mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: 𝕜 : Type u
inst✝⁷ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace 𝕜 E
F : Type w
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace 𝕜 F
inst✝² : CompleteSpace 𝕜
X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : FiniteDimensional 𝕜 E
f : X → E →L[𝕜] F
s : Set X
x : X
⊢ ContinuousWithinAt f s x ↔ ∀ (y : E), ContinuousWithinAt (fun q ↦ (f q) y) s x

Tactic:
refine ⟨fun h y ↦ (apply 𝕜 F y).continuous.continuousAt.comp_continuousWithinAt h, fun h ↦ ?_⟩
  let e : (E →L[𝕜] F) ≃L[𝕜] Fin (finrank 𝕜 E) → F :=
    ((ContinuousLinearEquiv.ofFinrankEq (finrank_fin_fun 𝕜).symm).arrowCongr
      (1 : F ≃L[𝕜] F)).trans (ContinuousLinearEquiv.piRing _)
  rw [e.toHomeomorph.isInducing.continuousWithinAt_iff]
  exact continuousWithinAt_pi.mpr fun i ↦ h _

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

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
hT : DominatedFinMeasAdditive μ T C
β : Type u_7
x✝ : MeasurableSpace β
μ' : Measure β
φ : α → β
T' : Set β → E →L[ℝ] F
hT' : DominatedFinMeasAdditive μ' T' C'
f : β → E
hf : Integrable (f ∘ φ) μ
hfm : AEStronglyMeasurable f (Measure.map φ μ)
hφ : Measurable φ
hμ' : μ' ≤ Measure.map φ μ
h : ∀ (s : Set β) (x : E), MeasurableSet s → (T' s) x = (T (φ ⁻¹' s)) x
⊢ setToFun μ' T' hT' f = setToFun μ T hT (f ∘ φ)

Tactic:
by
  let g := hfm.mk
  have A : setToFun μ' T' hT' f = setToFun μ' T' hT' g :=
    setToFun_congr_ae _ (ae_mono hμ' hfm.ae_eq_mk)
  have B : setToFun μ T hT (f ∘ φ) = setToFun μ T hT (g ∘ φ) := by
    apply setToFun_congr_ae
    exact ae_of_ae_map hφ.aemeasurable hfm.ae_eq_mk
  rw [A, B]
  exact setToFun_of_le_map_of_stronglyMeasurable _ _
    (hf.congr (ae_of_ae_map hφ.aemeasurable hfm.ae_eq_mk)) hfm.stronglyMeasurable_mk hφ hμ' h

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

Goal state before:
pp: case e_f
α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
hE : CompleteSpace E
inst✝ : Unique α
f : α → E
x : α
⊢ f x = f default

Tactic:
congr

Goal state after:
case e_f
α : Type u_1
E : Type u_2
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
m : MeasurableSpace α
μ : Measure α
hE : CompleteSpace E
inst✝ : Unique α
f : α → E
x : α
⊢ x = default
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
f : α → ℝ≥0 × ℝ≥0∞ := ⋯
sub : ℝ≥0 × ℝ≥0∞ → ℝ≥0∞ := ⋯
⊢ ENNReal.ofReal δ ≠ 0

Tactic:
norm_num [δ_pos]

Goal state after:
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 : α
⊢ ∀ (y : α), dist (thickenedIndicatorAux δ E x).toNNReal (thickenedIndicatorAux δ E y).toNNReal ≤ 2

Tactic:
y

Goal state after:
case h
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
⊢ dist (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
δ : ℝ
δ_pos : 0 < δ
E : Set α
x : α
⊢ (thickenedIndicator δ_pos E) x ≤ 1

Tactic:
rw [thickenedIndicator.coeFn_eq_comp]

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x : α
⊢ (ENNReal.toNNReal ∘ thickenedIndicatorAux δ E) x ≤ 1
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x : α
⊢ ∀ (y : α), edist ((thickenedIndicator δ_pos E) x) ((thickenedIndicator δ_pos E) y) ≤ ↑δ.toNNReal⁻¹ * edist x y

Tactic:
y

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
δ : ℝ
δ_pos : 0 < δ
E : Set α
x y : α
⊢ edist ((thickenedIndicator δ_pos E) x) ((thickenedIndicator δ_pos E) y) ≤ ↑δ.toNNReal⁻¹ * edist x y
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
E : Type u_2
F : Type u_3
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C C' : ℝ
μ' : Measure α
hT_add : DominatedFinMeasAdditive (μ + μ') T C'
hT : DominatedFinMeasAdditive μ T C
f : α → E
hf : Integrable f (μ + μ')
⊢ setToFun (μ + μ') T hT_add f = setToFun μ T hT f

Tactic:
refine setToFun_congr_measure_of_integrable 1 one_ne_top ?_ hT_add hT f hf
  rw [one_smul]
  nth_rw 1 [← add_zero μ]
  exact add_le_add le_rfl bot_le

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

Goal state before:
pp: case pos
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
c : ℝ≥0∞
hc_ne_top : c ≠ ∞
hT : DominatedFinMeasAdditive μ T C
hT_smul : DominatedFinMeasAdditive (c • μ) T C'
f : α → E
hc0 : c = 0
⊢ setToFun μ T hT f = setToFun (c • μ) T hT_smul f

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 C' : ℝ
c : ℝ≥0∞
hc_ne_top : c ≠ ∞
hT : DominatedFinMeasAdditive μ T C
hT_smul : DominatedFinMeasAdditive (c • μ) T C'
f : α → E
hc0 : ¬c = 0
⊢ setToFun μ T hT f = setToFun (c • μ) T hT_smul f

Tactic:
· simp [hc0] at hT_smul
    have h : ∀ s, MeasurableSet s → μ s < ∞ → T s = 0 := fun s hs _ => hT_smul.eq_zero hs
    rw [setToFun_zero_left' _ h, setToFun_measure_zero]
    simp [hc0]

Goal state after:
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 C' : ℝ
c : ℝ≥0∞
hc_ne_top : c ≠ ∞
hT : DominatedFinMeasAdditive μ T C
hT_smul : DominatedFinMeasAdditive (c • μ) T C'
f : α → E
hc0 : ¬c = 0
⊢ setToFun μ T hT f = setToFun (c • μ) T hT_smul f
Module: Mathlib.MeasureTheory.Integral.Marginal

Goal state before:
pp: δ : Type u_1
X : δ → Type u_3
inst✝² : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝¹ : DecidableEq δ
s : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
i : δ
hi : i ∉ s
⊢ ∫⋯∫⁻_s ∪ {i}, f ∂μ = ∫⋯∫⁻_s, fun x ↦ ∫⁻ (xᵢ : X i), f (update x i xᵢ) ∂μ i ∂μ

Tactic:
lmarginal_union (s := s) μ f hf (Finset.disjoint_singleton_right.mpr hi),

Goal state after:
δ : Type u_1
X : δ → Type u_3
inst✝² : (i : δ) → MeasurableSpace (X i)
μ : (i : δ) → Measure (X i)
inst✝¹ : DecidableEq δ
s : Finset δ
inst✝ : ∀ (i : δ), SigmaFinite (μ i)
f : ((i : δ) → X i) → ℝ≥0∞
hf : Measurable f
i : δ
hi : i ∉ s
⊢ ∫⋯∫⁻_s, ∫⋯∫⁻_{i}, f ∂μ ∂μ = ∫⋯∫⁻_s, fun x ↦ ∫⁻ (xᵢ : X i), f (update x i xᵢ) ∂μ i ∂μ
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x : α
⊢ ∀ ⦃b : α⦄, x ≤ b → leftLim f x ≤ leftLim f b

Tactic:
y

Goal state after:
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : ConditionallyCompleteLinearOrder β
inst✝¹ : TopologicalSpace β
inst✝ : OrderTopology β
f : α → β
hf : Monotone f
x y : α
⊢ x ≤ y → leftLim f x ≤ leftLim f y
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' : ∫⁻ (a : α), ∑' (i : ι), ‖f i a‖ₑ ∂μ ≠ ∞
hF : CompleteSpace F
hf'' : ∀ (i : ι), AEMeasurable (fun x ↦ ‖f i x‖ₑ) μ
⊢ ∀ᵐ (a : α) ∂μ, Summable fun n ↦ ↑‖f n a‖₊

Tactic:
refine (ae_lt_top' (AEMeasurable.tsum hf'') hf').mono ?_

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' : ∫⁻ (a : α), ∑' (i : ι), ‖f i a‖ₑ ∂μ ≠ ∞
hF : CompleteSpace F
hf'' : ∀ (i : ι), AEMeasurable (fun x ↦ ‖f i x‖ₑ) μ
⊢ ∀ (x : α), ∑' (i : ι), ‖f i x‖ₑ < ∞ → Summable fun n ↦ ↑‖f n x‖₊
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝² : LinearOrder R
inst✝¹ : TopologicalSpace R
f✝ : StieltjesFunction R
inst✝ : OrderTopology R
f : R → ℝ
hf : Monotone f
x : R
s : Set ℝ
hs : s ∈ 𝓝 (rightLim f x)
l u : ℝ
hlu : rightLim f x ∈ Ioo l u
lus : Ioo l u ⊆ s
hx : ∀ (y : R), y ≤ x
y : R
hy : x ≤ y
⊢ rightLim f y ∈ s

Tactic:
rw [show y = x by exact le_antisymm (hx y) hy]

Goal state after:
R : Type u_1
inst✝² : LinearOrder R
inst✝¹ : TopologicalSpace R
f✝ : StieltjesFunction R
inst✝ : OrderTopology R
f : R → ℝ
hf : Monotone f
x : R
s : Set ℝ
hs : s ∈ 𝓝 (rightLim f x)
l u : ℝ
hlu : rightLim f x ∈ Ioo l u
lus : Ioo l u ⊆ s
hx : ∀ (y : R), y ≤ x
y : R
hy : x ≤ y
⊢ rightLim f x ∈ s
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_5
m : MeasurableSpace X
μ : Measure X
s : Set X
hs : μ s ≠ ∞
⊢ ENNReal.ofReal (∫ (x : X) in s, 1 ∂μ) = ENNReal.ofReal (∫ (x : X) in s, ‖1‖ ∂μ)

Tactic:
by
      simp only [norm_one]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝ : TopologicalSpace G
μ : Content G
⊢ μ ⊥ = 0

Tactic:
by simpa [toNNReal_eq_zero_iff] using μ.sup_disjoint' ⊥ ⊥

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝ : TopologicalSpace G
μ : Content G
K : Compacts G
hK : ↑K ⊆ ↑⊥
⊢ μ K ≤ μ ⊥

Tactic:
have : K = ⊥ := by
    ext1
    rw [subset_empty_iff.mp hK, Compacts.coe_bot]

Goal state after:
G : Type w
inst✝ : TopologicalSpace G
μ : Content G
K : Compacts G
hK : ↑K ⊆ ↑⊥
this : K = ⊥
⊢ μ K ≤ μ ⊥
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ, x ∈ t → f x = 0
hf : AEStronglyMeasurable f (μ.restrict t)
⊢ ∀ᵐ (x : X) ∂μ, x ∈ t → AEStronglyMeasurable.mk f hf x = 0 x

Tactic:
filter_upwards [ae_imp_of_ae_restrict hf.ae_eq_mk, ht_eq] with x hx h'x h''x

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : X → E
t : Set X
μ : Measure X
ht_eq : ∀ᵐ (x : X) ∂μ, x ∈ t → f x = 0
hf : AEStronglyMeasurable f (μ.restrict t)
x : X
hx : x ∈ t → f x = AEStronglyMeasurable.mk f hf x
h'x : x ∈ t → f x = 0
h''x : x ∈ t
⊢ AEStronglyMeasurable.mk f hf x = 0 x
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this : Nonempty R
g : ℕ → R × R
hg : ∀ (i : ℕ), Icc a' b ∩ s i ⊆ Iotop (g i).1 (g i).2 ∧ ofReal (↑f (g i).2 - ↑f (g i).1) < f.length (s i) + ↑(ε' i)
I_subset : Icc a' b ⊆ ⋃ i, Iotop (g i).1 (g i).2
⊢ ∑' (i : ℕ), (f.length (s i) + ↑(ε' i)) + ↑δ = ∑' (i : ℕ), f.length (s i) + ∑' (i : ℕ), ↑(ε' i) + ↑δ

Tactic:
ENNReal.tsum_add

Goal state after:
R : Type u_1
inst✝⁴ : LinearOrder R
inst✝³ : TopologicalSpace R
f : StieltjesFunction R
inst✝² : OrderTopology R
inst✝¹ : CompactIccSpace R
inst✝ : DenselyOrdered R
a b : R
hab : a < b
s : ℕ → Set R
hs : Ioc a b ⊆ ⋃ i, s i
ε : ℝ≥0
εpos : 0 < ε
h : ∑' (i : ℕ), f.length (s i) < ∞
δ : ℝ≥0 := ε / 2
δpos : 0 < ↑δ
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑δ
a' : R
ha' : ↑f a' - ↑f a < ↑δ
aa' : a < a'
this : Nonempty R
g : ℕ → R × R
hg : ∀ (i : ℕ), Icc a' b ∩ s i ⊆ Iotop (g i).1 (g i).2 ∧ ofReal (↑f (g i).2 - ↑f (g i).1) < f.length (s i) + ↑(ε' i)
I_subset : Icc a' b ⊆ ⋃ i, Iotop (g i).1 (g i).2
⊢ ∑' (a : ℕ), f.length (s a) + ∑' (a : ℕ), ↑(ε' a) + ↑δ = ∑' (i : ℕ), f.length (s i) + ∑' (i : ℕ), ↑(ε' i) + ↑δ
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝⁶ : LinearOrder R
inst✝⁵ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁴ : OrderTopology R
inst✝³ : CompactIccSpace R
inst✝² : MeasurableSpace R
inst✝¹ : BorelSpace R
inst✝ : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ i, t i
ε : ℝ≥0
ε0 : 0 < ε
h : ∑' (i : ℕ), f.length (t i) < ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε
⊢ ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), f.outer t ≤ ∑' (i : ℕ), f.length (t i) + ∑' (i : ℕ), ↑(ε' i)

Tactic:
← ENNReal.tsum_add

Goal state after:
R : Type u_1
inst✝⁶ : LinearOrder R
inst✝⁵ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁴ : OrderTopology R
inst✝³ : CompactIccSpace R
inst✝² : MeasurableSpace R
inst✝¹ : BorelSpace R
inst✝ : DenselyOrdered R
s : Set R
t : ℕ → Set R
ht : s ⊆ ⋃ i, t i
ε : ℝ≥0
ε0 : 0 < ε
h : ∑' (i : ℕ), f.length (t i) < ∞
ε' : ℕ → ℝ≥0
ε'0 : ∀ (i : ℕ), 0 < ε' i
hε : ∑' (i : ℕ), ↑(ε' i) < ↑ε
⊢ ⨅ t, ⨅ (_ : s ⊆ t), ⨅ (_ : MeasurableSet t), f.outer t ≤ ∑' (a : ℕ), (f.length (t a) + ↑(ε' a))
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
x : α
⊢ x ∈ fundamentalInterior G s ↔ x ∈ s ∧ ∀ (g : G), g ≠ 1 → x ∉ g • s

Tactic:
by
  simp [fundamentalInterior]

Goal state after:
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
a b : G
hab : a ≠ b
⊢ ∀ ⦃a_1 : α⦄, a_1 ∈ (fun g ↦ g • fundamentalInterior G s) a → a_1 ∉ (fun g ↦ g • fundamentalInterior G s) b

Tactic:
rintro _ ⟨x, hx, rfl⟩ ⟨y, hy, hxy⟩

Goal state after:
G : Type u_1
α : Type u_3
inst✝¹ : Group G
inst✝ : MulAction G α
s : Set α
a b : G
hab : a ≠ b
x : α
hx : x ∈ fundamentalInterior G s
y : α
hy : y ∈ fundamentalInterior G s
hxy : (fun x ↦ b • x) y = (fun x ↦ a • x) x
⊢ False
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
mX : MeasurableSpace X
s : Set X
μ : Measure X
f : X → ℝ
hf : 0 ≤ᵐ[μ.restrict s] f
hfi : IntegrableOn f s μ
⊢ 0 < ∫ (x : X) in s, f x ∂μ ↔ 0 < μ (support f ∩ s)

Tactic:
rw [integral_pos_iff_support_of_nonneg_ae hf hfi, Measure.restrict_apply₀]
  rw [support_eq_preimage]
  exact hfi.aestronglyMeasurable.aemeasurable.nullMeasurable (measurableSet_singleton 0).compl

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Basic

Goal state before:
pp: case left.refine_1
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : (interior V).Nonempty
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V
h2s : s.card = index (K₁.carrier ∪ K₂.carrier) V
this : ∀ K ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' V, index K V ≤ {g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K).Nonempty}.card
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ (g₁ * g₂)⁻¹⁻¹ ∈ V

pp: case left.refine_2
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : (interior V).Nonempty
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V
h2s : s.card = index (K₁.carrier ∪ K₂.carrier) V
this : ∀ K ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' V, index K V ≤ {g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K).Nonempty}.card
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ g₂ * (g₁ * g₂)⁻¹ = g₁⁻¹

Tactic:
· simp only [inv_inv, h1g₂]

Goal state after:
case left.refine_2
G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₁ K₂ : Compacts G
V : Set G
hV : (interior V).Nonempty
h : Disjoint (K₁.carrier * V⁻¹) (K₂.carrier * V⁻¹)
s : Finset G
h1s : K₁.carrier ∪ K₂.carrier ⊆ ⋃ g ∈ s, (fun h ↦ g * h) ⁻¹' V
h2s : s.card = index (K₁.carrier ∪ K₂.carrier) V
this : ∀ K ⊆ ⋃ g ∈ s, (fun x ↦ g * x) ⁻¹' V, index K V ≤ {g ∈ s | ((fun x ↦ g * x) ⁻¹' V ∩ K).Nonempty}.card
g₁ : G
a✝ : g₁ ∈ s
g₂ : G
h2g₂ : g₂ ∈ K₁.carrier
g₃ : G
h2g₃ : g₃ ∈ K₂.carrier
h1g₃ : g₁ * g₃ ∈ V
h1g₂ : g₁ * g₂ ∈ V
⊢ g₂ * (g₁ * g₂)⁻¹ = g₁⁻¹
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : NullMeasurableSet s μ
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ ∫ (x : X) in toMeasurable μ s, f x ∂μ ≤ ∫ (x : X) in toMeasurable μ s, g x ∂μ

Tactic:
refine setIntegral_mono_on_ae ?_ ?_ ?_ ?_

Goal state after:
case refine_1
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : NullMeasurableSet s μ
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ IntegrableOn f (toMeasurable μ s) μ

case refine_2
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : NullMeasurableSet s μ
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ IntegrableOn g (toMeasurable μ s) μ

case refine_3
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : NullMeasurableSet s μ
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ MeasurableSet (toMeasurable μ s)

case refine_4
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : PartialOrder E
inst✝² : IsOrderedAddMonoid E
inst✝¹ : IsOrderedModule ℝ E
μ : Measure X
f g : X → E
s : Set X
inst✝ : ClosedIciTopology E
hf : IntegrableOn f s μ
hg : IntegrableOn g s μ
hs : NullMeasurableSet s μ
h : ∀ᵐ (x : X) ∂μ, x ∈ s → f x ≤ g x
⊢ ∀ᵐ (x : X) ∂μ, x ∈ toMeasurable μ s → f x ≤ g x
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x y : F
⊢ ⟪x - y, x - y⟫ = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, x⟫ + ⟪y, y⟫

Tactic:
by
  simp only [inner_sub_left, inner_sub_right]; ring

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Defs

Goal state before:
pp: 𝕜 : Type u_1
F : Type u_3
inst✝² : RCLike 𝕜
inst✝¹ : AddCommGroup F
inst✝ : Module 𝕜 F
c : PreInnerProductSpace.Core 𝕜 F
x : F
t : ℝ
⊢ re ⟪↑t • x, ↑t • x⟫ = normSq x * t * t

Tactic:
by
  simp [inner_smul_ofReal_left, inner_smul_ofReal_right, normSq]

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

Goal state before:
pp: X : Type u_1
F : Type u_4
mX : MeasurableSpace X
𝕜 : Type u_5
inst✝³ : NormedRing 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : Module 𝕜 F
inst✝ : IsBoundedSMul 𝕜 F
p : ℝ≥0∞
μ : Measure X
c : 𝕜
f : ↥(Lp F p μ)
s : Set X
⊢ ↑↑(MemLp.toLp ↑↑(c • f) ⋯) =ᵐ[μ.restrict s] ↑↑(c • MemLp.toLp ↑↑f ⋯)

Tactic:
refine (ae_restrict_of_ae (Lp.coeFn_smul c f)).mp ?_

Goal state after:
X : Type u_1
F : Type u_4
mX : MeasurableSpace X
𝕜 : Type u_5
inst✝³ : NormedRing 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : Module 𝕜 F
inst✝ : IsBoundedSMul 𝕜 F
p : ℝ≥0∞
μ : Measure X
c : 𝕜
f : ↥(Lp F p μ)
s : Set X
⊢ ∀ᵐ (x : X) ∂μ.restrict s, ↑↑(c • f) x = (c • ↑↑f) x → ↑↑(MemLp.toLp ↑↑(c • f) ⋯) x = ↑↑(c • MemLp.toLp ↑↑f ⋯) x
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
μ : Measure X
inst✝ : NormedSpace ℝ E
s : Set X
this : Fact (1 ≤ 1)
f : ↥(Lp E 1 μ)
⊢ ∫ (x : X) in s, ↑↑f x ∂μ = (integral (μ.restrict s) ∘ fun f ↦ ↑↑((LpToLpRestrictCLM X E ℝ μ 1 s) f)) f

Tactic:
Function.comp_apply,

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝¹ : NormedAddCommGroup E
μ : Measure X
inst✝ : NormedSpace ℝ E
s : Set X
this : Fact (1 ≤ 1)
f : ↥(Lp E 1 μ)
⊢ ∫ (x : X) in s, ↑↑f x ∂μ = integral (μ.restrict s) ↑↑((LpToLpRestrictCLM X E ℝ μ 1 s) f)
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ ⟪x - y, x - y⟫ = ⟪x, x⟫ - ⟪x, y⟫ - ⟪y, x⟫ + ⟪y, y⟫

Tactic:
by
  simp only [inner_sub_left, inner_sub_right]; ring

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ re ⟪x, y⟫ = (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - ‖x - y‖ * ‖x - y‖) / 2

Tactic:
@norm_sub_mul_self 𝕜

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ re ⟪x, y⟫ = (‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ - (‖x‖ * ‖x‖ - 2 * re ⟪x, y⟫ + ‖y‖ * ‖y‖)) / 2
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ re ⟪x, y⟫ = (‖x‖ * ‖x‖ + 2 * re ⟪x, y⟫ + ‖y‖ * ‖y‖ - ‖x - y‖ * ‖x - y‖) / 4

Tactic:
@norm_sub_mul_self 𝕜

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
⊢ re ⟪x, y⟫ = (‖x‖ * ‖x‖ + 2 * re ⟪x, y⟫ + ‖y‖ * ‖y‖ - (‖x‖ * ‖x‖ - 2 * re ⟪x, y⟫ + ‖y‖ * ‖y‖)) / 4
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
| ‖x‖ = ‖y‖

Tactic:
← mul_self_inj_of_nonneg (norm_nonneg _) (norm_nonneg _)

Goal state after:
F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
| ‖x‖ * ‖x‖ = ‖y‖ * ‖y‖
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ ⟪x + y, x - y⟫_ℝ = 0 ↔ ‖x‖ = ‖y‖

Tactic:
conv_rhs => rw [← mul_self_inj_of_nonneg (norm_nonneg _) (norm_nonneg _)]

Goal state after:
F : Type u_3
inst✝¹ : SeminormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
x y : F
⊢ ⟪x + y, x - y⟫_ℝ = 0 ↔ ‖x‖ * ‖x‖ = ‖y‖ * ‖y‖
Module: Mathlib.Analysis.InnerProductSpace.Basic

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v w : E
h : ⟪v, w⟫ = 0
⊢ ‖w - v‖ = ‖w + v‖

Tactic:
← mul_self_inj_of_nonneg (norm_nonneg _) (norm_nonneg _)

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : RCLike 𝕜
inst✝¹ : SeminormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
v w : E
h : ⟪v, w⟫ = 0
⊢ ‖w - v‖ * ‖w - v‖ = ‖w + v‖ * ‖w + v‖