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

Goal state before:
pp: case inr.const.inl
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_ne_top : p ≠ ∞
P : (α → E) → Prop
h0P :
  ∀ (c : E) ⦃s : Set α⦄,
    MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤ ε ∧ P g
h1P : ∀ (f g : α → E), P f → P g → P (f + g)
h2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ
f : α → E
hf : MemLp f p μ
ε✝ : ℝ≥0∞
hε : ε✝ ≠ 0
hp_pos : p ≠ 0
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
εpos : ε ≠ 0
Hs : MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p μ
g : α → E
hg : eLpNorm (-(g - ∅.indicator fun x ↦ 0)) p μ ≤ ε
Pg : P g
⊢ ∃ g, eLpNorm (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0)) - g) p μ ≤ ε ∧ P g

Tactic:
neg_sub

Goal state after:
case inr.const.inl
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_ne_top : p ≠ ∞
P : (α → E) → Prop
h0P :
  ∀ (c : E) ⦃s : Set α⦄,
    MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤ ε ∧ P g
h1P : ∀ (f g : α → E), P f → P g → P (f + g)
h2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ
f : α → E
hf : MemLp f p μ
ε✝ : ℝ≥0∞
hε : ε✝ ≠ 0
hp_pos : p ≠ 0
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
εpos : ε ≠ 0
Hs : MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0))) p μ
g : α → E
hg : eLpNorm ((∅.indicator fun x ↦ 0) - g) p μ ≤ ε
Pg : P g
⊢ ∃ g, eLpNorm (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α 0) (SimpleFunc.const α 0)) - g) p μ ≤ ε ∧ P g
Module: Mathlib.MeasureTheory.Function.SimpleFuncDenseLp

Goal state before:
pp: case inr.const.inr
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_ne_top : p ≠ ∞
P : (α → E) → Prop
h0P :
  ∀ (c : E) ⦃s : Set α⦄,
    MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤ ε ∧ P g
h1P : ∀ (f g : α → E), P f → P g → P (f + g)
h2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ
f : α → E
hf : MemLp f p μ
ε✝ : ℝ≥0∞
hε : ε✝ ≠ 0
hp_pos : p ≠ 0
c : E
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
εpos : ε ≠ 0
Hs : MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p μ
hc : c ≠ 0
this : μ s < ∞
g : α → E
hg : eLpNorm (-(g - s.indicator fun x ↦ c)) p μ ≤ ε
Pg : P g
⊢ ∃ g, eLpNorm (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) - g) p μ ≤ ε ∧ P g

Tactic:
neg_sub

Goal state after:
case inr.const.inr
α : Type u_1
E : Type u_4
inst✝¹ : MeasurableSpace α
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure α
hp_ne_top : p ≠ ∞
P : (α → E) → Prop
h0P :
  ∀ (c : E) ⦃s : Set α⦄,
    MeasurableSet s → μ s < ∞ → ∀ {ε : ℝ≥0∞}, ε ≠ 0 → ∃ g, eLpNorm (g - s.indicator fun x ↦ c) p μ ≤ ε ∧ P g
h1P : ∀ (f g : α → E), P f → P g → P (f + g)
h2P : ∀ (f : α → E), P f → AEStronglyMeasurable f μ
f : α → E
hf : MemLp f p μ
ε✝ : ℝ≥0∞
hε : ε✝ ≠ 0
hp_pos : p ≠ 0
c : E
s : Set α
hs : MeasurableSet s
ε : ℝ≥0∞
εpos : ε ≠ 0
Hs : MemLp (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0))) p μ
hc : c ≠ 0
this : μ s < ∞
g : α → E
hg : eLpNorm ((s.indicator fun x ↦ c) - g) p μ ≤ ε
Pg : P g
⊢ ∃ g, eLpNorm (⇑(SimpleFunc.piecewise s hs (SimpleFunc.const α c) (SimpleFunc.const α 0)) - g) p μ ≤ ε ∧ P g
Module: Mathlib.Topology.Algebra.ContinuousAffineMap

Goal state before:
pp: R : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝¹⁵ : Ring R
inst✝¹⁴ : AddCommGroup V
inst✝¹³ : Module R V
inst✝¹² : TopologicalSpace P
inst✝¹¹ : AddTorsor V P
inst✝¹⁰ : AddCommGroup W
inst✝⁹ : Module R W
inst✝⁸ : TopologicalSpace Q
inst✝⁷ : AddTorsor W Q
inst✝⁶ : TopologicalSpace R
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace W
inst✝³ : ContinuousSMul R V
inst✝² : ContinuousSMul R W
inst✝¹ : ContinuousVAdd V P
inst✝ : ContinuousVAdd W Q
f : P →ᴬ[R] Q
p₀ p₁ : P
c : R
⊢ f ((lineMap p₀ p₁) c) = (lineMap (f p₀) (f p₁)) c

Tactic:
coe_lineMap_eq,

Goal state after:
R : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝¹⁵ : Ring R
inst✝¹⁴ : AddCommGroup V
inst✝¹³ : Module R V
inst✝¹² : TopologicalSpace P
inst✝¹¹ : AddTorsor V P
inst✝¹⁰ : AddCommGroup W
inst✝⁹ : Module R W
inst✝⁸ : TopologicalSpace Q
inst✝⁷ : AddTorsor W Q
inst✝⁶ : TopologicalSpace R
inst✝⁵ : TopologicalSpace V
inst✝⁴ : TopologicalSpace W
inst✝³ : ContinuousSMul R V
inst✝² : ContinuousSMul R W
inst✝¹ : ContinuousVAdd V P
inst✝ : ContinuousVAdd W Q
f : P →ᴬ[R] Q
p₀ p₁ : P
c : R
⊢ f ((AffineMap.lineMap p₀ p₁) c) = (AffineMap.lineMap (f p₀) (f p₁)) c
Module: Mathlib.Topology.Algebra.ContinuousAffineMap

Goal state before:
pp: case refine_2
R : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : Module R V
inst✝⁹ : TopologicalSpace P
inst✝⁸ : AddTorsor V P
inst✝⁷ : AddCommGroup W
inst✝⁶ : Module R W
inst✝⁵ : TopologicalSpace Q
inst✝⁴ : AddTorsor W Q
inst✝³ : TopologicalSpace V
inst✝² : IsTopologicalAddTorsor P
inst✝¹ : TopologicalSpace W
inst✝ : IsTopologicalAddTorsor Q
f : P →ᴬ[R] Q
h : (↑f).linear = 0
x✝ : V
⊢ f.contLinear x✝ = 0 x✝

Tactic:
← coe_linear_eq_coe_contLinear,

Goal state after:
case refine_2
R : Type u_1
V : Type u_2
W : Type u_3
P : Type u_4
Q : Type u_5
inst✝¹² : Ring R
inst✝¹¹ : AddCommGroup V
inst✝¹⁰ : Module R V
inst✝⁹ : TopologicalSpace P
inst✝⁸ : AddTorsor V P
inst✝⁷ : AddCommGroup W
inst✝⁶ : Module R W
inst✝⁵ : TopologicalSpace Q
inst✝⁴ : AddTorsor W Q
inst✝³ : TopologicalSpace V
inst✝² : IsTopologicalAddTorsor P
inst✝¹ : TopologicalSpace W
inst✝ : IsTopologicalAddTorsor Q
f : P →ᴬ[R] Q
h : (↑f).linear = 0
x✝ : V
⊢ (↑f).linear x✝ = 0 x✝
Module: Mathlib.Topology.Algebra.AffineSubspace

Goal state before:
pp: R : Type u_1
V : Type u_2
P : Type u_3
W : Type u_4
Q : Type u_5
inst✝⁹ : Ring R
inst✝⁸ : AddCommGroup V
inst✝⁷ : Module R V
inst✝⁶ : TopologicalSpace P
inst✝⁵ : AddTorsor V P
inst✝⁴ : AddCommGroup W
inst✝³ : Module R W
inst✝² : TopologicalSpace Q
inst✝¹ : AddTorsor W Q
e : P ≃ᴬ[R] Q
s : AffineSubspace R P
inst✝ : Nonempty ↥s
⊢ Continuous ((↑e).affineSubspaceMap s).toFun

Tactic:
simpa [Topology.IsEmbedding.subtypeVal.continuous_iff] using!
      (e.continuous.comp continuous_subtype_val).congr fun _ => rfl

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

Goal state before:
pp: case neg
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ
r_pos : 0 < r
c : ℝ
f : E →ₗ[𝕜] 𝕜
h : ∀ z ∈ sphere 0 r, ‖f z‖ ≤ c
z : E
z_zero : ¬z = 0
⊢ ‖f z‖ ≤ c / r * ‖z‖

Tactic:
set z₁ := ((r : 𝕜) * (‖z‖ : 𝕜)⁻¹) • z with hz₁

Goal state after:
case neg
𝕜 : Type u_1
inst✝² : RCLike 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
r : ℝ
r_pos : 0 < r
c : ℝ
f : E →ₗ[𝕜] 𝕜
h : ∀ z ∈ sphere 0 r, ‖f z‖ ≤ c
z : E
z_zero : ¬z = 0
z₁ : E := (↑r * (↑‖z‖)⁻¹) • z
hz₁ : z₁ = (↑r * (↑‖z‖)⁻¹) • z
⊢ ‖f z‖ ≤ c / r * ‖z‖
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)
⊢ SimpleFunc.setToSimpleFunc T (toSimpleFunc (f + g)) =
    SimpleFunc.setToSimpleFunc T (toSimpleFunc f) + SimpleFunc.setToSimpleFunc T (toSimpleFunc g)

Tactic:
← SimpleFunc.setToSimpleFunc_add T h_add (SimpleFunc.integrable f)
      (SimpleFunc.integrable g)

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 →₁ₛ[μ] E)
⊢ SimpleFunc.setToSimpleFunc T (toSimpleFunc (f + g)) = SimpleFunc.setToSimpleFunc T (toSimpleFunc f + toSimpleFunc g)
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
𝕜 : Type u_6
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedSpace ℝ E
inst✝⁷ : NormedAddCommGroup F
inst✝⁶ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
inst✝⁵ : NormedRing 𝕜
inst✝⁴ : Module 𝕜 E
inst✝³ : Module 𝕜 F
inst✝² : IsBoundedSMul 𝕜 E
inst✝¹ : IsBoundedSMul 𝕜 F
inst✝ : CompleteSpace F
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
h_smul : ∀ (c : 𝕜) (s : Set α) (x : E), (T s) (c • x) = c • (T s) x
f : ↥(α →₁ₛ[μ] E)
⊢ DenseRange ⇑(coeToLp α E 𝕜)

Tactic:
exact simpleFunc.denseRange one_ne_top

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
A : ↥(Lp E 1 μ) →L[ℝ] F
hA : ∀ (f : ↥(α →₁ₛ[μ] E)), (setToL1SCLM α E μ hT) f = A ↑f
f : ↥(Lp E 1 μ)
⊢ setToL1 hT = A

Tactic:
apply ContinuousLinearMap.extend_unique

Goal state after:
case h_dense
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
A : ↥(Lp E 1 μ) →L[ℝ] F
hA : ∀ (f : ↥(α →₁ₛ[μ] E)), (setToL1SCLM α E μ hT) f = A ↑f
f : ↥(Lp E 1 μ)
⊢ DenseRange ⇑(coeToLp α E ℝ)

case h_e
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
A : ↥(Lp E 1 μ) →L[ℝ] F
hA : ∀ (f : ↥(α →₁ₛ[μ] E)), (setToL1SCLM α E μ hT) f = A ↑f
f : ↥(Lp E 1 μ)
⊢ IsUniformInducing ⇑(coeToLp α E ℝ)

case H
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
A : ↥(Lp E 1 μ) →L[ℝ] F
hA : ∀ (f : ↥(α →₁ₛ[μ] E)), (setToL1SCLM α E μ hT) f = A ↑f
f : ↥(Lp E 1 μ)
⊢ A ∘SL coeToLp α E ℝ = setToL1SCLM α E μ hT
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:
by rw [yy', ← smul_sub, norm_smul]

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

Goal state before:
pp: case indicatorConst
α : Type u_1
E : Type u_2
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
G'' : Type u_8
inst✝⁵ : NormedAddCommGroup G''
inst✝⁴ : PartialOrder G''
inst✝³ : IsOrderedAddMonoid G''
inst✝² : NormedSpace ℝ G''
inst✝¹ : CompleteSpace G''
inst✝ : OrderClosedTopology G''
T T' : Set α → E →L[ℝ] G''
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ T' C'
hTT' : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : E), (T s) x ≤ (T' s) x
c : E
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
⊢ (setToL1 hT) ↑(indicatorConst 1 hs ⋯ c) ≤ (setToL1 hT') ↑(indicatorConst 1 hs ⋯ c)

Tactic:
rw [setToL1_simpleFunc_indicatorConst hT hs hμs, setToL1_simpleFunc_indicatorConst hT' hs hμs]
    exact hTT' s hs hμs c

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

Goal state before:
pp: case hs
X : Type u_2
E : Type u_3
mX : MeasurableSpace X
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : CompleteSpace E
inst✝ : MeasurableSingletonClass X
μ : Measure X
f : X → E
s : Set X
hs : s.Countable
hf : Integrable f (μ.restrict s)
hi : Countable { x // x ∈ s }
⊢ MeasurableSet s

Tactic:
exact Countable.measurableSet hs

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

Goal state before:
pp: case neg
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 g : α → E
hT : DominatedFinMeasAdditive μ T C
h : f =ᵐ[μ] g
hF : CompleteSpace F
hfi : ¬Integrable f μ
⊢ setToFun μ T hT f = setToFun μ T hT g

Tactic:
· have hgi : ¬Integrable g μ := by rw [integrable_congr h] at hfi; exact hfi
    rw [setToFun_undef hT hfi, setToFun_undef hT hgi]

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
T : Set α → E →L[ℝ] F
C : ℝ
inst✝ : CompleteSpace F
hT : DominatedFinMeasAdditive μ T C
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E
⊢ setToFun μ T hT (s.indicator fun x_1 ↦ x) = (T s) x

Tactic:
by
  rw [setToFun_congr_ae hT (@indicatorConstLp_coeFn _ _ _ 1 _ _ _ hs hμs x).symm]
  rw [L1.setToFun_eq_setToL1 hT]
  exact L1.setToL1_indicatorConstLp hT hs hμs x

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

Goal state before:
pp: 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 : AEMeasurable φ μ
f : β → G
hfm : AEStronglyMeasurable f (Measure.map φ μ)
g : β → G := AEStronglyMeasurable.mk f hfm
⊢ ∫ (y : β), g y ∂Measure.map φ μ = ∫ (y : β), g y ∂Measure.map (AEMeasurable.mk φ hφ) μ

Tactic:
congr 1; exact Measure.map_congr hφ.ae_eq_mk

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

Goal state before:
pp: case inl
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetricSpace β
f g : α →ᵇ β
h✝ : IsEmpty α
⊢ dist f g = ⨆ x, dist (f x) (g x)

pp: case inr
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetricSpace β
f g : α →ᵇ β
h✝ : Nonempty α
⊢ dist f g = ⨆ x, dist (f x) (g x)

Tactic:
· rw [iSup_of_empty', Real.sSup_empty, dist_zero_of_empty]

Goal state after:
case inr
α : Type u
β : Type v
inst✝¹ : TopologicalSpace α
inst✝ : PseudoMetricSpace β
f g : α →ᵇ β
h✝ : Nonempty α
⊢ dist f g = ⨆ x, dist (f x) (g 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✝² : TopologicalSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : PseudoMetricSpace γ
f g : α →ᵇ β
x✝ : α
C : ℝ
s : Set α
hs : IsClopen s
x y : α
⊢ dist (s.indicator 1 x) (s.indicator 1 y) ≤ 1

Tactic:
by_cases hx : x ∈ s <;> by_cases hy : y ∈ s <;> simp [hx, hy]

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

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

Tactic:
have key (c : ℝ) (hc : 0 ≤ c) : ‖f + g‖ ≤ c ↔ max ‖f‖ ‖g‖ ≤ c := by
    simp_rw [norm_le hc, hfg', max_le_iff, norm_le hc, forall_and]

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

Goal state before:
pp: F : Type u_1
inst✝² : NormedRing F
inst✝¹ : NormOneClass F
inst✝ : NormMulClass F
k : ℕ
r : F
hr : ‖r‖ < 1
u : ℕ → F
hu : u =O[atTop] fun n ↦ ↑(n ^ k)
r' : ℝ
hrr' : ‖r‖ < r'
h : r' < 1
⊢ (fun n ↦ ‖u n * r ^ n‖) =O[atTop] fun x ↦ ‖r' ^ x‖

Tactic:
calc
  fun n ↦ ‖(u n) * r ^ n‖
  _ =O[atTop] fun n ↦ ‖u n‖ * ‖r‖ ^ n := by
      apply (IsBigOWith.of_bound (c := ‖(1 : ℝ)‖) ?_).isBigO
      filter_upwards [eventually_norm_pow_le r] with n hn
      simp
  _ =O[atTop] fun n ↦ ‖((n : F) ^ k)‖ * ‖r‖ ^ n := by
      simpa [Nat.cast_pow] using
      (isBigO_norm_left.mpr (isBigO_norm_right.mpr hu)).mul (isBigO_refl (fun n ↦ (‖r‖ ^ n)) atTop)
  _ =O[atTop] fun n ↦ ‖r' ^ n‖ := by
      convert!
        isBigO_norm_right.mpr
          (isBigO_norm_left.mpr
            (isLittleO_pow_const_mul_const_pow_const_pow_of_norm_lt k hrr').isBigO)
      simp only [norm_pow, norm_mul]

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

Goal state before:
pp: case pos.h_ind
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 C' : ℝ
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hfμ : Integrable f μ
hF : CompleteSpace F
h_int : ∀ (g : α → E), Integrable g μ → Integrable g μ'
c : E
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
hμ's : μ' s ≠ ∞
⊢ (T s) c = setToFun μ' T hT' (s.indicator fun x ↦ c)

Tactic:
setToFun_indicator_const hT' hs hμ's

Goal state after:
case pos.h_ind
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 C' : ℝ
μ' : Measure α
c' : ℝ≥0∞
hc' : c' ≠ ∞
hμ'_le : μ' ≤ c' • μ
hT : DominatedFinMeasAdditive μ T C
hT' : DominatedFinMeasAdditive μ' T C'
f : α → E
hfμ : Integrable f μ
hF : CompleteSpace F
h_int : ∀ (g : α → E), Integrable g μ → Integrable g μ'
c : E
s : Set α
hs : MeasurableSet s
hμs : μ s < ∞
hμ's : μ' s ≠ ∞
⊢ (T s) c = (T s) c
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: case h
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
⊢ |↑(thickenedIndicatorAux δ E x).toNNReal - ↑(thickenedIndicatorAux δ E y).toNNReal| ≤ 2

Tactic:
apply (abs_sub _ _).trans

Goal state after:
case h
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
⊢ |↑(thickenedIndicatorAux δ E x).toNNReal| + |↑(thickenedIndicatorAux δ E y).toNNReal| ≤ 2
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ₁ δ₂ : ℝ
δ₁_pos : 0 < δ₁
δ₂_pos : 0 < δ₂
hle : δ₁ ≤ δ₂
E : Set α
⊢ ⇑(thickenedIndicator δ₁_pos E) ≤ ⇑(thickenedIndicator δ₂_pos E)

Tactic:
by
  intro x
  apply (toNNReal_le_toNNReal (by finiteness) (by finiteness)).mpr
  apply thickenedIndicatorAux_mono hle

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

Goal state before:
pp: case insert
δ : Type u_1
δ' : Type u_2
X : δ → Type u_3
inst✝³ : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝² : DecidableEq δ
inst✝¹ : ∀ (i : δ), SigmaFinite (μ i)
inst✝ : DecidableEq δ'
e : δ' → δ
he : Injective e
f : ((i : δ') → X (e i)) → ℝ≥0∞
hf : Measurable f
h : Measurable fun x ↦ (fun {x_1} ↦ x) ∘' e
a✝ : δ'
s✝ : Finset δ'
hi : a✝ ∉ s✝
ih :
  ∀ (x : (i : δ) → X i),
    (∫⋯∫⁻_Finset.image e s✝, f ∘ fun x ↦ (fun {x_1} ↦ x) ∘' e ∂μ) x =
      (∫⋯∫⁻_s✝, f ∂(fun {x} ↦ μ) ∘' e) ((fun {x_1} ↦ x) ∘' e)
x : (i : δ) → X i
⊢ ∫⁻ (xᵢ : X (e a✝)), (∫⋯∫⁻_Finset.image e s✝, f ∘ fun x ↦ (fun {x_1} ↦ x) ∘' e ∂μ) (update x (e a✝) xᵢ) ∂μ (e a✝) =
    ∫⁻ (xᵢ : X (e a✝)), (∫⋯∫⁻_s✝, f ∂(fun {x} ↦ μ) ∘' e) (update ((fun {x_1} ↦ x) ∘' e) a✝ xᵢ) ∂((fun {x} ↦ μ) ∘' e) a✝

Tactic:
ih,

Goal state after:
case insert
δ : Type u_1
δ' : Type u_2
X : δ → Type u_3
inst✝³ : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝² : DecidableEq δ
inst✝¹ : ∀ (i : δ), SigmaFinite (μ i)
inst✝ : DecidableEq δ'
e : δ' → δ
he : Injective e
f : ((i : δ') → X (e i)) → ℝ≥0∞
hf : Measurable f
h : Measurable fun x ↦ (fun {x_1} ↦ x) ∘' e
a✝ : δ'
s✝ : Finset δ'
hi : a✝ ∉ s✝
ih :
  ∀ (x : (i : δ) → X i),
    (∫⋯∫⁻_Finset.image e s✝, f ∘ fun x ↦ (fun {x_1} ↦ x) ∘' e ∂μ) x =
      (∫⋯∫⁻_s✝, f ∂(fun {x} ↦ μ) ∘' e) ((fun {x_1} ↦ x) ∘' e)
x : (i : δ) → X i
⊢ ∫⁻ (xᵢ : X (e a✝)), (∫⋯∫⁻_s✝, f ∂(fun {x} ↦ μ) ∘' e) ((fun {x_1} ↦ update x (e a✝) xᵢ) ∘' e) ∂μ (e a✝) =
    ∫⁻ (xᵢ : X (e a✝)), (∫⋯∫⁻_s✝, f ∂(fun {x} ↦ μ) ∘' e) (update ((fun {x_1} ↦ x) ∘' e) a✝ xᵢ) ∂((fun {x} ↦ μ) ∘' e) a✝
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: case refine_2
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
T : Set α → E →L[ℝ] F✝
C : ℝ
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
inst✝ : Countable ι
F : ι → α → E
f : α → E
bound : ι → α → ℝ
hF_meas : ∀ (n : ι), AEStronglyMeasurable (F n) μ
h_bound : ∀ (n : ι), ∀ᵐ (a : α) ∂μ, ‖F n a‖ ≤ bound n a
bound_summable : ∀ᵐ (a : α) ∂μ, Summable fun n ↦ bound n a
bound_integrable : Integrable (fun a ↦ ∑' (n : ι), bound n a) μ
h_lim : ∀ᵐ (a : α) ∂μ, HasSum (fun n ↦ F n a) (f a)
hb_nonneg : ∀ᵐ (a : α) ∂μ, ∀ (n : ι), 0 ≤ bound n a
hb_le_tsum : ∀ (n : ι), bound n ≤ᵐ[μ] fun a ↦ ∑' (n : ι), bound n a
hF_integrable : ∀ (n : ι), Integrable (F n) μ
⊢ ∀ᶠ (n : Finset ι) in (SummationFilter.unconditional ι).filter, ∀ᵐ (a : α) ∂μ, ‖∑ i ∈ n, F i a‖ ≤ ∑' (n : ι), bound n a

Tactic:
· filter_upwards with s
    filter_upwards [eventually_countable_forall.2 h_bound, hb_nonneg, bound_summable]
      with a hFa ha0 has
    calc
      ‖∑ n ∈ s, F n a‖ ≤ ∑ n ∈ s, bound n a := norm_sum_le_of_le _ fun n _ => hFa n
      _ ≤ ∑' n, bound n a := has.sum_le_tsum _ (fun n _ => ha0 n)

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✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
inst✝¹ : CompleteSpace E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
inst✝ : Countable ι
f : ι → α → E
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf' : ∑' (i : ι), ∫⁻ (a : α), ‖f i a‖ₑ ∂μ ≠ ∞
hF : CompleteSpace F
hf'' : ∀ (i : ι), AEMeasurable (fun x ↦ ‖f i x‖ₑ) μ
hhh : ∀ᵐ (a : α) ∂μ, Summable fun n ↦ ↑‖f n a‖₊
this : ∫⁻ (a : α), ∑' (n : ι), ‖f n a‖ₑ ∂μ < ∞
⊢ (fun a ↦ ↑(∑' (a_1 : ι), ‖f a_1 a‖₊)) =ᵐ[μ] fun a ↦ ∑' (n : ι), ↑‖f n a‖₊

Tactic:
filter_upwards [hhh] with a ha

Goal state after:
α : Type u_1
E : Type u_2
F : Type u_3
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
inst✝¹ : CompleteSpace E
hT : DominatedFinMeasAdditive μ T C
ι : Type u_7
inst✝ : Countable ι
f : ι → α → E
hf : ∀ (i : ι), AEStronglyMeasurable (f i) μ
hf' : ∑' (i : ι), ∫⁻ (a : α), ‖f i a‖ₑ ∂μ ≠ ∞
hF : CompleteSpace F
hf'' : ∀ (i : ι), AEMeasurable (fun x ↦ ‖f i x‖ₑ) μ
hhh : ∀ᵐ (a : α) ∂μ, Summable fun n ↦ ↑‖f n a‖₊
this : ∫⁻ (a : α), ∑' (n : ι), ‖f n a‖ₑ ∂μ < ∞
a : α
ha : Summable fun n ↦ ↑‖f n a‖₊
⊢ ↑(∑' (a_1 : ι), ‖f a_1 a‖₊) = ∑' (n : ι), ↑‖f n a‖₊