mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
A : ∀ (f : ContinuousMultilinearMap 𝕜 E G × ContinuousMultilinearMap 𝕜 E G), ‖-f.1 + f.2‖ = ‖f.1 - f.2‖
x✝¹ : Set ((i : ι) → E i) × ℝ
s : Set ((i : ι) → E i)
r : ℝ
x✝ : Bornology.IsVonNBounded 𝕜 (s, r).1 ∧ 0 < (s, r).2
hs : Bornology.IsVonNBounded 𝕜 (s, r).1
hr : 0 < (s, r).2
ε : ℝ
hε : ∀ x ∈ (s, r).1, ‖x‖ ≤ ε
δ : ℝ
hδ₀ : 0 < δ
hδ : ε ^ Fintype.card ι * δ < (s, r).2
f : ContinuousMultilinearMap 𝕜 E G
x : (i : ι) → E i
hf : ‖f‖ ≤ δ
hx : ‖x‖ ≤ ε
⊢ ‖f‖ * ε ^ Fintype.card ι ≤ δ * ε ^ Fintype.card ι

Tactic:
by have := (norm_nonneg x).trans hx; gcongr

Goal state after:
Module: Mathlib.MeasureTheory.Constructions.Polish.StronglyMeasurable

Goal state before:
pp: case pos
X : Type u_4
E : Type u_5
ι : Type u_6
inst✝⁶ : MeasurableSpace X
inst✝⁵ : CommMonoid E
inst✝⁴ : TopologicalSpace E
inst✝³ : PseudoMetrizableSpace E
inst✝² : ContinuousMul E
L : SummationFilter ι
inst✝¹ : L.NeBot
inst✝ : L.filter.IsCountablyGenerated
μ : Measure X
f : ι → X → E
h : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hm : Multipliable (fun i ↦ f i) L
h✝¹ : L.HasSupport ∧ ((Function.mulSupport fun i ↦ f i) ∩ L.support).Finite
h✝ : Function.HasFiniteMulSupport ((L.support.mulIndicator fun i ↦ f i) ∘ PLift.down)
⊢ AEStronglyMeasurable (∏ i ∈ Finite.toFinset h✝, L.support.mulIndicator (fun i ↦ f i) i.down) μ

Tactic:
refine Finset.aestronglyMeasurable_prod _ (fun _ _ ↦ ?_)

Goal state after:
case pos
X : Type u_4
E : Type u_5
ι : Type u_6
inst✝⁶ : MeasurableSpace X
inst✝⁵ : CommMonoid E
inst✝⁴ : TopologicalSpace E
inst✝³ : PseudoMetrizableSpace E
inst✝² : ContinuousMul E
L : SummationFilter ι
inst✝¹ : L.NeBot
inst✝ : L.filter.IsCountablyGenerated
μ : Measure X
f : ι → X → E
h : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hm : Multipliable (fun i ↦ f i) L
h✝¹ : L.HasSupport ∧ ((Function.mulSupport fun i ↦ f i) ∩ L.support).Finite
h✝ : Function.HasFiniteMulSupport ((L.support.mulIndicator fun i ↦ f i) ∘ PLift.down)
x✝¹ : PLift ι
x✝ : x✝¹ ∈ Finite.toFinset h✝
⊢ AEStronglyMeasurable (L.support.mulIndicator (fun i ↦ f i) x✝¹.down) μ
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹¹ : Fintype ι'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁸ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁷ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁶ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁵ : SeminormedAddCommGroup G
inst✝⁴ : NormedSpace 𝕜 G
inst✝³ : SeminormedAddCommGroup G'
inst✝² : NormedSpace 𝕜 G'
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
f : (i : ι) → E i →L[𝕜] E₁ i
i : ι
a : 𝕜
f₀ : E i →L[𝕜] E₁ i
⊢ compContinuousLinearMapL (update f i (a • f₀)) = a • compContinuousLinearMapL (update f i f₀)

Tactic:
ext g x
    change (g fun j ↦ update f i (a • f₀) j <| x j) = a • g fun j ↦ update f i f₀ j (x j)
    convert! g.map_update_smul (fun j ↦ f j (x j)) i a (f₀ (x i)) <;>
      exact apply_update (fun (i : ι) (f : E i →L[𝕜] E₁ i) ↦ f (x i)) f i _ _

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

Goal state before:
pp: 𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : SeminormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
inst✝² : Fintype ι
α : Type u_1
inst✝¹ : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : α ≃ ↑s
inst✝ : DecidablePred fun x ↦ x ∈ s
x : (i : { a // a ∉ s }) → E₁ ↑i
⊢ ∀ (m₂ : α → (i : ι) → E₁ i), ‖((f.iteratedFDerivComponent e) x) m₂‖ ≤ (‖f‖ * ∏ i, ‖x i‖) * ∏ i, ‖m₂ i‖

Tactic:
m

Goal state after:
𝕜 : Type u
ι : Type v
ι' : Type v'
E : ι → Type wE
E₁ : ι → Type wE₁
E' : ι' → Type wE'
G : Type wG
G' : Type wG'
inst✝¹² : Fintype ι'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁹ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁶ : SeminormedAddCommGroup G
inst✝⁵ : NormedSpace 𝕜 G
inst✝⁴ : SeminormedAddCommGroup G'
inst✝³ : NormedSpace 𝕜 G'
inst✝² : Fintype ι
α : Type u_1
inst✝¹ : Fintype α
f : ContinuousMultilinearMap 𝕜 E₁ G
s : Set ι
e : α ≃ ↑s
inst✝ : DecidablePred fun x ↦ x ∈ s
x : (i : { a // a ∉ s }) → E₁ ↑i
m : α → (i : ι) → E₁ i
⊢ ‖((f.iteratedFDerivComponent e) x) m‖ ≤ (‖f‖ * ∏ i, ‖x i‖) * ∏ i, ‖m i‖
Module: Mathlib.Analysis.Normed.Operator.Extend

Goal state before:
pp: 𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : NontriviallyNormedField 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedAddCommGroup Eₗ
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace 𝕜 Eₗ
inst✝² : NormedSpace 𝕜₂ F
inst✝¹ : CompleteSpace F
f : E →SL[σ₁₂] F
e : E →L[𝕜] Eₗ
N : ℝ≥0
inst✝ : RingHomIsometric σ₁₂
h_dense : DenseRange ⇑e
h_e : ∀ (x : E), ‖x‖ ≤ ↑N * ‖e x‖
x : E
⊢ ‖f‖ * (↑N * ‖e x‖) ≤ ↑N * ‖f‖ * ‖e x‖

Tactic:
rw [mul_comm ↑N ‖f‖, mul_assoc]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
ι : Type u_6
f : α → G
hfi : AEStronglyMeasurable f μ
F : ι → α → G
l : Filter ι
hFi : ∀ᶠ (i : ι) in l, Integrable (F i) μ
hF : Tendsto (fun i ↦ eLpNorm (F i - f) 1 μ) l (𝓝 0)
s : Set α
⊢ Tendsto (fun i ↦ ∫ (x : α) in s, F i x ∂μ) l (𝓝 (∫ (x : α) in s, f x ∂μ))

Tactic:
refine tendsto_setIntegral_of_L1 f hfi hFi ?_ s
  simp_rw [eLpNorm_one_eq_lintegral_enorm, Pi.sub_apply] at hF
  exact hF

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

Goal state before:
pp: case neg
α : Type u_1
F : Type u_3
F' : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
T : Set α → F →L[ℝ] F'
hT_empty : T ∅ = 0
m : MeasurableSpace α
s : Set α
hs : MeasurableSet s
x : F
hs_empty : s.Nonempty
hs_univ : s ≠ Set.univ
hx0 : ¬x = 0
⊢ x ∉ {0}

pp: case neg
α : Type u_1
F : Type u_3
F' : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
T : Set α → F →L[ℝ] F'
hT_empty : T ∅ = 0
m : MeasurableSpace α
s : Set α
hs : MeasurableSet s
x : F
hs_empty : s.Nonempty
hs_univ : s ≠ Set.univ
hx0 : ¬x = 0
⊢ (T (⇑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x})) x +
      ∑ x_1 ∈ {0}, (T (⇑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x_1})) x_1 =
    (T s) x

Tactic:
· rw [Finset.mem_singleton]; exact hx0

Goal state after:
case neg
α : Type u_1
F : Type u_3
F' : Type u_4
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
inst✝¹ : NormedAddCommGroup F'
inst✝ : NormedSpace ℝ F'
T : Set α → F →L[ℝ] F'
hT_empty : T ∅ = 0
m : MeasurableSpace α
s : Set α
hs : MeasurableSet s
x : F
hs_empty : s.Nonempty
hs_univ : s ≠ Set.univ
hx0 : ¬x = 0
⊢ (T (⇑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x})) x +
      ∑ x_1 ∈ {0}, (T (⇑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x_1})) x_1 =
    (T s) x
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
inst✝ : NormedAddCommGroup β
u : α → β
l : Filter α
c : β
hu : u ~[l] const α c
h : ¬c = 0
⊢ Tendsto u l (𝓝 c)

Tactic:
exact (isEquivalent_const_iff_tendsto h).mp hu

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
inst✝ : ClosedIciTopology E
f : α → E
hf : f ≤ᵐ[μ] 0
⊢ ∫ (x : α), f x ∂μ ≤ 0

Tactic:
← neg_nonneg,

Goal state after:
α : Type u_1
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
inst✝ : ClosedIciTopology E
f : α → E
hf : f ≤ᵐ[μ] 0
⊢ 0 ≤ -∫ (x : α), f x ∂μ
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : α → ℝ
hf : 0 ≤ᵐ[μ] f
hfi : Integrable f μ
⊢ AEMeasurable (fun a ↦ ENNReal.ofReal (f a)) μ

Tactic:
· exact (ENNReal.measurable_ofReal.comp_aemeasurable hfi.1.aemeasurable)

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf : ∀ (n : ℕ), Integrable (f n) μ
hF : Integrable F μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
f' : ℕ → α → ℝ := fun n x ↦ f n x - f 0 x
hf'_nonneg : ∀ (i : ℕ), ∀ᵐ (a : α) ∂μ, 0 ≤ f' i a
hf'_meas : ∀ (n : ℕ), Integrable (f' n) μ
hF_ge : 0 ≤ᵐ[μ] fun x ↦ (F - f 0) x
h_cont : ContinuousAt ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal ((F - f 0) a) ∂μ)
x : α
hx : Monotone fun n ↦ f n x
n : ℕ
⊢ ∀ ⦃b : ℕ⦄, n ≤ b → (fun n ↦ ENNReal.ofReal (f' n x)) n ≤ (fun n ↦ ENNReal.ofReal (f' n x)) b

Tactic:
m

Goal state after:
α : Type u_1
m✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf : ∀ (n : ℕ), Integrable (f n) μ
hF : Integrable F μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
f' : ℕ → α → ℝ := fun n x ↦ f n x - f 0 x
hf'_nonneg : ∀ (i : ℕ), ∀ᵐ (a : α) ∂μ, 0 ≤ f' i a
hf'_meas : ∀ (n : ℕ), Integrable (f' n) μ
hF_ge : 0 ≤ᵐ[μ] fun x ↦ (F - f 0) x
h_cont : ContinuousAt ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal ((F - f 0) a) ∂μ)
x : α
hx : Monotone fun n ↦ f n x
n m : ℕ
⊢ n ≤ m → (fun n ↦ ENNReal.ofReal (f' n x)) n ≤ (fun n ↦ ENNReal.ofReal (f' n x)) m
Module: Mathlib.Topology.Algebra.ContinuousAffineEquiv

Goal state before:
pp: k : Type u_1
P₁ : Type u_2
P₂ : Type u_3
V₁ : Type u_6
V₂ : Type u_7
inst✝⁸ : Ring k
inst✝⁷ : AddCommGroup V₁
inst✝⁶ : Module k V₁
inst✝⁵ : AddTorsor V₁ P₁
inst✝⁴ : TopologicalSpace P₁
inst✝³ : AddCommGroup V₂
inst✝² : Module k V₂
inst✝¹ : AddTorsor V₂ P₂
inst✝ : TopologicalSpace P₂
e e' : P₁ ≃ᴬ[k] P₂
h : e.toContinuousAffineMap = e'.toContinuousAffineMap
p : P₁
⊢ e p = e' p

Tactic:
← coe_toContinuousAffineMap,

Goal state after:
k : Type u_1
P₁ : Type u_2
P₂ : Type u_3
V₁ : Type u_6
V₂ : Type u_7
inst✝⁸ : Ring k
inst✝⁷ : AddCommGroup V₁
inst✝⁶ : Module k V₁
inst✝⁵ : AddTorsor V₁ P₁
inst✝⁴ : TopologicalSpace P₁
inst✝³ : AddCommGroup V₂
inst✝² : Module k V₂
inst✝¹ : AddTorsor V₂ P₂
inst✝ : TopologicalSpace P₂
e e' : P₁ ≃ᴬ[k] P₂
h : e.toContinuousAffineMap = e'.toContinuousAffineMap
p : P₁
⊢ e.toContinuousAffineMap p = e'.toContinuousAffineMap p
Module: Mathlib.Topology.Algebra.ContinuousAffineMap

Goal state before:
pp: R : Type u_1
V : Type u_2
W : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module R V
inst✝⁵ : TopologicalSpace V
inst✝⁴ : AddCommGroup W
inst✝³ : Module R W
inst✝² : TopologicalSpace W
inst✝¹ : IsTopologicalAddGroup V
inst✝ : IsTopologicalAddGroup W
f : V →ᵃ[R] W
h : Continuous f.toFun
⊢ (⇑f.linear + fun x ↦ f 0) = ⇑f.linear + Function.const V (f 0)

Tactic:
← Function.const_def

Goal state after:
R : Type u_1
V : Type u_2
W : Type u_3
inst✝⁸ : Ring R
inst✝⁷ : AddCommGroup V
inst✝⁶ : Module R V
inst✝⁵ : TopologicalSpace V
inst✝⁴ : AddCommGroup W
inst✝³ : Module R W
inst✝² : TopologicalSpace W
inst✝¹ : IsTopologicalAddGroup V
inst✝ : IsTopologicalAddGroup W
f : V →ᵃ[R] W
h : Continuous f.toFun
⊢ (⇑f.linear + fun x ↦ f 0) = ⇑f.linear + fun x ↦ f 0
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf : ∀ (n : ℕ), Integrable (f n) μ
hF : Integrable F μ
h_mono : ∀ᵐ (x : α) ∂μ, Antitone fun n ↦ f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
x : α
hx : Antitone fun n ↦ f n x
n : ℕ
⊢ ∀ ⦃b : ℕ⦄, n ≤ b → (fun n ↦ -f n x) n ≤ (fun n ↦ -f n x) b

Tactic:
m

Goal state after:
α : Type u_1
m✝ : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf : ∀ (n : ℕ), Integrable (f n) μ
hF : Integrable F μ
h_mono : ∀ᵐ (x : α) ∂μ, Antitone fun n ↦ f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
x : α
hx : Antitone fun n ↦ f n x
n m : ℕ
⊢ n ≤ m → (fun n ↦ -f n x) n ≤ (fun n ↦ -f n x) m
Module: Mathlib.MeasureTheory.Integral.Bochner.Basic

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf_int : ∀ (n : ℕ), Integrable (f n) μ
hF_int : Integrable F μ
hf_tendsto : Tendsto (fun i ↦ ∫ (a : α), f i a ∂μ) atTop (𝓝 (∫ (a : α), F a ∂μ))
hf_mono : ∀ᵐ (a : α) ∂μ, Monotone fun i ↦ f i a
hf_bound : ∀ᵐ (a : α) ∂μ, ∀ (i : ℕ), f i a ≤ F a
f' : ℕ → α → ℝ≥0∞ := fun n a ↦ ENNReal.ofReal (f n a - f 0 a)
F' : α → ℝ≥0∞ := fun a ↦ ENNReal.ofReal (F a - f 0 a)
hf'_int_eq : ∀ (i : ℕ), ∫⁻ (a : α), f' i a ∂μ = ENNReal.ofReal (∫ (a : α), f i a ∂μ - ∫ (a : α), f 0 a ∂μ)
hF'_int_eq : ∫⁻ (a : α), F' a ∂μ = ENNReal.ofReal (∫ (a : α), F a ∂μ - ∫ (a : α), f 0 a ∂μ)
h_tendsto : Tendsto (fun i ↦ ∫⁻ (a : α), f' i a ∂μ) atTop (𝓝 (∫⁻ (a : α), F' a ∂μ))
a : α
ha_mono : Monotone fun i ↦ f i a
i j : ℕ
hij : i ≤ j
⊢ f i a - f 0 a ≤ f j a - f 0 a

Tactic:
simp [ha_mono hij]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.RieszLemma

Goal state before:
pp: case inl
F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : NormedSpace ℝ F
x : F
s : Set F
hx : x ∈ s
h₀ : infDist x sᶜ = 0
⊢ closedBall x (infDist x sᶜ) ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s

Tactic:
rw [h₀, closedBall_zero']
    exact closure_mono (singleton_subset_iff.2 hx)

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.SumMeasure

Goal state before:
pp: ι : Type u_1
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝ : NormedAddCommGroup E
μ : ι → Measure X
f : X → E
hf : Integrable f (Measure.sum μ)
i : ι
⊢ ∫ (x : X), ‖f x‖ ∂μ i = (∫⁻ (x : X), ‖f x‖ₑ ∂μ i).toReal

Tactic:
rw [← integral_toReal ?_ (by simp)]
    · simp
    · exact (hf.aestronglyMeasurable.mono_measure (Measure.le_sum _ i)).enorm

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.SumMeasure

Goal state before:
pp: ι : Type u_1
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
μ : ι → Measure X
f : X → E
inst✝ : NormedSpace ℝ E
hfi : ∀ (i : ι), Integrable f (μ i)
ε : ℝ≥0
ε0 : 0 < ↑ε
hf_lt : ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < ∞
hmem : ∀ᶠ (y : ℝ≥0∞) in 𝓝 (∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ), ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < y + ↑ε
s : Finset ι
ν : Measure X
hs : ∫⁻ (x : X), ‖f x‖ₑ ∂(∑ i ∈ s, μ i + ν) < ∑ b ∈ s, (fun i ↦ ∫⁻ (a : X), ‖f a‖ₑ ∂μ i) b + ↑ε
hf : Integrable f (∑ i ∈ s, μ i) ∧ Integrable f ν
hν : ∑ i ∈ s, μ i + ν = Measure.sum μ
⊢ ∫⁻ (x : X), ‖f x‖ₑ ∂ν < ↑ε

Tactic:
lintegral_add_measure,

Goal state after:
ι : Type u_1
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
μ : ι → Measure X
f : X → E
inst✝ : NormedSpace ℝ E
hfi : ∀ (i : ι), Integrable f (μ i)
ε : ℝ≥0
ε0 : 0 < ↑ε
hf_lt : ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < ∞
hmem : ∀ᶠ (y : ℝ≥0∞) in 𝓝 (∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ), ∫⁻ (x : X), ‖f x‖ₑ ∂Measure.sum μ < y + ↑ε
s : Finset ι
ν : Measure X
hs : ∫⁻ (a : X), ‖f a‖ₑ ∂∑ i ∈ s, μ i + ∫⁻ (a : X), ‖f a‖ₑ ∂ν < ∑ b ∈ s, (fun i ↦ ∫⁻ (a : X), ‖f a‖ₑ ∂μ i) b + ↑ε
hf : Integrable f (∑ i ∈ s, μ i) ∧ Integrable f ν
hν : ∑ i ∈ s, μ i + ν = Measure.sum μ
⊢ ∫⁻ (x : X), ‖f x‖ₑ ∂ν < ↑ε
Module: Mathlib.MeasureTheory.Integral.Bochner.SumMeasure

Goal state before:
pp: X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝⁴ : NormedAddCommGroup E
f : X → E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSingletonClass X
μ : Measure X
inst✝ : Countable X
hf : Integrable f μ
⊢ ∫ (x : X), f x ∂μ = ∑' (x : X), μ.real {x} • f x

Tactic:
rw [← Measure.sum_smul_dirac μ] at hf
  rw [← Measure.sum_smul_dirac μ, integral_sum_measure hf]
  congr 1 with a : 1
  rw [integral_smul_measure, integral_dirac, Measure.sum_smul_dirac, measureReal_def]

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.SumMeasure

Goal state before:
pp: X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝⁴ : NormedAddCommGroup E
f : X → E
inst✝³ : NormedSpace ℝ E
inst✝² : CompleteSpace E
inst✝¹ : MeasurableSingletonClass X
μ : Measure X
inst✝ : Fintype X
hf : Integrable f μ
⊢ ∫ (x : X), f x ∂μ = ∑ x, μ.real {x} • f x

Tactic:
rw [← setIntegral_finset .univ, Finset.coe_univ, Measure.restrict_univ]
  simp [Finset.coe_univ, hf]

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: α : Type u
β : Type v
inst✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
f g : α →ᵇ β
C : ℝ
inst✝ : CompactSpace α
C0 : 0 < C
h : ¬Nonempty α
⊢ sInf {C | 0 ≤ C ∧ ∀ (x : α), dist (f x) (g x) ≤ C} ≤ 0

Tactic:
exact csInf_le ⟨0, fun C => And.left⟩ ⟨le_rfl, fun x => False.elim (h (Nonempty.intro x))⟩

Goal state after:
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁸ : NormedAddCommGroup G''
inst✝⁷ : PartialOrder G''
inst✝⁶ : IsOrderedAddMonoid G''
inst✝⁵ : NormedSpace ℝ G''
inst✝⁴ : NormedAddCommGroup G'
inst✝³ : PartialOrder G'
inst✝² : NormedSpace ℝ G'
inst✝¹ : ClosedIciTopology G''
inst✝ : IsOrderedAddMonoid G'
T : Set α → G' →L[ℝ] G''
C : ℝ
hT : DominatedFinMeasAdditive μ T C
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f g : α → G'
hf : Integrable f μ
hg : Integrable g μ
hfg : f ≤ᵐ[μ] g
a : α
ha : f a ≤ g a
⊢ f a ≤ g a

Tactic:
exact ha

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type w
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
inst✝ : CompleteSpace 𝕜
n : ℕ
⊢ IsOpen {f | ↑n ≤ (↑f).rank}

Tactic:
by
  simp only [LinearMap.le_rank_iff_exists_linearIndependent_finset, ofPred_exists, ← exists_prop]
  refine isOpen_biUnion fun t _ => ?_
  have : Continuous fun f : E →L[𝕜] F => fun x : (t : Set E) => f x :=
    continuous_pi fun x => (ContinuousLinearMap.apply 𝕜 F (x : E)).continuous
  exact isOpen_setOfPred_linearIndependent.preimage this

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u
β : Type v
γ : Type w
inst✝¹ : TopologicalSpace α
inst✝ : SeminormedAddCommGroup β
f✝ g✝ : α →ᵇ β
x : α
C : ℝ
f g : α →ᵇ β
⊢ dist f g = ‖-f + g‖

Tactic:
simp only [norm_eq, dist_eq, dist_eq_norm_neg_add, add_apply, neg_apply]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
E : Type v
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
h : ¬FiniteDimensional 𝕜 E
s : Finset E
F : Submodule 𝕜 E := Submodule.span 𝕜 ↑s
hF : F.FG
this✝ : FiniteDimensional 𝕜 ↥F
Fclosed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
this : ∃ x, x ∉ F
x : E
xR : ‖x‖ ≤ R
hx : ∀ y ∈ F, 1 ≤ ‖x - y‖
⊢ ∃ x, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖

Tactic:
have hx' : ∀ y : E, y ∈ F → 1 ≤ ‖y - x‖ := by
    intro y hy
    rw [← norm_neg]
    simpa using hx y hy

Goal state after:
𝕜 : Type u
inst✝³ : NontriviallyNormedField 𝕜
E : Type v
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace 𝕜 E
inst✝ : CompleteSpace 𝕜
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
h : ¬FiniteDimensional 𝕜 E
s : Finset E
F : Submodule 𝕜 E := Submodule.span 𝕜 ↑s
hF : F.FG
this✝ : FiniteDimensional 𝕜 ↥F
Fclosed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
this : ∃ x, x ∉ F
x : E
xR : ‖x‖ ≤ R
hx : ∀ y ∈ F, 1 ≤ ‖x - y‖
hx' : ∀ y ∈ F, 1 ≤ ‖y - x‖
⊢ ∃ x, ‖x‖ ≤ R ∧ ∀ y ∈ s, 1 ≤ ‖y - x‖
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u
inst✝² : TopologicalSpace α
R : Type u_1
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
f g : α →ᵇ R
h : f * g = 0
hfg : ∀ (x : α), f x = 0 ∨ g x = 0
hfg' : ∀ (x : α), ‖(f + g) x‖ = max ‖f x‖ ‖g x‖
key : ∀ (c : ℝ), 0 ≤ c → (‖f + g‖ ≤ c ↔ max ‖f‖ ‖g‖ ≤ c)
⊢ ‖f + g‖ ≤ max ‖f‖ ‖g‖

Tactic:
by rw [key]; positivity

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Bounded.Normed

Goal state before:
pp: α : Type u
inst✝² : TopologicalSpace α
R : Type u_1
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
ι : Type u_2
f : ι → α →ᵇ R
s : Finset ι
h : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)
⊢ ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊

Tactic:
by
  classical
  induction s using Finset.induction_on with
  | empty => simp
  | insert j s hj ih =>
    suffices f j * ∑ i ∈ s, f i = 0 by simpa [hj, ← ih] using nnnorm_add_eq_max this
    simpa [Finset.mul_sum] using Finset.sum_eq_zero fun i hi ↦ h (by grind)

Goal state after:
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
R : Type u_4
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
ι : Type u_5
f : ι → C(α, R)
s : Finset ι
h : Pairwise ((fun x1 x2 ↦ x1 * x2 = 0) on f)
⊢ ‖∑ i ∈ s, f i‖₊ = s.sup fun x ↦ ‖f x‖₊

Tactic:
by
  classical
  induction s using Finset.induction_on with
  | empty => simp
  | insert j s hj ih =>
    suffices f j * ∑ i ∈ s, f i = 0 by simpa [hj, ← ih] using nnnorm_add_eq_max this
    simpa [Finset.mul_sum] using Finset.sum_eq_zero fun i hi ↦ h (by grind)

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
⊢ Continuous[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace, _] (thickenedIndicatorAux δ E)

Tactic:
unfold thickenedIndicatorAux
  let f := fun x : α => (⟨1, infEDist x E / ENNReal.ofReal δ⟩ : ℝ≥0 × ℝ≥0∞)
  let sub := fun p : ℝ≥0 × ℝ≥0∞ => (p.1 : ℝ≥0∞) - p.2
  rw [show (fun x : α => (1 : ℝ≥0∞) - infEDist x E / ENNReal.ofReal δ) = sub ∘ f by rfl]
  apply (@ENNReal.continuous_nnreal_sub 1).comp
  apply (ENNReal.continuous_div_const (ENNReal.ofReal δ) _).comp continuous_infEDist
  norm_num [δ_pos]

Goal state after: