mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: case neg
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝² : NormedAddCommGroup E
inst✝¹ : MeasurableSpace E
inst✝ : OpensMeasurableSpace E
R : ℝ≥0
p : ℝ≥0∞
f : ℕ → α → E
hfmeas : ∀ (n : ℕ), Measurable (f n)
hp0 : ¬p.toReal = 0
hp : p ≠ 0
hp' : p ≠ ∞
hbdd : ∀ (n : ℕ), (∫⁻ (x : α), ‖f n x‖ₑ ^ p.toReal ∂μ) ^ p.toReal⁻¹ ≤ ↑R
⊢ sSup {a | ∃ a_1, ∀ (b : ℕ), a_1 ≤ b → a ≤ ∫⁻ (a : α), ↑‖f b a‖₊ ^ p.toReal ∂μ} ≤ ↑R ^ p.toReal

Tactic:
exact
    sSup_le fun b ⟨a, ha⟩ =>
      (ha a le_rfl).trans ((ENNReal.rpow_inv_le_iff (ENNReal.toReal_pos hp hp')).1 (hbdd _))

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case neg
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
A : ∀ i ∈ s, z i = 0 → w i = 0
this : ∏ x ∈ s, rexp (log (z x) * w x) ≤ ∑ x ∈ s, w x * rexp (log (z x))
⊢ ∏ i ∈ s, z i ^ w i ≤ ∑ i ∈ s, w i * z i

Tactic:
convert! this using 1 <;> [apply prod_congr rfl; apply sum_congr rfl] <;> intro i hi

Goal state after:
case e'_3
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
A : ∀ i ∈ s, z i = 0 → w i = 0
this : ∏ x ∈ s, rexp (log (z x) * w x) ≤ ∑ x ∈ s, w x * rexp (log (z x))
i : ι
hi : i ∈ s
⊢ z i ^ w i = rexp (log (z i) * w i)

case e'_4
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
A : ∀ i ∈ s, z i = 0 → w i = 0
this : ∏ x ∈ s, rexp (log (z x) * w x) ≤ ∑ x ∈ s, w x * rexp (log (z x))
i : ι
hi : i ∈ s
⊢ w i * z i = w i * rexp (log (z i))
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 ≤ w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
⊢ ∏ i ∈ s, z i ^ w i < ∑ i ∈ s, w i * z i ↔ ∃ j ∈ s, w j ≠ 0 ∧ z j ≠ ∑ i ∈ s, w i * z i

Tactic:
have :
      ∏ i ∈ s with w i ≠ 0, z i ^ w i < ∑ i ∈ s with w i ≠ 0, w i * z i ↔
        ∃ j ∈ {x ∈ s | w x ≠ 0}, z j ≠ ∑ i ∈ s with w i ≠ 0, w i * z i :=
    geom_mean_lt_arith_mean_weighted_iff_of_pos' _ w z (by grind)
      (sum_filter_ne_zero _ |>.trans hw') (hz _ <| mem_of_mem_filter · ·)
  grind [prod_filter_of_ne, sum_filter_of_ne, rpow_zero]

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 < w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
h : ∏ i ∈ s, z i ^ w i < ∑ i ∈ s, w i * z i
h_contra : ∀ j ∈ s, ∀ k ∈ s, z j = z k
j : ι
hjs : j ∈ s
⊢ ∑ i ∈ s, w i * z j = ∑ i ∈ s, w i * z i

Tactic:
apply sum_congr rfl (fun x a => congrArg (HMul.hMul (w x)) (h_contra j hjs x a))

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: α : Type u_1
ε' : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace ε'
inst✝ : ESeminormedAddMonoid ε'
p q : ℝ≥0∞
f : α → ε'
s : Set α
hfq : MemLp f q (μ.restrict (toMeasurable μ s))
hf : ∀ x ∉ s, f x = 0
hs : μ s ≠ ∞
hpq : p ≤ q
this✝ : (toMeasurable μ s).indicator f = f
this : Fact (μ (toMeasurable μ s) < ∞)
⊢ MemLp f p (μ.restrict (toMeasurable μ s))

Tactic:
exact hfq.mono_exponent hpq

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case inr.calc_2
ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0
hp : 1 < p
⊢ (∑ i ∈ s, (w i ^ (1 - p⁻¹)) ^ (1 - p⁻¹)⁻¹) ^ (1 / (1 - p⁻¹)⁻¹) * (∑ i ∈ s, (w i ^ p⁻¹ * f i) ^ p) ^ (1 / p) =
    (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹

Tactic:
· have hp₀ : p ≠ 0 := by positivity
    have hp₁ : 1 - p⁻¹ ≠ 0 := by simp [sub_eq_zero, hp.ne']
    simp [mul_rpow, div_inv_eq_mul, one_mul, one_div, hp₀, hp₁]

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

Goal state before:
pp: case e'_6
α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
q : ℝ≥0∞
f : α → E
hf : AEStronglyMeasurable f μ
q_zero : q ≠ 0
q_top : q ≠ ∞
h : MemLp (fun x ↦ ‖f x‖ ^ q.toReal) (p / q) μ
x : α
⊢ ‖f x‖ = |‖f x‖| ^ (q.toReal * q⁻¹.toReal)

Tactic:
ENNReal.toReal_inv,

Goal state after:
case e'_6
α : Type u_1
E : Type u_4
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
q : ℝ≥0∞
f : α → E
hf : AEStronglyMeasurable f μ
q_zero : q ≠ 0
q_top : q ≠ ∞
h : MemLp (fun x ↦ ‖f x‖ ^ q.toReal) (p / q) μ
x : α
⊢ ‖f x‖ = |‖f x‖| ^ (q.toReal * q.toReal⁻¹)
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: p : ℝ≥0∞
α : Type u_6
E : Type u_7
F : Type u_8
K' : ℝ≥0
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
f : α → E
g : E → F
hL : MemLp (g ∘ f) p μ
hg : UniformContinuous g
hg' : AntilipschitzWith K' g
g0 : g 0 = 0
x : α
⊢ dist (f x) 0 ≤ ↑K' * dist (g (f x)) (g 0)

Tactic:
apply hg'.le_mul_dist

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

Goal state before:
pp: α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
f : ι → α → E
g : α → E
inst✝ : SeminormedAddCommGroup E
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
l : Filter ι
ε : ℝ≥0∞
hε : 0 < ε
hε_top : ε ≠ ∞
hfg : ∀ ε_1 > 0, ∀ᶠ (x : ι) in l, 1 / ε ^ p.toReal * eLpNorm (f x - g) p μ ^ p.toReal ≤ ε_1
⊢ ∀ ε_1 > 0, ∀ᶠ (x : ι) in l, μ {x_1 | ε ≤ edist (f x x_1) (g x_1)} ≤ ε_1

Tactic:
intro δ

Goal state after:
α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
p : ℝ≥0∞
f : ι → α → E
g : α → E
inst✝ : SeminormedAddCommGroup E
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
hf : ∀ (n : ι), StronglyMeasurable (f n)
hg : StronglyMeasurable g
l : Filter ι
ε : ℝ≥0∞
hε : 0 < ε
hε_top : ε ≠ ∞
hfg : ∀ ε_1 > 0, ∀ᶠ (x : ι) in l, 1 / ε ^ p.toReal * eLpNorm (f x - g) p μ ^ p.toReal ≤ ε_1
δ : ℝ≥0∞
⊢ δ > 0 → ∀ᶠ (x : ι) in l, μ {x_1 | ε ≤ edist (f x x_1) (g x_1)} ≤ δ
Module: Mathlib.Logic.Equiv.Embedding

Goal state before:
pp: case inl.inr
α : Type u_1
β : Type u_2
γ : Type u_3
x✝ : { f // Disjoint (Set.range ⇑f.1) (Set.range ⇑f.2) }
f : α ↪ γ
g : β ↪ γ
disj : Disjoint (Set.range ⇑(f, g).1) (Set.range ⇑(f, g).2)
a₁ : α
b₂ : β
f_eq : f a₁ = g b₂
⊢ Sum.inl a₁ = Sum.inr b₂

Tactic:
exfalso

Goal state after:
case inl.inr
α : Type u_1
β : Type u_2
γ : Type u_3
x✝ : { f // Disjoint (Set.range ⇑f.1) (Set.range ⇑f.2) }
f : α ↪ γ
g : β ↪ γ
disj : Disjoint (Set.range ⇑(f, g).1) (Set.range ⇑(f, g).2)
a₁ : α
b₂ : β
f_eq : f a₁ = g b₂
⊢ False
Module: Mathlib.Logic.Equiv.Embedding

Goal state before:
pp: case inr.inl
α : Type u_1
β : Type u_2
γ : Type u_3
x✝ : { f // Disjoint (Set.range ⇑f.1) (Set.range ⇑f.2) }
f : α ↪ γ
g : β ↪ γ
disj : Disjoint (Set.range ⇑(f, g).1) (Set.range ⇑(f, g).2)
b₁ : β
a₂ : α
f_eq : g b₁ = f a₂
⊢ Sum.inr b₁ = Sum.inl a₂

Tactic:
exfalso

Goal state after:
case inr.inl
α : Type u_1
β : Type u_2
γ : Type u_3
x✝ : { f // Disjoint (Set.range ⇑f.1) (Set.range ⇑f.2) }
f : α ↪ γ
g : β ↪ γ
disj : Disjoint (Set.range ⇑(f, g).1) (Set.range ⇑(f, g).2)
b₁ : β
a₂ : α
f_eq : g b₁ = f a₂
⊢ False
Module: Mathlib.Analysis.Normed.Ring.Units

Goal state before:
pp: R : Type u_1
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
I : Ideal R
hI : I ≠ ⊤
⊢ I.closure ≠ ⊤

Tactic:
by
  have h := closure_minimal (coe_subset_nonunits hI) nonunits.isClosed
  simpa only [I.closure.eq_top_iff_one, Ne] using! mt (@h 1) one_notMem_nonunits

Goal state after:
Module: Mathlib.Topology.Maps.Strict.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
f : X → Y
g : Y → Z
g_emb : IsEmbedding g
⊢ IsStrictMap f ↔ IsStrictMap (g ∘ f)

Tactic:
set Φ : Quotient (Setoid.ker (g ∘ f)) ≃ₜ Quotient (Setoid.ker (f)) :=
    Homeomorph.Quotient.congrRight (fun _ _ ↦ by simp [g_emb.injective.eq_iff])

Goal state after:
X : Type u_1
Y : Type u_2
Z : Type u_3
inst✝² : TopologicalSpace X
inst✝¹ : TopologicalSpace Y
inst✝ : TopologicalSpace Z
f : X → Y
g : Y → Z
g_emb : IsEmbedding g
Φ : Quotient (ker (g ∘ f)) ≃ₜ Quotient (ker f) := Homeomorph.Quotient.congrRight ⋯
⊢ IsStrictMap f ↔ IsStrictMap (g ∘ f)
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s₁ s₂ : Set α
h : μ.real (s₁ ∩ s₂) = 0
h' : μ s₁ ≠ ∞
⊢ (μ (s₁ \ s₂)).toReal = (μ s₁).toReal

Tactic:
rw [measure_sdiff_null']

Goal state after:
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s₁ s₂ : Set α
h : μ.real (s₁ ∩ s₂) = 0
h' : μ s₁ ≠ ∞
⊢ μ (s₁ ∩ s₂) = 0
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
ε : Type u_3
mα : MeasurableSpace α
f : α → ε
s : Set α
μ ν : Measure α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
inst✝ : PseudoMetrizableSpace ε
hμ : IntegrableOn f s μ
hν : IntegrableOn f s ν
⊢ Integrable f (μ.restrict s + ν.restrict s)

Tactic:
exact hμ.integrable.add_measure hν

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : MultilinearMap 𝕜 E G
C : ℝ
D : ℝ := max C 1
D_pos : 0 ≤ D
H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ D * ∏ i, ‖m i‖
⊢ Continuous[Pi.topologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑f

Tactic:
refine continuous_iff_continuousAt.2 fun m => ?_

Goal state after:
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
f : MultilinearMap 𝕜 E G
C : ℝ
D : ℝ := max C 1
D_pos : 0 ≤ D
H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ D * ∏ i, ‖m i‖
m : (i : ι) → E i
⊢ ContinuousAt (⇑f) m
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
G : Type wG
G' : Type wG'
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : SeminormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : SeminormedAddCommGroup G'
inst✝² : NormedSpace 𝕜 G'
inst✝¹ : Fintype ι
inst✝ : Subsingleton ι
i : ι
f : G →L[𝕜] G'
⊢ ‖(ofSubsingleton 𝕜 G G' i) f‖ = ‖f‖

Tactic:
let : Unique ι := uniqueOfSubsingleton i
  simp [norm_def, ContinuousLinearMap.norm_def, (Equiv.funUnique _ _).symm.surjective.forall]

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

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
b : α
hb : μ {b} ≠ ∞
hb' : ‖f b‖ₑ ≠ ∞
⊢ IntegrableOn f (Iic b) μ ↔ IntegrableOn f (Iio b) μ

Tactic:
← Iio_union_right,

Goal state after:
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
b : α
hb : μ {b} ≠ ∞
hb' : ‖f b‖ₑ ≠ ∞
⊢ IntegrableOn f (Iio b ∪ {b}) μ ↔ IntegrableOn f (Iio b) μ
Module: Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable

Goal state before:
pp: X : Type u_4
E : Type u_5
ι : Type u_6
inst✝⁶ : MeasurableSpace X
inst✝⁵ : CommMonoid E
inst✝⁴ : TopologicalSpace E
inst✝³ : PseudoMetrizableSpace E
inst✝² : ContinuousMul E
L : SummationFilter ι
inst✝¹ : L.NeBot
inst✝ : L.filter.IsCountablyGenerated
μ : Measure X
f : ι → X → E
h : ∀ (i : ι), AEStronglyMeasurable (f i) μ
⊢ AEStronglyMeasurable (∏'[L] (i : ι), f i) μ

Tactic:
by
  rw [tprod_def, finprod_def']
  split_ifs with hm
  any_goals exact aestronglyMeasurable_one
  · refine Finset.aestronglyMeasurable_prod _ (fun _ _ ↦ ?_)
    rw [Set.mulIndicator]
    split_ifs <;> fun_prop
  · apply aestronglyMeasurable_of_tendsto_ae L.filter (f := fun s => ∏ i ∈ s, f i) (by fun_prop)
    exact .of_forall fun x ↦ hm.choose_spec.apply_nhds x

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

Goal state before:
pp: case refine_3
X : Type u_1
ε : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
T : Set (Set X)
T_count : T.Countable
T_open : ∀ u ∈ T, IsOpen[inst✝³] u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s) μ
T' : Set (Set X) := insert ∅ T
T'_count : T'.Countable
T'_ne : T'.Nonempty
u : ℕ → Set X
hu : T' = range u
⊢ ∀ (n : ℕ), IntegrableOn f (u n ∩ s) μ

Tactic:
intro n
    have : u n ∈ T' := by rw [hu]; exact mem_range_self n
    rcases mem_insert_iff.1 this with h | h
    · simp only [h, empty_inter, integrableOn_empty]
    · exact hT _ h

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

Goal state before:
pp: X : Type u_1
ε : Type u_3
inst✝⁵ : MeasurableSpace X
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace ε
inst✝² : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝¹ : PseudoMetrizableSpace ε
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
u : ℕ → Set X
su : s ⊆ ⋃ n, u n
hu : ∀ (n : ℕ), IntegrableOn f (u n ∩ s) μ
this : s = ⋃ n, u n ∩ s
⊢ ∀ (i : ℕ), AEStronglyMeasurable f (μ.restrict (u i ∩ s))

Tactic:
exact fun i : ℕ => (hu i).aestronglyMeasurable

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

Goal state before:
pp: case inr
X : Type u_1
E : Type u_6
inst✝⁹ : MeasurableSpace X
inst✝⁸ : TopologicalSpace X
inst✝⁷ : NormedAddCommGroup E
μ : Measure X
s : Set X
inst✝⁶ : BorelSpace X
inst✝⁵ : ConditionallyCompleteLinearOrder X
inst✝⁴ : ConditionallyCompleteLinearOrder E
inst✝³ : OrderTopology X
inst✝² : OrderTopology E
inst✝¹ : SecondCountableTopology E
p : ℝ≥0∞
f : X → E
inst✝ : IsFiniteMeasureOnCompacts μ
hs : IsCompact s
hmono : MonotoneOn f s
h : s.Nonempty
⊢ MemLp f p (μ.restrict s)

Tactic:
exact hmono.memLp_of_measure_ne_top (hs.isLeast_sInf h) (hs.isGreatest_sSup h)
      hs.measure_lt_top.ne hs.measurableSet

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

Goal state before:
pp: β : Type u_2
E : Type u_4
inst✝⁴ : MeasurableSpace β
inst✝³ : MeasurableSpace E
inst✝² : NormedAddCommGroup E
inst✝¹ : OpensMeasurableSpace E
f : β → E
hf : Measurable f
s : Set E
y₀ : E
h₀ : y₀ ∈ s
inst✝ : SeparableSpace ↑s
μ : Measure β
hμ : ∀ᵐ (x : β) ∂μ, f x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
hi : HasFiniteIntegral (fun x ↦ f x - y₀) μ
⊢ eLpNorm (fun x ↦ f x - y₀) 1 μ < ∞

Tactic:
by simpa [eLpNorm_one_eq_lintegral_enorm] using! hi

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

Goal state before:
pp: case neg
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
hfp : ∀ x ∈ f.range, x ≠ 0 → μ (⇑f ⁻¹' {x}) ≠ ∞
a : α
hb : f a ∈ f.range
h0 : ¬g (f a) = 0
h_left_eq : (T (⇑(map g f) ⁻¹' {g (f a)})) (g (f a)) = (T (⇑f ⁻¹' ↑({b ∈ f.range | g b = g (f a)}))) (g (f a))
⊢ (T (⇑(map g f) ⁻¹' {g (f a)})) (g (f a)) = ∑ j ∈ f.range with g j = g (f a), (T (⇑f ⁻¹' {j})) (g j)

Tactic:
h_left_eq

Goal state after:
case neg
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
hfp : ∀ x ∈ f.range, x ≠ 0 → μ (⇑f ⁻¹' {x}) ≠ ∞
a : α
hb : f a ∈ f.range
h0 : ¬g (f a) = 0
h_left_eq : (T (⇑(map g f) ⁻¹' {g (f a)})) (g (f a)) = (T (⇑f ⁻¹' ↑({b ∈ f.range | g b = g (f a)}))) (g (f a))
⊢ (T (⇑f ⁻¹' ↑({b ∈ f.range | g b = g (f a)}))) (g (f a)) = ∑ j ∈ f.range with g j = g (f a), (T (⇑f ⁻¹' {j})) (g j)
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
F : Type u_3
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
G'' : Type u_8
inst✝³ : NormedAddCommGroup G''
inst✝² : PartialOrder G''
inst✝¹ : IsOrderedAddMonoid G''
inst✝ : NormedSpace ℝ G''
m : MeasurableSpace α
T T' : Set α → F →L[ℝ] G''
hTT' : ∀ (s : Set α) (x : F), (T s) x ≤ (T' s) x
f : α →ₛ F
⊢ setToSimpleFunc T f ≤ setToSimpleFunc T' f

Tactic:
simp_rw [setToSimpleFunc]; gcongr; apply hTT'

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

Goal state before:
pp: case inr.inr
α : Type u_1
F : Type u_3
F' : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
T : Set α → F →L[ℝ] F'
hT_empty : T ∅ = 0
m : MeasurableSpace α
s : Set α
hs : MeasurableSet s
x : F
hs_empty : s.Nonempty
hs_univ : s ≠ Set.univ
⊢ ∑ x_1 ∈ (piecewise s hs (const α x) (const α 0)).range,
      (T (⇑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x_1})) x_1 =
    (T s) x

Tactic:
rw [range_indicator hs hs_empty hs_univ]

Goal state after:
case inr.inr
α : Type u_1
F : Type u_3
F' : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
T : Set α → F →L[ℝ] F'
hT_empty : T ∅ = 0
m : MeasurableSpace α
s : Set α
hs : MeasurableSet s
x : F
hs_empty : s.Nonempty
hs_univ : s ≠ Set.univ
⊢ ∑ x_1 ∈ {x, 0}, (T (⇑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x_1})) x_1 = (T s) x
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
E : Type u_2
𝕜 : Type u_3
inst✝² : NormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
a b : α → 𝕜
u v : α → E
l : Filter α
hab : (a - b) =o[l] b
huv : (u - v) =o[l] v
φ : α → 𝕜
hφ : Tendsto φ l (𝓝 1)
habφ : a =ᶠ[l] φ * b
this : ((fun x ↦ a x • u x) - fun x ↦ b x • v x) =ᶠ[l] fun x ↦ b x • (φ x • u x - v x)
C : ℝ
hC : C > 0
hCuv : IsBigOWith C l u v
⊢ (fun x ↦ φ x • u x - v x) =o[l] v

Tactic:
isLittleO_iff

Goal state after:
α : Type u_1
E : Type u_2
𝕜 : Type u_3
inst✝² : NormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
a b : α → 𝕜
u v : α → E
l : Filter α
hab : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : α) in l, ‖(a - b) x‖ ≤ c * ‖b x‖
huv : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : α) in l, ‖(u - v) x‖ ≤ c * ‖v x‖
φ : α → 𝕜
hφ : Tendsto φ l (𝓝 1)
habφ : a =ᶠ[l] φ * b
this : ((fun x ↦ a x • u x) - fun x ↦ b x • v x) =ᶠ[l] fun x ↦ b x • (φ x • u x - v x)
C : ℝ
hC : C > 0
hCuv : IsBigOWith C l u v
⊢ ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x : α) in l, ‖φ x • u x - v x‖ ≤ c * ‖v x‖
Module: Mathlib.Analysis.Normed.Group.AddTorsor

Goal state before:
pp: α : Type u_1
V : Type u_2
P : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁵ : SeminormedAddCommGroup V
inst✝⁴ : PseudoMetricSpace P
inst✝³ : NormedAddTorsor V P
inst✝² : SeminormedAddCommGroup W
inst✝¹ : PseudoMetricSpace Q
inst✝ : NormedAddTorsor W Q
c : V
x y : P
⊢ dist (c +ᵥ x) (c +ᵥ y) = dist x y

Tactic:
by
    simp [NormedAddTorsor.dist_eq_norm']

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

Goal state before:
pp: α : Type u_1
V✝ : Type u_2
P✝ : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁷ : SeminormedAddCommGroup V✝
inst✝⁶ : PseudoMetricSpace P✝
inst✝⁵ : NormedAddTorsor V✝ P✝
inst✝⁴ : SeminormedAddCommGroup W
inst✝³ : PseudoMetricSpace Q
inst✝² : NormedAddTorsor W Q
V : Type u_6
P : Type u_7
inst✝¹ : SeminormedAddCommGroup V
inst✝ : AddTorsor V P
x y : P
⊢ ‖x -ᵥ y‖ = ‖y -ᵥ x‖

Tactic:
simp only [← neg_vsub_eq_vsub_rev y x, norm_neg]

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

Goal state before:
pp: α : Type u_1
V✝ : Type u_2
P✝ : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁷ : SeminormedAddCommGroup V✝
inst✝⁶ : PseudoMetricSpace P✝
inst✝⁵ : NormedAddTorsor V✝ P✝
inst✝⁴ : SeminormedAddCommGroup W
inst✝³ : PseudoMetricSpace Q
inst✝² : NormedAddTorsor W Q
V : Type u_6
P : Type u_7
inst✝¹ : NormedAddCommGroup V
inst✝ : AddTorsor V P
x y : P
⊢ ‖x -ᵥ y‖ = ‖y -ᵥ x‖

Tactic:
simp only [← neg_vsub_eq_vsub_rev y x, norm_neg]

Goal state after:
Module: Mathlib.Topology.Algebra.ContinuousAffineEquiv

Goal state before:
pp: k : Type u_1
P₁ : Type u_2
P₂ : Type u_3
V₁ : Type u_6
V₂ : Type u_7
inst✝⁸ : Ring k
inst✝⁷ : AddCommGroup V₁
inst✝⁶ : Module k V₁
inst✝⁵ : AddTorsor V₁ P₁
inst✝⁴ : TopologicalSpace P₁
inst✝³ : AddCommGroup V₂
inst✝² : Module k V₂
inst✝¹ : AddTorsor V₂ P₂
inst✝ : TopologicalSpace P₂
e e' : P₁ ≃ᴬ[k] P₂
h : e.toContinuousAffineMap = e'.toContinuousAffineMap
⊢ e = e'

Tactic:
ext p

Goal state after:
k : Type u_1
P₁ : Type u_2
P₂ : Type u_3
V₁ : Type u_6
V₂ : Type u_7
inst✝⁸ : Ring k
inst✝⁷ : AddCommGroup V₁
inst✝⁶ : Module k V₁
inst✝⁵ : AddTorsor V₁ P₁
inst✝⁴ : TopologicalSpace P₁
inst✝³ : AddCommGroup V₂
inst✝² : Module k V₂
inst✝¹ : AddTorsor V₂ P₂
inst✝ : TopologicalSpace P₂
e e' : P₁ ≃ᴬ[k] P₂
h : e.toContinuousAffineMap = e'.toContinuousAffineMap
p : P₁
⊢ e p = e' p
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: 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
h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0
h_add : FinMeasAdditive μ T
f g : ↥(α →₁ₛ[μ] E)
⊢ setToL1S T (f - g) = setToL1S T f - setToL1S T g

Tactic:
by
  rw [sub_eq_add_neg, setToL1S_add T h_zero h_add, setToL1S_neg h_zero h_add, sub_eq_add_neg]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.RieszLemma

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
r : ℝ
hr : r < 1
x : E
hx : x ∉ F
d : ℝ := infDist x ↑F
hFn : (↑F).Nonempty
hdp : 0 < d
r' : ℝ := max r 2⁻¹
hr' : r' < 1
hlt : 0 < r'
hdlt : d < d / r'
y₀ : E
hy₀F : y₀ ∈ F
hxy₀ : dist x y₀ < d / r'
h : x - y₀ ∈ F
⊢ False

Tactic:
have : x - y₀ + y₀ ∈ F := F.add_mem h hy₀F

Goal state after:
𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
r : ℝ
hr : r < 1
x : E
hx : x ∉ F
d : ℝ := infDist x ↑F
hFn : (↑F).Nonempty
hdp : 0 < d
r' : ℝ := max r 2⁻¹
hr' : r' < 1
hlt : 0 < r'
hdlt : d < d / r'
y₀ : E
hy₀F : y₀ ∈ F
hxy₀ : dist x y₀ < d / r'
h : x - y₀ ∈ F
this : x - y₀ + y₀ ∈ F
⊢ False
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_6
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : NormedRing 𝕜
inst✝⁴ : Module 𝕜 E
inst✝³ : Module 𝕜 F
inst✝² : IsBoundedSMul 𝕜 E
inst✝¹ : IsBoundedSMul 𝕜 F
inst✝ : CompleteSpace F
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x
c : 𝕜
f : ↥(Lp E 1 μ)
⊢ (setToL1 hT) (c • f) = c • (setToL1 hT) f

Tactic:
by
  rw [setToL1_eq_setToL1' hT h_smul, setToL1_eq_setToL1' hT h_smul]
  exact map_smul _ _ _

Goal state after:
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) y = ‖2 • (x -ᵥ y)‖

Tactic:
pointReflection_apply,

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 ((x -ᵥ y) +ᵥ x) y = ‖2 • (x -ᵥ y)‖