mathlib-initiative/mathlib-tactics huggingface dataset:
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✝
this : ∀ᶠ (x : E) in 𝓝 0, ∀ (k : κ), (q i) ((f k) x) ≤ 1
bdd : BddAbove (Set.range fun k ↦ (q i).comp (f k))
⊢ (⨆ i_1, (q i).comp (f i_1)).closedBall 0 1 ∈ 𝓝 0

Tactic:
filter_upwards [this] with x hx

Goal state after:
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✝
this : ∀ᶠ (x : E) in 𝓝 0, ∀ (k : κ), (q i) ((f k) x) ≤ 1
bdd : BddAbove (Set.range fun k ↦ (q i).comp (f k))
x : E
hx : ∀ (k : κ), (q i) ((f k) x) ≤ 1
⊢ x ∈ (⨆ i_1, (q i).comp (f i_1)).closedBall 0 1
Module: Mathlib.Analysis.Normed.Operator.Basic

Goal state before:
pp: case refine_1
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
A : ∀ (f : (E →SL[σ₁₂] F) × (E →SL[σ₁₂] F)), ‖-f.1 + f.2‖ = ‖f.1 - f.2‖
c : 𝕜
hc : 1 < ‖c‖
⊢ ∃ r, ContinuousLinearMap.seminorm.closedBall 0 r ∈ 𝓝 0

Tactic:
refine ⟨‖c‖, ContinuousLinearMap.hasBasis_nhds_zero.mem_iff.2
      ⟨(closedBall 0 1, closedBall 0 1), ?_⟩⟩

Goal state after:
case refine_1
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
A : ∀ (f : (E →SL[σ₁₂] F) × (E →SL[σ₁₂] F)), ‖-f.1 + f.2‖ = ‖f.1 - f.2‖
c : 𝕜
hc : 1 < ‖c‖
⊢ (IsVonNBounded 𝕜 (closedBall 0 1, closedBall 0 1).1 ∧ (closedBall 0 1, closedBall 0 1).2 ∈ 𝓝 0) ∧
    {f | ∀ x ∈ (closedBall 0 1, closedBall 0 1).1, f x ∈ (closedBall 0 1, closedBall 0 1).2} ⊆
      ContinuousLinearMap.seminorm.closedBall 0 ‖c‖
Module: Mathlib.Analysis.Normed.Operator.Basic

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
F : Type u_5
G : Type u_7
inst✝¹¹ : SeminormedAddCommGroup E
inst✝¹⁰ : SeminormedAddCommGroup F
inst✝⁹ : SeminormedAddCommGroup G
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NontriviallyNormedField 𝕜₂
inst✝⁶ : NontriviallyNormedField 𝕜₃
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜₂ F
inst✝³ : NormedSpace 𝕜₃ G
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹ : RingHomIsometric σ₁₂
inst✝ : RingHomIsometric σ₂₃
h : F →SL[σ₂₃] G
f : E →SL[σ₁₂] F
x : E
⊢ ‖(h ∘SL f) x‖ ≤ ‖h‖ * ‖f‖ * ‖x‖

Tactic:
mul_assoc

Goal state after:
𝕜 : Type u_1
𝕜₂ : Type u_2
𝕜₃ : Type u_3
E : Type u_4
F : Type u_5
G : Type u_7
inst✝¹¹ : SeminormedAddCommGroup E
inst✝¹⁰ : SeminormedAddCommGroup F
inst✝⁹ : SeminormedAddCommGroup G
inst✝⁸ : NontriviallyNormedField 𝕜
inst✝⁷ : NontriviallyNormedField 𝕜₂
inst✝⁶ : NontriviallyNormedField 𝕜₃
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜₂ F
inst✝³ : NormedSpace 𝕜₃ G
σ₁₂ : 𝕜 →+* 𝕜₂
σ₂₃ : 𝕜₂ →+* 𝕜₃
σ₁₃ : 𝕜 →+* 𝕜₃
inst✝² : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃
inst✝¹ : RingHomIsometric σ₁₂
inst✝ : RingHomIsometric σ₂₃
h : F →SL[σ₂₃] G
f : E →SL[σ₁₂] F
x : E
⊢ ‖(h ∘SL f) x‖ ≤ ‖h‖ * (‖f‖ * ‖x‖)
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
ι : Type u_9
ι' : Type u_10
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
p : SeminormFamily 𝕜 E ι
t : TopologicalSpace E
hp : WithSeminorms p
e : ι' ≃ ι
⊢ WithSeminorms (p ∘ ⇑e)

Tactic:
by
  refine hp.congr ?_ ?_ <;>
  intro i <;>
  [use {e i}, 1; use {e.symm i}, 1] <;>
  simp

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NontriviallyNormedField 𝕜₂
inst✝⁴ : SeminormedAddCommGroup E
inst✝³ : SeminormedAddCommGroup F
inst✝² : NormedSpace 𝕜 E
inst✝¹ : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝ : RingHomIsometric σ₁₂
f : E →SL[σ₁₂] F
r : ℝ
hr₀ : 0 ≤ r
hr : r < ‖f‖
⊢ ∃ x, r * ‖x‖ < ‖f x‖

Tactic:
lift r to ℝ≥0 using hr₀
  exact f.exists_mul_lt_apply_of_lt_opNNNorm hr

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

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_4
F : Type u_5
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : SeminormedAddCommGroup F
inst✝⁵ : DenselyNormedField 𝕜
inst✝⁴ : NontriviallyNormedField 𝕜₂
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedSpace 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹ : RingHomIsometric σ₁₂
inst✝ : NormedAlgebra ℝ 𝕜
f : E →SL[σ₁₂] F
⊢ sSup ((fun x ↦ ‖f x‖₊) '' sphere 0 1) = ‖f‖₊

Tactic:
cases subsingleton_or_nontrivial E
  · simp [sphere_eq_empty_of_subsingleton one_ne_zero]
  have : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
  refine csSup_eq_of_forall_le_of_forall_lt_exists_gt
      ((NormedSpace.sphere_nonempty.mpr zero_le_one).image _) ?_ fun ub hub => ?_
  · rintro - ⟨x, hx, rfl⟩
    simpa only [mul_one] using! f.le_opNorm_of_le (mem_sphere_zero_iff_norm.1 hx).le
  · obtain ⟨x, hx, hxf⟩ := f.exists_nnnorm_eq_one_lt_apply_of_lt_opNNNorm hub
    exact ⟨_, ⟨x, by simpa using! congrArg NNReal.toReal hx, rfl⟩, hxf⟩

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

Goal state before:
pp: 𝕜₂ : Type u_3
E : Type u_6
F : Type u_7
inst✝⁹ : AddCommGroup E
inst✝⁸ : NormedField 𝕜₂
inst✝⁷ : AddCommGroup F
inst✝⁶ : Module 𝕜₂ F
inst✝⁵ : TopologicalSpace F
𝕜 : Type u_11
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : Module 𝕜 E
inst✝² : TopologicalSpace E
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝¹ : RingHomIsometric σ₁₂
p : Seminorm 𝕜 E
hp : Continuous[inst✝², _] ⇑p
inst✝ : PolynormableSpace 𝕜₂ F
f : E →ₛₗ[σ₁₂] F
hf : IsInducing ⇑f
s : Finset { p // Continuous[inst✝⁵, _] ⇑p }
C : ℝ≥0
hqC : p ≤ (C • s.sup fun p ↦ ↑p).comp f
this : IsTopologicalAddGroup F
⊢ ∃ p₂, Continuous[inst✝⁵, _] ⇑p₂ ∧ p ≤ p₂.comp f

Tactic:
exact ⟨_, Continuous.const_smul (continuous_finsetSup fun i _ => i.2) C, hqC⟩

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

Goal state before:
pp: case mpr
α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
ι : Type u₃
p : Filter ι
F : ι → C(α, β)
f : C(α, β)
⊢ (∀ (K : Set α), IsCompact K → TendstoUniformlyOn (fun i a ↦ (F i) a) (⇑f) p K) →
    ∀ (K : Set α),
      IsCompact K →
        ∀ (U : Set β), IsOpen[inst✝.toTopologicalSpace] U → MapsTo (⇑f) K U → ∀ᶠ (a : ι) in p, MapsTo (⇑(F a)) K U

Tactic:
intro h K hK U hU hf

Goal state after:
case mpr
α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
ι : Type u₃
p : Filter ι
F : ι → C(α, β)
f : C(α, β)
h : ∀ (K : Set α), IsCompact K → TendstoUniformlyOn (fun i a ↦ (F i) a) (⇑f) p K
K : Set α
hK : IsCompact K
U : Set β
hU : IsOpen[inst✝.toTopologicalSpace] U
hf : MapsTo (⇑f) K U
⊢ ∀ᶠ (a : ι) in p, MapsTo (⇑(F a)) K U
Module: Mathlib.Order.Filter.ENNReal

Goal state before:
pp: α : Type u_1
f : Filter α
inst✝ : CountableInterFilter f
u : α → ℝ≥0∞
a : ℝ≥0∞
ha_top : a = ∞
hu : u =ᶠ[f] 0
x : α
hx : u x = 0 x
⊢ (fun x ↦ a * u x) x = 0 x

Tactic:
by simp [hx]

Goal state after:
Module: Mathlib.Probability.UniformOn

Goal state before:
pp: Ω : Type u_2
inst✝ : DecidableEq Ω
x✝ : MeasurableSpace Ω
s t : Finset Ω
hs : MeasurableSet ↑s
ht : MeasurableSet ↑t
⊢ (uniformOn ↑s) ↑t = ↑(#(s ∩ t)) / ↑(#s)

Tactic:
by
  rw [uniformOn, cond_apply hs, Measure.count_apply_finset' hs, ← coe_inter,
    Measure.count_apply_finset']
  · rw [div_eq_mul_inv, mul_comm]
  rw [coe_inter]
  exact hs.inter ht

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]; exact one_ne_zero

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

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace β
f g : α →ₘ[μ] β
H : f.toGerm = g.toGerm
⊢ ↑↑f = ↑↑g

Tactic:
← toGerm_eq,

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace β
f g : α →ₘ[μ] β
H : f.toGerm = g.toGerm
⊢ f.toGerm = ↑↑g
Module: Mathlib.Probability.UniformOn

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s u t : Set Ω
hs : s.Finite
| (uniformOn s) t

Tactic:
(by simp : s = s ∩ u ∪ s ∩ uᶜ),

Goal state after:
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s u t : Set Ω
hs : s.Finite
| (uniformOn (s ∩ u ∪ s ∩ uᶜ)) t
Module: Mathlib.MeasureTheory.Function.AEEqFun

Goal state before:
pp: case empty
α : Type u_1
γ : Type u_3
inst✝³ : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace γ
inst✝¹ : CommMonoid γ
inst✝ : ContinuousMul γ
ι : Type u_5
f : ι → α →ₘ[μ] γ
⊢ ↑(∏ i ∈ ∅, f i) =ᵐ[μ] ∏ i ∈ ∅, ↑(f i)

Tactic:
| empty =>

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Indicator

Goal state before:
pp: α : Type u_1
ε : Type u_2
inst✝¹ : TopologicalSpace ε
inst✝ : ESeminormedAddMonoid ε
s t : Set α
h : s ⊆ t
f : α → ε
a : α
⊢ ‖s.indicator f a‖ₑ ≤ ‖t.indicator f a‖ₑ

Tactic:
simp only [enorm_indicator_eq_indicator_enorm]
  grw [h]

Goal state after:
Module: Mathlib.Analysis.Normed.Group.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝ : SeminormedAddGroup E
s : Set α
f : α → E
a : α
⊢ ‖s.indicator f a‖ ≤ ‖f a‖

Tactic:
rw [norm_indicator_eq_indicator_norm]
  apply indicator_norm_le_norm_self

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

Goal state before:
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
c : ε
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ 0
hp : p ≠ 0
hp_top : ¬p = ∞
⊢ eLpNorm (s.indicator fun x ↦ c) p μ = ‖c‖ₑ * μ s ^ (1 / p.toReal)

Tactic:
· exact eLpNorm_indicator_const hs hp hp_top

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

Goal state before:
pp: α : Type u_1
ε : Type u_2
ε' : Type u_3
m0 : MeasurableSpace α
q : ℝ
μ : Measure α
inst✝¹ : ENorm ε
inst✝ : ENorm ε'
f : α → ε
g : α → ε'
hfg : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ = ‖g x‖ₑ
x : α
hx : ‖f x‖ₑ = ‖g x‖ₑ
⊢ (fun x ↦ ‖f x‖ₑ ^ q) x = (fun x ↦ ‖g x‖ₑ ^ q) x

Tactic:
by simp [hx]

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

Goal state before:
pp: s : ℝ
hs✝ : -1 ≤ s
p : ℝ
hp : 1 < p
hp' : 0 < p
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs' : p = 0 ∨ s = 0
⊢ s = 0

Tactic:
eq_false_intro hp'.ne',

Goal state after:
s : ℝ
hs✝ : -1 ≤ s
p : ℝ
hp : 1 < p
hp' : 0 < p
hs : -1 < s
hs1 : 0 < 1 + s
hs2 : 0 < 1 + p * s
hs3 : 1 + s ≠ 1
hs' : False ∨ s = 0
⊢ s = 0
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: case e'_2.e'_5
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
x✝ : α
⊢ ‖f x✝‖ₑ ^ q = (ENNReal.ofReal ∘ fun x ↦ ‖f x‖ ^ q) x✝

Tactic:
rw [Function.comp_apply, ← ofReal_norm]

Goal state after:
case e'_2.e'_5
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
x✝ : α
⊢ ENNReal.ofReal ‖f x✝‖ ^ q = ENNReal.ofReal (‖f x✝‖ ^ q)
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: p q r : ℝ
h : p.HolderTriple q r
this : 0 < p
⊢ 0 < p / r

Tactic:
have := h.pos'

Goal state after:
p q r : ℝ
h : p.HolderTriple q r
this✝ : 0 < p
this : 0 < r
⊢ 0 < p / r
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: p q r : ℝ
h : p.HolderTriple q r
this : 0 < q
⊢ 0 < q / r

Tactic:
have := h.pos'

Goal state after:
p q r : ℝ
h : p.HolderTriple q r
this✝ : 0 < q
this : 0 < r
⊢ 0 < q / r
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: p q r : ℝ≥0
h : p.HolderTriple q r
this : 0 < p
⊢ 0 < p / r

Tactic:
have := h.pos'

Goal state after:
p q r : ℝ≥0
h : p.HolderTriple q r
this✝ : 0 < p
this : 0 < r
⊢ 0 < p / r
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: p q r : ℝ≥0
h : p.HolderTriple q r
this : 0 < q
⊢ 0 < q / r

Tactic:
have := h.pos'

Goal state after:
p q r : ℝ≥0
h : p.HolderTriple q r
this✝ : 0 < q
this : 0 < r
⊢ 0 < q / r
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: p q r : ℝ≥0∞
h : p.toReal.HolderTriple q.toReal r.toReal
⊢ p.HolderTriple q r

Tactic:
have hp := h.pos

Goal state after:
p q r : ℝ≥0∞
h : p.toReal.HolderTriple q.toReal r.toReal
hp : 0 < p.toReal
⊢ p.HolderTriple q r
Module: Mathlib.Algebra.Order.Monovary

Goal state before:
pp: ι : Type u_1
α : Type u_2
inst✝² : Ring α
inst✝¹ : LinearOrder α
inst✝ : IsStrictOrderedRing α
f g : ι → α
⊢ Antivary f g ↔ ∀ (i j : ι), f i * g i + f j * g j ≤ f i * g j + f j * g i

Tactic:
by
  simp only [smul_eq_mul, antivary_iff_smul_rearrangement]

Goal state after:
Module: Mathlib.Analysis.MeanInequalitiesPow

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

Tactic:
lift b to NNReal using hb

Goal state after:
p : ℝ
hp1 : 1 ≤ p
a b : ℝ≥0
⊢ ↑a ^ p + ↑b ^ p ≤ (↑a + ↑b) ^ p
Module: Mathlib.Analysis.MeanInequalitiesPow

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

Tactic:
lift b to NNReal using hb

Goal state after:
p : ℝ
hp1 : 1 ≤ p
a b : ℝ≥0
⊢ (↑a ^ p + ↑b ^ p) ^ (1 / p) ≤ ↑a + ↑b
Module: Mathlib.Analysis.MeanInequalitiesPow

Goal state before:
pp: p q b : ℝ
hb : 0 ≤ b
hp_pos : 0 < p
hpq : p ≤ q
a : ℝ≥0
⊢ (↑a ^ q + b ^ q) ^ (1 / q) ≤ (↑a ^ p + b ^ p) ^ (1 / p)

Tactic:
lift b to NNReal using hb

Goal state after:
p q : ℝ
hp_pos : 0 < p
hpq : p ≤ q
a b : ℝ≥0
⊢ (↑a ^ q + ↑b ^ q) ^ (1 / q) ≤ (↑a ^ p + ↑b ^ p) ^ (1 / p)
Module: Mathlib.Analysis.MeanInequalitiesPow

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

Tactic:
lift b to NNReal using hb

Goal state after:
p : ℝ
hp : 0 ≤ p
hp1 : p ≤ 1
a b : ℝ≥0
⊢ (↑a + ↑b) ^ p ≤ ↑a ^ p + ↑b ^ p
Module: Mathlib.MeasureTheory.Integral.MeanInequalities

Goal state before:
pp: case empty
α : Type u_2
ι : Type u_3
inst✝ : MeasurableSpace α
μ : Measure α
f : ι → α → ℝ≥0∞
hf : ∀ i ∈ ∅, AEMeasurable (f i) μ
p : ι → ℝ
hp : ∑ i ∈ ∅, p i = 1
h2p : ∀ i ∈ ∅, 0 ≤ p i
⊢ ∫⁻ (a : α), ∏ i ∈ ∅, f i a ^ p i ∂μ ≤ ∏ i ∈ ∅, (∫⁻ (a : α), f i a ∂μ) ^ p i

Tactic:
| empty =>

Goal state after:
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
h2p : ∀ i ∈ s, p i = 0
⊢ ∫⁻ (a : α), f i₀ a ^ p i₀ * ∏ i ∈ s, f i a ^ p i ∂μ = ∫⁻ (a : α), f i₀ a ^ p i₀ * ∏ i ∈ s, 1 ∂μ

Tactic:
congr! 3 with x

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
h2p : ∀ i ∈ s, p i = 0
x : α
⊢ ∏ i ∈ s, f i x ^ p i = ∏ i ∈ s, 1
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
⊢ ∫⁻ (a : α), (f a + g a) ^ p ∂μ ≤ ∫⁻ (a : α), 2 ^ (p - 1) * f a ^ p + 2 ^ (p - 1) * g a ^ p ∂μ

Tactic:
refine lintegral_mono fun a => ?_
      simpa [mul_add] using ENNReal.rpow_add_le_mul_rpow_add_rpow (f a) (g a) hp1

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u_1
s : Finset ι
hs : s.Nonempty
w z : ι → ℝ
hw : ∀ i ∈ s, 0 < w i
hz : ∀ i ∈ s, 0 < z i
n : ℝ := ∑ i ∈ s, w i
hw' : 0 < n
this : (∑ i ∈ s, w i / n / z i)⁻¹ ≤ ∏ i ∈ s, z i ^ (w i / n)
⊢ ∏ i ∈ s, (z i ^ w i) ^ n⁻¹ = ∏ i ∈ s, z i ^ (w i / n)

Tactic:
refine Finset.prod_congr rfl (fun i hi => ?_)

Goal state after:
ι : Type u_1
s : Finset ι
hs : s.Nonempty
w z : ι → ℝ
hw : ∀ i ∈ s, 0 < w i
hz : ∀ i ∈ s, 0 < z i
n : ℝ := ∑ i ∈ s, w i
hw' : 0 < n
this : (∑ i ∈ s, w i / n / z i)⁻¹ ≤ ∏ i ∈ s, z i ^ (w i / n)
i : ι
hi : i ∈ s
⊢ (z i ^ w i) ^ n⁻¹ = z i ^ (w i / n)
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 = ∞
hq_eq_zero : q = 0
⊢ p ≤ 0

Tactic:
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 ≤ 0
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
⊢ p ≤ 0
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 : ℝ≥0∞
f : α → E
g : α → F
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‖₊
⊢ ↑c * eLpNorm (fun i ↦ ‖f i‖ * ‖g i‖) p μ ≤ ↑c * eLpNorm f ∞ μ * eLpNorm g p μ

Tactic:
mul_assoc

Goal state after:
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 : ℝ≥0∞
f : α → E
g : α → F
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‖₊
⊢ ↑c * eLpNorm (fun i ↦ ‖f i‖ * ‖g i‖) p μ ≤ ↑c * (eLpNorm f ∞ μ * eLpNorm g p μ)
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case h₂.inr.inl
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
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‖₊
⊢ eLpNorm (fun i ↦ ‖f i‖ * ‖g i‖) ∞ μ ≤ eLpNorm (fun x ↦ ‖f x‖) ∞ μ * eLpNorm g ∞ μ

Tactic:
← eLpNorm_norm g

Goal state after:
case h₂.inr.inl
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
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‖₊
⊢ eLpNorm (fun i ↦ ‖f i‖ * ‖g i‖) ∞ μ ≤ eLpNorm (fun x ↦ ‖f x‖) ∞ μ * eLpNorm (fun x ↦ ‖g x‖) ∞ μ
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : 0 < ∑ i ∈ s, f i ^ p
hg : 0 < ∑ i ∈ s, g i ^ q
f' : ι → ℝ≥0 := fun i ↦ f i / (∑ i ∈ s, f i ^ p) ^ (1 / p)
g' : ι → ℝ≥0 := fun i ↦ g i / (∑ i ∈ s, g i ^ q) ^ (1 / q)
this : ∑ x ∈ s, f x / (∑ i ∈ s, f i ^ p) ^ (1 / p) * g' x ≤ 1
⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)

Tactic:
g',

Goal state after:
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : 0 < ∑ i ∈ s, f i ^ p
hg : 0 < ∑ i ∈ s, g i ^ q
f' : ι → ℝ≥0 := fun i ↦ f i / (∑ i ∈ s, f i ^ p) ^ (1 / p)
g' : ι → ℝ≥0 := fun i ↦ g i / (∑ i ∈ s, g i ^ q) ^ (1 / q)
this : ∑ x ∈ s, f x / (∑ i ∈ s, f i ^ p) ^ (1 / p) * (g x / (∑ i ∈ s, g i ^ q) ^ (1 / q)) ≤ 1
⊢ ∑ i ∈ s, f i * g i ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) * (∑ i ∈ s, g i ^ q) ^ (1 / q)
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case inr.inr.refine_2
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : 0 < ∑ i ∈ s, f i ^ p
hg : 0 < ∑ i ∈ s, g i ^ q
f' : ι → ℝ≥0 := fun i ↦ f i / (∑ i ∈ s, f i ^ p) ^ (1 / p)
g' : ι → ℝ≥0 := fun i ↦ g i / (∑ i ∈ s, g i ^ q) ^ (1 / q)
⊢ ∑ i ∈ s, g' i ^ q = 1

Tactic:
g',

Goal state after:
case inr.inr.refine_2
ι : Type u
s : Finset ι
f g : ι → ℝ≥0
p q : ℝ
hpq : p.HolderConjugate q
hf : 0 < ∑ i ∈ s, f i ^ p
hg : 0 < ∑ i ∈ s, g i ^ q
f' : ι → ℝ≥0 := fun i ↦ f i / (∑ i ∈ s, f i ^ p) ^ (1 / p)
g' : ι → ℝ≥0 := fun i ↦ g i / (∑ i ∈ s, g i ^ q) ^ (1 / q)
⊢ ∑ x ∈ s, (g x / (∑ i ∈ s, g i ^ q) ^ (1 / q)) ^ q = 1
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
f : α → E
g : α → F
p q r : ℝ
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‖₊
hro_lt : 0 < r
hrp : r < p
hpqr : 1 / r = 1 / p + 1 / q
⊢ ↑c * eLpNorm' (fun i ↦ ‖f i‖ * ‖g i‖) r μ ≤ ↑c * eLpNorm' f p μ * eLpNorm' g q μ

Tactic:
mul_assoc

Goal state after:
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
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‖₊
hro_lt : 0 < r
hrp : r < p
hpqr : 1 / r = 1 / p + 1 / q
⊢ ↑c * eLpNorm' (fun i ↦ ‖f i‖ * ‖g i‖) r μ ≤ ↑c * (eLpNorm' f p μ * eLpNorm' g q μ)
Module: Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp

Goal state before:
pp: case empty
ι : Type u_1
α : Type u_2
𝕜 : Type u_3
x✝ : MeasurableSpace α
inst✝ : NormedCommRing 𝕜
μ : Measure α
f : ι → α → 𝕜
p : ι → ℝ≥0∞
s : Finset ι
hf : ∀ i ∈ ∅, MemLp (f i) (p i) μ
⊢ MemLp (∏ i ∈ ∅, f i) (∑ i ∈ ∅, (p i)⁻¹)⁻¹ μ

Tactic:
| empty =>

Goal state after: