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

Goal state before:
pp: 𝕝 : Type u_4
𝕝₂ : Type u_5
E : Type u_6
F : Type u_7
ι' : Type u_10
inst✝⁹ : AddCommGroup E
inst✝⁸ : NormedField 𝕝
inst✝⁷ : Module 𝕝 E
inst✝⁶ : AddCommGroup F
inst✝⁵ : NormedField 𝕝₂
inst✝⁴ : Module 𝕝₂ F
τ₁₂ : 𝕝 →+* 𝕝₂
inst✝³ : RingHomIsometric τ₁₂
q : SeminormFamily 𝕝₂ F ι'
inst✝² : TopologicalSpace E
inst✝¹ : IsTopologicalAddGroup E
inst✝ : TopologicalSpace F
hq : WithSeminorms q
f : E →ₛₗ[τ₁₂] F
hf : ∀ (i : ι'), Continuous[inst✝², _] ⇑((q i).comp f)
⊢ Continuous[inst✝², inst✝] ⇑f

Tactic:
by
  have : IsTopologicalAddGroup F := hq.topologicalAddGroup
  refine continuous_of_continuousAt_zero f ?_
  simp_rw [ContinuousAt, f.map_zero, q.withSeminorms_iff_nhds_eq_iInf.mp hq, Filter.tendsto_iInf,
    Filter.tendsto_comap_iff]
  intro i
  convert! (hf i).continuousAt.tendsto
  exact (map_zero _).symm

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

Goal state before:
pp: case neg
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜₂
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝ : RingHomIsometric σ₁₂
f : E →SL[σ₁₂] F
ε C : ℝ
ε_pos : 0 < ε
hC : 0 ≤ C
c : 𝕜
hc : 1 < ‖c⁻¹‖
hf : ∀ (x : E), ε * ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖
h0 : ¬c = 0
⊢ ‖f‖ ≤ C

Tactic:
refine opNorm_le_of_shell ε_pos hC hc ?_

Goal state after:
case neg
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁶ : SeminormedAddCommGroup E
inst✝⁵ : SeminormedAddCommGroup F
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NontriviallyNormedField 𝕜₂
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝ : RingHomIsometric σ₁₂
f : E →SL[σ₁₂] F
ε C : ℝ
ε_pos : 0 < ε
hC : 0 ≤ C
c : 𝕜
hc : 1 < ‖c⁻¹‖
hf : ∀ (x : E), ε * ‖c‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖
h0 : ¬c = 0
⊢ ∀ (x : E), ε / ‖c⁻¹‖ ≤ ‖x‖ → ‖x‖ < ε → ‖f x‖ ≤ C * ‖x‖
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
𝕜₂ : Type u_3
E : Type u_6
F : Type u_7
ι' : Type u_10
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : AddCommGroup E
inst✝⁷ : Module 𝕜 E
inst✝⁶ : NormedField 𝕜₂
inst✝⁵ : AddCommGroup F
inst✝⁴ : Module 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝³ : RingHomIsometric σ₁₂
κ : Type u_11
q : SeminormFamily 𝕜₂ F ι'
inst✝² : UniformSpace E
inst✝¹ : IsUniformAddGroup E
inst✝ : ContinuousSMul 𝕜 E
f : κ → E →ₛₗ[σ₁₂] F
i : ι'
x✝¹ : SeminormedAddCommGroup F := (q i).toSeminormedAddCommGroup
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
x✝ : UniformEquicontinuous (DFunLike.coe ∘ f)
H : UniformEquicontinuous (DFunLike.coe ∘ f) := x✝
x : E
hx : ∀ (i : κ), dist ((DFunLike.coe ∘ f) i 0) ((DFunLike.coe ∘ f) i x) < 1
k : κ
⊢ (q i) ((f k) x) ≤ 1

Tactic:
simpa using! (hx k).le

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
δ : Type u_4
l : Filter α
f✝¹ g h : α → β
M : Type u_5
N : Type u_6
R : Type u_7
inst✝¹ : Monoid M
inst✝ : MulAction M β
f✝ : l.Germ β
f : α → β
⊢ 1 • ↑f = ↑f

Tactic:
by simp only [← coe_one, ← coe_smul', one_smul]

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

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : AddCommGroup E
inst✝² : Module 𝕜 E
inst✝¹ : Countable ι
p : SeminormFamily 𝕜 E ι
inst✝ : TopologicalSpace E
hp : WithSeminorms p
this✝ : IsTopologicalAddGroup E
x✝ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this : IsUniformAddGroup E
⊢ (⨅ i, comap (⇑(p i)) (𝓝 0)).IsCountablyGenerated

Tactic:
exact Filter.iInf.isCountablyGenerated _

Goal state after:
Module: Mathlib.Probability.UniformOn

Goal state before:
pp: case neg.h0
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
hst : ¬s ∩ t = ∅
⊢ Measure.count (s ∩ t) ≠ 0

Tactic:
rwa [← Measure.count_eq_zero_iff] at hst

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

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : ConditionallyCompleteLinearOrder β
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : FirstCountableTopology β
inst✝ : OrderTopology β
hf : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (ae μ) f
⊢ μ {y | essSup f μ < f y} = 0

Tactic:
simp_rw [← not_le]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : ConditionallyCompleteLinearOrder β
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : FirstCountableTopology β
inst✝ : OrderTopology β
hf : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) (ae μ) f
⊢ μ {y | ¬f y ≤ essSup f μ} = 0
Module: Mathlib.MeasureTheory.Function.EssSup

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : ConditionallyCompleteLinearOrder β
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : FirstCountableTopology β
inst✝ : OrderTopology β
hf : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) (ae μ) f
⊢ μ {y | f y < essInf f μ} = 0

Tactic:
simp_rw [← not_le]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝³ : ConditionallyCompleteLinearOrder β
f : α → β
inst✝² : TopologicalSpace β
inst✝¹ : FirstCountableTopology β
inst✝ : OrderTopology β
hf : IsBoundedUnder (fun x1 x2 ↦ x1 ≥ x2) (ae μ) f
⊢ μ {y | ¬essInf f μ ≤ f y} = 0
Module: Mathlib.Probability.UniformOn

Goal state before:
pp: case inr.inr
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ (Measure.count (s ∩ u) + Measure.count (t ∩ u)) * (Measure.count (s ∪ t))⁻¹ =
    (Measure.count (s ∪ t))⁻¹ * Measure.count ((s ∪ t) ∩ u)

pp: case inr.inr.h0
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count t ≠ 0

pp: case inr.inr.ht
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count t ≠ ⊤

pp: case inr.inr.h0
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count s ≠ 0

pp: case inr.inr.ht
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count s ≠ ⊤

Tactic:
← measure_union,

Goal state after:
case inr.inr
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count (s ∩ u ∪ t ∩ u) * (Measure.count (s ∪ t))⁻¹ = (Measure.count (s ∪ t))⁻¹ * Measure.count ((s ∪ t) ∩ u)

case inr.inr.hd
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Disjoint (s ∩ u) (t ∩ u)

case inr.inr.h
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ MeasurableSet (t ∩ u)

case inr.inr.h0
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count t ≠ 0

case inr.inr.ht
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count t ≠ ⊤

case inr.inr.h0
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count s ≠ 0

case inr.inr.ht
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t u : Set Ω
hs : s.Finite
ht : t.Finite
hst : Disjoint s t
hs' : s.Nonempty
ht' : t.Nonempty
⊢ Measure.count s ≠ ⊤
Module: Mathlib.Probability.UniformOn

Goal state before:
pp: Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : MeasurableSingletonClass Ω
ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : Finite Ω
f t : ι → Set Ω
ht : ∀ (i : ι), MeasurableSet (t i)
⊢ (uniformOn (Set.univ.pi f)) (Set.univ.pi t) = ∏ i, (uniformOn (f i)) (t i)

Tactic:
lift f to ι → Finset Ω using by simp [Set.toFinite]

Goal state after:
Ω : Type u_1
inst✝³ : MeasurableSpace Ω
inst✝² : MeasurableSingletonClass Ω
ι : Type u_2
inst✝¹ : Fintype ι
inst✝ : Finite Ω
t : ι → Set Ω
ht : ∀ (i : ι), MeasurableSet (t i)
f : ι → Finset Ω
⊢ (uniformOn (Set.univ.pi fun i ↦ ↑(f i))) (Set.univ.pi t) = ∏ i, (uniformOn ((fun i ↦ ↑(f i)) i)) (t i)
Module: Mathlib.MeasureTheory.Function.AEEqFun

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace β
ι : Type u_5
inst✝¹ : T2Space β
g h : α → β
f : ι → α → β
l : Filter ι
inst✝ : l.NeBot
hg : ∀ᵐ (ω : α) ∂μ, Tendsto (fun i ↦ f i ω) l (𝓝 (g ω))
hh : ∀ᵐ (ω : α) ∂μ, Tendsto (fun i ↦ f i ω) l (𝓝 (h ω))
⊢ g =ᵐ[μ] h

Tactic:
filter_upwards [hg, hh] with ω hg1 hh1 using tendsto_nhds_unique hg1 hh1

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

Goal state before:
pp: α : Type u_1
ε' : Type u_3
m0 : MeasurableSpace α
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
p : ℝ≥0∞
μ : Measure α
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
f : α → ε'
hf : AEStronglyMeasurable f μ
ε : ℝ≥0∞
x : α
⊢ ε ≤ ‖f x‖ₑ ↔ ε ^ p.toReal ≤ ‖f x‖ₑ ^ p.toReal

Tactic:
rw [ENNReal.rpow_le_rpow_iff (ENNReal.toReal_pos hp_ne_zero hp_ne_top)]

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

Goal state before:
pp: case pos
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
s : Set α
hs : MeasurableSet s
hp_zero : ¬p = 0
hp_top : p = ∞
⊢ eLpNorm (s.indicator f) p μ = eLpNorm f p (μ.restrict s)

pp: case neg
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
s : Set α
hs : MeasurableSet s
hp_zero : ¬p = 0
hp_top : ¬p = ∞
⊢ eLpNorm (s.indicator f) p μ = eLpNorm f p (μ.restrict s)

Tactic:
· simp_rw [hp_top, eLpNorm_exponent_top, eLpNormEssSup_eq_essSup_enorm,
       enorm_indicator_eq_indicator_enorm, ENNReal.essSup_indicator_eq_essSup_restrict hs]

Goal state after:
case neg
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
f : α → ε
s : Set α
hs : MeasurableSet s
hp_zero : ¬p = 0
hp_top : ¬p = ∞
⊢ eLpNorm (s.indicator f) p μ = eLpNorm f p (μ.restrict s)
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✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
c : ε
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ 0
hp : p ≠ 0
⊢ eLpNorm (s.indicator fun x ↦ c) p μ = ‖c‖ₑ * μ s ^ (1 / p.toReal)

Tactic:
by
  by_cases hp_top : p = ∞
  · simp [hp_top, eLpNormEssSup_indicator_const_eq s c hμs]
  · exact eLpNorm_indicator_const hs hp hp_top

Goal state after:
Module: Mathlib.Data.ENNReal.Holder

Goal state before:
pp: case mpr
p q : ℝ≥0∞
inst✝ : p.HolderConjugate q
⊢ q = 1 → p = ∞

Tactic:
· rintro rfl
    rw [← inv_inv p, ← one_sub_inv 1 p]
    simp

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

Goal state before:
pp: α : Type u_1
F : Type u_5
m0 : MeasurableSpace α
p : ℝ≥0∞
q : ℝ
μ : Measure α
inst✝ : NormedAddCommGroup F
f : α → F
hq_pos : 0 < q
⊢ ∀ᵐ (x : α) ∂?m.48, 0 ≤ ‖f x‖ ^ q

Tactic:
filter_upwards with x using by positivity

Goal state after:
Module: Mathlib.Analysis.Convex.SpecificFunctions.Basic

Goal state before:
pp: case inr.inl
s : ℝ
hs✝ : -1 ≤ s
hs'✝ : s ≠ 0
p : ℝ
hp1 : 0 < p
hp2 : p < 1
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs4 : 1 + p * s ≠ 1
hs' : s < 0
⊢ log (1 + s) * p < log (1 + p * s)

Tactic:
rw [← lt_div_iff₀ hp1, ← div_lt_div_right_of_neg hs']
    convert! strictConcaveOn_log_Ioi.secant_strict_mono (zero_lt_one' ℝ) hs1 hs2 hs3 hs4 _ using 1
    · rw [add_sub_cancel_left, div_div, log_one, sub_zero]
    · rw [add_sub_cancel_left, log_one, sub_zero]
    · gcongr
      exact lt_mul_of_lt_one_left hs' hp2

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
p : ℝ≥0∞
hp_ne_zero : p ≠ 0
hp_ne_top : p ≠ ∞
f : α → ε
c : ℝ≥0∞
⊢ eLpNorm f p (c • μ) = c ^ (1 / p).toReal • eLpNorm f p μ

Tactic:
by
  simp_rw [eLpNorm_eq_eLpNorm' hp_ne_zero hp_ne_top]
  rw [eLpNorm'_smul_measure ENNReal.toReal_nonneg]
  congr
  simp_rw [one_div]
  rw [ENNReal.toReal_inv]

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

Goal state before:
pp: case inr
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 : 𝕜
hx : x ∈ s
hxy : x < y
hxy' : f x < f y
u : 𝕜
hu : u ∈ s ∩ Set.Ici y
v : 𝕜
hv : v ∈ s ∩ Set.Ici y
huv : u < v
step1 : ∀ {z : 𝕜}, z ∈ s ∩ Set.Ioi y → f y < f z
hu2 : y < u
⊢ f u < f v

Tactic:
refine hf.lt_right_of_left_lt ?_ hv.1 ?_ (step1 ⟨hu.1, hu2⟩)
    · apply hf.1.segment_subset hx hu.1
      rw [segment_eq_Icc (hxy.le.trans hu.2)]
      exact ⟨hxy.le, hu.2⟩
    · rw [openSegment_eq_Ioo (hu2.trans huv)]
      exact ⟨hu2, huv⟩

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
hp0 : 0 < p
f : α → ℝ≥0∞
a : α
⊢ ((∫⁻ (c : α), f c ^ p ∂μ) ^ (1 / p))⁻¹ ^ p = (∫⁻ (c : α), f c ^ p ∂μ)⁻¹

Tactic:
inv_rpow,

Goal state after:
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
hp0 : 0 < p
f : α → ℝ≥0∞
a : α
⊢ (((∫⁻ (c : α), f c ^ p ∂μ) ^ (1 / p)) ^ p)⁻¹ = (∫⁻ (c : α), f c ^ p ∂μ)⁻¹
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: p : ℝ
a b : ℝ≥0
hp1 : 1 ≤ p
⊢ 0 < p⁻¹

Tactic:
by simp [lt_of_lt_of_le zero_lt_one hp1]

Goal state after:
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: p : ℝ
a b : ℝ≥0
hp1 : 1 ≤ p
⊢ a ^ p + b ^ p ≤ (a + b) ^ p

Tactic:
exact add_rpow_le_rpow_add _ _ hp1

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
G : Type u_4
inst✝¹⁸ : CommRing 𝕜
inst✝¹⁷ : LinearOrder 𝕜
inst✝¹⁶ : IsStrictOrderedRing 𝕜
inst✝¹⁵ : CommRing E
inst✝¹⁴ : LinearOrder E
inst✝¹³ : IsStrictOrderedRing E
inst✝¹² : AddCommGroup F
inst✝¹¹ : LinearOrder F
inst✝¹⁰ : IsOrderedAddMonoid F
inst✝⁹ : AddCommGroup G
inst✝⁸ : Module 𝕜 G
inst✝⁷ : Module 𝕜 E
inst✝⁶ : Module 𝕜 F
inst✝⁵ : Module E F
inst✝⁴ : IsScalarTower 𝕜 E F
inst✝³ : SMulCommClass 𝕜 E F
inst✝² : IsOrderedModule 𝕜 F
inst✝¹ : IsStrictOrderedModule E F
s : Set G
f : G → E
g : G → F
inst✝ : IsOrderedModule 𝕜 E
hf : ConcaveOn 𝕜 s f
hg : ConcaveOn 𝕜 s g
hf₀ : ∀ ⦃x : G⦄, x ∈ s → 0 ≤ f x
hg₀ : ∀ ⦃x : G⦄, x ∈ s → 0 ≤ g x
hfg : AntivaryOn f g s
x : G
hx : x ∈ s
y : G
hy : y ∈ s
a b : 𝕜
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
⊢ (a * a) • f x • g x + (b * b) • f y • g y + ((a * b) • f x • g y + (a * b) • f y • g x) =
    (a • f x) • a • g x + (b • f y) • a • g x + ((a • f x) • b • g y + (b • f y) • b • g y)

Tactic:
rw [← smul_smul_smul_comm a, ← smul_smul_smul_comm b, ← smul_smul_smul_comm a b,
    ← smul_smul_smul_comm b b, smul_eq_mul, smul_eq_mul, smul_eq_mul, smul_eq_mul, mul_comm b a,
    add_comm ((a * b) • f x • g y), add_comm ((a * b) • f x • g y), add_add_add_comm]

Goal state after:
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: p : ℝ
a b : ℝ≥0∞
hp1 : 1 ≤ p
⊢ a ^ p + b ^ p ≤ (a + b) ^ p

Tactic:
by
  have hp_pos : 0 < p := by positivity
  by_cases h_top : a + b = ⊤
  · rw [← @ENNReal.rpow_eq_top_iff_of_pos (a + b) p hp_pos] at h_top
    rw [h_top]
    exact le_top
  obtain ⟨ha_top, hb_top⟩ := add_ne_top.mp h_top
  lift a to ℝ≥0 using ha_top
  lift b to ℝ≥0 using hb_top
  simpa [ENNReal.coe_rpow_of_nonneg _ hp_pos.le] using
    ENNReal.coe_le_coe.2 (NNReal.add_rpow_le_rpow_add a b hp1)

Goal state after:
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: p : ℝ
a b : ℝ≥0∞
hp1 : 1 ≤ p
⊢ 0 < p⁻¹

Tactic:
by simp [lt_of_lt_of_le zero_lt_one hp1]

Goal state after:
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: p : ℝ
a b : ℝ≥0∞
hp1 : 1 ≤ p
⊢ a ^ p + b ^ p ≤ (a + b) ^ p

Tactic:
exact add_rpow_le_rpow_add _ _ hp1

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

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

Tactic:
suffices ∏ i ∈ s, z k ^ w i = ∑ i ∈ s, w i * z k by convert this using 3 <;> grind

Goal state after:
ι : Type u
s : Finset ι
w z : ι → ℝ
hw : ∀ i ∈ s, 0 < w i
hw' : ∑ i ∈ s, w i = 1
hz : ∀ i ∈ s, 0 ≤ z i
h : ∀ j ∈ s, ∀ k ∈ s, z j = z k
k : ι
hk : k ∈ s
⊢ ∏ i ∈ s, z k ^ w i = ∑ i ∈ s, w i * z k
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
  (∀ i ∈ s, AEMeasurable (f i) μ) →
    ∀ {p : ι → ℝ},
      ∑ i ∈ s, p i = 1 →
        (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ ≠ 1
hpi₀ : 0 ≤ 1 - p i₀
h2pi₀ : 1 - p i₀ ≠ 0
q : ι → ℝ := fun i ↦ p i / (1 - p i₀)
⊢ ∑ i ∈ s, q i = 1

Tactic:
← Finset.sum_div,

Goal state after:
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
i₀ : ι
s : Finset ι
hi₀ : i₀ ∉ s
ih :
  (∀ i ∈ s, AEMeasurable (f i) μ) →
    ∀ {p : ι → ℝ},
      ∑ i ∈ s, p i = 1 →
        (∀ i ∈ s, 0 ≤ p i) → ∫⁻ (a : α), ∏ i ∈ s, f i a ^ p i ∂μ ≤ ∏ i ∈ s, (∫⁻ (a : α), f i a ∂μ) ^ p i
hf : ∀ i ∈ insert i₀ s, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ insert i₀ s, p i = 1
h2p : ∀ i ∈ insert i₀ s, 0 ≤ p i
h2i₀ : p i₀ ≠ 1
hpi₀ : 0 ≤ 1 - p i₀
h2pi₀ : 1 - p i₀ ≠ 0
q : ι → ℝ := fun i ↦ p i / (1 - p i₀)
⊢ (∑ i ∈ s, p i) / (1 - p i₀) = 1
Module: Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality

Goal state before:
pp: α : Type u_1
ε : Type u_3
m : MeasurableSpace α
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
μ : Measure α
f g : α → ε
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
p : ℝ≥0∞
⊢ eLpNorm (f + g) p μ ≤ p.LpAddConst * (eLpNorm f p μ + eLpNorm g p μ)

Tactic:
rcases eq_or_ne p 0 with (rfl | hp)

Goal state after:
case inl
α : Type u_1
ε : Type u_3
m : MeasurableSpace α
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
μ : Measure α
f g : α → ε
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
⊢ eLpNorm (f + g) 0 μ ≤ LpAddConst 0 * (eLpNorm f 0 μ + eLpNorm g 0 μ)

case inr
α : Type u_1
ε : Type u_3
m : MeasurableSpace α
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
μ : Measure α
f g : α → ε
hf : AEStronglyMeasurable f μ
hg : AEStronglyMeasurable g μ
p : ℝ≥0∞
hp : p ≠ 0
⊢ eLpNorm (f + g) p μ ≤ p.LpAddConst * (eLpNorm f p μ + eLpNorm g p μ)
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞
hg_top : ∫⁻ (a : α), g a ^ p ∂μ < ∞
hp1 : 1 ≤ p
⊢ 2 ^ (p - 1) * ∫⁻ (a : α), f a ^ p ∂μ < ∞ ∧ 2 ^ (p - 1) * ∫⁻ (a : α), g a ^ p ∂μ < ∞

pp: case hf
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞
hg_top : ∫⁻ (a : α), g a ^ p ∂μ < ∞
hp1 : 1 ≤ p
⊢ AEMeasurable (fun a ↦ 2 ^ (p - 1) * f a ^ p) μ

Tactic:
· constructor <;> finiteness

Goal state after:
case hf
α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞
hg_top : ∫⁻ (a : α), g a ^ p ∂μ < ∞
hp1 : 1 ≤ p
⊢ AEMeasurable (fun a ↦ 2 ^ (p - 1) * f a ^ p) μ
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: 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
p q r : ℝ≥0∞
f : α → E
g : α → F
b : E → F → G
c : ℝ≥0
hf : MemLp f p μ
hg : MemLp g q μ
h : AEStronglyMeasurable (fun x ↦ b (f x) (g x)) μ
hb : ∀ᵐ (x : α) ∂μ, ‖b (f x) (g x)‖₊ ≤ c * ‖f x‖₊ * ‖g x‖₊
hpqr : p.HolderTriple q r
⊢ ↑c * eLpNorm f p μ * eLpNorm g q μ < ∞

Tactic:
finiteness [hf.2, hg.2]

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

Goal state before:
pp: α : Type u_1
inst✝ : MeasurableSpace α
μ : Measure α
p : ℝ
f g : α → ℝ≥0∞
hf : AEMeasurable f μ
hg : AEMeasurable g μ
hp1 : 1 ≤ p
hp_pos : 0 < p
hf_top : ∫⁻ (a : α), f a ^ p ∂μ < ∞
hg_top : ∫⁻ (a : α), g a ^ p ∂μ < ∞
h1 : ¬p = 1
hp1_lt : 1 < p
hpq : p.HolderConjugate p.conjExponent
h0 : ∫⁻ (a : α), (f + g) a ^ p ∂μ = 0
⊢ 0 < 1 / p

Tactic:
by simp [lt_of_lt_of_le zero_lt_one hp1]

Goal state after:
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✝² : NormedAddCommGroup E✝
inst✝¹ : NormedAddCommGroup F
α : Type ?u.18
E : Type u_6
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f : α →ₘ[μ] E
hf : f ∈ {f | eLpNorm (↑f) p μ < ∞}
⊢ eLpNorm (↑(-f)) p μ < ∞

Tactic:
eLpNorm_congr_ae (AEEqFun.coeFn_neg f),

Goal state after:
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✝² : NormedAddCommGroup E✝
inst✝¹ : NormedAddCommGroup F
α : Type ?u.18
E : Type u_6
m : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f : α →ₘ[μ] E
hf : f ∈ {f | eLpNorm (↑f) p μ < ∞}
⊢ eLpNorm (-↑f) p μ < ∞
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≥0
hfC : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖ ≤ ↑C
⊢ ‖f‖ ≤ ↑(measureUnivNNReal μ) ^ p.toReal⁻¹ * ↑C

Tactic:
have := nnnorm_le_of_ae_bound hfC

Goal state after:
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≥0
hfC : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖ ≤ ↑C
this : ‖f‖₊ ≤ measureUnivNNReal μ ^ p.toReal⁻¹ * C
⊢ ‖f‖ ≤ ↑(measureUnivNNReal μ) ^ p.toReal⁻¹ * ↑C