mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.LinearAlgebra.AffineSpace.Independent

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p : ι → P
h : AffineIndependent k p
w w₁ w₂ : ι → k
s : Finset ι
x✝ : ∑ i ∈ s, w i = 1
hw₁ : ∑ i ∈ s, w₁ i = 1
hw₂ : ∑ i ∈ s, w₂ i = 1
⊢ (s.weightedVSub p) (w - w₁) ∈ vectorSpan k {(affineCombination k s p) w₁, (affineCombination k s p) w₂} ↔
    ∃ r, ∀ i ∈ s, w i = r * (w₂ i - w₁ i) + w₁ i

Tactic:
Set.pair_comm,

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝³ : Ring k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p : ι → P
h : AffineIndependent k p
w w₁ w₂ : ι → k
s : Finset ι
x✝ : ∑ i ∈ s, w i = 1
hw₁ : ∑ i ∈ s, w₁ i = 1
hw₂ : ∑ i ∈ s, w₂ i = 1
⊢ (s.weightedVSub p) (w - w₁) ∈ vectorSpan k {(affineCombination k s p) w₂, (affineCombination k s p) w₁} ↔
    ∃ r, ∀ i ∈ s, w i = r * (w₂ i - w₁ i) + w₁ i
Module: Mathlib.Analysis.Convex.Topology

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_3
inst✝⁷ : Field 𝕜
inst✝⁶ : PartialOrder 𝕜
inst✝⁵ : AddCommGroup E
inst✝⁴ : Module 𝕜 E
inst✝³ : TopologicalSpace E
inst✝² : IsTopologicalAddGroup E
inst✝¹ : ContinuousConstSMul 𝕜 E
inst✝ : AddRightMono 𝕜
s : Set E
hs : Convex 𝕜 s
x y : E
hx : x ∈ closure s
hy : y ∈ interior s
t : 𝕜
ht : t ∈ Ioc 0 1
⊢ x + t • (y - x) ∈ interior s

Tactic:
simpa only [sub_smul, smul_sub, one_smul, add_sub, add_comm] using
    hs.combo_interior_closure_mem_interior hy hx ht.1 (sub_nonneg.mpr ht.2)
      (add_sub_cancel _ _)

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

Goal state before:
pp: case mpr.inl
𝕜 : Type u_4
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
x : 𝕜
hs : Convex 𝕜 {x}
hx : x ∈ interior {x}
⊢ {x}.Nontrivial

pp: case mpr.inr
𝕜 : Type u_4
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
s : Set 𝕜
hs : Convex 𝕜 s
x : 𝕜
hx : x ∈ interior s
h : s.Nontrivial
⊢ s.Nontrivial

Tactic:
· rw [interior_singleton] at hx
      exact hx.elim

Goal state after:
case mpr.inr
𝕜 : Type u_4
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
s : Set 𝕜
hs : Convex 𝕜 s
x : 𝕜
hx : x ∈ interior s
h : s.Nontrivial
⊢ s.Nontrivial
Module: Mathlib.Analysis.Convex.Topology

Goal state before:
pp: 𝕜 : Type u_4
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
s : Set 𝕜
hs : Convex 𝕜 s
a : 𝕜
has : a ∈ closure[inst✝¹] s
h_Iio : (s ∩ Iio a).Nonempty
h_Ioi : (s ∩ Ioi a).Nonempty
⊢ 𝓝[s \ {a}] a = 𝓝[≠] a

Tactic:
rw [sdiff_eq, ← Iio_union_Ioi, inter_union_distrib_left, nhdsWithin_union, nhdsWithin_union]

Goal state after:
𝕜 : Type u_4
inst✝⁴ : Field 𝕜
inst✝³ : LinearOrder 𝕜
inst✝² : IsStrictOrderedRing 𝕜
inst✝¹ : TopologicalSpace 𝕜
inst✝ : OrderTopology 𝕜
s : Set 𝕜
hs : Convex 𝕜 s
a : 𝕜
has : a ∈ closure[inst✝¹] s
h_Iio : (s ∩ Iio a).Nonempty
h_Ioi : (s ∩ Ioi a).Nonempty
⊢ 𝓝[s ∩ Iio a] a ⊔ 𝓝[s ∩ Ioi a] a = 𝓝[<] a ⊔ 𝓝[>] a
Module: Mathlib.Analysis.Convex.Combination

Goal state before:
pp: R : Type u_1
E : Type u_3
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
s : Finset E
hs : s.Nonempty
⊢ ↑(#s) ≠ 0

Tactic:
simp [Finset.nonempty_iff_ne_empty.mp hs]

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Independent

Goal state before:
pp: case refine_1
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Ring k
inst✝⁴ : LinearOrder k
inst✝³ : IsStrictOrderedRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p : ι → P
h : AffineIndependent k p
w : ι → k
s : Finset ι
hw : ∑ i ∈ s, w i = 1
i₁ i₂ i₃ : ι
h₁ : i₁ ∈ s
h₂ : i₂ ∈ s
h₃ : i₃ ∈ s
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
c : k
hc0 : 0 < c
hc1 : c < 1
hs :
  (affineCombination k s p) w ∈
    line[k, (affineCombination k s p) (Pi.single i₁ 1),
      (affineCombination k s p) (affineCombinationLineMapWeights i₂ i₃ c)]
⊢ ∑ i ∈ s, Pi.single i₁ 1 i = 1

pp: case refine_2
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Ring k
inst✝⁴ : LinearOrder k
inst✝³ : IsStrictOrderedRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p : ι → P
h : AffineIndependent k p
w : ι → k
s : Finset ι
hw : ∑ i ∈ s, w i = 1
i₁ i₂ i₃ : ι
h₁ : i₁ ∈ s
h₂ : i₂ ∈ s
h₃ : i₃ ∈ s
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
c : k
hc0 : 0 < c
hc1 : c < 1
hs :
  (affineCombination k s p) w ∈
    line[k, (affineCombination k s p) (Pi.single i₁ 1),
      (affineCombination k s p) (affineCombinationLineMapWeights i₂ i₃ c)]
⊢ SignType.sign (affineCombinationLineMapWeights i₂ i₃ c i₂) =
    SignType.sign (affineCombinationLineMapWeights i₂ i₃ c i₃)

Tactic:
· rw [Finset.sum_pi_single', if_pos h₁]

Goal state after:
case refine_2
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : Ring k
inst✝⁴ : LinearOrder k
inst✝³ : IsStrictOrderedRing k
inst✝² : AddCommGroup V
inst✝¹ : Module k V
inst✝ : AffineSpace V P
ι : Type u_4
p : ι → P
h : AffineIndependent k p
w : ι → k
s : Finset ι
hw : ∑ i ∈ s, w i = 1
i₁ i₂ i₃ : ι
h₁ : i₁ ∈ s
h₂ : i₂ ∈ s
h₃ : i₃ ∈ s
h₁₂ : i₁ ≠ i₂
h₁₃ : i₁ ≠ i₃
h₂₃ : i₂ ≠ i₃
c : k
hc0 : 0 < c
hc1 : c < 1
hs :
  (affineCombination k s p) w ∈
    line[k, (affineCombination k s p) (Pi.single i₁ 1),
      (affineCombination k s p) (affineCombinationLineMapWeights i₂ i₃ c)]
⊢ SignType.sign (affineCombinationLineMapWeights i₂ i₃ c i₂) =
    SignType.sign (affineCombinationLineMapWeights i₂ i₃ c i₃)
Module: Mathlib.Analysis.Convex.Combination

Goal state before:
pp: case refine_2
R : Type u_1
E : Type u_3
inst✝⁴ : Field R
inst✝³ : AddCommGroup E
inst✝² : Module R E
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
ι : Type u_8
b : AffineBasis ι R E
x : E
hx : x ∈ {x | ∀ (i : ι), 0 ≤ (b.coord i) x}
⊢ x ∈ {x | ∃ s w, (∀ i ∈ s, 0 ≤ w i) ∧ s.sum w = 1 ∧ (affineCombination R s ⇑b) w = x}

Tactic:
have hx' : x ∈ affineSpan R (range b) := by
      rw [b.tot]
      exact AffineSubspace.mem_top R E x
    obtain ⟨s, w, hw₁, rfl⟩ := (mem_affineSpan_iff_eq_affineCombination R E).mp hx'
    refine ⟨s, w, ?_, hw₁, rfl⟩
    intro i hi
    specialize hx i
    rw [b.coord_apply_combination_of_mem hi hw₁] at hx
    exact hx

Goal state after:
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
ι : Type u_6
p : Filter ι
inst✝¹ : UniformSpace F
inst✝ : IsUniformAddGroup F
𝔖 : Set (Set E)
a : ι → E →SLᵤ[σ, 𝔖] F
a₀ : E →SLᵤ[σ, 𝔖] F
⊢ Tendsto a p (𝓝 a₀) ↔ ∀ s ∈ 𝔖, TendstoUniformlyOn (fun x1 x2 ↦ (a x1) x2) (⇑a₀) p s

Tactic:
rw [(isEmbedding_coeFn σ F 𝔖).tendsto_nhds_iff, UniformOnFun.tendsto_iff_tendstoUniformlyOn]

Goal state after:
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
ι : Type u_6
p : Filter ι
inst✝¹ : UniformSpace F
inst✝ : IsUniformAddGroup F
𝔖 : Set (Set E)
a : ι → E →SLᵤ[σ, 𝔖] F
a₀ : E →SLᵤ[σ, 𝔖] F
⊢ (∀ s ∈ 𝔖,
      TendstoUniformlyOn (⇑(UniformOnFun.toFun 𝔖) ∘ (⇑(UniformOnFun.ofFun 𝔖) ∘ DFunLike.coe) ∘ a)
        ((UniformOnFun.toFun 𝔖) ((⇑(UniformOnFun.ofFun 𝔖) ∘ DFunLike.coe) a₀)) p s) ↔
    ∀ s ∈ 𝔖, TendstoUniformlyOn (fun x1 x2 ↦ (a x1) x2) (⇑a₀) p s
Module: Mathlib.Analysis.LocallyConvex.WithSeminorms

Goal state before:
pp: 𝕜 : Type u_2
E : Type u_6
ι : Type u_9
inst✝² : NormedField 𝕜
inst✝¹ : AddCommGroup E
inst✝ : Module 𝕜 E
κ : ι → Type u_11
p : (i : ι) → SeminormFamily 𝕜 E (κ i)
t : ι → TopologicalSpace E
hp : ∀ (i : ι), WithSeminorms (p i)
⊢ WithSeminorms (SeminormFamily.sigma p)

Tactic:
by
  have : ∀ i, @IsTopologicalAddGroup E (t i) _ :=
    fun i ↦ @WithSeminorms.topologicalAddGroup _ _ _ _ _ _ (t i) _ (hp i)
  have : @IsTopologicalAddGroup E (⨅ i, t i) _ := topologicalAddGroup_iInf inferInstance
  simp_rw [@SeminormFamily.withSeminorms_iff_topologicalSpace_eq_iInf _ _ _ _ _ _ _ (_)] at hp ⊢
  rw [iInf_sigma]
  exact iInf_congr hp

Goal state after:
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u
inst✝ : TopologicalSpace X
h :
  ∀ (s : Set X),
    (∀ (K : Type u) [inst : TopologicalSpace K] [CompactSpace K] (f : K → X),
        Continuous[_, inst✝] f → IsOpen (f ⁻¹' s)) →
      IsOpen[inst✝] s
A : Set X
K : Type u
x✝¹ : TopologicalSpace K
x✝ : CompactSpace K
f : K → X
hA : IsOpen[instTopologicalSpaceSubtype] (range f ↓∩ A)
hf : Continuous[x✝¹, inst✝] f
this : IsOpen[x✝¹] (codRestrict f (range f) ⋯ ⁻¹' range f ↓∩ A)
⊢ IsOpen[x✝¹] (f ⁻¹' A)

Tactic:
← preimage_comp

Goal state after:
X : Type u
inst✝ : TopologicalSpace X
h :
  ∀ (s : Set X),
    (∀ (K : Type u) [inst : TopologicalSpace K] [CompactSpace K] (f : K → X),
        Continuous[_, inst✝] f → IsOpen (f ⁻¹' s)) →
      IsOpen[inst✝] s
A : Set X
K : Type u
x✝¹ : TopologicalSpace K
x✝ : CompactSpace K
f : K → X
hA : IsOpen[instTopologicalSpaceSubtype] (range f ↓∩ A)
hf : Continuous[x✝¹, inst✝] f
this : IsOpen[x✝¹] (Subtype.val ∘ codRestrict f (range f) ⋯ ⁻¹' A)
⊢ IsOpen[x✝¹] (f ⁻¹' A)
Module: Mathlib.Topology.Compactness.CompactlyCoherentSpace

Goal state before:
pp: X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
U : Set (𝐤X)
hU :
  ∀ (K : Type u_1) [inst : TopologicalSpace K] [CompactSpace K] (f : K → 𝐤X),
    Continuous[_, instTopologicalSpace] f → IsOpen (f ⁻¹' U)
K : Set X
hK : IsCompact K
⊢ IsOpen[instTopologicalSpaceSubtype] (K ↓∩ ⇑(CompactCoherentification.mk X) ⁻¹' U)

Tactic:
← preimage_comp

Goal state after:
X : Type u_1
Y : Type u_2
inst✝¹ : TopologicalSpace X
inst✝ : TopologicalSpace Y
U : Set (𝐤X)
hU :
  ∀ (K : Type u_1) [inst : TopologicalSpace K] [CompactSpace K] (f : K → 𝐤X),
    Continuous[_, instTopologicalSpace] f → IsOpen (f ⁻¹' U)
K : Set X
hK : IsCompact K
⊢ IsOpen[instTopologicalSpaceSubtype] (⇑(CompactCoherentification.mk X) ∘ Subtype.val ⁻¹' U)
Module: Mathlib.Topology.UniformSpace.CompactConvergence

Goal state before:
pp: α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
K : Set α
V : Set (β × β)
f✝ f : C(α, β)
l : Filter C(α, β)
⊢ (∀ (K : Set α), IsCompact K → TendstoUniformlyOn (fun i a ↦ (id i) a) (⇑f) l K) ↔
    Tendsto id l (Filter.comap toUniformOnFunIsCompact (𝓝 f.toUniformOnFunIsCompact))

Tactic:
tendsto_comap_iff,

Goal state after:
α : Type u₁
β : Type u₂
inst✝¹ : TopologicalSpace α
inst✝ : UniformSpace β
K : Set α
V : Set (β × β)
f✝ f : C(α, β)
l : Filter C(α, β)
⊢ (∀ (K : Set α), IsCompact K → TendstoUniformlyOn (fun i a ↦ (id i) a) (⇑f) l K) ↔
    Tendsto (toUniformOnFunIsCompact ∘ id) l (𝓝 f.toUniformOnFunIsCompact)
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:
simp [hx]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
g : β → γ
hg : Continuous[inst✝¹, inst✝] g
f : α →ₘ[μ] β
⊢ ↑(comp g hg f) =ᵐ[μ] g ∘ ↑f

Tactic:
by
  rw [comp_eq_mk]
  apply coeFn_mk

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case cons
δ : Type u_4
X : δ → Type u_5
inst✝¹ : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
s : (i : δ) → Set (X i)
a : δ
l : List δ
ih : (Measure.tprod l μ) (Set.tprod l s) = (List.map (fun i ↦ (μ i) (s i)) l).prod
⊢ (Measure.tprod (a :: l) μ) (Set.tprod (a :: l) s) = (List.map (fun i ↦ (μ i) (s i)) (a :: l)).prod

Tactic:
rw [tprod_cons, Set.tprod]

Goal state after:
case cons
δ : Type u_4
X : δ → Type u_5
inst✝¹ : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
s : (i : δ) → Set (X i)
a : δ
l : List δ
ih : (Measure.tprod l μ) (Set.tprod l s) = (List.map (fun i ↦ (μ i) (s i)) l).prod
⊢ ((μ a).prod (Measure.tprod l μ)) (s a ×ˢ Set.tprod l s) = (List.map (fun i ↦ (μ i) (s i)) (a :: l)).prod
Module: Mathlib.MeasureTheory.Function.AEEqFun

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace β
inst✝ : NeZero μ
b : β
x : α
h : ∃ b_1, const α b = mk (Function.const α b_1) ⋯
b' : β := Classical.choose h
this : ∀ᵐ (x : α) ∂μ, b = Function.const α b' x
⊢ Function.const α b' x = b

Tactic:
← const_def,

Goal state after:
case pos
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace β
inst✝ : NeZero μ
b : β
x : α
h : ∃ b_1, const α b = mk (Function.const α b_1) ⋯
b' : β := Classical.choose h
this : ∀ᵐ (x : α) ∂μ, b = b'
⊢ Function.const α b' x = b
Module: Mathlib.MeasureTheory.Function.AEEqFun

Goal state before:
pp: case pos
α : Type u_1
β : Type u_2
inst✝² : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace β
inst✝ : NeZero μ
b : β
x : α
h : ∃ b_1, const α b = mk (Function.const α b_1) ⋯
b' : β := Classical.choose h
this : b = b'
⊢ Function.const α b' x = b

Tactic:
rw [Function.const, this]

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

Goal state before:
pp: 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
hμsc : c = 0 ∨ μ s ≠ ∞
⊢ MemLp (s.indicator fun x ↦ c) p μ

Tactic:
rw [memLp_indicator_iff_restrict hs]

Goal state after:
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
hμsc : c = 0 ∨ μ s ≠ ∞
⊢ MemLp (fun x ↦ c) p (μ.restrict s)
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Indicator

Goal state before:
pp: case pos
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝² : TopologicalSpace ε
inst✝¹ : ESeminormedAddMonoid ε
s : Set α
f g : α → ε
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
x : α
hx : x ∈ s
⊢ ‖s.piecewise f g x‖ₑ = s.piecewise (fun x ↦ ‖f x‖ₑ) (fun x ↦ ‖g x‖ₑ) x

Tactic:
simp [hx]

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

Goal state before:
pp: case neg
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝² : TopologicalSpace ε
inst✝¹ : ESeminormedAddMonoid ε
s : Set α
f g : α → ε
inst✝ : DecidablePred fun x ↦ x ∈ s
hs : MeasurableSet s
x : α
hx : x ∉ s
⊢ ‖s.piecewise f g x‖ₑ = s.piecewise (fun x ↦ ‖f x‖ₑ) (fun x ↦ ‖g x‖ₑ) x

Tactic:
simp [hx]

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)
hp_zero : ¬p = 0
hp_top : p ≠ ∞
x : α
hx : x ∈ s
⊢ ‖s.piecewise f g x‖ₑ ^ p.toReal = ‖f x‖ₑ ^ p.toReal

Tactic:
simp [hx]

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

Goal state before:
pp: α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
s : Set α
β : Type u_9
inst✝ : NormedAddCommGroup β
μ : Measure α
hp : p ≠ ∞
hs : MeasurableSet s
c : ℝ
hc : 0 ≤ c
f g : α → β
h : ∀ (x : α), dist (f x) (g x) ≤ c
hs₁ : Function.support f ⊆ s
hs₂ : Function.support g ⊆ s
⊢ s.indicator (f - g) = f - g

Tactic:
rw [Set.indicator_eq_self]

Goal state after:
α : Type u_1
m0 : MeasurableSpace α
p : ℝ≥0∞
s : Set α
β : Type u_9
inst✝ : NormedAddCommGroup β
μ : Measure α
hp : p ≠ ∞
hs : MeasurableSet s
c : ℝ
hc : 0 ≤ c
f g : α → β
h : ∀ (x : α), dist (f x) (g x) ≤ c
hs₁ : Function.support f ⊆ s
hs₂ : Function.support g ⊆ s
⊢ Function.support (f - g) ⊆ s
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:
simp [hx]

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

Goal state before:
pp: p q : ℝ≥0∞
hq : q ≠ 0
inst✝ : p.HolderTriple q 0
hp : p ≠ 0
⊢ p⁻¹ + q⁻¹ < ∞ + ∞

Tactic:
simp [hp, hq, pos_iff_ne_zero]

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

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

Tactic:
rcases eq_or_ne c 0 with rfl | hc
  · simp
  · exact (eLpNorm_smul_measure_of_ne_zero hc f p μ).le

Goal state after:
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: p q r : ℝ≥0∞
hp : 0 < p.toNNReal
hq : 0 < q.toNNReal
⊢ p.toNNReal.HolderTriple q.toNNReal r.toNNReal ↔ p.HolderTriple q r

Tactic:
simp_rw [← NNReal.holderTriple_coe_iff, coe_toNNReal_eq_toReal]

Goal state after:
p q r : ℝ≥0∞
hp : 0 < p.toNNReal
hq : 0 < q.toNNReal
⊢ p.toReal.HolderTriple q.toReal r.toReal ↔ p.HolderTriple q r
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: p q : ℝ≥0∞
hp : 1 < p.toReal
⊢ p.toReal.HolderConjugate q.toReal ↔ p.HolderConjugate q

Tactic:
by
  refine ⟨of_toReal, fun h ↦ ?_⟩
  have hq : 0 < q.toReal := by
    rw [toReal_pos_iff]
    refine ⟨pos q p, lt_top_iff_one_lt q p |>.mpr ?_⟩
    contrapose! hp
    exact toReal_mono one_ne_top hp
  simpa using HolderTriple.toReal 1 (zero_lt_one.trans hp) hq

Goal state after:
Module: Mathlib.Data.Real.ConjExponents

Goal state before:
pp: p q : ℝ≥0∞
hp : 1 < p.toNNReal
⊢ p.toNNReal.HolderConjugate q.toNNReal ↔ p.HolderConjugate q

Tactic:
simp_rw [← NNReal.holderTriple_coe_iff, coe_toNNReal_eq_toReal]

Goal state after:
p q : ℝ≥0∞
hp : 1 < p.toNNReal
⊢ p.toReal.HolderTriple q.toReal ↑1 ↔ p.HolderConjugate q
Module: Mathlib.Analysis.Convex.Slope

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
s : Set 𝕜
f : 𝕜 → 𝕜
hf : ConcaveOn 𝕜 s f
x y : 𝕜
hy : y ∈ s
hxy : x < y
hxy' : f x < f y
⊢ StrictMonoOn f (s ∩ Set.Iic x)

Tactic:
simpa using (neg_convexOn_iff.mpr hf |>.strictAntiOn hy hxy <| neg_lt_neg hxy').neg

Goal state after:
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
hf_norm : ∫⁻ (a : α), f a ^ p ∂μ = 1
hg_norm : ∫⁻ (a : α), g a ^ q ∂μ = 1
⊢ ∫⁻ (a : α), f a ^ p * (ENNReal.ofReal p)⁻¹ + g a ^ q * (ENNReal.ofReal q)⁻¹ ∂μ = 1

Tactic:
rw [lintegral_add_left']

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
hf_norm : ∫⁻ (a : α), f a ^ p ∂μ = 1
hg_norm : ∫⁻ (a : α), g a ^ q ∂μ = 1
⊢ ∫⁻ (a : α), f a ^ p * (ENNReal.ofReal p)⁻¹ ∂μ + ∫⁻ (a : α), g a ^ q * (ENNReal.ofReal q)⁻¹ ∂μ = 1

case hf
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
hf_norm : ∫⁻ (a : α), f a ^ p ∂μ = 1
hg_norm : ∫⁻ (a : α), g a ^ q ∂μ = 1
⊢ AEMeasurable (fun a ↦ f a ^ p * (ENNReal.ofReal p)⁻¹) μ
Module: Mathlib.Analysis.Convex.Mul

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : Field 𝕜
inst✝¹ : LinearOrder 𝕜
inst✝ : IsStrictOrderedRing 𝕜
n : ℕ
⊢ ConvexOn 𝕜 (Ioi 0) fun x ↦ x ^ -[n+1]

Tactic:
simp_rw [zpow_negSucc, ← inv_pow]
    refine (convexOn_iff_forall_pos.2 ⟨convex_Ioi _, ?_⟩).pow (fun x (hx : 0 < x) ↦ by positivity) _
    rintro x (hx : 0 < x) y (hy : 0 < y) a b ha hb hab
    simp only [smul_eq_mul]
    field_simp
    have H : 0 ≤ a * b * (x - y) ^ 2 := by positivity
    linear_combination H - x * y * (a + b + 1) * hab

Goal state after:
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: case mp
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
⊢ ∏ i ∈ s, z i ^ w i < ∑ i ∈ s, w i * z i → ∃ j ∈ s, ∃ k ∈ s, z j ≠ z k

pp: case mpr
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
⊢ (∃ j ∈ s, ∃ k ∈ s, z j ≠ z k) → ∏ i ∈ s, z i ^ w i < ∑ i ∈ s, w i * z i

Tactic:
· intro h
    by_contra! h_contra
    rw [(geom_mean_eq_arith_mean_weighted_iff_of_pos' s w z hw hw' hz).mpr ?_] at h
    · exact (lt_self_iff_false _).mp h
    · intro j hjs
      rw [← arith_mean_weighted_of_constant s w (fun _ => z j) (z j) hw' fun _ _ => congrFun rfl]
      apply sum_congr rfl (fun x a => congrArg (HMul.hMul (w x)) (h_contra j hjs x a))

Goal state after:
case mpr
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
⊢ (∃ j ∈ s, ∃ k ∈ s, z j ≠ z k) → ∏ i ∈ s, z i ^ w i < ∑ i ∈ s, w i * z i
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: a b p q : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hpq : p.HolderConjugate q
⊢ a * b ≤ a ^ p / p + b ^ q / q

Tactic:
simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, div_eq_inv_mul] using
    geom_mean_le_arith_mean2_weighted hpq.inv_nonneg hpq.symm.inv_nonneg
      (rpow_nonneg ha p) (rpow_nonneg hb q) hpq.inv_add_inv_eq_one

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

Goal state before:
pp: case pos
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
h_zero : (f + g) a = 0
⊢ (f + g) a ^ p ≤ (f + g) a * (f + g) a ^ (p - 1)

pp: case neg
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
h_zero : ¬(f + g) a = 0
⊢ (f + g) a ^ p ≤ (f + g) a * (f + g) a ^ (p - 1)

Tactic:
· rw [h_zero, ENNReal.zero_rpow_of_pos hpq.pos]
        exact zero_le

Goal state after:
case neg
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
h_zero : ¬(f + g) a = 0
⊢ (f + g) a ^ p ≤ (f + g) a * (f + g) a ^ (p - 1)
Module: Mathlib.Analysis.MeanInequalities

Goal state before:
pp: ι : Type u_1
s : Finset ι
f g : ι → ℝ
p q r : ℝ
hpqr : p.HolderTriple q r
hf : ∀ i ∈ s, 0 ≤ f i
hg : ∀ i ∈ s, 0 ≤ g i
⊢ ∑ i ∈ s, (f i * g i) ^ r ≤ (∑ i ∈ s, f i ^ p) ^ (r / p) * (∑ i ∈ s, g i ^ q) ^ (r / q)

Tactic:
convert Lr_rpow_le_Lp_mul_Lq s f g hpqr with i hi

Goal state after:
case e'_3.a
ι : Type u_1
s : Finset ι
f g : ι → ℝ
p q r : ℝ
hpqr : p.HolderTriple q r
hf : ∀ i ∈ s, 0 ≤ f i
hg : ∀ i ∈ s, 0 ≤ g i
i : ι
hi : i ∈ s
⊢ f i * g i = |f i * g i|

case e'_4.e'_5.a
ι : Type u_1
s : Finset ι
f g : ι → ℝ
p q r : ℝ
hpqr : p.HolderTriple q r
hf : ∀ i ∈ s, 0 ≤ f i
hg : ∀ i ∈ s, 0 ≤ g i
x✝ : ι
a✝ : x✝ ∈ s
⊢ f x✝ = |f x✝|

case e'_4.e'_6.a
ι : Type u_1
s : Finset ι
f g : ι → ℝ
p q r : ℝ
hpqr : p.HolderTriple q r
hf : ∀ i ∈ s, 0 ≤ f i
hg : ∀ i ∈ s, 0 ≤ g i
x✝ : ι
a✝ : x✝ ∈ s
⊢ g x✝ = |g x✝|
Module: Mathlib.MeasureTheory.Function.ConvergenceInMeasure

Goal state before:
pp: α : Type u_1
ι : Type u_2
E : Type u_4
m : MeasurableSpace α
μ : Measure α
inst✝ : EDist E
l : Filter ι
f f' : ι → α → E
g g' : α → E
h_left : ∀ᶠ (i : ι) in l, f i =ᵐ[μ] f' i
h_right : g =ᵐ[μ] g'
h_tendsto : TendstoInMeasure μ f l g
ε : ℝ≥0∞
hε : 0 < ε
this : (fun i ↦ μ {x | ε ≤ edist (f' i x) (g' x)}) =ᶠ[l] fun i ↦ μ {x | ε ≤ edist (f i x) (g x)}
⊢ Tendsto (fun i ↦ μ {x | ε ≤ edist (f' i x) (g' x)}) l (𝓝 0)

Tactic:
by
    rw [tendsto_congr' this]
    exact h_tendsto ε hε

Goal state after: