mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.MeasureTheory.Measure.Real

Goal state before:
pp: α : Type u_1
ι : Type u_3
x✝ : MeasurableSpace α
μ : Measure α
inst✝ : IsFiniteMeasure μ
s : Finset ι
t : ι → Set α
h : ∀ i ∈ s, MeasurableSet (t i)
H : μ.real univ < ∑ i ∈ s, μ.real (t i)
⊢ ∃ i ∈ s, ∃ j ∈ s, ∃ (_ : i ≠ j), (t i ∩ t j).Nonempty

Tactic:
by
  apply exists_nonempty_inter_of_measure_univ_lt_sum_measure μ
    (fun i mi ↦ (h i mi).nullMeasurableSet)
  simp only [Measure.real] at H
  apply (ENNReal.toReal_lt_toReal (by finiteness) _).1
  · convert! H
    rw [ENNReal.toReal_sum (by finiteness)]
  · exact (ENNReal.sum_lt_top.mpr (fun i hi ↦ measure_lt_top ..)).ne

Goal state after:
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: case neg
α : Type u_1
ε : Type u_5
m : MeasurableSpace α
μ : Measure α
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
inst✝ : IsFiniteMeasure μ
f : α → ε
p : ℝ≥0∞
hf : MemLp f p μ
h_zero : ¬p = 0
h_top : ¬p = ∞
⊢ Integrable (fun x ↦ ‖f x‖ₑ ^ p.toReal) μ

Tactic:
exact hf.integrable_enorm_rpow h_zero h_top

Goal state after:
Module: Mathlib.MeasureTheory.Function.L1Space.Integrable

Goal state before:
pp: α : Type u_1
ε : Type u_5
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
inst✝ : PseudoMetrizableSpace ε
ι : Type u_8
m : MeasurableSpace α
f : α → ε
μ : ι → Measure α
s : Finset ι
⊢ Integrable f (∑ i ∈ s, μ i) ↔ ∀ i ∈ s, Integrable f (μ i)

Tactic:
classical
  induction s using Finset.induction_on <;> simp [*]

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

Goal state before:
pp: case neg
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
t : TopologicalSpace 𝕜
h₁ : IsTopologicalAddGroup 𝕜
h₂ : ContinuousSMul 𝕜 𝕜
h₃ : T2Space 𝕜
ε : ℝ
hε : 0 < ε
ξ₀ : 𝕜
hξ₀ : 0 < ‖ξ₀‖
hξ₀ε : ‖ξ₀‖ < ε
this✝ : {ξ₀}ᶜ ∈ 𝓝 0
this : balancedCore 𝕜 {ξ₀}ᶜ ∈ 𝓝 0
ξ : 𝕜
hξ : ξ ∈ balancedCore 𝕜 {ξ₀}ᶜ
hξ0 : ¬ξ = 0
h : ε < ‖ξ‖
⊢ ‖ξ₀‖ * ‖ξ‖⁻¹ ≤ 1

Tactic:
mul_inv_le_iff₀ (norm_pos_iff.mpr hξ0),

Goal state after:
case neg
𝕜 : Type u
hnorm : NontriviallyNormedField 𝕜
t : TopologicalSpace 𝕜
h₁ : IsTopologicalAddGroup 𝕜
h₂ : ContinuousSMul 𝕜 𝕜
h₃ : T2Space 𝕜
ε : ℝ
hε : 0 < ε
ξ₀ : 𝕜
hξ₀ : 0 < ‖ξ₀‖
hξ₀ε : ‖ξ₀‖ < ε
this✝ : {ξ₀}ᶜ ∈ 𝓝 0
this : balancedCore 𝕜 {ξ₀}ᶜ ∈ 𝓝 0
ξ : 𝕜
hξ : ξ ∈ balancedCore 𝕜 {ξ₀}ᶜ
hξ0 : ¬ξ = 0
h : ε < ‖ξ‖
⊢ ‖ξ₀‖ ≤ 1 * ‖ξ‖
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: case inl
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
hf : Continuous[Pi.topologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑f
h✝ : IsEmpty ι
m : (i : ι) → E i
⊢ ‖f m‖ ≤ (‖f 0‖ + 1) * ∏ i, ‖m i‖

Tactic:
obtain rfl : m = 0 := funext (IsEmpty.elim ‹_›)

Goal state after:
case inl
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
hf : Continuous[Pi.topologicalSpace, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ⇑f
h✝ : IsEmpty ι
⊢ ‖f 0‖ ≤ (‖f 0‖ + 1) * ∏ i, ‖0 i‖
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: case pos
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : DecidableEq 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
s : Finset ι
his : i ∉ s
Hrec : ‖f m₁ - f (s.piecewise m₂ m₁)‖ ≤ C * ∑ i ∈ s, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖
A : (insert i s).piecewise m₂ m₁ = update (s.piecewise m₂ m₁) i (m₂ i)
B : s.piecewise m₂ m₁ = update (s.piecewise m₂ m₁) i (m₁ i)
j : ι
a✝ : j ∈ univ
h : j = i
⊢ ‖update (s.piecewise m₂ m₁) i (m₁ i - m₂ i) j‖ ≤ if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖

Tactic:
rw [h]

Goal state after:
case pos
𝕜 : Type u
ι : Type v
E : ι → Type wE
G : Type wG
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁴ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝³ : SeminormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : Fintype ι
inst✝ : DecidableEq 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
s : Finset ι
his : i ∉ s
Hrec : ‖f m₁ - f (s.piecewise m₂ m₁)‖ ≤ C * ∑ i ∈ s, ∏ j, if j = i then ‖m₁ i - m₂ i‖ else max ‖m₁ j‖ ‖m₂ j‖
A : (insert i s).piecewise m₂ m₁ = update (s.piecewise m₂ m₁) i (m₂ i)
B : s.piecewise m₂ m₁ = update (s.piecewise m₂ m₁) i (m₁ i)
j : ι
a✝ : j ∈ univ
h : j = i
⊢ ‖update (s.piecewise m₂ m₁) i (m₁ i - m₂ i) i‖ ≤ if i = i then ‖m₁ i - m₂ i‖ else max ‖m₁ i‖ ‖m₂ i‖
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: case h
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‖
c : 𝕜
hc : 1 < ‖c‖
hc₀ : 0 < ‖c‖
f : ContinuousMultilinearMap 𝕜 E G
hf : ∀ (x : (i : ι) → E i), ‖x‖ ≤ ‖c‖ → ‖f x‖ ≤ 1
⊢ ‖f‖ ≤ 1

Tactic:
refine opNorm_le_bound (by positivity) <|
      f.1.bound_of_shell_of_continuous f.2 (fun _ ↦ hc₀) (fun _ ↦ hc) fun x hcx hx ↦ ?_

Goal state after:
case h
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‖
c : 𝕜
hc : 1 < ‖c‖
hc₀ : 0 < ‖c‖
f : ContinuousMultilinearMap 𝕜 E G
hf : ∀ (x : (i : ι) → E i), ‖x‖ ≤ ‖c‖ → ‖f x‖ ≤ 1
x : (i : ι) → E i
hcx : ∀ (i : ι), ‖c‖ / ‖c‖ ≤ ‖x i‖
hx : ∀ (i : ι), ‖x i‖ < ‖c‖
⊢ ‖f.toMultilinearMap x‖ ≤ 1 * ∏ i, ‖x i‖
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‖
c : 𝕜
hc : 1 < ‖c‖
hc₀ : 0 < ‖c‖
f : ContinuousMultilinearMap 𝕜 E G
hf : ∀ (x : (i : ι) → E i), ‖x‖ ≤ ‖c‖ → ‖f x‖ ≤ 1
x : (i : ι) → E i
hcx : ∀ (i : ι), ‖c‖ / ‖c‖ ≤ ‖x i‖
hx : ∀ (i : ι), ‖x i‖ < ‖c‖
⊢ ∏ i, 1 ≤ ∏ i, ‖x i‖

Tactic:
gcongr with i

Goal state after:
case h1
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‖
c : 𝕜
hc : 1 < ‖c‖
hc₀ : 0 < ‖c‖
f : ContinuousMultilinearMap 𝕜 E G
hf : ∀ (x : (i : ι) → E i), ‖x‖ ≤ ‖c‖ → ‖f x‖ ≤ 1
x : (i : ι) → E i
hcx : ∀ (i : ι), ‖c‖ / ‖c‖ ≤ ‖x i‖
hx : ∀ (i : ι), ‖x i‖ < ‖c‖
i : ι
a✝ : i ∈ univ
⊢ 1 ≤ ‖x i‖
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
μ : Measure X
f : ι → X → E
h : ∀ (i : ι), AEStronglyMeasurable (f i) μ
⊢ AEStronglyMeasurable (fun x ↦ ∏'[L] (i : ι), f i x) μ

Tactic:
by
  choose g hg_meas hg_eq_f using h
  use (fun x => ∏'[L] i, g i x), StronglyMeasurable.tprod hg_meas
  filter_upwards [ae_all_iff.mpr hg_eq_f] with x h_eq using tprod_congr h_eq

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

Goal state before:
pp: case refine_1.inl
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 : Set (Set X)
T_count : T.Countable
T_open : ∀ u ∈ T, IsOpen[inst✝³] u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s) μ
T' : Set (Set X) := insert ∅ T
T'_count : T'.Countable
T'_ne : T'.Nonempty
u : ℕ → Set X
hu : T' = range u
n : ℕ
this : u n ∈ T'
h : u n = ∅
⊢ IsOpen[inst✝³] (u n)

Tactic:
rw [h]

Goal state after:
case refine_1.inl
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 : Set (Set X)
T_count : T.Countable
T_open : ∀ u ∈ T, IsOpen[inst✝³] u
sT : s ⊆ ⋃ u ∈ T, u
hT : ∀ u ∈ T, IntegrableOn f (u ∩ s) μ
T' : Set (Set X) := insert ∅ T
T'_count : T'.Countable
T'_ne : T'.Nonempty
u : ℕ → Set X
hu : T' = range u
n : ℕ
this : u n ∈ T'
h : u n = ∅
⊢ IsOpen[inst✝³] ∅
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 : ↥(Lp β 1 μ)
hf : Integrable (↑↑f) μ
⊢ toL1 (↑↑f) hf = f

Tactic:
by
  simp [Integrable.toL1]

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✝¹⁰ : Fintype ι'
inst✝⁹ : NontriviallyNormedField 𝕜
inst✝⁸ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁷ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝⁴ : SeminormedAddCommGroup G
inst✝³ : NormedSpace 𝕜 G
inst✝² : SeminormedAddCommGroup G'
inst✝¹ : NormedSpace 𝕜 G'
inst✝ : Fintype ι
f : ContinuousMultilinearMap 𝕜 E (G →L[𝕜] G')
x : G
m : (i : ι) → E i
⊢ ‖({ toFun := fun x ↦ { toFun := fun m ↦ (f m) x, map_update_add' := ⋯, map_update_smul' := ⋯ }, map_add' := ⋯,
            map_smul' := ⋯ }
          x)
        m‖ ≤
    ‖f‖ * ‖x‖ * ∏ i, ‖m i‖

Tactic:
rw [LinearMap.coe_mk, AddHom.coe_mk, MultilinearMap.coe_mk, mul_right_comm]
    apply ((f m).le_opNorm x).trans
    gcongr
    apply f.le_opNorm

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

Goal state before:
pp: 𝕜 : Type u
ι : Type v
E : ι → Type wE
E₁ : ι → Type wE₁
G : Type wG
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
g : ContinuousMultilinearMap 𝕜 E₁ G
f : (i : ι) → E i →L[𝕜] E₁ i
m : (i : ι) → E i
⊢ ‖g‖ * ∏ i, ‖(f i) (m i)‖ ≤ ‖g‖ * ∏ i, ‖f i‖ * ‖m i‖

Tactic:
gcongr with i

Goal state after:
case h1
𝕜 : Type u
ι : Type v
E : ι → Type wE
E₁ : ι → Type wE₁
G : Type wG
inst✝⁷ : NontriviallyNormedField 𝕜
inst✝⁶ : (i : ι) → SeminormedAddCommGroup (E i)
inst✝⁵ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝⁴ : (i : ι) → SeminormedAddCommGroup (E₁ i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E₁ i)
inst✝² : SeminormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
inst✝ : Fintype ι
g : ContinuousMultilinearMap 𝕜 E₁ G
f : (i : ι) → E i →L[𝕜] E₁ i
m : (i : ι) → E i
i : ι
a✝ : i ∈ univ
⊢ ‖(f i) (m i)‖ ≤ ‖f i‖ * ‖m i‖
Module: Mathlib.Analysis.Normed.Module.Multilinear.Basic

Goal state before:
pp: case h₂
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
⊢ ∏ a, ‖if h : ↑a ∈ s then m (e.symm ⟨↑a, ⋯⟩) ↑a else x ⟨↑a, ⋯⟩‖ ≤ ∏ i, ‖m (e.symm i)‖

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
m : α → (i : ι) → E₁ i
⊢ Fintype (Subtype (Membership.mem s))

Tactic:
gcongr with i

Goal state after:
case h1
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
i : Subtype (Membership.mem s)
a✝ : i ∈ univ
⊢ ‖if h : ↑i ∈ s then m (e.symm ⟨↑i, ⋯⟩) ↑i else x ⟨↑i, ⋯⟩‖ ≤ ‖m (e.symm i)‖
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✝ → (toSimpleFunc (f + g)) a✝ = (toSimpleFunc f) a✝ + (toSimpleFunc g) a✝

Tactic:
rw [h]

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.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✝ = (toSimpleFunc f) a✝ + (toSimpleFunc g) a✝

Tactic:
rw [h]

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✝ + (toSimpleFunc g) a✝

Tactic:
rw [h]

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.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 + ↑↑g) a✝ = ↑↑↑f a✝ + ↑↑↑g a✝

Tactic:
rw [h]

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✝ → (toSimpleFunc (-f)) a✝ = -(toSimpleFunc f) a✝

Tactic:
rw [h]

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.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✝ = -(toSimpleFunc f) a✝

Tactic:
rw [h]

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.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:
rw [h]

Goal state after:
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
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
h : ∀ (x : E), ‖f x‖ ≤ C * ‖g x‖
x : Eₗ
y : E
hxy : g y = x
⊢ ‖(g.ker.liftQ f ⋯ ∘ₛₗ ↑g.quotKerEquivRange.symm) ⟨x, ⋯⟩‖ ≤ C * ‖⟨x, ⋯⟩‖

Tactic:
simpa [← hxy] using h y

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✝ → (toSimpleFunc (f - g)) a✝ = (toSimpleFunc f) a✝ - (toSimpleFunc g) a✝

Tactic:
rw [h]

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.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✝ = (toSimpleFunc f) a✝ - (toSimpleFunc g) a✝

Tactic:
rw [h]

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✝ - (toSimpleFunc g) a✝

Tactic:
rw [h]

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.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 - ↑↑g) a✝ = ↑↑↑f a✝ - ↑↑↑g a✝

Tactic:
rw [h]

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

Goal state before:
pp: α : Type u_1
E : Type u_4
𝕜 : Type u_6
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
k : 𝕜
f : ↥(simpleFunc E p μ)
a✝ : α
h : (toSimpleFunc (k • f)) a✝ = ↑↑(k • ↑f) a✝
⊢ (toSimpleFunc f) a✝ = ↑↑↑f a✝ → ↑↑(k • ↑f) a✝ = k • ↑↑↑f a✝ → (toSimpleFunc (k • f)) a✝ = k • (toSimpleFunc f) a✝

Tactic:
rw [h]

Goal state after:
α : Type u_1
E : Type u_4
𝕜 : Type u_6
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
k : 𝕜
f : ↥(simpleFunc E p μ)
a✝ : α
h : (toSimpleFunc (k • f)) a✝ = ↑↑(k • ↑f) a✝
⊢ (toSimpleFunc f) a✝ = ↑↑↑f a✝ → ↑↑(k • ↑f) a✝ = k • ↑↑↑f a✝ → ↑↑(k • ↑f) a✝ = k • (toSimpleFunc f) a✝
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: α : Type u_1
E : Type u_4
𝕜 : Type u_6
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
k : 𝕜
f : ↥(simpleFunc E p μ)
a✝ : α
h✝ : (toSimpleFunc (k • f)) a✝ = ↑↑(k • ↑f) a✝
h : (toSimpleFunc f) a✝ = ↑↑↑f a✝
⊢ ↑↑(k • ↑f) a✝ = k • ↑↑↑f a✝ → ↑↑(k • ↑f) a✝ = k • (toSimpleFunc f) a✝

Tactic:
rw [h]

Goal state after:
α : Type u_1
E : Type u_4
𝕜 : Type u_6
inst✝⁴ : MeasurableSpace α
inst✝³ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
inst✝² : NormedRing 𝕜
inst✝¹ : Module 𝕜 E
inst✝ : IsBoundedSMul 𝕜 E
k : 𝕜
f : ↥(simpleFunc E p μ)
a✝ : α
h✝ : (toSimpleFunc (k • f)) a✝ = ↑↑(k • ↑f) a✝
h : (toSimpleFunc f) a✝ = ↑↑↑f a✝
⊢ ↑↑(k • ↑f) a✝ = k • ↑↑↑f a✝ → ↑↑(k • ↑f) a✝ = k • ↑↑↑f a✝