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

Goal state before:
pp: α : Type u_1
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSingletonClass α
f : α → E
a : α
s : Set α
inst✝ : Decidable (a ∈ s)
⊢ ∫ (x : α) in s, f x ∂Measure.dirac a = if a ∈ s then f a else 0

Tactic:
rw [restrict_dirac]

Goal state after:
α : Type u_1
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
hE : CompleteSpace E
inst✝² : MeasurableSpace α
inst✝¹ : MeasurableSingletonClass α
f : α → E
a : α
s : Set α
inst✝ : Decidable (a ∈ s)
⊢ (∫ (x : α), f x ∂if a ∈ s then Measure.dirac a else 0) = if a ∈ s then f a else 0
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‖
c : ℝ
hc : 0 ≤ c
⊢ (∀ (x : α), ‖f x‖ ≤ c ∧ ‖g x‖ ≤ c) ↔ (∀ (x : α), ‖f x‖ ≤ c) ∧ ∀ (x : α), ‖g x‖ ≤ c

Tactic:
forall_and

Goal state after:
Module: Mathlib.Analysis.Normed.Module.FiniteDimension

Goal state before:
pp: case inl
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : Nontrivial E
x : E
K : Set E
hK : IsCompact K
hx : x ∈ K
hx' : ∃ i, 0 < i ∧ closedBall x i ⊆ K
⊢ ∃ y ∈ frontier K, infDist x Kᶜ = dist x y

Tactic:
rcases hx' with ⟨r, hr₀, hrK⟩

Goal state after:
case inl
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : Nontrivial E
x : E
K : Set E
hK : IsCompact K
hx : x ∈ K
r : ℝ
hr₀ : 0 < r
hrK : closedBall x r ⊆ K
⊢ ∃ y ∈ frontier K, infDist x Kᶜ = dist x y
Module: Mathlib.Topology.ContinuousMap.Compact

Goal state before:
pp: α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : SeminormedAddCommGroup E
x y : C(α, E)
⊢ ‖-mkOfCompact x + mkOfCompact y‖ = ‖mkOfCompact (-x) + mkOfCompact y‖

Tactic:
mkOfCompact_neg

Goal state after:
α : Type u_1
β : Type u_2
E : Type u_3
inst✝³ : TopologicalSpace α
inst✝² : CompactSpace α
inst✝¹ : PseudoMetricSpace β
inst✝ : SeminormedAddCommGroup E
x y : C(α, E)
⊢ ‖-mkOfCompact x + mkOfCompact y‖ = ‖-mkOfCompact x + mkOfCompact y‖
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
δseq_lim : Tendsto δseq atTop (𝓝 0)
E : Set α
⊢ Tendsto (fun n ↦ thickenedIndicatorAux (δseq n) E) atTop
    (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator fun x ↦ 1))

Tactic:
rw [tendsto_pi_nhds]

Goal state after:
α : Type u_1
inst✝ : PseudoEMetricSpace α
δseq : ℕ → ℝ
δseq_lim : Tendsto δseq atTop (𝓝 0)
E : Set α
⊢ ∀ (x : α),
    Tendsto (fun i ↦ thickenedIndicatorAux (δseq i) E x) atTop
      (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).indicator (fun x ↦ 1) x))
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝² : PseudoEMetricSpace α
β : Type u_2
inst✝¹ : One β
inst✝ : TopologicalSpace β
f : α → β
E : Set α
⊢ Tendsto (fun δ ↦ (thickening δ E).mulIndicator f) (𝓝[>] 0)
    (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f))

Tactic:
rw [tendsto_pi_nhds]

Goal state after:
α : Type u_1
inst✝² : PseudoEMetricSpace α
β : Type u_2
inst✝¹ : One β
inst✝ : TopologicalSpace β
f : α → β
E : Set α
⊢ ∀ (x : α),
    Tendsto (fun i ↦ (thickening i E).mulIndicator f x) (𝓝[>] 0)
      (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x))
Module: Mathlib.Topology.MetricSpace.ThickenedIndicator

Goal state before:
pp: α : Type u_1
inst✝² : PseudoEMetricSpace α
β : Type u_2
inst✝¹ : One β
inst✝ : TopologicalSpace β
f : α → β
E : Set α
⊢ Tendsto (fun δ ↦ (cthickening δ E).mulIndicator f) (𝓝 0)
    (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f))

Tactic:
rw [tendsto_pi_nhds]

Goal state after:
α : Type u_1
inst✝² : PseudoEMetricSpace α
β : Type u_2
inst✝¹ : One β
inst✝ : TopologicalSpace β
f : α → β
E : Set α
⊢ ∀ (x : α),
    Tendsto (fun i ↦ (cthickening i E).mulIndicator f x) (𝓝 0)
      (𝓝 ((closure[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] E).mulIndicator f x))
Module: Mathlib.Topology.Order.LeftRightLim

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → β
a : α
⊢ (𝓝 (f a) ⊓ map f (𝓝[≤] a)).NeBot

Tactic:
refine inf_neBot_iff.mpr (fun s hs s' hs' ↦ ?_)

Goal state after:
α : Type u_1
β : Type u_2
inst✝³ : LinearOrder α
inst✝² : TopologicalSpace β
inst✝¹ : TopologicalSpace α
inst✝ : OrderTopology α
f : α → β
a : α
s : Set β
hs : s ∈ 𝓝 (f a)
s' : Set β
hs' : s' ∈ map f (𝓝[≤] a)
⊢ (s ∩ s').Nonempty
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 y : α
h : x < y
⊢ f x ≤ leftLim f y

Tactic:
by
  let : TopologicalSpace α := Preorder.topology α
  have : OrderTopology α := ⟨rfl⟩
  rcases eq_or_neBot (𝓝[<] y) with h' | h'
  · rw [leftLim_eq_of_eq_bot _ h']
    exact hf h.le
  rw [leftLim_eq_sSup hf]
  refine le_csSup ⟨f y, ?_⟩ (mem_image_of_mem _ h)
  simp only [upperBounds, mem_image, mem_Iio, forall_exists_index, and_imp,
    forall_apply_eq_imp_iff₂, mem_ofPred_eq]
  intro z hz
  exact hf hz.le

Goal state after:
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
⊢ Monotone (leftLim f)

Tactic:
by
  intro x y h
  rcases eq_or_lt_of_le h with (rfl | hxy)
  · exact le_rfl
  · exact (hf.leftLim_le le_rfl).trans (hf.le_leftLim hxy)

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
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
⊢ Tendsto (fun s ↦ setToFun μ T hT fun a ↦ ∑ i ∈ s, F i a) (SummationFilter.unconditional ι).filter
    (𝓝 (setToFun μ T hT f))

Tactic:
refine tendsto_setToFun_filter_of_dominated_convergence _
      (fun a => ∑' n, bound n a) ?_ ?_ bound_integrable h_lim

Goal state after:
case refine_1
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, AEStronglyMeasurable (fun a ↦ ∑ i ∈ n, F i a) μ

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
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: R : Type u_1
inst✝ : LinearOrder R
x y : R
h : y ∈ botSet
⊢ y ≤ x

Tactic:
exact h x

Goal state after:
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
cv : Icc a b ⊆ ⋃ i ∈ ↑s, Iotop (c i) (d i)
ab : a ≤ b
i : ℕ
is : i ∈ s
bcd : b ∈ Iotop (c i) (d i)
⊢ ofReal (↑f b - ↑f a) ≤ ∑ i ∈ s, ofReal (↑f (d i) - ↑f (c i))

Tactic:
rw [← Finset.insert_erase is] at cv ⊢

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✝⁴ : Group G
inst✝³ : MulAction G α
inst✝² : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : Countable G
inst✝ : SMulInvariantMeasure G α μ
hμ : μ ≠ 0
h : IsFundamentalDomain G s μ
hc : μ s = 0
⊢ μ (⋃ g, g • s) ≤ 0

Tactic:
refine le_trans (measure_iUnion_le _) ?_

Goal state after:
G : Type u_1
α : Type u_3
inst✝⁴ : Group G
inst✝³ : MulAction G α
inst✝² : MeasurableSpace α
s : Set α
μ : Measure α
inst✝¹ : Countable G
inst✝ : SMulInvariantMeasure G α μ
hμ : μ ≠ 0
h : IsFundamentalDomain G s μ
hc : μ s = 0
⊢ ∑' (i : G), μ (i • s) ≤ 0
Module: Mathlib.MeasureTheory.Measure.Content

Goal state before:
pp: G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
A : Set G
hA : MeasurableSet A
r : ℝ≥0∞
U : Set G
hUo : IsOpen U
hAU : A ⊆ U
hr : μ.outerMeasure U < r
⊢ ∃ U ⊇ A, IsOpen U ∧ μ.measure U < r

Tactic:
← μ.measure_apply hUo.measurableSet

Goal state after:
G : Type w
inst✝² : TopologicalSpace G
μ : Content G
inst✝¹ : R1Space G
S : MeasurableSpace G
inst✝ : BorelSpace G
A : Set G
hA : MeasurableSet A
r : ℝ≥0∞
U : Set G
hUo : IsOpen U
hAU : A ⊆ U
hr : μ.measure U < r
⊢ ∃ U ⊇ A, IsOpen U ∧ μ.measure U < r
Module: Mathlib.MeasureTheory.Integral.Bochner.Set

Goal state before:

Tactic:
∫ x in t, f x ∂μ = ∫ x in t, f' x ∂μ := integral_congr_ae h.1.ae_eq_mk
    _ = ∫ x in s, f' x ∂μ := by
      apply
        setIntegral_eq_of_subset_of_ae_sdiff_eq_zero_aux hts _ h.1.stronglyMeasurable_mk
          (h.congr h.1.ae_eq_mk)
      filter_upwards [h't, ae_imp_of_ae_restrict h.1.ae_eq_mk] with x hx h'x h''x
      rw [← h'x h''x.1, hx h''x]
    _ = ∫ x in s, f x ∂μ := by
      apply integral_congr_ae
      apply ae_restrict_of_ae_restrict_of_subset hts
      exact h.1.ae_eq_mk.symm

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

Goal state before:
pp: 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 : R
ha : ¬IsBot a
b : R
hb : b < a
u : ℕ → R
u_mono : StrictMono u
u_lim : Tendsto u atTop (𝓝 a)
u_lt_a : ∀ (n : ℕ), u n < a
A : {a} = ⋂ n, Ioc (u n) a
⊢ Tendsto (fun n ↦ f.measure (Ioc (u n) a)) atTop (𝓝 (f.measure {a}))

Tactic:
A

Goal state after:
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 : R
ha : ¬IsBot a
b : R
hb : b < a
u : ℕ → R
u_mono : StrictMono u
u_lim : Tendsto u atTop (𝓝 a)
u_lt_a : ∀ (n : ℕ), u n < a
A : {a} = ⋂ n, Ioc (u n) a
⊢ Tendsto (fun n ↦ f.measure (Ioc (u n) a)) atTop (𝓝 (f.measure (⋂ n, Ioc (u n) a)))
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inr
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
l : ℝ
hf : Tendsto (↑f) atBot (𝓝 l)
x : R
this : Nonempty R
val✝ : NoBotOrder R
⊢ f.measure (Iic x) = ofReal (↑f x - l)

Tactic:
have : NoMinOrder R := NoBotOrder.to_noMinOrder R

Goal state after:
case inr
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
l : ℝ
hf : Tendsto (↑f) atBot (𝓝 l)
x : R
this✝ : Nonempty R
val✝ : NoBotOrder R
this : NoMinOrder R
⊢ f.measure (Iic x) = ofReal (↑f x - l)
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inr
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
hf : Tendsto (↑f) atBot atBot
x : R
this : Nonempty R
val✝ : NoBotOrder R
⊢ f.measure (Iio x) = ∞

Tactic:
have : NoMinOrder R := NoBotOrder.to_noMinOrder R

Goal state after:
case inr
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
hf : Tendsto (↑f) atBot atBot
x : R
this✝ : Nonempty R
val✝ : NoBotOrder R
this : NoMinOrder R
⊢ f.measure (Iio x) = ∞
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inl.hq
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
g : StieltjesFunction R
y : R
hfg : f.measure = g.measure
hy : ↑f y = ↑g y
x : R
hxy : x ≤ y
hf : ofReal (↑g y - ↑g x) = ofReal (↑f y - ↑f x)
⊢ 0 ≤ ↑f y - ↑f x

Tactic:
rw [sub_nonneg]
      exact f.mono hxy

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

Goal state before:
pp: case inr.hq
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
g : StieltjesFunction R
y : R
hfg : f.measure = g.measure
hy : ↑f y = ↑g y
x : R
hxy : y ≤ x
hf : ofReal (↑g x - ↑g y) = ofReal (↑f x - ↑f y)
⊢ 0 ≤ ↑f x - ↑f y

Tactic:
rw [sub_nonneg]
      exact f.mono hxy

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

Goal state before:
pp: X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure X
f g : ↥(Lp E p μ)
s : Set X
⊢ ∀ᵐ (x : X) ∂μ.restrict s,
    ↑↑(MemLp.toLp ↑↑g ⋯) x = ↑↑g x →
      ↑↑(MemLp.toLp ↑↑f ⋯) x = ↑↑f x →
        ↑↑(MemLp.toLp ↑↑f ⋯ + MemLp.toLp ↑↑g ⋯) x = (↑↑(MemLp.toLp ↑↑f ⋯) + ↑↑(MemLp.toLp ↑↑g ⋯)) x →
          ↑↑(f + g) x = (↑↑f + ↑↑g) x → ↑↑(MemLp.toLp ↑↑(f + g) ⋯) x = ↑↑(MemLp.toLp ↑↑f ⋯ + MemLp.toLp ↑↑g ⋯) x

Tactic:
refine (MemLp.coeFn_toLp ((Lp.memLp (f + g)).restrict s)).mono fun x hx1 hx2 hx3 hx4 hx5 => ?_

Goal state after:
X : Type u_1
E : Type u_3
mX : MeasurableSpace X
inst✝ : NormedAddCommGroup E
p : ℝ≥0∞
μ : Measure X
f g : ↥(Lp E p μ)
s : Set X
x : X
hx1 : ↑↑(MemLp.toLp ↑↑(f + g) ⋯) x = ↑↑(f + g) x
hx2 : ↑↑(MemLp.toLp ↑↑g ⋯) x = ↑↑g x
hx3 : ↑↑(MemLp.toLp ↑↑f ⋯) x = ↑↑f x
hx4 : ↑↑(MemLp.toLp ↑↑f ⋯ + MemLp.toLp ↑↑g ⋯) x = (↑↑(MemLp.toLp ↑↑f ⋯) + ↑↑(MemLp.toLp ↑↑g ⋯)) x
hx5 : ↑↑(f + g) x = (↑↑f + ↑↑g) x
⊢ ↑↑(MemLp.toLp ↑↑(f + g) ⋯) x = ↑↑(MemLp.toLp ↑↑f ⋯ + MemLp.toLp ↑↑g ⋯) x
Module: Mathlib.MeasureTheory.Measure.Stieltjes

Goal state before:
pp: case inr
R : Type u_1
inst✝⁷ : LinearOrder R
inst✝⁶ : TopologicalSpace R
inst✝⁵ : OrderTopology R
inst✝⁴ : CompactIccSpace R
inst✝³ : MeasurableSpace R
inst✝² : BorelSpace R
inst✝¹ : SecondCountableTopology R
inst✝ : DenselyOrdered R
f g : StieltjesFunction R
a b : R
h : a ≤ b
ha : (𝓝[<] a).NeBot
⊢ leftLim (↑(f + g)) a = leftLim (↑f) a + leftLim (↑g) a

Tactic:
exact tendsto_nhds_unique ((f + g).mono.tendsto_leftLim a)
        ((f.mono.tendsto_leftLim a).add (g.mono.tendsto_leftLim a))

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

Goal state before:
pp: G : Type u_1
inst✝⁴ : Group G
inst✝³ : TopologicalSpace G
inst✝² : IsTopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
g : G
⊢ ∀ (A : Set G), MeasurableSet A → (haarMeasure K₀) ((fun h ↦ g * h) ⁻¹' A) = (haarMeasure K₀) A

Tactic:
A

Goal state after:
G : Type u_1
inst✝⁴ : Group G
inst✝³ : TopologicalSpace G
inst✝² : IsTopologicalGroup G
inst✝¹ : MeasurableSpace G
inst✝ : BorelSpace G
K₀ : PositiveCompacts G
g : G
A : Set G
⊢ MeasurableSet A → (haarMeasure K₀) ((fun h ↦ g * h) ⁻¹' A) = (haarMeasure K₀) A
Module: Mathlib.Analysis.InnerProductSpace.Continuous

Goal state before:
pp: E : Type u_4
𝕜 : Type u_7
inst✝² : RCLike 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace 𝕜 E
x y : E
S : Set E
hS : Dense S
h : ∀ v ∈ S, ⟪x, v⟫ = ⟪y, v⟫
⊢ x = y

Tactic:
rw [← sub_eq_zero]; exact hS.eq_zero_of_inner_left 𝕜 (by simpa [inner_sub_left, sub_eq_zero])

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h : Convex ℝ K
u v : F
hv : v ∈ K
this : Nonempty ↑K := Nonempty.intro ⟨v, hv⟩
eq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
w : F
hw : w ∈ K
δ : ℝ := ⨅ w, ‖u - ↑w‖
p : ℝ := ⟪u - v, w - v⟫_ℝ
q : ℝ := ‖w - v‖ ^ 2
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
θ : ℝ
hθ₁ : 0 < θ
hθ₂ : θ ≤ 1
⊢ ‖u - v‖ * ‖u - v‖ ≤ ‖u - (θ • w + (1 - θ) • v)‖ * ‖u - (θ • w + (1 - θ) • v)‖

Tactic:
apply mul_self_le_mul_self (norm_nonneg _)

Goal state after:
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Set F
h : Convex ℝ K
u v : F
hv : v ∈ K
this : Nonempty ↑K := ⋯
eq : ‖u - v‖ = ⨅ w, ‖u - ↑w‖
w : F
hw : w ∈ K
δ : ℝ := ⋯
p : ℝ := ⋯
q : ℝ := ⋯
δ_le : ∀ (w : ↑K), δ ≤ ‖u - ↑w‖
δ_le' : ∀ w ∈ K, δ ≤ ‖u - w‖
θ : ℝ
hθ₁ : 0 < θ
hθ₂ : θ ≤ 1
⊢ ‖u - v‖ ≤ ‖u - (θ • w + (1 - θ) • v)‖
Module: Mathlib.Analysis.InnerProductSpace.Orthonormal

Goal state before:
pp: case mpr
𝕜 : Type u_1
E : Type u_2
inst✝³ : RCLike 𝕜
inst✝² : SeminormedAddCommGroup E
inst✝¹ : InnerProductSpace 𝕜 E
ι : Type u_4
inst✝ : DecidableEq ι
v : ι → E
⊢ (∀ (i j : ι), ⟪v i, v j⟫ = if i = j then 1 else 0) → Orthonormal 𝕜 v

Tactic:
· intro h
    constructor
    · intro i
      have h' : ‖v i‖ ^ 2 = 1 ^ 2 := by
        rw [@norm_sq_eq_re_inner 𝕜, h i i]; simp
      have h₁ : 0 ≤ ‖v i‖ := norm_nonneg _
      have h₂ : (0 : ℝ) ≤ 1 := zero_le_one
      rwa [sq_eq_sq₀ h₁ h₂] at h'
    · intro i j hij
      simpa [hij] using h i j

Goal state after:
Module: Mathlib.Analysis.InnerProductSpace.Projection.Minimal

Goal state before:
pp: F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : InnerProductSpace ℝ F
K : Submodule ℝ F
u v : F
hv : v ∈ K
⊢ ‖u - v‖ = ⨅ w, ‖u - ↑w‖ → ∀ w ∈ K, ⟪u - v, w⟫_ℝ = 0

Tactic:
by
      intro h
      have h : ∀ w ∈ K, ⟪u - v, w - v⟫_ℝ ≤ 0 := by
        rwa [norm_eq_iInf_iff_real_inner_le_zero] at h
        exacts [K.convex, hv]
      intro w hw
      have le : ⟪u - v, w⟫_ℝ ≤ 0 := by
        let w' := w + v
        have : w' ∈ K := Submodule.add_mem _ hw hv
        have h₁ := h w' this
        have h₂ : w' - v = w := by
          simp only [w', add_neg_cancel_right, sub_eq_add_neg]
        rw [h₂] at h₁
        exact h₁
      have ge : ⟪u - v, w⟫_ℝ ≥ 0 := by
        let w'' := -w + v
        have : w'' ∈ K := Submodule.add_mem _ (Submodule.neg_mem _ hw) hv
        have h₁ := h w'' this
        have h₂ : w'' - v = -w := by
          simp only [w'', add_neg_cancel_right, sub_eq_add_neg]
        rw [h₂, inner_neg_right] at h₁
        linarith
      exact le_antisymm le ge

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

Goal state before:
pp: case pos
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
α : ι → Type u_6
v_family : (i : ι) → α i → G i
hv_family : ∀ (i : ι), Orthonormal 𝕜 (v_family i)
i : ι
v : α i
j : ι
w : α j
hvw : ⟨i, v⟩ ≠ ⟨j, w⟩
hij : i = j
⊢ ⟪(fun a ↦ (V a.fst) (v_family a.fst a.snd)) ⟨i, v⟩, (fun a ↦ (V a.fst) (v_family a.fst a.snd)) ⟨j, w⟩⟫ = 0

pp: case neg
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
α : ι → Type u_6
v_family : (i : ι) → α i → G i
hv_family : ∀ (i : ι), Orthonormal 𝕜 (v_family i)
i : ι
v : α i
j : ι
w : α j
hvw : ⟨i, v⟩ ≠ ⟨j, w⟩
hij : ¬i = j
⊢ ⟪(fun a ↦ (V a.fst) (v_family a.fst a.snd)) ⟨i, v⟩, (fun a ↦ (V a.fst) (v_family a.fst a.snd)) ⟨j, w⟩⟫ = 0

Tactic:
· subst hij
    have : v ≠ w := fun h => by
      subst h
      exact hvw rfl
    simpa only [LinearIsometry.inner_map_map] using (hv_family i).2 this

Goal state after:
case neg
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : RCLike 𝕜
inst✝³ : SeminormedAddCommGroup E
inst✝² : InnerProductSpace 𝕜 E
ι : Type u_4
G : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (G i)
inst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)
V : (i : ι) → G i →ₗᵢ[𝕜] E
hV : OrthogonalFamily 𝕜 G V
α : ι → Type u_6
v_family : (i : ι) → α i → G i
hv_family : ∀ (i : ι), Orthonormal 𝕜 (v_family i)
i : ι
v : α i
j : ι
w : α j
hvw : ⟨i, v⟩ ≠ ⟨j, w⟩
hij : ¬i = j
⊢ ⟪(fun a ↦ (V a.fst) (v_family a.fst a.snd)) ⟨i, v⟩, (fun a ↦ (V a.fst) (v_family a.fst a.snd)) ⟨j, w⟩⟫ = 0
Module: Mathlib.Topology.Baire.CompleteMetrizable

Goal state before:
pp: X : Type u_1
inst✝¹ : TopologicalSpace X
inst✝ : IsCompletelyPseudoMetrizableSpace X
x✝ : UpgradedIsCompletelyPseudoMetrizableSpace X := upgradeIsCompletelyPseudoMetrizable X
f : ℕ → Set X
ho : ∀ (n : ℕ), IsOpen (f n)
hd : ∀ (n : ℕ), Dense (f n)
B : ℕ → ℝ≥0∞ := fun n ↦ 1 / 2 ^ n
Bpos : ∀ (n : ℕ), 0 < B n
center : ℕ → X → ℝ≥0∞ → X
radius : ℕ → X → ℝ≥0∞ → ℝ≥0∞
Hpos : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → 0 < radius n x δ
HB : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → radius n x δ ≤ B (n + 1)
Hball : ∀ (n : ℕ) (x : X) (δ : ℝ≥0∞), δ ≠ 0 → closedEBall (center n x δ) (radius n x δ) ⊆ closedEBall x δ ∩ f n
x : X
ε : ℝ≥0∞
εpos : 0 < ε
F : ℕ → X × ℝ≥0∞ := fun n ↦ Nat.recOn n (x, min ε (B 0)) fun n p ↦ (center n p.1 p.2, radius n p.1 p.2)
c : ℕ → X := fun n ↦ (F n).1
r : ℕ → ℝ≥0∞ := fun n ↦ (F n).2
n : ℕ
⊢ 0 < r n

Tactic:
induction n with

Goal state after: