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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
hp : Fact (1 ≤ p)
H :
  ∀ (f : ℕ → α → E),
    (∀ (n : ℕ), MemLp (f n) p μ) →
      ∀ (B : ℕ → ℝ≥0∞),
        ∑' (i : ℕ), B i < ∞ →
          (∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → eLpNorm (f n - f m_1) p μ < B N) →
            ∃ f_lim, MemLp f_lim p μ ∧ Tendsto (fun n ↦ eLpNorm (f n - f_lim) p μ) atTop (𝓝 0)
B : ℕ → ℝ := fun n ↦ (1 / 2) ^ n
hB_pos : ∀ (n : ℕ), 0 < B n
f : ℕ → ↥(Lp E p μ)
hf : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → dist (f n) (f m_1) < B N
M : ℝ
hB : HasSum B M
B1 : ℕ → ℝ≥0∞ := fun n ↦ ENNReal.ofReal (B n)
⊢ ∃ f_lim, MemLp f_lim p μ ∧ Tendsto (fun n ↦ eLpNorm (↑↑(f n) - f_lim) p μ) atTop (𝓝 0)

Tactic:
have hB1_has : HasSum B1 (ENNReal.ofReal M) := by
    have h_tsum_B1 : ∑' i, B1 i = ENNReal.ofReal M := by
      change (∑' n : ℕ, ENNReal.ofReal (B n)) = ENNReal.ofReal M
      rw [← hB.tsum_eq]
      exact (ENNReal.ofReal_tsum_of_nonneg (fun n => le_of_lt (hB_pos n)) hB.summable).symm
    have h_sum := (@ENNReal.summable _ B1).hasSum
    rwa [h_tsum_B1] at h_sum

Goal state after:
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
E : Type u_3
inst✝ : NormedAddCommGroup E
hp : Fact (1 ≤ p)
H :
  ∀ (f : ℕ → α → E),
    (∀ (n : ℕ), MemLp (f n) p μ) →
      ∀ (B : ℕ → ℝ≥0∞),
        ∑' (i : ℕ), B i < ∞ →
          (∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → eLpNorm (f n - f m_1) p μ < B N) →
            ∃ f_lim, MemLp f_lim p μ ∧ Tendsto (fun n ↦ eLpNorm (f n - f_lim) p μ) atTop (𝓝 0)
B : ℕ → ℝ := fun n ↦ (1 / 2) ^ n
hB_pos : ∀ (n : ℕ), 0 < B n
f : ℕ → ↥(Lp E p μ)
hf : ∀ (N n m_1 : ℕ), N ≤ n → N ≤ m_1 → dist (f n) (f m_1) < B N
M : ℝ
hB : HasSum B M
B1 : ℕ → ℝ≥0∞ := fun n ↦ ENNReal.ofReal (B n)
hB1_has : HasSum B1 (ENNReal.ofReal M)
⊢ ∃ f_lim, MemLp f_lim p μ ∧ Tendsto (fun n ↦ eLpNorm (↑↑(f n) - f_lim) p μ) atTop (𝓝 0)
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✝¹ : NormedAddCommGroup E
inst✝ : IsFiniteMeasure μ
f : ↥(Lp E p μ)
C : ℝ
hC : 0 ≤ C
hfC : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖ ≤ C
⊢ ‖f‖ ≤ ↑(measureUnivNNReal μ) ^ p.toReal⁻¹ * C

Tactic:
by
  lift C to ℝ≥0 using hC
  have := nnnorm_le_of_ae_bound hfC
  rwa [← NNReal.coe_le_coe, NNReal.coe_mul, NNReal.coe_rpow] at 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‖‖ =
    ∑ x ∈ Finset.range (n + 1), ENNReal.ofReal ‖f (x + 1) a - f x a‖

Tactic:
rw [← ENNReal.ofReal_sum_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‖)

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
⊢ ∀ i ∈ Finset.range (n + 1), 0 ≤ ‖f (i + 1) a - f i a‖
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
f : α → E
hf : MemLp f p μ
q : ℝ≥0∞
⊢ MemLp (fun x ↦ ‖f x‖ ^ q.toReal) (p / q) μ

Tactic:
refine ⟨(hf.1.norm.aemeasurable.pow_const q.toReal).aestronglyMeasurable, ?_⟩
  by_cases q_top : q = ∞
  · simp [q_top]
  by_cases q_zero : q = 0
  · simp only [q_zero, ENNReal.toReal_zero, Real.rpow_zero]
    by_cases p_zero : p = 0
    · simp [p_zero]
    rw [ENNReal.div_zero p_zero]
    exact (memLp_top_const (1 : ℝ)).2
  rw [eLpNorm_norm_rpow _ (ENNReal.toReal_pos q_zero q_top)]
  apply ENNReal.rpow_lt_top_of_nonneg ENNReal.toReal_nonneg
  rw [ENNReal.ofReal_toReal q_top, div_eq_mul_inv, mul_assoc, ENNReal.inv_mul_cancel q_zero q_top,
    mul_one]
  exact hf.2.ne

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

Goal state before:
pp: α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k ↦ {x | 2⁻¹ ^ k ≤ edist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
⊢ μ s = 0

Tactic:
refine measure_limsup_atTop_eq_zero (ne_top_of_le_ne_top ?_ (ENNReal.tsum_le_tsum hμS_le))

Goal state after:
α : Type u_1
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : PseudoEMetricSpace E
f : ℕ → α → E
g : α → E
hfg : TendstoInMeasure μ f atTop g
h_lt_ε_real : ∀ (ε : ℝ≥0∞), 0 < ε → ∃ k, 2 * 2⁻¹ ^ k < ε
ns : ℕ → ℕ := ExistsSeqTendstoAe.seqTendstoAeSeq hfg
S : ℕ → Set α := fun k ↦ {x | 2⁻¹ ^ k ≤ edist (f (ns k) x) (g x)}
hμS_le : ∀ (k : ℕ), μ (S k) ≤ 2⁻¹ ^ k
s : Set α := limsup S atTop
hs : s = limsup S atTop
⊢ ∑' (a : ℕ), 2⁻¹ ^ a ≠ ∞
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, 1 < ‖x‖

Tactic:
by simpa using h₁

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
t : R
h : ‖t‖ < ‖↑x⁻¹‖⁻¹
a✝ : Nontrivial R
⊢ ‖-(↑x⁻¹ * t)‖ < 1

Tactic:
have hpos : 0 < ‖(↑x⁻¹ : R)‖ := Units.norm_pos x⁻¹

Goal state after:
R : Type u_1
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : Rˣ
t : R
h : ‖t‖ < ‖↑x⁻¹‖⁻¹
a✝ : Nontrivial R
hpos : 0 < ‖↑x⁻¹‖
⊢ ‖-(↑x⁻¹ * t)‖ < 1
Module: Mathlib.Analysis.Normed.Ring.Units

Goal state before:
pp: R : Type u_1
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : Rˣ
n : ℕ
hzero : Tendsto (fun x_1 ↦ -↑x⁻¹ * x_1) (𝓝 0) (𝓝 0)
t : R
ht : (↑x + t)⁻¹ʳ = (1 + ↑x⁻¹ * t)⁻¹ʳ * ↑x⁻¹
ht' : (1 + ↑x⁻¹ * t)⁻¹ʳ = ∑ i ∈ range n, (-(↑x⁻¹ * t)) ^ i + (-(↑x⁻¹ * t)) ^ n * (1 + ↑x⁻¹ * t)⁻¹ʳ
⊢ (∑ i ∈ range n, (-↑x⁻¹ * t) ^ i) * ↑x⁻¹ + (-↑x⁻¹ * t) ^ n * ((1 + ↑x⁻¹ * t)⁻¹ʳ * ↑x⁻¹) =
    (∑ i ∈ range n, (-↑x⁻¹ * t) ^ i) * ↑x⁻¹ + (-↑x⁻¹ * t) ^ n * (↑x + t)⁻¹ʳ

Tactic:
rw [ht]

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

Goal state before:
pp: case mp
𝕜 : Type u_1
𝕜₂ : Type u_3
E : Type u_5
F : Type u_6
ι : Type u_9
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁴ : RingHomIsometric σ₁₂
inst✝³ : SeminormedAddCommGroup E
inst✝² : SeminormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 E
inst✝ : NormedSpace 𝕜₂ F
f : ι → E →SL[σ₁₂] F
tfae_1_to_3 : EquicontinuousAt (DFunLike.coe ∘ f) 0 → UniformEquicontinuous (DFunLike.coe ∘ f)
tfae_3_to_2 : UniformEquicontinuous (DFunLike.coe ∘ f) → Equicontinuous (DFunLike.coe ∘ f)
tfae_2_to_1 : Equicontinuous (DFunLike.coe ∘ f) → EquicontinuousAt (DFunLike.coe ∘ f) 0
tfae_4_iff_5 : (∃ C, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖) ↔ ∃ C ≥ 0, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖
tfae_5_iff_7 : (∃ C ≥ 0, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖) ↔ ∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C
tfae_7_iff_8 : (∃ C ≥ 0, ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (Set.range fun x ↦ ‖f x‖)
tfae_6_iff_8 : (∃ C, ∀ (i : ι), ‖f i‖ ≤ C) ↔ BddAbove (Set.range fun x ↦ ‖f x‖)
tfae_8_iff_9 : BddAbove (Set.range fun x ↦ ‖f x‖) ↔ ⨆ i, ↑‖f i‖₊ < ⊤
p : Seminorm 𝕜 E
hp : Continuous[PseudoMetricSpace.toUniformSpace.toTopologicalSpace, _] ⇑p
hpf : ∀ (k : ι), (normSeminorm 𝕜₂ F).comp ↑(f k) ≤ p
C : ℝ
hC : ∀ (x : E), p x ≤ C * ‖x‖
⊢ ∃ C, ∀ (i : ι) (x : E), ‖(f i) x‖ ≤ C * ‖x‖

Tactic:
exact ⟨C, fun i x ↦ (hpf i x).trans (hC x)⟩

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

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹¹ : NormedField 𝕜
inst✝¹⁰ : (i : ι) → TopologicalSpace (E i)
inst✝⁹ : (i : ι) → AddCommGroup (E i)
inst✝⁸ : (i : ι) → Module 𝕜 (E i)
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜 F
inst✝⁵ : TopologicalSpace F
inst✝⁴ : IsTopologicalAddGroup F
E₁ : ι → Type u_5
inst✝³ : (i : ι) → TopologicalSpace (E₁ i)
inst✝² : ContinuousConstSMul 𝕜 F
inst✝¹ : (i : ι) → AddCommGroup (E₁ i)
inst✝ : (i : ι) → Module 𝕜 (E₁ i)
f : (i : ι) → E i →L[𝕜] E₁ i
aux : ContinuousMultilinearMap 𝕜 E₁ F →ₗ[𝕜] ContinuousMultilinearMap 𝕜 E F :=
  { toFun := fun g ↦ g.compContinuousLinearMap f, map_add' := ⋯, map_smul' := ⋯ }
⊢ Continuous[instTopologicalSpace, instTopologicalSpace] aux.toFun

Tactic:
by
      apply continuous_of_tendsto_nhds_zero aux
      rw [hasBasis_nhds_zero.tendsto_iff hasBasis_nhds_zero]
      rintro ⟨U, V⟩ ⟨hU, hV⟩
      set φ : (∀ i, E i) →L[𝕜] (∀ i, E₁ i) := .piMap f
      exact ⟨(φ '' U, V), ⟨hU.image φ, hV⟩, fun g hg ↦ hg.comp (mapsTo_image _ _)⟩

Goal state after:
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
⊢ IsStrictMap f ↔ IsEmbedding (kerLift f)

Tactic:
simp only [isStrictMap_iff_isHomeomorph_quotientKerEquivRange,
    isHomeomorph_iff_isEmbedding_surjective, Equiv.surjective, and_true]
  exact (IsEmbedding.of_comp_iff .subtypeVal).symm

Goal state after:
Module: Mathlib.MeasureTheory.Function.StronglyMeasurable.Lemmas

Goal state before:
pp: case mpr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g g' : α → E
g'meas : StronglyMeasurable g'
hg' : (fun x ↦ ↑(f x) • g x) =ᵐ[μ] g'
⊢ g =ᵐ[μ.withDensity fun x ↦ ↑(f x)] fun x ↦ (↑(f x))⁻¹ • g' x

Tactic:
rw [EventuallyEq, ae_withDensity_iff hf.coe_nnreal_ennreal]

Goal state after:
case mpr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
E : Type u_4
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
hf : Measurable f
g g' : α → E
g'meas : StronglyMeasurable g'
hg' : (fun x ↦ ↑(f x) • g x) =ᵐ[μ] g'
⊢ ∀ᵐ (x : α) ∂μ, ↑(f x) ≠ 0 → g x = (↑(f x))⁻¹ • g' x
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✝ : T1Space ↥p
h : IsTopCompl p q
⊢ IsClosed[inst✝³] ↑q

Tactic:
← ker_projectionOntoL h

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✝ : T1Space ↥p
h : IsTopCompl p q
⊢ IsClosed[inst✝³] ↑(↑(p.projectionOntoL q h)).ker
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[inst✝³] (⇑(p.projectionOntoL q h) ⁻¹' {0})

Tactic:
← ker_projectionOntoL h

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] ↑(↑(p.projectionOntoL q h)).ker
⊢ IsClosed[inst✝³] (⇑(p.projectionOntoL q h) ⁻¹' {0})
Module: Mathlib.MeasureTheory.Function.LpSpace.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : E
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
⊢ ‖indicatorConstLp p hs hμs c‖ = ‖c‖ * μ.real s ^ (1 / p.toReal)

Tactic:
by
  rw [Lp.norm_def, eLpNorm_congr_ae indicatorConstLp_coeFn,
    eLpNorm_indicator_const hs hp_ne_zero hp_ne_top, ENNReal.toReal_mul, measureReal_def,
    ENNReal.toReal_rpow, toReal_enorm]

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

Goal state before:
pp: α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
⊢ MemLp f 1 μ ↔ AEStronglyMeasurable f μ ∧ HasFiniteIntegral f μ

Tactic:
hasFiniteIntegral_iff_enorm,

Goal state after:
α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
⊢ MemLp f 1 μ ↔ AEStronglyMeasurable f μ ∧ ∫⁻ (a : α), ‖f a‖ₑ ∂μ < ∞
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
c : ε
hc : ‖c‖ₑ ≠ ∞
this : AEStronglyMeasurable (fun x ↦ c) μ
⊢ HasFiniteIntegral (fun x ↦ c) μ ↔ ‖c‖ₑ = 0 ∨ IsFiniteMeasure μ

Tactic:
hasFiniteIntegral_const_iff_enorm hc

Goal state after:
α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
c : ε
hc : ‖c‖ₑ ≠ ∞
this : AEStronglyMeasurable (fun x ↦ c) μ
⊢ ‖c‖ₑ = 0 ∨ IsFiniteMeasure μ ↔ ‖c‖ₑ = 0 ∨ IsFiniteMeasure μ
Module: Mathlib.MeasureTheory.Function.LpSpace.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
inst✝¹ : NormedAddCommGroup E
inst✝ : IsFiniteMeasure μ
c : E
⊢ ‖(Lp.const p μ) c‖ ≤ ‖c‖ * μ.real Set.univ ^ (1 / p.toReal)

Tactic:
by
  rw [← indicatorConstLp_univ]
  exact norm_indicatorConstLp_le

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
hf₀ : ∀ {m : (i : ι) → E i} {i : ι}, ‖m i‖ = 0 → ‖f m‖ = 0
ε : ι → ℝ
C : ℝ
hε : ∀ (i : ι), 0 < ε i
c : ι → 𝕜
hc : ∀ (i : ι), 1 < ‖c i‖
hf : ∀ (m : (i : ι) → E i), (∀ (i : ι), ε i / ‖c i‖ ≤ ‖m i‖) → (∀ (i : ι), ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i, ‖m i‖
m : (i : ι) → E i
⊢ ‖f m‖ ≤ C * ∏ i, ‖m i‖

Tactic:
by_cases! hm : ∃ i, ‖m i‖ = 0
  · rcases hm with ⟨i, hi⟩
    rw [hf₀ hi, prod_eq_zero (mem_univ i) hi, mul_zero]
  choose δ hδ0 hδm_lt hle_δm _ using fun i => rescale_to_shell_semi_normed (hc i) (hε i) (hm i)
  have hδ0 : 0 < ∏ i, ‖δ i‖ := prod_pos fun i _ => norm_pos_iff.2 (hδ0 i)
  simpa [map_smul_univ, norm_smul, prod_mul_distrib, mul_left_comm C, hδ0] using
    hf (fun i => δ i • m i) hle_δm hδm_lt

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

Goal state before:
pp: α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ ν : Measure α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
inst✝ : PseudoMetrizableSpace ε
f : α → ε
hμ : MemLp f 1 μ
hν : MemLp f 1 ν
⊢ eLpNorm f 1 μ < ∞ ∧ eLpNorm f 1 ν < ∞

Tactic:
exact ⟨hμ.eLpNorm_lt_top, hν.eLpNorm_lt_top⟩

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_8
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
c : ℝ≥0∞
h₁ : c ≠ 0
h₂ : c ≠ ∞
⊢ c⁻¹ ≠ ∞

Tactic:
by simpa using h₁

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

Goal state before:
pp: case inl
α : Type u_1
m : MeasurableSpace α
ε : Type u_8
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
h : Integrable f 0
⊢ Integrable f ((0 univ)⁻¹ • 0)

Tactic:
rwa [smul_zero]

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

Goal state before:
pp: case inr
α : Type u_1
m : MeasurableSpace α
μ : Measure α
ε : Type u_8
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
h : Integrable f μ
hne : μ ≠ 0
⊢ Integrable f ((μ univ)⁻¹ • μ)

Tactic:
· apply h.smul_measure
    simpa

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

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : TopologicalSpace ε'
inst✝ : ESeminormedAddMonoid ε'
f : α → ε'
hs : MeasurableSet s
⊢ AEStronglyMeasurable (s.indicator f) μ ∧ HasFiniteIntegral (s.indicator f) μ ↔
    AEStronglyMeasurable f (μ.restrict s) ∧ HasFiniteIntegral f (μ.restrict s)

Tactic:
hasFiniteIntegral_iff_enorm,

Goal state after:
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : TopologicalSpace ε'
inst✝ : ESeminormedAddMonoid ε'
f : α → ε'
hs : MeasurableSet s
⊢ AEStronglyMeasurable (s.indicator f) μ ∧ ∫⁻ (a : α), ‖s.indicator f a‖ₑ ∂μ < ∞ ↔
    AEStronglyMeasurable f (μ.restrict s) ∧ ∫⁻ (a : α) in s, ‖f a‖ₑ ∂μ < ∞
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : TopologicalSpace ε'
inst✝ : ESeminormedAddMonoid ε'
f : α → ε'
hf : IntegrableOn f s μ
h's : ∀ x ∈ s, ‖f x‖ₑ ≠ 0
u : ℕ → ℝ≥0∞
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), u n ∈ Ioo 0 ∞
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → Set α := fun n ↦ toMeasurable (μ.restrict s) {x | u n ≤ ‖f x‖ₑ}
A : ∀ (n : ℕ), μ (s ∩ v n) ≠ ∞
x : α
hx : x ∈ s
⊢ x ∈ ⋃ n, v n

Tactic:
obtain ⟨n, hn⟩ : ∃ n, u n < ‖f x‖ₑ :=
    ((tendsto_order.1 u_lim).2 _ (pos_of_ne_zero (h's x hx))).exists

Goal state after:
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : TopologicalSpace ε'
inst✝ : ESeminormedAddMonoid ε'
f : α → ε'
hf : IntegrableOn f s μ
h's : ∀ x ∈ s, ‖f x‖ₑ ≠ 0
u : ℕ → ℝ≥0∞
left✝ : StrictAnti u
u_pos : ∀ (n : ℕ), u n ∈ Ioo 0 ∞
u_lim : Tendsto u atTop (𝓝 0)
v : ℕ → Set α := fun n ↦ toMeasurable (μ.restrict s) {x | u n ≤ ‖f x‖ₑ}
A : ∀ (n : ℕ), μ (s ∩ v n) ≠ ∞
x : α
hx : x ∈ s
n : ℕ
hn : u n < ‖f x‖ₑ
⊢ x ∈ ⋃ n, v n
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

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
ε : ℝ
hε : 0 < ε
x : α
⊢ ε ≤ ‖f x‖ ↔ ENNReal.ofReal ε ≤ ‖f x‖ₑ

Tactic:
rw [← Real.enorm_of_nonneg hε.le, enorm_le_iff_norm_le, Real.norm_of_nonneg hε.le]

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
A : ∀ (f : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G), ‖-f.1 + f.2‖ = ‖f.1 - f.2‖
c : 𝕜
hc : 1 < ‖c‖
hc₀ : 0 < ‖c‖
f : ContinuousMultilinearMap 𝕜 E G
hf : ∀ (x : (i : ι) → E i), ‖x‖ ≤ ‖c‖ → ‖f x‖ ≤ 1
x : (i : ι) → E i
hcx : ∀ (i : ι), ‖c‖ / ‖c‖ ≤ ‖x i‖
hx : ∀ (i : ι), ‖x i‖ < ‖c‖
⊢ ∏ i, 1 ≤ ∏ i, ‖x i‖

Tactic:
by gcongr with i; simpa only [div_self hc₀.ne'] using hcx i

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
𝕜 : Type u_8
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : CompleteSpace 𝕜
E : Type u_9
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : α → 𝕜
c : E
hc : c ≠ 0
⊢ AEStronglyMeasurable f μ → (HasFiniteIntegral (fun x ↦ f x • c) μ ↔ HasFiniteIntegral f μ)

Tactic:
hasFiniteIntegral_iff_enorm,

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
𝕜 : Type u_8
inst✝³ : NontriviallyNormedField 𝕜
inst✝² : CompleteSpace 𝕜
E : Type u_9
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : α → 𝕜
c : E
hc : c ≠ 0
⊢ AEStronglyMeasurable f μ → (∫⁻ (a : α), ‖f a • c‖ₑ ∂μ < ∞ ↔ ∫⁻ (a : α), ‖f a‖ₑ ∂μ < ∞)
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : CompleteSpace 𝕜
inst✝¹ : T2Space E
inst✝ : FiniteDimensional 𝕜 E
f : E →ₗ[𝕜] E
g : E →L[𝕜] E
⊢ toContinuousLinearMap f = g ↔ f = ↑g

Tactic:
simp [ContinuousLinearMap.ext_iff, LinearMap.ext_iff]

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

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
E : Type v
inst✝⁷ : AddCommGroup E
inst✝⁶ : Module 𝕜 E
inst✝⁵ : TopologicalSpace E
inst✝⁴ : IsTopologicalAddGroup E
inst✝³ : ContinuousSMul 𝕜 E
inst✝² : CompleteSpace 𝕜
inst✝¹ : T2Space E
inst✝ : FiniteDimensional 𝕜 E
g : E →L[𝕜] E
f : E →ₗ[𝕜] E
⊢ ↑g = f ↔ g = toContinuousLinearMap f

Tactic:
simp [ContinuousLinearMap.ext_iff, LinearMap.ext_iff]

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

Goal state before:
pp: case hbc
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✝¹⁰ : Fintype ι'
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : SeminormedAddCommGroup G'
inst✝¹ : NormedSpace 𝕜 G'
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E (G →L[𝕜] G')
x : G
m : (i : ι) → E i
⊢ ‖f m‖ ≤ ‖f‖ * ∏ i, ‖m i‖

Tactic:
apply f.le_opNorm

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

Goal state before:
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✝¹⁰ : Fintype ι'
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : SeminormedAddCommGroup G'
inst✝¹ : NormedSpace 𝕜 G'
inst✝ : Fintype ι
⊢ (G →L[𝕜] ContinuousMultilinearMap 𝕜 E G') ≃L[𝕜] ContinuousMultilinearMap 𝕜 E (G →L[𝕜] G')

Tactic:
refine (flipMultilinearEquivₗ 𝕜 E G G').toContinuousLinearEquivOfBounds 1 1 ?_ ?_

Goal state after:
case refine_1
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✝¹⁰ : Fintype ι'
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : SeminormedAddCommGroup G'
inst✝¹ : NormedSpace 𝕜 G'
inst✝ : Fintype ι
⊢ ∀ (x : G →L[𝕜] ContinuousMultilinearMap 𝕜 E G'), ‖(flipMultilinearEquivₗ 𝕜 E G G') x‖ ≤ 1 * ‖x‖

case refine_2
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✝¹⁰ : Fintype ι'
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : SeminormedAddCommGroup G'
inst✝¹ : NormedSpace 𝕜 G'
inst✝ : Fintype ι
⊢ ∀ (x : ContinuousMultilinearMap 𝕜 E (G →L[𝕜] G')), ‖(flipMultilinearEquivₗ 𝕜 E G G').symm x‖ ≤ 1 * ‖x‖