mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Module.Spaces.UniformConvergenceCLM

Goal state before:
pp: 𝕜₁ : Type u_1
𝕜₂ : Type u_2
inst✝⁸ : NormedField 𝕜₁
inst✝⁷ : NormedField 𝕜₂
σ : 𝕜₁ →+* 𝕜₂
E : Type u_3
F : Type u_4
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜₁ E
inst✝⁴ : TopologicalSpace E
inst✝³ : AddCommGroup F
inst✝² : Module 𝕜₂ F
inst✝¹ : TopologicalSpace F
inst✝ : IsTopologicalAddGroup F
𝔖 : Set (Set E)
ι : Type u_6
p : ι → Prop
b : ι → Set F
h : (𝓝 0).HasBasis p b
⊢ 𝓝 0 = ⨅ s ∈ 𝔖, ⨅ i, ⨅ (_ : p i), 𝓟 {f | MapsTo (⇑f) s (b i)}

Tactic:
by
  let : UniformSpace F := IsTopologicalAddGroup.rightUniformSpace F
  have : IsUniformAddGroup F := isUniformAddGroup_of_addCommGroup
  rw [(isEmbedding_coeFn σ F 𝔖).isInducing.nhds_eq_comap,
    UniformOnFun.nhds_eq_of_basis _ _ h.uniformity_of_nhds_zero]
  simp [MapsTo]

Goal state after:
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : AddCommGroup E
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : ContinuousConstSMul 𝕜 E
p : Seminorm 𝕜 E
h : WithSeminorms fun x ↦ p
i : Fin 1
⊢ ∃ r > 0, ∀ x ∈ p.ball 0 1, p x < r

Tactic:
exact ⟨1, zero_lt_one, by simp⟩

Goal state after:
Module: Mathlib.Order.Filter.Germ.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
l : Filter α
f : l.Germ β
lc : Filter γ
g : γ → α
hf : f.IsConstant
hg : Tendsto g lc l
⊢ (f.compTendsto g hg).IsConstant

Tactic:
induction f using Quotient.inductionOn with | _ f => ?_

Goal state after:
case h
α : Type u_1
β : Type u_2
γ : Type u_3
l : Filter α
lc : Filter γ
g : γ → α
hg : Tendsto g lc l
f : α → β
hf : IsConstant ⟦f⟧
⊢ (compTendsto ⟦f⟧ g hg).IsConstant
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
t₁ t₂ : TopologicalSpace E
ht₁ : PolynormableSpace 𝕜 E
ht₂ : PolynormableSpace 𝕜 E
⊢ PolynormableSpace 𝕜 E

Tactic:
exact .sInf (by simp [ht₁, ht₂])

Goal state after:
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝² : TopologicalSpace α
inst✝¹ : UniformSpace β
ι : Type u₃
δ : ι → Type u_1
inst✝ : (i : ι) → TopologicalSpace (δ i)
φ : (i : ι) → C(δ i, α)
h_proper : ∀ (i : ι), IsProperMap ⇑(φ i)
h_lf : LocallyFinite fun i ↦ range ⇑(φ i)
h_cover : ⋃ i, range ⇑(φ i) = univ
𝔖 : Set (Set α) := {K | IsCompact K}
𝔗 : (i : ι) → Set (Set (δ i)) := fun i ↦ {K | IsCompact K}
h_image : ∀ (i : ι), MapsTo (fun x ↦ ⇑(φ i) '' x) (𝔗 i) 𝔖
h_preimage : ∀ (i : ι), MapsTo (fun x ↦ ⇑(φ i) ⁻¹' x) 𝔖 (𝔗 i)
S : Set α
hS : S ∈ 𝔖
⊢ ⋃ i, ⋃ (_ : (range ⇑(φ i) ∩ S).Nonempty), range ⇑(φ i) ∩ S = S

Tactic:
iUnion_nonempty_self,

Goal state after:
α : Type u₁
β : Type u₂
inst✝² : TopologicalSpace α
inst✝¹ : UniformSpace β
ι : Type u₃
δ : ι → Type u_1
inst✝ : (i : ι) → TopologicalSpace (δ i)
φ : (i : ι) → C(δ i, α)
h_proper : ∀ (i : ι), IsProperMap ⇑(φ i)
h_lf : LocallyFinite fun i ↦ range ⇑(φ i)
h_cover : ⋃ i, range ⇑(φ i) = univ
𝔖 : Set (Set α) := {K | IsCompact K}
𝔗 : (i : ι) → Set (Set (δ i)) := fun i ↦ {K | IsCompact K}
h_image : ∀ (i : ι), MapsTo (fun x ↦ ⇑(φ i) '' x) (𝔗 i) 𝔖
h_preimage : ∀ (i : ι), MapsTo (fun x ↦ ⇑(φ i) ⁻¹' x) 𝔖 (𝔗 i)
S : Set α
hS : S ∈ 𝔖
⊢ ⋃ i, range ⇑(φ i) ∩ S = S
Module: Mathlib.Topology.ContinuousMap.Algebra

Goal state before:
pp: case inr
α : Type u_1
inst✝⁴ : TopologicalSpace α
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalSemiring R
inst✝ : Subsingleton α
s₁ s₂ : Subalgebra R C(α, R)
h✝ : Nonempty α
inhabited_h : Inhabited α
f : C(α, R)
h : f = (algebraMap R C(α, R)) (f default)
⊢ f ∈ s₁ ↔ f ∈ s₂

Tactic:
rw [h]

Goal state after:
case inr
α : Type u_1
inst✝⁴ : TopologicalSpace α
R : Type u_2
inst✝³ : CommSemiring R
inst✝² : TopologicalSpace R
inst✝¹ : IsTopologicalSemiring R
inst✝ : Subsingleton α
s₁ s₂ : Subalgebra R C(α, R)
h✝ : Nonempty α
inhabited_h : Inhabited α
f : C(α, R)
h : f = (algebraMap R C(α, R)) (f default)
⊢ (algebraMap R C(α, R)) (f default) ∈ s₁ ↔ (algebraMap R C(α, R)) (f default) ∈ s₂
Module: Mathlib.Topology.ContinuousMap.Algebra

Goal state before:
pp: α : Type u_1
inst✝⁸ : TopologicalSpace α
R : Type u_2
inst✝⁷ : Semiring R
inst✝⁶ : TopologicalSpace R
M : Type u_3
inst✝⁵ : TopologicalSpace M
inst✝⁴ : AddCommMonoid M
inst✝³ : Module R M
inst✝² : ContinuousSMul R M
inst✝¹ : IsTopologicalSemiring R
inst✝ : ContinuousAdd M
f : C(α, M)
⊢ 0 • f = 0

Tactic:
by ext x; exact zero_smul _ _

Goal state after:
Module: Mathlib.Probability.UniformOn

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t : Set Ω
h : (uniformOn s) t = 1
⊢ (uniformOn ?m.14) ?m.15 ≠ 0

Tactic:
rw [h]

Goal state after:
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t : Set Ω
h : (uniformOn s) t = 1
⊢ 1 ≠ 0
Module: Mathlib.MeasureTheory.Function.AEEqFun

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
δ : Type u_4
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace δ
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
g : β → γ → δ
hg : Continuous[instTopologicalSpaceProd, inst✝²] (uncurry g)
f₁ : α →ₘ[μ] β
f₂ : α →ₘ[μ] γ
⊢ ↑(comp₂ g hg f₁ f₂) =ᵐ[μ] fun a ↦ g (↑f₁ a) (↑f₂ a)

Tactic:
by
  rw [comp₂_eq_mk]
  apply coeFn_mk

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

Goal state before:
pp: α : Type u_1
x✝ : MeasurableSpace α
f : α → ℝ
μ : Measure α
hf : AEMeasurable (fun x ↦ rexp (f x)) μ
⊢ AEMeasurable f μ

Tactic:
have : f = fun x ↦ log (exp (f x)) := by ext; rw [log_exp]
  rw [this]
  exact measurable_log.comp_aemeasurable hf

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

Goal state before:
pp: case inl
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
p : ℝ≥0∞
hs : MeasurableSet s
⊢ MemLp (fun x ↦ 0) p (μ.restrict s)

pp: case inr
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
p : ℝ≥0∞
hs : MeasurableSet s
c : E
hμ : μ s ≠ ∞
⊢ MemLp (fun x ↦ c) p (μ.restrict s)

Tactic:
· exact MemLp.zero

Goal state after:
case inr
α : Type u_1
E : Type u_4
m0 : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
p : ℝ≥0∞
hs : MeasurableSet s
c : E
hμ : μ s ≠ ∞
⊢ MemLp (fun x ↦ c) p (μ.restrict s)
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι✝ : Type u_1
ι' : Type u_2
α : ι✝ → Type u_3
inst✝⁷ : Fintype ι✝
m : (i : ι✝) → OuterMeasure (α i)
inst✝⁶ : (i : ι✝) → MeasurableSpace (α i)
μ✝ : (i : ι✝) → Measure (α i)
inst✝⁵ : ∀ (i : ι✝), SigmaFinite (μ✝ i)
ι : Type u_4
inst✝⁴ : Fintype ι
X : ι → Type u_5
inst✝³ : (i : ι) → PseudoMetricSpace (X i)
inst✝² : (i : ι) → MeasurableSpace (X i)
μ : (i : ι) → Measure (X i)
inst✝¹ : ∀ (i : ι), SigmaFinite (μ i)
inst✝ : ∀ (i : ι), IsUnifLocDoublingMeasure (μ i)
⊢ IsUnifLocDoublingMeasure (Measure.pi μ)

Tactic:
use ∏ i, IsUnifLocDoublingMeasure.doublingConstant (μ i)
  filter_upwards [Filter.eventually_all.mpr fun i ↦
      IsUnifLocDoublingMeasure.eventually_measure_le_doublingConstant_mul (μ i),
    eventually_mem_nhdsWithin] with r hr (hr₀ : 0 < r) x
  simpa (disch := positivity) [Finset.prod_mul_distrib, closedBall_pi, pi_pi]
    using Fintype.prod_mono' fun i ↦ hr i (x i)

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝² : TopologicalSpace ε
inst✝¹ : ESeminormedAddMonoid ε
s : Set α
f : α → ε
inst✝ : DecidablePred fun x ↦ x ∈ s
g : α → ε
hs : MeasurableSet s
hf : MemLp f p (μ.restrict s)
hg : MemLp g p (μ.restrict sᶜ)
⊢ MemLp (s.piecewise f g) p μ

Tactic:
by_cases hp_zero : p = 0
  · simp only [hp_zero, memLp_zero_iff_aestronglyMeasurable]
    exact AEStronglyMeasurable.piecewise hs hf.1 hg.1
  refine ⟨AEStronglyMeasurable.piecewise hs hf.1 hg.1, ?_⟩
  obtain rfl | hp_top := eq_or_ne p ∞
  · rw [eLpNorm_top_piecewise f g hs]
    exact max_lt hf.2 hg.2
  rw [eLpNorm_lt_top_iff_lintegral_rpow_enorm_lt_top hp_zero hp_top, ← lintegral_add_compl _ hs,
    ENNReal.add_lt_top]
  constructor
  · have h (x) (hx : x ∈ s) : ‖Set.piecewise s f g x‖ₑ ^ p.toReal = ‖f x‖ₑ ^ p.toReal := by
      simp [hx]
    rw [setLIntegral_congr_fun hs h]
    exact lintegral_rpow_enorm_lt_top_of_eLpNorm_lt_top hp_zero hp_top hf.2
  · have h (x) (hx : x ∈ sᶜ) : ‖Set.piecewise s f g x‖ₑ ^ p.toReal = ‖g x‖ₑ ^ p.toReal := by
      have hx' : x ∉ s := hx
      simp [hx']
    rw [setLIntegral_congr_fun hs.compl h]
    exact lintegral_rpow_enorm_lt_top_of_eLpNorm_lt_top hp_zero hp_top hg.2

Goal state after:
Module: Mathlib.Analysis.Convex.Slope

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
hxy' : 0 < y - x
hyz' : 0 < z - y
hxz' : 0 < z - x
ha : 0 ≤ (z - y) / (z - x)
hb : 0 ≤ (y - x) / (z - x)
key : f (((z - y) / (z - x)) • x + ((y - x) / (z - x)) • z) ≤ ((z - y) / (z - x)) • f x + ((y - x) / (z - x)) • f z
⊢ (z - x) * f y ≤ (z - y) * f x + (y - x) * f z

pp: case refine_1
𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
hxy' : 0 < y - x
hyz' : 0 < z - y
hxz' : 0 < z - x
ha : 0 ≤ (z - y) / (z - x)
hb : 0 ≤ (y - x) / (z - x)
⊢ (z - y) / (z - x) + (y - x) / (z - x) = 1

Tactic:
· simp only [smul_eq_mul] at key
    ring_nf at key
    field_simp at key
    linear_combination key

Goal state after:
case refine_1
𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
hxy' : 0 < y - x
hyz' : 0 < z - y
hxz' : 0 < z - x
ha : 0 ≤ (z - y) / (z - x)
hb : 0 ≤ (y - x) / (z - x)
⊢ (z - y) / (z - x) + (y - x) / (z - x) = 1
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: s : ℝ
hs : -1 ≤ s
p : ℝ
hp : 1 ≤ p
⊢ 1 + p * s ≤ (1 + s) ^ p

Tactic:
rcases eq_or_lt_of_le hp with (rfl | hp)
  · simp
  by_cases hs' : s = 0
  · simp [hs']
  exact (one_add_mul_self_lt_rpow_one_add hs hs' hp).le

Goal state after:
Module: Mathlib.Analysis.Convex.Slope

Goal state before:
pp: case refine_2
𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : StrictConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
hxy' : 0 < y - x
hyz' : 0 < z - y
hxz' : 0 < z - x
ha : 0 < (z - y) / (z - x)
hb : 0 < (y - x) / (z - x)
key : f (((z - y) / (z - x)) • x + ((y - x) / (z - x)) • z) < ((z - y) / (z - x)) • f x + ((y - x) / (z - x)) • f z
⊢ (z - x) * f y < (z - y) * f x + (y - x) * f z

pp: case refine_1
𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : StrictConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
hxy' : 0 < y - x
hyz' : 0 < z - y
hxz' : 0 < z - x
ha : 0 < (z - y) / (z - x)
hb : 0 < (y - x) / (z - x)
⊢ (z - y) / (z - x) + (y - x) / (z - x) = 1

Tactic:
· simp only [smul_eq_mul] at key
    ring_nf at key
    field_simp at key
    linear_combination key

Goal state after:
case refine_1
𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : StrictConvexOn 𝕜 s f
x y z : 𝕜
hx : x ∈ s
hz : z ∈ s
hxy : x < y
hyz : y < z
hxy' : 0 < y - x
hyz' : 0 < z - y
hxz' : 0 < z - x
ha : 0 < (z - y) / (z - x)
hb : 0 < (y - x) / (z - x)
⊢ (z - y) / (z - x) + (y - x) / (z - x) = 1
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: p q : ℝ≥0
hp : 1 < p
⊢ p.HolderConjugate q ↔ q = p / (p - 1)

Tactic:
rw [← holderConjugate_coe_iff, Real.holderConjugate_iff_eq_conjExponent (by exact_mod_cast hp),
    ← coe_one, ← NNReal.coe_sub hp.le]

Goal state after:
p q : ℝ≥0
hp : 1 < p
⊢ ↑q = ↑p / ↑(p - 1) ↔ q = p / (p - 1)
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
hp0_lt : 0 < p
f : α → ℝ≥0∞
hf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ∞
⊢ ∫⁻ (c : α), funMulInvSnorm f p μ c ^ p ∂μ = 1

Tactic:
funMulInvSnorm_rpow hp0_lt

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
hp0_lt : 0 < p
f : α → ℝ≥0∞
hf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0
hf_top : ∫⁻ (a : α), f a ^ p ∂μ ≠ ∞
⊢ ∫⁻ (c : α), f c ^ p * (∫⁻ (c : α), f c ^ p ∂μ)⁻¹ ∂μ = 1
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case pos.inl
a b : ℝ≥0∞
p q : ℝ
hpq : p.HolderConjugate q
h : a = ∞
⊢ ∞ = a ^ p * (ENNReal.ofReal p)⁻¹ + b ^ q * (ENNReal.ofReal q)⁻¹

Tactic:
rw [h]

Goal state after:
case pos.inl
a b : ℝ≥0∞
p q : ℝ
hpq : p.HolderConjugate q
h : a = ∞
⊢ ∞ = ∞ ^ p * (ENNReal.ofReal p)⁻¹ + b ^ q * (ENNReal.ofReal q)⁻¹
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ∞
hp0 : p ≠ 0
hp_top : ¬p = ∞
hp_pos : 0 < p.toReal
hq_top : ¬q = ∞
⊢ q ≠ 0

Tactic:
by_contra hq_eq_zero

Goal state after:
α : Type u_1
ε : Type u_2
m : MeasurableSpace α
μ : Measure α
f : α → ε
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
p q : ℝ≥0∞
inst✝ : IsFiniteMeasure μ
hpq : p ≤ q
hfq_m : AEStronglyMeasurable f μ
hfq_lt_top : eLpNorm f q μ < ∞
hp0 : p ≠ 0
hp_top : ¬p = ∞
hp_pos : 0 < p.toReal
hq_top : ¬q = ∞
hq_eq_zero : q = 0
⊢ False
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case pos.inr
a b : ℝ≥0∞
p q : ℝ
hpq : p.HolderConjugate q
h : b = ∞
⊢ ∞ = a ^ p * (ENNReal.ofReal p)⁻¹ + b ^ q * (ENNReal.ofReal q)⁻¹

Tactic:
rw [h]

Goal state after:
case pos.inr
a b : ℝ≥0∞
p q : ℝ
hpq : p.HolderConjugate q
h : b = ∞
⊢ ∞ = a ^ p * (ENNReal.ofReal p)⁻¹ + ∞ ^ q * (ENNReal.ofReal q)⁻¹
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : p.HolderConjugate q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
⊢ ∫⁻ (a : α), f a * g a ^ (p - 1) ∂μ ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / q)

Tactic:
refine (ENNReal.lintegral_mul_le_Lp_mul_Lq μ hpq hf (hg.pow_const _)).trans_eq ?_

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p q : ℝ
hpq : p.HolderConjugate q
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
⊢ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), (g a ^ (p - 1)) ^ q ∂μ) ^ (1 / q) =
    (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) * (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / q)
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case inr.inr.inr.inr
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
p q r : ℝ≥0∞
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hpqr✝ : p.HolderTriple q r
hpqr : 1 / r = 1 / p + 1 / q
hp : 0 < p.toReal
hq : 0 < q.toReal
hp₁ : 0 < p
hp₂ : p < ∞
hq₁ : 0 < q
hq₂ : q < ∞
hpqr' : 1 / r.toReal = 1 / p.toReal + 1 / q.toReal
hr : 0 < r.toReal
⊢ eLpNorm (fun x ↦ b (f x) (g x)) r μ ≤ ↑c * eLpNorm f p μ * eLpNorm g q μ

Tactic:
obtain ⟨hr₁, hr₂⟩ := ENNReal.toReal_pos_iff.mp hr

Goal state after:
case inr.inr.inr.inr
α : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
m : MeasurableSpace α
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedAddCommGroup G
μ : Measure α
f : α → E
g : α → F
p q r : ℝ≥0∞
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
b : E → F → G
c : ℝ≥0
h : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hpqr✝ : p.HolderTriple q r
hpqr : 1 / r = 1 / p + 1 / q
hp : 0 < p.toReal
hq : 0 < q.toReal
hp₁ : 0 < p
hp₂ : p < ∞
hq₁ : 0 < q
hq₂ : q < ∞
hpqr' : 1 / r.toReal = 1 / p.toReal + 1 / q.toReal
hr : 0 < r.toReal
hr₁ : 0 < r
hr₂ : r < ∞
⊢ eLpNorm (fun x ↦ b (f x) (g x)) r μ ≤ ↑c * eLpNorm f p μ * eLpNorm g q μ
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case pos
ι : Type u
s : Finset ι
f g : ι → ℝ≥0∞
p q : ℝ
hpq : p.HolderConjugate q
H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0
this : ∀ i ∈ s, f i * g i = 0
⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)

Tactic:
simp [sum_eq_zero this]

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‖| ^ 1

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
⊢ q.toReal ≠ 0

Tactic:
abs_of_nonneg (norm_nonneg _),

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‖ ^ 1

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
⊢ q.toReal ≠ 0
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case pos
ι : Type u
s : Finset ι
p : ℝ
hp✝ : 1 ≤ p
w f : ι → ℝ≥0∞
hp : 1 < p
hp₀ : 0 < p
hp₁ : p⁻¹ < 1
H : (∀ i ∈ s, w i = 0) ∨ ∀ i ∈ s, w i = 0 ∨ f i = 0
this : ∀ i ∈ s, w i * f i = 0
⊢ ∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹

Tactic:
simp [sum_eq_zero this]

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

Goal state before:
pp: case h
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
hμs : μ s = 0
h_tendsto : ∀ x ∈ sᶜ, Tendsto (fun i ↦ f (ns i) x) atTop (𝓝 (g x))
x : α
⊢ ¬Tendsto (fun i ↦ f (ns i) x) atTop (𝓝 (g x)) → ¬x ∉ s

Tactic:
not_imp_not

Goal state after:
case h
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
hμs : μ s = 0
h_tendsto : ∀ x ∈ sᶜ, Tendsto (fun i ↦ f (ns i) x) atTop (𝓝 (g x))
x : α
⊢ x ∉ s → Tendsto (fun i ↦ f (ns i) x) atTop (𝓝 (g x))
Module: Mathlib.Topology.Algebra.Module.Multilinear.Bounded

Goal state before:
pp: case inr
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
i₀ : ι
y : (i : ι) → E i := I.piecewise (fun i ↦ c i • x i) x
i : ι
hi : i ∈ ↑I
hne : i ≠ i₀
⊢ I.piecewise (fun i ↦ c i • x i) x i ∈ t i

Tactic:
I.piecewise_eq_of_mem _ _ hi

Goal state after:
case inr
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
i₀ : ι
y : (i : ι) → E i := I.piecewise (fun i ↦ c i • x i) x
i : ι
hi : i ∈ ↑I
hne : i ≠ i₀
⊢ c i • x i ∈ t i
Module: Mathlib.Analysis.Normed.Ring.Units

Goal state before:
pp: R : Type u_1
inst✝¹ : NormedRing R
inst✝ : HasSummableGeomSeries R
x : Rˣ
⊢ (fun t ↦ (↑x + t)⁻¹ʳ - ↑x⁻¹ + ↑x⁻¹ * t * ↑x⁻¹) =O[𝓝 0] fun t ↦ ‖t‖ ^ 2

Tactic:
convert! inverse_add_norm_diff_nth_order x 2 using 2
  simp only [sum_range_succ, sum_range_zero, zero_add, pow_zero, pow_one, add_mul, one_mul,
    ← sub_sub, neg_mul, sub_neg_eq_add]

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
R : Type u_3
inst✝⁴ : SeminormedRing R
inst✝³ : NormedAlgebra 𝕜 R
inst✝² : Module R E
inst✝¹ : IsBoundedSMul R E
inst✝ : IsScalarTower 𝕜 R E
⊢ ‖lsmul 𝕜 R‖ₑ ≤ 1

Tactic:
rw [enorm_eq_nnnorm]
  simpa using opNNNorm_lsmul_le

Goal state after:
Module: Mathlib.Topology.Algebra.Module.ContinuousLinearMap.Idempotent

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : Ring R
inst✝² : TopologicalSpace M
inst✝¹ : AddCommGroup M
inst✝ : Module R M
f T : M →L[R] M
hf : IsIdempotentElem f
⊢ (↑f).ker ∈ Module.End.invtSubmodule ↑T ↔ f ∘SL T ∘SL f = f ∘SL T

Tactic:
simpa [← toLinearMap_comp] using
    LinearMap.IsIdempotentElem.ker_mem_invtSubmodule_iff (T := T) hf.toLinearMap

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

Goal state before:
pp: case mp.ht
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' : g =ᵐ[μ.withDensity fun x ↦ ↑(f x)] g'
A : MeasurableSet {x | f x ≠ 0}
⊢ ∀ᵐ (x : α) ∂μ.restrict {x | f x ≠ 0}, (fun x ↦ ↑(f x) • g x) x = (fun x ↦ ↑(f x) • g' x) x

Tactic:
rw [EventuallyEq, ae_withDensity_iff hf.coe_nnreal_ennreal] at hg'
      rw [ae_restrict_iff' A]
      filter_upwards [hg'] with a ha h'a
      have : (f a : ℝ≥0∞) ≠ 0 := by simpa only [Ne, ENNReal.coe_eq_zero] using h'a
      rw [ha this]

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

Goal state before:
pp: case inl
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s₁ s₂ : Set α
h : μ (s₁ ∪ s₂) = ∞
⊢ μ.real (s₁ ∪ s₂) ≤ μ.real s₁ + μ.real s₂

Tactic:
simp only [Measure.real, h, ENNReal.toReal_top]

Goal state after:
case inl
α : Type u_1
x✝ : MeasurableSpace α
μ : Measure α
s₁ s₂ : Set α
h : μ (s₁ ∪ s₂) = ∞
⊢ 0 ≤ (μ s₁).toReal + (μ s₂).toReal