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

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf : ∀ (n : ℕ), Integrable (f n) μ
hF : Integrable F μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
f' : ℕ → α → ℝ := fun n x ↦ f n x - f 0 x
⊢ Tendsto (fun n ↦ ∫ (x : α), f n x ∂μ) atTop (𝓝 (∫ (x : α), F x ∂μ))

Tactic:
have hf'_nonneg : ∀ᵐ x ∂μ, ∀ n, 0 ≤ f' n x := by
    filter_upwards [h_mono] with a ha n
    simp [f', ha zero_le]

Goal state after:
α : Type u_1
m : MeasurableSpace α
μ : Measure α
f : ℕ → α → ℝ
F : α → ℝ
hf : ∀ (n : ℕ), Integrable (f n) μ
hF : Integrable F μ
h_mono : ∀ᵐ (x : α) ∂μ, Monotone fun n ↦ f n x
h_tendsto : ∀ᵐ (x : α) ∂μ, Tendsto (fun n ↦ f n x) atTop (𝓝 (F x))
f' : ℕ → α → ℝ := fun n x ↦ f n x - f 0 x
hf'_nonneg : ∀ᵐ (x : α) ∂μ, ∀ (n : ℕ), 0 ≤ f' n x
⊢ Tendsto (fun n ↦ ∫ (x : α), f n x ∂μ) atTop (𝓝 (∫ (x : α), F x ∂μ))
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:
simp_rw [coe_lineMap_eq, apply_lineMap]

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

Goal state before:
pp: case neg
α : Type u_1
G : Type u_5
inst✝ : NormedAddCommGroup G
m : MeasurableSpace α
μ : Measure α
f : ↥(α →₁ₛ[μ] G)
h_eq : ∀ (a : α), (SimpleFunc.map (fun x ↦ ‖x‖ₑ) (toSimpleFunc f)) a = ‖(toSimpleFunc f) a‖ₑ
x : G
a✝ : x ∈ (toSimpleFunc f).range
hx0 : ¬x = 0
⊢ ‖x‖ₑ * μ (⇑(toSimpleFunc f) ⁻¹' {x}) ≠ ∞

Tactic:
· finiteness [SimpleFunc.measure_preimage_lt_top_of_integrable _ (SimpleFunc.integrable f) hx0]

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

Goal state before:
pp: 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 : StrongDual 𝕜 E
h : ∀ z ∈ closedBall 0 r, ‖f z‖ ≤ c
⊢ ‖f‖ ≤ c / r

Tactic:
apply ContinuousLinearMap.opNorm_le_bound
  · apply div_nonneg _ r_pos.le
    exact
      (norm_nonneg _).trans
        (h 0 (by simp only [norm_zero, mem_closedBall, dist_zero_left, r_pos.le]))
  apply LinearMap.bound_of_ball_bound' r_pos
  exact fun z hz => h z hz

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁷ : RCLike 𝕜
𝓕 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace 𝕜 F
inst✝² : FunLike 𝓕 E F
inst✝¹ : AddMonoidHomClass 𝓕 E F
inst✝ : MulActionHomClass 𝓕 𝕜 E F
f : 𝓕
K : NNReal
h : ∀ (x : E), ‖x‖ = 1 → 1 ≤ ↑K * ‖f x‖
x : E
⊢ ‖x‖ ≤ ↑K * ‖f x‖

Tactic:
obtain rfl | hx := eq_or_ne x 0

Goal state after:
case inl
𝕜 : Type u_1
inst✝⁷ : RCLike 𝕜
𝓕 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace 𝕜 F
inst✝² : FunLike 𝓕 E F
inst✝¹ : AddMonoidHomClass 𝓕 E F
inst✝ : MulActionHomClass 𝓕 𝕜 E F
f : 𝓕
K : NNReal
h : ∀ (x : E), ‖x‖ = 1 → 1 ≤ ↑K * ‖f x‖
⊢ ‖0‖ ≤ ↑K * ‖f 0‖

case inr
𝕜 : Type u_1
inst✝⁷ : RCLike 𝕜
𝓕 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace 𝕜 E
inst✝³ : NormedSpace 𝕜 F
inst✝² : FunLike 𝓕 E F
inst✝¹ : AddMonoidHomClass 𝓕 E F
inst✝ : MulActionHomClass 𝓕 𝕜 E F
f : 𝓕
K : NNReal
h : ∀ (x : E), ‖x‖ = 1 → 1 ≤ ↑K * ‖f x‖
x : E
hx : x ≠ 0
⊢ ‖x‖ ≤ ↑K * ‖f x‖
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:
norm_smul

Goal state after:
𝕜 : Type u_1
inst✝² : NormedField 𝕜
E : Type u_2
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
c : 𝕜
hc : 1 < ‖c‖
R : ℝ
hR : ‖c‖ < R
F : Subspace 𝕜 E
hFc : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑F
hF : ∃ x, x ∉ F
Rpos : 0 < R
this : ‖c‖ / R < 1
x : E
xF : x ∉ F
hx : ∀ y ∈ F, ‖c‖ / R * ‖x‖ ≤ ‖x - y‖
x0 : x ≠ 0
d : 𝕜
d0 : d ≠ 0
dxlt : ‖d • x‖ < R
ledx : R / ‖c‖ ≤ ‖d • x‖
y : E
hy : y ∈ F
y' : E := d⁻¹ • y
yy' : y = d • y'
⊢ ‖d‖ * ‖x - y'‖ = ‖d‖ * ‖x - y'‖
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
s : Set α
hs : MeasurableSet s
hμs : μ s ≠ ∞
x : E
⊢ (setToL1 hT) (indicatorConstLp 1 hs hμs x) = (T s) x

Tactic:
by
  rw [← Lp.simpleFunc.coe_indicatorConst hs hμs x]
  exact setToL1_simpleFunc_indicatorConst hT hs hμs.lt_top x

Goal state after:
Module: Mathlib.Topology.Bornology.BoundedOperation

Goal state before:
pp: R : Type u_1
inst✝ : SeminormedAddCommGroup R
x : R
s : Set R
hs : s ∈ cobounded R
⊢ (fun x_1 ↦ x_1 + x) ⁻¹' s ∈ cobounded R

Tactic:
rw [← isCobounded_def, ← isBounded_compl_iff] at hs ⊢

Goal state after:
R : Type u_1
inst✝ : SeminormedAddCommGroup R
x : R
s : Set R
hs : Bornology.IsBounded sᶜ
⊢ Bornology.IsBounded ((fun x_1 ↦ x_1 + x) ⁻¹' s)ᶜ
Module: Mathlib.Topology.Bornology.BoundedOperation

Goal state before:
pp: R : Type u_1
inst✝ : SeminormedAddCommGroup R
x : R
s : Set R
hs : s ∈ cobounded R
⊢ (fun x_1 ↦ x + x_1) ⁻¹' s ∈ cobounded R

Tactic:
rw [← isCobounded_def, ← isBounded_compl_iff] at hs ⊢

Goal state after:
R : Type u_1
inst✝ : SeminormedAddCommGroup R
x : R
s : Set R
hs : Bornology.IsBounded sᶜ
⊢ Bornology.IsBounded ((fun x_1 ↦ x + x_1) ⁻¹' s)ᶜ
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: case mp
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type w
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace 𝕜
inst✝ : FiniteDimensional 𝕜 E
f : E →ₗ[𝕜] F
⊢ Injective ⇑f → ∃ K > 0, AntilipschitzWith K ⇑f

Tactic:
← LinearMap.ker_eq_bot

Goal state after:
case mp
𝕜 : Type u
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type v
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type w
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
inst✝¹ : CompleteSpace 𝕜
inst✝ : FiniteDimensional 𝕜 E
f : E →ₗ[𝕜] F
⊢ f.ker = ⊥ → ∃ K > 0, AntilipschitzWith K ⇑f
Module: Mathlib.MeasureTheory.Integral.SetToL1

Goal state before:
pp: α : Type u_1
m : MeasurableSpace α
μ : Measure α
G' : Type u_7
G'' : Type u_8
inst✝⁷ : NormedAddCommGroup G''
inst✝⁶ : PartialOrder G''
inst✝⁵ : IsOrderedAddMonoid G''
inst✝⁴ : NormedSpace ℝ G''
inst✝³ : NormedAddCommGroup G'
inst✝² : PartialOrder G'
inst✝¹ : NormedSpace ℝ G'
inst✝ : ClosedIciTopology G''
T : Set α → G' →L[ℝ] G''
C : ℝ
hT : DominatedFinMeasAdditive μ T C
hT_nonneg : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → ∀ (x : G'), 0 ≤ x → 0 ≤ (T s) x
f : α → G'
hf : 0 ≤ᵐ[μ] f
⊢ 0 ≤ setToFun μ T hT f

Tactic:
by_cases hG'' : CompleteSpace G''; swap
  · simp [setToFun, hG'']
  by_cases hfi : Integrable f μ
  · simp_rw [setToFun_eq _ hfi]
    exact L1.setToL1_nonneg hT hT_nonneg hf
  · simp_rw [setToFun_undef _ hfi, le_rfl]

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

Goal state before:
pp: F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
inst✝¹ : PseudoMetricSpace γ
f✝ g : α →ᵇ β
x : α
C : ℝ
inst✝ : CompleteSpace β
f : ℕ → α →ᵇ β
hf : CauchySeq f
b : ℕ → ℝ
b0 : ∀ (n : ℕ), 0 ≤ b n
b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N
b_lim : Tendsto b atTop (𝓝 0)
f_bdd : ∀ (x : α) (n m N : ℕ), N ≤ n → N ≤ m → dist ((f n) x) ((f m) x) ≤ b N
fx_cau : ∀ (x : α), CauchySeq fun n ↦ (f n) x
⊢ ∃ a, Tendsto f atTop (𝓝 a)

Tactic:
choose F hF using fun x => cauchySeq_tendsto_of_complete (fx_cau x)

Goal state after:
F✝ : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
inst✝¹ : PseudoMetricSpace γ
f✝ g : α →ᵇ β
x : α
C : ℝ
inst✝ : CompleteSpace β
f : ℕ → α →ᵇ β
hf : CauchySeq f
b : ℕ → ℝ
b0 : ∀ (n : ℕ), 0 ≤ b n
b_bound : ∀ (n m N : ℕ), N ≤ n → N ≤ m → dist (f n) (f m) ≤ b N
b_lim : Tendsto b atTop (𝓝 0)
f_bdd : ∀ (x : α) (n m N : ℕ), N ≤ n → N ≤ m → dist ((f n) x) ((f m) x) ≤ b N
fx_cau : ∀ (x : α), CauchySeq fun n ↦ (f n) x
F : α → β
hF : ∀ (x : α), Tendsto (fun n ↦ (f n) x) atTop (𝓝 (F x))
⊢ ∃ a, Tendsto f atTop (𝓝 a)
Module: Mathlib.Topology.ContinuousMap.Bounded.Basic

Goal state before:
pp: case refine_3
α : Type u
β : Type v
inst✝³ : TopologicalSpace α
inst✝² : PseudoMetricSpace β
δ : Type u_2
inst✝¹ : TopologicalSpace δ
inst✝ : DiscreteTopology δ
f : α ↪ δ
g₁ g₂ : α →ᵇ β
h₁ h₂ : δ →ᵇ β
⊢ dist (h₁.domRestrict (range ⇑f)ᶜ) (h₂.domRestrict (range ⇑f)ᶜ) ≤ dist (extend f g₁ h₁) (extend f g₂ h₂)

Tactic:
· refine (dist_le dist_nonneg).2 fun x => ?_
    calc
      dist (h₁ x) (h₂ x) = dist (extend f g₁ h₁ x) (extend f g₂ h₂ x) := by
        rw [extend_apply' x.coe_prop, extend_apply' x.coe_prop]
      _ ≤ _ := dist_coe_le_dist _

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✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : NormedAddCommGroup F
inst✝³ : NormedSpace ℝ F
m : MeasurableSpace α
μ : Measure α
T : Set α → E →L[ℝ] F
C : ℝ
hT : DominatedFinMeasAdditive μ T C
inst✝² : MeasurableSpace E
inst✝¹ : BorelSpace E
f : α → E
fmeas : Measurable f
hf : Integrable f μ
s : Set E
inst✝ : SeparableSpace ↑s
hs : range f ∪ {0} ⊆ s
⊢ Tendsto (fun n ↦ setToFun μ T hT ⇑(SimpleFunc.approxOn f fmeas s 0 ⋯ n)) atTop (𝓝 (setToFun μ T hT f))

Tactic:
refine tendsto_setToFun_approxOn_of_measurable hT hf fmeas ?_ _ (integrable_zero _ _ _)
  exact Eventually.of_forall fun x => subset_closure (hs (Set.mem_union_left _ (mem_range_self _)))

Goal state after:
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
x_mem : x ∈ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
⊢ (thickenedIndicator δ_pos E) x = 1

Tactic:
by
  rw [thickenedIndicator_apply, thickenedIndicatorAux_one_of_mem_closure δ E x_mem, toNNReal_one]

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
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:
by
  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 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
hT : DominatedFinMeasAdditive μ T C
f : ↥(Lp E 1 μ)
hC : 0 ≤ C
hF : ¬CompleteSpace F
⊢ ‖setToFun μ T hT ↑↑f‖ ≤ C * ‖f‖

Tactic:
simp only [setToFun, hF, ↓reduceDIte, norm_zero]
    positivity

Goal state after:
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: case pos
α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
x_mem_closure : x ∈ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
⊢ ∀ᶠ (δ : ℝ) in 𝓝 0,
    (cthickening δ E).mulIndicator f x =
      (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x

pp: case neg
α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
x_mem_closure : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
⊢ ∀ᶠ (δ : ℝ) in 𝓝 0,
    (cthickening δ E).mulIndicator f x =
      (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x

Tactic:
· filter_upwards [univ_mem] with δ _
    have obs : x ∈ cthickening δ E := closure_subset_cthickening δ E x_mem_closure
    rw [mulIndicator_of_mem obs f, mulIndicator_of_mem x_mem_closure f]

Goal state after:
case neg
α : Type u_1
inst✝¹ : PseudoEMetricSpace α
β : Type u_2
inst✝ : One β
f : α → β
E : Set α
x : α
x_mem_closure : x ∉ closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E
⊢ ∀ᶠ (δ : ℝ) in 𝓝 0,
    (cthickening δ E).mulIndicator f x =
      (closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x
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
hT : DominatedFinMeasAdditive μ T C
f : ↥(Lp E 1 μ)
hF : ¬CompleteSpace F
⊢ ‖setToFun μ T hT ↑↑f‖ ≤ max C 0 * ‖f‖

Tactic:
simp only [setToFun, hF, ↓reduceDIte, norm_zero]
    positivity

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 : α → E
hT : DominatedFinMeasAdditive μ T C
hf : Integrable f μ
hC : 0 ≤ C
hF : ¬CompleteSpace F
⊢ ‖setToFun μ T hT f‖ ≤ C * ‖Integrable.toL1 f hf‖

Tactic:
simp only [setToFun, hF, ↓reduceDIte, norm_zero]
    positivity

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 : α → E
hT : DominatedFinMeasAdditive μ T C
hf : Integrable f μ
hF : ¬CompleteSpace F
⊢ ‖setToFun μ T hT f‖ ≤ max C 0 * ‖Integrable.toL1 f hf‖

Tactic:
simp only [setToFun, hF, ↓reduceDIte, norm_zero]
    positivity

Goal state after:
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: case inr
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
h : x < y
this✝ : TopologicalSpace α := Preorder.topology α
this : OrderTopology α
h' : (𝓝[<] y).NeBot
z : α
hz : z < y
⊢ f z ≤ f y

Tactic:
exact hf hz.le

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

Goal state before:
pp: R : Type u_1
inst✝² : LinearOrder R
inst✝¹ : TopologicalSpace R
inst✝ : OrderTopology R
f : StieltjesFunction R
x : R
⊢ rightLim (↑f) x = ↑f x

Tactic:
← f.mono.continuousWithinAt_Ioi_iff_rightLim_eq,

Goal state after:
R : Type u_1
inst✝² : LinearOrder R
inst✝¹ : TopologicalSpace R
inst✝ : OrderTopology R
f : StieltjesFunction R
x : R
⊢ ContinuousWithinAt (↑f) (Ioi x) x
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inr
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
ab : a ≤ b
i : ℕ
cv : Icc a b ⊆ ⋃ i_1 ∈ ↑(insert i (s.erase i)), Iotop (c i_1) (d i_1)
is : i ∈ s
bcd : b ∈ Iotop (c i) (d i)
⊢ ofReal (↑f b - ↑f a) ≤ ∑ i ∈ insert i (s.erase i), ofReal (↑f (d i) - ↑f (c i))

Tactic:
Finset.coe_insert,

Goal state after:
case inr
R : Type u_1
inst✝³ : LinearOrder R
inst✝² : TopologicalSpace R
f : StieltjesFunction R
inst✝¹ : OrderTopology R
inst✝ : CompactIccSpace R
a : R
c d : ℕ → R
s✝ s : Finset ℕ
IH :
  ∀ t ⊂ s,
    ∀ (b : R), Icc a b ⊆ ⋃ i ∈ ↑t, Iotop (c i) (d i) → ofReal (↑f b - ↑f a) ≤ ∑ i ∈ t, ofReal (↑f (d i) - ↑f (c i))
b : R
ab : a ≤ b
i : ℕ
cv : Icc a b ⊆ ⋃ i_1 ∈ insert i ↑(s.erase i), Iotop (c i_1) (d i_1)
is : i ∈ s
bcd : b ∈ Iotop (c i) (d i)
⊢ ofReal (↑f b - ↑f a) ≤ ∑ i ∈ insert i (s.erase i), ofReal (↑f (d i) - ↑f (c i))
Module: Mathlib.MeasureTheory.Group.FundamentalDomain

Goal state before:
pp: G : Type u_1
α : Type u_3
inst✝² : Group G
inst✝¹ : MulAction G α
inst✝ : MeasurableSpace α
s : Set α
μ : Measure α
h_meas : NullMeasurableSet s μ
h_exists : ∀ (x : α), ∃! g, g • x ∈ s
a b : G
hab : a ≠ b
x : α
hxa : x ∈ (fun g ↦ g • s) a
hxb : x ∈ (fun g ↦ g • s) b
⊢ False

Tactic:
by
    rw [mem_smul_set_iff_inv_smul_mem] at hxa hxb
    exact hab (inv_injective <| (h_exists x).unique hxa hxb)

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

Goal state before:
pp: case e'_3.e'_4
G : Type u_1
α : Type u_3
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
s t : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
A : Set α
hA₀ : MeasurableSet A
hA : ∀ (g : G), (fun x ↦ g • x) ⁻¹' A = A
g : G
x : α
⊢ A = (fun x ↦ g • x) ⁻¹' A

Tactic:
hA g

Goal state after:
case e'_3.e'_4
G : Type u_1
α : Type u_3
inst✝⁵ : Group G
inst✝⁴ : MulAction G α
inst✝³ : MeasurableSpace α
s t : Set α
μ : Measure α
inst✝² : MeasurableConstSMul G α
inst✝¹ : SMulInvariantMeasure G α μ
inst✝ : Countable G
hs : IsFundamentalDomain G s μ
ht : IsFundamentalDomain G t μ
A : Set α
hA₀ : MeasurableSet A
hA : ∀ (g : G), (fun x ↦ g • x) ⁻¹' A = A
g : G
x : α
⊢ A = A
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inl
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
a b : R
hab : a ≤ b
⊢ f.measure (Icc a b) = ofReal (↑f b - leftLim (↑f) a)

Tactic:
have A : Disjoint {a} (Ioc a b) := by simp

Goal state after:
case inl
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
f : StieltjesFunction R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
a b : R
hab : a ≤ b
A : Disjoint {a} (Ioc a b)
⊢ f.measure (Icc a b) = ofReal (↑f b - leftLim (↑f) a)
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:
pp: 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

pp: 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

pp: 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)

pp: 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

Tactic:
· rwa [integrableOn_congr_set_ae hs.toMeasurable_ae_eq]

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

Goal state before:
pp: G : Type u_1
inst✝² : Group G
inst✝¹ : TopologicalSpace G
inst✝ : IsTopologicalGroup G
K₀ : PositiveCompacts G
U : Set G
hU : (interior U).Nonempty
K₁ K₂ : Compacts G
h : ↑K₁ ⊆ K₂.carrier
⊢ prehaar (↑K₀) U K₁ ≤ prehaar (↑K₀) U K₂

Tactic:
by
  simp only [prehaar]; rw [div_le_div_iff_of_pos_right]
  · exact mod_cast index_mono K₂.2 h hU
  · exact mod_cast index_pos K₀ hU

Goal state after: