mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: case refine_2.a
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 : ι
hi : a ∈ s
hai : a = i
⊢ False

Tactic:
]

Goal state after:
case refine_2.a
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 : ι
hi : a ∈ s
hai : a = i
⊢ False
Module: Mathlib.Analysis.Normed.Operator.Extend

Goal state before:
pp: case refine_2
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
Fₗ : Type u_6
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 g : E →SL[σ₁₂] F
inst✝ : CompleteSpace F
e : E →L[𝕜] Eₗ
h : DenseRange ⇑e ∧ IsUniformInducing ⇑e
cont : Continuous (⋯.extend ⇑f)
eq : ∀ (b : E), ⋯.extend (⇑f) (e b) = f b
k : 𝕜
b : Eₗ
x : E
⊢ ⋯.extend (⇑f) (e (k • x)) = σ₁₂ k • ⋯.extend (⇑f) (e x)

Tactic:
]

Goal state after:
case refine_2
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
Fₗ : Type u_6
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 g : E →SL[σ₁₂] F
inst✝ : CompleteSpace F
e : E →L[𝕜] Eₗ
h : DenseRange ⇑e ∧ IsUniformInducing ⇑e
cont : Continuous (⋯.extend ⇑f)
eq : ∀ (b : E), ⋯.extend (⇑f) (e b) = f b
k : 𝕜
b : Eₗ
x : E
⊢ ⋯.extend (⇑f) (e (k • x)) = σ₁₂ k • ⋯.extend (⇑f) (e x)
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
T : Set α → β
inst✝ : AddGroup β
hT : FinMeasAdditive μ T
s t : Set α
hs : MeasurableSet s
ht : MeasurableSet t
hμs : μ s ≠ ∞
hμt : μ t ≠ ∞
hst : Disjoint s t
⊢ T (t ∪ s) = T (t ∪ s)

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
k : 𝕜
f : ↥(Lp E p μ)
s : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
hs : AEEqFun.mk ⇑s ⋯ = ↑f
⊢ k • ↑f = ↑(k • ↑⟨f, ⋯⟩)

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
k : 𝕜
f : ↥(Lp E p μ)
s : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
hs : AEEqFun.mk ⇑s ⋯ = ↑f
⊢ k • ↑f = ↑(k • ↑⟨f, ⋯⟩)
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
β : Type u_7
inst✝ : SeminormedAddCommGroup β
C : ℝ
m : MeasurableSpace α
μ : Measure α
hC : 0 ≤ C
s : Set α
x✝¹ : MeasurableSet s
x✝ : μ s < ∞
⊢ 0 ≤ C * μ.real s

Tactic:
]

Goal state after:
α : Type u_1
β : Type u_7
inst✝ : SeminormedAddCommGroup β
C : ℝ
m : MeasurableSpace α
μ : Measure α
hC : 0 ≤ C
s : Set α
x✝¹ : MeasurableSet s
x✝ : μ s < ∞
⊢ 0 ≤ C * μ.real s
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_8
inst✝ : NormedAddCommGroup β
T : Set α → β
C : ℝ
hT : DominatedFinMeasAdditive μ T C
s : Set α
hs : MeasurableSet s
hs_zero : μ s = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: α : Type u_1
β : Type u_2
ι : Type u_3
E : Type u_4
F : Type u_5
𝕜 : Type u_6
inst✝⁵ : MeasurableSpace α
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedAddCommGroup F
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
f : ↥(simpleFunc E p μ)
⊢ 0 • f = 0

Tactic:
ext1; exact zero_smul _ _

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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : SeminormedAddCommGroup β
T T' : Set α → β
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ ‖T s + T' s‖ ≤ C * μ.real s + C' * μ.real s

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : SeminormedAddCommGroup β
T T' : Set α → β
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ ‖T s + T' s‖ ≤ C * μ.real s + C' * μ.real s
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f : α →ₛ E
hfi : MemLp (⇑f) p μ
⊢ AEEqFun.mk ⇑(toSimpleFunc (f.toLp hfi)) ?m.37 = AEEqFun.mk ⇑f ?m.38

pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f : α →ₛ E
hfi : MemLp (⇑f) p μ
⊢ AEStronglyMeasurable (⇑(toSimpleFunc (f.toLp hfi))) μ

pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f : α →ₛ E
hfi : MemLp (⇑f) p μ
⊢ AEStronglyMeasurable (⇑f) μ

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f : α →ₛ E
hfi : MemLp (⇑f) p μ
⊢ AEEqFun.mk ⇑(toSimpleFunc (f.toLp hfi)) ?m.37 = AEEqFun.mk ⇑f ?m.38

α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f : α →ₛ E
hfi : MemLp (⇑f) p μ
⊢ AEStronglyMeasurable (⇑(toSimpleFunc (f.toLp hfi))) μ

α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f : α →ₛ E
hfi : MemLp (⇑f) p μ
⊢ AEStronglyMeasurable (⇑f) μ
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
a✝¹ : α
h₁ : (toSimpleFunc 0) a✝¹ = ↑↑↑0 a✝¹
a✝ : ↑↑0 a✝¹ = 0 a✝¹
⊢ ↑↑↑0 a✝¹ = 0 a✝¹

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
a✝¹ : α
h₁ : (toSimpleFunc 0) a✝¹ = ↑↑↑0 a✝¹
a✝ : ↑↑0 a✝¹ = 0 a✝¹
⊢ ↑↑↑0 a✝¹ = 0 a✝¹
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
𝕜 : Type u_6
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝³ : SeminormedAddCommGroup β
T : Set α → β
C : ℝ
inst✝² : SeminormedAddGroup 𝕜
inst✝¹ : DistribSMul 𝕜 β
inst✝ : IsBoundedSMul 𝕜 β
hT : DominatedFinMeasAdditive μ T C
c : 𝕜
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ ‖c‖ * ‖T s‖ ≤ ‖c‖ * (C * μ.real s)

Tactic:
]

Goal state after:
α : Type u_1
𝕜 : Type u_6
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝³ : SeminormedAddCommGroup β
T : Set α → β
C : ℝ
inst✝² : SeminormedAddGroup 𝕜
inst✝¹ : DistribSMul 𝕜 β
inst✝ : IsBoundedSMul 𝕜 β
hT : DominatedFinMeasAdditive μ T C
c : 𝕜
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ ‖c‖ * ‖T s‖ ≤ ‖c‖ * (C * μ.real s)
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f g : ↥(simpleFunc E p μ)
a✝ : α
h : (toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝
⊢ (toSimpleFunc f) a✝ = ↑↑↑f a✝ →
    (toSimpleFunc g) a✝ = ↑↑↑g a✝ →
      ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ → ↑(↑↑f + ↑↑g) a✝ = (toSimpleFunc f) a✝ + (toSimpleFunc g) a✝

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f g : ↥(simpleFunc E p μ)
a✝ : α
h : (toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝
⊢ (toSimpleFunc f) a✝ = ↑↑↑f a✝ →
    (toSimpleFunc g) a✝ = ↑↑↑g a✝ →
      ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ → ↑(↑↑f + ↑↑g) a✝ = (toSimpleFunc f) a✝ + (toSimpleFunc g) a✝
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:
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.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f g : ↥(simpleFunc E p μ)
a✝ : α
h✝ : (toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝
h : (toSimpleFunc f) a✝ = ↑↑↑f a✝
⊢ (toSimpleFunc g) a✝ = ↑↑↑g a✝ → ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ → ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + (toSimpleFunc g) a✝

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f g : ↥(simpleFunc E p μ)
a✝ : α
h✝ : (toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝
h : (toSimpleFunc f) a✝ = ↑↑↑f a✝
⊢ (toSimpleFunc g) a✝ = ↑↑↑g a✝ → ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ → ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + (toSimpleFunc g) a✝
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f g : ↥(simpleFunc E p μ)
a✝ : α
h✝¹ : (toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝
h✝ : (toSimpleFunc f) a✝ = ↑↑↑f a✝
h : (toSimpleFunc g) a✝ = ↑↑↑g a✝
⊢ ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ → ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f g : ↥(simpleFunc E p μ)
a✝ : α
h✝¹ : (toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝
h✝ : (toSimpleFunc f) a✝ = ↑↑↑f a✝
h : (toSimpleFunc g) a✝ = ↑↑↑g a✝
⊢ ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝ → ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
β : Type u_7
inst✝ : SeminormedAddCommGroup β
T T' : Set α → β
C C' : ℝ
μ ν : Measure α
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive ν T' C'
s : Set α
hs : MeasurableSet s
hsf : (μ + ν) s < ∞
hμs : μ s < ∞
hνs : ν s < ∞
⊢ ‖T s + T' s‖ ≤ max C C' * μ.real s + max C C' * ν.real s

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
β : Type u_7
inst✝ : SeminormedAddCommGroup β
T T' : Set α → β
C C' : ℝ
μ ν : Measure α
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive ν T' C'
s : Set α
hs : MeasurableSet s
hsf : (μ + ν) s < ∞
hμs : μ s < ∞
hνs : ν s < ∞
⊢ ‖T s + T' s‖ ≤ max C C' * μ.real s + max C C' * ν.real s
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f g : ↥(simpleFunc E p μ)
a✝ : α
h✝² : (toSimpleFunc (f + g)) a✝ = ↑(↑↑f + ↑↑g) a✝
h✝¹ : (toSimpleFunc f) a✝ = ↑↑↑f a✝
h✝ : (toSimpleFunc g) a✝ = ↑↑↑g a✝
h : ↑(↑↑f + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝
⊢ ↑↑↑f a✝ + ↑↑↑g a✝ = ↑↑↑f a✝ + ↑↑↑g a✝

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Normed.Operator.Extend

Goal state before:
pp: case refine_3
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 x‖ ≤ ↑N * ‖f‖ * ‖e x‖

Tactic:
]

Goal state after:
case refine_3
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 x‖ ≤ ↑N * ‖f‖ * ‖e x‖
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‖) ≤ ‖f‖ * (↑N * ‖e x‖)

Tactic:
]

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

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f : ↥(simpleFunc E p μ)
a✝ : α
h : (toSimpleFunc (-f)) a✝ = ↑(-↑↑f) a✝
⊢ (toSimpleFunc f) a✝ = ↑↑↑f a✝ → ↑(-↑↑f) a✝ = -↑↑↑f a✝ → ↑(-↑↑f) a✝ = -(toSimpleFunc f) a✝

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f : ↥(simpleFunc E p μ)
a✝ : α
h : (toSimpleFunc (-f)) a✝ = ↑(-↑↑f) a✝
⊢ (toSimpleFunc f) a✝ = ↑↑↑f a✝ → ↑(-↑↑f) a✝ = -↑↑↑f a✝ → ↑(-↑↑f) a✝ = -(toSimpleFunc f) a✝
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
Fₗ : Type u_6
inst✝⁷ : DivisionRing 𝕜
inst✝⁶ : DivisionRing 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁵ : AddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : SeminormedAddCommGroup Eₗ
inst✝² : Module 𝕜 E
inst✝¹ : Module 𝕜₂ F
inst✝ : Module 𝕜 Eₗ
f : E →ₛₗ[σ₁₂] F
g : E →ₗ[𝕜] Eₗ
C : ℝ
x : E
h : ‖f x‖ ≤ C * ‖0‖
hx : x ∈ g.ker
⊢ x ∈ f.ker

Tactic:
]

Goal state after:
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
Fₗ : Type u_6
inst✝⁷ : DivisionRing 𝕜
inst✝⁶ : DivisionRing 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁵ : AddCommGroup E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : SeminormedAddCommGroup Eₗ
inst✝² : Module 𝕜 E
inst✝¹ : Module 𝕜₂ F
inst✝ : Module 𝕜 Eₗ
f : E →ₛₗ[σ₁₂] F
g : E →ₗ[𝕜] Eₗ
C : ℝ
x : E
h : ‖f x‖ ≤ C * ‖0‖
hx : x ∈ g.ker
⊢ x ∈ f.ker
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f : ↥(simpleFunc E p μ)
a✝ : α
h✝ : (toSimpleFunc (-f)) a✝ = ↑(-↑↑f) a✝
h : (toSimpleFunc f) a✝ = ↑↑↑f a✝
⊢ ↑(-↑↑f) a✝ = -↑↑↑f a✝ → ↑(-↑↑f) a✝ = -↑↑↑f a✝

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f : ↥(simpleFunc E p μ)
a✝ : α
h✝ : (toSimpleFunc (-f)) a✝ = ↑(-↑↑f) a✝
h : (toSimpleFunc f) a✝ = ↑↑↑f a✝
⊢ ↑(-↑↑f) a✝ = -↑↑↑f a✝ → ↑(-↑↑f) a✝ = -↑↑↑f a✝
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : SeminormedAddCommGroup β
T : Set α → β
C : ℝ
c : ℝ≥0∞
hc_ne_top : c ≠ ∞
hT : DominatedFinMeasAdditive (c • μ) T C
h : ∀ (s : Set α), MeasurableSet s → c • μ s = ∞ → μ s = ∞
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
hcμs : c * μ s ≠ ∞
⊢ ‖T s‖ ≤ c.toReal * C * μ.real s

Tactic:
]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
β : Type u_7
inst✝ : SeminormedAddCommGroup β
T : Set α → β
C : ℝ
c : ℝ≥0∞
hc_ne_top : c ≠ ∞
hT : DominatedFinMeasAdditive (c • μ) T C
h : ∀ (s : Set α), MeasurableSet s → c • μ s = ∞ → μ s = ∞
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
hcμs : c * μ s ≠ ∞
⊢ ‖T s‖ ≤ c.toReal * C * μ.real s
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f : ↥(simpleFunc E p μ)
a✝ : α
h✝¹ : (toSimpleFunc (-f)) a✝ = ↑(-↑↑f) a✝
h✝ : (toSimpleFunc f) a✝ = ↑↑↑f a✝
h : ↑(-↑↑f) a✝ = -↑↑↑f a✝
⊢ -↑↑↑f a✝ = -↑↑↑f a✝

Tactic:
]

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

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f g : ↥(simpleFunc E p μ)
a✝ : α
h : (toSimpleFunc (f - g)) a✝ = ↑(↑↑f - ↑↑g) a✝
⊢ (toSimpleFunc f) a✝ = ↑↑↑f a✝ →
    (toSimpleFunc g) a✝ = ↑↑↑g a✝ →
      ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝ → ↑(↑↑f - ↑↑g) a✝ = (toSimpleFunc f) a✝ - (toSimpleFunc g) a✝

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f g : ↥(simpleFunc E p μ)
a✝ : α
h : (toSimpleFunc (f - g)) a✝ = ↑(↑↑f - ↑↑g) a✝
⊢ (toSimpleFunc f) a✝ = ↑↑↑f a✝ →
    (toSimpleFunc g) a✝ = ↑↑↑g a✝ →
      ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝ → ↑(↑↑f - ↑↑g) a✝ = (toSimpleFunc f) a✝ - (toSimpleFunc g) a✝
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: case neg
α : Type u_1
E : Type u_2
F' : Type u_4
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 < ∞ → T s = 0
f : α →ₛ E
hf : Integrable (⇑f) μ
x : E
x✝ : x ∈ f.range
hx0 : ¬x = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f g : ↥(simpleFunc E p μ)
a✝ : α
h✝ : (toSimpleFunc (f - g)) a✝ = ↑(↑↑f - ↑↑g) a✝
h : (toSimpleFunc f) a✝ = ↑↑↑f a✝
⊢ (toSimpleFunc g) a✝ = ↑↑↑g a✝ → ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝ → ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - (toSimpleFunc g) a✝

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f g : ↥(simpleFunc E p μ)
a✝ : α
h✝ : (toSimpleFunc (f - g)) a✝ = ↑(↑↑f - ↑↑g) a✝
h : (toSimpleFunc f) a✝ = ↑↑↑f a✝
⊢ (toSimpleFunc g) a✝ = ↑↑↑g a✝ → ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝ → ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - (toSimpleFunc g) a✝
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
F : Type u_3
F' : Type u_4
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : DecidablePred fun x ↦ x ≠ 0
m : MeasurableSpace α
T : Set α → F →L[ℝ] F'
f : α →ₛ F
x : F
x✝ : x ∈ f.range
hx0 : x = 0
⊢ (T (⇑f ⁻¹' {0})) 0 = 0

Tactic:
]

Goal state after:
α : Type u_1
F : Type u_3
F' : Type u_4
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
inst✝ : DecidablePred fun x ↦ x ≠ 0
m : MeasurableSpace α
T : Set α → F →L[ℝ] F'
f : α →ₛ F
x : F
x✝ : x ∈ f.range
hx0 : x = 0
⊢ (T (⇑f ⁻¹' {0})) 0 = 0
Module: Mathlib.Analysis.Normed.Operator.Extend

Goal state before:
pp: case H
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
inst✝¹⁰ : NormedDivisionRing 𝕜
inst✝⁹ : NormedDivisionRing 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁸ : AddCommGroup E
inst✝⁷ : SeminormedAddCommGroup Eₗ
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : Module 𝕜 E
inst✝⁴ : Module 𝕜₂ F
inst✝³ : IsBoundedSMul 𝕜₂ F
inst✝² : Module 𝕜 Eₗ
inst✝¹ : IsBoundedSMul 𝕜 Eₗ
inst✝ : CompleteSpace F
f : E →ₛₗ[σ₁₂] F
e : E →ₗ[𝕜] Eₗ
h_dense : DenseRange ⇑e
C : ℝ
h_norm : ∀ (x : E), ‖f x‖ ≤ C * ‖e x‖
g : Eₗ →SL[σ₁₂] F
H : ↑g ∘ₛₗ e = f
y : Eₗ
x : E
hxy : e x = y
⊢ (g ∘SL e.range.subtypeL) ⟨y, ⋯⟩ = f x

Tactic:
]

Goal state after:
case H
𝕜 : Type u_1
𝕜₂ : Type u_2
E : Type u_3
Eₗ : Type u_4
F : Type u_5
inst✝¹⁰ : NormedDivisionRing 𝕜
inst✝⁹ : NormedDivisionRing 𝕜₂
σ₁₂ : 𝕜 →+* 𝕜₂
inst✝⁸ : AddCommGroup E
inst✝⁷ : SeminormedAddCommGroup Eₗ
inst✝⁶ : NormedAddCommGroup F
inst✝⁵ : Module 𝕜 E
inst✝⁴ : Module 𝕜₂ F
inst✝³ : IsBoundedSMul 𝕜₂ F
inst✝² : Module 𝕜 Eₗ
inst✝¹ : IsBoundedSMul 𝕜 Eₗ
inst✝ : CompleteSpace F
f : E →ₛₗ[σ₁₂] F
e : E →ₗ[𝕜] Eₗ
h_dense : DenseRange ⇑e
C : ℝ
h_norm : ∀ (x : E), ‖f x‖ ≤ C * ‖e x‖
g : Eₗ →SL[σ₁₂] F
H : ↑g ∘ₛₗ e = f
y : Eₗ
x : E
hxy : e x = y
⊢ (g ∘SL e.range.subtypeL) ⟨y, ⋯⟩ = f x
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f g : ↥(simpleFunc E p μ)
a✝ : α
h✝¹ : (toSimpleFunc (f - g)) a✝ = ↑(↑↑f - ↑↑g) a✝
h✝ : (toSimpleFunc f) a✝ = ↑↑↑f a✝
h : (toSimpleFunc g) a✝ = ↑↑↑g a✝
⊢ ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝ → ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝

Tactic:
]

Goal state after:
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
f g : ↥(simpleFunc E p μ)
a✝ : α
h✝¹ : (toSimpleFunc (f - g)) a✝ = ↑(↑↑f - ↑↑g) a✝
h✝ : (toSimpleFunc f) a✝ = ↑↑↑f a✝
h : (toSimpleFunc g) a✝ = ↑↑↑g a✝
⊢ ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝ → ↑(↑↑f - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝
Module: Mathlib.MeasureTheory.Integral.FinMeasAdditive

Goal state before:
pp: α : Type u_1
F : Type u_3
F' : Type u_4
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
m : MeasurableSpace α
inst✝ : DecidablePred fun x ↦ x ≠ 0
T : Set α → F →L[ℝ] F'
hT : T ∅ = 0
f : α →ₛ F
s : Finset F
hs : {x ∈ f.range | x ≠ 0} ⊆ s
⊢ ∑ x ∈ s, (T (⇑f ⁻¹' {x})) x = ∑ x ∈ s, (T (⇑f ⁻¹' {x})) x

pp: α : Type u_1
F : Type u_3
F' : Type u_4
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
m : MeasurableSpace α
inst✝ : DecidablePred fun x ↦ x ≠ 0
T : Set α → F →L[ℝ] F'
hT : T ∅ = 0
f : α →ₛ F
s : Finset F
hs : {x ∈ f.range | x ≠ 0} ⊆ s
⊢ ∀ x ∈ s, x ∉ {x ∈ f.range | x ≠ 0} → (T (⇑f ⁻¹' {x})) x = 0

Tactic:
]

Goal state after:
α : Type u_1
F : Type u_3
F' : Type u_4
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
inst✝² : NormedAddCommGroup F'
inst✝¹ : NormedSpace ℝ F'
m : MeasurableSpace α
inst✝ : DecidablePred fun x ↦ x ≠ 0
T : Set α → F →L[ℝ] F'
hT : T ∅ = 0
f : α →ₛ F
s : Finset F
hs : {x ∈ f.range | x ≠ 0} ⊆ s
⊢ ∀ x ∈ s, x ∉ {x ∈ f.range | x ≠ 0} → (T (⇑f ⁻¹' {x})) x = 0