mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case left
ι : Type u
s : Finset ι
f : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
⊢ (∑ i ∈ s, f i ^ p) ^ (1 / p) ∈ (fun g ↦ ∑ i ∈ s, f i * g i) '' {g | ∑ i ∈ s, g i ^ q ≤ 1}

Tactic:
use fun i => f i ^ p / f i / (∑ i ∈ s, f i ^ p) ^ (1 / q)
    obtain hf | hf := eq_zero_or_pos (∑ i ∈ s, f i ^ p)
    · simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
    · have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
      have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
        refine fun y => mul_div_cancel_left_of_imp fun h => ?_
        simp [h, hpq.ne_zero]
      simp only [Set.mem_ofPred_eq, div_rpow, ← sum_div, ← rpow_mul,
        div_mul_cancel₀ _ hpq.symm.ne_zero, rpow_one, div_le_iff₀ hf, one_mul, hpq.mul_eq_add, ←
        rpow_sub' A, add_sub_cancel_right, le_refl, true_and, ← mul_div_assoc, B]
      rw [div_eq_iff, ← rpow_add hf.ne', one_div, one_div, hpq.inv_add_inv_eq_one, rpow_one]
      simpa [hpq.symm.ne_zero] using hf.ne'

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

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_2
E : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝³ : NormedRing 𝕜
inst✝² : NormedAddCommGroup E
inst✝¹ : MulActionWithZero 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
p q r : ℝ
f : α → E
hf : AEStronglyMeasurable f μ
φ : α → 𝕜
hφ : AEStronglyMeasurable φ μ
hp0_lt : 0 < p
hpq : p < q
hpqr : 1 / p = 1 / q + 1 / r
⊢ eLpNorm' (φ • f) p μ ≤ eLpNorm' φ q μ * eLpNorm' f r μ

Tactic:
simpa using! eLpNorm'_le_eLpNorm'_mul_eLpNorm' hφ hf (· • ·) 1
    (.of_forall fun _ => by simpa using! nnnorm_smul_le _ _)
    hp0_lt hpq hpqr

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
f : ι → ℝ
p : ℝ
hp : 1 ≤ p
⊢ (∑ i ∈ s, |f i|) ^ p ≤ ↑(#s) ^ (p - 1) * ∑ i ∈ s, |f i| ^ p

Tactic:
by
  have :=
    NNReal.coe_le_coe.2
      (NNReal.rpow_sum_le_const_mul_sum_rpow s (fun i => ⟨_, abs_nonneg (f i)⟩) hp)
  push_cast at this
  exact this

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
f : ι → ℝ
p : ℝ
hp : 1 ≤ p
hf : ∀ i ∈ s, 0 ≤ f i
⊢ (∑ i ∈ s, f i) ^ p ≤ ↑(#s) ^ (p - 1) * ∑ i ∈ s, f i ^ p

Tactic:
by
  convert! rpow_sum_le_const_mul_sum_rpow s f hp using 2 <;> apply sum_congr rfl <;> intro i hi <;>
    simp only [abs_of_nonneg, hf i hi]

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

Goal state before:
pp: case neg
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_6
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
hf : MemLp f p μ
q : ℝ≥0∞
q_top : ¬q = ∞
q_zero : ¬q = 0
⊢ eLpNorm f (p * q⁻¹ * q) μ ≠ ∞

Tactic:
mul_assoc,

Goal state after:
case neg
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_6
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
hf : MemLp f p μ
q : ℝ≥0∞
q_top : ¬q = ∞
q_zero : ¬q = 0
⊢ eLpNorm f (p * (q⁻¹ * q)) μ ≠ ∞
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: case neg
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∞
q_top : ¬q = ∞
q_zero : ¬q = 0
⊢ eLpNorm f (p * q⁻¹ * q) μ ≠ ∞

Tactic:
mul_assoc,

Goal state after:
case neg
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∞
q_top : ¬q = ∞
q_zero : ¬q = 0
⊢ eLpNorm f (p * (q⁻¹ * q)) μ ≠ ∞
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: case e'_7
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_6
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
q : ℝ≥0∞
f : α → ε
hf : AEStronglyMeasurable f μ
q_zero : q ≠ 0
q_top : q ≠ ∞
h : MemLp (fun x ↦ ‖f x‖ₑ ^ q.toReal) (p / q) μ
⊢ p = p * q⁻¹ * q

Tactic:
mul_assoc,

Goal state after:
case e'_7
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_6
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
q : ℝ≥0∞
f : α → ε
hf : AEStronglyMeasurable f μ
q_zero : q ≠ 0
q_top : q ≠ ∞
h : MemLp (fun x ↦ ‖f x‖ₑ ^ q.toReal) (p / q) μ
⊢ p = p * (q⁻¹ * q)
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: case e'_7
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
⊢ p = p * q⁻¹ * q

Tactic:
mul_assoc,

Goal state after:
case e'_7
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
⊢ p = p * (q⁻¹ * q)
Module: Mathlib.MeasureTheory.Function.LpSpace.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
f : ↥(Lp ℝ p μ)
⊢ ∀ᵐ (a : α) ∂μ, ↑↑(negPart f) a = max (-↑↑f a) 0

Tactic:
negPart

Goal state after:
α : Type u_1
m : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
f : ↥(Lp ℝ p μ)
⊢ ∀ᵐ (a : α) ∂μ, ↑↑(posPart (-f)) a = max (-↑↑f a) 0
Module: Mathlib.Topology.Algebra.Module.Multilinear.Topology

Goal state before:
pp: case inr
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : (i : ι) → TopologicalSpace (E i)
inst✝⁸ : (i : ι) → AddCommGroup (E i)
inst✝⁷ : (i : ι) → Module 𝕜 (E i)
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 F
inst✝⁴ : UniformSpace F
inst✝³ : IsUniformAddGroup F
inst✝² : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
inst✝¹ : ContinuousConstSMul 𝕜 F
inst✝ : CompleteSpace F
h : IsCoherentWith {s | IsVonNBounded 𝕜 s}
this :
  ∀ {F : Type u_4} [inst : AddCommGroup F] [inst_1 : Module 𝕜 F] [inst_2 : UniformSpace F]
    [inst_3 : IsUniformAddGroup F] [ContinuousConstSMul 𝕜 F] [CompleteSpace F],
    T2Space F → CompleteSpace (ContinuousMultilinearMap 𝕜 E F)
hF : ¬T2Space F
⊢ CompleteSpace (ContinuousMultilinearMap 𝕜 E F)

Tactic:
(isUniformInducing_postcomp (SeparationQuotient.mkCLM _ _)
      SeparationQuotient.isUniformInducing_mk).completeSpace_congr

Goal state after:
case inr
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : (i : ι) → TopologicalSpace (E i)
inst✝⁸ : (i : ι) → AddCommGroup (E i)
inst✝⁷ : (i : ι) → Module 𝕜 (E i)
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 F
inst✝⁴ : UniformSpace F
inst✝³ : IsUniformAddGroup F
inst✝² : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
inst✝¹ : ContinuousConstSMul 𝕜 F
inst✝ : CompleteSpace F
h : IsCoherentWith {s | IsVonNBounded 𝕜 s}
this :
  ∀ {F : Type u_4} [inst : AddCommGroup F] [inst_1 : Module 𝕜 F] [inst_2 : UniformSpace F]
    [inst_3 : IsUniformAddGroup F] [ContinuousConstSMul 𝕜 F] [CompleteSpace F],
    T2Space F → CompleteSpace (ContinuousMultilinearMap 𝕜 E F)
hF : ¬T2Space F
⊢ CompleteSpace (ContinuousMultilinearMap 𝕜 E (SeparationQuotient F))

case inr
𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
F : Type u_4
inst✝¹⁰ : NormedField 𝕜
inst✝⁹ : (i : ι) → TopologicalSpace (E i)
inst✝⁸ : (i : ι) → AddCommGroup (E i)
inst✝⁷ : (i : ι) → Module 𝕜 (E i)
inst✝⁶ : AddCommGroup F
inst✝⁵ : Module 𝕜 F
inst✝⁴ : UniformSpace F
inst✝³ : IsUniformAddGroup F
inst✝² : ∀ (i : ι), ContinuousSMul 𝕜 (E i)
inst✝¹ : ContinuousConstSMul 𝕜 F
inst✝ : CompleteSpace F
h : IsCoherentWith {s | IsVonNBounded 𝕜 s}
this :
  ∀ {F : Type u_4} [inst : AddCommGroup F] [inst_1 : Module 𝕜 F] [inst_2 : UniformSpace F]
    [inst_3 : IsUniformAddGroup F] [ContinuousConstSMul 𝕜 F] [CompleteSpace F],
    T2Space F → CompleteSpace (ContinuousMultilinearMap 𝕜 E F)
hF : ¬T2Space F
⊢ Surjective (SeparationQuotient.mkCLM 𝕜 F).compContinuousMultilinearMap
Module: Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Idempotent

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : Ring R
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsTopologicalAddGroup M
f T : M →L[R] M
hT : IsUnit T
hf : IsIdempotentElem f
⊢ Commute f T ↔ Submodule.map (↑T) (↑f).range = (↑f).range ∧ Submodule.map (↑T) (↑f).ker = (↑f).ker

Tactic:
have := hT.map ContinuousLinearMap.toLinearMapRingHom

Goal state after:
R : Type u_1
M : Type u_2
inst✝⁴ : Ring R
inst✝³ : TopologicalSpace M
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : IsTopologicalAddGroup M
f T : M →L[R] M
hT : IsUnit T
hf : IsIdempotentElem f
this : IsUnit (toLinearMapRingHom T)
⊢ Commute f T ↔ Submodule.map (↑T) (↑f).range = (↑f).range ∧ Submodule.map (↑T) (↑f).ker = (↑f).ker
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s t : Set α
hs : MeasurableSet s
hst : s ⊆ t
ε : ℝ
h : μ.real t < μ.real s + ε
ht' : μ t ≠ ∞
⊢ μ.real (t \ s) < ε

Tactic:
rw [measureReal_sdiff hst hs ht']; linarith

Goal state after:
Module: Mathlib.Order.Filter.IndicatorFunction

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : One β
l : Filter α
f : α → β
hf : ∀ᶠ (x : α) in l, f x ≠ 1
s✝ t : Set α
h : s✝.mulIndicator f =ᶠ[l] t.mulIndicator f
s : Set α
⊢ Function.mulSupport (s.mulIndicator f) =ᶠ[l] s

Tactic:
mulSupport_mulIndicator

Goal state after:
α : Type u_1
β : Type u_2
inst✝ : One β
l : Filter α
f : α → β
hf : ∀ᶠ (x : α) in l, f x ≠ 1
s✝ t : Set α
h : s✝.mulIndicator f =ᶠ[l] t.mulIndicator f
s : Set α
⊢ s ∩ Function.mulSupport f =ᶠ[l] s
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: case neg
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
t : TopologicalSpace 𝕜
h₁ : IsTopologicalAddGroup 𝕜
h₂ : ContinuousSMul 𝕜 𝕜
h₃ : T2Space 𝕜
ε : ℝ
hε : 0 < ε
ξ₀ : 𝕜
hξ₀ : 0 < ‖ξ₀‖
hξ₀ε : ‖ξ₀‖ < ε
this✝ : {ξ₀}ᶜ ∈ 𝓝 0
this : balancedCore 𝕜 {ξ₀}ᶜ ∈ 𝓝 0
ξ : 𝕜
hξ : ξ ∈ balancedCore 𝕜 {ξ₀}ᶜ
hξ0 : ¬ξ = 0
⊢ ξ ∈ Metric.closedBall 0 ε

Tactic:
rw [mem_closedBall_zero_iff]

Goal state after:
case neg
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
t : TopologicalSpace 𝕜
h₁ : IsTopologicalAddGroup 𝕜
h₂ : ContinuousSMul 𝕜 𝕜
h₃ : T2Space 𝕜
ε : ℝ
hε : 0 < ε
ξ₀ : 𝕜
hξ₀ : 0 < ‖ξ₀‖
hξ₀ε : ‖ξ₀‖ < ε
this✝ : {ξ₀}ᶜ ∈ 𝓝 0
this : balancedCore 𝕜 {ξ₀}ᶜ ∈ 𝓝 0
ξ : 𝕜
hξ : ξ ∈ balancedCore 𝕜 {ξ₀}ᶜ
hξ0 : ¬ξ = 0
⊢ ‖ξ‖ ≤ ε
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
t : TopologicalSpace 𝕜
h₁ : IsTopologicalAddGroup 𝕜
h₂ : ContinuousSMul 𝕜 𝕜
h₃ : T2Space 𝕜
ε : ℝ
hε : 0 < ε
ξ₀ : 𝕜
hξ₀ : 0 < ‖ξ₀‖
hξ₀ε : ‖ξ₀‖ < ε
this✝¹ : {ξ₀}ᶜ ∈ 𝓝 0
this✝ : balancedCore 𝕜 {ξ₀}ᶜ ∈ 𝓝 0
ξ : 𝕜
hξ : ξ ∈ balancedCore 𝕜 {ξ₀}ᶜ
hξ0 : ¬ξ = 0
h : ε < ‖ξ‖
this : ξ₀ * ξ⁻¹ * ξ ∈ balancedCore 𝕜 {ξ₀}ᶜ
⊢ False

Tactic:
mul_assoc,

Goal state after:
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
t : TopologicalSpace 𝕜
h₁ : IsTopologicalAddGroup 𝕜
h₂ : ContinuousSMul 𝕜 𝕜
h₃ : T2Space 𝕜
ε : ℝ
hε : 0 < ε
ξ₀ : 𝕜
hξ₀ : 0 < ‖ξ₀‖
hξ₀ε : ‖ξ₀‖ < ε
this✝¹ : {ξ₀}ᶜ ∈ 𝓝 0
this✝ : balancedCore 𝕜 {ξ₀}ᶜ ∈ 𝓝 0
ξ : 𝕜
hξ : ξ ∈ balancedCore 𝕜 {ξ₀}ᶜ
hξ0 : ¬ξ = 0
h : ε < ‖ξ‖
this : ξ₀ * (ξ⁻¹ * ξ) ∈ balancedCore 𝕜 {ξ₀}ᶜ
⊢ False
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 g : α → β
hf : Integrable f μ
hg : Integrable g μ
⊢ ∫⁻ (a : α), edist (f a) (0 a) ∂μ < ∞ ∧ ∫⁻ (a : α), edist (g a) (0 a) ∂μ < ∞

Tactic:
by
      simp_rw [Pi.zero_apply, ← hasFiniteIntegral_iff_edist]
      exact ⟨hf.hasFiniteIntegral, hg.hasFiniteIntegral⟩

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

Goal state before:
pp: case mp
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
l l' : Filter α
⊢ IntegrableAtFilter f (l ⊔ l') μ → IntegrableAtFilter f l μ ∧ IntegrableAtFilter f l' μ

pp: case mpr
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
l l' : Filter α
⊢ IntegrableAtFilter f l μ ∧ IntegrableAtFilter f l' μ → IntegrableAtFilter f (l ⊔ l') μ

Tactic:
· exact fun h => ⟨h.filter_mono le_sup_left, h.filter_mono le_sup_right⟩

Goal state after:
case mpr
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
l l' : Filter α
⊢ IntegrableAtFilter f l μ ∧ IntegrableAtFilter f l' μ → IntegrableAtFilter f (l ⊔ l') μ
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
ε : Type u_5
ε' : Type u_6
ε'' : Type u_7
m : MeasurableSpace α
μ ν : Measure α
inst✝⁹ : MeasurableSpace δ
inst✝⁸ : NormedAddCommGroup β
inst✝⁷ : NormedAddCommGroup γ
inst✝⁶ : TopologicalSpace ε
inst✝⁵ : ContinuousENorm ε
inst✝⁴ : TopologicalSpace ε'
inst✝³ : ContinuousENorm ε'
inst✝² : ENorm ε''
E : Type u_8
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : α → ℝ≥0
f_meas : Measurable f
u v : ↥(Lp E 1 (μ.withDensity fun x ↦ ↑(f x)))
x : α
hu : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) x = f x • ↑↑u x
hv : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯) x = f x • ↑↑v x
huv : ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑(u + v) x) ⋯) x = f x • ↑↑(u + v) x
h' :
  ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯ + MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯) x =
    (↑↑(MemLp.toLp (fun x ↦ f x • ↑↑u x) ⋯) + ↑↑(MemLp.toLp (fun x ↦ f x • ↑↑v x) ⋯)) x
h'' : ↑(f x) ≠ 0 → ↑↑(u + v) x = (↑↑u + ↑↑v) x
hx : f x ≠ 0
⊢ f x • ↑↑(u + v) x = f x • ↑↑u x + f x • ↑↑v x

Tactic:
rw [h'' _, Pi.add_apply, smul_add]
      simpa only [Ne, ENNReal.coe_eq_zero] using hx

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✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : Fintype ι
𝕜' : Type u_1
inst✝³ : SeminormedRing 𝕜'
inst✝² : Module 𝕜' G
inst✝¹ : IsBoundedSMul 𝕜' G
inst✝ : SMulCommClass 𝕜 𝕜' G
c : 𝕜'
f : ContinuousMultilinearMap 𝕜 E G
m : (i : ι) → E i
⊢ ‖c‖ * ‖f m‖ ≤ ‖c‖ * ‖f‖ * ∏ i, ‖m i‖

Tactic:
mul_assoc,

Goal state after:
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : Fintype ι
𝕜' : Type u_1
inst✝³ : SeminormedRing 𝕜'
inst✝² : Module 𝕜' G
inst✝¹ : IsBoundedSMul 𝕜' G
inst✝ : SMulCommClass 𝕜 𝕜' G
c : 𝕜'
f : ContinuousMultilinearMap 𝕜 E G
m : (i : ι) → E i
⊢ ‖c‖ * ‖f m‖ ≤ ‖c‖ * (‖f‖ * ∏ i, ‖m i‖)
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
h : SecondCountableTopologyEither α β
inst✝¹ : OpensMeasurableSpace α
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : MeasurableSet s
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
h✝ : SecondCountableTopology α
⊢ IsSeparable (f '' s)

Tactic:
image_eq_range

Goal state after:
case inl
α : Type u_1
β : Type u_2
mα : MeasurableSpace α
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
h : SecondCountableTopologyEither α β
inst✝¹ : OpensMeasurableSpace α
inst✝ : PseudoMetrizableSpace β
f : α → β
s : Set α
μ : Measure α
hf : ContinuousOn f s
hs : MeasurableSet s
this✝¹ : MeasurableSpace β := borel β
this✝ : BorelSpace β
h✝ : SecondCountableTopology α
⊢ IsSeparable (range fun x ↦ f ↑x)
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hfm : AEStronglyMeasurable f μ
hf : 0 ≤ᵐ[μ] f
⊢ ∫⁻ (a : α), ENNReal.ofReal (f a) ∂μ ≠ ∞ ↔ Integrable f μ

Tactic:
rw [Integrable, hasFiniteIntegral_iff_ofReal hf]
  simp [hfm]

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

Goal state before:
pp: case refine_2
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 : ContinuousMultilinearMap 𝕜 E 𝕜
z : G
⊢ ‖f‖₊ * ‖z‖₊ ≤ ‖f.smulRight z‖₊

Tactic:
obtain hz | hz := eq_zero_or_pos ‖z‖₊
    · simp [hz]
    rw [← le_div_iff₀ hz, opNNNorm_le_iff]
    intro m
    rw [div_mul_eq_mul_div, le_div_iff₀ hz]
    refine le_trans ?_ ((f.smulRight z).le_opNNNorm m)
    rw [smulRight_apply, nnnorm_smul]

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

Goal state before:
pp: case hxy
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 = ↑M

Tactic:
lift c to 𝕜ˣ using isUnit_iff_ne_zero.mpr hc_ne

Goal state after:
case hxy
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
F : Set Eᵤ
hF_finite : F.Finite
M : Submodule 𝕜 Eᵤ := Submodule.span 𝕜 F
this : FiniteDimensional 𝕜 ↥M := Finite.span_of_finite 𝕜 hF_finite
n : ℕ
c : 𝕜ˣ
hc0 : 0 < ‖↑c‖
hc1 : ‖↑c‖ < 1
hc_ne : ↑c ≠ 0
hF_cover : U ⊆ ⋃ y ∈ F, y +ᵥ ↑c • U
h_cover : U ⊆ ↑M + ↑c • U
ih : U ⊆ ↑M + ↑c ^ n • U
⊢ ↑M + ↑c ^ n • ↑M = ↑M
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✝³ : NormedAddCommGroup β
𝕜 : Type u_5
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 β
inst✝ : IsBoundedSMul 𝕜 β
c : 𝕜
f : α →ₘ[μ] β
_f : α → β
hfm : AEStronglyMeasurable _f μ
hfi : (mk _f hfm).Integrable
⊢ MeasureTheory.Integrable ((fun x ↦ c • x) ∘ ↑⟨_f, hfm⟩) μ

Tactic:
by simpa using! ((integrable_mk hfm).1 hfi).smul c

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

Goal state before:
pp: case mpr
X : Type u_1
ε : Type u_3
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace ε
inst✝⁵ : ContinuousENorm ε
f : X → ε
μ : Measure X
inst✝⁴ : PseudoMetrizableSpace ε
a : X
inst✝³ : LinearOrder X
inst✝² : CompactIccSpace X
inst✝¹ : NoMinOrder X
inst✝ : OrderTopology X
⊢ IntegrableAtFilter f atBot μ ∧ IntegrableAtFilter f (𝓝[<] a) μ ∧ LocallyIntegrableOn f (Iio a) μ →
    IntegrableOn f (Iio a) μ

Tactic:
intro ⟨hbot, ⟨s, hsl, hs⟩, hlocal⟩

Goal state after:
case mpr
X : Type u_1
ε : Type u_3
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace ε
inst✝⁵ : ContinuousENorm ε
f : X → ε
μ : Measure X
inst✝⁴ : PseudoMetrizableSpace ε
a : X
inst✝³ : LinearOrder X
inst✝² : CompactIccSpace X
inst✝¹ : NoMinOrder X
inst✝ : OrderTopology X
hbot : IntegrableAtFilter f atBot μ
s : Set X
hsl : s ∈ 𝓝[<] a
hs : IntegrableOn f s μ
hlocal : LocallyIntegrableOn f (Iio a) μ
⊢ IntegrableOn f (Iio a) μ
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 : ContinuousOn g K
hg' : IntegrableOn g' A μ
hK : IsCompact K
hA : MeasurableSet A
hAK : A ⊆ K
C : ℝ
hC : ∀ x ∈ K, ‖g x‖ ≤ C
⊢ IntegrableOn (fun x ↦ g x * g' x) A μ

Tactic:
exact hg'.bdd_mul ((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: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
T : Set α → β
inst✝ : AddMonoid β
c : ℝ≥0∞
hc_ne_top : c ≠ ∞
hT : FinMeasAdditive (c • μ) T
s : Set α
x✝ : MeasurableSet s
hμs : ¬c = 0 ∧ μ s = ∞
⊢ μ s = ∞

Tactic:
exact hμs.2

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

Goal state before:
pp: case refine_1
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
Fₗ : Type u_6
inst✝¹⁷ : AddCommGroup E
inst✝¹⁶ : UniformSpace E
inst✝¹⁵ : IsUniformAddGroup E
inst✝¹⁴ : AddCommGroup F
inst✝¹³ : UniformSpace F
inst✝¹² : IsUniformAddGroup F
inst✝¹¹ : T0Space F
inst✝¹⁰ : AddCommMonoid Eₗ
inst✝⁹ : UniformSpace Eₗ
inst✝⁸ : ContinuousAdd Eₗ
inst✝⁷ : Semiring 𝕜
inst✝⁶ : Semiring 𝕜₂
inst✝⁵ : Module 𝕜 E
inst✝⁴ : Module 𝕜₂ F
inst✝³ : Module 𝕜 Eₗ
inst✝² : ContinuousConstSMul 𝕜 Eₗ
inst✝¹ : ContinuousConstSMul 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
f g : E →SL[σ₁₂] F
inst✝ : CompleteSpace F
e : E →L[𝕜] Eₗ
h : DenseRange ⇑e ∧ IsUniformInducing ⇑e
cont : Continuous (⋯.extend ⇑f)
eq : ∀ (b : E), ⋯.extend (⇑f) (e b) = f b
k : 𝕜
b : Eₗ
⊢ IsClosed {b | ⋯.extend (⇑f) (k • b) = σ₁₂ k • ⋯.extend (⇑f) b}

pp: case refine_2
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
Fₗ : Type u_6
inst✝¹⁷ : AddCommGroup E
inst✝¹⁶ : UniformSpace E
inst✝¹⁵ : IsUniformAddGroup E
inst✝¹⁴ : AddCommGroup F
inst✝¹³ : UniformSpace F
inst✝¹² : IsUniformAddGroup F
inst✝¹¹ : T0Space F
inst✝¹⁰ : AddCommMonoid Eₗ
inst✝⁹ : UniformSpace Eₗ
inst✝⁸ : ContinuousAdd Eₗ
inst✝⁷ : Semiring 𝕜
inst✝⁶ : Semiring 𝕜₂
inst✝⁵ : Module 𝕜 E
inst✝⁴ : Module 𝕜₂ F
inst✝³ : Module 𝕜 Eₗ
inst✝² : ContinuousConstSMul 𝕜 Eₗ
inst✝¹ : ContinuousConstSMul 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
f g : E →SL[σ₁₂] F
inst✝ : CompleteSpace F
e : E →L[𝕜] Eₗ
h : DenseRange ⇑e ∧ IsUniformInducing ⇑e
cont : Continuous (⋯.extend ⇑f)
eq : ∀ (b : E), ⋯.extend (⇑f) (e b) = f b
k : 𝕜
b : Eₗ
⊢ ∀ (a : E), ⋯.extend (⇑f) (k • e a) = σ₁₂ k • ⋯.extend (⇑f) (e a)

Tactic:
· exact isClosed_eq (cont.comp (continuous_const_smul _))
          ((continuous_const_smul _).comp cont)

Goal state after:
case refine_2
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
Fₗ : Type u_6
inst✝¹⁷ : AddCommGroup E
inst✝¹⁶ : UniformSpace E
inst✝¹⁵ : IsUniformAddGroup E
inst✝¹⁴ : AddCommGroup F
inst✝¹³ : UniformSpace F
inst✝¹² : IsUniformAddGroup F
inst✝¹¹ : T0Space F
inst✝¹⁰ : AddCommMonoid Eₗ
inst✝⁹ : UniformSpace Eₗ
inst✝⁸ : ContinuousAdd Eₗ
inst✝⁷ : Semiring 𝕜
inst✝⁶ : Semiring 𝕜₂
inst✝⁵ : Module 𝕜 E
inst✝⁴ : Module 𝕜₂ F
inst✝³ : Module 𝕜 Eₗ
inst✝² : ContinuousConstSMul 𝕜 Eₗ
inst✝¹ : ContinuousConstSMul 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
f g : E →SL[σ₁₂] F
inst✝ : CompleteSpace F
e : E →L[𝕜] Eₗ
h : DenseRange ⇑e ∧ IsUniformInducing ⇑e
cont : Continuous (⋯.extend ⇑f)
eq : ∀ (b : E), ⋯.extend (⇑f) (e b) = f b
k : 𝕜
b : Eₗ
⊢ ∀ (a : E), ⋯.extend (⇑f) (k • e a) = σ₁₂ k • ⋯.extend (⇑f) (e a)
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
f : ↥(simpleFunc E p μ)
⊢ 1 • f = f

Tactic:
ext1; exact one_smul _ _

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
f : α →ₛ ℝ
⊢ map norm f.negPart = f.negPart

Tactic:
negPart

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
f : α →ₛ ℝ
⊢ map norm (-f).posPart = (-f).posPart