mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Function.LpSpace.Indicator

Goal state before:
pp: α : Type u_1
E : Type u_2
m : MeasurableSpace α
μ : Measure α
inst✝ : NormedAddCommGroup E
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
c : E
hμs_ne_zero : μ s ≠ 0
⊢ ‖indicatorConstLp ∞ hs hμs c‖ = ‖c‖

Tactic:
rw [Lp.norm_def, eLpNorm_congr_ae indicatorConstLp_coeFn,
    eLpNorm_indicator_const' hs hμs_ne_zero ENNReal.top_ne_zero, ENNReal.toReal_top,
    _root_.div_zero, ENNReal.rpow_zero, mul_one, toReal_enorm]

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

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : TopologicalSpace ε'
inst✝ : ESeminormedAddMonoid ε'
C : ε'
hC : ‖C‖ₑ ≠ ∞
⊢ ‖C‖ₑ = 0 ∨ IsFiniteMeasure (μ.restrict s) ↔ ‖C‖ₑ = 0 ∨ μ s < ∞

Tactic:
isFiniteMeasure_restrict,

Goal state after:
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : TopologicalSpace ε'
inst✝ : ESeminormedAddMonoid ε'
C : ε'
hC : ‖C‖ₑ ≠ ∞
⊢ ‖C‖ₑ = 0 ∨ μ s ≠ ∞ ↔ ‖C‖ₑ = 0 ∨ μ s < ∞
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
f : α → ε'
x : α
inst✝ : MeasurableSingletonClass α
hfx : ‖f x‖ₑ ≠ ∞
this : f =ᵐ[μ.restrict {x}] fun x_1 ↦ f x
⊢ ‖f x‖ₑ = 0 ∨ IsFiniteMeasure (μ.restrict {x}) ↔ ‖f x‖ₑ = 0 ∨ μ {x} < ∞

pp: α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
f : α → ε'
x : α
inst✝ : MeasurableSingletonClass α
hfx : ‖f x‖ₑ ≠ ∞
this : f =ᵐ[μ.restrict {x}] fun x_1 ↦ f x
⊢ ‖f x‖ₑ ≠ ∞

Tactic:
isFiniteMeasure_restrict,

Goal state after:
α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
f : α → ε'
x : α
inst✝ : MeasurableSingletonClass α
hfx : ‖f x‖ₑ ≠ ∞
this : f =ᵐ[μ.restrict {x}] fun x_1 ↦ f x
⊢ ‖f x‖ₑ = 0 ∨ μ {x} ≠ ∞ ↔ ‖f x‖ₑ = 0 ∨ μ {x} < ∞

α : Type u_1
ε' : Type u_4
mα : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε'
inst✝¹ : ESeminormedAddMonoid ε'
f : α → ε'
x : α
inst✝ : MeasurableSingletonClass α
hfx : ‖f x‖ₑ ≠ ∞
this : f =ᵐ[μ.restrict {x}] fun x_1 ↦ f x
⊢ ‖f x‖ₑ ≠ ∞
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup β
inst✝ : MeasurableSingletonClass α
f : α → β
hs : Summable fun x ↦ ‖f x‖
hs' : (Function.support f).Countable
this : MeasurableSpace β := borel β
⊢ AEStronglyMeasurable f Measure.count

Tactic:
have : BorelSpace β := ⟨rfl⟩

Goal state after:
α : Type u_1
β : Type u_2
m : MeasurableSpace α
inst✝¹ : NormedAddCommGroup β
inst✝ : MeasurableSingletonClass α
f : α → β
hs : Summable fun x ↦ ‖f x‖
hs' : (Function.support f).Countable
this✝ : MeasurableSpace β := borel β
this : BorelSpace β
⊢ AEStronglyMeasurable f Measure.count
Module: Mathlib.MeasureTheory.Integral.IntegrableOn

Goal state before:
pp: case neg
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
ha : μ {a} ≠ ∞
ha' : ‖f a‖ₑ ≠ ∞
hab : ¬a ≤ b
⊢ IntegrableOn f ∅ μ ↔ IntegrableOn f (Ioc a b) μ

Tactic:
Ioc_eq_empty

Goal state after:
case neg
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
ha : μ {a} ≠ ∞
ha' : ‖f a‖ₑ ≠ ∞
hab : ¬a ≤ b
⊢ IntegrableOn f ∅ μ ↔ IntegrableOn f ∅ μ

case neg
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
ha : μ {a} ≠ ∞
ha' : ‖f a‖ₑ ≠ ∞
hab : ¬a ≤ b
⊢ ¬a < b
Module: Mathlib.Topology.Algebra.Module.FiniteDimension

Goal state before:
pp: case succ
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
H₁ : ∀ (s : Submodule 𝕜 E), finrank 𝕜 ↥s = n → IsClosed[inst✝⁴] ↑s
H₂ : ∀ (f : E →ₗ[𝕜] 𝕜), Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑f
i : ι
⊢ Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑(ξ.coord i)

Tactic:
exact H₂ (ξ.coord i)

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

Goal state before:
pp: 𝕜 : Type u
ι : Type v
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : Fintype ι
A : Type u_1
inst✝² : NormedCommRing A
inst✝¹ : NormedAlgebra 𝕜 A
inst✝ : Nonempty ι
m : ι → A
⊢ ‖∏ i, m i‖ ≤ ∏ i, ‖m i‖

Tactic:
exact norm_prod_le' _ univ_nonempty _

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

Goal state before:
pp: X : Type u_1
ε : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
⊢ ∃ T, T.Countable ∧ (∀ u ∈ T, IsOpen[inst✝³] u) ∧ s ⊆ ⋃ u ∈ T, u ∧ ∀ u ∈ T, IntegrableOn f (u ∩ s) μ

Tactic:
have : ∀ x : s, ∃ u, IsOpen u ∧ x.1 ∈ u ∧ IntegrableOn f (u ∩ s) μ := by
    rintro ⟨x, hx⟩
    rcases hf x hx with ⟨t, ht, h't⟩
    rcases mem_nhdsWithin.1 ht with ⟨u, u_open, x_mem, u_sub⟩
    exact ⟨u, u_open, x_mem, h't.mono_set u_sub⟩

Goal state after:
X : Type u_1
ε : Type u_3
inst✝⁴ : MeasurableSpace X
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
f : X → ε
μ : Measure X
s : Set X
inst✝ : SecondCountableTopology X
hf : LocallyIntegrableOn f s μ
this : ∀ (x : ↑s), ∃ u, IsOpen[inst✝³] u ∧ ↑x ∈ u ∧ IntegrableOn f (u ∩ s) μ
⊢ ∃ T, T.Countable ∧ (∀ u ∈ T, IsOpen[inst✝³] u) ∧ s ⊆ ⋃ u ∈ T, u ∧ ∀ u ∈ T, IntegrableOn f (u ∩ s) μ
Module: Mathlib.MeasureTheory.Function.L1Space.AEEqFun

Goal state before:
pp: α : Type u_1
ε : Type u_3
m : MeasurableSpace α
μ : Measure α
inst✝¹ : TopologicalSpace ε
inst✝ : ContinuousENorm ε
f : α →ₘ[μ] ε
⊢ MeasureTheory.Integrable (↑f) μ ↔ f.Integrable

Tactic:
rw [← integrable_mk f.aestronglyMeasurable, mk_coeFn]

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
⊢ ∀ (m₁ : (i : { a // a ∉ s }) → E₁ ↑i) (m₂ : α → (i : ι) → E₁ i),
    ‖((f.iteratedFDerivComponent e) m₁) m₂‖ ≤ (‖f‖ * ∏ i, ‖m₁ i‖) * ∏ i, ‖m₂ i‖

Tactic:
intro x 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.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: case refine_2.hSp
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : AddCommMonoid β
T : Set α → β
T_empty : T ∅ = 0
h_add : FinMeasAdditive μ T
ι : Type u_8
S : ι → Set α
sι : Finset ι
hS_meas : ∀ (i : ι), MeasurableSet (S i)
a : ι
s : Finset ι
has : a ∉ s
h : (∀ i ∈ s, μ (S i) ≠ ∞) → (∀ i ∈ s, ∀ j ∈ s, i ≠ j → Disjoint (S i) (S j)) → T (⋃ i ∈ s, S i) = ∑ i ∈ s, T (S i)
hps : ∀ i ∈ insert a s, μ (S i) ≠ ∞
h_disj : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → Disjoint (S i) (S j)
⊢ ∀ i ∈ s, μ (S i) ≠ ∞

pp: case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : AddCommMonoid β
T : Set α → β
T_empty : T ∅ = 0
h_add : FinMeasAdditive μ T
ι : Type u_8
S : ι → Set α
sι : Finset ι
hS_meas : ∀ (i : ι), MeasurableSet (S i)
a : ι
s : Finset ι
has : a ∉ s
h : (∀ i ∈ s, μ (S i) ≠ ∞) → (∀ i ∈ s, ∀ j ∈ s, i ≠ j → Disjoint (S i) (S j)) → T (⋃ i ∈ s, S i) = ∑ i ∈ s, T (S i)
hps : ∀ i ∈ insert a s, μ (S i) ≠ ∞
h_disj : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → Disjoint (S i) (S j)
⊢ T (⋃ i ∈ insert a s, S i) = T (S a) + T (⋃ i ∈ s, S i)

pp: case refine_2.h_disj
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : AddCommMonoid β
T : Set α → β
T_empty : T ∅ = 0
h_add : FinMeasAdditive μ T
ι : Type u_8
S : ι → Set α
sι : Finset ι
hS_meas : ∀ (i : ι), MeasurableSet (S i)
a : ι
s : Finset ι
has : a ∉ s
h : (∀ i ∈ s, μ (S i) ≠ ∞) → (∀ i ∈ s, ∀ j ∈ s, i ≠ j → Disjoint (S i) (S j)) → T (⋃ i ∈ s, S i) = ∑ i ∈ s, T (S i)
hps : ∀ i ∈ insert a s, μ (S i) ≠ ∞
h_disj : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → Disjoint (S i) (S j)
⊢ ∀ i ∈ s, ∀ j ∈ s, i ≠ j → Disjoint (S i) (S j)

Tactic:
· exact fun i hi => hps i (Finset.mem_insert_of_mem hi)

Goal state after:
case refine_2
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : AddCommMonoid β
T : Set α → β
T_empty : T ∅ = 0
h_add : FinMeasAdditive μ T
ι : Type u_8
S : ι → Set α
sι : Finset ι
hS_meas : ∀ (i : ι), MeasurableSet (S i)
a : ι
s : Finset ι
has : a ∉ s
h : (∀ i ∈ s, μ (S i) ≠ ∞) → (∀ i ∈ s, ∀ j ∈ s, i ≠ j → Disjoint (S i) (S j)) → T (⋃ i ∈ s, S i) = ∑ i ∈ s, T (S i)
hps : ∀ i ∈ insert a s, μ (S i) ≠ ∞
h_disj : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → Disjoint (S i) (S j)
⊢ T (⋃ i ∈ insert a s, S i) = T (S a) + T (⋃ i ∈ s, S i)

case refine_2.h_disj
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : AddCommMonoid β
T : Set α → β
T_empty : T ∅ = 0
h_add : FinMeasAdditive μ T
ι : Type u_8
S : ι → Set α
sι : Finset ι
hS_meas : ∀ (i : ι), MeasurableSet (S i)
a : ι
s : Finset ι
has : a ∉ s
h : (∀ i ∈ s, μ (S i) ≠ ∞) → (∀ i ∈ s, ∀ j ∈ s, i ≠ j → Disjoint (S i) (S j)) → T (⋃ i ∈ s, S i) = ∑ i ∈ s, T (S i)
hps : ∀ i ∈ insert a s, μ (S i) ≠ ∞
h_disj : ∀ i ∈ insert a s, ∀ j ∈ insert a s, i ≠ j → Disjoint (S i) (S j)
⊢ ∀ i ∈ s, ∀ j ∈ s, i ≠ j → Disjoint (S i) (S j)
Module: Mathlib.Analysis.Normed.Operator.Extend

Goal state before:

Tactic:
‖f x‖ ≤ ‖f‖ * ‖x‖ := le_opNorm _ _
      _ ≤ ‖f‖ * (N * ‖e x‖) := by gcongr; exact h_e x
      _ ≤ N * ‖f‖ * ‖e x‖ := by rw [mul_comm ↑N ‖f‖, mul_assoc]

Goal state after:
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 : 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 →ₛ E
hf : Integrable (⇑f) μ
h : μ {a | ¬f a = g a} = 0
x y : E
hxy : x ≠ y
z : α
⊢ z ∈ ⇑f ⁻¹' {x} ∩ ⇑g ⁻¹' {y} → z ∈ {a | ¬f a = g a}

Tactic:
simp_rw [Set.mem_inter_iff, Set.mem_ofPred_eq, Set.mem_preimage, Set.mem_singleton_iff]

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 : 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 →ₛ E
hf : Integrable (⇑f) μ
h : μ {a | ¬f a = g a} = 0
x y : E
hxy : x ≠ y
z : α
⊢ f z = x ∧ g z = y → ¬f z = g z
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: case pos
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
motive : (α → E) → Prop
indicator : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → motive f → motive g → motive (f + g)
closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive ↑↑f}
ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MemLp f p μ → motive f → motive g
c : E
s : Set α
hs : MeasurableSet s
h : MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p μ
hc : c = 0
⊢ motive ⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))

pp: case neg
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
motive : (α → E) → Prop
indicator : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → motive f → motive g → motive (f + g)
closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive ↑↑f}
ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MemLp f p μ → motive f → motive g
c : E
s : Set α
hs : MeasurableSet s
h : MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p μ
hc : ¬c = 0
⊢ motive ⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))

Tactic:
· subst hc; convert! indicator 0 MeasurableSet.empty (by simp) using 1; ext; simp

Goal state after:
case neg
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
_i : Fact (1 ≤ p)
hp_ne_top : p ≠ ∞
motive : (α → E) → Prop
indicator : ∀ (c : E) ⦃s : Set α⦄, MeasurableSet s → μ s < ∞ → motive (s.indicator fun x ↦ c)
add :
  ∀ ⦃f g : α → E⦄, Disjoint (support f) (support g) → MemLp f p μ → MemLp g p μ → motive f → motive g → motive (f + g)
closed : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] {f | motive ↑↑f}
ae : ∀ ⦃f g : α → E⦄, f =ᵐ[μ] g → MemLp f p μ → motive f → motive g
c : E
s : Set α
hs : MeasurableSet s
h : MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p μ
hc : ¬c = 0
⊢ motive ⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: case pos
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_1 ∈ {x, 0}, (T (⇑(piecewise s hs (const α x) (const α 0)) ⁻¹' {x_1})) x_1 = (T s) x

Tactic:
simp_rw [hx0]

Goal state after:
case pos
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, 0}, (T (⇑(piecewise s hs (const α 0) (const α 0)) ⁻¹' {x})) x = (T s) 0
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝ : NormedField β
u v : α → β
l : Filter α
⊢ u ~[l] v ↔ ∃ φ, ∃ (_ : Tendsto φ l (𝓝 1)), u =ᶠ[l] φ * v

Tactic:
rw [IsEquivalent, isLittleO_iff_exists_eq_mul]
  constructor <;> rintro ⟨φ, hφ, h⟩ <;> [refine ⟨φ + 1, ?_, ?_⟩; refine ⟨φ - 1, ?_, ?_⟩]
  · conv in 𝓝 _ => rw [← zero_add (1 : β)]
    exact hφ.add tendsto_const_nhds
  · convert! h.fun_add (EventuallyEq.refl l v) <;> simp [add_mul]
  · conv in 𝓝 _ => rw [← sub_self (1 : β)]
    exact hφ.sub tendsto_const_nhds
  · convert! h.fun_sub (EventuallyEq.refl l v); simp [sub_mul]

Goal state after:
Module: Mathlib.Analysis.Asymptotics.AsymptoticEquivalent

Goal state before:
pp: case nil
α : Type u_1
ι : Type u_2
β : Type u_3
inst✝ : NormedField β
l : Filter α
f g : ι → α → β
h : ∀ i ∈ [], f i ~[l] g i
⊢ (fun x ↦ (List.map (fun x_1 ↦ f x_1 x) []).prod) ~[l] fun x ↦ (List.map (fun x_1 ↦ g x_1 x) []).prod

Tactic:
| nil =>

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

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : PartialOrder E
inst✝⁴ : IsOrderedAddMonoid E
inst✝³ : IsOrderedModule ℝ E
inst✝² : ClosedIciTopology E
β : Type u_6
inst✝¹ : AddCommMonoid β
inst✝ : Module ℝ β
f : α → β → E
s : Set β
hs : Convex ℝ s
hf_conv : ∀ᵐ (x : α) ∂μ, ConvexOn ℝ s (f x)
hf_int : ∀ a ∈ s, Integrable (fun x ↦ f x a) μ
⊢ ConvexOn ℝ s fun b ↦ ∫ (x : α), f x b ∂μ

Tactic:
refine ⟨hs, ?_⟩
  intro a ha b hb p q hp hq hpq
  calc ∫ x, f x (p • a + q • b) ∂μ ≤ ∫ x, p • f x a + q • f x b ∂μ := by
                  refine integral_mono_ae ?lhs ?rhs ?ae_le
                  case lhs =>
                    refine hf_int _ ?_
                    rw [convex_iff_add_mem] at hs
                    exact hs ha hb hp hq hpq
                  case rhs => fun_prop (disch := aesop)
                  case ae_le =>
                    filter_upwards [hf_conv] with x hx
                    exact hx.2 ha hb hp hq hpq
            _ = ∫ x, p • f x a ∂μ + ∫ x, q • f x b ∂μ := by
                  apply integral_add
                  all_goals fun_prop (disch := aesop)
            _ = p • ∫ x, f x a ∂μ + q • ∫ x, f x b ∂μ := by simp [integral_smul]

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

Goal state before:
pp: case inr
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup V
inst✝⁵ : Module R V
inst✝⁴ : TopologicalSpace P
inst✝³ : AddTorsor V P
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddTorsor P
inst✝ : T1Space V
s : AffineSubspace R P
x : P
hx : x ∈ ↑s
⊢ IsClosed ↑s.direction ↔ IsClosed ↑s

Tactic:
← (Homeomorph.vaddConst x).symm.isClosed_image,

Goal state after:
case inr
R : Type u_1
V : Type u_2
P : Type u_3
inst✝⁷ : Ring R
inst✝⁶ : AddCommGroup V
inst✝⁵ : Module R V
inst✝⁴ : TopologicalSpace P
inst✝³ : AddTorsor V P
inst✝² : TopologicalSpace V
inst✝¹ : IsTopologicalAddTorsor P
inst✝ : T1Space V
s : AffineSubspace R P
x : P
hx : x ∈ ↑s
⊢ IsClosed ↑s.direction ↔ IsClosed (⇑(Homeomorph.vaddConst x).symm '' ↑s)
Module: Mathlib.Analysis.Normed.Module.RieszLemma

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
r : ℝ
hr : r < 1
x : E
hx : x ∉ F
d : ℝ := infDist x ↑F
hFn : (↑F).Nonempty
hdp : 0 < d
r' : ℝ := max r 2⁻¹
hr' : r' < 1
hlt : 0 < r'
hdlt : d < d / r'
y₀ : E
hy₀F : y₀ ∈ F
hxy₀ : dist x y₀ < d / r'
x_ne_y₀ : x - y₀ ∉ F
y : E
hy : y ∈ F
hy₀y : y₀ + y ∈ F
⊢ r * ‖x - y₀‖ < ‖x - y₀ - y‖

Tactic:
calc
    r * ‖x - y₀‖ ≤ r' * ‖x - y₀‖ := by gcongr; apply le_max_left
    _ < d := by
      rw [← dist_eq_norm]
      exact (lt_div_iff₀' hlt).1 hxy₀
    _ ≤ dist x (y₀ + y) := Metric.infDist_le_dist_of_mem hy₀y
    _ = ‖x - y₀ - y‖ := by rw [sub_sub, dist_eq_norm]

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✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝ : CompleteSpace F
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
h_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → T s = 0
f : ↥(Lp E 1 μ)
⊢ ∀ (f : ↥(α →₁ₛ[μ] E)), (setToL1SCLM α E μ hT) f = 0 ↑f

Tactic:
by simp [setToL1SCLM_zero_left' hT h_zero]

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
hF : ∃ x, x ∉ F
Rpos : 0 < R
this : ‖c‖ / R < 1
x : E
xF : x ∉ F
hx : ∀ y ∈ F, ‖c‖ / R * ‖x‖ ≤ ‖x - y‖
x0 : x ≠ 0
d : 𝕜
d0 : d ≠ 0
dxlt : ‖d • x‖ < R
ledx : R / ‖c‖ ≤ ‖d • x‖
y : E
hy : y ∈ F
y' : E := d⁻¹ • y
yy' : y = d • y'
⊢ ‖d‖ * ‖x - y'‖ = ‖d • x - y‖

Tactic:
yy',

Goal state after:
𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
hF : ∃ x, x ∉ F
Rpos : 0 < R
this : ‖c‖ / R < 1
x : E
xF : x ∉ F
hx : ∀ y ∈ F, ‖c‖ / R * ‖x‖ ≤ ‖x - y‖
x0 : x ≠ 0
d : 𝕜
d0 : d ≠ 0
dxlt : ‖d • x‖ < R
ledx : R / ‖c‖ ≤ ‖d • x‖
y : E
hy : y ∈ F
y' : E := d⁻¹ • y
yy' : y = d • y'
⊢ ‖d‖ * ‖x - y'‖ = ‖d • x - d • y'‖