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

Goal state before:
pp: case neg
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
inst✝ : IsFiniteMeasure μ
hf : ∀ (n : ℕ), Measurable fun a ↦ edist (f n a) (g a)
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : 0 < ε
δ : ℝ≥0
hδ : 0 < ↑δ
t : Set α
left✝ : MeasurableSet t
ht : μ t ≤ ENNReal.ofReal ↑δ
hunif : TendstoUniformlyOn f g atTop tᶜ
⊢ ∃ N, ∀ n ≥ N, μ {x | ε ≤ edist (f n x) (g x)} ≤ ↑δ

Tactic:
rw [ENNReal.ofReal_coe_nnreal] at ht

Goal state after:
case neg
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝¹ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
inst✝ : IsFiniteMeasure μ
hf : ∀ (n : ℕ), Measurable fun a ↦ edist (f n a) (g a)
hfg : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (g x))
ε : ℝ≥0∞
hε : 0 < ε
δ : ℝ≥0
hδ : 0 < ↑δ
t : Set α
left✝ : MeasurableSet t
ht : μ t ≤ ↑δ
hunif : TendstoUniformlyOn f g atTop tᶜ
⊢ ∃ N, ∀ n ≥ N, μ {x | ε ≤ edist (f n x) (g x)} ≤ ↑δ
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
𝕜 : Type u_2
𝕜' : Type u_3
E : Type u_4
F : Type u_5
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
hp : Fact (1 ≤ p)
f g : ↥(Lp E p μ)
this : ‖f + g‖ₑ ≤ ‖f‖ₑ + ‖g‖ₑ
⊢ ‖f + g‖ ≤ ‖f‖ + ‖g‖

Tactic:
by
            -- Squeezed for performance reasons
            simpa only [ge_iff_le, enorm, ← ENNReal.coe_add, ENNReal.coe_le_coe] using! this

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

Goal state before:
pp: case e_a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
n : ℕ
hp_pos : 0 < p
hn : (∫⁻ (a : α), ‖∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖ₑ ^ p ∂μ) ^ p⁻¹ ≤ ∑' (i : ℕ), B i
a : α
⊢ ENNReal.ofReal ‖∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖ =
    ENNReal.ofReal (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖)

Tactic:
Real.norm_of_nonneg _

Goal state after:
case e_a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
n : ℕ
hp_pos : 0 < p
hn : (∫⁻ (a : α), ‖∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖ₑ ^ p ∂μ) ^ p⁻¹ ≤ ∑' (i : ℕ), B i
a : α
⊢ ENNReal.ofReal (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖) =
    ENNReal.ofReal (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖)

α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
n : ℕ
hp_pos : 0 < p
hn : (∫⁻ (a : α), ‖∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖ₑ ^ p ∂μ) ^ p⁻¹ ≤ ∑' (i : ℕ), B i
a : α
⊢ 0 ≤ ∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖
Module: Mathlib.MeasureTheory.Function.LpSpace.Complete

Goal state before:
pp: case e_a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
f : ℕ → α → E
p : ℝ
hp1 : 1 ≤ p
B : ℕ → ℝ≥0∞
n : ℕ
hp_pos : 0 < p
hn : (∫⁻ (a : α), ‖∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖ₑ ^ p ∂μ) ^ p⁻¹ ≤ ∑' (i : ℕ), B i
a : α
⊢ ENNReal.ofReal ‖∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖‖ =
    ENNReal.ofReal (∑ i ∈ Finset.range (n + 1), ‖f (i + 1) a - f i a‖)

Tactic:
rw [Real.norm_of_nonneg _]
      exact Finset.sum_nonneg fun x _ => norm_nonneg _

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

Goal state before:
pp: 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
⊢ MemLp f p μ

Tactic:
apply (memLp_norm_iff hf).1

Goal state after:
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
⊢ MemLp (fun x ↦ ‖f x‖) p μ
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
δ : ℝ≥0∞
⊢ δ > 0 → ∀ᶠ (x : ι) in l, μ {x_1 | ε ≤ edist (f x x_1) (g x_1)} ≤ δ

Tactic:
hδ

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∞
hδ : δ > 0
⊢ ∀ᶠ (x : ι) in l, μ {x_1 | ε ≤ edist (f x x_1) (g x_1)} ≤ δ
Module: Mathlib.Data.Fintype.CardEmbedding

Goal state before:
pp: case refine_3
β : Type u_2
inst✝ : Fintype β
γ : Type u_1
h : Fintype γ
ih : ‖γ ↪ β‖ = ‖β‖.descFactorial ‖γ‖
⊢ ‖Option γ ↪ β‖ = ‖β‖.descFactorial ‖Option γ‖

Tactic:
rw [card_option, Nat.descFactorial_succ, card_congr (Embedding.optionEmbeddingEquiv γ β),
        card_sigma, ← ih]
    simp only [Fintype.card_compl_set, Fintype.card_range, Finset.sum_const, Finset.card_univ,
      Nat.nsmul_eq_mul, mul_comm]

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

Goal state before:
pp: case pos.refine_2
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : CompleteSpace E
f : ℕ → α → E
hf : ∀ (n : ℕ), AEStronglyMeasurable (f n) μ
B : ℕ → ℝ≥0∞
hB : ∑' (i : ℕ), B i ≠ ∞
hp : 1 ≤ ∞
hp_top : True
h_cau_ae : ∀ᵐ (x : α) ∂μ, ∀ (N n m : ℕ), N ≤ n → N ≤ m → ‖(f n - f m) x‖ₑ < B N
h_cau : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → essSup (fun x ↦ ‖(f n - f m_1) x‖ₑ) μ < B N
x : α
hx : ∀ (N n m : ℕ), N ≤ n → N ≤ m → ‖(f n - f m) x‖ₑ < B N
⊢ Tendsto (fun n ↦ (B n).toReal) atTop (𝓝 (ENNReal.toReal 0))

Tactic:
exact
        Tendsto.comp (g := ENNReal.toReal) (ENNReal.tendsto_toReal ENNReal.zero_ne_top)
          (ENNReal.tendsto_atTop_zero_of_tsum_ne_top hB)

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Multilinear.Basic

Goal state before:
pp: R : Type u
ι : Type v
M : Type u_1
inst✝⁷ : Fintype ι
inst✝⁶ : CommRing R
inst✝⁵ : AddCommMonoid M
inst✝⁴ : Module R M
inst✝³ : TopologicalSpace R
inst✝² : TopologicalSpace M
inst✝¹ : ContinuousMul R
inst✝ : ContinuousSMul R M
⊢ ((ContinuousMultilinearMap.mkPiRing R ι 0) fun x ↦ 1) = 0 fun x ↦ 1

Tactic:
rw [mkPiRing_apply, smul_zero, zero_apply]

Goal state after:
Module: Mathlib.Topology.Algebra.Module.Multilinear.Bounded

Goal state before:
pp: ι : Type u_1
𝕜 : Type u_2
F : Type u_3
E : ι → Type u_4
inst✝⁷ : NormedField 𝕜
inst✝⁶ : (i : ι) → AddCommGroup (E i)
inst✝⁵ : (i : ι) → Module 𝕜 (E i)
inst✝⁴ : (i : ι) → TopologicalSpace (E i)
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜 F
inst✝¹ : TopologicalSpace F
inst✝ : Nonempty ι
s : Set ((i : ι) → E i)
hs : IsVonNBounded 𝕜 s
f : ContinuousMultilinearMap 𝕜 E F
V : Set F
hV : V ∈ 𝓝 0
h₁ : ¬∀ (c : 𝕜), ‖c‖ ≤ 1
x✝ : NontriviallyNormedField 𝕜 := { toNormedField := inst✝⁷, non_trivial := ⋯ }
I : Finset ι
t : (i : ι) → Set (E i)
ht₀ : ∀ (i : ι), t i ∈ 𝓝 0
hft : (↑I).pi t ⊆ ⇑f ⁻¹' V
c : ι → 𝕜
hc₀ : ∀ (i : ι), c i ≠ 0
hc : ∀ (i : ι) (c' : 𝕜), ‖c'‖ ≤ ‖c i‖ → ∀ x ∈ s, c' • x i ∈ t i
hc₀' : ∏ i ∈ I, c i ≠ 0
a : 𝕜
ha : a ∈ {y | ‖y‖ < ‖∏ i ∈ I, c i‖}
x : (i : ι) → E i
hx : x ∈ s
y : (i : ι) → E i := I.piecewise (fun i ↦ c i • x i) x
i : ι
hi : i ∈ ↑I
⊢ ‖∏ i ∈ I, c i‖ / ‖∏ i ∈ I, c i‖ * ‖c i‖ ≤ 1 * ‖c i‖

Tactic:
by gcongr; apply div_self_le_one

Goal state after:
Module: Mathlib.Analysis.Normed.Ring.Units

Goal state before:
pp: R : Type u_1
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : Rˣ
⊢ ContinuousAt inverse ↑x

Tactic:
by
  have h_is_o : (fun t : R => (↑x + t)⁻¹ʳ - ↑x⁻¹) =o[𝓝 0] (fun _ => 1 : R → ℝ) :=
    (inverse_add_norm_diff_first_order x).trans_isLittleO (isLittleO_id_const one_ne_zero).norm_left
  have h_lim : Tendsto (fun y : R => y - x) (𝓝 x) (𝓝 0) := by
    refine tendsto_zero_iff_norm_tendsto_zero.mpr ?_
    exact tendsto_iff_norm_sub_tendsto_zero.mp tendsto_id
  rw [ContinuousAt, tendsto_iff_norm_sub_tendsto_zero, inverse_unit]
  simpa [Function.comp_def] using h_is_o.norm_left.tendsto_div_nhds_zero.comp h_lim

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

Goal state before:
pp: 𝕜✝ : Type u_1
E : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜✝
inst✝⁹ : SeminormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜✝ E
R✝ : Type u_3
inst✝⁷ : NonUnitalSeminormedRing R✝
inst✝⁶ : NormedSpace 𝕜✝ R✝
inst✝⁵ : IsScalarTower 𝕜✝ R✝ R✝
inst✝⁴ : SMulCommClass 𝕜✝ R✝ R✝
𝕜 : Type u_4
R : Type u_5
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : SeminormedRing R
inst✝¹ : NormedAlgebra 𝕜 R
inst✝ : NormOneClass R
x : R
⊢ ‖x‖ ≤ ‖(mul 𝕜 R) x‖

Tactic:
convert! ratio_le_opNorm ((mul 𝕜 R) x) (1 : R)

Goal state after:
case e'_3
𝕜✝ : Type u_1
E : Type u_2
inst✝¹⁰ : NontriviallyNormedField 𝕜✝
inst✝⁹ : SeminormedAddCommGroup E
inst✝⁸ : NormedSpace 𝕜✝ E
R✝ : Type u_3
inst✝⁷ : NonUnitalSeminormedRing R✝
inst✝⁶ : NormedSpace 𝕜✝ R✝
inst✝⁵ : IsScalarTower 𝕜✝ R✝ R✝
inst✝⁴ : SMulCommClass 𝕜✝ R✝ R✝
𝕜 : Type u_4
R : Type u_5
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : SeminormedRing R
inst✝¹ : NormedAlgebra 𝕜 R
inst✝ : NormOneClass R
x : R
⊢ ‖x‖ = ‖((mul 𝕜 R) x) 1‖ / ‖1‖
Module: Mathlib.Analysis.Normed.Operator.NormedSpace

Goal state before:
pp: case h
𝕜 : Type u_1
E : Type u_5
Fₗ : Type u_7
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup Fₗ
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 Fₗ
r : ℝ
r_pos : 0 < r
c : ℝ
f : E →ₗ[𝕜] Fₗ
h : ∀ z ∈ ball 0 r, ‖f z‖ ≤ c
k : 𝕜
hk : 1 < ‖k‖
z : E
⊢ ‖f z‖ ≤ c * (‖k‖ / r) * ‖z‖

Tactic:
refine bound_of_shell _ r_pos hk (fun x hko hxo => ?_) _

Goal state after:
case h
𝕜 : Type u_1
E : Type u_5
Fₗ : Type u_7
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup Fₗ
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜 Fₗ
r : ℝ
r_pos : 0 < r
c : ℝ
f : E →ₗ[𝕜] Fₗ
h : ∀ z ∈ ball 0 r, ‖f z‖ ≤ c
k : 𝕜
hk : 1 < ‖k‖
z x : E
hko : r / ‖k‖ ≤ ‖x‖
hxo : ‖x‖ < r
⊢ ‖f x‖ ≤ c * (‖k‖ / r) * ‖x‖
Module: Mathlib.Topology.Maps.Strict.Basic

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsStrictMap f
⊢ Continuous[inst✝¹, inst✝] f

Tactic:
rw [isStrictMap_iff_isQuotientMap_rangeFactorization] at hf

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
f : X → Y
hf : IsQuotientMap (rangeFactorization f)
⊢ Continuous[inst✝¹, inst✝] f
Module: Mathlib.Topology.Algebra.Module.Complement

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : IsTopologicalAddGroup M
h : IsTopCompl p q
hq : IsClosed[inst✝³] ↑q
⊢ IsClosed[instTopologicalSpaceSubtype] {0}

Tactic:
← (isQuotientMap_projectionOntoL h).isClosed_preimage

Goal state after:
R : Type u_1
inst✝⁴ : Ring R
M : Type u_2
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
p q : Submodule R M
inst✝ : IsTopologicalAddGroup M
h : IsTopCompl p q
hq : IsClosed[inst✝³] ↑q
⊢ IsClosed[inst✝³] (⇑(p.projectionOntoL q h) ⁻¹' {0})
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
f : α → ε
μ : Measure α
inst✝³ : TopologicalSpace ε
inst✝² : ContinuousENorm ε
inst✝¹ : PseudoMetrizableSpace ε
inst✝ : Finite β
t : β → Set α
⊢ IntegrableOn f (⋃ i, t i) μ ↔ ∀ (i : β), IntegrableOn f (t i) μ

Tactic:
cases nonempty_fintype β
  simpa using integrableOn_finset_iUnion (f := f) (μ := μ) (s := Finset.univ) (t := t)

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

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
inst✝⁷ : CompleteSpace 𝕜
n : ℕ
IH :
  ∀ {E : Type v} [inst : AddCommGroup E] [inst_1 : Module 𝕜 E] [inst_2 : TopologicalSpace E]
    [inst_3 : IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E] [T2Space E] {ι : Type v} [inst_6 : Finite ι]
    (ξ : Basis ι 𝕜 E) (this : Fintype ι),
    let this_1 := IsTopologicalAddGroup.rightUniformSpace E;
    let this_2 := ⋯;
    Fintype.card ι = n → Continuous[_, Pi.topologicalSpace] ⇑ξ.equivFun
E : Type v
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : T2Space E
ι : Type v
inst✝ : Finite ι
ξ : Basis ι 𝕜 E
this✝² : Fintype ι
this✝¹ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this✝ : IsUniformAddGroup E := isUniformAddGroup_of_addCommGroup
hn : Fintype.card ι = n + 1
this : FiniteDimensional 𝕜 E
s : Submodule 𝕜 E
s_dim : finrank 𝕜 ↥s = n
⊢ IsClosed[inst✝⁴] ↑s

Tactic:
let : IsUniformAddGroup s := s.toAddSubgroup.isUniformAddGroup

Goal state after:
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
inst✝⁷ : CompleteSpace 𝕜
n : ℕ
IH :
  ∀ {E : Type v} [inst : AddCommGroup E] [inst_1 : Module 𝕜 E] [inst_2 : TopologicalSpace E]
    [inst_3 : IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E] [T2Space E] {ι : Type v} [inst_6 : Finite ι]
    (ξ : Basis ι 𝕜 E) (this : Fintype ι),
    let this_1 := IsTopologicalAddGroup.rightUniformSpace E;
    let this_2 := ⋯;
    Fintype.card ι = n → Continuous[_, Pi.topologicalSpace] ⇑ξ.equivFun
E : Type v
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : T2Space E
ι : Type v
inst✝ : Finite ι
ξ : Basis ι 𝕜 E
this✝³ : Fintype ι
this✝² : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this✝¹ : IsUniformAddGroup E := isUniformAddGroup_of_addCommGroup
hn : Fintype.card ι = n + 1
this✝ : FiniteDimensional 𝕜 E
s : Submodule 𝕜 E
s_dim : finrank 𝕜 ↥s = n
this : IsUniformAddGroup ↥s := AddSubgroup.isUniformAddGroup s.toAddSubgroup
⊢ IsClosed[inst✝⁴] ↑s
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
a b : α
ha : μ {a} ≠ ∞
ha' : ‖f a‖ₑ ≠ ∞
⊢ IntegrableOn f (Ico a b) μ ↔ IntegrableOn f (Ioo a b) μ

Tactic:
by
  by_cases hab : a < b
  · rw [← Ioo_union_left hab, integrableOn_union,
      eq_true (integrableOn_singleton ha'), and_true]
  · rw [Ioo_eq_empty hab, Ico_eq_empty hab]

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

Goal state before:
pp: 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, T.Countable ∧ (∀ u ∈ T, IsOpen[inst✝³] u) ∧ s ⊆ ⋃ u ∈ T, u ∧ ∀ u ∈ T, IntegrableOn f (u ∩ s) μ

Tactic:
by
  have : ∀ x : s, ∃ u, IsOpen u ∧ x.1 ∈ u ∧ IntegrableOn f (u ∩ s) μ := by
    rintro ⟨x, hx⟩
    rcases hf x hx with ⟨t, ht, h't⟩
    rcases mem_nhdsWithin.1 ht with ⟨u, u_open, x_mem, u_sub⟩
    exact ⟨u, u_open, x_mem, h't.mono_set u_sub⟩
  choose u u_open xu hu using this
  obtain ⟨T, T_count, hT⟩ : ∃ T : Set s, T.Countable ∧ s ⊆ ⋃ i ∈ T, u i := by
    have : s ⊆ ⋃ x : s, u x := fun y hy => mem_iUnion_of_mem ⟨y, hy⟩ (xu ⟨y, hy⟩)
    obtain ⟨T, hT_count, hT_un⟩ := isOpen_iUnion_countable u u_open
    exact ⟨T, hT_count, by rwa [hT_un]⟩
  refine ⟨u '' T, T_count.image _, ?_, by rwa [biUnion_image], ?_⟩
  · rintro v ⟨w, -, rfl⟩
    exact u_open _
  · rintro v ⟨w, -, rfl⟩
    exact hu _

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

Goal state before:
pp: 𝕜 : Type u_4
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : CompleteSpace 𝕜
Eᵤ : Type u_6
inst✝⁵ : AddCommGroup Eᵤ
inst✝⁴ : Module 𝕜 Eᵤ
inst✝³ : UniformSpace Eᵤ
inst✝² : T2Space Eᵤ
inst✝¹ : IsUniformAddGroup Eᵤ
inst✝ : ContinuousSMul 𝕜 Eᵤ
U : Set Eᵤ
hU_nhds : U ∈ 𝓝 0
hU_tb : TotallyBounded U
c : 𝕜
hc0 : 0 < ‖c‖
hc1 : ‖c‖ < 1
hc_ne : c ≠ 0
F : Set Eᵤ
hF_finite : F.Finite
hF_cover : U ⊆ ⋃ y ∈ F, y +ᵥ c • U
M : Submodule 𝕜 Eᵤ := Submodule.span 𝕜 F
this : FiniteDimensional 𝕜 ↥M := Finite.span_of_finite 𝕜 hF_finite
h_cover : U ⊆ ↑M + c • U
n : ℕ
ih : U ⊆ ↑M + c ^ n • U
⊢ ↑M + (c ^ n • ↑M + (c ^ n * c) • U) ⊆ ↑M + c ^ (n + 1) • U

Tactic:
pow_succ,

Goal state after:
𝕜 : Type u_4
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : CompleteSpace 𝕜
Eᵤ : Type u_6
inst✝⁵ : AddCommGroup Eᵤ
inst✝⁴ : Module 𝕜 Eᵤ
inst✝³ : UniformSpace Eᵤ
inst✝² : T2Space Eᵤ
inst✝¹ : IsUniformAddGroup Eᵤ
inst✝ : ContinuousSMul 𝕜 Eᵤ
U : Set Eᵤ
hU_nhds : U ∈ 𝓝 0
hU_tb : TotallyBounded U
c : 𝕜
hc0 : 0 < ‖c‖
hc1 : ‖c‖ < 1
hc_ne : c ≠ 0
F : Set Eᵤ
hF_finite : F.Finite
hF_cover : U ⊆ ⋃ y ∈ F, y +ᵥ c • U
M : Submodule 𝕜 Eᵤ := Submodule.span 𝕜 F
this : FiniteDimensional 𝕜 ↥M := Finite.span_of_finite 𝕜 hF_finite
h_cover : U ⊆ ↑M + c • U
n : ℕ
ih : U ⊆ ↑M + c ^ n • U
⊢ ↑M + (c ^ n • ↑M + (c ^ n * c) • U) ⊆ ↑M + (c ^ n * c) • U
Module: Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable

Goal state before:
pp: case neg
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
f : ι → X → E
h : ∀ (i : ι), StronglyMeasurable (f i)
hm : Multipliable (fun i ↦ f i) L
h✝¹ : ¬(L.HasSupport ∧ ((Function.mulSupport fun i ↦ f i) ∩ L.support).Finite)
h✝ : ¬HasProd (fun i ↦ f i) 1 L
⊢ StronglyMeasurable (Exists.choose hm)

Tactic:
· exact stronglyMeasurable_of_tendsto L.filter (by fun_prop) hm.choose_spec

Goal state after:
Module: Mathlib.MeasureTheory.Function.L1Space.AEEqFun

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hf : Integrable f μ
⊢ (∫⁻ (a : α), edist (f a) 0 ∂μ).toReal = (∫⁻ (a : α), ENNReal.ofReal ‖f a‖ ∂μ).toReal

Tactic:
lintegral_norm_eq_lintegral_edist

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f : α → β
hf : Integrable f μ
⊢ (∫⁻ (a : α), edist (f a) 0 ∂μ).toReal = (∫⁻ (a : α), edist (f a) 0 ∂μ).toReal
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E : ι → Type wE
E₁ : ι → Type wE₁
G : Type wG
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
g : ContinuousMultilinearMap 𝕜 E₁ G
f : (i : ι) → E i ≃ₗᵢ[𝕜] E₁ i
⊢ ‖g.compContinuousLinearMap fun i ↦ ↑↑(f i)‖ = ‖g‖

Tactic:
by
  apply le_antisymm (g.norm_compContinuous_linearIsometry_le fun i => (f i).toLinearIsometry)
  have : g = (g.compContinuousLinearMap fun i => (f i : E i →L[𝕜] E₁ i)).compContinuousLinearMap
      fun i => ((f i).symm : E₁ i →L[𝕜] E i) := by
    ext1 m
    simp
  conv_lhs => rw [this]
  apply (g.compContinuousLinearMap fun i =>
    (f i : E i →L[𝕜] E₁ i)).norm_compContinuous_linearIsometry_le
      fun i => (f i).symm.toLinearIsometry

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

Goal state before:
pp: case h₂
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : SeminormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
inst✝² : Fintype ι
α : Type u_1
inst✝¹ : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : α ≃ ↑s
inst✝ : DecidablePred fun x ↦ x ∈ s
x : (i : { a // a ∉ s }) → E₁ ↑i
m : α → (i : ι) → E₁ i
⊢ ∏ a, ‖if h : ↑a ∈ s then m (e.symm ⟨↑a, ⋯⟩) ↑a else x ⟨↑a, ⋯⟩‖ ≤ ∏ i, ‖m i‖

pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : SeminormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
inst✝² : Fintype ι
α : Type u_1
inst✝¹ : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : α ≃ ↑s
inst✝ : DecidablePred fun x ↦ x ∈ s
x : (i : { a // a ∉ s }) → E₁ ↑i
m : α → (i : ι) → E₁ i
⊢ Fintype (Subtype (Membership.mem s))

Tactic:
← Equiv.prod_comp e.symm

Goal state after:
case h₂
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : SeminormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
inst✝² : Fintype ι
α : Type u_1
inst✝¹ : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : α ≃ ↑s
inst✝ : DecidablePred fun x ↦ x ∈ s
x : (i : { a // a ∉ s }) → E₁ ↑i
m : α → (i : ι) → E₁ i
⊢ ∏ a, ‖if h : ↑a ∈ s then m (e.symm ⟨↑a, ⋯⟩) ↑a else x ⟨↑a, ⋯⟩‖ ≤ ∏ i, ‖m (e.symm i)‖

𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : SeminormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
inst✝² : Fintype ι
α : Type u_1
inst✝¹ : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : α ≃ ↑s
inst✝ : DecidablePred fun x ↦ x ∈ s
x : (i : { a // a ∉ s }) → E₁ ↑i
m : α → (i : ι) → E₁ i
⊢ Fintype (Subtype (Membership.mem s))
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
R : Type u_8
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
μ : Measure X
inst✝² : OpensMeasurableSpace X
A K : Set X
inst✝¹ : NormedRing R
inst✝ : SecondCountableTopologyEither X R
g g' : X → R
hg : IntegrableOn g A μ
hg' : ContinuousOn g' K
hA : MeasurableSet A
hK : IsCompact K
hAK : A ⊆ K
⊢ IntegrableOn (fun x ↦ g x * g' x) A μ

Tactic:
by
  rcases IsCompact.exists_bound_of_continuousOn hK hg' with ⟨C, hC⟩
  exact hg.mul_bdd ((hg'.mono hAK).aestronglyMeasurable hA)
    (ae_restrict_of_forall_mem hA fun x hx => hC x (hAK hx))

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

Goal state before:
pp: case refine_2.a
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : AddCommMonoid β
T : Set α → β
T_empty : T ∅ = 0
h_add : FinMeasAdditive μ T
ι : Type u_8
S : ι → Set α
sι : Finset ι
hS_meas : ∀ (i : ι), MeasurableSet (S i)
a : ι
s : Finset ι
has : a ∉ s
h : (∀ i ∈ s, μ (S i) ≠ ∞) → (∀ i ∈ s, ∀ j ∈ s, i ≠ j → Disjoint (S i) (S j)) → T (⋃ i ∈ s, S i) = ∑ i ∈ s, T (S i)
hps : ∀ i ∈ insert a s, μ (S i) ≠ ∞
h_disj : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → Disjoint (S i) (S j)
⊢ μ (⋃ i ∈ s, S i) ≠ ∞

Tactic:
exact (measure_biUnion_lt_top s.finite_toSet fun i hi ↦
      (hps i <| Finset.mem_insert_of_mem hi).lt_top).ne

Goal state after: