mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
inst✝⁷ : CompleteSpace 𝕜
n : ℕ
IH :
  ∀ {E : Type v} [inst : AddCommGroup E] [inst_1 : Module 𝕜 E] [inst_2 : TopologicalSpace E]
    [inst_3 : IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E] [T2Space E] {ι : Type v} [inst_6 : Finite ι]
    (ξ : Basis ι 𝕜 E) (this : Fintype ι),
    let this_1 := IsTopologicalAddGroup.rightUniformSpace E;
    let this_2 := ⋯;
    Fintype.card ι = n → Continuous[_, Pi.topologicalSpace] ⇑ξ.equivFun
E : Type v
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : T2Space E
ι : Type v
inst✝ : Finite ι
ξ : Basis ι 𝕜 E
this✝² : Fintype ι
this✝¹ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this✝ : IsUniformAddGroup E := isUniformAddGroup_of_addCommGroup
hn : Fintype.card ι = n + 1
this : FiniteDimensional 𝕜 E
s : Submodule 𝕜 E
⊢ finrank 𝕜 ↥s = n → IsClosed[inst✝⁴] ↑s

Tactic:
s_dim

Goal state after:
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
inst✝⁷ : CompleteSpace 𝕜
n : ℕ
IH :
  ∀ {E : Type v} [inst : AddCommGroup E] [inst_1 : Module 𝕜 E] [inst_2 : TopologicalSpace E]
    [inst_3 : IsTopologicalAddGroup E] [ContinuousSMul 𝕜 E] [T2Space E] {ι : Type v} [inst_6 : Finite ι]
    (ξ : Basis ι 𝕜 E) (this : Fintype ι),
    let this_1 := IsTopologicalAddGroup.rightUniformSpace E;
    let this_2 := ⋯;
    Fintype.card ι = n → Continuous[_, Pi.topologicalSpace] ⇑ξ.equivFun
E : Type v
inst✝⁶ : AddCommGroup E
inst✝⁵ : Module 𝕜 E
inst✝⁴ : TopologicalSpace E
inst✝³ : IsTopologicalAddGroup E
inst✝² : ContinuousSMul 𝕜 E
inst✝¹ : T2Space E
ι : Type v
inst✝ : Finite ι
ξ : Basis ι 𝕜 E
this✝² : Fintype ι
this✝¹ : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
this✝ : IsUniformAddGroup E := isUniformAddGroup_of_addCommGroup
hn : Fintype.card ι = n + 1
this : FiniteDimensional 𝕜 E
s : Submodule 𝕜 E
s_dim : finrank 𝕜 ↥s = n
⊢ IsClosed[inst✝⁴] ↑s
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
f : MultilinearMap 𝕜 E G
C : ℝ
hC : 0 ≤ C
H : ∀ (m : (i : ι) → E i), ‖f m‖ ≤ C * ∏ i, ‖m i‖
m₁ m₂ : (i : ι) → E i
i : ι
⊢ ‖m₁ - m₂‖ * ∏ x ∈ univ \ {i}, max ‖m₁‖ ‖m₂‖ = ‖m₁ - m₂‖ * max ‖m₁‖ ‖m₂‖ ^ (Fintype.card ι - 1)

Tactic:
simp [card_univ_sdiff]

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

Goal state before:
pp: α : Type u_1
E : Type u_5
mα : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : TopologicalSpace α
inst✝² : SecondCountableTopologyEither α E
inst✝¹ : OpensMeasurableSpace α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
f : α → E
hf : Continuous[inst✝³, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] f
a : α
⊢ IntegrableAtFilter f (𝓝 a) μ

Tactic:
by
  rw [← nhdsWithin_univ]
  exact hf.continuousOn.integrableAt_nhdsWithin MeasurableSet.univ (mem_univ a)

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

Goal state before:
pp: 𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
n : ℕ
A : Type u_1
inst✝¹ : SeminormedRing A
inst✝ : NormedAlgebra 𝕜 A
m : Fin n.succ → A
⊢ ‖(List.map m (List.finRange n.succ)).prod‖ ≤ (List.map (fun i ↦ ‖m i‖) (List.finRange n.succ)).prod

Tactic:
refine (List.norm_prod_le' ?_).trans_eq ?_

Goal state after:
case refine_1
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
n : ℕ
A : Type u_1
inst✝¹ : SeminormedRing A
inst✝ : NormedAlgebra 𝕜 A
m : Fin n.succ → A
⊢ List.map m (List.finRange n.succ) ≠ []

case refine_2
𝕜 : Type u
inst✝² : NontriviallyNormedField 𝕜
n : ℕ
A : Type u_1
inst✝¹ : SeminormedRing A
inst✝ : NormedAlgebra 𝕜 A
m : Fin n.succ → A
⊢ (List.map norm (List.map m (List.finRange n.succ))).prod = (List.map (fun i ↦ ‖m i‖) (List.finRange n.succ)).prod
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: case pos
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
inst✝⁴ : PartialOrder α
inst✝³ : MeasurableSingletonClass α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
inst✝ : PseudoMetrizableSpace ε'
f : α → ε'
μ : Measure α
a b : α
hb : μ {b} ≠ ∞
hb' : ‖f b‖ₑ ≠ ∞
hab : a < b
⊢ IntegrableOn f (Ioc a b) μ ↔ IntegrableOn f (Ioo a b) μ

Tactic:
rw [← Ioo_union_right hab, integrableOn_union, eq_true (integrableOn_singleton hb'), and_true]

Goal state after:
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : CompleteSpace 𝕜
inst✝⁷ : AddCommGroup F
inst✝⁶ : TopologicalSpace F
inst✝⁵ : IsTopologicalAddGroup F
inst✝⁴ : Module 𝕜 F
inst✝³ : ContinuousSMul 𝕜 F
E : Type u_4
inst✝² : AddCommGroup E
inst✝¹ : Module 𝕜 E
f : E →ₗ[𝕜] F
s : Submodule 𝕜 E
inst✝ : s.CoFG
h : IsClosed[inst✝⁶] ↑(Submodule.map f s)
⊢ IsClosed[inst✝⁶] ↑f.range

Tactic:
by
  obtain ⟨t, s_compl_t⟩ := Submodule.exists_isCompl s
  have : FiniteDimensional 𝕜 t := .of_fg <| Submodule.CoFG.fg_of_isCompl s_compl_t inferInstance
  rw [← Submodule.map_top, ← s_compl_t.sup_eq_top, Submodule.map_sup]
  exact Submodule.isClosed_sup_finiteDimensional _ _ h

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

Goal state before:
pp: X : Type u_4
E✝ : Type u_5
ι : Type u_6
inst✝⁷ : MeasurableSpace X
inst✝⁶ : CommMonoid E✝
inst✝⁵ : TopologicalSpace E✝
inst✝⁴ : ContinuousMul E✝
L : SummationFilter ι
inst✝³ : L.NeBot
inst✝² : L.filter.IsCountablyGenerated
inst✝¹ : IsCompletelyPseudoMetrizableSpace E✝
inst✝ : Countable ι
f : ι → X → E✝
h : ∀ (i : ι), StronglyMeasurable (f i)
E : Set X := {x | Multipliable (fun x_1 ↦ f x_1 x) L}
⊢ StronglyMeasurable fun x ↦ ∏'[L] (i : ι), f i x

Tactic:
have hE : MeasurableSet E := StronglyMeasurable.measurableSet_exists_tendsto (by fun_prop)

Goal state after:
X : Type u_4
E✝ : Type u_5
ι : Type u_6
inst✝⁷ : MeasurableSpace X
inst✝⁶ : CommMonoid E✝
inst✝⁵ : TopologicalSpace E✝
inst✝⁴ : ContinuousMul E✝
L : SummationFilter ι
inst✝³ : L.NeBot
inst✝² : L.filter.IsCountablyGenerated
inst✝¹ : IsCompletelyPseudoMetrizableSpace E✝
inst✝ : Countable ι
f : ι → X → E✝
h : ∀ (i : ι), StronglyMeasurable (f i)
E : Set X := {x | Multipliable (fun x_1 ↦ f x_1 x) L}
hE : MeasurableSet E
⊢ StronglyMeasurable fun x ↦ ∏'[L] (i : ι), f i x
Module: Mathlib.MeasureTheory.Function.L1Space.AEEqFun

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : α → β
hf : Integrable f μ
hg : Integrable g μ
⊢ edist (toL1 f hf) (toL1 g hg) = ∫⁻ (a : α), edist (f a) (g a) ∂μ

Tactic:
simp only [toL1, Lp.edist_toLp_toLp, eLpNorm, one_ne_zero, eLpNorm'_eq_lintegral_enorm,
    Pi.sub_apply, toReal_one, ENNReal.rpow_one, ne_eq, not_false_eq_true, div_self, ite_false]

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup β
f g : α → β
hf : Integrable f μ
hg : Integrable g μ
⊢ (if 1 = ∞ then eLpNormEssSup (f - g) μ else ∫⁻ (a : α), ‖f a - g a‖ₑ ∂μ) = ∫⁻ (a : α), edist (f a) (g a) ∂μ
Module: Mathlib.MeasureTheory.Function.LocallyIntegrable

Goal state before:
pp: X : Type u_1
E : Type u_6
F : Type u_7
inst✝³ : MeasurableSpace X
inst✝² : TopologicalSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedAddCommGroup F
μ : Measure X
f : X → E
hf : LocallyIntegrable f μ
g : X → F
hg : AEStronglyMeasurable g μ
h : ∀ᵐ (x : X) ∂μ, ‖g x‖ ≤ ‖f x‖
⊢ LocallyIntegrable g μ

Tactic:
by
  rw [← locallyIntegrableOn_univ] at hf ⊢
  exact hf.mono hg h

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

Goal state before:
pp: case mpr
X : Type u_1
ε : Type u_3
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace ε
inst✝⁵ : ContinuousENorm ε
f : X → ε
μ : Measure X
inst✝⁴ : PseudoMetrizableSpace ε
a : X
inst✝³ : LinearOrder X
inst✝² : CompactIccSpace X
inst✝¹ : NoMinOrder X
inst✝ : OrderTopology X
hbot : IntegrableAtFilter f atBot μ
s : Set X
hsl : s ∈ 𝓝[<] a
hs : IntegrableOn f s μ
hlocal : LocallyIntegrableOn f (Iio a) μ
⊢ IntegrableOn f (Iio a) μ

Tactic:
obtain ⟨s', ⟨hs'_mono, hs'⟩⟩ := mem_nhdsLT_iff_exists_Ioo_subset.mp hsl

Goal state after:
case mpr
X : Type u_1
ε : Type u_3
inst✝⁸ : MeasurableSpace X
inst✝⁷ : TopologicalSpace X
inst✝⁶ : TopologicalSpace ε
inst✝⁵ : ContinuousENorm ε
f : X → ε
μ : Measure X
inst✝⁴ : PseudoMetrizableSpace ε
a : X
inst✝³ : LinearOrder X
inst✝² : CompactIccSpace X
inst✝¹ : NoMinOrder X
inst✝ : OrderTopology X
hbot : IntegrableAtFilter f atBot μ
s : Set X
hsl : s ∈ 𝓝[<] a
hs : IntegrableOn f s μ
hlocal : LocallyIntegrableOn f (Iio a) μ
s' : X
hs'_mono : s' ∈ Iio a
hs' : Ioo s' a ⊆ s
⊢ IntegrableOn f (Iio a) μ
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✝¹⁷ : AddCommGroup E
inst✝¹⁶ : UniformSpace E
inst✝¹⁵ : IsUniformAddGroup E
inst✝¹⁴ : AddCommGroup F
inst✝¹³ : UniformSpace F
inst✝¹² : IsUniformAddGroup F
inst✝¹¹ : T0Space F
inst✝¹⁰ : AddCommMonoid Eₗ
inst✝⁹ : UniformSpace Eₗ
inst✝⁸ : ContinuousAdd Eₗ
inst✝⁷ : Semiring 𝕜
inst✝⁶ : Semiring 𝕜₂
inst✝⁵ : Module 𝕜 E
inst✝⁴ : Module 𝕜₂ F
inst✝³ : Module 𝕜 Eₗ
inst✝² : ContinuousConstSMul 𝕜 Eₗ
inst✝¹ : ContinuousConstSMul 𝕜₂ F
σ₁₂ : 𝕜 →+* 𝕜₂
f : E →SL[σ₁₂] F
inst✝ : CompleteSpace F
e : E →L[𝕜] Eₗ
h_dense : DenseRange ⇑e
h_e : IsUniformInducing ⇑e
g : Eₗ →SL[σ₁₂] F
H : g ∘SL e = f
⊢ f.extend e = g

Tactic:
by
  simp only [extend, h_dense, h_e, and_self, ↓reduceDIte]
  exact ContinuousLinearMap.coeFn_injective <|
    uniformly_extend_unique h_e h_dense (ContinuousLinearMap.ext_iff.1 H) g.continuous

Goal state after:
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
s t : Set α
hst : μ s = μ t
⊢ weightedSMul μ s = weightedSMul μ t

Tactic:
ext1 x; simp_rw [weightedSMul_apply, measureReal_def]; congr 2

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

Goal state before:
pp: 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'
m : MeasurableSpace α
T T' : Set α → F →L[ℝ] F'
f : α →ₛ F
⊢ ∑ x ∈ f.range, ((T + T') (⇑f ⁻¹' {x})) x = ∑ x ∈ f.range, (T (⇑f ⁻¹' {x})) x + ∑ x ∈ f.range, (T' (⇑f ⁻¹' {x})) x

Tactic:
Pi.add_apply

Goal state after:
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'
m : MeasurableSpace α
T T' : Set α → F →L[ℝ] F'
f : α →ₛ F
⊢ ∑ x ∈ f.range, (T (⇑f ⁻¹' {x}) + T' (⇑f ⁻¹' {x})) x =
    ∑ x ∈ f.range, (T (⇑f ⁻¹' {x})) x + ∑ x ∈ f.range, (T' (⇑f ⁻¹' {x})) x
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedSpace ℝ E
f : ↥(α →₁ₛ[μ] E)
⊢ ‖MeasureTheory.SimpleFunc.integral μ (toSimpleFunc f)‖ ≤ ‖f‖

Tactic:
norm_eq_integral

Goal state after:
α : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedSpace ℝ E
f : ↥(α →₁ₛ[μ] E)
⊢ ‖MeasureTheory.SimpleFunc.integral μ (toSimpleFunc f)‖ ≤
    MeasureTheory.SimpleFunc.integral μ (SimpleFunc.map norm (toSimpleFunc f))
Module: Mathlib.MeasureTheory.Integral.Bochner.L1

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedSpace ℝ E
f : ↥(α →₁ₛ[μ] E)
⊢ ‖integral f‖ ≤ ‖f‖

Tactic:
rw [integral, norm_eq_integral]

Goal state after:
α : Type u_1
E : Type u_2
inst✝¹ : NormedAddCommGroup E
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedSpace ℝ E
f : ↥(α →₁ₛ[μ] E)
⊢ ‖MeasureTheory.SimpleFunc.integral μ (toSimpleFunc f)‖ ≤
    MeasureTheory.SimpleFunc.integral μ (SimpleFunc.map norm (toSimpleFunc f))
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T T' T'' : Set α → E →L[ℝ] F
h_add : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T'' s = T s + T' s
f : α →ₛ E
hf : Integrable (⇑f) μ
this : ∀ x ∈ {x ∈ f.range | x ≠ 0}, T'' (⇑f ⁻¹' {x}) = T (⇑f ⁻¹' {x}) + T' (⇑f ⁻¹' {x})
x : E
hx : x ∈ {x ∈ f.range | x ≠ 0}
⊢ (⇑(T (⇑f ⁻¹' {x})) + ⇑(T' (⇑f ⁻¹' {x}))) x = (T (⇑f ⁻¹' {x})) x + (T' (⇑f ⁻¹' {x})) x

Tactic:
Pi.add_apply

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T T' T'' : Set α → E →L[ℝ] F
h_add : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T'' s = T s + T' s
f : α →ₛ E
hf : Integrable (⇑f) μ
this : ∀ x ∈ {x ∈ f.range | x ≠ 0}, T'' (⇑f ⁻¹' {x}) = T (⇑f ⁻¹' {x}) + T' (⇑f ⁻¹' {x})
x : E
hx : x ∈ {x ∈ f.range | x ≠ 0}
⊢ (T (⇑f ⁻¹' {x})) x + (T' (⇑f ⁻¹' {x})) x = (T (⇑f ⁻¹' {x})) x + (T' (⇑f ⁻¹' {x})) x
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
F : Type u_3
𝕜 : Type u_6
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
E : Type u_7
inst✝³ : NormedAddCommGroup E
inst✝² : SMulZeroClass 𝕜 E
inst✝¹ : NormedSpace ℝ E
inst✝ : DistribSMul 𝕜 F
T : Set α → E →L[ℝ] F
h_add : FinMeasAdditive μ T
h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x
c : 𝕜
f : α →ₛ E
hf : Integrable (⇑f) μ
b : E
x✝ : b ∈ f.range
⊢ (T (⇑f ⁻¹' {b})) (c • b) = c • (T (⇑f ⁻¹' {b})) b

Tactic:
h_smul

Goal state after:
α : Type u_1
F : Type u_3
𝕜 : Type u_6
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
E : Type u_7
inst✝³ : NormedAddCommGroup E
inst✝² : SMulZeroClass 𝕜 E
inst✝¹ : NormedSpace ℝ E
inst✝ : DistribSMul 𝕜 F
T : Set α → E →L[ℝ] F
h_add : FinMeasAdditive μ T
h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x
c : 𝕜
f : α →ₛ E
hf : Integrable (⇑f) μ
b : E
x✝ : b ∈ f.range
⊢ c • (T (⇑f ⁻¹' {b})) b = c • (T (⇑f ⁻¹' {b})) b
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
f g : α → G
hf : Integrable f μ
hg : Integrable g μ
⊢ ∫ (a : α), f a - g a ∂μ = ∫ (a : α), f a ∂μ - ∫ (a : α), g a ∂μ

Tactic:
simp only [integral_eq_setToFun]
  exact setToFun_sub (dominatedFinMeasAdditive_weightedSMul μ) hf hg

Goal state after:
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 : P₁ ≃ᴬ[k] P₂
s : Set P₂
⊢ ⇑e.symm '' s = ⇑e ⁻¹' s

Tactic:
by
  rw [e.symm.image_eq_preimage_symm, e.symm_symm]

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

Goal state before:
pp: α : Type u_1
G : Type u_5
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
f : ↥(α →₁ₛ[μ] G)
⊢ ‖f‖ = ∑ x ∈ (toSimpleFunc f).range, μ.real (⇑(toSimpleFunc f) ⁻¹' {x}) * ‖x‖

Tactic:
by
  rw [norm_toSimpleFunc, eLpNorm_one_eq_lintegral_enorm]
  have h_eq := SimpleFunc.map_apply (‖·‖ₑ) (toSimpleFunc f)
  simp_rw [← h_eq, measureReal_def]
  rw [SimpleFunc.lintegral_eq_lintegral, SimpleFunc.map_lintegral, ENNReal.toReal_sum]
  · congr
    ext1 x
    rw [ENNReal.toReal_mul, mul_comm, ← ofReal_norm,
      ENNReal.toReal_ofReal (norm_nonneg _)]
  · intro x _
    by_cases hx0 : x = 0
    · rw [hx0]; simp
    · finiteness [SimpleFunc.measure_preimage_lt_top_of_integrable _ (SimpleFunc.integrable f) hx0]

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

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
h_zero : ∀ (s : Set α), MeasurableSet s → μ s = 0 → T s = 0
h_add : FinMeasAdditive μ T
f g : ↥(α →₁ₛ[μ] E)
⊢ setToL1S T (f - g) = setToL1S T f - setToL1S T g

Tactic:
rw [sub_eq_add_neg, setToL1S_add T h_zero h_add, setToL1S_neg h_zero h_add, sub_eq_add_neg]

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

Goal state before:
pp: case inr
𝕜 : Type u_1
inst✝⁷ : RCLike 𝕜
𝓕 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace 𝕜 F
inst✝² : FunLike 𝓕 E F
inst✝¹ : AddMonoidHomClass 𝓕 E F
inst✝ : MulActionHomClass 𝓕 𝕜 E F
f : 𝓕
K : NNReal
h : ∀ (x : E), ‖x‖ = 1 → 1 ≤ ↑K * ‖f x‖
x : E
hx : x ≠ 0
⊢ ‖x‖ ≤ ↑K * ‖f x‖

Tactic:
simpa [norm_smul, field] using h ((‖x‖⁻¹ : 𝕜) • x) (norm_smul_inv_norm hx)

Goal state after:
Module: Mathlib.Analysis.Normed.Affine.Isometry

Goal state before:
pp: 𝕜 : Type u_1
V : Type u_2
V₂ : Type u_5
P : Type u_10
P₂ : Type u_11
inst✝⁸ : NormedField 𝕜
inst✝⁷ : SeminormedAddCommGroup V
inst✝⁶ : NormedSpace 𝕜 V
inst✝⁵ : PseudoMetricSpace P
inst✝⁴ : NormedAddTorsor V P
inst✝³ : SeminormedAddCommGroup V₂
inst✝² : NormedSpace 𝕜 V₂
inst✝¹ : PseudoMetricSpace P₂
inst✝ : NormedAddTorsor V₂ P₂
f : P →ᵃⁱ[𝕜] P₂
x y : P
⊢ nndist (f x) (f y) = nndist x y

Tactic:
simp [nndist_dist]

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

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
f : α → E
hT : DominatedFinMeasAdditive μ T C
hf : ¬Integrable f μ
⊢ setToFun μ T hT f = 0

Tactic:
by
  by_cases hF : CompleteSpace F
  · simp [setToFun, hF, hf]
  · simp [setToFun, hF]

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

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
s : Finset ι
f : ι → α → E
hf : ∀ i ∈ s, Integrable (f i) μ
⊢ setToFun μ T hT (∑ i ∈ s, f i) = ∑ i ∈ s, setToFun μ T hT (f i)

Tactic:
classical
  revert hf
  refine Finset.induction_on s ?_ ?_
  · intro _
    simp only [setToFun_zero, Finset.sum_empty]
  · intro i s his ih hf
    simp only [his, Finset.sum_insert, not_false_iff]
    rw [setToFun_add hT (hf i (Finset.mem_insert_self i s)) _]
    · rw [ih fun i hi => hf i (Finset.mem_insert_of_mem hi)]
    · convert! integrable_finsetSum s fun i hi => hf i (Finset.mem_insert_of_mem hi) with x
      simp

Goal state after:
Module: Mathlib.Topology.Bornology.BoundedOperation

Goal state before:
pp: s t : Set ℝ≥0
hs : Bornology.IsBounded s
ht : Bornology.IsBounded t
Af : ℝ
hAf : s ⊆ closedBall 0 Af
Ag : ℝ
hAg : t ⊆ closedBall 0 Ag
key : IsCompact (closedBall 0 Af ×ˢ closedBall 0 Ag)
a✝ x : ℝ≥0
x_in_s : x ∈ s
y : ℝ≥0
y_in_t : y ∈ t
xy_eq : (fun x1 x2 ↦ x1 * x2) x y = a✝
⊢ (x, y) ∈ closedBall 0 Af ×ˢ closedBall 0 Ag ∧ (fun p ↦ p.1 * p.2) (x, y) = a✝

Tactic:
by simpa only [Set.mem_prod] using ⟨⟨hAf x_in_s, hAg y_in_t⟩, xy_eq⟩

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

Goal state before:
pp: case pos
α : Type u_1
G : Type u_5
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
β : Type u_6
inst✝ : MeasurableSpace β
φ : α → β
hφ : Measurable φ
f : β → G
hfm : StronglyMeasurable f
hfi : Integrable f (Measure.map φ μ)
⊢ setToFun (Measure.map φ μ) (weightedSMul (Measure.map φ μ)) ⋯ f = setToFun μ (weightedSMul μ) ⋯ fun a ↦ f (φ a)

Tactic:
apply setToFun_of_le_map_of_stronglyMeasurable _ _
    ((integrable_map_measure hfm.aestronglyMeasurable hφ.aemeasurable).1 hfi) hfm hφ le_rfl

Goal state after:
case pos
α : Type u_1
G : Type u_5
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace ℝ G
m : MeasurableSpace α
μ : Measure α
β : Type u_6
inst✝ : MeasurableSpace β
φ : α → β
hφ : Measurable φ
f : β → G
hfm : StronglyMeasurable f
hfi : Integrable f (Measure.map φ μ)
⊢ ∀ (s : Set β) (x : G), MeasurableSet s → (weightedSMul (Measure.map φ μ) s) x = (weightedSMul μ (φ ⁻¹' s)) x
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: F✝ : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
inst✝¹ : PseudoMetricSpace γ
f✝ g : α →ᵇ β
x : α
C : ℝ
inst✝ : CompleteSpace β
f : ℕ → α →ᵇ β
hf : CauchySeq f
b : ℕ → ℝ
b0 : ∀ (n : ℕ), 0 ≤ b n
b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N
b_lim : Tendsto b atTop (𝓝 0)
f_bdd : ∀ (x : α) (n m N : ℕ), N ≤ n → N ≤ m → dist ((f n) x) ((f m) x) ≤ b N
fx_cau : ∀ (x : α), CauchySeq fun n ↦ (f n) x
F : α → β
hF : ∀ (x : α), Tendsto (fun n ↦ (f n) x) atTop (𝓝 (F x))
fF_bdd : ∀ (x : α) (N : ℕ), dist ((f N) x) (F x) ≤ b N
⊢ TendstoUniformly (fun n x ↦ (f n) x) F atTop

Tactic:
refine Metric.tendstoUniformly_iff.2 fun ε ε0 => ?_

Goal state after:
F✝ : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
inst✝¹ : PseudoMetricSpace γ
f✝ g : α →ᵇ β
x : α
C : ℝ
inst✝ : CompleteSpace β
f : ℕ → α →ᵇ β
hf : CauchySeq f
b : ℕ → ℝ
b0 : ∀ (n : ℕ), 0 ≤ b n
b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N
b_lim : Tendsto b atTop (𝓝 0)
f_bdd : ∀ (x : α) (n m N : ℕ), N ≤ n → N ≤ m → dist ((f n) x) ((f m) x) ≤ b N
fx_cau : ∀ (x : α), CauchySeq fun n ↦ (f n) x
F : α → β
hF : ∀ (x : α), Tendsto (fun n ↦ (f n) x) atTop (𝓝 (F x))
fF_bdd : ∀ (x : α) (N : ℕ), dist ((f N) x) (F x) ≤ b N
ε : ℝ
ε0 : ε > 0
⊢ ∀ᶠ (n : ℕ) in atTop, ∀ (x : α), dist (F x) ((f n) x) < ε
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: case intro
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type v
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CompleteSpace 𝕜
ι : Type u_1
inst✝ : Finite ι
f : ι → E
val✝ : Fintype ι
hf : ((LinearMap.lsum 𝕜 (fun x ↦ 𝕜) ℕ) fun i ↦ LinearMap.id.smulRight (f i)).ker = ⊥
K : ℝ≥0
K0 : K > 0
hK : AntilipschitzWith K ⇑((LinearMap.lsum 𝕜 (fun x ↦ 𝕜) ℕ) fun i ↦ LinearMap.id.smulRight (f i))
this : Tendsto (fun g ↦ ∑ i, ‖g i - f i‖) (𝓝 f) (𝓝 0)
g : ι → E
hg : ∑ i, ‖g i - f i‖₊ < K⁻¹
⊢ Function.Injective ⇑((LinearMap.lsum 𝕜 (fun x ↦ 𝕜) ℕ) fun i ↦ LinearMap.id.smulRight (g i))

Tactic:
refine (hK.add_sub_lipschitzWith (LipschitzWith.of_dist_le_mul fun v u => ?_) hg).injective

Goal state after:
case intro
𝕜 : Type u
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type v
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CompleteSpace 𝕜
ι : Type u_1
inst✝ : Finite ι
f : ι → E
val✝ : Fintype ι
hf : ((LinearMap.lsum 𝕜 (fun x ↦ 𝕜) ℕ) fun i ↦ LinearMap.id.smulRight (f i)).ker = ⊥
K : ℝ≥0
K0 : K > 0
hK : AntilipschitzWith K ⇑((LinearMap.lsum 𝕜 (fun x ↦ 𝕜) ℕ) fun i ↦ LinearMap.id.smulRight (f i))
this : Tendsto (fun g ↦ ∑ i, ‖g i - f i‖) (𝓝 f) (𝓝 0)
g : ι → E
hg : ∑ i, ‖g i - f i‖₊ < K⁻¹
v u : ι → 𝕜
⊢ dist
      ((⇑((LinearMap.lsum 𝕜 (fun x ↦ 𝕜) ℕ) fun i ↦ LinearMap.id.smulRight (g i)) -
          ⇑((LinearMap.lsum 𝕜 (fun x ↦ 𝕜) ℕ) fun i ↦ LinearMap.id.smulRight (f i)))
        v)
      ((⇑((LinearMap.lsum 𝕜 (fun x ↦ 𝕜) ℕ) fun i ↦ LinearMap.id.smulRight (g i)) -
          ⇑((LinearMap.lsum 𝕜 (fun x ↦ 𝕜) ℕ) fun i ↦ LinearMap.id.smulRight (f i)))
        u) ≤
    ↑(∑ i, ‖g i - f i‖₊) * dist v u
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
f g : C(α, R)
h : f * g = 0
⊢ ‖f + g‖ = max ‖f‖ ‖g‖

Tactic:
replace h : mkOfCompact f * mkOfCompact g = 0 := by ext x; simpa using! congr($h x)

Goal state after:
α : Type u_1
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
R : Type u_4
inst✝¹ : NonUnitalSeminormedRing R
inst✝ : IsCancelMulZero R
f g : C(α, R)
h : mkOfCompact f * mkOfCompact g = 0
⊢ ‖f + g‖ = max ‖f‖ ‖g‖