mathlib-initiative/mathlib-tactics huggingface dataset:
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
⊢ inferInstance = ⨅ i, UniformSpace.comap (fun x ↦ x.comp (φ i)) inferInstance

Tactic:
by
  -- We check the analogous result for `UniformOnFun` using
  -- `UniformOnFun.uniformSpace_eq_iInf_precomp_of_cover`...
  set 𝔖 : Set (Set α) := {K | IsCompact K}
  set 𝔗 : Π i, Set (Set (δ i)) := fun i ↦ {K | IsCompact K}
  have h_image : ∀ i, MapsTo (φ i '' ·) (𝔗 i) 𝔖 := fun i K hK ↦ hK.image (φ i).continuous
  have h_preimage : ∀ i, MapsTo (φ i ⁻¹' ·) 𝔖 (𝔗 i) := fun i K ↦ (h_proper i).isCompact_preimage
  have h_cover' : ∀ S ∈ 𝔖, ∃ I : Set ι, I.Finite ∧ S ⊆ ⋃ i ∈ I, range (φ i) := fun S hS ↦ by
    refine ⟨{i | (range (φ i) ∩ S).Nonempty}, h_lf.finite_nonempty_inter_compact hS,
      inter_eq_right.mp ?_⟩
    simp_rw [iUnion₂_inter, mem_ofPred, iUnion_nonempty_self, ← iUnion_inter, h_cover, univ_inter]
  -- ... and we just pull it back.
  simp_rw +zetaDelta [compactConvergenceUniformSpace, replaceTopology_eq,
    UniformOnFun.uniformSpace_eq_iInf_precomp_of_cover _ _ _ h_image h_preimage h_cover',
    UniformSpace.comap_iInf, ← UniformSpace.comap_comap]
  rfl

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

Goal state before:
pp: case pos
α : Type u₁
β : Type u₂
inst✝² : TopologicalSpace α
inst✝¹ : UniformSpace β
inst✝ : CompleteSpace C(α, β)
f : α → β
s : Set α
l : Filter C(α, β)
hlc : Cauchy l
hlf : l ≤ 𝓟 {g | EqOn (⇑g) f s}
f' : C(α, β)
hf' : l ≤ 𝓝 f'
this : l.NeBot
H₁ : ∀ x ∈ s, Inseparable (f x) (f' x)
x : α
hx : x ∈ s
⊢ Inseparable (s.piecewise f (⇑f') x) (f' x)

pp: case neg
α : Type u₁
β : Type u₂
inst✝² : TopologicalSpace α
inst✝¹ : UniformSpace β
inst✝ : CompleteSpace C(α, β)
f : α → β
s : Set α
l : Filter C(α, β)
hlc : Cauchy l
hlf : l ≤ 𝓟 {g | EqOn (⇑g) f s}
f' : C(α, β)
hf' : l ≤ 𝓝 f'
this : l.NeBot
H₁ : ∀ x ∈ s, Inseparable (f x) (f' x)
x : α
hx : x ∉ s
⊢ Inseparable (s.piecewise f (⇑f') x) (f' x)

Tactic:
<;>

Goal state after:
case pos
α : Type u₁
β : Type u₂
inst✝² : TopologicalSpace α
inst✝¹ : UniformSpace β
inst✝ : CompleteSpace C(α, β)
f : α → β
s : Set α
l : Filter C(α, β)
hlc : Cauchy l
hlf : l ≤ 𝓟 {g | EqOn (⇑g) f s}
f' : C(α, β)
hf' : l ≤ 𝓝 f'
this : l.NeBot
H₁ : ∀ x ∈ s, Inseparable (f x) (f' x)
x : α
hx : x ∈ s
⊢ Inseparable (s.piecewise f (⇑f') x) (f' x)

case neg
α : Type u₁
β : Type u₂
inst✝² : TopologicalSpace α
inst✝¹ : UniformSpace β
inst✝ : CompleteSpace C(α, β)
f : α → β
s : Set α
l : Filter C(α, β)
hlc : Cauchy l
hlf : l ≤ 𝓟 {g | EqOn (⇑g) f s}
f' : C(α, β)
hf' : l ≤ 𝓝 f'
this : l.NeBot
H₁ : ∀ x ∈ s, Inseparable (f x) (f' x)
x : α
hx : x ∉ s
⊢ Inseparable (s.piecewise f (⇑f') x) (f' x)
Module: Mathlib.Topology.ContinuousMap.Algebra

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : CommGroup β
inst✝ : IsTopologicalGroup β
this✝ : UniformSpace β := IsTopologicalGroup.rightUniformSpace β
this : IsUniformGroup β
f g : C(α, β)
⊢ ContinuousAt (fun p ↦ p.1 * p.2) (f, g)

Tactic:
ContinuousAt,

Goal state after:
α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : CommGroup β
inst✝ : IsTopologicalGroup β
this✝ : UniformSpace β := IsTopologicalGroup.rightUniformSpace β
this : IsUniformGroup β
f g : C(α, β)
⊢ Filter.Tendsto (fun p ↦ p.1 * p.2) (nhds (f, g)) (nhds ((f, g).1 * (f, g).2))
Module: Mathlib.Topology.ContinuousMap.Algebra

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : CommGroup β
inst✝ : IsTopologicalGroup β
this✝ : UniformSpace β := IsTopologicalGroup.rightUniformSpace β
this : IsUniformGroup β
f : C(α, β)
⊢ ContinuousAt (fun a ↦ a⁻¹) f

Tactic:
ContinuousAt,

Goal state after:
α : Type u_1
β : Type u_2
inst✝³ : TopologicalSpace α
inst✝² : TopologicalSpace β
inst✝¹ : CommGroup β
inst✝ : IsTopologicalGroup β
this✝ : UniformSpace β := IsTopologicalGroup.rightUniformSpace β
this : IsUniformGroup β
f : C(α, β)
⊢ Filter.Tendsto (fun a ↦ a⁻¹) (nhds f) (nhds f⁻¹)
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 : α → β
hf : AEStronglyMeasurable f μ
⊢ ↑(mk f hf) =ᵐ[μ] f

Tactic:
← mk_eq_mk (hf := AEEqFun.aestronglyMeasurable ..) (hg := hf),

Goal state after:
α : Type u_1
β : Type u_2
inst✝¹ : MeasurableSpace α
μ : Measure α
inst✝ : TopologicalSpace β
f : α → β
hf : AEStronglyMeasurable f μ
⊢ mk ↑(mk f hf) ⋯ = mk f hf
Module: Mathlib.Probability.UniformOn

Goal state before:
pp: Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t : Set Ω
hsf : s.Finite
h : Measure.count (s ∩ t) = Measure.count s
⊢ s ⊆ t

Tactic:
Measure.count_apply_finite _ hsf,

Goal state after:
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
s t : Set Ω
hsf : s.Finite
h : Measure.count (s ∩ t) = ↑(#hsf.toFinset)
⊢ s ⊆ t
Module: Mathlib.Probability.UniformOn

Goal state before:
pp: case inl
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
t u : Set Ω
ht : t.Finite
hs : ∅.Finite
hst : Disjoint ∅ t
⊢ (uniformOn ∅) u * (uniformOn (∅ ∪ t)) ∅ + (uniformOn t) u * (uniformOn (∅ ∪ t)) t = (uniformOn (∅ ∪ t)) u

pp: case 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
⊢ (uniformOn s) u * (uniformOn (s ∪ t)) s + (uniformOn t) u * (uniformOn (s ∪ t)) t = (uniformOn (s ∪ t)) u

Tactic:
<;>

Goal state after:
case inl
Ω : Type u_1
inst✝¹ : MeasurableSpace Ω
inst✝ : MeasurableSingletonClass Ω
t u : Set Ω
ht : t.Finite
hs : ∅.Finite
hst : Disjoint ∅ t
⊢ (uniformOn ∅) u * (uniformOn (∅ ∪ t)) ∅ + (uniformOn t) u * (uniformOn (∅ ∪ t)) t = (uniformOn (∅ ∪ t)) u

case 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
⊢ (uniformOn s) u * (uniformOn (s ∪ t)) s + (uniformOn t) u * (uniformOn (s ∪ t)) t = (uniformOn (s ∪ t)) u
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Defs

Goal state before:
pp: α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
inst✝ : ENorm ε
μ : Measure α
f : α → ε
⊢ (∫⁻ (x : α), ‖f x‖ₑ ^ ENNReal.toReal 1 ∂μ) ^ (1 / ENNReal.toReal 1) = ∫⁻ (x : α), ‖f x‖ₑ ∂μ

Tactic:
ENNReal.toReal_one,

Goal state after:
α : Type u_1
ε : Type u_2
m0 : MeasurableSpace α
inst✝ : ENorm ε
μ : Measure α
f : α → ε
⊢ (∫⁻ (x : α), ‖f x‖ₑ ^ 1 ∂μ) ^ (1 / 1) = ∫⁻ (x : α), ‖f x‖ₑ ∂μ
Module: Mathlib.MeasureTheory.Function.EssSup

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
s : Set α
f : α → ℝ≥0∞
hsf : Function.support f ⊆ s
⊢ essSup f (μ.restrict s) = essSup f μ

Tactic:
by
  apply le_antisymm (essSup_mono_measure' Measure.restrict_le_self)
  apply le_of_forall_lt (fun c hc ↦ ?_)
  obtain ⟨d, cd, hd⟩ : ∃ d, c < d ∧ d < essSup f μ := exists_between hc
  let t := {x | d < f x}
  have A : 0 < (μ.restrict t) t := by
    simp only [Measure.restrict_apply_self]
    rw [essSup_eq_sInf] at hd
    have : d ∉ {a | μ {x | a < f x} = 0} := notMem_of_lt_csInf hd (OrderBot.bddBelow _)
    exact bot_lt_iff_ne_bot.2 this
  have B : 0 < (μ.restrict s) t := by
    have : μ.restrict t ≤ μ.restrict s := by
      apply Measure.restrict_mono _ le_rfl
      apply subset_trans _ hsf
      intro x (hx : d < f x)
      exact (lt_of_le_of_lt bot_le hx).ne'
    exact lt_of_lt_of_le A (this _)
  apply cd.trans_le
  rw [essSup_eq_sInf]
  apply le_sInf (fun b hb ↦ ?_)
  contrapose! hb
  exact ne_of_gt (B.trans_le (measure_mono (fun x hx ↦ hb.trans hx)))

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

Goal state before:
pp: case _hl
ι : Type u_1
α : ι → Type u_3
inst✝³ : Fintype ι
inst✝² : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝¹ : Encodable ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
s : (i : ι) → Set (α i)
⊢ (sortedUniv ι).Nodup

pp: case hnd
ι : Type u_1
α : ι → Type u_3
inst✝³ : Fintype ι
inst✝² : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝¹ : Encodable ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
s : (i : ι) → Set (α i)
⊢ (sortedUniv ι).Nodup

pp: case hnd
ι : Type u_1
α : ι → Type u_3
inst✝³ : Fintype ι
inst✝² : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝¹ : Encodable ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
s : (i : ι) → Set (α i)
⊢ (sortedUniv ι).Nodup

Tactic:
<;>

Goal state after:
case _hl
ι : Type u_1
α : ι → Type u_3
inst✝³ : Fintype ι
inst✝² : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝¹ : Encodable ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
s : (i : ι) → Set (α i)
⊢ (sortedUniv ι).Nodup

case hnd
ι : Type u_1
α : ι → Type u_3
inst✝³ : Fintype ι
inst✝² : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝¹ : Encodable ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
s : (i : ι) → Set (α i)
⊢ (sortedUniv ι).Nodup

case hnd
ι : Type u_1
α : ι → Type u_3
inst✝³ : Fintype ι
inst✝² : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
inst✝¹ : Encodable ι
inst✝ : ∀ (i : ι), SigmaFinite (μ i)
s : (i : ι) → Set (α i)
⊢ (sortedUniv ι).Nodup
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case refine_1
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝¹ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
C : (i : ι) → Set (Set (α i))
hμ : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)
this✝ : ∀ (i : ι), SigmaFinite (μ i)
this : Encodable ι
n : ℕ
⊢ (univ.pi fun i ↦ (hμ i).set ((decode n).getD default i)) ∈ univ.pi '' univ.pi C

pp: case refine_2
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝¹ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
C : (i : ι) → Set (Set (α i))
hμ : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)
this✝ : ∀ (i : ι), SigmaFinite (μ i)
this : Encodable ι
n : ℕ
⊢ (Measure.pi μ) (univ.pi fun i ↦ (hμ i).set ((decode n).getD default i)) < ∞

pp: case refine_3
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝¹ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
C : (i : ι) → Set (Set (α i))
hμ : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)
this✝ : ∀ (i : ι), SigmaFinite (μ i)
this : Encodable ι
⊢ (⋃ i, univ.pi fun i_1 ↦ (hμ i_1).set ((decode i).getD default i_1)) = univ

Tactic:
<;>

Goal state after:
case refine_1
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝¹ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
C : (i : ι) → Set (Set (α i))
hμ : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)
this✝ : ∀ (i : ι), SigmaFinite (μ i)
this : Encodable ι
n : ℕ
⊢ (univ.pi fun i ↦ (hμ i).set ((decode n).getD default i)) ∈ univ.pi '' univ.pi C

case refine_2
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝¹ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
C : (i : ι) → Set (Set (α i))
hμ : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)
this✝ : ∀ (i : ι), SigmaFinite (μ i)
this : Encodable ι
n : ℕ
⊢ (Measure.pi μ) (univ.pi fun i ↦ (hμ i).set ((decode n).getD default i)) < ∞

case refine_3
ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝¹ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
C : (i : ι) → Set (Set (α i))
hμ : (i : ι) → (μ i).FiniteSpanningSetsIn (C i)
this✝ : ∀ (i : ι), SigmaFinite (μ i)
this : Encodable ι
⊢ (⋃ i, univ.pi fun i_1 ↦ (hμ i_1).set ((decode i).getD default i_1)) = univ
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u_1
ι' : Type u_2
α : ι → Type u_3
inst✝¹ : Fintype ι
m : (i : ι) → OuterMeasure (α i)
inst✝ : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
C : (i : ι) → Set (Set (α i))
i : ι
⊢ MeasurableSpace (α i)

Tactic:
by apply_assumption

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
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 [eLpNorm_eq_lintegral_rpow_enorm_toReal hp_zero hp_top]

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 = ∞
⊢ (∫⁻ (x : α), ‖s.indicator f x‖ₑ ^ p.toReal ∂μ) ^ (1 / p.toReal) =
    (∫⁻ (x : α) in s, ‖f x‖ₑ ^ p.toReal ∂μ) ^ (1 / p.toReal)
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: case none
ι : Type u_1
inst✝² : Fintype ι
β : Option ι → Type u_4
inst✝¹ : (i : Option ι) → MeasurableSpace (β i)
μ : (i : Option ι) → Measure (β i)
inst✝ : ∀ (i : Option ι), SigmaFinite (μ i)
s : (i : Option ι) → Set (β i)
x✝¹ : ∀ (i : Option ι), MeasurableSet (s i)
e_meas : ((i : ι) → β (some i)) × β none ≃ᵐ ((i : Option ι) → β i) := (MeasurableEquiv.piOptionEquivProd β).symm
me : MeasurableEmbedding ⇑e_meas
x : ((i : ι) → β (some i)) × β none
x✝ : (∀ (i : ι), x.1 i ∈ s (some i)) ∧ x.2 ∈ s none
⊢ e_meas x none ∈ s none

pp: case some
ι : Type u_1
inst✝² : Fintype ι
β : Option ι → Type u_4
inst✝¹ : (i : Option ι) → MeasurableSpace (β i)
μ : (i : Option ι) → Measure (β i)
inst✝ : ∀ (i : Option ι), SigmaFinite (μ i)
s : (i : Option ι) → Set (β i)
x✝¹ : ∀ (i : Option ι), MeasurableSet (s i)
e_meas : ((i : ι) → β (some i)) × β none ≃ᵐ ((i : Option ι) → β i) := (MeasurableEquiv.piOptionEquivProd β).symm
me : MeasurableEmbedding ⇑e_meas
x : ((i : ι) → β (some i)) × β none
x✝ : (∀ (i : ι), x.1 i ∈ s (some i)) ∧ x.2 ∈ s none
val✝ : ι
⊢ e_meas x (some val✝) ∈ s (some val✝)

Tactic:
<;>

Goal state after:
case none
ι : Type u_1
inst✝² : Fintype ι
β : Option ι → Type u_4
inst✝¹ : (i : Option ι) → MeasurableSpace (β i)
μ : (i : Option ι) → Measure (β i)
inst✝ : ∀ (i : Option ι), SigmaFinite (μ i)
s : (i : Option ι) → Set (β i)
x✝¹ : ∀ (i : Option ι), MeasurableSet (s i)
e_meas : ((i : ι) → β (some i)) × β none ≃ᵐ ((i : Option ι) → β i) := (MeasurableEquiv.piOptionEquivProd β).symm
me : MeasurableEmbedding ⇑e_meas
x : ((i : ι) → β (some i)) × β none
x✝ : (∀ (i : ι), x.1 i ∈ s (some i)) ∧ x.2 ∈ s none
⊢ e_meas x none ∈ s none

case some
ι : Type u_1
inst✝² : Fintype ι
β : Option ι → Type u_4
inst✝¹ : (i : Option ι) → MeasurableSpace (β i)
μ : (i : Option ι) → Measure (β i)
inst✝ : ∀ (i : Option ι), SigmaFinite (μ i)
s : (i : Option ι) → Set (β i)
x✝¹ : ∀ (i : Option ι), MeasurableSet (s i)
e_meas : ((i : ι) → β (some i)) × β none ≃ᵐ ((i : Option ι) → β i) := (MeasurableEquiv.piOptionEquivProd β).symm
me : MeasurableEmbedding ⇑e_meas
x : ((i : ι) → β (some i)) × β none
x✝ : (∀ (i : ι), x.1 i ∈ s (some i)) ∧ x.2 ∈ s none
val✝ : ι
⊢ e_meas x (some val✝) ∈ s (some val✝)
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

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:
<;>

Goal state after:
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

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
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Basic

Goal state before:
pp: case pos
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε' : Type u_7
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
c : ε'
hc : ‖c‖ₑ ≠ ∞
h : μ = 0
⊢ eLpNorm (fun x ↦ c) ∞ μ < ∞

pp: case neg
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε' : Type u_7
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
c : ε'
hc : ‖c‖ₑ ≠ ∞
h : ¬μ = 0
⊢ eLpNorm (fun x ↦ c) ∞ μ < ∞

Tactic:
<;>

Goal state after:
case pos
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε' : Type u_7
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
c : ε'
hc : ‖c‖ₑ ≠ ∞
h : μ = 0
⊢ eLpNorm (fun x ↦ c) ∞ μ < ∞

case neg
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε' : Type u_7
inst✝¹ : TopologicalSpace ε'
inst✝ : ContinuousENorm ε'
c : ε'
hc : ‖c‖ₑ ≠ ∞
h : ¬μ = 0
⊢ eLpNorm (fun x ↦ c) ∞ μ < ∞
Module: Mathlib.MeasureTheory.Constructions.Pi

Goal state before:
pp: ι : Type u
α : ι → Type v
inst✝¹ : Fintype ι
inst✝ : IsEmpty ι
m : (i : ι) → MeasurableSpace (α i)
μ : (i : ι) → Measure (α i)
e : ((i : ι) → α i) ≃ᵐ Unit := MeasurableEquiv.ofUniqueOfUnique ((i : ι) → α i) Unit
⊢ Measure.map (⇑e) (Measure.pi μ) = Measure.dirac ()

Tactic:
rw [Measure.pi_of_empty, Measure.map_dirac' e.measurable]

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

Goal state before:
pp: case e'_3
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
a x y : 𝕜
ha : a ∈ s
hx : x ∈ s
hy : y ∈ s
hxa✝ : x ≠ a
hya✝ : y ≠ a
hxy✝ : x ≤ y
hxy : x < y
hxa : x < a
hya : y < a
⊢ (f x - f a) / (x - a) = (f a - f x) / (a - x)

pp: case e'_4
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
a x y : 𝕜
ha : a ∈ s
hx : x ∈ s
hy : y ∈ s
hxa✝ : x ≠ a
hya✝ : y ≠ a
hxy✝ : x ≤ y
hxy : x < y
hxa : x < a
hya : y < a
⊢ (f y - f a) / (y - a) = (f a - f y) / (a - y)

Tactic:
<;>

Goal state after:
case e'_3
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
a x y : 𝕜
ha : a ∈ s
hx : x ∈ s
hy : y ∈ s
hxa✝ : x ≠ a
hya✝ : y ≠ a
hxy✝ : x ≤ y
hxy : x < y
hxa : x < a
hya : y < a
⊢ (f x - f a) / (x - a) = (f a - f x) / (a - x)

case e'_4
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
a x y : 𝕜
ha : a ∈ s
hx : x ∈ s
hy : y ∈ s
hxa✝ : x ≠ a
hya✝ : y ≠ a
hxy✝ : x ≤ y
hxy : x < y
hxa : x < a
hya : y < a
⊢ (f y - f a) / (y - a) = (f a - f y) / (a - y)
Module: Mathlib.Analysis.Convex.Slope

Goal state before:
pp: case e'_3
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
a x y : 𝕜
ha : a ∈ s
hx : x ∈ s
hy : y ∈ s
hxa✝ : x ≠ a
hya✝ : y ≠ a
hxy✝ : x ≤ y
hxy : x < y
hxa : x < a
hya : y < a
⊢ -(f x - f a) / -(x - a) = (f a - f x) / (a - x)

pp: case e'_4
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
a x y : 𝕜
ha : a ∈ s
hx : x ∈ s
hy : y ∈ s
hxa✝ : x ≠ a
hya✝ : y ≠ a
hxy✝ : x ≤ y
hxy : x < y
hxa : x < a
hya : y < a
⊢ -(f y - f a) / -(y - a) = (f a - f y) / (a - y)

Tactic:
<;>

Goal state after:
case e'_3
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
a x y : 𝕜
ha : a ∈ s
hx : x ∈ s
hy : y ∈ s
hxa✝ : x ≠ a
hya✝ : y ≠ a
hxy✝ : x ≤ y
hxy : x < y
hxa : x < a
hya : y < a
⊢ -(f x - f a) / -(x - a) = (f a - f x) / (a - x)

case e'_4
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
a x y : 𝕜
ha : a ∈ s
hx : x ∈ s
hy : y ∈ s
hxa✝ : x ≠ a
hya✝ : y ≠ a
hxy✝ : x ≤ y
hxy : x < y
hxa : x < a
hya : y < a
⊢ -(f y - f a) / -(y - a) = (f a - f y) / (a - y)
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
f : α → ε
c : ℝ≥0∞
⊢ c ^ (1 / 1).toReal • eLpNorm f 1 μ = c * eLpNorm f 1 μ

pp: case hp_ne_top
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
c : ℝ≥0∞
⊢ 1 ≠ ∞

Tactic:
<;>

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

case hp_ne_top
α : Type u_1
m0 : MeasurableSpace α
μ : Measure α
ε : Type u_7
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α → ε
c : ℝ≥0∞
⊢ 1 ≠ ∞
Module: Mathlib.Analysis.Convex.Slope

Goal state before:
pp: case e'_3
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
a x y : 𝕜
ha : a ∈ s
hx : x ∈ s
hy : y ∈ s
hxa✝ : x ≠ a
hya✝ : y ≠ a
hxy : x < y
hxa : x < a
hya : y < a
⊢ (f x - f a) / (x - a) = (f a - f x) / (a - x)

pp: case e'_4
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
a x y : 𝕜
ha : a ∈ s
hx : x ∈ s
hy : y ∈ s
hxa✝ : x ≠ a
hya✝ : y ≠ a
hxy : x < y
hxa : x < a
hya : y < a
⊢ (f y - f a) / (y - a) = (f a - f y) / (a - y)

Tactic:
<;>

Goal state after:
case e'_3
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
a x y : 𝕜
ha : a ∈ s
hx : x ∈ s
hy : y ∈ s
hxa✝ : x ≠ a
hya✝ : y ≠ a
hxy : x < y
hxa : x < a
hya : y < a
⊢ (f x - f a) / (x - a) = (f a - f x) / (a - x)

case e'_4
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
a x y : 𝕜
ha : a ∈ s
hx : x ∈ s
hy : y ∈ s
hxa✝ : x ≠ a
hya✝ : y ≠ a
hxy : x < y
hxa : x < a
hya : y < a
⊢ (f y - f a) / (y - a) = (f a - f y) / (a - y)
Module: Mathlib.Analysis.Convex.Slope

Goal state before:
pp: case e'_3
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
a x y : 𝕜
ha : a ∈ s
hx : x ∈ s
hy : y ∈ s
hxa✝ : x ≠ a
hya✝ : y ≠ a
hxy : x < y
hxa : x < a
hya : y < a
⊢ -(f x - f a) / -(x - a) = (f a - f x) / (a - x)

pp: case e'_4
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
a x y : 𝕜
ha : a ∈ s
hx : x ∈ s
hy : y ∈ s
hxa✝ : x ≠ a
hya✝ : y ≠ a
hxy : x < y
hxa : x < a
hya : y < a
⊢ -(f y - f a) / -(y - a) = (f a - f y) / (a - y)

Tactic:
<;>

Goal state after:
case e'_3
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
a x y : 𝕜
ha : a ∈ s
hx : x ∈ s
hy : y ∈ s
hxa✝ : x ≠ a
hya✝ : y ≠ a
hxy : x < y
hxa : x < a
hya : y < a
⊢ -(f x - f a) / -(x - a) = (f a - f x) / (a - x)

case e'_4
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
a x y : 𝕜
ha : a ∈ s
hx : x ∈ s
hy : y ∈ s
hxa✝ : x ≠ a
hya✝ : y ≠ a
hxy : x < y
hxa : x < a
hya : y < a
⊢ -(f y - f a) / -(y - a) = (f a - f y) / (a - y)