4
G : Type u_5
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
T : Set α → F →L[ℝ] F'
h_add : FinMeasAdditive μ T
f : α →ₛ G
hf : Integrable (⇑f) μ
g : G → F
hg : g 0 = 0
T_empty : T ∅ = 0
⊢ setToSimpleFunc T (map g f) = ∑ x ∈ f.range, (T (⇑f ⁻¹' {x})) (g x)
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
inst✝² : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
G : Type u_7
inst✝¹ : NormedAddCommGroup G
inst✝ : PartialOrder G
f g : ↥(simpleFunc G p μ)
⊢ ↑↑↑f ≤ᵐ[μ] ↑↑↑g ↔ f ≤ g

Tactic:
rw [← Subtype.coe_le_coe, ← Lp.coeFn_le]

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

Goal state before:
pp: α : Type u_1
inst✝² : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
G : Type u_7
inst✝¹ : NormedAddCommGroup G
inst✝ : PartialOrder G
hp : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
g : { g // 0 ≤ g }
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
hg_memLp : MemLp (↑↑↑g) p μ
zero_mem : 0 ∈ (Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}
this : SeparableSpace ↑((Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y})
g_meas : Measurable ↑↑↑g
x : ℕ → α →ₛ G := fun n ↦ approxOn (↑↑↑g) g_meas ((Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}) 0 zero_mem n
hx_nonneg : ∀ (n : ℕ), 0 ≤ x n
hx_memLp : ∀ (n : ℕ), MemLp (⇑(x n)) p μ
⊢ ∃ x, (∀ (n : ℕ), x n ∈ Set.range (coeSimpleFuncNonnegToLpNonneg p μ G)) ∧ Tendsto x atTop (𝓝 g)

Tactic:
have h_toLp := fun n => MemLp.coeFn_toLp (hx_memLp n)

Goal state after:
α : Type u_1
inst✝² : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
G : Type u_7
inst✝¹ : NormedAddCommGroup G
inst✝ : PartialOrder G
hp : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
g : { g // 0 ≤ g }
this✝¹ : MeasurableSpace G := borel G
this✝ : BorelSpace G
hg_memLp : MemLp (↑↑↑g) p μ
zero_mem : 0 ∈ (Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}
this : SeparableSpace ↑((Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y})
g_meas : Measurable ↑↑↑g
x : ℕ → α →ₛ G := fun n ↦ approxOn (↑↑↑g) g_meas ((Set.range ↑↑↑g ∪ {0}) ∩ {y | 0 ≤ y}) 0 zero_mem n
hx_nonneg : ∀ (n : ℕ), 0 ≤ x n
hx_memLp : ∀ (n : ℕ), MemLp (⇑(x n)) p μ
h_toLp : ∀ (n : ℕ), ↑↑(MemLp.toLp ⇑(x n) ⋯) =ᵐ[μ] ⇑(x n)
⊢ ∃ x, (∀ (n : ℕ), x n ∈ Set.range (coeSimpleFuncNonnegToLpNonneg p μ G)) ∧ Tendsto x atTop (𝓝 g)
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: case neg
α : Type u_1
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝² : PartialOrder F
inst✝¹ : IsOrderedAddMonoid F
inst✝ : IsOrderedModule ℝ F
ν : Measure α
f : α →ₛ F
hf : 0 ≤ᵐ[ν] ⇑f
hμν : μ ≤ ν
hfν : Integrable (⇑f) ν
x : α
hx : ¬0 ≤ f x
⊢ ν (⇑f ⁻¹' {f x}) = 0

Tactic:
← nonpos_iff_eq_zero,

Goal state after:
case neg
α : Type u_1
F : Type u_3
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝² : PartialOrder F
inst✝¹ : IsOrderedAddMonoid F
inst✝ : IsOrderedModule ℝ F
ν : Measure α
f : α →ₛ F
hf : 0 ≤ᵐ[ν] ⇑f
hμν : μ ≤ ν
hfν : Integrable (⇑f) ν
x : α
hx : ¬0 ≤ f x
⊢ ν (⇑f ⁻¹' {f x}) ≤ 0
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case e'_3
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_7
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
f : α → E
fmeas : Measurable f
hf : Integrable f μ
inst✝ : SeparableSpace ↑(range f ∪ {0})
n : ℕ
⊢ ∫ (x : α), ‖(approxOn f fmeas (range f ∪ {0}) 0 ⋯ n) x - f x‖ ∂μ =
    (ENNReal.toReal ∘ fun n ↦ ∫⁻ (x : α), ‖(approxOn f fmeas (range f ∪ {0}) 0 ⋯ n) x - f x‖ₑ ∂μ) n

Tactic:
rw [integral_norm_eq_lintegral_enorm]

Goal state after:
case e'_3
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_7
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
f : α → E
fmeas : Measurable f
hf : Integrable f μ
inst✝ : SeparableSpace ↑(range f ∪ {0})
n : ℕ
⊢ (∫⁻ (x : α), ‖(approxOn f fmeas (range f ∪ {0}) 0 ⋯ n) x - f x‖ₑ ∂μ).toReal =
    (ENNReal.toReal ∘ fun n ↦ ∫⁻ (x : α), ‖(approxOn f fmeas (range f ∪ {0}) 0 ⋯ n) x - f x‖ₑ ∂μ) n

case e'_3
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_7
inst✝³ : NormedAddCommGroup E
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
f : α → E
fmeas : Measurable f
hf : Integrable f μ
inst✝ : SeparableSpace ↑(range f ∪ {0})
n : ℕ
⊢ AEStronglyMeasurable (fun x ↦ (approxOn f fmeas (range f ∪ {0}) 0 ⋯ n) x - f x) μ
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable 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
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a ↦ ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a ↦ ENNReal.ofReal (F a - f 0 a)
hf'_int_eq : ∀ (i : ℕ), ∫⁻ (a : α), f' i a ∂μ = ENNReal.ofReal (∫ (a : α), f i a ∂μ - ∫ (a : α), f 0 a ∂μ)
a : α
⊢ (∀ (i : ℕ), f i a ≤ F a) → 0 a ≤ F a - f 0 a

Tactic:
h_bound

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable 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
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a ↦ ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a ↦ ENNReal.ofReal (F a - f 0 a)
hf'_int_eq : ∀ (i : ℕ), ∫⁻ (a : α), f' i a ∂μ = ENNReal.ofReal (∫ (a : α), f i a ∂μ - ∫ (a : α), f 0 a ∂μ)
a : α
h_bound : ∀ (i : ℕ), f i a ≤ F a
⊢ 0 a ≤ F a - f 0 a
Module: Mathlib.Analysis.Normed.Affine.Isometry

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
P : Type u_10
inst✝⁴ : NormedField 𝕜
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace 𝕜 V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
x y : P
⊢ dist ((pointReflection 𝕜 x) y) x = dist y x

Tactic:
← (pointReflection 𝕜 x).dist_map y x,

Goal state after:
𝕜 : Type u_1
V : Type u_2
P : Type u_10
inst✝⁴ : NormedField 𝕜
inst✝³ : SeminormedAddCommGroup V
inst✝² : NormedSpace 𝕜 V
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor V P
x y : P
⊢ dist ((pointReflection 𝕜 x) y) x = dist ((pointReflection 𝕜 x) y) ((pointReflection 𝕜 x) x)
Module: Mathlib.MeasureTheory.Integral.Bochner.SumMeasure

Goal state before:
pp: ι : Type u_1
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝ : NormedAddCommGroup E
μ : ι → Measure X
f : X → E
hf : Integrable f (Measure.sum μ)
⊢ ∑' (x : ι), ∫⁻ (x : X), ‖f x‖ₑ ∂μ x ≠ ∞

Tactic:
← lintegral_sum_measure

Goal state after:
ι : Type u_1
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝ : NormedAddCommGroup E
μ : ι → Measure X
f : X → E
hf : Integrable f (Measure.sum μ)
⊢ ∫⁻ (a : X), ‖f a‖ₑ ∂Measure.sum μ ≠ ∞
Module: Mathlib.MeasureTheory.Integral.Bochner.SumMeasure

Goal state before:
pp: ι : Type u_1
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
μ : ι → Measure X
f : X → E
inst✝ : NormedSpace ℝ E
hf : Integrable f (Measure.sum μ)
hfi : ∀ (i : ι), Integrable f (μ i)
⊢ Tendsto (fun s ↦ ∫ (x : X), f x ∂∑ i ∈ s, μ i) (SummationFilter.unconditional ι).filter
    (𝓝 (∫ (x : X), f x ∂Measure.sum μ))

Tactic:
refine Metric.nhds_basis_ball.tendsto_right_iff.mpr fun ε ε0 ↦ ?_

Goal state after:
ι : Type u_1
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
μ : ι → Measure X
f : X → E
inst✝ : NormedSpace ℝ E
hf : Integrable f (Measure.sum μ)
hfi : ∀ (i : ι), Integrable f (μ i)
ε : ℝ
ε0 : 0 < ε
⊢ ∀ᶠ (x : Finset ι) in (SummationFilter.unconditional ι).filter,
    ∫ (x : X), f x ∂∑ i ∈ x, μ i ∈ Metric.ball (∫ (x : X), f x ∂Measure.sum μ) ε
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case cons
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
ι : Type u_6
m : MeasurableSpace α
f : α → G
μ : ι → Measure α
a✝ : ι
s✝ : Finset ι
h : a✝ ∉ s✝
ih : (∀ i ∈ s✝, Integrable f (μ i)) → ∫ (a : α), f a ∂∑ i ∈ s✝, μ i = ∑ i ∈ s✝, ∫ (a : α), f a ∂μ i
hf : Integrable f (μ a✝) ∧ ∀ x ∈ s✝, Integrable f (μ x)
⊢ ∫ (a : α), f a ∂(μ a✝ + ∑ x ∈ s✝, μ x) = ∫ (a : α), f a ∂μ a✝ + ∑ x ∈ s✝, ∫ (a : α), f a ∂μ x

Tactic:
← ih hf.2

Goal state after:
case cons
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
ι : Type u_6
m : MeasurableSpace α
f : α → G
μ : ι → Measure α
a✝ : ι
s✝ : Finset ι
h : a✝ ∉ s✝
ih : (∀ i ∈ s✝, Integrable f (μ i)) → ∫ (a : α), f a ∂∑ i ∈ s✝, μ i = ∑ i ∈ s✝, ∫ (a : α), f a ∂μ i
hf : Integrable f (μ a✝) ∧ ∀ x ∈ s✝, Integrable f (μ x)
⊢ ∫ (a : α), f a ∂(μ a✝ + ∑ x ∈ s✝, μ x) = ∫ (a : α), f a ∂μ a✝ + ∫ (a : α), f a ∂∑ i ∈ s✝, μ i
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ ν : Measure α
f : α → G
h₁ : Integrable f μ
h₂ : Integrable f ν
⊢ ↑(nndist (∫ (x : α), f x ∂μ + ∫ (x : α), f x ∂ν) (∫ (x : α), f x ∂μ)) ≤ ∫⁻ (x : α), ‖f x‖ₑ ∂ν

Tactic:
nndist_eq_nnnorm,

Goal state after:
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ ν : Measure α
f : α → G
h₁ : Integrable f μ
h₂ : Integrable f ν
⊢ ↑‖∫ (x : α), f x ∂μ + ∫ (x : α), f x ∂ν - ∫ (x : α), f x ∂μ‖₊ ≤ ∫⁻ (x : α), ‖f x‖ₑ ∂ν
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: case inl
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
f : α → G
⊢ ∫ (x : α), f x ∂∞ • μ = 0 • ∫ (x : α), f x ∂μ

Tactic:
zero_smul,

Goal state after:
case inl
α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
f : α → G
⊢ ∫ (x : α), f x ∂∞ • μ = 0
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: case refine_2
α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
δ : Type u_2
inst✝¹ : TopologicalSpace δ
inst✝ : DiscreteTopology δ
f : α ↪ δ
g₁ g₂ : α →ᵇ β
h₁ h₂ : δ →ᵇ β
⊢ dist g₁ g₂ ≤ dist (extend f g₁ h₁) (extend f g₂ h₂)

Tactic:
refine (dist_le dist_nonneg).2 fun x => ?_
    rw [← extend_apply f g₁ h₁, ← extend_apply f g₂ h₂]
    exact dist_coe_le_dist _

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝⁴ : TopologicalSpace α
inst✝³ : PseudoMetricSpace β
inst✝² : AddMonoid β
inst✝¹ : BoundedAdd β
inst✝ : ContinuousAdd β
f : α →ᵇ β
⊢ ⇑0 = 0 • ⇑f

Tactic:
zero_smul,

Goal state after:
α : Type u
β : Type v
inst✝⁴ : TopologicalSpace α
inst✝³ : PseudoMetricSpace β
inst✝² : AddMonoid β
inst✝¹ : BoundedAdd β
inst✝ : ContinuousAdd β
f : α →ᵇ β
⊢ ⇑0 = 0
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : SeminormedAddCommGroup β
inst✝ : CompactSpace α
f : α →ᵇ β
M : ℝ
M0 : 0 < M
⊢ ‖f‖ < M ↔ ∀ (x : α), ‖f x‖ < M

Tactic:
by
  simp_rw [norm_def, ← dist_zero_right]
  exact dist_lt_iff_of_compact M0

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case pos.h_add
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hfμ : Integrable f μ
hF : CompleteSpace F
h_int : ∀ (g : α → E), Integrable g μ → Integrable g μ'
f₂ g₂ : α → E
a✝ : Disjoint (Function.support f₂) (Function.support g₂)
hf₂ : Integrable f₂ μ
hg₂ : Integrable g₂ μ
h_eq_f : setToFun μ T hT f₂ = setToFun μ' T hT' f₂
h_eq_g : setToFun μ T hT g₂ = setToFun μ' T hT' g₂
⊢ setToFun μ T hT f₂ + setToFun μ T hT g₂ = setToFun μ' T hT' (f₂ + g₂)

Tactic:
setToFun_add hT' (h_int f₂ hf₂) (h_int g₂ hg₂),

Goal state after:
case pos.h_add
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hfμ : Integrable f μ
hF : CompleteSpace F
h_int : ∀ (g : α → E), Integrable g μ → Integrable g μ'
f₂ g₂ : α → E
a✝ : Disjoint (Function.support f₂) (Function.support g₂)
hf₂ : Integrable f₂ μ
hg₂ : Integrable g₂ μ
h_eq_f : setToFun μ T hT f₂ = setToFun μ' T hT' f₂
h_eq_g : setToFun μ T hT g₂ = setToFun μ' T hT' g₂
⊢ setToFun μ T hT f₂ + setToFun μ T hT g₂ = setToFun μ' T hT' f₂ + setToFun μ' T hT' g₂
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : (𝓝[<] x).NeBot
⊢ sSup (f '' Iio x) ≤ f y

Tactic:
refine csSup_le ?_ ?_

Goal state after:
case inr.refine_1
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : (𝓝[<] x).NeBot
⊢ (f '' Iio x).Nonempty

case inr.refine_2
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
h : x ≤ y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : (𝓝[<] x).NeBot
⊢ ∀ b ∈ f '' Iio x, b ≤ f y
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: case hf
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
μ : Measure X
ι : Type u_5
t : Finset ι
s : ι → Set X
hs : ∀ i ∈ t, MeasurableSet (s i)
hf : ∀ i ∈ t, IntegrableOn f (s i) μ
A : ∀ u ∈ {x ∈ t.powerset | x.Nonempty}, MeasurableSet (⋂ i ∈ u, s i)
this :
  ∑ x ∈ t.powerset with x.Nonempty, ∫ (a : X) in ⋂ i ∈ x, s i, (-1) ^ (#x + 1) • f a ∂μ =
    ∑ x ∈ t.powerset with x.Nonempty, ∫ (a : X), (⋂ i ∈ x, s i).indicator (fun a ↦ (-1) ^ (#x + 1) • f a) a ∂μ
⊢ ∀ i ∈ {x ∈ t.powerset | x.Nonempty}, Integrable ((⋂ i_1 ∈ i, s i_1).indicator fun a ↦ (-1) ^ (#i + 1) • f a) μ

Tactic:
intro u hu
    rw [integrable_indicator_iff (A u hu)]
    apply Integrable.smul
    simp only [Finset.mem_filter, Finset.mem_powerset] at hu
    rcases hu.2 with ⟨i, hi⟩
    exact (hf i (hu.1 hi)).mono (biInter_subset_of_mem hi) le_rfl

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝ : TopologicalSpace G
μ : Content G
⊢ μ.innerContent ⊥ = 0

Tactic:
← nonpos_iff_eq_zero,

Goal state after:
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
⊢ μ.innerContent ⊥ ≤ 0
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this : Nonempty R
i : ℕ
p q' : R
spq : s i \ botSet ⊆ Ioc p q'
hq' : ofReal (↑f q' - ↑f p) < f.length (s i) + ↑(ε' i)
A : Icc a' b ∩ s i ⊆ Ioc p q'
htq' : ¬IsTop q'
⊢ (𝓝[>] q').NeBot

Tactic:
simp [Filter.neBot_iff, nhdsGT_eq_bot_iff, htq', not_covBy]

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
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
ht : IntegrableOn f t μ
⊢ ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ

Tactic:
by_cases H : IntegrableOn f (s ∪ t) μ

Goal state after:
case pos
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
ht : IntegrableOn f t μ
H : IntegrableOn f (s ∪ t) μ
⊢ ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ

case neg
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
s t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ.restrict t, f x = 0
ht : IntegrableOn f t μ
H : ¬IntegrableOn f (s ∪ t) μ
⊢ ∫ (x : X) in s ∪ t, f x ∂μ = ∫ (x : X) in s, f x ∂μ
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inr.inl
R : Type u_1
inst✝¹ : LinearOrder R
inst✝ : TopologicalSpace R
f : StieltjesFunction R
c : R
t : Set R
this : Nonempty R
a b : R
h : t \ botSet ⊆ Ioc a b
hac : c ≤ a
hbc : b ≤ c
⊢ f.length (Ioc a b ∩ Ioi c) + f.length (Ioc a b \ Ioi c) ≤ ofReal (↑f b - ↑f a)

Tactic:
simp only [hbc, le_refl, Ioc_eq_empty, Ioc_inter_Ioi, min_eq_left, Ioc_sdiff_Ioi,
      f.length_empty, zero_add, or_true, le_sup_iff, f.length_Ioc, not_lt]

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

Goal state before:
pp: G : Type u_1
inst✝¹ : Group G
inst✝ : TopologicalSpace G
K₀ : PositiveCompacts G
U : Set G
⊢ ↑(index ∅ U) / ↑(index (↑K₀) U) = 0

Tactic:
index_empty,

Goal state after:
G : Type u_1
inst✝¹ : Group G
inst✝ : TopologicalSpace G
K₀ : PositiveCompacts G
U : Set G
⊢ ↑0 / ↑(index (↑K₀) U) = 0
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
s t : Set X
μ : Measure X
inst✝ : CompleteSpace E
p : ℝ≥0∞
hs : MeasurableSet s
ht : MeasurableSet t
hμt : μ t ≠ ∞
e : E
⊢ ∫ (x : X) in s, ↑↑(indicatorConstLp p ht hμt e) x ∂μ = ∫ (x : X) in s, t.indicator (fun x ↦ e) x ∂μ

Tactic:
rw [setIntegral_congr_ae hs (indicatorConstLp_coeFn.mono fun x hx _ => hx)]

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

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝⁷ : Group G
inst✝⁶ : MulAction G α
inst✝⁵ : MeasurableSpace α
ν : Measure α
inst✝⁴ : SMulInvariantMeasure G α ν
inst✝³ : Countable G
inst✝² : MeasurableConstSMul G α
μ : Measure (Quotient α_mod_G)
inst✝¹ : QuotientMeasureEqMeasurePreimage ν μ
inst✝ : IsFiniteMeasure μ
⊢ covolume G α ν < ∞

Tactic:
by
  by_cases hasFun : HasFundamentalDomain G α ν
  · obtain ⟨𝓕, h𝓕⟩ := hasFun.ExistsIsFundamentalDomain
    have H : μ univ < ∞ := IsFiniteMeasure.measure_univ_lt_top
    rw [h𝓕.projection_respects_measure_apply (μ := μ) MeasurableSet.univ] at H
    simpa [h𝓕.covolume_eq_volume ν] using H
  · simp [covolume, hasFun]

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

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
⊢ ⟪x, -y⟫ = -⟪x, y⟫

Tactic:
rw [← inner_conj_symm, inner_neg_left]; simp only [map_neg, inner_conj_symm]

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

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
this : discrim (normSq x) (2 * ‖⟪x, y⟫‖) (normSq y) ≤ 0
⊢ ‖⟪x, y⟫‖ * ‖⟪x, y⟫‖ ≤ re ⟪x, x⟫ * re ⟪y, y⟫

Tactic:
discrim,

Goal state after:
𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
this : (2 * ‖⟪x, y⟫‖) ^ 2 - 4 * normSq x * normSq y ≤ 0
⊢ ‖⟪x, y⟫‖ * ‖⟪x, y⟫‖ ≤ re ⟪x, x⟫ * re ⟪y, y⟫
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
⊢ ↑‖⟪x, x⟫‖ = ⟪x, x⟫

Tactic:
← inner_self_re_eq_norm

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x : E
⊢ ↑(re ⟪x, x⟫) = ⟪x, x⟫
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ ⟪x, -y⟫ = -⟪x, y⟫

Tactic:
rw [← inner_conj_symm, inner_neg_left]; simp only [map_neg, inner_conj_symm]

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

Goal state before:
pp: F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ ⟪x + y, x + y⟫_ℝ = ⟪x, x⟫_ℝ + 2 * ⟪x, y⟫_ℝ + ⟪y, y⟫_ℝ

Tactic:
have : ⟪y, x⟫_ℝ = ⟪x, y⟫_ℝ := by rw [← inner_conj_symm]; rfl
  simp only [inner_add_add_self, this, add_left_inj]
  ring

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ ‖⟪x, y⟫‖ ≤ ‖x‖ * ‖y‖

Tactic:
norm_eq_sqrt_re_inner (𝕜 := 𝕜) x,

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ ‖⟪x, y⟫‖ ≤ √(re ⟪x, x⟫) * ‖y‖
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:
inner_eq_norm_mul_iff_div h₀,

Goal state after:
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
⊢ (↑‖y‖ / ↑‖x‖) • x = y ↔ ↑‖y‖ • x = ↑‖x‖ • y
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: case mp
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h : Convex ℝ K
u v : F
hv : v ∈ K
this✝ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
eq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
w : F
hw : w ∈ K
δ : ℝ := ⨅ w, ‖u - ↑w‖
p : ℝ := ⟪u - v, w - v⟫_ℝ
q : ℝ := ‖w - v‖ ^ 2
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
this : ∀ (θ : ℝ), 0 < θ → θ ≤ 1 → 2 * p ≤ θ * q
⊢ ⟪u - v, w - v⟫_ℝ ≤ 0

Tactic:
by_cases hq : q = 0

Goal state after:
case pos
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h : Convex ℝ K
u v : F
hv : v ∈ K
this✝ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
eq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
w : F
hw : w ∈ K
δ : ℝ := ⨅ w, ‖u - ↑w‖
p : ℝ := ⟪u - v, w - v⟫_ℝ
q : ℝ := ‖w - v‖ ^ 2
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
this : ∀ (θ : ℝ), 0 < θ → θ ≤ 1 → 2 * p ≤ θ * q
hq : q = 0
⊢ ⟪u - v, w - v⟫_ℝ ≤ 0

case neg
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h : Convex ℝ K
u v : F
hv : v ∈ K
this✝ : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
eq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
w : F
hw : w ∈ K
δ : ℝ := ⨅ w, ‖u - ↑w‖
p : ℝ := ⟪u - v, w - v⟫_ℝ
q : ℝ := ‖w - v‖ ^ 2
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
this : ∀ (θ : ℝ), 0 < θ → θ ≤ 1 → 2 * p ≤ θ * q
hq : ¬q = 0
⊢ ⟪u - v, w - v⟫_ℝ ≤ 0